Resonant-Superlinear and Resonant-Sublinear Dirichlet Problems

Abstract

In this paper, we study elliptic equations in which the reaction (right hand side) exhibits an asymmetric behavior as $x\rightarrow \pm \infty$. More precisely, we assume that we have resonance as $x\rightarrow -\infty$, while as $x\rightarrow +\infty$ the equation is superlinear. Using variational tools combined with the theory of critical groups, we prove several multiplicity theorems for nonlinear, nonhomogeneous equations and for semilinear equations (driven by the Laplacian).

1 Introduction

Let $\Omega \subseteq {\mathbb {R}}^N (N\ge 2)$ be a bounded domain with a $C^2$-boundary $\partial \Omega$. In this paper, we study the following Dirichlet (p, q)-equation

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta _pu(z)-\Delta _qu(z)= {\hat{\lambda }}_1(p)|u(z)|^{p-2}u(z)+f(z,u(z)) \quad \textrm{in}\ \Omega , \\ u|_{\partial \Omega }=0, \quad 1<q<p. \end{array}\right\} . \end{aligned}$$

(1)

For $r\in (1,\infty )$, by $\Delta _r$ we denote the r-Laplace differential operator defined by

$$\begin{aligned} \Delta _ru=\text {div}(|Du|^{r-2}Du) \quad \text {for all } u\in W_0^{1,r}(\Omega ). \end{aligned}$$

Equation (1) is driven by the sum of two such operators with distinct exponents (double phase problem with balanced growth). So, the differential operator in (1) is not homogeneous. In the reaction (right hand side) of (1), we have a resonant term $u\rightarrow {\hat{\lambda }}_1(p)|u|^{p-2}u$ with ${\hat{\lambda }}_1(p)>0$ being the principal eigenvalue of ($-\Delta _p,W_0^{1,p}(\Omega )$) and a Carathéodory perturbation f(z, x) (that is, for all $x\in {\mathbb {R}}~ z\rightarrow f(z,x)$ is measurable and for a.e. $z\in \Omega ,~x\rightarrow f(z,x)$ is continuous) which exhibits asymmetric behavior as $x\rightarrow \pm \infty$. Our work here was motivated by that of Domingos da Silva-Ribeiro [8], who investigated the “resonant-superlinear” case for semilinear equations driven by the Dirichlet Laplacian. Similar problems were considered earlier by Cuesta–de Figueiredo–Srikanth [5] and Cuesta–De Coster [6]. Other versions of asymmetric equations can be found in the works of Recova–Rumbos [24] (semilinear equations), Motreanu–Motreanu–Papageorgiou [17] (nonlinear equations driven by the p-Laplacian) and Gasiński–Papageorgiou [12], Papageorgiou–Winkert [22] ((p, 2)-equations).

Here, in addition to the “resonant-superlinear” case (that is, the equation is resonant as $x\rightarrow -\infty$ and superlinear as $x\rightarrow +\infty$), we examine also the “resonant-sublinear” case which has not been considered in the literature. For both cases, we prove multiplicity results.

2 Mathematical Background

The main spaces in the analysis of problem (1) are the Sobolev space $W^{1,p}_{0}(\Omega )$ and the Banach space $C^1_0(\bar{\Omega })=\{u\in C^1(\bar{\Omega })$: $u|_{\partial \Omega =0}\}$.

On account of the Poincaré inequality, the norm of $W^{1,p}_{0}(\Omega )$ is given by

$$\begin{aligned} \Vert u\Vert =\Vert Du\Vert _p~~~\text {for all}~~u\in W^{1,p}_{0}(\Omega ). \end{aligned}$$

The space $C^{1}_{0}(\bar{\Omega })$ is an ordered Banach space with positive (order) cone $C_+=\{u\in C^{1}_{0}(\bar{\Omega }):0\le u(z)~\text {for all~} z\in \bar{\Omega }\}$. This cone has a nonempty interior given by

$$\begin{aligned} \text {int~} C_+=\big \{u\in C_+: 0<u(z)~\text {for all~} z\in \Omega ,~\frac{\partial u}{\partial n}\bigg |_{\partial \Omega }<0\big \} \end{aligned}$$

with $n(\cdot )$ being the outward unit normal on $\partial \Omega$ and $\frac{\partial u}{\partial n}=(Du, n)_{\mathbb {R}^N}$.

For $r\in (1,\infty )$, let $A_r$: $W^{1,r}_{0}(\Omega )\rightarrow W^{-1,r^{'}}(\Omega )$ $=W_{0}^{1,r}(\Omega )^{*}$ $(\frac{1}{r}+\frac{1}{r^{\prime }}=1)$ be the nonlinear operator defined by

$$\begin{aligned} \langle A_r(u), h\rangle =\int _{\Omega } |Du|^{r-2}(Du,Dh)_{\mathbb {R}^N}dz\text {~~for all~}u, h\in W^{1,r}_{0}(\Omega ). \end{aligned}$$

If $r=2$, then we write $A=A_2\in \mathcal L(H^1_0(\Omega ),H^{-1}(\Omega ))$. We set $V=A_p+A_q:W^{1,p}_{0}(\Omega )\rightarrow W^{-1,p'}(\Omega )$ $(\frac{1}{p}+\frac{1}{p'}=1)$. This operator has the following properties (see Gasiński–Papageorgiou [10], Problem 2.192, p.279).

