Multiplicity results for some nonlinear elliptic problems with asymptotically ${{\varvec{p}}}$-linear terms

Abstract

Taking any $p > 1$, we consider the asymptotically p-linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} - {{\mathrm{div}}}(a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ \lambda ^\infty |u|^{p-2}u + g^\infty (x,u) &{}\quad \hbox {in}\;\Omega ,\\ u\ = \ 0 &{}\quad \hbox {on}\;\partial \Omega , \end{array} \right. \end{aligned}$$

where $\Omega$ is a bounded domain in $\mathbb R^N$, $N\ge 2$, $A(x,t,\xi )$ is a real function on $\Omega \times \mathbb R\times \mathbb R^N$ which grows with power p with respect to $\xi$ and has partial derivatives $A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )$, $a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )$. If $A(x,t,\xi ) \rightarrow A^\infty (x,t)$ and $\frac{g^\infty (x,t)}{|t|^{p-1}} \rightarrow 0$ as $|t| \rightarrow +\infty$, suitable assumptions, variational methods and either the cohomological index theory or its related pseudo-index one, allow us to prove the existence of multiple nontrivial bounded solutions in the non-resonant case, i.e. if $\lambda ^\infty$ is not an eigenvalue of the operator associated to $\nabla _\xi A^\infty (x,\xi )$. In particular, while in [14] the model problem $A(x,t,\xi ) = \mathcal{A}(x,t) |\xi |^p$ with $p > N$ is studied, here our goal is twofold: extending such results not only to a more general family of functions $A(x,t,\xi )$, but also to the more difficult case $1 < p \le N$.

1 Introduction

Let us consider the nonlinear problem

$$\begin{aligned} (GP)\qquad \left\{ \begin{array}{ll} - {{\mathrm{div}}}(a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ f(x,u) &{}\quad \hbox {in}\;\Omega ,\\ u\ = \ 0 &{}\quad \hbox {on}\;\partial \Omega , \end{array} \right. \end{aligned}$$

where $\Omega \subset \mathbb R^N$ is a bounded domain, $N\ge 2$, f(x, t) is a given real function on $\Omega \times \mathbb R$ and $A(x,t,\xi )$ is a real function on $\Omega \times \mathbb R\times \mathbb R^N$, with $A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )$, $a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )$.

If we set $F(x,t) = \int _0^t f(x,s) ds$, problem (GP) can be associated to the functional

$$\begin{aligned} \mathcal{J}(u)\ =\ \int _\Omega A(x,u,\nabla u) dx\ -\ \int _\Omega F(x,u)\ dx. \end{aligned}$$

(1.1)

If $A(x,t,\xi )$ depends on t, the derivative $d\mathcal{J}$ is not defined in the Sobolev space $W^{1, p}_0(\Omega )$ and its natural domain contains $X := W^{1,p}_0(\Omega ) \cap L^\infty (\Omega )$ where it is also continuous (see Proposition 3.5). Moreover, u is a weak solution of (GP) if

$$\begin{aligned} \int _\Omega (a(x,u,\nabla u)\cdot \nabla v + A_t(x,u,\nabla u)v) dx - \int _\Omega f(x,u)v dx = 0 \quad \hbox {for all }v \in X; \end{aligned}$$

thus, we prove that u belongs to X and is a critical point of $\mathcal{J}$. Hence, in order to solve (GP), we can use variational tools.

Model problems can be written by considering

$$\begin{aligned} A_2(x,t,\xi )\ =\ \sum _{i,j=1}^N a_{i,j}(x,t) \xi _i \xi _j\quad \hbox {or}\quad A_p(x,t,\xi )\ =\ \left( A_2(x,t,\xi )\right) ^{\frac{p}{2}}, \end{aligned}$$

where $(a_{i,j}(x,t))_{1\le i,j\le N}$ is an elliptic matrix.

An example is given by $A(x,t,\xi ) = \mathcal{A}(x,t) |\xi |^p$ with $p > 1$, so that the equation in (GP) is reduced to the quasi-p-linear equation

$$\begin{aligned} \qquad \qquad - {{\mathrm{div}}}(\mathcal{A}(x,u)\, |\nabla u|^{p-2}\, \nabla u) + \dfrac{1}{p}\, \mathcal{A}_t(x,u)\, |\nabla u|^p\ =\ f(x,u) \qquad \hbox {in }\Omega , \end{aligned}$$

(P)

which is studied in [13, 14] if $p > N$. In this setting, the related functional is

$$\begin{aligned} J_\mathcal{A}(u)\ =\ \frac{1}{p}\ \int _\Omega \mathcal{A}(x,u)\ |\nabla u|^p\, dx\ -\ \int _\Omega F(x,u)\ dx. \end{aligned}$$

(1.2)

Roughly speaking, we say that problem (GP) is asymptotically p-linear, if both $A(x,t,\xi )$ and F(x, t) admit the limit as $|t| \rightarrow +\infty$, so that, taking

$$\begin{aligned} A^\infty (x,\xi ) = \lim _{|t| \rightarrow +\infty }A(x,t,\xi ) \quad \hbox {and}\quad F(x,t) = \frac{\lambda ^\infty }{p} |t|^p + G^\infty (x,t), \end{aligned}$$

(1.3)

$A^\infty (x,\xi )$ is a positively p-homogeneous function with respect to $\xi$ equivalent to $|\xi |^p$, while $G^\infty (x,t)$ is at worst an infinity of lower order with respect to $|t|^p$ (for more details, see Sect. 3).

Thus, our aim is to investigate the existence of weak solutions of the nonlinear elliptic problem (GP) when it is asymptotically p-linear but in the non-resonant case, i.e., when $\lambda ^\infty$ in (1.3) is not an eigenvalue of the operator associated to $\nabla _\xi A^\infty (x,\xi )$.

For $A(x,t,\xi ) = |\xi |^p$ or, at worst, for $A(x,t,\xi ) = {\bar{A}}(x) |\xi |^p$, i.e., for $\mathcal{A}(x,t)\equiv 1$ or $\mathcal{A}(x,t) \equiv {\bar{A}}(x)$ independent of t, a variational approach was first used for $p=2$ in the asymptotically linear case (see the seminal papers [1, 5]) and then if $p\ne 2$ (see, e.g., [2, 4, 6, 16, 17, 20,21,22, 25], or the survey in the book [24]). Furthermore, in the model case (P) some multiplicity results have already been proved if $p > N$ (see [13, 14]).

We want to prove that the multiplicity results in [14] can also be stated in the general case (GP) and not only if $p > N$, but also in the complementary condition $1 < p \le N$ (for the complete statements, see Sect. 5).

We note that if $1 < p \le N$ and $\lambda ^\infty$ is not an eigenvalue of the operator associated to $\nabla _\xi A^\infty (x,\xi )$, in quite general suitable assumptions, a Palais–Smale sequence of $\mathcal{J}$ in X can have subsequences converging in $W^{1,p}_0(\Omega )$, but not in $L^\infty (\Omega )$ (see Example 4.3). Therefore, the classical Palais–Smale condition does not hold. This is why the geometric conditions are given by making use of the topology of $W^{1,p}_0(\Omega )$. As typical of such a problem, we consider subsets of neighbourhoods of zero and of infinity, but with respect to the norm in $W^{1,p}_0(\Omega )$ and not to that in X. Hence, in both cases we have no information about the $L^\infty$-norm for the elements of such sets.

In any case, we prove the existence of multiple nontrivial solutions according to the behaviour of F(x, t), both in zero and at infinity, and by considering $A^0(x,\xi ) = A(x,0,\xi )$ for the geometric conditions in zero and the limit function $A^\infty (x,\xi )$ for those at infinity (see Theorems 5.6, 5.7 and 5.8).

Finally, let us point out that no global p-homogeneity assumption on function $A(x,t,\xi )$ is required, but only that $A^0(x,\xi )$ and $A^\infty (x,\xi )$ have to be positively p-homogeneous with respect to $\xi$. Moreover, also in the non-resonant case, the proof of the boundedness of the Palais–Smale sequences is rather hard and our results imply the previous ones obtained when the term $A(x,t,\xi )$ does not depend on t.

This paper is organized as follows. In Sect. 2 we introduce the weak Palais–Smale condition and prove the related abstract multiplicity results, both with the cohomological index and the related pseudo-index. In Sect. 3 we introduce the hypotheses for (GP) and prove the first properties of $\mathcal{J}$ in X, while the weak Palais–Smale condition is proved in Sect. 4. In Sect. 5 the main results are stated (see Theorems 5.6, 5.7 and 5.8) and, once the geometric conditions have been checked, their proofs are given in Sect. 6, for solutions with negative critical levels via the index theory, and in Sect. 7, for solutions with positive critical levels via the related pseudo-index.

2 The abstract variational setting

We denote $\mathbb N= \{1, 2, \dots \}$ and, throughout this section, let us assume that:

• $(X, \Vert \cdot \Vert _X)$ is a Banach space with dual $(X',\Vert \cdot \Vert _{X'})$,

• $(W,\Vert \cdot \Vert _W)$ is another Banach space such that $X \hookrightarrow W$ continuously, i.e. $X \subset W$ and a constant $\sigma _0 > 0$ exists such that

  $$\begin{aligned} \Vert u\Vert _W \ \le \ \sigma _0\ \Vert u\Vert _X\qquad \hbox {for all }u \in X, \end{aligned}$$

  (2.1)

• $J {:} \mathcal{D} \subset W \rightarrow \mathbb R$ and $J \in C^1(X,\mathbb R)$ with $X \subset \mathcal{D}$.

Furthermore, fixing $\beta$, $\beta _1$, $\beta _2 \in \mathbb R$ and a set $\mathcal{C}\subset X$, let us denote

• $K^J = \{u \in X:\ dJ(u) = 0\}$ the set of the critical points of J in X,

• $K^J_\beta = \{u \in X:\ J(u) = \beta ,\ dJ(u) = 0\}$ the set of the critical points of J in X at level $\beta$,

• $J^\beta = \{u\in X:\ J(u) \le \beta \}$ the sublevel of J with respect to level $\beta$,

• $J^{\beta _2}_{\beta _1} = \{u\in X:\ \beta _1 \le J(u) \le \beta _2\}$ the closed “strip” between $\beta _1$ and $\beta _2$,

• $I\mathcal{C}= \{su \in X: u \in \mathcal{C}, s\in [0,1]\}$ the cone with base $\mathcal{C}$,

while, taking $u_0 \in X$, $r > 0$, by pointing out the two different norms $\Vert \cdot \Vert _W$ and $\Vert \cdot \Vert _X$, for $\ddagger = W$ or $\ddagger = X$ we put

• $B^\ddagger _r(u_0)\ =\ \{u \in X:\ \Vert u - u_0\Vert _\ddagger \ <\ r\}$,

• $\bar{B}^\ddagger _r(u_0)\ =\ \{u \in X:\ \Vert u - u_0\Vert _\ddagger \ \le \ r\}$,

• $\displaystyle d_\ddagger (u,{\mathcal{C}})\ =\ \inf _{v \in \mathcal{C}} \Vert u - v\Vert _\ddagger$,

• $N^\ddagger _r(\mathcal{C})\ =\ \{u \in X:\ d_\ddagger (u,\mathcal{C}) \le \ r\}$.

In any case, in order to avoid any ambiguity and to simplify, where possible, the notations, from now on we denote by X the space equipped with its given norm while, if a different norm is involved, we write it down explicitely. Accordingly, we denote by $\overline{\mathcal{C}}$ the closure of a set $\mathcal{C}\subset X$ with respect to the norm $\Vert \cdot \Vert _X$.

For investigating the number of critical points of a $C^1$ functional J in the Banach space X, let us introduce suitable variational tools.

For simplicity, taking $\beta \in \mathbb R$, we say that a sequence $(u_n)_n\subset X$ is a Palais–Smale sequence at level $\beta$, briefly $(PS)_\beta$-sequence, if

$$\begin{aligned} \lim _{n \rightarrow +\infty }J(u_n) = \beta \quad \text{ and }\quad \lim _{n \rightarrow +\infty }\Vert dJ(u_n)\Vert _{X'} = 0. \end{aligned}$$

(2.2)

Hence, the functional J satisfies the in X, briefly $(PS)_\beta$, if every $(PS)_\beta$-sequence converges in $(X,\Vert \cdot \Vert _X)$, up to subsequences (see [23]).

Different versions of the (classical) Palais–Smale condition can be introduced (see, e.g., [9, 12, 15]). In particular, as in [9], we say that the functional J satisfies the , if the following statement holds:

“If a $(PS)_\beta$-sequence exists, then $\beta$ is a critical value”.

Unfortunately, in order to find multiple solutions to our model problem (P), both the previous definitions are not useful. In fact, for the Palais–Smale condition, the convergence of a sequence in the intersection space $W^{1,p}_0(\Omega ) \cap L^\infty (\Omega )$ requires the convergence not only in the norm of $W^{1,p}_0(\Omega )$, but also in that of $L^\infty (\Omega )$, which may not hold (see Example 4.3). However, even if the Brézis–Coron–Nirenberg condition allows us to prove some existence results (see [9, Theorem 2]), contrary to the classical Palais–Smale it is not sufficient for finding multiple critical points if they occur at the same critical level.

For this reason, following some ideas developed in [12], in our setting we introduce a new condition, which considers both the involved norms and is weaker than the Palais–Smale but stronger than the Brézis–Coron–Nirenberg.

Definition 2.1

The functional J satisfies a weak version of the Palais–Smale condition at level $\beta$ ($\beta \in \mathbb R$), briefly $(wPS)_\beta$, if, for every $(PS)_\beta$-sequence $(u_{n})_n$, $u \in X$ exists, such that

• (i) $\lim \limits _{n\rightarrow +\infty } \Vert u_n - u\Vert _W = 0\quad$ (up to subsequences),

• (ii) $J(u) = \beta$, $\; dJ(u) = 0$.

If J satisfies $(wPS)_\beta$ at each level $\beta \in I$, I real interval, we say that J satisfies (wPS) in I.

The following lemmas are direct consequences of Definition 2.1.

Lemma 2.2

If J satisfies $(wPS)_\beta$ at a level $\beta \in \mathbb R$, then $K^J_\beta$ is compact with respect to $\Vert \cdot \Vert _W$.

Lemma 2.3

If J satisfies $(wPS)_\beta$ at level $\beta \in \mathbb R$, then, for each $\varrho > 0$, some $\varepsilon _\varrho$, $\mu _\varrho > 0$ exist, such that

$$\begin{aligned} u \in J^{\beta +\varepsilon _\varrho }_{\beta -\varepsilon _\varrho },\quad d_W(u,K^J_\beta ) \ge \varrho \qquad \Longrightarrow \qquad \Vert dJ(u)\Vert _{X'} \ge \mu _\varrho . \end{aligned}$$

Proof

Arguing by contradiction, we assume that $\bar{\varrho } > 0$ and a sequence $(u_n)_n \subset X$ exist, so that (2.2) holds and

$$\begin{aligned} d_W(u_n,K^J_\beta ) \ge \bar{\varrho }\quad \hbox {for all }n \in \mathbb N. \end{aligned}$$

(2.3)

However, by $(wPS)_{\beta }$ $u \in X$ exists, such that $\Vert u_n - u\Vert _W \rightarrow 0$ (up to subsequences) and $u \in K^J_\beta$, in contradiction with (2.3). $\square$

Now our aim is to generalize the classical Deformation Lemma (see, e.g., [26, Theorem A.4] or [27, Theorem 3.3.4]), when the Palais–Smale condition is replaced by its weak version in Definition 2.1.

Proposition 2.4

Let J be a $C^1$ functional which satisfies (wPS) in $\mathbb R$. Taking $\beta \in \mathbb R$, for any fixed $\varrho > 0$ and $\varepsilon _0 > 0$ a constant $\varepsilon ^* > 0$, $2\varepsilon ^* < \varepsilon _0$, exists, such that for each $\varepsilon \in \ ]0,\varepsilon ^*]$ a homeomorphism $\Psi : X \rightarrow X$ exists which satisfies the following conditions:

• (i) $\Psi (u) = u$ for all $u \not \in J_{\beta -\varepsilon _0}^{\beta +\varepsilon _0}$;

• (ii) $\Psi (J^{\beta + \varepsilon } {\setminus } N^W_{\varrho }(K^J_\beta )) \subset J^{\beta - \varepsilon }\quad$ and $\quad \Psi (J^{\beta + \varepsilon }) \subset J^{\beta - \varepsilon } \cup N^W_{\varrho }(K^J_\beta )$.

Furthermore, $\Psi$ is odd if J is even.

Proof

The proof is essentially similar to the classical one in [26, Theorem A.4] but checking carefully the change of norm when necessary. Thus, here we just outline the differences with respect to such a proof.

From Lemma 2.3 $\varepsilon _\varrho$, $\mu _\varrho > 0$ exist, such that

$$\begin{aligned} u \in J^{\beta +\varepsilon _\varrho }_{\beta -\varepsilon _\varrho },\quad d_W(u,K^J_\beta ) \ge \frac{\varrho }{8}\qquad \Longrightarrow \qquad \Vert d J(u)\Vert _{X'} \ge \mu _\varrho . \end{aligned}$$

Moreover, as J is a $C^1$ functional on X, then $V : X \rightarrow X$ pseudogradient vector field of J exists, such that $V(u) = 0$ if $u \in K^J$ and

$$\begin{aligned} \Vert d J(u)\Vert _{X'} \le \Vert V(u)\Vert _X \le 2 \Vert d J(u)\Vert _{X'},\quad \langle d J(u),V(u) \rangle \ge \Vert d J(u)\Vert _{X'}^2 \end{aligned}$$

(2.4)

for all $u \in X$, and V can be chosen odd if J is even (see [27, Chapter II]).

Now, taking $\varepsilon ^* > 0$, such that

$$\begin{aligned} 2\varepsilon ^* < \min \left\{ \varepsilon _0, \varepsilon _\varrho , \frac{\varrho \ \mu _\varrho }{4 \sigma _0}\right\} , \end{aligned}$$

with $\sigma _0$ as in (2.1), for any $\varepsilon \in \ ]0,\varepsilon ^*]$, we can define a Lipschitz continuous cut-off function $\chi _\varepsilon : X \rightarrow [0,1]$, such that

$$\begin{aligned} \chi _\varepsilon (u)\ =\ \left\{ \begin{array}{ll} 0 &{}\hbox {if }u \not \in J_{\beta -2\varepsilon }^{\beta +2\varepsilon }\\ 1 &{}\hbox {if }u \in J_{\beta -\varepsilon }^{\beta +\varepsilon } \end{array} \right. . \end{aligned}$$

(2.5)

On the other hand, taking

$$\begin{aligned} N^W = N^W_{\varrho /8}(K^J_\beta ), \quad C^W = \{u \in X:\, d_W(u,K^J_\beta ) \ge \frac{\varrho }{4}\}, \end{aligned}$$

(both closed also with respect to $\Vert \cdot \Vert _X$) with $N^W \cap C^W = \emptyset$, we can define

$$\begin{aligned} \vartheta (u)\ =\ \frac{d_W(u,N^W)}{d_W(u,N^W) + d_W(u,C^W)}. \end{aligned}$$

By direct computations and from (2.1) it follows that $\vartheta : X \rightarrow [0,1]$ is a Lipschitz continuous function, such that

$$\begin{aligned} \vartheta (u)\ =\ \left\{ \begin{array}{ll} 0 &{}\hbox {if }u \in N^W\\ 1 &{}\hbox {if }u \in C^W \end{array} \right. . \end{aligned}$$

(2.6)

Defining

$$\begin{aligned} V_\varepsilon (u)\ =\ \left\{ \begin{array}{ll} \displaystyle -\ \chi _\varepsilon (u)\ \vartheta (u)\ \frac{V(u)}{\Vert V(u)\Vert _X} &{}\quad \hbox {if }u \not \in K^J\\ 0 &{}\quad \hbox {if }u \in K^J \end{array} \right. , \end{aligned}$$

(2.7)

for any “initial point” $u \in X$, we consider the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial \eta }{\partial s}(s;u)\ =\ V_\varepsilon (\eta (s;u)),\\ \eta (0;u) = u. \end{array} \right. \end{aligned}$$

(2.8)

By construction, the function $V_\varepsilon$ is locally Lipschitz continuous and bounded with respect to $\Vert \cdot \Vert _X$; hence, for each $u \in X$ a unique $C^1$ function $\eta (\cdot ;u): \mathbb R\rightarrow X$ exists which solves (2.8). Moreover, $\eta (s;\cdot ): X \rightarrow X$ is a homeomorphism for each $s \in \mathbb R$ and is odd if J is even.

We note that, from definitions (2.5) and (2.6), $\eta (s;u) = u$ not only if $s = 0$ for all $u \in X$ (initial datum in (2.8)), but also for all $s \in \mathbb R$, if $u \not \in J_{\beta -2\varepsilon }^{\beta +2\varepsilon }$ or $u \in N^W_{\varrho /8}(K^J_\beta )$. In particular, if $u \not \in \mathcal{J}_{\beta -\varepsilon _0}^{\beta +\varepsilon _0}$, $\eta (s;u) \equiv u$ for all $s \in \mathbb R$.