Proposition 1

$V:W^{1,p}_{0}(\Omega )\rightarrow W^{-1,p'}(\Omega )$ is bounded (maps bounded sets to bounded ones), continuous, strictly monotone (thus, it is maximal monotone too) and of type $(S)_{+}$, that is,

$''\text {if}~u_n{\mathop {\longrightarrow }\limits ^{w}} u$ in $W^{1,p}_{0}(\Omega )$  and  $\limsup _{n\rightarrow \infty }\langle V(u_{n}),u_{n}-u\rangle \le 0$, then $u_{n}\rightarrow u$ in $W^{1,p}_{0}(\Omega ).''$

We will need some facts about the spectrum of $(-\Delta _{p},W^{1,p}_{0}(\Omega ) )$ and of $(-\Delta ,H^1_0(\Omega ) )$. First we consider the nonlinear eigenvalue problem:

$$\begin{aligned} -\Delta _{r}u(z)=\widehat{\lambda } |u|^{r-2}u\text {~~in~} \Omega , u|_{\partial \Omega }=0, 1<r<\infty . \end{aligned}$$

(2)

We say that $\widehat{\lambda }\in \mathbb {R}$ is an eigenvalue of (2), if the problem has a nontrivial solution $\widehat{u}\in W^{1,r}_{0}(\Omega )$ known as an eigenfunction corresponding to the eigenvalue $\widehat{\lambda }$. The set of eigenvalues is denoted by $\widehat{\sigma }(r)$. Acting on (2) with $\widehat{u}$, we see that $\widehat{\sigma }(r)\subseteq \mathbb {R}_+=[0,\infty )$. In fact $\widehat{\sigma }(r)$ has a smallest element $\widehat{\lambda _1}(r)$ which has the following properties:

• (a) $\widehat{\lambda _1}(r)>0.$

• (b) $\widehat{\lambda _1}(r)$ is isolated in $\widehat{\sigma }(r)$ (that is, we can find $\varepsilon >0$ such that $(\widehat{\lambda _1}(r),\widehat{\lambda _1}(r)+\varepsilon )\cap \widehat{\sigma }(r)=\emptyset$).

• (c) $\widehat{\lambda _1}(r)$ is simple (that is, if $\widehat{u},\widetilde{u}$ are two eigenfunctions corresponding to $\widehat{\lambda _1}(r)$, then $\widehat{u}=\vartheta \widetilde{u}$ for some $\vartheta \in \mathbb {R}\setminus \{0\}$, that is the corresponding eigenspace is a one-dimensional vector space).

  $$\begin{aligned}&\quad (d)\quad \widehat{\lambda }_1(r)= \text {inf}\bigg [\frac{||Du||^r_r}{||u||^r_r}: u\in W^{1,r}_{0}(\Omega ), u\ne 0 \bigg ].&\end{aligned}$$

  (3)

The infimum in (3) is realized on the corresponding one dimensional eigenspace. Since in (3) u can be replaced by |u|, we see that the elements of this first eigenspace have fixed sign. In fact, $\hat{\lambda }_1(r)$ is the only eigenvalue with eigenfunctions of fixed sign. All other eigenvalues have eigenfunctions which are nodal (sign-changing). By $\widehat{u}_{1}(r)$ we denote the positive, $L^{r}$-normalized (that is, $\Vert \widehat{u}_{1}(r)\Vert _{r}=1$) eigenfunction. From Ladyzhenskaya–Uraltseva [14](p.286), we have that $\widehat{u}_{1}(r)\in L^{\infty }(\Omega )$ and then the nonlinear regularity theory of Lieberman [16] implies that $\widehat{u}_{1}(r)\in C_{+}\setminus \{0\}$. In fact the nonlinear Hopf maximum principle (see Gasiński–Papageorgiou [9] and Pucci–Serrin [23]) implies that $\widehat{u}_{1}(r)\in \text {int} C_+$. The set $\widehat{\sigma }(r)\subseteq (0,\infty )$ is closed and so the second eigenvalue of $(-\Delta _r,W_0^{1,r}(\Omega ))$ is defined by

$$\begin{aligned} \widehat{\lambda }_2(r)=\text {inf}~\big [\widehat{\lambda }\in \widehat{\sigma }(r):\widehat{\lambda }>\widehat{\lambda }_{1}(r)\big ]. \end{aligned}$$

Note that using the Lusternik–Schnirelmann minimax scheme (see Gasiński–Papageorgiou [9]), we can generate a whole sequence $\{\widetilde{\lambda }_k(r)\}_{k\in \mathbb {N}}$ of eigenvalues of $(-\Delta _r,W_0^{1,r}(\Omega ))$, known as “variational eigenvalues”, such that $\widetilde{\lambda }_k(r)\rightarrow +\infty$ as $k\in +\infty$. We have $\widetilde{\lambda }_{1}(r)=\widehat{\lambda }_{1}(r)$ and $\widetilde{\lambda }_2(r)=\widehat{\lambda }_2(r)$, but we do not know if the sequence of variational eigenvalues exhausts $\widehat{\sigma }(r)$. This is the case in the linear eigenvalue problem (that is, $r=2$). So, we consider the following linear eigenvalue problem

$$\begin{aligned} -\Delta u=\widehat{\lambda }u \text { in } \Omega ,u|_{\partial \Omega }=0. \end{aligned}$$

(4)

The spectrum $\widehat{\sigma }$(2) of (4) is a sequence $\{\widehat{\lambda }_k(2)\}_{k\in \mathbb {N}}$ of eigenvalues such that $\widehat{\lambda }_k(2)\rightarrow +\infty$ as $k\rightarrow \infty$ and the corresponding eigenspaces $E(\widehat{\lambda }_k(2)),k\in \mathbb {N}$ are all linear spaces and we have

$$\begin{aligned} H^1_0(\Omega )=\overline{\bigoplus _{k\in \mathbb {N}}E(\widehat{\lambda }_k(2))}. \end{aligned}$$

Each eigenspace $E(\widehat{\lambda }_k(2))$ has the unique continuation property; that is, if $u\in E(\widehat{\lambda }_k(2)) (k\in \mathbb {N})$ vanishes on a set of positive Lebesgue measure, then $u\equiv 0$. Note that $E(\widehat{\lambda }_k(2))\subseteq C^1_0(\overline{\Omega })$.

In this case, all eigenvalues have variational characterizations. So, for $m\in \mathbb {N}$, let

$$\begin{aligned} \overline{H}_m=\bigoplus ^m_{k=1}E(\widehat{\lambda }_k(2)).\text { and } \widehat{H}_m=\overline{\bigoplus _{k\ge m}E(\widehat{\lambda }_k(2))}. \end{aligned}$$

We have

$$\begin{aligned} \hat{\lambda }_1(2)=\text {inf}\bigg [\frac{||Du||^2_2}{||u||^2_2}: u\in H^{1}_{0}(\Omega ), u\ne 0 \bigg ] \end{aligned}$$

(5)

and for $m\in \mathbb {N}\setminus \{1\}$ (that is, $m\ge 2$), we have

$$\begin{aligned} \hat{\lambda }_m(2)=&\text {sup}\bigg [\frac{||Du||^2_2}{||u||^2_2}: u\in \overline{H}_{m}, u\ne 0\bigg ]\nonumber \\ =&\text {inf}\bigg [\frac{||Du||^2_2}{||u||^2_2}: u\in \widehat{H}_{m}, u\ne 0 \bigg ]. \end{aligned}$$

(6)

Note that (5) is a particular case of (3) (when $r=2$) and the infimum is realized on $E(\widehat{\lambda }_1(2))$. In (6), both the supremum and the infimum are realized on $E(\widehat{\lambda }_m(2))$.

Using (5),(6) and the unique continuation property, we can have the following basic inequalities.

Proposition 2

(a) If $\vartheta \in L^{\infty }(\Omega )$ and $\vartheta (z)\ge \widehat{\lambda }_{m}(2)$ for a.e. $z\in \Omega ,\vartheta \not \equiv \widehat{\lambda }_{m}(2)$, then there exists $c_1>0$ such that

$$\begin{aligned} \Vert Du\Vert ^{2}_{2}-\int _{\Omega }\vartheta (z)|u|^{2}dz\le -c_1\Vert u\Vert ^{2}~~~\text {for all}~u\in \overline{H}_m. \end{aligned}$$

(b) If $\vartheta \in L^{\infty }(\Omega )$ and $\vartheta (z)\le \widehat{\lambda }_{m}(2)$ for a.e. $z\in \Omega ,\vartheta \not \equiv \widehat{\lambda }_{m}(2)$, then there exists $c_2>0$ such that

$$\begin{aligned} \Vert Du\Vert ^{2}_{2}-\int _{\Omega }\vartheta (z)|u|^{2}dz\ge c_2\Vert u\Vert ^{2}~~~\text {for all}~u\in \widehat{H}_m. \end{aligned}$$

We will also consider a weighted version of (4). So, let $\eta \in L^{\infty }(\Omega )\setminus \{0\},\eta (z)\ge 0$ for a.e. $z\in \Omega$ and consider the following linear eigenvalue problem

$$\begin{aligned} -\Delta u=\widetilde{\lambda }\eta (z)u \text { in } \Omega ,u|_{\partial \Omega }=0. \end{aligned}$$

(7)

The spectrum of this eigenvalue problem is a sequence $\{\widetilde{\lambda }_k(\eta ,2)\}_{k\in \mathbb {N}}$ of distinct eigenvalues such that $\widetilde{\lambda }_k(\eta ,2)\rightarrow \infty$ as $k\rightarrow \infty$. Again we have variational characterizations for all the eigenvalues using the Rayleigh quotient $\frac{\Vert Du\Vert ^2_2}{\int _{\Omega }\eta (z)u^2dz}$.