From (2.4), (2.7) and (2.8) it follows that

$$\begin{aligned} s \in \mathbb R\ \mapsto \ J(\eta (s;u)) \in \mathbb R\quad \hbox {is decreasing for any fixed }u \in X, \end{aligned}$$

(2.9)

thus, $J(\eta (s;u)) \le J(u)$ for all $s \ge 0$.

We point out that, taking any $u \in X$, (2.1) and (2.8) imply that

$$\begin{aligned} \Vert \eta (s_1;u) - \eta (s_2;u)\Vert _W \ \le \ \sigma _0 \Vert \eta (s_1;u) - \eta (s_2;u)\Vert _X \ \le \ \sigma _0 |s_1 - s_2|; \end{aligned}$$

whence

$$\begin{aligned} \Vert \eta (s;u) - u\Vert _W \ \le \ \sigma _0 s\quad \hbox {for all }s \ge 0, \end{aligned}$$

so, fixing $s^* = \frac{\varrho }{2 \sigma _0}$, we have that

$$\begin{aligned} \Vert \eta (s^*;u) - u\Vert _W \ \le \ \frac{\varrho }{2}. \end{aligned}$$

(2.10)

Now, let $u \in J^{\beta + \varepsilon }$. If $u \not \in N^W_{\varrho }(K^J_\beta )$, $s \in [0,s^*]$ exists, such that $\eta (s;u) \in J^{\beta -\varepsilon }$, then (2.9) implies $\eta (s^*;u) \in J^{\beta -\varepsilon }$. On the contrary, if $u \in N^W_{\varrho }(K^J_\beta )$, either $s \in ]0,s^*]$ exists, such that $\eta (s;u) \in N^W_{\varrho /2}(K^J_\beta )$, thus (2.10) implies that $\eta (s^*;u) \in N^W_{\varrho }(K^J_\beta )$, or $\eta (s^*;u) \in J^{\beta -\varepsilon }$. Hence, we choose $\Psi = \eta (s^*;\cdot )$. $\square$

Now, we assume that J is even and $J(0) = 0$ and, in order to obtain multiple critical points, we quote the main tools on the $\mathbb Z_2$-cohomological index on a Banach space X, as introduced by Fadell and Rabinowitz in [18].

Firstly, let us recall the definition and some basic properties of the cohomological index $i(\cdot )$, defined in the Banach space $(X,\Vert \cdot \Vert _X)$.

Taking

$$\begin{aligned} \mathcal{P}\ =\ \{P \subset X {\setminus } \{0\}:\ P \ \hbox {symmetric, i.e.} -u \in P \hbox { if }u \in P\}, \end{aligned}$$

(2.11)

for $P \in \mathcal{P}$ we denote by

• $\; {\tilde{P}} = P/\mathbb Z_2$ the quotient space of P with each u and $-u$ identified,

• $\; f : {\tilde{P}} \rightarrow \mathbb R\text {P}^\infty$ the classifying map of ${\tilde{P}}$,

• $\; f^*: H^*(\mathbb R\text {P}^\infty ) \rightarrow H^*({\tilde{P}})$ the induced homomorphism of the Alexander–Spanier cohomology rings.

Then the cohomological index of $P \in \mathcal{P}$ is defined by

$$\begin{aligned} i(P)\ =\ {\left\{ \begin{array}{ll} \sup \big \{m \ge 1 : f^*(\omega ^{m-1}) \ne 0\big \} &{}\hbox {if}\, P \ne \emptyset ,\\ 0 &{}\hbox {if}\, P = \emptyset , \end{array}\right. } \end{aligned}$$

where $\omega \in H^1(\mathbb R\text {P}^\infty )$ is the generator of the polynomial ring $H^*(\mathbb R\text {P}^\infty ) = \mathbb Z_2[\omega ]$.

Here we list the basic properties of the cohomological index (see, e.g., [24, Proposition 2.12]).

Proposition 2.5

Index $i : \mathcal{P}\rightarrow \mathbb N\cup \{0,+\infty \}$ has the following properties:

• $(i_{1})$ Definiteness: taking $P \in \mathcal{P}$, $i(P) = 0$ if and only if $P = \emptyset$;

• $(i_{2})$ Monotonicity: let B be a topological space and let $\eta : X \rightarrow B$ be an odd continuous map, then $i(P) \le i(\eta (P))$ for any $P \in \mathcal{P}$. Hence, the equality holds when the map is an odd homeomorphism. In particular, if $P_1$, $P_2 \in \mathcal{P}$ are such that $P_1 \subset P_2$, then $i(P_1) \le i(P_2)$;

• $(i_{3})$ Dimension: taking any finite dimensional space $X_0 \subset X$ and $P \in \mathcal{P}$, such that $P \subset X_0$, then $i(P) \le \dim X_0$;

• $(i_{4})$ Continuity: If $P \in \mathcal{P}$ is closed, then there is a closed neighbourhood $U \in \mathcal{P}$ of P, such that $i(U) = i(P)$. When P is compact, then U may be chosen to be a $\varrho$-neighbourhood $\displaystyle N^X_\varrho (P)$;

• $(i_{5})$ Subadditivity: If $P_1, P_2 \in \mathcal{P}$ are closed, then $i(P_1 \cup P_2) \le i(P_1) + i(P_2)$;

• $(i_{6})$ Stability: taking $P \in \mathcal{P}$, if SP is the suspension of $P \ne \emptyset$, obtained as the quotient space of $P \times [-1,1]$ with $P \times \{1\}$ and $P \times \{-1\}$ collapsed at different points, then $SP \in \mathcal{P}$ and $i(SP) = i(P) + 1$;

• $(i_{7})$ Piercing property: If $P, P_0, P_1 \in \mathcal{P}$ are closed and $\varphi : P \times [0,1] \rightarrow P_0 \cup P_1$ is a continuous map, such that $\varphi (-u,s) = - \varphi (u,s)$ for all $(u,s) \in P \times [0,1]$, $\varphi (P \times [0,1])$ is closed, $\varphi (P \times \{0\}) \subset P_0$, and $\varphi (P \times \{1\}) \subset P_1$, then $i(\varphi (P \times [0,1]) \cap P_0 \cap P_1) \ge i(P)$;

• $(i_{8})$ Neighbourhood of zero: If U is a bounded closed symmetric neighbourhood of 0 contained in a finite dimensional subspace $X_0 \subset X$, then $\partial {U} \in \mathcal{P}$ and $i(\partial {U}) = \dim X_0$.

Remark 2.6

Since in our setting X is continuously imbedded in W, namely a continuous map $j_W: X \rightarrow W$ exists, for simplicity we put $i_W(P) = i(j_W(P))$ if $P \in \mathcal{P}$. Thus, from the monotonicity property $(i_{2})$ it follows that

$$\begin{aligned} i(P) \le i_W(P)\qquad \hbox {for all }P \in \mathcal{P}. \end{aligned}$$

(2.12)

We point out that our problem deals with a functional $J:X\rightarrow \mathbb R$, which is $C^1$ with respect to $\Vert \cdot \Vert _X$, but cannot satisfy the Palais–Smale condition in the same Banach space, as such a norm is “too strong”. Hence, the classical multiplicity theorem in [24, Proposition 3.36] cannot be applied and has to be generalized. Here we state an abstract multiplicity theorem by working with the stronger norm $\Vert \cdot \Vert _X$, but assuming (wPS), so that Proposition 2.4 holds.

To this aim, for any integer $k \in \mathbb N$ let us define

$$\begin{aligned} c_k := \inf _{P \in \mathcal{P}_k}\, \sup _{u \in P}\, J(u), \end{aligned}$$

(2.13)

with

$$\begin{aligned} \mathcal{P}_k = \{P \subset X{\setminus } \{0\} :\ P\, \text {symmetric and compact in }X \text { with }i(P) \ge k\}. \end{aligned}$$

Since $\mathcal{P}_{k+1} \subset \mathcal{P}_k$, then

$$\begin{aligned} c_k \le c_{k+1}. \end{aligned}$$

Furthermore, for any k-dimensional subspace $X_0$ of X and $\delta > 0$, from $(i_{8})$ we have $\partial {B^X_\delta (0)} \cap X_0 \in \mathcal{P}_k$, while from the continuity of J in $(X,\Vert \cdot \Vert _X)$ we have

$$\begin{aligned} \sup _{u \in \partial {B^X_\delta (0)}} J(u)\ \rightarrow \ J(0)\quad \hbox {as } \delta \rightarrow 0; \end{aligned}$$

hence,

$$\begin{aligned} c_k\ \le \ J(0)\ =\ 0. \end{aligned}$$

(2.14)

Theorem 2.7

Let $J :X \rightarrow \mathbb R$ be an even functional of class $C^1$, such that $J(0) = 0$, which satisfies (wPS) in $\mathbb R$. If h, $m \in \mathbb N$ exist, such that

$$\begin{aligned} -\infty< c_h \le \ \cdots \le c_{h+m-1} < 0, \end{aligned}$$

(2.15)

then J has at least m distinct pairs of nontrivial critical points in X with a negative critical level. Furthermore, if

$$\begin{aligned} -\infty< c_k < 0\qquad \hbox {for all sufficiently large }k, \end{aligned}$$

(2.16)

then $c_k\ \nearrow \ 0$ and J has infinitely many distinct pairs of nontrivial critical points in X.

Proof

The proof can be essentially split into three parts.

Step 1. If $k \in \mathbb N$ is such that $-\infty< c_k < 0$, then level $c_k$ is critical. In fact, otherwise Proposition 2.4, with $N_\varrho ^W(K^J_{\beta }) = \emptyset$, $\beta = c_k$, yields a contradiction.

Step 2. If (2.15) holds, from Step 1. it is enough to prove that, if $k\in \{h,\dots ,h+m-2\}$ and $j \in \mathbb N$ exist, such that $\beta = c_k =\dots = c_{k+j}$, then the critical point set $K^J$ has infinitely many elements.

From Lemma 2.2, $K^J_\beta \in \mathcal{P}$ is compact in $(W,\Vert \cdot \Vert _W)$, then the continuity property $(i_{4})$ in Proposition 2.5 implies the existence of a radious $\varrho > 0$, such that

$$\begin{aligned} i_W(N_\varrho ^W(K^J_\beta ))\ =\ i_W(K^J_\beta ). \end{aligned}$$

(2.17)

Fixing $\varepsilon _0 > 0$, such that $\beta + \varepsilon _0 < 0$, from Proposition 2.4 for $\varepsilon \in ]0,\varepsilon _0[$ small enough an odd homeomorphism $\Psi : X \rightarrow X$ exists, such that (i) and (ii) in Proposition 2.4 hold. Thus, from $c_{k+j} < \beta + \varepsilon$ a set $P_\varepsilon \in \mathcal{P}_{k+j}$ exists, such that $P_\varepsilon \subset J^{\beta + \varepsilon }$ and from $(i_{5})$ in Proposition 2.5, (2.12) and (2.17) it follows that

$$\begin{aligned} \begin{aligned} k+j\,&\le \ i(P_\varepsilon ) \le i(P_\varepsilon {\setminus } N^W_{\varrho }(K^J_\beta )) + i(N^W_{\varrho }(K^J_\beta ))\\&\le \ i(\overline{\Psi (P_\varepsilon {\setminus } N^W_{\varrho }(K^J_\beta ))}) + i_W(N^W_{\varrho }(K^J_\beta )) \le \ k -1 + i_W(K^J_\beta ), \end{aligned} \end{aligned}$$

as $\beta -\varepsilon < c_k$ implies that $\overline{\Psi (P_\varepsilon {\setminus } N^W_{\varrho }(K^J_\beta ))}$ is a compact symmetric subset of $X{\setminus } \{0\}$ but it is not in $\mathcal{P}_k$. Hence, $i_W(K^J_\beta ) \ge j+1 \ge 2$ and then $K^J_\beta$ has infinitely many elements.

Step 3. Now, let us assume condition (2.16). Since a large enough $k \in \mathbb N$ can be fixed so that condition (2.15) holds for all $m \in \mathbb N$, then Step 2. implies that J has infinitely many distinct pairs of nontrivial critical points in X. So, we have only to prove that the increasing sequence of critical levels $(c_k)_k$ goes to zero. Arguing by contradiction, we assume that $c_k\ \nearrow \ \bar{c}$ with $\displaystyle \bar{c} = \sup _{k \in \mathbb N} c_k < 0$. By reasoning as in Step 1., it follows that $\bar{c}$ is also a critical level of J in X; hence, from Lemma 2.2 and property $(i_{4})$ in Proposition 2.5, a radious $\varrho > 0$ exists, such that (2.17) holds with $\beta = \bar{c}$. Fixing $\varepsilon _0 > 0$, such that $\bar{c} + \varepsilon _0 < 0$, from Proposition 2.4 for $\varepsilon \in ]0,\varepsilon _0[$ small enough an odd homeomorphism $\Psi : X \rightarrow X$ exists, such that (i) and (ii) in Proposition 2.4 hold. Moreover, a large enough integer k exists, so that

$$\begin{aligned} \bar{c} -\varepsilon< c_k \le c_{k+j} \le \bar{c}< \bar{c}+\varepsilon < 0\quad \hbox {for all }j \in \mathbb N. \end{aligned}$$

Hence, reasoning as in Step 2. with $\beta = \bar{c}$, we prove that $i_W(K^J_\beta ) \ge j+1$ for all $j \in \mathbb N$, i.e. $i_W(K^J_{\bar{c}}) = +\infty$ in contradiction with Lemma 2.2. $\square$

Remark 2.8

Theorem 2.7 holds also if the assumption of compactness is weakened, i.e., $\mathcal{P}_k$ is the set of symmetric subsets of $X{\setminus } \{0\}$, which are closed in X with $i(P) \ge k$.

Since all the critical levels defined as in (2.13), by using the cohomological index, are non-positive (see (2.14)), in order to deal with positive levels we have to replace the cohomological index $i(\cdot )$ with the related pseudo-index introduced by Benci in [7]. So, we recall the definition of the pseudo-index and some of its basic properties (here, we consider the pseudo-index when X is equipped with $\Vert \cdot \Vert _X$).

Let $\mathcal{P}^*$ denote the class of symmetric subsets of X, let $\mathcal{M}\in \mathcal{P}$ be closed in X (see (2.11)), and define

$$\begin{aligned} \begin{aligned} \mathcal{H}\ =\ \{\gamma :X \rightarrow X\ :\,&\,\,\gamma \hbox { is an odd homeomorphism, such that }\\&\gamma (u) = u \hbox { for all }u \in J^0\}. \end{aligned} \end{aligned}$$

Then, the pseudo-index of $P \in \mathcal{P}^*$ related to $i(\cdot )$, $\mathcal{M}$ and $\mathcal{H}$ is defined by

$$\begin{aligned} i^*(P) = \min _{\gamma \in \mathcal{H}}\, i(\gamma (P) \cap \mathcal{M}). \end{aligned}$$

(2.18)

Proposition 2.9

The pseudo-index $i^*: \mathcal{P}^*\rightarrow \mathbb N\cup \{0,+\infty \}$ has the following properties:

• $({i^*_{1}})$ if $P_1$, $P_2 \in \mathcal{P}^*$ are such that $P_1 \subset P_2$, then $i^*(P_1) \le i^*(P_2)$;

• $(i^*_{2})$ if $\eta \in \mathcal{H}$ and $P \in \mathcal{P}^*$, then $i^*(P) = i^*(\eta (P))$;

• $(i^*_{3})$ if $P \in \mathcal{P}^*$ and $B \in \mathcal{P}$ are closed, then $i^*(P \cup B) \le i^*(P) + i(B)$.

As already pointed out, here we want to apply the pseudo-index theory to our setting and we have to generalize the classical statement in [24, Proposition 3.42].

For any integer $k \ge 1$, such that $k \le i(\mathcal{M})$, let

$$\begin{aligned} \mathcal{P}_k^*\ =\ \{P \subset X :\ P\; \text {symmetric and compact in }X \text { with }i^*(P) \ge k\} \end{aligned}$$

and set

$$\begin{aligned} c_k^*\ :=\ \inf _{P \in \mathcal{P}_k^*}\, \sup _{u \in P}\, J(u). \end{aligned}$$

From $\mathcal{P}_{k+1}^*\subset \mathcal{P}_k^*$, it follows that $c_k^*\le c_{k+1}^*$.

Theorem 2.10

Let $J :X \rightarrow \mathbb R$ be an even functional of class $C^1$ which satisfies (wPS) in $\mathbb R$ and is such that $J(0) = 0$. If h, $m \in \mathbb N$ exist, such that

$$\begin{aligned} 0< c_h^*\ \le \ \cdots \ \le \ c_{h+m-1}^*< +\infty , \end{aligned}$$

then J has at least m distinct pairs of nontrivial critical points in X with a positive critical level.

Proof

Firstly, we claim that each $\beta = c_k^*$, $k \in \{h,\dots , h+m-1\}$, is a critical level of J in X. Otherwise, fixing $\varepsilon _0 > 0$ such that $\beta - \varepsilon _0 >0$, from Proposition 2.4 for small enough $\varepsilon < \varepsilon _0$ a map $\Psi \in \mathcal{H}$ exists, such that $\Psi (J^{\beta + \varepsilon }) \subset J^{\beta - \varepsilon }$. On the other hand, from definition $P^*_\varepsilon \in \mathcal{P}^*_k$ exists, such that $P^*_\varepsilon \in J^{\beta + \varepsilon }$. Hence, the properties of $\Psi$ imply that not only $\Psi (P^*_\varepsilon ) \in \mathcal{P}^*_k$, but also $\Psi (P^*_\varepsilon ) \subset J^{\beta - \varepsilon }$, i.e. $\beta \le \sup J(\Psi (P^*_\varepsilon )) \le \beta - \varepsilon$: a contradiction.

Now, in order to complete the proof, it is sufficient to investigate what happens if $k\in \{h,\dots ,h+m-2\}$ and $j \in \mathbb N$ exist, such that $\beta = c^*_k =\dots = c^*_{k+j} > 0$. Accordingly to these assumptions, by reasoning as in the proof of Theorem 2.7, $\varrho > 0$ exists, such that (2.17) holds. Then, fixing $\varepsilon _0 > 0$, such that $\beta + \varepsilon _0 > 0$, from Proposition 2.4 for small enough $\varepsilon \in ]0,\varepsilon _0[$ a map $\Psi \in \mathcal{H}$ exists, such that (ii) in Proposition 2.4 holds. Thus, from Proposition 2.9, (2.12) and (2.17) it follows that

$$\begin{aligned} \begin{aligned} i^*(J^{\beta + \varepsilon })&\le i^*(J^{\beta + \varepsilon } {\setminus } N^W_{\varrho }(K^J_\beta )) + i(N^W_{\varrho }(K^J_\beta ))\\&\le i^*(\Psi (J^{\beta + \varepsilon } {\setminus } N^W_{\varrho }(K^J_\beta ))) + i_W(N^W_{\varrho }(K^J_\beta ))\\&\le i^*(\overline{\Psi (J^{\beta + \varepsilon } {\setminus } N^W_{\varrho }(K^J_\beta ))}) + i_W(K^J_\beta ), \end{aligned} \end{aligned}$$

and, as in Theorem 2.7, it has to be $i_W(K^J_\beta ) \ge 2$. $\square$

Remark 2.11

Theorem 2.10 holds also if $\mathcal{P}^*_k$ is the set of the symmetric subsets which are closed in X with $i^*(P) \ge k$.

3 Hypotheses and first properties

From now on, we investigate the existence of weak solutions of the nonlinear problem (GP), where $\Omega \subset \mathbb R^N$ is a bounded domain, $N\ge 2$, so the notations introduced for the abstract setting at the beginning of Sect. 2 are referred to our problem with $(X,\Vert \cdot \Vert _X)$ the Banach space defined as

$$\begin{aligned} X := W^{1,p}_0(\Omega ) \cap L^\infty (\Omega ),\qquad \Vert u\Vert _X = \Vert u\Vert _W + |u|_\infty , \end{aligned}$$

(3.1)

with

$$\begin{aligned} \Vert u\Vert _W \ =\ \left( \int _\Omega |\nabla u|^p\, dx\right) ^{\frac{1}{p}}, \quad |u|_{\infty }\ =\ \mathop {{{\mathrm{ess \; sup}}}}\limits _{x\in \Omega } |u(x)| \end{aligned}$$

(here and in the following, $|\cdot |$ will denote the standard norm on any Euclidean space as the dimension of the considered vector is clear and no ambiguity arises), while $(W,\Vert \cdot \Vert _W) = (W^{1,p}_0(\Omega ),\Vert \cdot \Vert _W)$, and $J = \mathcal{J}$ the functional in (1.1). Moreover, we denote by

• $(W^{-1,p'}(\Omega ),\Vert \cdot \Vert _{W^{-1}})$ the dual space of $(W^{1,p}_0(\Omega ),\Vert \cdot \Vert _W)$,

• $L^q(\Omega )$ the Lebesgue space equipped with the canonical norm $|\cdot |_q$ for any $1 \le q \le +\infty$,

• $\mathrm{meas}( \cdot )$ the usual Lebesgue measure in $\mathbb R^N$,

• $\Omega ^u_r = \{x \in \Omega : |u(x)| > r\}$ if $u : \Omega \rightarrow \mathbb R$, $r > 0$,

and let us recall that, from the Sobolev Imbedding Theorem, $\sigma _p > 0$ exists, such that

$$\begin{aligned} \int _\Omega |u|^p dx\ \le \ \sigma _p \int _\Omega |\nabla u|^p dx \quad \hbox {for all }u \in W^{1,p}_0(\Omega ) \end{aligned}$$

(3.2)

and the imbedding $W^{1,p}_0(\Omega ) \hookrightarrow \hookrightarrow L^{p}(\Omega )$ is compact.