Proposition 3

If $\eta ,\widehat{\eta }\in L^{\infty }(\Omega ){\setminus }\{0\}, \eta (z)\le \widehat{\eta }(z)$ for a.e. $z\in \Omega ,\eta \ne \widehat{\eta },$ then $\widetilde{\lambda }_1(\widehat{\eta },2)<\widetilde{\lambda }_1(\eta ,2)$.

Let X be a Banach and $\varphi \in C^1(X),c\in \mathbb {R}$. We set

$$\begin{aligned}&K_{\varphi }=\{u\in X:\varphi '(u)=0\},\\&\varphi ^{c}=\{u\in X:\varphi (u)\le c\}. \end{aligned}$$

Also, if $Y_{2}\subseteq Y_{1}\subseteq X$ and $k\in \mathbb {N}_{0}$, then by $H_{k}(Y_{1},Y_{2})$ we denote the $k{\mathop {=}\limits ^{th}}$-relative singular homology group with integer coefficients. Given $u\in K_{\varphi }$ isolated with $c=\varphi (u)$, then the critical groups of $\varphi$ at u are defined by

$$\begin{aligned} C_k(\varphi , u)=H_k(\varphi ^c\cap U, \varphi ^c\cap U\backslash \{u\}),~\text {for all}~k\in \mathbb {N}_{0}, \end{aligned}$$

with U being a neighborhood of u such that $K_{\varphi }\cap \varphi ^c\cap U=\{u\}$(isolating neighborhood). The excision property of singular homology implies that the above definition of critical groups is independent of the isolating neighborhood U.

We say that $\varphi \in C^{1}(X)$ satisfies the C-condition, if every sequence $\{u_n\}_{n\in \mathbb {N}}\subseteq X$ such that $\{\varphi (u_n)\}_{n\in \mathbb {N}}\subseteq \mathbb {R}$ is bounded and $(1+\Vert u_n\Vert _X)\varphi '(u_n)\rightarrow 0$ in $X^*$ as $n\rightarrow \infty$, admits a strongly convergent subsequence. Suppose that $\varphi \in C^{1}(X)$ satisfies the C-condition and $-\infty < \text { inf } \varphi (K_{\varphi })$. Let $c < \text { inf } \varphi (K_{\varphi })$. Then, the critical groups of $\varphi (\cdot )$ at infinity are defined by

$$\begin{aligned} C_k(\varphi , \infty )=H_k(X,\varphi ^{c}),~\text {for all}~k\in \mathbb {N}_{0}. \end{aligned}$$

Using the second deformation theorem (see [20], p.386), we see that this definition is independent of the choice of the level $c<\inf \varphi (K_{\varphi })$.

Suppose $K_{\varphi }$ is finite. We introduce the following series in $t\in \mathbb {R}$.

$$\begin{aligned}&M(t,u)=\sum _{k\in \mathbb {N}_{0}}~ \text {rank}~C_{k}(\varphi ,u)t^{k},~ \text {for all}~u\in K_{\varphi },\nonumber \\&P(t,\infty )=\sum _{k\in \mathbb {N}_{0}}~\text {rank}~C_{k}(\varphi ,\infty )t^{k}. \end{aligned}$$

These two series are related by the Morse identity

$$\begin{aligned} \sum _{u\in K_{\varphi }}M(t,u)=P(t,\infty )+(1+t)Q(t)~\text {for all}~t\in \mathbb {R}. \end{aligned}$$

(8)

Here, $Q(t)=\sum _{k\in \mathbb {N}_{0}}\alpha _{k}t^{k}$ is a formal series in $t\in \mathbb {R}$ with nonnegative integer coefficients $\alpha _k$. For details, we refer to [20].

If $u:\Omega \rightarrow \mathbb {R}$ is measurable, then for all $z\in \Omega$ we define $u^{\pm }(z)=max\{\pm u(z),0\}$. Evidently $z\rightarrow u^{\pm }(z)$ are both measurable and $u=u^{+}-u^{-},|u|=u^{+}+u^{-}$. If $u\in W^{1,p}_0(\Omega )$, then $u^{\pm }\in W^{1,p}_0(\Omega )$. Finally if $1<r<\infty$, then

$$\begin{aligned} r^*=\left\{ \begin{array}{lll} \frac{Nr}{N-r} \qquad &{}\text {if}~r< N,\\ \\ +\infty \qquad &{}\text {if}~N\le r. \end{array}\right. \end{aligned}$$

3 Resonant Superlinear Equations

The hypotheses on the data of (1) are the following:

$H_0$: If $q<N,$ then $N\le \frac{pq}{p-q}$.

$H_{1}$: f: $\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ is a Carathéodory function such that $f(z, 0)=0$ for a.e. $z\in \Omega$ and

(i) $|f(z,x)|\le a(z)[1+|x|^{r-1}]$ for a.e. $z\in \Omega$, all $x\in \mathbb {R}$, with $a\in L^{\infty }(\Omega )$, $p<r<p^*$;

(ii) If $F(z, x)=\int ^x_0f(z, s)ds$, then $\lim _{x\rightarrow +\infty }\dfrac{F(z, x)}{x^{p}}=+\infty$ uniformly for a.e. $z\in \Omega$ and there exist $\mu \in \big ((r-p)\max \{1,\frac{N}{p}\},p^{*}\big )$ such that

$$\begin{aligned} 0<\beta _{0}\le \liminf _{x\rightarrow +\infty } \frac{f(z,x)x-pF(z,x)}{x^{\mu }}~ \text {uniformly for}~a.e.~z\in \Omega ; \end{aligned}$$

(iii) $\lim _{x\rightarrow -\infty }\dfrac{f(z, x)}{|x|^{p-2}x}=0$ uniformly for a.e. $z\in \Omega$;

(iv) there exist $\theta \in L^{\infty }(\Omega )$ and $\widehat{\theta },\widehat{\eta }>0$ such that

$$\begin{aligned}&\widehat{\lambda }_1(q)\le \theta (z) ~\text {for}~ a.e.~z\in \Omega , \theta \not \equiv \widehat{\lambda }_1(q),\\&\theta (z)\le \liminf _{x\rightarrow 0}\frac{f(z,x)}{|x|^{q-2}x} \le \limsup _{x\rightarrow 0}\frac{f(z,x)}{|x|^{q-2}x}\le \widehat{\theta } ~\text {uniformly for}~a.e.~z\in \Omega ,\\&e(z,x)=f(z,x)x-pF(z,x)\ge -\widehat{\eta } ~ \text {for}~a.e.~z\in \Omega ,~\text {all}~x\le 0; \end{aligned}$$

(v) for every $\rho >0$, there exists $\widehat{\xi }_{\rho }>0$ such that for $a.e.z\in \Omega$, the function $x\rightarrow f(z,x)+\widehat{\xi }_{\rho }|x|^{p-2}x$ is nondecreasing on $[-\rho ,0]$.

Remarks: Hypothesis $H_{1}(ii)$ implies that

$$\begin{aligned} \lim _{x\rightarrow +\infty }\frac{f(z,x)}{x^{p-1}}=+\infty ~\text {uniformly for}~a.e.z\in \Omega . \end{aligned}$$

So, the perturbation $f(z,\cdot )$ is $(p-1)$ superlinear as $x\rightarrow +\infty$. However $f(z,\cdot )$ need not satisfy the Ambrosetti–Rabinowitz condition(see [2]), which is common in the literature when studying superlinear problems. Hypothesis $H_1(iii)$ implies that problem (1) is resonant with respect to the principal eigenvalue $\widehat{\lambda }_1(\rho )$ when $x\rightarrow -\infty$. So, the reaction of problem (1) exhibits an asymmetric behavior asymptotically as $x\rightarrow \pm \infty$.

The following two functions satisfy hypotheses $H_1$. For the sake of simplicity, we drop the z-dependence

$$\begin{aligned} f_1(x)=\left\{ \begin{array}{lll} (({\widehat{\theta }}+1) |x|^{\tau -2}-|x|^{s-2})x &{} \text {if }~ x<-1 \\ {\widehat{\theta }}|x|^{q-2}x &{}\text {if }~-1\le x\le 1\\ {\widehat{\theta }} x^{r-1}&{}\text {if }~1<x \end{array}\right. \end{aligned}$$

with ${\widehat{\theta }}>{\widehat{\lambda }}_1(q), 1<\tau<s<p<r$,

$$\begin{aligned} f_2(x)=\left\{ \begin{array}{lll} (({\widehat{\theta }}+1) |x|^{\tau -2}-|x|^{s-2})x &{} \text {if }~ x<-1 \\ {\widehat{\theta }}|x|^{q-2}x &{}\text {if }~-1\le x\le 1\\ {\widehat{\theta }} (\ln x+1)x^{p-1}&{}\text {if }~1<x \end{array}\right. \end{aligned}$$

with ${\widehat{\theta }}>{\widehat{\lambda }}_1(q), 1<\tau<s<p$.

Note that $f_1(\cdot )$ satisfies the AR-condition but $f_2(\cdot )$ does not.