From definition, $X \hookrightarrow W^{1,p}_0(\Omega )$ and $X \hookrightarrow L^\infty (\Omega )$ with continuous imbeddings and (2.1) holds with $\sigma _0 = 1$. Moreover, in the stronger assumption $p > N$, we have

$$\begin{aligned} |u|_\infty \ \le \ \sigma _\infty \ \Vert u\Vert _W \qquad \hbox {for all }u \in W^{1,p}_0(\Omega ); \end{aligned}$$

(3.3)

hence, in this case $X = W^{1,p}_0(\Omega )$ and the two norms $\Vert \cdot \Vert _X$ and $\Vert \cdot \Vert _W$ are equivalent.

Here and in the following, let us consider problem (GP), where

$$\begin{aligned} A : (x,t,\xi ) \in \Omega \times \mathbb R\times \mathbb R^N \mapsto A(x,t,\xi )\in \mathbb R\end{aligned}$$

is a Carathéodory function of class $C^1$, i.e. measurable with respect to x in $\Omega$ for all $(t,\xi )\in \mathbb R\times \mathbb R^N$ and $C^1$ with respect to $(t,\xi )$ in $\mathbb R\times \mathbb R^N$ for a.e. $x \in \Omega$, with $A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )$, $a(x,t,\xi ) = \nabla _\xi A(x,t,\xi ) = (\frac{\partial A}{\partial \xi _1}(x,t,\xi ), \dots ,\frac{\partial A}{\partial \xi _N}(x,t,\xi ))$, which satisfies the following conditions:

• $(H_1)$ $p > 1$ exists and some positive continuous functions $\Phi _j: \mathbb R\rightarrow \mathbb R$, $j \in \{0,1\}$, and $\phi _i : \mathbb R\rightarrow \mathbb R$, $i \in \{0,1,2\}$, are such that

  $$\begin{aligned} |A(x,t,\xi )|\le & {} \Phi _0(t) + \phi _0(t)\ |\xi |^p, \end{aligned}$$

  (3.4)
  $$\begin{aligned} |a(x,t,\xi )|\le & {} \Phi _1(t) + \phi _1(t)\ |\xi |^{p-1}, \end{aligned}$$

  (3.5)
  $$\begin{aligned} |A_t(x,t,\xi )|\le & {} \phi _2(t)\ |\xi |^p \end{aligned}$$

  (3.6)

  for a.e. $x \in \Omega$ and all $(t,\xi ) \in \mathbb R\times \mathbb R^N$;

• $(H_2)$ $\alpha _0 > 0$ exists, such that

  $$\begin{aligned} a(x,t,\xi )\cdot \xi \ge \alpha _0 |\xi |^p\qquad \hbox {a.e. in }\Omega , \hbox { for all }(t,\xi ) \in \mathbb R\times \mathbb R^N; \end{aligned}$$

• $(H_3)$ $\alpha _1 > 0$ exists, such that

  $$\begin{aligned} A(x,t,\xi )\ \ge \ \alpha _1 |\xi |^p\qquad \hbox {a.e. in }\Omega ,\text { for all }(t,\xi ) \in \mathbb R\times \mathbb R^N; \end{aligned}$$

• $(H_4)$ a (Carathéodory) function

  $$\begin{aligned} A^\infty : (x,\xi ) \in \Omega \times \mathbb R^N\ \mapsto \ A^\infty (x,\xi ) \in \mathbb R\end{aligned}$$

  exists, which is positively p-homogeneous in $\xi$ for a.e. $x \in \Omega$, and satisfies the following condition: for all $\varepsilon > 0$ a constant $r_\varepsilon > 0$ exists, such that

  $$\begin{aligned} |t| \ge r_\varepsilon \;\Longrightarrow \; \big |A(x,t,\xi ) - A^\infty (x,\xi )\big | \le \varepsilon |\xi |^{p} \; \text {for a.e. }x \in \Omega ,\text { all }\xi \in \mathbb R^N; \end{aligned}$$

• $(H_5)$ a (Carathéodory) vector field

  $$\begin{aligned} a^\infty : (x,\xi ) \in \Omega \times \mathbb R^N\ \mapsto \ a^\infty (x,\xi ) = (a_1^\infty (x,\xi ),\dots , a_N^\infty (x,\xi )) \in \mathbb R^N \end{aligned}$$

  exists, which satisfies the following condition: for all $\varepsilon > 0$ a constant $r_\varepsilon > 0$ exists, such that

  $$\begin{aligned} |t| \ge r_\varepsilon \, \Longrightarrow \, \big |a(x,t,\xi ) - a^\infty (x,\xi )\big | \le \varepsilon |\xi |^{p-1} \; \text {for a.e. }x \in \Omega ,\hbox { all }\xi \in \mathbb R^N; \end{aligned}$$

• $(H_6)$ for all $\xi$, $\xi ^* \in \mathbb R^N$, $\xi \ne \xi ^*$,

  $$\begin{aligned}{}[a(x,t,\xi ) - a(x,t,\xi ^*)]\cdot [\xi - \xi ^*] > 0 \quad \hbox {a.e. in }\Omega ,\hbox { for all }t\in \mathbb R. \end{aligned}$$

Remark 3.1

Since $a(x,t,\cdot )$ is continuous for a.e. $x \in \Omega$ and all $t \in \mathbb R$, hypothesis $(H_2)$ implies that

$$\begin{aligned} a(x,t,0)\ =\ 0\quad \text {for a.e. }x \in \Omega ,\hbox { all }t \in \mathbb R\end{aligned}$$

(3.7)

(it is sufficient to fix any $\xi \ne 0$ and apply $(H_2)$ once to $s\xi$, then to $- s \xi$, and in both passing to the limit as $s \rightarrow 0^+$).

Moreover, from (3.5), $(H_5)$ and direct computations, it follows that $M_1$, $M_2 > 0$ exist, such that

$$\begin{aligned} \big |a^\infty (x,\xi )\big |\ \le \ M_1 + M_2 |\xi |^{p-1} \quad \text {for a.e. }x \in \Omega ,\hbox { all }\xi \in \mathbb R^N. \end{aligned}$$

(3.8)

On the other hand, (3.4), $(H_3)$ and $(H_4)$ imply that $\alpha _2 >0$ exists, such that

$$\begin{aligned} \alpha _1 |\xi |^{p}\ \le \ A^\infty (x,\xi )\ \le \ \alpha _2 |\xi |^{p} \quad \text {for a.e. }x \in \Omega , \mathrm{all}\,\,\xi \in \mathbb R^N. \end{aligned}$$

(3.9)

If (3.8) holds, we can consider the nonlinear operator associated to $a^\infty (x,\xi )$, namely,

$$\begin{aligned} A_p^\infty : u \in W^{1,p}_0(\Omega ) \mapsto A_p^\infty u \in W^{-1,p'}(\Omega ), \end{aligned}$$

such that

$$\begin{aligned} \langle A_p^\infty u,v\rangle \ =\ \int _\Omega a^\infty (x,\nabla u)\cdot \nabla v\ dx \qquad \hbox {for any }u, v \in W^{1,p}_0(\Omega ), \end{aligned}$$

(3.10)

and denote its spectrum by $\sigma (A_p^\infty )$.

By definition, $\lambda \in \sigma (A_p^\infty )$ if some $u \in W^{1,p}_0(\Omega )$, $u \not \equiv 0$, exist, such that

$$\begin{aligned} \int _\Omega a^\infty (x,\nabla u) \cdot \nabla \varphi \ dx \ =\ \lambda \int _\Omega |u|^{p-2}u\varphi \ dx \qquad \hbox {for all }\varphi \in W^{1,p}_0(\Omega ). \end{aligned}$$

Lemma 3.2

Assume that $A(x,t,\xi )$ and its gradient with respect to $\xi$, namely $a(x,t,\xi )$, satisfy the growth estimate (3.5) and the hypotheses $(H_2)$, $(H_4)$–$(H_6)$. Then, the nonlinear operator $A_p^\infty$ defined in (3.10) is:

• (i) continuous from the reflexive Banach space $W^{1,p}_0(\Omega )$ to its dual $W^{-1,p'}(\Omega )$;

• (ii) a potential operator such as $a^\infty (x,\xi ) = \nabla _\xi A^\infty (x,\xi )$ for a.e. $x \in \Omega$, all $\xi \in \mathbb R^N$;

• (iii) $(p-1)$-homogeneous and it is odd if $A^\infty (x,\cdot )$ is even for a.e. $x \in \Omega$;

• (iv) uniformly positive as

  $$\begin{aligned} a^\infty (x,\xi )\cdot \xi \ \ge \ \alpha _0 |\xi |^p \quad \hbox {a.e. in }\Omega ,\hbox { for all }\xi \in \mathbb R^N; \end{aligned}$$

• (v) of type (S): if $(u_n)_n\subset W^{1,p}_0(\Omega )$ and $u \in W^{1,p}_0(\Omega )$ are such that

  $$\begin{aligned} u_n \rightharpoonup u\ \hbox {weakly in }W^{1,p}_0(\Omega ),\quad \langle A_p^\infty u_n,u_n-u\rangle \rightarrow 0, \end{aligned}$$

  then $u_n \rightarrow u$ strongly in $W^{1,p}_0(\Omega )$, up to subsequences.

Proof

• (i) The proof follows from the growth estimate (3.8) of the Carathéodory vector field $a^\infty (x,\xi )$ and the properties of the related Nemitskii operator.

• (ii) Fixing $i \in \{1,\dots ,N\}$, $\xi \in \mathbb R^N$, $h \in \mathbb R$, for a.e. $x \in \Omega$, we have that

  $$\begin{aligned} A(x,t,\xi +h e_i) - A(x,t,\xi )\ =\ h\ \int _0^1 a_i(x,t,\xi + \theta h e_i)d\theta , \end{aligned}$$

  for all $t \in \mathbb R$, where $(H_4)$ implies

  $$\begin{aligned} A(x,t,\xi +h e_i) - A(x,t,\xi ) \rightarrow A^\infty (x,\xi +h e_i) - A^\infty (x,\xi )\quad \hbox {as }|t| \rightarrow +\infty , \end{aligned}$$

  while from $(H_5)$ it follows that

  $$\begin{aligned} a_i(x,t,\xi + \theta h e_i) \rightarrow a_i^\infty (x,\xi + \theta h e_i)\quad \hbox {as }|t| \rightarrow +\infty \end{aligned}$$

  uniformly with respect to $\theta \in [0,1]$, hence,

  $$\begin{aligned} \int _0^1 a_i(x,t,\xi + \theta h e_i)d\theta \ \rightarrow \ \int _0^1 a_i^\infty (x,\xi + \theta h e_i)d\theta \quad \hbox {as }|t| \rightarrow +\infty . \end{aligned}$$

  On the other hand, as $A^\infty (x,\cdot )$ is p-homogeneous with $p > 1$, then it is differentiable and $\frac{\partial A^\infty }{\partial \xi _i}(x,\xi )$ exists for a.e. $x \in \Omega$, all $\xi \in \mathbb R^N$, while the continuity of $a_i^\infty (x,\cdot )$, (3.8) and the Lebesgue’s dominated convergence theorem imply

  $$\begin{aligned} \int _0^1 a_i^\infty (x,\xi + \theta h e_i)d\theta \ \rightarrow \ a_i^\infty (x,\xi ) \quad \hbox {as }h \rightarrow 0. \end{aligned}$$

  Hence, $\frac{\partial A^\infty }{\partial \xi _i}(x,\xi ) = a_i^\infty (x,\xi )$.

• (iii) It follows from (ii) and the properties of homogeneous functions.

• (iv) It is a direct consequence of $(H_2)$ and $(H_5)$.

• (v) From assumption $(H_6)$, it is a direct consequence of [8, Lemma 5] (see also [10, pp. 27]).

$\square$

Remark 3.3

From assumptions $(H_1)$, $(H_4)$, $(H_5)$, the properties of homogeneous functions and direct computations it follows that some constants $M_0$, $M_1$, $M_2 > 0$ exist, such that

$$\begin{aligned} |a^\infty (x,\xi )|\le & {} M_0 |\xi |^{p-1} \quad \hbox {for a.e. }x \in \Omega ,\hbox { all }\xi \in \mathbb R^N, \end{aligned}$$

(3.11)

$$\begin{aligned} |A(x,t,\xi )|\le & {} M_1 + M_2 |\xi |^p \quad \hbox {for a.e. }x \in \Omega ,\hbox { all }(t,\xi ) \in \mathbb R\times \mathbb R^N, \end{aligned}$$

(3.12)

$$\begin{aligned} |a(x,t,\xi )|\le & {} M_1 + M_2 |\xi |^{p-1} \quad \hbox {for a.e. }x \in \Omega ,\hbox { all }(t,\xi ) \in \mathbb R\times \mathbb R^N. \end{aligned}$$

(3.13)

Moreover, suppose that

$$\begin{aligned} f : (x,t) \in \Omega \times \mathbb R\mapsto f(x,t) \in \mathbb R\end{aligned}$$

is a Carthéodory function, i.e. measurable with respect to x in $\Omega$ for all $t\in \mathbb R$ and continuous with respect to t in $\mathbb R$ for a.e. $x \in \Omega$, which satisfies the hypotheses:

• $(h_1)$ for any $r > 0$ we have

  $$\begin{aligned} \sup _{|t| \le r} |f(\cdot ,t)| \in L^\infty (\Omega ); \end{aligned}$$

• $(h_2)$ $\lambda ^\infty \in \mathbb R$ and a (Carathéodory) function $g^\infty : \Omega \times \mathbb R\rightarrow \mathbb R$ exist, such that

  $$\begin{aligned} f(x,t)\ =\ \lambda ^\infty \ |t|^{p-2}\ t + g^\infty (x,t), \end{aligned}$$

  (3.14)

  where

  $$\begin{aligned} \lim _{|t| \rightarrow + \infty }\, \frac{g^\infty (x,t)}{|t|^{p-1}} = 0 \quad \text {uniformly a.e. in } \Omega . \end{aligned}$$

  (3.15)

Now and in the following, we set

$$\begin{aligned} F(x,t)\ =\ \int _0^t f(x,s) ds. \end{aligned}$$

Clearly, from the assumptions on f(x, t), it follows that both the functionals associated to f and F are continuous in $L^p(\Omega )$.

Remark 3.4

From (3.14) it follows that

$$\begin{aligned} F(x,t)\ =\ \frac{\lambda ^\infty }{p}\ |t|^p + G^\infty (x,t),\quad \hbox {with}\; G^\infty (x,t) = \int _0^t g^\infty (x,s) ds. \end{aligned}$$

(3.16)

Furthermore, $(h_1)$ and (3.14) imply that

$$\begin{aligned} \sup _{|t| \le r} |g^\infty (\cdot ,t)| \in L^\infty (\Omega )\qquad \hbox {for any }r > 0; \end{aligned}$$

(3.17)

while (3.17), respectively (3.15), implies that

$$\begin{aligned}&\sup _{|t| \le r} |G^\infty (\cdot ,t)| \in L^\infty (\Omega )\qquad \hbox {for any }r > 0, \end{aligned}$$

(3.18)

$$\begin{aligned}&\lim _{|t| \rightarrow + \infty }\, \frac{G^\infty (x,t)}{|t|^{p}} = 0 \quad \text {uniformly a.e. in } \Omega \end{aligned}$$

(3.19)

and then

$$\begin{aligned} \lim _{|t| \rightarrow + \infty }\, \frac{F(x,t)}{|t|^{p}}\ =\ \frac{\lambda ^\infty }{p} \quad \text {uniformly a.e. in } \Omega . \end{aligned}$$

(3.20)

Hence, (3.15) and (3.17), respectively (3.18) and (3.19), imply that for any $\varepsilon > 0$ a constant $L_\varepsilon > 0$ exists, such that

$$\begin{aligned} |g^\infty (x,t)|\le & {} \varepsilon |t|^{p-1} + L_\varepsilon \qquad \hbox {for a.e. }x \in \Omega ,\hbox { all }t \in \mathbb R, \end{aligned}$$

(3.21)

$$\begin{aligned} |G^\infty (x,t)|\le & {} \varepsilon |t|^p + L_\varepsilon \qquad \hbox {for a.e. }x \in \Omega ,\hbox { all }t \in \mathbb R; \end{aligned}$$

(3.22)

so (3.14) and (3.21) imply

$$\begin{aligned} |f(x,t)|\ \le \ \left( |\lambda ^\infty | + \varepsilon \right) |t|^{p-1} + L_\varepsilon \qquad \hbox {for a.e. }x \in \Omega ,\hbox { all }t \in \mathbb R, \end{aligned}$$

(3.23)

while (3.16) and (3.22) imply

$$\begin{aligned} |F(x,t)|\ \le \ \left( \frac{|\lambda ^\infty |}{p} + \varepsilon \right) |t|^p + L_\varepsilon \qquad \hbox {for a.e. }x \in \Omega ,\hbox { all }t \in \mathbb R. \end{aligned}$$

(3.24)

Hence, in particular, suitable constants $D_1$, $D_2 > 0$ exist, such that

$$\begin{aligned} |F(x,t)|\ \le \ D_1 |t|^p + D_2 \qquad \hbox {for a.e. }x \in \Omega , \hbox {all }t \in \mathbb R. \end{aligned}$$

(3.25)

Firstly, we need to prove that problem (GP) has a variational structure, since, effectively, its weak solutions are critical points of $\mathcal{J}$ in the Banach space X.

The following proposition can be stated (the proof is essentially the same as in [11, Proposition 3.1]).

Proposition 3.5

Let us assume that $A(x,t,\xi )$ satisfies the growth estimates (3.5), (3.6) and (3.12), while f(x, t) is such that $(h_1)$–$(h_2)$ hold. If $(u_n)_n \subset X$, $u \in X$ are such that

$$\begin{aligned} \Vert u_n - u\Vert _W \rightarrow 0 \qquad \hbox {then}\qquad \mathcal{J}(u_n) \rightarrow \mathcal{J}(u)\quad \hbox {as }n\rightarrow +\infty . \end{aligned}$$

Furthermore, if $r > 0$ exists so that

$$\begin{aligned} |u_n|_\infty \le r\qquad \hbox {for all }n \in \mathbb N, \end{aligned}$$

then also

$$\begin{aligned} \Vert d\mathcal{J}(u_n) - d\mathcal{J}(u)\Vert _{X'} \rightarrow 0\quad \hbox {as }\ n\rightarrow +\infty . \end{aligned}$$

In particular, $\mathcal{J}$ is continuous on X equipped with $\Vert \cdot \Vert _W$, while $C^1$ on X equipped with the stronger norm $\Vert \cdot \Vert _X$, and its derivative $d\mathcal{J}: X \rightarrow X'$ is such that

$$\begin{aligned} \langle d\mathcal{J}(u),v\rangle \ =\ \int _\Omega (a(x,u,\nabla u)\cdot \nabla v + A_t(x,u,\nabla u)v) dx - \int _\Omega f(x,u)v dx \end{aligned}$$

(3.26)

for any $u, v \in X$.

In order to apply variational methods to the study of critical points of $\mathcal{J}$ in the asymptotically p-linear case, we introduce the following further conditions:

• $(H_7)$ for all $\varepsilon > 0$ a constant $r_\varepsilon > 0$ exists, such that

  $$\begin{aligned} |t| \ge r_\varepsilon \, \Longrightarrow \, |A_t(x,t,\xi ) t| \ \le \ \varepsilon |\xi |^p \quad \hbox {for a.e. }x\in \Omega ,\hbox { all }\xi \in \mathbb R^N; \end{aligned}$$

• $(H_8)$ $\alpha _3 > 0$, $\alpha _3 \le 1$, exists, such that

  $$\begin{aligned} a(x,t,\xi )\cdot \xi + A_t(x,t,\xi ) t \ge \alpha _3\ a(x,t,\xi )\cdot \xi \end{aligned}$$

  (3.27)

  for a.e. $x \in \Omega$, all $(t,\xi ) \in \mathbb R\times \mathbb R^N$.

Remark 3.6

From hypothesis $(H_4)$ it follows that $r_0 > 0$ exists, such that

$$\begin{aligned} A(x,t,0)\ =\ 0\quad \text {for a.e. }x \in \Omega \hbox { if }|t| \ge r_0, \end{aligned}$$

(3.28)

while (3.6) implies

$$\begin{aligned} A_t(x,t,0)\ =\ 0\quad \text {for a.e. }x \in \Omega ,\hbox { all }t \in \mathbb R. \end{aligned}$$

(3.29)

Furthermore, (3.6) and $(H_7)$ imply that $L > 0$ exists, such that

$$\begin{aligned} \big |A_t(x,t,\xi )\big |\ \le \ L |\xi |^{p} \quad \text {for a.e. }x \in \Omega ,\hbox { all } (t,\xi ) \in \mathbb R\times \mathbb R^N. \end{aligned}$$

(3.30)

As useful in the following, taking any $r > 0$ we define the truncation function

$$\begin{aligned} T_r : \mathbb R\rightarrow \mathbb R,\quad \hbox {such that}\qquad T_rt = \left\{ \begin{array}{ll} t&{}\hbox {if }|t| \le r\\ r\frac{t}{|t|}&{}\hbox {if }|t| > r \end{array}\right. \end{aligned}$$

(3.31)

and its remainder

$$\begin{aligned} R_rt\ =\ t - T_rt\ =\ \left\{ \begin{array}{ll} 0 &{}\hbox {if }|t| \le r\\ t - r\ \frac{t}{|t|} &{}\hbox {if }|t| > r \end{array}\right. . \end{aligned}$$

(3.32)

Remark 3.7

The properties of $T_r$ and $R_r$ and direct computations imply that not only their Nemitskii operators are continuous from the Lebesgue space $(L^p(\Omega ),|\cdot |_p)$ in itself, but also $T_r$, $R_r: W^{1,p}_0(\Omega ) \rightarrow W^{1,p}_0(\Omega )$ are continuous with respect to $\Vert \cdot \Vert _W$; hence, $T_r$, $R_r: X \rightarrow X$ are continuous. Furthermore, if $(u_n)_n \subset W^{1,p}_0(\Omega )$, $u \in W^{1,p}_0(\Omega )$ are such that $u_n \rightharpoonup u$ weakly in $W^{1,p}_0(\Omega )$, then $T_ru_n \rightharpoonup T_ru$ weakly in $W^{1,p}_0(\Omega )$, too.

4 The weak Palais–Smale condition

In hypotheses (3.5), (3.6), (3.12) and $(h_1)$–$(h_2)$ we can consider the $C^1$ functional $\mathcal{J}$ in (1.1) on the Banach space X defined in (3.1) (see Proposition 3.5). Taking any $\beta \in \mathbb R$, the aim of this section is to exploit the properties of the $(PS)_\beta$-sequences of $\mathcal{J}$ in X, i.e. sequences $(u_n)_n \subset X$, such that

$$\begin{aligned} \lim _{n \rightarrow +\infty }\mathcal{J}(u_n) = \beta \quad \text{ and }\quad \lim _{n \rightarrow +\infty }\Vert d\mathcal{J}(u_n)\Vert _{X'} = 0. \end{aligned}$$

(4.1)

Proposition 4.1

Assume that the hypotheses $(H_1)$–$(H_8)$, $(h_1)$–$(h_2)$ hold and $\lambda ^\infty \not \in \sigma (A_p^\infty )$. Then, for all $\beta \in \mathbb R$, each $(PS)_\beta$-sequence of $\mathcal{J}$ in X is bounded in the $W^{1,p}_0$-norm. Hence, the critical point set $K^\mathcal{J}_\beta$ is bounded in $W^{1,p}_0(\Omega )$.

Furthermore, some strictly positive constants ${\bar{R}}$, ${\bar{\varepsilon }}$, ${\bar{\mu }}$ exist, such that

$$\begin{aligned} u \in \mathcal{J}^{\beta +{\bar{\varepsilon }}}_{\beta -{\bar{\varepsilon }}}, \quad \Vert u\Vert _W \ge {\bar{R}}\qquad \Longrightarrow \qquad \Vert d\mathcal{J}(u)\Vert _{X'} \ge 2{\bar{\mu }}. \end{aligned}$$

(4.2)

Proof

Let $(u_n)_n \subset X$ be a $(PS)_\beta$-sequence. Arguing by contradiction, we suppose that $\Vert u_n\Vert _W \ \rightarrow \ +\infty$ and define $\displaystyle v_n \ =\ \frac{u_n}{\Vert u_n\Vert _W}$. As $\Vert v_n\Vert _W = 1$ for each $n\in \mathbb N$, then $v \in W^{1,p}_0(\Omega )$ exists, such that, up to subsequences, we have that $v_n \rightharpoonup v$ weakly in $W^{1,p}_0(\Omega )$, $v_n \rightarrow v$ strongly in $L^{s}(\Omega )$ for each $s \in [1,p^*[$ and $v_n(x) \rightarrow v(x)$ a.e. in $\Omega$. Rearranging conveniently the arguments developed in [14, Proposition 3.5] for the model case (1.2), we prove that

• 1. $v \not \equiv 0$;

• 2. a constant $b > 0$ exists, such that for any $\mu > 0$ an integer $n_\mu \in \mathbb N$ exists, such that

  $$\begin{aligned} \int _{\Omega {\setminus }\Omega _\mu ^n} |\nabla v_n|^p dx\ \le \ b\ \max \{\mu , \mu ^2\}\qquad \hbox {for all }n\ge n_\mu , \end{aligned}$$

  with $\Omega _\mu ^n = \{x \in \Omega : |v_n(x)| > \mu \}$;

• 3. for all $\varepsilon > 0$ an integer $n_\varepsilon \in \mathbb N$ exists, such that for all $n \ge n_\varepsilon$ we have that

  $$\begin{aligned} \left| \int _{\Omega } a^\infty (x,\nabla v_n) \cdot \nabla \varphi \ dx - \lambda ^\infty \int _{\Omega } |v_n|^{p-2}v_n \varphi \ dx\right| \le \varepsilon \Vert \varphi \Vert _X \; \hbox {for all }\varphi \in X; \end{aligned}$$

  hence, fixing any $\varphi \in X$ we obtain

  $$\begin{aligned} \int _{\Omega } a^\infty (x,\nabla v_n) \cdot \nabla \varphi \ dx - \lambda ^\infty \int _{\Omega } |v_n|^{p-2}v_n \varphi \ dx\ \rightarrow \ 0\quad \hbox {as }n \rightarrow +\infty . \end{aligned}$$

  (4.3)

Now, taking any $r > 0$ and the corresponding truncation function $T_r$ in (3.31), we have that $\Vert T_r v_n - T_r v\Vert _X \le 2r + \Vert v\Vert _W$ for all $n \in \mathbb N$; hence, from the previous Step 3 with $\varphi = T_r v_n - T_r v$, it follows that

$$\begin{aligned} \int _{\Omega } a^\infty (x,\nabla v_n) \cdot \nabla (T_r v_n - T_r v) \ dx - \lambda ^\infty \int _{\Omega } |v_n|^{p-2}v_n (T_r v_n - T_r v)\ dx\ \rightarrow \ 0, \end{aligned}$$

which implies

$$\begin{aligned} \int _{\Omega } a^\infty (x,\nabla v_n) \cdot \nabla (T_r v_n - T_r v) \ dx\ \rightarrow \ 0, \end{aligned}$$

as $T_rv_n \rightarrow T_rv$ strongly in $L^p(\Omega )$. On the other hand, we note that

$$\begin{aligned} \begin{aligned} \int _{\Omega } a^\infty (x,\nabla v_n) \cdot \nabla (T_r v_n - T_r v) \ dx\ =&\int _{\Omega {\setminus } \Omega _r^n} a^\infty (x,\nabla v_n) \cdot \nabla (v_n - T_r v) \ dx\\&- \int _{\Omega _r^n{\setminus } \Omega _r^v} a^\infty (x,\nabla v_n) \cdot \nabla v \ dx, \end{aligned} \end{aligned}$$

where (3.11), Hölder inequality and $\mathrm{meas}(\Omega _r^n{\setminus } \Omega _r^v) \rightarrow 0$ imply

$$\begin{aligned} \int _{\Omega _r^n{\setminus } \Omega _r^v} a^\infty (x,\nabla v_n) \cdot \nabla v \ dx\ \rightarrow \ 0; \end{aligned}$$

hence,

$$\begin{aligned} \int _{\Omega {\setminus } \Omega _r^n} a^\infty (x,\nabla v_n) \cdot \nabla (v_n - T_r v) \ dx\ \rightarrow \ 0. \end{aligned}$$

Then, as $v_n = T_r v_n$ in $\Omega {\setminus } \Omega _r^n$, while $a^\infty (x,\nabla (T_rv_n)) = a^\infty (x,0) = 0$ in a.a. $\Omega _r^n$, we have

$$\begin{aligned} \int _{\Omega } a^\infty (x,\nabla (T_rv_n)) \cdot \nabla (T_rv_n - T_r v) \ dx\ \rightarrow \ 0 \quad \hbox {as }n\rightarrow +\infty , \end{aligned}$$

with $T_rv_n \rightharpoonup T_rv$ weakly in $W^{1,p}_0(\Omega )$; whence, $T_rv_n \rightarrow T_rv$ strongly in $W^{1,p}_0(\Omega )$ from Lemma 3.2 (v). From the arbitrariness of $r > 0$ we have that $v_n \rightarrow v$ strongly in $W^{1,p}_0(\Omega )$ too, and passing to the limit as $n\rightarrow +\infty$ in (4.3) we obtain $\lambda ^\infty \in \sigma (A^\infty _p)$ in contradiction with the hypotheses.

Finally, if (4.2) does not hold, a $(PS)_\beta$-sequence $(u_n)_n \subset X$ exists, such that $\Vert u_n\Vert _W \rightarrow +\infty$, in contradiction with the first part of this proof. $\square$

Proposition 4.2

Assume $p > N$ and that the hypotheses of Proposition 4.1 hold. Then the functional $\mathcal{J}$ satisfies $(PS)_\beta$ in $W^{1,p}_0(\Omega )$ at each level $\beta \in \mathbb R$.

Proof

For the proof, it is sufficient to conveniently rearrange the arguments developed in [14, Proposition 3.6] for the model case (1.2) to our general setting. $\square$

Unfortunately, if $p < N$ the same statement cannot hold as Palais–Smale sequences of $\mathcal{J}$ exist which converge in $\Vert \cdot \Vert _W$ but not in $\Vert \cdot \Vert _X$.

Example 4.3

Suppose that the hypotheses $(H_1)$, $(H_4)$–$(H_5)$, $(H_7)$, $(h_1)$–$(h_2)$ are satisfied and, without loss of generality, assume that the closed unit ball of $\mathbb R^N$, namely $\overline{B}_1(0) = \{x \in \mathbb R^N : |x| \le 1\}$, is contained in $\Omega$. Taking $u \in X$, such that $d\mathcal{J}(u) = 0$, put $\beta = \mathcal{J}(u)$ and consider a smooth function $v \in C^\infty _0(\mathbb R^N)$, such that

$$\begin{aligned} v(x) \ge 0\; \hbox {in }\mathbb R^N,\quad v(x) = 0\; \hbox {if }|x| \ge 1,\quad v(0) > 0\quad \hbox {and}\quad \int _{B_1(0)} |\nabla v|^p dx = 1. \end{aligned}$$

If $p < N$, then $\theta > 0$ exists, such that $\frac{N}{p} - 1 - \theta > 0$ and for each $n \in \mathbb N$ we define

$$\begin{aligned} v_n (x) = n^{\frac{N}{p} - 1 - \theta } v(nx)\quad \hbox {and}\quad u_n (x) = u(x) + v_n (x). \end{aligned}$$

By definition, $v_n \in W^{1,p}_0(\Omega )$ and $v_n(x) = 0$ for each $x \in \Omega {\setminus } \{0\}$ if n is large enough; hence, $u_n \in W^{1,p}_0(\Omega )$ and $u_n(x) \rightarrow u(x)$ and $\nabla u_n(x) \rightarrow \nabla u(x)$ for a.e. $x \in \Omega$. Moreover, direct computations imply that

$$\begin{aligned} \Vert v_n\Vert _W = n^{-\theta }; \quad \hbox {hence }, \Vert u_n - u\Vert _W \rightarrow 0\hbox { as } n \rightarrow +\infty . \end{aligned}$$

On the other hand, since the functionals associated to f and F are continuous in $L^p(\Omega )$, from (3.12), (3.13) and the Lebesgue’s Dominated Convergence Theorem, it follows that $\mathcal{J}(u_n) \rightarrow \beta$ and $\Vert d\mathcal{J}(u_n)\Vert _{X'} \rightarrow 0$, i.e. $(u_n)_n$ is a $(PS)_\beta$-sequence. In any case,

$$\begin{aligned} |u_n - u|_\infty \ =\ |v_n|_\infty \ \ge \ n^{\frac{N}{p} - 1 - \theta } v(0)\ \rightarrow \ +\infty , \end{aligned}$$

so $(u_n)_n$ has no converging subsequence in X.

As already mentioned, the proof of Proposition 4.2 strongly requires the assumption $p > N$, but if $p \le N$ we can prove that the weaker condition $(wPS)_\beta$ in Definition 2.1 holds. To this aim, firstly we need to find sufficient conditions for the boundedness of a $W^{1,p}_0$-function.

Lemma 4.4

Let $1 < p \le N$ and take $u \in W^{1,p}_0(\Omega )$. If $b_0 >0$ and $k_0\in \mathbb N$ exist, such that the inequality

$$\begin{aligned} \int _{\Omega ^u_r}|\nabla u|^p dx \le b_0 \left( r^p\ \mathrm{meas}(\Omega ^u_r) + \int _{\Omega ^u_r} |u|^p dx\right) \end{aligned}$$

(4.4)

holds for all $r \ge k_0$, then $u \in L^\infty (\Omega )$, with $|u|_\infty$ bounded from above by a positive constant which can be chosen so that it depends only on $\mathrm{meas}(\Omega )$, N, p, $b_0$, $k_0$, $\Vert u\Vert _W$.

Proof

It is a direct consequence of [19, Theorem 5.1 in Chapter 2] (see [11, Lemma 4.5]). $\square$

Proposition 4.5

Let $p > 1$ and assume that the hypotheses $(H_1)$–$(H_8)$, $(h_1)$–$(h_2)$ hold. Then, if $\lambda ^\infty \not \in \sigma (A_p^\infty )$, functional $\mathcal{J}$ satisfies $(wPS)_\beta$ in X at each level $\beta \in \mathbb R$.

Proof

Fixing $\beta \in \mathbb R$, let $(u_n)_n \subset X$ be a $(PS)_\beta$-sequence of $\mathcal{J}$ in X, i.e. (4.1) holds. Then, from Proposition 4.1 a constant $L > 0$ exists, such that

$$\begin{aligned} \Vert u_n\Vert _W\ \le \ L \qquad \text{ for } \text{ all } \ n\in \mathbb N\text{. } \end{aligned}$$

(4.5)

Hence, $u \in W^{1,p}_0(\Omega )$ exists, such that, up to subsequences, we have

$$\begin{aligned}&u_n \rightharpoonup u\ \hbox {weakly in }W^{1,p}_0(\Omega ), \end{aligned}$$

(4.6)

$$\begin{aligned}&u_n \rightarrow u\ \hbox {strongly in }L^p(\Omega ), \end{aligned}$$

(4.7)

therefore,

$$\begin{aligned} u_n \rightarrow u\ \hbox {a.e. in }\Omega \hbox { and in measure}, \end{aligned}$$

(4.8)

and a positive function $\nu \in L^p(\Omega )$ exists, such that

$$\begin{aligned} |u_n(x)| \le \nu (x)\quad \hbox {for a.e. } x\in \Omega ,\hbox { all }n \in \mathbb N. \end{aligned}$$

(4.9)

For simplicity, our proof is divided into several steps:

• 1. $u \in L^\infty (\Omega )$;

• 2. fixing $r \ge |u|_\infty + 1$, we have that

  $$\begin{aligned} \int _{\Omega _r^n}|\nabla u_n|^p dx\ \rightarrow \ 0\quad \hbox {as }n \rightarrow +\infty , \end{aligned}$$

  (4.10)

  where, in general, we put $\Omega ^n_\mu = \{x \in \Omega : |u_n(x)| > \mu \}$ for any $\mu \ge 0$;

• 3. taking $r \ge \max \{|u|_\infty + 1,r_0\}$, with $r_0 > 0$ as in (3.28), as $n \rightarrow +\infty$ we have

  $$\begin{aligned} \mathcal{J}(T_ru_n) \rightarrow \beta \qquad \hbox {and} \qquad \Vert d\mathcal{J}(T_ru_n)\Vert _{X'} \rightarrow 0 \end{aligned}$$

  (4.11)

  and

  $$\begin{aligned} T_r u_n \rightarrow u \quad \hbox {strongly in }W^{1,p}_0(\Omega ), \end{aligned}$$

  (4.12)

  with the truncation function $T_r$ as in (3.31);

• 4. $u_n \rightarrow u$ strongly in $W^{1,p}_0(\Omega )$ and $\mathcal{J}(u) = \beta$, $d\mathcal{J}(u) = 0$.

For simplicity, here and in the following we use the notation $(\varepsilon _n)_n$ for any infinitesimal sequence depending only on $(u_n)_n$, while $(\varepsilon _{\mu ,n})_n$ for any infinitesimal sequence depending not only on $(u_n)_n$, but also on some fixed real number $\mu$.

Step 1. From the Sobolev Imbedding Theorem, the proof is required only if $p \le N$. So, under this assumption, taking $r > 0$, any $\rho > r$ and considering the truncation function $T_\rho$, as in (3.31), and the remainder function $R_r$, as in (3.32), define the new sequence of functions $\varphi ^{n}_{r,\rho }(x) = T_\rho (R_r(u_n(x)))$, namely,

$$\begin{aligned} \varphi ^{n}_{r,\rho }(x) \ =\ \left\{ \begin{array}{ll} 0 &{}\hbox {if }|u_n(x)| \le r,\\ u_n(x)\ - r \ \frac{u_n(x)}{|u_n(x)|}&{}\hbox {if }r < |u_n(x)| \le \rho + r,\\ \rho \ \frac{u_n(x)}{|u_n(x)|} &{}\hbox {if }|u_n(x)| > \rho + r. \end{array}\right. \end{aligned}$$

(4.13)

By definition

$$\begin{aligned} \nabla \varphi ^{n}_{r,\rho }(x) \ =\ \left\{ \begin{array}{ll} 0 &{}\hbox {a.e. in }\Omega {\setminus } \Omega ^{n}_{r,\rho },\\ \nabla u_n(x) &{}\hbox {a.e. in }\Omega ^{n}_{r,\rho }, \end{array}\right. \end{aligned}$$

(4.14)

with $\Omega ^{n}_{r,\rho } = \{x \in \Omega : r < |u_n(x)| \le \rho + r\}$; hence, from (4.5) and the properties of $T_\rho$ and $R_r$ it follows that $\varphi ^{n}_{r,\rho } \in X$ with

$$\begin{aligned} \Vert \varphi ^{n}_{r,\rho }\Vert _X\ \le \ L + \rho \qquad \text{ for } \text{ all } n\in \mathbb N. \end{aligned}$$

(4.15)

On the one hand, from (4.1) and (4.15) it follows that

$$\begin{aligned} |\langle d\mathcal{J}(u_n),\varphi ^{n}_{r,\rho }\rangle |\ \le \ \varepsilon _n (L + \rho )\qquad \text{ for } \text{ all } n\in \mathbb N. \end{aligned}$$

(4.16)

On the other hand, from (3.26), (4.13), (4.14), $(H_2)$, $(H_8)$ with $\alpha _3 \le 1$, and direct computations we prove that

$$\begin{aligned} \begin{aligned}&\langle d\mathcal{J}(u_n),\varphi ^{n}_{r,\rho }\rangle = \int _{\Omega ^{n}_{r,\rho }}\left( 1- \frac{r}{|u_n|}\right) \big (a(x,u_n,\nabla u_n) \cdot \nabla u_n + A_t(x,u_n,\nabla u_n) u_n\big ) dx\\&\qquad + \int _{\Omega ^{n}_{r,\rho }} \frac{r}{|u_n|}\ a(x,u_n,\nabla u_n) \cdot \nabla u_n\ dx + \int _{\Omega ^{n}_{\rho + r}} A_t(x,u_n,\nabla u_n) \rho \frac{u_n}{|u_n|}\ dx\\&\qquad - \int _{\Omega } f(x,u_n) \varphi ^{n}_{r,\rho } dx\\&\quad \ge \ \alpha _3 \int _{\Omega ^{n}_{r,\rho }} a(x,u_n,\nabla u_n) \cdot \nabla u_n\ dx + \int _{\Omega ^{n}_{\rho + r}} \frac{\rho }{|u_n|} A_t(x,u_n,\nabla u_n)u_n\ dx\\&\qquad - \int _{\Omega } f(x,u_n) \varphi ^{n}_{r,\rho } dx\\&\quad \ge \ \alpha _0 \alpha _3 \int _{\Omega ^{n}_{r,\rho }} |\nabla u_n|^p\ dx - \int _{\Omega ^{n}_{\rho + r}} |A_t(x,u_n,\nabla u_n) u_n|\ dx\\&\qquad - \int _{\Omega } f(x,u_n) \varphi ^{n}_{r,\rho } dx. \end{aligned} \end{aligned}$$