Let $\varphi :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}$ be the energy functional for problem (1) defined by

$$\begin{aligned} \varphi (u)=\frac{1}{p}\Vert Du\Vert ^p_p+\frac{1}{q} \Vert Du\Vert ^q_q-\frac{\widehat{\lambda }_1(p)}{p}\Vert u\Vert ^p_p-\int _{\Omega }F(z,u)dz \text { for all }u\in W^{1,p}_0(\Omega ). \end{aligned}$$

Clearly, $\varphi \in C^1(W^{1,p}_0(\Omega ))$. Also, let $\varphi _{-}:W^{1,p}_0(\Omega )\rightarrow \mathbb {R}$ be the “negative” truncation of $\varphi (\cdot )$ defined by

$$\begin{aligned} \varphi _{-}(u)=\frac{1}{p}\Vert Du\Vert ^p_p+\frac{1}{q}\Vert Du\Vert ^q_q- \frac{\widehat{\lambda }_1(p)}{p}\Vert u^-\Vert ^p_p-\int _{\Omega }F(z,-u^-)dz \text { for all }u\in W^{1,p}_0(\Omega ). \end{aligned}$$

Proposition 4

If hypotheses $H_{1}$ hold, then the functional $\varphi _-(\cdot )$ is coercive.

Proof

We argue by contradiction. So, suppose that $\varphi _-(\cdot )$ is not coercive. Then, we can find $\{u_n\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} \Vert u_{n}\Vert \rightarrow \infty ~\text { and } \varphi _{-}(u_{n})\le c_{3}~\text {for some}~c_{3}>0,~\text {all}~n\in \mathbb {N}. \end{aligned}$$

(9)

We have

$$\begin{aligned} \frac{1}{p}\Vert Du_{n}\Vert ^{p}_{p}+\frac{1}{q}\Vert Du_{n}\Vert ^{q}_{q}\le c_3+\frac{\widehat{\lambda }_1(p)}{p}\Vert u^{-}_n\Vert ^p_p+\int _{\Omega }F(z,-u^-_n)dz \text { for all }n\in \mathbb {N}. \end{aligned}$$

(10)

If $\{u_n^-\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0(\Omega )$ is bounded, then from (10) and hypothesis $H_1(i)$ we see that $\{u_n\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0(\Omega )$ is bounded, which contradicts (9).

So, we may assume that $\Vert u^{-}_{n}\Vert \rightarrow \infty$. Let $v_{n}=\frac{u^{-}_{n}}{\Vert u^{-}_{n}\Vert },~n\in \mathbb {N}$. Then, $\Vert v_{n}\Vert =1,~v_{n}\ge 0$ for all $n\in \mathbb {N}$ and we may assume that

$$\begin{aligned} v_n{\mathop {\longrightarrow }\limits ^{w}} v~\text {in}~W^{1,p}_{0}(\Omega ),v_{n}\rightarrow v~\text {in}~L^{p}(\Omega ),v\ge 0. \end{aligned}$$

(11)

From (10), we have

$$\begin{aligned}&\frac{1}{p}\Vert Dv_{n}\Vert ^{p}_{p}+\frac{1}{q\Vert u^-_n\Vert ^{p-q}} \Vert Dv_{n}\Vert ^{q}_{q}\le \frac{c_1}{\Vert u^-_n\Vert ^p}\nonumber \\ {}&+\frac{\widehat{\lambda }_1(p)}{p} \Vert v^{-}_n\Vert ^p_p-\int _{\Omega }\frac{F(z,-u^-_n)}{\Vert u^-_n\Vert ^p}dz \text { for all }n\in \mathbb {N}. \end{aligned}$$

(12)

Hypotheses $H_1(i),(iii)$ imply that given $\varepsilon >0$, we can find $c_4=c_4(\varepsilon )>0$ such that

$$\begin{aligned} |F(z,x)|\le c_4+\varepsilon |p|^p~ \text { for}~a.e.z\in \Omega ,\text {all}~x\le 0. \end{aligned}$$

(13)

From (13), we infer that

$$\begin{aligned} \bigg \{\frac{F(\cdot ,-u_{n}^{-}(\cdot ))}{\Vert u^{-}_{n}\Vert ^{p}}\bigg \}_{n\in \mathbb {N}}\subseteq L^{1}(\Omega )~\text {is uniformly integrable}. \end{aligned}$$

The Dunford–Pettis theorem and hypothesis $H_{1}$(iii) imply that at least for a subsequence we have that

$$\begin{aligned} \frac{F(\cdot ,-u_{n}^{-}(\cdot ))}{\Vert u^{-}_{n} \Vert ^{p}}{\mathop {\longrightarrow }\limits ^{w}} 0~\text {in}~L^{1}(\Omega ). \end{aligned}$$

(14)

So, if in (12) we pass to the limit as $n\rightarrow \infty$ and use (11) and (14), we obtain

$$\begin{aligned}&\Vert Dv\Vert ^p_p\le \widehat{\lambda }_1(p)\Vert v\Vert ^p_p,\\ \Rightarrow&\Vert Dv\Vert ^p_p=\widehat{\lambda }_1 (p)\Vert v\Vert ^p_p~~(\text {see }(3)),\\ \Rightarrow&v=\vartheta \widehat{u}_1(p)~ \text { for some }\vartheta \ge 0~~(\text {see } (11)). \end{aligned}$$

If $\vartheta =0$, then $v=0$ and so we have

$$\begin{aligned} \Vert Dv_n\Vert ^p_p\rightarrow 0 \Rightarrow v_n\rightarrow 0 ~\text { in } W^{1,p}_{0}(\Omega ), \end{aligned}$$

which contradicts the fact that $\Vert v_n\Vert =1$ for all $n\in \mathbb {N}$.