Hence, by summing up these last estimates and (4.16), for all $\rho > r$ and all $n \in \mathbb N$, we have that

$$\begin{aligned} \begin{aligned} \alpha _0 \alpha _3 \int _{\Omega ^{n}_{r,\rho }} |\nabla u_n|^p\ dx&\le \varepsilon _n (L + \rho ) + \int _{\Omega ^{n}_{\rho +r}} |A_t(x,u_n,\nabla u_n) u_n|\ dx\\&\quad + \int _{\Omega } f(x,u_n) \varphi ^{n}_{r,\rho } dx. \end{aligned} \end{aligned}$$

(4.17)

Now, fix any $\varepsilon > 0$ and $r \ge 1$. From $(H_7)$, a constant $\rho _\varepsilon > r$ exists, such that for any $\rho \ge \rho _\varepsilon$ we have

$$\begin{aligned} |A_t(x,u_n,\nabla u_n) u_n|\ \le \ \frac{\varepsilon }{L^p} |\nabla u_n|^p \quad \hbox {for a.e. }x \in \Omega ^{n}_{\rho +r},\hbox { all }n \in \mathbb N; \end{aligned}$$

hence, from (4.5) it follows

$$\begin{aligned} \limsup _{n\rightarrow +\infty } \int _{\Omega ^{n}_{\rho + r}} |A_t(x,u_n,\nabla u_n) u_n|\ dx\ \le \ \varepsilon \quad \hbox {for any }\rho \ge \rho _\varepsilon . \end{aligned}$$

(4.18)

Furthermore, taking any $\rho \ge \rho _\varepsilon$, from (4.8), (4.9) and the continuity of the involved maps, the Lebesgue’s Dominated Convergence Theorem applies and we have

$$\begin{aligned} \lim _{n\rightarrow +\infty } \int _{\Omega } f(x,u_n) \varphi ^{n}_{r,\rho } dx \ =\ \int _{\Omega } f(x,u) \varphi _{r,\rho } dx, \end{aligned}$$

(4.19)

where $\varphi _{r,\rho }(x) = T_\rho (R_r(u(x)))$ and from (3.23) and direct computations a constant $b_0 = b_0(\lambda ^\infty ) > 0$ exists, such that

$$\begin{aligned} \left| \int _{\Omega } f(x,u) \varphi _{r,\rho } dx\right| \ \le \ b_0 \int _{\Omega _r^u} |u|^p dx\quad \hbox {for all }\rho > r. \end{aligned}$$

(4.20)

Finally, (4.6) and the weak lower semicontinuity of the norm $\Vert \cdot \Vert _W$ imply

$$\begin{aligned} \int _{\Omega _{r,\rho }} |\nabla u|^p\ dx \ \le \ \liminf _{n\rightarrow +\infty }\int _{\Omega ^{n}_{r,\rho }} |\nabla u_n|^p\ dx, \end{aligned}$$

(4.21)

with $\Omega _{r,\rho } = \{x \in \Omega : r < |u(x)| \le \rho + r\}$. Hence, summing up, as $n \rightarrow +\infty$ in (4.17), from (4.18)–(4.21) it results that

$$\begin{aligned} \alpha _0 \alpha _3 \int _{\Omega _{r,\rho }} |\nabla u|^p\ dx \ \le \ \varepsilon + b_0 \int _{\Omega _r^u} |u|^p dx \quad \hbox {for all }\rho \ge \rho _\varepsilon , \end{aligned}$$

(4.22)

thus, as $\varepsilon$ is arbitrary small, (4.4) holds for all $r \ge 1$. Then, by applying Lemma 4.4, we have that $u \in L^\infty (\Omega )$ and $|u|_\infty$ is smaller than a constant which depends only on $\mathrm{meas}(\Omega )$, N, p, $\alpha _0$, $\alpha _3$, $\lambda ^\infty$ and $\Vert u\Vert _W$.

Step 2. Now, let $r \ge |u|_\infty + 1$ in all the formulae of the proof of Step 1. With this choice the limit in (4.19) becomes

$$\begin{aligned} \lim _{n\rightarrow +\infty } \int _{\Omega } f(x,u_n) \varphi ^{n}_{r,\rho } dx \ =\ 0 \quad \hbox {for any }\rho > r, \end{aligned}$$

thus for any $\varepsilon > 0$ a constant $\rho _\varepsilon > r$ exists, such that, by passing to the maximum limit as $n\rightarrow +\infty$ in (4.17), from (4.18) it follows that

$$\begin{aligned} \alpha _0 \alpha _3 \limsup _{n\rightarrow +\infty }\int _{\Omega ^{n}_{r,\rho }} |\nabla u_n|^p\ dx \ \le \ \varepsilon \quad \hbox {for all }\rho \ge \rho _\varepsilon ; \end{aligned}$$

hence, (4.10) holds.

Step 3. As $r > |u|_\infty$, $T_r u=u$; hence, from Remark 3.7 and (4.6) we have that

$$\begin{aligned} T_r u_n \rightharpoonup u\ \hbox {weakly in }W^{1,p}_0(\Omega ) \end{aligned}$$

(4.23)

and also, from (4.7),

$$\begin{aligned}&T_r u_n \rightarrow u\ \hbox { and }\; T_r u_n - u_n \rightarrow 0\ \hbox {strongly in } L^p(\Omega ), \end{aligned}$$

(4.24)

$$\begin{aligned}&\mathrm{meas}(\Omega ^n_r) \rightarrow 0,\nonumber \\&T_r u_n \rightarrow u\ \hbox { a.e. in }\Omega . \end{aligned}$$

(4.25)

Since $r \ge r_0$, from (3.28) and the definitions (1.1) and (3.31) it follows that

$$\begin{aligned} \begin{aligned} \mathcal{J}(T_ru_n) =&\mathcal{J}(u_n)\ -\ \int _{\Omega ^n_r} A(x,u_n,\nabla u_n) dx \\&-\ \int _{\Omega ^n_r} (F(x,T_ru_n) - F(x,u_n)) dx, \end{aligned} \end{aligned}$$

where (3.12), (4.10) and (4.25) imply that

$$\begin{aligned} \int _{\Omega ^n_r} A(x,u_n,\nabla u_n) dx\ =\ \varepsilon _{r,n}, \end{aligned}$$

while from the continuity of the functional associated to F in $L^p(\Omega )$ and (4.24) we have that

$$\begin{aligned} \int _{\Omega ^n_r} (F(x,T_ru_n) - F(x,u_n)) dx\ =\ \varepsilon _{r,n}. \end{aligned}$$

Hence,

$$\begin{aligned} \mathcal{J}(T_r u_n)\ =\ \mathcal{J}(u_n) + \varepsilon _{r,n} \end{aligned}$$

and the first limit in (4.11) follows from (4.1).

Now, taking $\varphi \in X$, from (3.7), (3.29) and direct computations it follows that

$$\begin{aligned} \begin{aligned}&|\langle d\mathcal{J}(T_r u_n),\varphi \rangle |\ \le \ \Vert d\mathcal{J}(u_n)\Vert _{X'} \Vert \varphi \Vert _X + \int _{\Omega ^{n}_{r}}|a(x,u_n,\nabla u_n)| |\nabla \varphi | dx\\&\qquad + \int _{\Omega ^{n}_{r}} |A_t(x,u_n,\nabla u_n)| |\varphi | dx + \int _{\Omega ^n_r} |f(x,T_r u_n) - f(x,u_n)| |\varphi | dx. \end{aligned} \end{aligned}$$

From (3.13), (4.10), (4.25), the Hölder inequality and direct computations it follows that

$$\begin{aligned} \int _{\Omega ^{n}_{r}}|a(x,u_n,\nabla u_n)| |\nabla \varphi | dx\ \le \ \varepsilon _{r,n} \Vert \varphi \Vert _W. \end{aligned}$$

Furthermore, (3.30) and (4.10) imply

$$\begin{aligned} \int _{\Omega ^{n}_{r}} |A_t(x,u_n,\nabla u_n)| |\varphi | dx \ \le \ \varepsilon _{r,n} |\varphi |_\infty , \end{aligned}$$

while from the continuity of the functional associated to f in $L^p(\Omega )$ and (4.24) it follows that

$$\begin{aligned} \int _{\Omega ^n_r} |f(x,T_r u_n) - f(x,u_n)| |\varphi | dx \ \le \ \varepsilon _{r,n} |\varphi |_\infty . \end{aligned}$$

Hence,

$$\begin{aligned} |\langle d\mathcal{J}(T_r u_n),\varphi \rangle |\ \le \ (\Vert d\mathcal{J}(u_n)\Vert _{X'} + \varepsilon _{r,n})\ \Vert \varphi \Vert _X\quad \hbox {for all }\varphi \in X, \end{aligned}$$

and the second limit in (4.11) follows from (4.1).

Finally, (4.12) follows from (4.10), the second limit in (4.11), (4.23), the given set of hypotheses once we repeat the proof of Step 4 in [11, Proposition 4.6].

Step 4. Since $u_n = T_ru_n + R_ru_n$, with $\displaystyle \Vert R_ru_n\Vert _W^p = \int _{\Omega _r^n}|\nabla u_n|^p dx$, then (4.10) and (4.12) imply $u_n \rightarrow u$ strongly in $W^{1,p}_0(\Omega )$. Hence, as $|T_ru_n|_\infty \le r$ for all $n \in \mathbb N$, Proposition 3.5 implies

$$\begin{aligned} \mathcal{J}(T_ru_n) \rightarrow \mathcal{J}(u),\quad \Vert d\mathcal{J}(T_ru_n) - d\mathcal{J}(u)\Vert _{X'} \rightarrow 0. \end{aligned}$$

Then the end of the proof follows from (4.11). $\square$

Essentially following some of the ideas introduced in the proof of Step 1 of Proposition 4.5, but replacing the global condition $(H_8)$ with (3.27), only if t is large enough we can prove a boundedness result similar to [3, Lemma 1.4], but with different hypotheses.

Proposition 4.6

Assume that the hypotheses $(H_2)$, $(H_7)$, $(h_1)$, $(h_2)$ hold and that $\rho _0 \ge 0$ exists, such that (3.27) holds for a.e. $x \in \Omega$, all $\xi \in \mathbb R^N$ if $|t| \ge \rho _0$. If $u \in W^{1,p}_0(\Omega )$ is such that

$$\begin{aligned} \int _\Omega a(x,u,\nabla u) \cdot \nabla \varphi dx + \int _\Omega A_t(x,u,\nabla u)\varphi dx - \int _\Omega f(x,u)\varphi dx = 0 \end{aligned}$$

(4.26)

for all $\varphi \in X$, then a positive constant $r_u > 0$ exists, which depends only on $\mathrm{meas}(\Omega )$, N, p, $\alpha _0$, $\alpha _3$, $\lambda ^\infty$ and $\Vert u\Vert _W$, such that $|u|_\infty \le r_u$. Hence, $u \in X$.

Proof

From the Sobolev Imbedding Theorem, the proof is required only if $p \le N$. So, by fixing $r \ge \rho _0 + 1$, taking any $\rho > r$, using $\varphi _{r,\rho }(x) = T_\rho (R_r(u(x)))$ as the test function in (4.26), and reasoning as in the proof of Step 1 in Proposition 4.5, we obtain

$$\begin{aligned} \alpha _0 \alpha _3 \int _{\Omega _{r,\rho }} |\nabla u|^p\ dx \ \le \ \int _{\Omega ^u_{\rho +r}} |A_t(x,u,\nabla u) u|\ dx + \int _{\Omega } f(x,u) \varphi _{r,\rho } dx, \end{aligned}$$

with $\Omega _{r,\rho } = \{x \in \Omega : r < |u(x)| \le \rho + r\}$, as

$$\begin{aligned} a(x,u,\nabla u) \cdot \nabla u + A_t(x,u,\nabla u) u \ge a(x,u,\nabla u) \cdot \nabla u\quad \hbox {a.e. in } \Omega _{r,\rho }. \end{aligned}$$

Thus, (4.22) is satisfied, since (4.20) still holds while from $(H_7)$ for all $\varepsilon > 0$, a constant $\rho _\varepsilon > r$ exists, such that

$$\begin{aligned} \int _{\Omega ^u_{\rho + r}} |A_t(x,u,\nabla u) u|\ dx\ \le \ \varepsilon \quad \hbox {for any }\rho \ge \rho _\varepsilon . \end{aligned}$$

Therefore, (4.4) follows and Lemma 4.4 applies. $\square$

Corollary 4.7

In the hypotheses of Proposition 4.1, if $\beta \in \mathbb R$ is such that $K^\mathcal{J}_\beta \ne \emptyset$, then a constant $r_\beta > 0$ exists, such that $|u|_\infty \le r_\beta$ for all $u \in K^\mathcal{J}_\beta$. Hence, the critical point set $K^\mathcal{J}_\beta$ is compact with respect to the $W^{1,p}_0$-norm, while it is bounded with respect to the $L^\infty$-norm.

Proof

If $p > N$ the statement is a direct consequence of Proposition 4.1 and (3.3). On the other hand, if $p \le N$, from Proposition 4.6 each $u \in K^\mathcal{J}_\beta$ is bounded by a constant which depends on $\Vert u\Vert _W$, while from Proposition 4.1 it follows that $K^\mathcal{J}_\beta$ is bounded with respect to $\Vert \cdot \Vert _W$. $\square$

5 Main results

In addition to the hypotheses $(H_1)$–$(H_8)$, $(h_1)$–$(h_2)$, we assume that:

• $(H_9)$ $A^0(x,\xi )$ is positively p-homogeneous in $\xi$ for a.e. $x \in \Omega$, with

  $$\begin{aligned} A^0(x,\xi ) = A(x,0,\xi ); \end{aligned}$$

• $(H_{10})$ $A(x,t,\xi )$ is even in $(t,\xi )$ for a.e. $x \in \Omega$;

• $(h_3)$ $\lambda ^0 \in \mathbb R$ and a (Carathéodory) function $g^0 : \Omega \times \mathbb R\rightarrow \mathbb R$ exist, such that

  $$\begin{aligned} f(x,t)\ =\ \lambda ^0\ |t|^{p-2}\, t + g^0(x,t), \end{aligned}$$

  with

  $$\begin{aligned} \lim _{t \rightarrow 0}\, \frac{g^0(x,t)}{|t|^{p-1}}\ =\ 0 \quad \text {uniformly a.e. in } \Omega ; \end{aligned}$$

• $(h_4)$ $f(x,\cdot )$ is odd for a.e. $x \in \Omega$.

Remark 5.1

Conditions (3.12), $(H_3)$ and $(H_9)$ imply that $\alpha _4 > 0$ exists, such that

$$\begin{aligned} \alpha _1 |\xi |^{p}\ \le \ A^0(x,\xi ) \ \le \ \alpha _4 |\xi |^{p} \qquad \hbox {for a.e. }x \in \Omega ,\hbox { all }\xi \in \mathbb R^N. \end{aligned}$$

(5.1)

Moreover, from (3.6) it follows that

$$\begin{aligned} |A(x,t,\xi ) - A^0(x,\xi )| \le \big | \int _0^t \phi _2(s) ds\big |\ |\xi |^{p} \quad \hbox {for a.e. }x \in \Omega ,\hbox { all } (t,\xi ) \in \mathbb R\times \mathbb R^N; \end{aligned}$$

hence, for all $\varepsilon > 0$ a constant $r_\varepsilon > 0$ exists, such that

$$\begin{aligned} |t| \le r_\varepsilon \;\Longrightarrow \; \big |A(x,t,\xi ) - A^0(x,\xi )\big | \le \varepsilon |\xi |^{p} \quad \text {for a.e. }x \in \Omega ,\hbox { all } \xi \in \mathbb R^N. \end{aligned}$$

(5.2)

Remark 5.2

From $(h_3)$ it follows that

$$\begin{aligned} F(x,t)\ =\ \frac{\lambda ^0}{p}\ |t|^{p} + G^0(x,t),\quad \hbox {with}\; G^0(x,t) = \int _\Omega g^0(x,s)ds, \end{aligned}$$

(5.3)

and

$$\begin{aligned} \lim _{t \rightarrow 0}\, \frac{G^0(x,t)}{|t|^{p}}\ =\ 0 \; \hbox {uniformly a.e. in }\Omega ; \end{aligned}$$

(5.4)

thus,

$$\begin{aligned} \lim _{t \rightarrow 0}\, \frac{F(x,t)}{|t|^{p}}\ =\ \frac{\lambda ^0}{p} \quad \text {uniformly a.e. in } \Omega . \end{aligned}$$

(5.5)

Furthermore, from $(h_1)$, (3.20), (5.5) and direct computations it follows that $D > 0$ exists, such that

$$\begin{aligned} |F(x,t)|\ \le \ D |t|^{p} \quad \text {for a.e. }x \in \Omega ,\hbox { all }t \in \mathbb R. \end{aligned}$$

(5.6)

Defining $a^0(x,\xi ) = a(x,0,\xi )$, as in (3.10), we introduce the operator

$$\begin{aligned} A^0_p : u \in W^{1,p}_0(\Omega ) \mapsto A_p^0 u \in W^{-1,p'}(\Omega ) \end{aligned}$$

as

$$\begin{aligned} \langle A_p^0 u,v\rangle \ =\ \int _\Omega a^0(x,\nabla u)\cdot \nabla v\ dx \qquad \hbox {for any }u, v \in W^{1,p}_0(\Omega ), \end{aligned}$$

(5.7)

and denote its spectrum by $\sigma (A^0_p)$.

Remark 5.3

If the hypotheses (3.5), $(H_2)$, $(H_6)$, $(H_9)$–$(H_{10})$ hold, from direct computations and [8, Lemma 5] the nonlinear operator $A^0_p$ in (5.7) has the following properties:

• (i) it is continuous from the reflexive Banach space $W^{1,p}_0(\Omega )$ to its dual $W^{-1,p'}(\Omega )$,

• (ii) it admits a potential operator, as it is $a^0(x,\xi ) = \nabla _\xi A^0(x,\xi )$ for a.e. $x \in \Omega$;

• (iii) by assumption, it is $(p-1)$-homogeneous and odd;

• (iv) it is uniformly positive, as $a^0(x,\xi )\cdot \xi \ \ge \ \alpha _0 |\xi |^p \$ a.e. in $\Omega$, for all $\xi \in \mathbb R^N$;

• (v) it is of type (S): if $(u_n)_n\subset X$ and $u \in X$ are such that

  $$\begin{aligned} u_n \rightharpoonup u\ \hbox {weakly in }W^{1,p}_0(\Omega )\quad \hbox {and}\quad \langle A_p^0 u_n,u_n-u\rangle \rightarrow 0, \end{aligned}$$

  then $u_n \rightarrow u$ strongly in $W^{1,p}_0(\Omega )$, up to subsequences.

For each $j\ge 1$ let us define:

$$\begin{aligned} \lambda ^0_j= & {} \inf _{Q \in \mathcal{S}_j}\ \max _{u \in Q}\ \frac{\displaystyle \int _\Omega A^0(x,\nabla u) dx}{\displaystyle \frac{1}{p} \int _\Omega |u|^p dx}, \end{aligned}$$

(5.8)

$$\begin{aligned} \lambda ^\infty _j= & {} \inf _{Q \in \mathcal{S}_j}\ \max _{u \in Q}\ \frac{\displaystyle \int _\Omega A^\infty (x,\nabla u) dx}{\displaystyle \frac{1}{p} \int _\Omega |u|^p dx}, \end{aligned}$$

(5.9)

with

$$\begin{aligned} \mathcal{S}_j = \{Q \subset \mathcal{S}^W:\; Q \hbox { symmetric and compact in } W^{1,p}_0(\Omega ),\hbox { with }i_W(Q) \ge j \}, \end{aligned}$$

where

$$\begin{aligned} \mathcal{S}^W = \{u \in W^{1,p}_0(\Omega ): \Vert u\Vert _W = 1 \} \end{aligned}$$

and $i_W(\cdot )$ is the cohomological index defined on the Banach space $W^{1,p}_0(\Omega )$.

As direct consequence of Lemma 3.2 and Remark 5.3 we have

$$\begin{aligned} \langle A_p^\infty u,u\rangle \ =\ p A^\infty (x,\nabla u)\; \hbox {and}\; \langle A_p^0 u,u\rangle \ =\ p A^0(x,\nabla u) \quad \hbox {if }u \in W^{1,p}_0(\Omega ). \end{aligned}$$

Moreover, [24, Theorem 4.6] applies and the following proposition can be pointed out.

Proposition 5.4

Assume that the hypotheses (3.5), $(H_2)$, $(H_4)$–$(H_6)$ and $(H_9)$–$(H_{10})$ hold. Then, taking $\natural = 0,\infty$, we have that $(\lambda ^\natural _j)_j$ is a nondecreasing sequence of eigenvalues of the nonlinear operator $A^\natural _p$, such that $\lambda ^\natural _j\nearrow +\infty$, as $j \rightarrow +\infty$ and the smallest eigenvalue, called the first eigenvalue, is

$$\begin{aligned} \lambda ^\natural _1 = \min _{u \in W^{1,p}_0(\Omega ) {\setminus }\{0\}}\ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla u) dx}{\displaystyle \frac{1}{p} \int _\Omega |u|^p dx}\ >\ 0. \end{aligned}$$

Remark 5.5

Since both if $\natural = \infty$ and if $\natural = 0$ the function $A^\natural (x,\xi )$ is positively p-homogeneous in $\xi$ (from $(H_4)$, respectively $(H_9)$), the eigenvalues $(\lambda ^\natural _j)_j$ can be characterized as

$$\begin{aligned} \lambda ^\natural _j\ = \ \inf _{Q \in \mathcal{W}_j}\ \max _{u \in Q}\ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla u) dx}{\displaystyle \frac{1}{p} \int _\Omega |u|^p dx}, \end{aligned}$$

(5.10)

where

$$\begin{aligned}\begin{aligned} \mathcal{W}_j\ =\ \{Q \subset W^{1,p}_0(\Omega ) {\setminus }\{0\}:\;&Q \hbox { symmetric and compact in }W^{1,p}_0(\Omega ),\\&\hbox {with }i_W(Q) \ge j \}.\end{aligned} \end{aligned}$$

In fact, taking $j \in \mathbb N$, firstly we note that $\mathcal{S}_j \subset \mathcal{W}_j$ implies

$$\begin{aligned} \inf _{Q \in \mathcal{W}_j}\ \max _{u \in Q}\ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla u) dx}{\displaystyle \frac{1}{p} \int _\Omega |u|^p dx} \ \le \ \lambda ^\natural _j. \end{aligned}$$

On the contrary, since the radial projection

$$\begin{aligned} \pi : u \in W^{1,p}_0(\Omega ){\setminus }\{0\}\ \mapsto \ \pi (u) = \frac{u}{\Vert u\Vert _W} \in S^W \end{aligned}$$

is odd and continuous, property $(i_{2})$ in Proposition 2.5 implies that if $Q \in \mathcal{W}_j$, then $\pi (Q) \in \mathcal{S}_j$, with

$$\begin{aligned} \max _{u \in Q} \frac{\displaystyle \int _\Omega A^\natural (x,\nabla \pi (u)) dx}{\displaystyle \frac{1}{p}\int _\Omega |\pi (u)|^p dx} \ =\ \max _{u \in Q} \frac{\displaystyle \int _\Omega A^\natural (x,\nabla u) dx}{\displaystyle \frac{1}{p}\int _\Omega |u|^p dx}; \end{aligned}$$

hence, (5.10) holds.

Now, we are able to state our main results.

Theorem 5.6

Assume that $(H_1)$–$(H_{10})$ and $(h_1)$–$(h_4)$ hold for some $p > 1$. If $\lambda ^\infty \not \in \sigma (A_p^\infty )$ and $k, h \in \mathbb N$ exist, such that

$$\begin{aligned} \lambda ^0_k< \lambda ^0,\qquad \lambda ^\infty < \lambda ^\infty _h,\qquad \hbox {with }k > h - 1, \end{aligned}$$

(5.11)

then problem (GP) has at least $k - h + 1$ distinct pairs of nontrivial bounded solutions with strictly negative critical levels.

Theorem 5.7

Assume that $(H_1)$–$(H_{10})$ and $(h_1)$–$(h_4)$ hold for some $p > 1$. If $\lambda ^\infty \not \in \sigma (A_p^\infty )$ and $k, h \in \mathbb N$ exist, such that

$$\begin{aligned} \lambda ^0< \lambda ^0_k,\qquad \lambda ^\infty _h< \lambda ^\infty ,\qquad \hbox { with }k < h + 1, \end{aligned}$$

(5.12)

then problem (GP) has at least $h - k + 1$ distinct pairs of nontrivial bounded solutions with strictly positive critical levels.

In the same hypotheses of the previous Theorem 5.6, but replacing conditions $(h_3)$ and (5.11) with F super-p-linear at zero, we are able to prove that (GP) has infinitely many solutions.

Theorem 5.8

Assume that $(H_1)$–$(H_{10})$, $(h_1)$–$(h_2)$ and $(h_4)$ hold for some $p > 1$. If $\lambda ^\infty \not \in \sigma (A_p^\infty )$ and

$$\begin{aligned} \lim _{t \rightarrow 0}\, \frac{F(x,t)}{|t|^{p}}\ =\ +\infty \quad \text {uniformly a.e. in } \Omega , \end{aligned}$$

(5.13)

then problem (GP) has an infinite number of distinct pairs of nontrivial bounded solutions $(u_k)_k \subset X$, with negative critical levels, such that $\mathcal{J}(u_k) \nearrow 0$.

Before going on with the proofs of our main theorems, we note that in the definitions of the eigenvalues $(\lambda ^0_j)_j$ and $(\lambda ^\infty _j)_j$ (see (5.8), respectively (5.9)) or in their characterization (5.10), only the space $W^{1,p}_0(\Omega )$ is involved. In any case, our natural setting is X, so we need to characterize them accordingly.

Proposition 5.9

Assume that (3.5), $(H_2)$–$(H_6)$ and $(H_9)$–$(H_{10})$ hold. Taking $\natural = 0,\infty$, for each $j\ge 1$ we define

$$\begin{aligned} {\tilde{\lambda }}^\natural _j\ =\ \inf _{Q \in {\tilde{\mathcal{W}}}_j}\ \max _{u \in Q}\ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla u) dx}{\displaystyle \frac{1}{p} \int _\Omega |u|^p dx}, \end{aligned}$$

with

$$\begin{aligned} {\tilde{\mathcal{W}}}_j = \{Q \in \mathcal{W}_j :\, \exists \, V\hbox { subspace of }X, \dim V < +\infty ,\hbox { such that }Q \subset V\}. \end{aligned}$$

Then,

$$\begin{aligned} {\tilde{\lambda }}^\natural _j\ =\ \lambda ^\natural _j. \end{aligned}$$

Proof

Taking any $j\ge 1$, by definition it is ${\tilde{\mathcal{W}}}_j \subset \mathcal{W}_j$, then (5.10) implies $\lambda ^\natural _j \le {\tilde{\lambda }}^\natural _j$. Now, we have just to prove that

$$\begin{aligned} {\tilde{\lambda }}^\natural _j \ \le \ \lambda ^\natural _j. \end{aligned}$$

(5.14)

To this aim, fixing $Q \in \mathcal{S}_j$ and any $\varepsilon \in ]0,1[$, we split our proof into two steps:

• 1. $Q_\varepsilon \in \mathcal{W}_j$ exists, such that $Q_\varepsilon$ is a bounded subset of X and

  $$\begin{aligned} \max _{v \in Q_\varepsilon } \ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla v) dx}{\displaystyle \frac{1}{p} \int _\Omega |v|^p dx}\ \le \ \frac{1}{1 - \varepsilon }\ \max _{u \in Q}\ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla u) dx}{\displaystyle \frac{1}{p} \int _\Omega |u|^p dx}, \end{aligned}$$

  (5.15)

• 2. ${\tilde{Q}}_\varepsilon \in {\tilde{\mathcal{W}}}_j$ exists, such that

  $$\begin{aligned} \max _{w \in {\tilde{Q}}_\varepsilon } \ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla w) dx}{\displaystyle \frac{1}{p} \int _\Omega |w|^p dx}\ \le \ \varepsilon \ \max _{v \in Q_\varepsilon } \ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla v) dx }{\displaystyle \frac{1}{p} \int _\Omega |v|^p dx}. \end{aligned}$$

  (5.16)

Step 1. Firstly we note that, by definition, Q is a subset of $W^{1,p}_0(\Omega )$ contained in $S^W$, so from the Sobolev Imbedding Theorem, Q is compact in $L^{p}(\Omega )$, with $0 \not \in Q$; hence, not only the minimum of the $L^p$-norm in Q is strictly positive, i.e. $b > 0$ exists, such that

$$\begin{aligned} \frac{1}{p} \int _\Omega |u|^p dx\ \ge \ b\quad \hbox {for all }u \in Q, \end{aligned}$$

(5.17)

but also from (3.2) and (5.17) a constant $\delta _{1,\varepsilon } > 0$ exists, such that for any measurable subset $E \subset \Omega$ with $\mathrm{meas}(E) < \delta _{1,\varepsilon }$ we have

$$\begin{aligned} \int _E |u|^p dx\ \le \ \varepsilon |u|_p^p\quad \hbox {for all }u \in Q. \end{aligned}$$

(5.18)

On the other hand, from (3.2) we have $|u|_p^p \le \sigma _p$ for all $u \in S^W$; hence, $r_\varepsilon = r_\varepsilon (\delta _{1,\varepsilon }) > 0$ exists, so that

$$\begin{aligned} \mathrm{meas}(\Omega ^u_{r_\varepsilon }) < \delta _{1,\varepsilon } \quad \hbox {for any }u \in S^W. \end{aligned}$$

(5.19)

Thus, from (5.18) and (5.19) it follows that

$$\begin{aligned} \int _{\Omega ^u_{r_\varepsilon }} |u|^p dx\ \le \ \varepsilon |u|_p^p\quad \hbox {for all }u \in Q, \end{aligned}$$

which implies that

$$\begin{aligned} \int _\Omega |T_{r_\varepsilon }u|^p dx \ \ge \ (1 - \varepsilon ) \int _\Omega |u|^p dx \quad \hbox {for all }u \in Q, \end{aligned}$$

(5.20)

with $T_{r_\varepsilon }$ as in (3.31).

Now, defining $Q_\varepsilon = T_{r_\varepsilon }(Q)$, by construction it is not only $Q_\varepsilon \subset L^\infty (\Omega )$, but also

$$\begin{aligned} |v|_\infty \le r_\varepsilon \quad \hbox {for all }v \in Q_\varepsilon , \end{aligned}$$

while the properties of $T_{r_\varepsilon }$ and property $(i_{2})$ in Proposition 2.5 also imply that $Q_\varepsilon \in \mathcal{W}_j$.

On the other hand, since

$$\begin{aligned} \int _\Omega A^\natural (x,\nabla (T_{r_\varepsilon }u)) dx \ \le \ \int _\Omega A^\natural (x,\nabla u) dx, \end{aligned}$$

from (5.20) we have that

$$\begin{aligned} \frac{\displaystyle \int _\Omega A^\natural (x,\nabla (T_{r_\varepsilon }u)) dx}{\displaystyle \frac{1}{p} \int _\Omega |T_{r_\varepsilon }u|^p dx}\ \le \ \frac{1}{1 - \varepsilon }\ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla u) dx}{\displaystyle \frac{1}{p} \int _\Omega |u|^p dx}\qquad \hbox {for all }u \in Q, \end{aligned}$$

which implies (5.15).

Step 2. From Step 1. two constants $L_1$, $L_2$ exist, such that

$$\begin{aligned} 0 < L_1 \le \Vert v\Vert _W \le L_2\quad \hbox {for all }v \in Q_\varepsilon . \end{aligned}$$

(5.21)

Moreover, since the map

$$\begin{aligned} w\ \mapsto \ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla w) dx}{\displaystyle \frac{1}{p} \int _\Omega |w|^p dx} \end{aligned}$$

is continuous on $W^{1,p}_0(\Omega ) {\setminus } \{0\}$ and $Q_\varepsilon$ is compact in the same space, $\delta _{2,\varepsilon } >0$ exists (without loss of generality, $\delta _{2,\varepsilon } < L_1$), such that for all $v \in Q_\varepsilon$, $w \in W^{1,p}_0(\Omega ) {\setminus } \{0\}$, we have

$$\begin{aligned} \left| \frac{\displaystyle \int _\Omega A^\natural (x,\nabla v) dx}{\displaystyle \frac{1}{p} \int _\Omega |v|^p dx}\ -\ \frac{\displaystyle \int _\Omega A^\natural (x,\nabla w) dx}{\displaystyle \frac{1}{p} \int _\Omega |w|^p dx}\right|< \varepsilon \quad \hbox {if }\Vert v - w\Vert _W < \delta _{2,\varepsilon }. \end{aligned}$$

(5.22)

On the other hand, again the compactness of $Q_\varepsilon$ in $W^{1,p}_0(\Omega )$ and $Q_\varepsilon \subset X$ imply that $v_1$, $\dots$, $v_l \in Q_\varepsilon$ exist, such that

$$\begin{aligned} Q_\varepsilon \subset \bigcup _{i=1}^{l} B^W_{\delta _{2,\varepsilon }}(v_i). \end{aligned}$$

(5.23)

Let V be the finite dimensional subspace of X generated by $\{v_1, \dots , v_l\}$. From (5.23) it follows that

$$\begin{aligned} d_W(v, V)< \delta _{2,\varepsilon } < L_1\quad \hbox {for all }v \in Q_\varepsilon . \end{aligned}$$

(5.24)

We claim that for each $v \in Q_\varepsilon$, one and only one $w(v) \in V {\setminus }\{0\}$ exists, such that $\Vert v - w(v)\Vert _W = d_W(v, V)$.

In fact, from (5.24) we can take a sequence $(w_n)_n \subset V$, such that

$$\begin{aligned} \Vert v - w_n\Vert _W \rightarrow d_W(v, V)\quad \hbox {and} \quad \Vert v - w_n\Vert _W \le \delta _{2,\varepsilon }\quad \hbox {for each }n. \end{aligned}$$

Hence, (5.21) implies that $(w_n)_n \subset \{w \in V: \Vert w\Vert _W \le \delta _{2,\varepsilon } + L_2\}$ which is compact in V, thus $\bar{w} \in V$ exists, such that

$$\begin{aligned} \Vert w_n - \bar{w}\Vert _W \rightarrow 0\; \hbox {(up to subsequences) and} \; \Vert v - \bar{w}\Vert _W = d_W(v, V). \end{aligned}$$

The uniqueness follows from the strong convexity of the space $W^{1,p}_0(\Omega )$. Furthermore, from (5.21) and (5.24), $\Vert w(v)\Vert _W \ge L_1 - \delta _{2,\varepsilon } > 0$.

Then, the map

$$\begin{aligned} \varphi _\varepsilon :\ v \in Q_\varepsilon \ \mapsto \ w(v) \in V {\setminus }\{0\} \end{aligned}$$

is well defined, odd and continuous with respect to $\Vert \cdot \Vert _W$. Hence, ${\tilde{Q}}_\varepsilon = \varphi _\varepsilon (Q_\varepsilon )$ is symmetric and compact both in $W^{1,p}_0(\Omega )$ and in X, with

$$\begin{aligned} j \le i_W(Q_\varepsilon ) \le i_W({\tilde{Q}}_\varepsilon ) \end{aligned}$$

from the monotonicity of the index; thus, ${\tilde{Q}}_\varepsilon \in \mathcal{W}_j$ and, by construction, ${\tilde{Q}}_\varepsilon \subset V$. Moreover, (5.22) implies (5.16).

Finally, (5.14) is a direct consequence of the previous steps for (5.8) and (5.9), respectively, and the arbitrariness of $\varepsilon$ and $Q \in \mathcal{S}_j$. $\square$

6 Looking for negative critical levels

Throughout this section, we assume that $(H_1)$–$(H_{10})$, $(h_1)$–$(h_2)$ and $(h_4)$ hold. Moreover, suppose $\mathcal{J}(0) = 0$ (true from $(H_9)$ if either $(h_3)$ or (5.13) is satisfied).

Remark 6.1

If $\lambda ^\infty \le 0$, from $(H_3)$, (3.2), (3.16) and (3.22) with $\varepsilon$ small enough, it follows that

$$\begin{aligned} \inf \mathcal{J}(X) > -\infty . \end{aligned}$$

Since from $(H_{10})$ and $(h_4)$ it follows that $\mathcal{J}$ is an even functional in X, we can use the cohomological index theory and its related pseudo-index, as stated in Sect. 2. To this aim, for all $j \in \mathbb N$, we define

$$\begin{aligned} \mathcal{P}_j = \{P \subset X{\setminus } \{0\}:\; P \hbox { symmetric and compact in } X\hbox { with }i(P) \ge j \}, \end{aligned}$$

where $i(\cdot )$ is the cohomological index on $(X,\Vert \cdot \Vert _X)$, and, as in (2.13), but with X as in (3.1), $W = W^{1,p}_0(\Omega )$ and $J = \mathcal{J}$, we take

$$\begin{aligned} c_j \ =\ \inf _{P \in \mathcal{P}_j}\ \max _{u \in P}\mathcal{J}(u). \end{aligned}$$

(6.1)

Remark 6.2

Taking any $j \in \mathbb N$, firstly let us point out that, if ${\tilde{\mathcal{W}}}_j$ is as in Proposition 5.9, then

$$\begin{aligned} Q \in {\tilde{\mathcal{W}}}_j\Longrightarrow Q \in \mathcal{P}_j. \end{aligned}$$

(6.2)

In fact, if $Q \in {\tilde{\mathcal{W}}}_j$, a finite dimensional subspace V of X exists, such that $Q \subset V$; hence, $Q \subset X {\setminus } \{0\}$ is symmetric and compact in X, with $i(Q) = i_W(Q) \ge j$.

On the other hand, either if $\natural = 0$ or if $\natural = \infty$, from (2.12) it follows that, if $P \in \mathcal{P}_j$, then $P \in \mathcal{W}_j$, hence (5.10) implies that

$$\begin{aligned} \max _{u \in P} \frac{\displaystyle \int _\Omega A^\natural (x,\nabla u) dx}{\displaystyle \frac{1}{p}\int _\Omega |u|^p dx} \ \ge \ \lambda _j^\natural \qquad \hbox {for any }P \in \mathcal{P}_j. \end{aligned}$$

(6.3)

Finally, denoting

$$\begin{aligned} \partial B^\natural _1\ =\ \{u \in X: \int _\Omega A^\natural (x,\nabla u) dx = 1 \}, \end{aligned}$$

(6.4)

we have that the projection

$$\begin{aligned} \pi ^\natural : u \in X{\setminus }\{0\}\ \mapsto \ \pi ^\natural (u) = \frac{u}{\displaystyle \left( \int _\Omega A^\natural (x,\nabla u) dx\right) ^{\frac{1}{p}}} \in \partial B^\natural _1 \end{aligned}$$

(6.5)

is odd and continuous; then, from property $(i_{2})$ in Proposition 2.5, it follows that

$$\begin{aligned} P \in \mathcal{P}_j \Longrightarrow \pi ^\natural (P) \in \mathcal{P}_j. \end{aligned}$$

(6.6)

In order to prove Theorems 5.6 and 5.8 by applying Theorem 2.7, firstly we have to show that $c_h > -\infty$ for some $h \in N$ (here $c_h$ is as in (6.1)). To this aim, we note that, fixing any $\tau > 0$ and $P\in \mathcal{P}_h$, from (3.2) and (3.24), direct computations imply the existence of a constant $c(\tau ) \in \mathbb R$, so that, if $u_0 \in P$ exists with $\Vert u_0\Vert _W \le \tau$, then $\max \mathcal{J}(P) \ge c(\tau )$. Thus, the existence of a uniform lower bound has to be proved only for the maxima of $\mathcal{J}$ on sets P, such that $\Vert u\Vert _W > \tau$ for all $u \in P$. In order to achieve this, it is quite natural to approximate $A(x,t,\xi )$ with $A^\infty (x,\xi )$, but such a replacement is allowed only if $|t| > \bar{r}$ with $\bar{r}$ large enough, while for every $u \in W^{1,p}_0(\Omega )$ set $\{x \in \Omega :\ |u(x)| \le r\}$ is nontrivial for all $r > 0$. Hence, we have to split $\Omega$ into two parts: the set in which $R_r(\pi ^\infty (u(x))) \ne 0$ (with the remainder $R_r$ as in (3.32)) and its complementary set. Obviously, taking $\bar{r}$ and a suitable $r >0$, a related $\tau$ can be fixed, so that $\Vert u\Vert _W > \tau$ and $|\pi ^\infty (u(x))| > r$ imply $|u(x)| >\bar{r}$; hence, the approximating scheme can be used.

More precisely, even if both assumption $(h_3)$ and (5.13) do not hold, the following statement can be proved.

Proposition 6.3

If $h \in \mathbb N$ is such that $\lambda ^\infty < \lambda _h^\infty$, then

$$\begin{aligned} \inf _{P \in \mathcal{P}_h}\ \max _{u \in P}\mathcal{J}(u) > -\infty . \end{aligned}$$

(6.7)

Proof

If $\lambda ^\infty \le 0$, then (6.7) follows from Remark 6.1.