If $\vartheta >0$, then $v\in ~\text {int}C_+$ and so

$$\begin{aligned} v^-_n(z)\rightarrow +\infty ~\text {for}~a.e.z\in \Omega . \end{aligned}$$

(15)

We have

$$\begin{aligned} \frac{d}{dx}\bigg [\frac{F(z,x)}{|x|^{p}}\bigg ]=&\frac{f(z,x)x-pF(z,x)}{|x|^{p}x}\nonumber \\ \le&\frac{-\widehat{\eta }}{|x|^{p-2}x}\text {for a.e. } z\in \Omega , \text { all }x\le 0~~\text {(see hypothesis }H_1(iv)),\\ \Rightarrow&\frac{F(z,s)}{|s|^{p}}-\frac{F(z,x)}{|x|^{p}} \le -\frac{\widehat{\eta }}{p}\bigg [\frac{1}{|x|^{p}}- \frac{1}{|s|^{p}}\bigg ]~~\text {for}~a.e.~z\in \Omega ,~\text {all}~x<s<0. \end{aligned}$$

We pass to the limit as $x\rightarrow -\infty$ and use hypothesis $H_1(iii)$. We obtain

$$\begin{aligned}&\frac{F(z,s)}{|s|^{p}}\le \frac{\widehat{\eta }}{p} \frac{1}{|s|^p}~\text {for}~a.e.~z\in \Omega ,~ \text {all}~s<0.\nonumber \\ \Rightarrow&-\widehat{\eta }\le -pF(z,s)~~ \text {for}~a.e.~z\in \Omega ,~\text {all}~s\le 0. \end{aligned}$$

(16)

From (8) we have

$$\begin{aligned}&\frac{1}{q}\Vert Du^-_{n}\Vert ^{q}_{q}-\int _{\Omega }F(z,-u^-_n)dz \le c_1~\text {for all }n\in \mathbb {N}, \nonumber \\ \Rightarrow&\frac{1}{q}\widehat{\lambda }_1(q)\Vert u^-_{n} \Vert ^{q}_{q}\le c_5~~\text {for some }c_5>0,~\text {all } n\in \mathbb {N}~~(\text {see}(16)). \end{aligned}$$

(17)

Fatou’s lemma and (15) imply that

$$\begin{aligned} \Vert u^-_n\Vert _q\rightarrow +\infty ,\text { which contradicts } (17). \end{aligned}$$

Therefore, we infer that

$$\begin{aligned} \{u_n^-\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0 (\Omega )\text { is bounded.} \end{aligned}$$

Then, from (10) it follows that

$$\begin{aligned} \{u_n\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0(\Omega )\text { is bounded.} \end{aligned}$$

which contradicts (9). This proves that $\varphi _-(\cdot )$ is coercive. $\square$

Using the above proposition, we can generate a negative solution for problem (1).

Proposition 5

If hypotheses $H_{1}$ hold, then problem (1) admits a negative solution $v_{0}\in -~\text {int}~C_{+}$ which is a local minimizer of the energy functional $\varphi (\cdot )$.

Proof

From Proposition 4, we know that $\varphi _-(\cdot )$ is coercive. Also, using the Sobolev embedding theorem, we see that $\varphi _-(\cdot )$ is sequentially weakly lower semicontinuous. Therefore, by the Weierstrass–Tonelli theorem, we can find $v_0\in W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} \varphi _-(v_0)=\text {inf}\bigg [\varphi _-(u):u \in W^{1,p}_0(\Omega )\bigg ]. \end{aligned}$$

(18)

On account of hypothesis $H_1(iv)$, given $\varepsilon >0$, we can find $\delta =\delta (\varepsilon )>0$ such that

$$\begin{aligned} \frac{1}{q}[\theta (z)-\varepsilon ]|x|^q\le F(z,x)~~\text {for}~a.e.~z\in \Omega ,~\text {all}~|x|\le \delta . \end{aligned}$$

(19)

Recall that $\widehat{u}_1(q)\in ~\text {int}C_+$. So, we can find $t\in (0,1)$ small such that

$$\begin{aligned} t\widehat{u}_1(q) (z)\in [0,\delta ]~ \text { for all}~z\in \overline{\Omega }. \end{aligned}$$

(20)

Then, we have

$$\begin{aligned}&~\varphi _{-}(-t\hat{u}_{1}(q))\le \frac{t^{p}}{p} \Vert D\hat{u}_{1}(q)\Vert ^{p}_{p} +\frac{t^{q}}{q}\bigg [\int _{\Omega }(\widehat{\lambda }_1(q) -\theta (z))\hat{u}_{1}(q)^{q}dz+\epsilon \bigg ]\nonumber \\&~(\text {see } (19),({20}) \text { and recall } \Vert \hat{u}_{1}(q)\Vert _{q}=1). \end{aligned}$$