Now, we assume that $\lambda ^\infty > 0$. As $0< \lambda ^\infty < \lambda _h^\infty$, fix $\varepsilon > 0$, such that

$$\begin{aligned} \varepsilon< & {} \min \left\{ 1, \left( \frac{\alpha _1}{\alpha _2 D_1 \mathrm{meas}(\Omega )}\right) ^{\frac{1}{p}}\right\} , \end{aligned}$$

(6.8)

$$\begin{aligned} \lambda ^\infty + \varepsilon p< & {} (1-\varepsilon )\ \lambda _h^\infty \ \left( 1 - \varepsilon p \left( \frac{\alpha _2 D_1 \mathrm{meas}(\Omega )}{\alpha _1}\right) ^{\frac{1}{p}}\right) , \end{aligned}$$

(6.9)

with $\alpha _1$ as in $(H_3)$, $\alpha _2$ as in (3.9) and $D_1$ as in (3.25).

Taking $P \in \mathcal{P}_h$, two cases may occur:

• (i) $u_0 \in P$ exists, such that $|\pi ^\infty (u_0)|_p \le \left( \frac{\alpha _1}{\alpha _2 D_1}\right) ^{\frac{1}{p}}$,

• (ii) for all $u \in P$: $|\pi ^\infty (u)|_p > \left( \frac{\alpha _1}{\alpha _2 D_1}\right) ^{\frac{1}{p}}$,

where $\pi ^\infty$ is as in (6.5).

Case (i) From $(H_3)$, (3.9) and (3.25) it follows that

$$\begin{aligned}\begin{aligned} \max _{u\in P} \mathcal{J}(u)&\ge \ \mathcal{J}(u_0)\\&\ge \ \left( \frac{\alpha _1}{\alpha _2} - D_1 |\pi ^\infty (u_0)|_p^p\right) \ \int _\Omega A^\infty (x,\nabla u) dx\ - D_2 \mathrm{meas}(\Omega )\\&\ge \ - D_2 \mathrm{meas}(\Omega ). \end{aligned} \end{aligned}$$

Case (ii) In this case, (6.8) and direct computations imply that

$$\begin{aligned} |v|_\infty \ >\ \varepsilon \qquad \hbox {for all }v \in \pi ^\infty (P), \end{aligned}$$

(6.10)

while, from (3.2) and (3.9) it follows that

$$\begin{aligned} \int _\Omega |v|^p dx \ \le \ \frac{\sigma _p}{\alpha _1}\qquad \hbox {for all }v \in \pi ^\infty (P). \end{aligned}$$

(6.11)

Now, since from (6.6) we have $\pi ^\infty (P) \in \mathcal{P}_h$, from definition (3.32), Remark 3.7 and the monotonicity property $(i_{2})$ in Proposition 2.5 it follows that also $R_\varepsilon (\pi ^\infty (P)) \in \mathcal{P}_h$; hence, (6.3) implies that

$$\begin{aligned} \max _{v \in \pi ^\infty (P)} \frac{\displaystyle \int _\Omega A^\infty (x,\nabla R_\varepsilon (v)) dx}{\displaystyle \frac{1}{p} \int _\Omega |R_\varepsilon (v)|^p dx} \ \ge \ \lambda _h^\infty . \end{aligned}$$

Thus, $u_\varepsilon \in P$ exists, such that, if $v_\varepsilon = \pi ^\infty (u_\varepsilon )$, we have

$$\begin{aligned} \int _\Omega A^\infty (x,\nabla R_\varepsilon (v_\varepsilon )) dx \ \ge \ \frac{\lambda _h^\infty }{p}\ \int _\Omega |R_\varepsilon (v_\varepsilon )|^p dx. \end{aligned}$$

(6.12)

We note that $(H_4)$ and (3.9) imply that a constant $r_\varepsilon > 0$ exists, such that

$$\begin{aligned} |t| > r_\varepsilon \; \Longrightarrow \; A(x,t,\xi ) \ge (1 - \varepsilon ) A^\infty (x,\xi ) \quad \hbox {for a.e. }x\in \Omega ,\hbox { all }\xi \in \mathbb R^N. \end{aligned}$$

(6.13)

Moreover, (3.24) holds for a suitable $L_\varepsilon > 0$, while as $|t|^p$ is a primitive of $p\ |t|^{p-1} t$, direct computations imply that

$$\begin{aligned} |t|^p\ \le \ |R_\varepsilon (t)|^p + \varepsilon p\ |t|^{p-1} \quad \hbox {for all }t \in \mathbb R. \end{aligned}$$

(6.14)

For simplicity, we put $\varrho _\varepsilon := \left( \int _\Omega A^\infty (x,\nabla u_\varepsilon ) dx\right) ^{\frac{1}{p}}$, so $u_\varepsilon = \varrho _\varepsilon v_\varepsilon$ and two cases may occur:

• (a) $\; \varepsilon \varrho _\varepsilon \ \le \ r_\varepsilon$,

• (b) $\; \varepsilon \varrho _\varepsilon \ >\ r_\varepsilon$.

Case (a) Since $A(x,t,\xi )$ is positive, from (3.25) and (6.11) it follows that

$$\begin{aligned} \mathcal{J}(u_\varepsilon ) \ \ge \ - D_1 |u_\varepsilon |_p^p - D_2 \mathrm{meas}(\Omega )\ \ge \ - \frac{D_1 \sigma _p}{\alpha _1} \left( \frac{r_\varepsilon }{\varepsilon }\right) ^p - D_2 \mathrm{meas}(\Omega ). \end{aligned}$$

Case (b) Taking $\Omega _\varepsilon = \Omega _\varepsilon ^{v_\varepsilon } = \{x \in \Omega : \ |v_\varepsilon (x)| > \varepsilon \}$, from (6.10) we have $\mathrm{meas}(\Omega _\varepsilon ) > 0$ and

$$\begin{aligned} |u_\varepsilon (x)|\ > \ r_\varepsilon \quad \hbox {for all }x \in \Omega _\varepsilon . \end{aligned}$$

Therefore, since $A(x,t,\xi )$ is positive, from (3.24), (6.13), and then $(H_4)$, (3.32), and (6.12), (6.14) we have that

$$\begin{aligned} \begin{aligned} \mathcal{J}(u_\varepsilon )&\ge (1-\varepsilon )\ \int _{\Omega _\varepsilon } A^\infty (x,\nabla u_\varepsilon ) dx\ -\ \left( \frac{\lambda ^\infty }{p} + \varepsilon \right) \int _{\Omega } |u_\varepsilon |^p dx \ -\ L_\varepsilon \mathrm{meas}(\Omega )\\&= (1-\varepsilon )\ \varrho _\varepsilon ^p\ \int _{\Omega } A^\infty (x,\nabla R_\varepsilon (v_\varepsilon )) dx\ -\ \left( \frac{\lambda ^\infty }{p} + \varepsilon \right) \varrho _\varepsilon ^p\ \int _{\Omega } |v_\varepsilon |^p dx - L_\varepsilon \mathrm{meas}(\Omega )\\&\ge (1-\varepsilon )\ \frac{\lambda _h^\infty }{p}\ \varrho _\varepsilon ^p \ \int _{\Omega } |R_\varepsilon (v_\varepsilon )|^p dx\ -\ \left( \frac{\lambda ^\infty }{p} + \varepsilon \right) \varrho _\varepsilon ^p\ \int _{\Omega } |v_\varepsilon |^p dx \\&\quad - L_\varepsilon \mathrm{meas}(\Omega )\\&\ge (1-\varepsilon )\ \frac{\lambda _h^\infty }{p}\ \varrho _\varepsilon ^p \ \left( \int _{\Omega } |v_\varepsilon |^p dx - \varepsilon p \int _{\Omega } |v_\varepsilon |^{p-1} dx\right) \\&\quad - \left( \frac{\lambda ^\infty }{p} + \varepsilon \right) \varrho _\varepsilon ^p\ \int _{\Omega } |v_\varepsilon |^p dx \ -\ L_\varepsilon \mathrm{meas}(\Omega ). \end{aligned} \end{aligned}$$

Summing up, from Hölder inequality, condition (ii) and direct computations it follows that

$$\begin{aligned} \begin{aligned} \mathcal{J}(u_\varepsilon )&\ge \frac{\varrho _\varepsilon ^p}{p}\ |v_\varepsilon |_p^p \left( (1-\varepsilon ) \lambda _h^\infty \left( 1 - \varepsilon p \left( \frac{\alpha _2 D_1 \mathrm{meas}(\Omega )}{\alpha _1}\right) ^{\frac{1}{p}}\right) \ -\ \left( \lambda ^\infty + \varepsilon p\right) \right) \\&\quad - L_\varepsilon \mathrm{meas}(\Omega ); \end{aligned} \end{aligned}$$

hence, (6.9) implies $\mathcal{J}(u_\varepsilon )\ \ge \ -\ L_\varepsilon \mathrm{meas}(\Omega )$.

Thus, the thesis follows. $\square$

Remark 6.4

The statement in Proposition 6.3 is optimal as, if $\lambda ^\infty > \lambda _h^\infty$, condition (6.7) does not hold (for more details, see Remark 7.4).

Proposition 6.5

If $(h_3)$ holds and $k \in \mathbb N$ is such that $\lambda _k^0 < \lambda ^0$, then we have

$$\begin{aligned} \inf _{P \in \mathcal{P}_k}\ \max _{u \in P}\mathcal{J}(u) < 0. \end{aligned}$$

Proof

Firstly, let us point out that from Lagrange’s Theorem and (3.6) it follows that a constant $b_1 > 0$ exists, such that

$$\begin{aligned} |t| \le 1\; \Longrightarrow \; |A(x,t,\xi ) - A^0(x,\xi )| \le b_1 |t| |\xi |^p \; \text {for a.e. }x \in \Omega ,\hbox { all }\xi \in \mathbb R^N. \end{aligned}$$

(6.15)

Now, being $\lambda _k^0 < \lambda ^0$, we can choose $\varepsilon > 0$ so small that

$$\begin{aligned} \left( 1 + \varepsilon \frac{b_1}{\alpha _1}\right) (\lambda ^0_k +\varepsilon ) + \varepsilon < \lambda ^0, \end{aligned}$$

(6.16)

with $\alpha _1$ as in $(H_3)$. Then, from (5.4) a constant $r_\varepsilon > 0$ exists, such that

$$\begin{aligned} |t| \le r_\varepsilon \quad \Longrightarrow \quad |G^0(x,t)|\ \le \ \frac{\varepsilon }{p}\ |t|^p \quad \text {a.e. in }\Omega . \end{aligned}$$

(6.17)

On the other hand, from Proposition 5.9 a set $Q_0 \in {\tilde{\mathcal{W}}}_k$ exists, such that

$$\begin{aligned} \max _{u \in Q_0} \frac{\displaystyle \int _\Omega A^0(x,\nabla u) dx}{\displaystyle \frac{1}{p}\int _\Omega |u|^p dx} < \lambda ^0_k + \varepsilon \end{aligned}$$

(6.18)

and, since from (6.2) we have $Q_0 \in \mathcal{P}_k$, $Q_0$ is also compact in X and a constant $\varrho _0 > 0$ exists, such that $|u|_\infty < \varrho _0$ for all $u \in Q_0$. Thus, we can define

$$\begin{aligned} P_\varepsilon ^0\ =\ \{v = \varrho _\varepsilon u \in X\ :\ u \in Q_0\}\quad \hbox {with}\quad \varrho _\varepsilon = \frac{1}{\varrho _0}\ \min \{1,\varepsilon ,r_\varepsilon \}. \end{aligned}$$

Since the map $u \in X \mapsto \varrho _\varepsilon u \in X$ is an odd homeomorphism on $(X,\Vert \cdot \Vert _X)$, then property $(i_{2})$ in Proposition 2.5 and (6.2) also imply that $P_\varepsilon ^0 \in \mathcal{P}_k$.

Taking any $v= \varrho _\varepsilon u \in P_\varepsilon ^0$, $u \in Q_0$, from $|v|_\infty < \min \{1,\varepsilon ,r_\varepsilon \}$ and (5.3), (6.15), (6.17), hypotheses $(H_3)$, $(H_9)$ and (6.18), it follows that

$$\begin{aligned} \mathcal{J}(v)\le & {} \ \int _\Omega A^0(x,\nabla v) dx + b_1 |v|_\infty \int _\Omega |\nabla v|^p dx - \frac{\lambda ^0 - \varepsilon }{p}\ \int _\Omega |v|^p dx\\\le & {} \left( 1 + \frac{b_1}{\alpha _1}\ \varepsilon \right) \varrho _\varepsilon ^p\ \int _\Omega A^0(x,\nabla u) dx - \left( \lambda ^0 - \varepsilon \right) \frac{\varrho _\varepsilon ^p}{p}\ \int _\Omega |u|^p dx\\\le & {} \left( \left( 1 + \frac{b_1}{\alpha _1} \varepsilon \right) (\lambda ^0_k +\varepsilon ) - (\lambda ^0 - \varepsilon ) \right) \frac{\varrho _\varepsilon ^p}{p} \int _\Omega |u|^p dx. \end{aligned}$$

Therefore, since the minimum of the $L^p$-norm in the compact set $Q_0$ is strictly positive, assumption (6.16) implies that $\displaystyle \max _{u \in P^0_\varepsilon } \mathcal{J}(u) < 0$ and the thesis is true. $\square$

Proof of Theorem 5.6

The hypotheses imply that $\mathcal{J}$ is an even functional, such that $\mathcal{J}(0) = 0$. Moreover, from Propositions 3.5 and 4.5 we have that $\mathcal{J}$ is $C^1$ in $(X,\Vert \cdot \Vert _X)$ and satisfies the (wPS) condition in $\mathbb R$. Then, assumption (5.11) allows us to apply Propositions 6.3 and 6.5, so definition (6.1) implies that (2.15) holds with $m = k - h +1$. Hence, the thesis follows from the first statement of Theorem 2.7. $\square$

Proof of Theorem 5.8

As in the proof of Theorem 5.6, in order to apply the second statement of Theorem 2.7, we have just to prove that $h \in \mathbb N$ exists, so that $-\infty< c_k < 0$ for all $k \ge h$, with $c_k$ as in (6.1).

To this aim, firstly we note that, taking any $k \in \mathbb N$, if we fix ${\bar{\lambda }} > \lambda _k^0$, from (5.13) a constant $\bar{r} > 0$ exists, such that

$$\begin{aligned} |t| \le \bar{r} \ \Longrightarrow \ F(x,t) \ge {\bar{\lambda }} |t|^p\quad \hbox {for a.e. }x\in \Omega . \end{aligned}$$

Then, reasoning as in the proof of Proposition 6.5, but with ${\bar{\lambda }}$ in the place of $\lambda ^0$ and $\bar{r}$ in the place of $r_\varepsilon$, we can find a subset $P_\varepsilon ^0 \in \mathcal{P}_k$, such that

$$\begin{aligned} c_k \le \max _{u \in P_\varepsilon ^0} \mathcal{J}(u) < 0. \end{aligned}$$

On the other hand, from Proposition 5.4 an integer $h \in \mathbb N$ exists, such that $\lambda _{h}^\infty > \lambda ^\infty$. Hence, Proposition 6.3 implies that $c_k \ge c_{h} > -\infty$ for all $k \ge h$. $\square$

7 Looking for positive critical levels

Throughout this section, we assume that $(H_1)$–$(H_{10})$ and $(h_1)$–$(h_4)$ hold and, in order to find critical points with positive critical level by applying Theorem 2.10, we require some useful information for the pseudo-index theory. To this aim, we need to evaluate the maximum of $\mathcal{J}$ in a family of subsets of X, which are part of a neighbourhood of the origin in $W^{1,p}_0(\Omega )$, so that it is quite natural to approximate $A(x,t,\xi )$ with $A^0(x,\xi )$. Unfortunately, for the $L^\infty$-norm of the elements of such sets no uniform a priori bound is given. So, for a suitable $K>0$ we split $\Omega$ into two parts: one where we can evaluate the truncation map $T_K$ (defined as in (3.31)) to apply the approximating scheme and its complementary set, where the remainder is small enough.

More precisely, the following statement can be proved.

Proposition 7.1

If $\lambda _h^0 > \lambda ^0$, then a suitable $\varepsilon _0 > 0$ can be found, such that for all $\varepsilon \in \ ]0,\varepsilon _0]$, a radius $\tau =\tau (\varepsilon ) >0$ and a constant $c_\varepsilon > 0$ exist, so that

$$\begin{aligned} \max _{u \in P}\mathcal{J}(u) \ge c_\varepsilon > 0 \quad \hbox {for all }P \in \mathcal{P}_h \hbox { such that } P \subset \partial B^0_\tau , \end{aligned}$$

(7.1)

with

$$\begin{aligned} \partial B^0_{\tau } \ =\ \big \{ u \in X:\ \int _\Omega A^0(x,\nabla u) dx = \tau ^p \big \}. \end{aligned}$$

Proof

Fix $\varepsilon \in ]0, \varepsilon _0]$ with $\varepsilon _0 > 0$ and

$$\begin{aligned} \varepsilon _0 < \min \left\{ 1, \frac{\alpha _1}{2 \sigma _p + \alpha _1}, \frac{\lambda _h^0 - \lambda ^0}{4 (\lambda _h^0 + p)}, \frac{2 (\lambda _h^0 - \lambda ^0) \alpha _1}{3 (\lambda _h^0 -\lambda ^0) \alpha _1 + 4 D \lambda ^0_h \sigma _p}\right\} , \end{aligned}$$

(7.2)

where $\alpha _1$ is as in $(H_3)$, $\sigma _p$ as in (3.2), $\alpha _4$ as in (5.1) and D as in (5.6).

Then, we note that from (5.1) and (5.2), respectively (5.4), a constant $r_\varepsilon > 0$ exists, such that

$$\begin{aligned} |t| \le r_\varepsilon \quad \Longrightarrow \quad A(x,t,\xi ) \ge (1 - \varepsilon ) A^0(x,\xi ) \quad \hbox {and}\quad |G^0(x,t)| \le \varepsilon |t|^p \end{aligned}$$

(7.3)

for a.e. $x \in \Omega$, all $\xi \in \mathbb R^N$.

Moreover, from (5.1) the set $\partial B^0_1$ is bounded in $W^{1,p}_0(\Omega )$, so it is compact in $L^p(\Omega )$ and $K_\varepsilon > 0$ exists, such that

$$\begin{aligned} \int _{\Omega _{K_\varepsilon }^v} |v|^p dx < \varepsilon \ \frac{\alpha _1}{2 \alpha _4 D}\qquad \hbox {for all }v \in \partial B^0_1. \end{aligned}$$

(7.4)

Now, put

$$\begin{aligned} \tau \ =\ \frac{r_\varepsilon }{K_\varepsilon }. \end{aligned}$$

(7.5)

Taking $P \in \mathcal{P}_h$ such that $P \subset \partial B^0_\tau$, two cases may occur:

• (i) $u_0 \in P$ exists such that $|\pi ^0(u_0)|_p \le \left( \frac{\alpha _1}{2 \alpha _4 D}\right) ^{\frac{1}{p}}$,

• (ii) for all $u \in P$: $|\pi ^0(u)|_p > \left( \frac{\alpha _1}{2 \alpha _4 D}\right) ^{\frac{1}{p}}$,

where $\pi ^0$ is as in (6.5).

Case (i) From definitions (6.4) and (6.5) we have $\pi ^0(u_0) \in \partial B^0_1$ with $u_0 = \tau \pi ^0(u_0)$, then from $(H_3)$, (5.1) and (5.6), assumption (i) implies that

$$\begin{aligned} \begin{aligned} \max _{u\in P} \mathcal{J}(u)&\ge \mathcal{J}(u_0)\ \ge \ \frac{\alpha _1}{\alpha _4} \int _\Omega A^0(x,\nabla u_0) dx - D \int _\Omega |u_0|^p dx\\&=\ \tau ^p \left( \frac{\alpha _1}{\alpha _4} - D \int _\Omega |\pi ^0(u_0)|^p dx\right) \ \ge \ \tau ^p \frac{\alpha _1}{2 \alpha _4}. \end{aligned} \end{aligned}$$

(7.6)

Case (ii) Firstly, suppose $\lambda ^0 \le 0$. From (6.4) and (6.5), for each $u \in \partial B^0_\tau$ we have $u = \tau v$ with $v = \pi ^0(u) \in \partial B^0_1$. We note that from (5.1) we have

$$\begin{aligned} \int _\Omega |\nabla v|^p dx \ \ge \ \frac{1}{\alpha _4}. \end{aligned}$$

(7.7)

Moreover, (7.5) implies

$$\begin{aligned} |u(x)|\ \le \ r_\varepsilon \quad \hbox {for a.e. }x \in \Omega {\setminus }\Omega _{K_\varepsilon }^{v}. \end{aligned}$$

From $(H_3)$, (5.3), (5.6), then (7.3), and from (3.2), (7.4) we have

$$\begin{aligned}\begin{aligned} \mathcal{J}(u)&\ge \alpha _1 \int _\Omega |\nabla u|^p dx - \int _{\Omega {\setminus }\Omega _{K_\varepsilon }^{v}} G^0(x,u) dx - D \int _{\Omega _{K_\varepsilon }^{v}} |u|^p dx\\&\ge \alpha _1 \tau ^p \int _\Omega |\nabla v|^p dx - \varepsilon \tau ^p \int _{\Omega {\setminus }\Omega _{K_\varepsilon }^{v}} |v|^p dx - D \tau ^p \int _{\Omega _{K_\varepsilon }^{v}} |v|^p dx\\&\ge (\alpha _1 - \varepsilon \sigma _p)\ \tau ^p \int _\Omega |\nabla v|^p dx - \varepsilon \tau ^p \frac{\alpha _1}{2 \alpha _4}. \end{aligned} \end{aligned}$$

Thus, summing up, estimate (7.2) and (7.7) imply

$$\begin{aligned} \max _{u \in P} \mathcal{J}(u)\ \ge \ \inf _{u \in \partial B^0_\tau } \mathcal{J}(u)\ \ge \ \tau ^p\ \frac{\alpha _1}{2 \alpha _4}. \end{aligned}$$

(7.8)

On the contrary, consider $\lambda ^0 > 0$. We point out that from (6.5) and (6.6), it follows that $\pi ^0(P) \in \mathcal{P}_h$ and $\pi ^0(P) \subset \partial B^0_1$. Hence, if we consider the truncation map $T_{K_\varepsilon }$ as in (3.31), from Remark 3.7 and property $(i_{2})$ in Proposition 2.5 we also have $T_{K_\varepsilon }(\pi ^0(P)) \in \mathcal{P}_h$, and (6.3) implies

$$\begin{aligned} \max _{v \in \pi ^0(P)} \frac{\displaystyle \int _\Omega A^0(x,\nabla T_{K_\varepsilon }(v)) dx}{\displaystyle \frac{1}{p} \int _\Omega |T_{K_\varepsilon }(v)|^p dx} \ \ge \ \lambda _h^0. \end{aligned}$$

Thus, $u_\varepsilon \in P$ exists, such that $u_\varepsilon = \tau v_\varepsilon$ with $v_\varepsilon = \pi ^0(u_\varepsilon )$, and

$$\begin{aligned} \int _\Omega A^0(x,\nabla T_{K_\varepsilon }(v_\varepsilon )) dx \ \ge \ \frac{\lambda _h^0}{p}\ \int _\Omega |T_{K_\varepsilon }(v_\varepsilon )|^p dx. \end{aligned}$$

(7.9)

If we define $\Omega _\varepsilon = \Omega _{K_\varepsilon }^{v_\varepsilon } = \{x \in \Omega :\ |v_\varepsilon (x)| > K_\varepsilon \}$, from (7.5) it follows that

$$\begin{aligned} |u_\varepsilon (x)|\ \le \ r_\varepsilon \quad \hbox {for a.e. }x \in \Omega {\setminus }\Omega _\varepsilon . \end{aligned}$$

Furthermore, in case (ii), estimate (7.4) implies

$$\begin{aligned} \int _{\Omega _\varepsilon } |v_\varepsilon |^p dx < \varepsilon \ |\pi ^0(u_\varepsilon )|^p_p. \end{aligned}$$

(7.10)

Then, since $A(x,t,\xi )$ is positive, from (5.3), (5.6) and (7.3), assumption $(H_9)$ and (3.31) imply that

$$\begin{aligned}\begin{aligned} \mathcal{J}(u_\varepsilon )&\ge (1-\varepsilon ) \int _{\Omega {\setminus }\Omega _\varepsilon } A^0(x,\nabla u_\varepsilon ) dx - \left( \frac{\lambda ^0}{p} + \varepsilon \right) \int _{\Omega {\setminus }\Omega _\varepsilon } |u_\varepsilon |^p dx\\&\quad - D \int _{\Omega _\varepsilon } |u_\varepsilon |^p dx\\&\ge (1-\varepsilon ) \tau ^p \int _{\Omega } A^0(x,\nabla T_{K_\varepsilon }(v_\varepsilon )) dx - \left( \frac{\lambda ^0}{p} + \varepsilon \right) \tau ^p \int _{\Omega } |T_{K_\varepsilon }(v_\varepsilon )|^p dx\\&\quad - D \tau ^p \int _{\Omega _\varepsilon } |v_\varepsilon |^p dx. \end{aligned} \end{aligned}$$

We note that, from (7.2), inequality (7.9) implies that

$$\begin{aligned} \begin{aligned}&(1-\varepsilon ) \int _{\Omega } A^0(x,\nabla T_{K_\varepsilon }(v_\varepsilon )) dx - \left( \frac{\lambda ^0}{p} + \varepsilon \right) \int _{\Omega } |T_{K_\varepsilon }(v_\varepsilon )|^p dx\\&\qquad \qquad \ge \ \frac{3}{4} \ \frac{\lambda ^0_h - \lambda ^0}{\lambda ^0_h}\ \int _{\Omega } A^0(x,\nabla T_{K_\varepsilon }(v_\varepsilon )) dx, \end{aligned} \end{aligned}$$

where from (5.1) and also (3.2), (3.31) and the characterization of $v_\varepsilon$, it follows that

$$\begin{aligned} \begin{aligned} \int _{\Omega } A^0(x,\nabla T_{K_\varepsilon }(v_\varepsilon )) dx&\ge \frac{\alpha _1}{\sigma _p} \int _{\Omega {\setminus }\Omega _\varepsilon } |v_\varepsilon |^p dx\\&=\ \frac{\alpha _1}{\sigma _p} \left( |\pi ^0(u_\varepsilon )|_p^p - \int _{\Omega _\varepsilon } |v_\varepsilon |^p dx\right) . \end{aligned} \end{aligned}$$

Hence, summing up, from (7.2), (7.10) and direct computations, it follows that

$$\begin{aligned} \begin{aligned} \mathcal{J}(u_\varepsilon )&\ge \ \tau ^p\ \left( (1 - \varepsilon )\ \frac{3 \alpha _1 (\lambda ^0_h - \lambda ^0)}{4 \lambda ^0_h \sigma _p} \ -\ \varepsilon D \right) |\pi ^0(u_\varepsilon )|_p^p\\&\ge \ \tau ^p\ \frac{\alpha _1 (\lambda ^0_h - \lambda ^0)}{4 \lambda ^0_h \sigma _p} \ |\pi ^0(u_\varepsilon )|_p^p; \end{aligned} \end{aligned}$$

hence, condition (ii) implies

$$\begin{aligned} \max _{u\in P} \mathcal{J}(u)\ \ge \ \mathcal{J}(u_\varepsilon ) \ge \ \tau ^p\ \frac{\alpha _1^2 (\lambda ^0_h - \lambda ^0)}{8 \alpha _4 \lambda ^0_h D\sigma _p}. \end{aligned}$$

(7.11)

Thus, taking

$$\begin{aligned} c_\varepsilon \ =\ \tau ^p\ \min \left\{ \frac{\alpha _1}{2 \alpha _4}, \frac{\alpha _1^2 (\lambda ^0_h - \lambda ^0)}{8 \alpha _4 \lambda ^0_h D\sigma _p}\right\} > 0, \end{aligned}$$

(7.1) follows from (7.6), (7.8) and (7.11). $\square$

In order to obtain ‘information at infinity’, we need the following technical lemma (for more details, see Step (a) in the proof of [14, Lemma 4.3]).

Lemma 7.2

If P is a compact subset of $W^{1,p}_0(\Omega )$, taking any $\varepsilon > 0$ a costant $\rho = \rho (P,\varepsilon ) > 0$ exists, such that

$$\begin{aligned} \int _{\Omega {\setminus } \Omega ^u_{\rho }} |\nabla u|^p dx < \varepsilon \quad \hbox {for all }u \in P. \end{aligned}$$

Proposition 7.3

If $\lambda _k^\infty < \lambda ^\infty$, then a suitable $\varepsilon _\infty > 0$ exists, such that for all $\varepsilon \in \ ]0,\varepsilon _\infty ]$ a constant $R^*_\varepsilon > 0$ can be found, such that for each $R \ge R^*_\varepsilon$ a suitable subset $P_\varepsilon ^R$ exists, so that

$$\begin{aligned} \max _{u \in P_\varepsilon ^R}\mathcal{J}(u) \ \le \ 0 \qquad \hbox {with }P_\varepsilon ^R \in \mathcal{P}_k\hbox { such that } P_\varepsilon ^R \subset \partial B^\infty _R, \end{aligned}$$

(7.12)

where

$$\begin{aligned} \partial B^\infty _{R} \ =\ \big \{ u \in X:\ \int _\Omega A^\infty (x,\nabla u) dx = R^p \big \}. \end{aligned}$$

Proof

Since $\lambda _k^\infty < \lambda ^\infty$, then $\varepsilon _\infty > 0$ exists, such that

$$\begin{aligned} (\lambda ^\infty _k + \varepsilon ) (1 + \varepsilon (M_2 + 1)) + \varepsilon p< \lambda ^\infty \quad \hbox {for all }0 < \varepsilon \le \varepsilon _\infty , \end{aligned}$$

(7.13)

with $M_2 > 0$ as in (3.12). Fixing any $0 < \varepsilon \le \varepsilon _\infty$, from $(H_4)$ and (3.9), a constant $r_\varepsilon > 0$ exists, such that

$$\begin{aligned} |t| > r_\varepsilon \; \Longrightarrow \; A(x,t,\xi ) \le (1 + \varepsilon ) A^\infty (x,\xi )\quad \hbox {for a.e. }x \in \Omega ,\hbox { all }\xi \in \mathbb R^N. \end{aligned}$$

(7.14)

Moreover, from Proposition 5.9 a set $Q_\varepsilon \in {\tilde{\mathcal{W}}}_k$ exists, such that

$$\begin{aligned} \max _{v \in Q_\varepsilon }\frac{\displaystyle \int _\Omega A^\infty (x,\nabla v) dx}{\displaystyle \frac{1}{p}\ \int _\Omega |v|^p dx} < \lambda ^\infty _k + \varepsilon . \end{aligned}$$

(7.15)

From (6.2) we have $Q_\varepsilon \in \mathcal{P}_k$ and, for simplicity, we can define $P_\varepsilon = \pi ^\infty (Q_\varepsilon ) \in \partial B^\infty _1$, with $\pi ^\infty$ as in (6.5). By definition, we have

$$\begin{aligned} \frac{\displaystyle \int _\Omega A^\infty (x,\nabla v) dx}{\displaystyle \frac{1}{p}\ \int _\Omega |v|^p dx} \ =\ \frac{1}{\displaystyle \frac{1}{p}\ \int _\Omega |\pi ^\infty (v)|^p dx} \quad \hbox {for all }v \in Q_\varepsilon ; \end{aligned}$$

hence, (6.6) and (7.15) imply

$$\begin{aligned} \frac{1}{\lambda ^\infty _k + \varepsilon }\ \le \ \frac{1}{p}\ \int _\Omega |w|^p dx \quad \hbox {for all }w \in P_\varepsilon ,\hbox { with }P_\varepsilon \in \mathcal{P}_k. \end{aligned}$$

(7.16)

Moreover, since $P_\varepsilon$ is also compact in $W^{1,p}_0(\Omega )$, from Lemma 7.2 $\rho = \rho (\varepsilon ) > 0$ exists, such that

$$\begin{aligned} \int _{\Omega {\setminus } \Omega ^w_{\rho }} |\nabla w|^p dx < \varepsilon \quad \hbox {for all }w \in P_\varepsilon . \end{aligned}$$

(7.17)

Then, fixing any $R > R_\varepsilon$ with $R_\varepsilon = \frac{r_\varepsilon }{\rho }$, we can define

$$\begin{aligned} P_\varepsilon ^R\ = \ \{u = R w \in X: \ w \in P_\varepsilon \}. \end{aligned}$$

(7.18)

Since the map $w\in P_\varepsilon \mapsto R w \in P_\varepsilon ^R$ is an odd homeomorphism from X in itself, from $(i_{2})$ in Proposition 2.5 it follows that $P_\varepsilon ^R \in \mathcal{P}_k$, while $P_\varepsilon ^R \subset \partial B^\infty _{R}$ from $(H_4)$.

Now, we note that, if $u = R w \in P_\varepsilon ^R$ with $w \in P_\varepsilon$, then

$$\begin{aligned} |u(x)| > r_\varepsilon \quad \hbox {if }x \in \Omega ^w_\rho , \end{aligned}$$

so, (3.12), (3.16), (3.22), (7.14), $(H_4)$ and direct computations imply that

$$\begin{aligned} \begin{aligned} \mathcal{J}(u)&\le (1 + \varepsilon ) R^p \int _{\Omega ^w_\rho } A^\infty (x,\nabla w) dx + M_2 R^p \int _{\Omega {\setminus }\Omega ^w_\rho } |\nabla w|^p dx\\&\quad - (\lambda ^\infty - \varepsilon p) \frac{R^p }{p} \ \int _{\Omega } |w|^p dx + b_{1,\varepsilon }. \end{aligned} \end{aligned}$$

with $b_{1,\varepsilon } = (M_1 + L_\varepsilon ) \mathrm{meas}(\Omega )$.

As $P_\varepsilon \subset \partial B^\infty _1$ and (7.13) imply $\lambda ^\infty - \varepsilon p > 0$, from (7.16), (7.17) and the last inequality, it follows that

$$\begin{aligned} \mathcal{J}(u)\ \le \ b_{2,\varepsilon } \ R^p\ + b_{1,\varepsilon } \qquad \hbox {for all }u \in P_\varepsilon ^R, \end{aligned}$$

(7.19)

with $b_{2,\varepsilon } = 1 + \varepsilon (1 + M_2) - \frac{\lambda ^\infty - \varepsilon p}{\lambda ^\infty _k + \varepsilon }$. Thus, since (7.13) implies $b_{2,\varepsilon } < 0$, a large enough $R_\varepsilon ^* > \frac{r_\varepsilon }{\rho }$ can be choosen, so that the second term of (7.19) is negative for all $R \ge R^*_\varepsilon$; hence, (7.12) holds. $\square$

Remark 7.4

If $k \in \mathbb N$ is such that $\lambda _k^\infty < \lambda ^\infty$, then

$$\begin{aligned} \inf _{P \in \mathcal{P}_k}\ \max _{u \in P}\mathcal{J}(u) = -\infty . \end{aligned}$$

In fact, reasoning as in the proof of Proposition 7.3 and taking $\varepsilon \le \varepsilon _\infty$ as in (7.13), for any $R > R_\varepsilon$ we can consider $P_\varepsilon ^R$ as in (7.18), so that $P_\varepsilon ^R \in \mathcal{P}_k$ and inequality (7.19) hold with $b_{2,\varepsilon } < 0$; hence,

$$\begin{aligned} \lim _{R \rightarrow +\infty }\max _{u \in P_\varepsilon ^R}\mathcal{J}(u)\ =\ -\infty . \end{aligned}$$

Proof of Theorem 5.7

The hypotheses imply that $\mathcal{J}$ is an even functional such that $\mathcal{J}(0) = 0$. Moreover, from Proposition 3.5 and Theorem 4.5 we have that $\mathcal{J}$ is $C^1$ in $(X,\Vert \cdot \Vert _X)$ and satisfies the (wPS) condition in $\mathbb R$.

Now, in order to prove Theorem 5.7, let h, $k \in \mathbb N$ be such that (5.12) holds, so we can fix $0 < \varepsilon \le \min \{\varepsilon _0, \varepsilon _\infty \}$, with $\varepsilon _0$ as in Proposition 7.1 and $\varepsilon _\infty$ as in Proposition 7.3, and take $\tau =\tau (\varepsilon ) >0$ and $\partial B^0_\tau$ as in Proposition 7.1. Then, we consider the pseudo-index $i^*(\cdot )$ related to the cohomological index $i(\cdot )$ on X, the set $\mathcal{M}= \partial B^0_\tau$ (closed in $W^{1,p}_0(\Omega )$ and then in X) and

$$\begin{aligned} \mathcal{H}\ =\ \{\gamma : X \rightarrow X:\ \gamma \hbox { odd homeomorphism, such that }\gamma (u) = u\,\,\forall u \in \mathcal{J}^0\}, \end{aligned}$$

which is defined as in (2.18) on any symmetric subset $P \subset X$.

Thus, as in Sect. 2, for all $j \in \mathbb N$, we define

$$\begin{aligned} \mathcal{P}_j^*\ =\ \{P \subset X:\;\ P\hbox { symmetric and compact in }X \hbox { with } i^*(P) \ge j \} \end{aligned}$$

and

$$\begin{aligned} c^*_j\ =\ \inf _{P \in \mathcal{P}_j^*} \max _{u \in P} \mathcal{J}(u). \end{aligned}$$

Firstly, taking any $P \in \mathcal{P}^*_h$, since the identity map on X is in $\mathcal{H}$, from definition (2.18), it follows that $h \le i^*(P) \le i(P\cap \mathcal{M})$. Thus, the properties of P and $\mathcal{M}$ imply that $P\cap \mathcal{M}\in \mathcal{P}_h$ and $P\cap \mathcal{M}\subset \partial B^0_\tau$; hence, from Proposition 7.1 we have that

$$\begin{aligned} \max _{u \in P}\mathcal{J}(u) \ \ge \ \max _{u \in P\cap \mathcal{M}}\mathcal{J}(u) \ge c_\varepsilon \end{aligned}$$

and, from the arbitrariness of P,

$$\begin{aligned} c^*_h \ge c_\varepsilon > 0. \end{aligned}$$

(7.20)

Now, take

$$\begin{aligned} R\ >\ \max \left\{ R^*_\varepsilon , \left( \frac{\alpha _2}{\alpha _1}\right) ^{\frac{1}{p}} \tau \right\} , \end{aligned}$$

(7.21)

with $R^*_\varepsilon >0$ as in Proposition 7.3 and $\alpha _1$ as in $(H_3)$, $\alpha _2$ as in (3.9), and consider $P_\varepsilon ^R$, such that (7.12) holds. By defining

$$\begin{aligned} IP_\varepsilon ^R\ =\ \{u = sv \in X:\; s \in [0,1],\, v \in P_\varepsilon ^R\}, \end{aligned}$$

the properties of $P_\varepsilon ^R$ imply that $IP_\varepsilon ^R$ is a symmetric compact subset of X.

We claim that

$$\begin{aligned} i^*(IP_\varepsilon ^R) \ge k;\qquad \hbox { hence }, IP_\varepsilon ^R \in \mathcal{P}^*_k. \end{aligned}$$

(7.22)

To this aim, let us consider the closed subsets

$$\begin{aligned} P_0 = \{u \in X:\, \int _\Omega A^0(x,u) dx \le \tau ^p\}, \quad P_1 = \overline{X {\setminus } P_0}, \end{aligned}$$

and, from (3.9), (5.1), (7.12) and (7.21), it follows that

$$\begin{aligned} P_\varepsilon ^R \subset \partial B^\infty _R \subset P_1. \end{aligned}$$

(7.23)

Taking any $\gamma \in \mathcal{H}$, from (7.12) the restriction of $\gamma$ to $P_\varepsilon ^R$ is the identity. Thus, the map

$$\begin{aligned} \varphi : (v,s)\in P_\varepsilon ^R \times [0,1] \mapsto \gamma (s v) \in X = P_0 \cup P_1 \end{aligned}$$

is continuous, odd in v, and such that

$$\begin{aligned} \varphi (P_\varepsilon ^R \times [0,1]) = \gamma (IP_\varepsilon ^R) \end{aligned}$$

is closed, while (7.23) implies that

$$\begin{aligned} \varphi (P_\varepsilon ^R \times \{0\}) = \{0\} \subset P_0,\quad \varphi (P_\varepsilon ^R \times \{1\}) = \gamma (P_\varepsilon ^R) = P_\varepsilon ^R \subset P_1. \end{aligned}$$

So, the piercing property $(i_{7})$ in Proposition 2.5 applies, and then from (7.12) it follows that

$$\begin{aligned} k\ \le \ i(P_\varepsilon ^R)\ \le \ i(\varphi (P_\varepsilon ^R \times [0,1]) \cap P_0 \cap P_1)\ =\ i(\gamma (IP_\varepsilon ^R) \cap \partial B^0_\tau ). \end{aligned}$$

Thus, (7.22) follows from the arbitrariness of $\gamma \in \mathcal{H}$ and (2.18). Finally, since (7.22) implies

$$\begin{aligned} c_k^* \le \max _{u \in IP_\varepsilon ^R} \mathcal{J}(u) < + \infty , \end{aligned}$$

(7.24)

the thesis follows from (7.20), (7.24) and Theorem 2.10, with $m = k+1 -h$. $\square$