(21)

Since $\hat{u}_1(q)\in ~\text {int} C_+$, using the hypothesis on $\theta (\cdot )$(see hypothesis $H_1(iv)$), we have

$$\begin{aligned} \int _{\Omega }[\theta (z)-\widehat{\lambda }_1(q)]\hat{u}_{1}(q)dz=\beta >0. \end{aligned}$$

Therefore, choosing $\varepsilon \in (0,\beta )$, from (21) we obtain

$$\begin{aligned} \varphi _{-}(-t\hat{u}_{1}(q))\le c_{6}t^{p}-c_{7}t^{q}~\text {for some } c_{6},c_{7}>0 \text { and } t\in (0,1) \text { small}. \end{aligned}$$

Recall that $q<p$. So, choosing $t\in (0,1)$ even smaller if necessary we have

$$\begin{aligned}&\varphi _{-}(-t\hat{u}_{1}(q))<0,\\ \Rightarrow&\varphi _-(v_0)<0= \varphi _-(0)~\text {(see (18))}\\ \Rightarrow&v_{0}\ne 0. \end{aligned}$$

From (18), we have

$$\begin{aligned}&\varphi {'}_{-}(v_0)=0,\nonumber \\ \Rightarrow&\langle V(v_0),h\rangle = \widehat{\lambda }_1(p)\int _{\Omega }(v^-_0)^{p-1} hdz+\int _{\Omega }f(z,v^-_0)hdz \text { for all } h\in W^{1,p}_0(\Omega ). \end{aligned}$$

(22)

In (22), we use the test function $h=v^+_0\in W^{1,p}_0(\Omega )$ and obtain

$$\begin{aligned}&\Vert Dv^+_0\Vert ^p_p\le 0,\nonumber \\ \Rightarrow&v_0\le 0,v_0\ne 0. \end{aligned}$$

(23)

From (22) and (23), it follows that

$$\begin{aligned} -\Delta _pv_0-\Delta _qv_0=\widehat{\lambda }_1(p)| v_0|^{p-2}v_0+f(z,v_0)~\text {in}~\Omega . \end{aligned}$$

Theorem 7.1, p.286, of Ladyzhenskaya–Uraltseva [14] implies that $v_0\in L^{\infty }(\Omega )$. Then, using the nonlinear regularity theory of Lieberman [16], we have $v_0\in (-C_+){\setminus }\{0\}$. Let $\rho =\Vert v_0\Vert _{\infty }$ and let $\hat{\xi }_\rho >0$ be as postulated by hypothesis $H_1(v)$. We have

$$\begin{aligned}&\Delta _p(-v_0)+\Delta _q(-v_0)\le \hat{\xi }_\rho (-v_0)^{p-1},\nonumber \\ \Rightarrow&v_0\in \text {-int} C_+~\text { (see Pucci--Serrin [{23}])}. \end{aligned}$$

(24)

Note that

$$\begin{aligned} \varphi \bigg |_{-C_+}=\varphi _-\bigg |_{-C_+}. \end{aligned}$$

So, from (24) it follows that $v_0$ is a local $C^1_0(\overline{\Omega })$-minimizer of $\varphi (\cdot )$. From Gasiński–Papageorgiou [11] it follows that $v_0$ is a local $W^{1,p}_0(\Omega )$-minimizer of $\varphi (\cdot )$. $\square$

Using this constant sign solution $v_0\in -~\text {int}~C_+$ together with variational tools and critical groups, we will generate a second nontrivial smooth solution and have the first multiplicity theorem for problem (1). To this end, we need to strengthen hypothesis $H_1(iv)$ (the behavior of the perturbation $f(z,\cdot )$ near zero). The new hypotheses on the perturbation f(z, x) are the following:

$H_{2}$: f: $\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ is a Carathéodory function such that $f(z, 0)=0$ for a.e. $z\in \Omega$, hypotheses $H_2(i),(ii),(iii),(v)$ are the same as the corresponding hypotheses $H_1(i),(ii),(iii),(v)$ and the new condition is (iv) there exist $\hat{\theta }\in (\hat{\lambda }_2(q),\infty ){\setminus } \hat{\sigma }(q)$ and $\hat{\eta }>0$ such that

$$\begin{aligned}&\lim _{x\rightarrow 0}\frac{f(z,x)}{|x|^{q-2}x}=\hat{\theta }~ \text {uniformly for a.e.}z\in \Omega ,\\&e(z,x)=f(z,x)x-pF(z,x)\ge -\hat{\eta }~\text { for a.e.} z\in \Omega ,~\text { all }x \le 0. \end{aligned}$$

The examples illustrating hypotheses $H_1$ (see functions $f_1(\cdot )$ and $f_2(\cdot )$) work also here, only now ${\widehat{\theta }}>{\widehat{\lambda }}_2(q), {\widehat{\theta }}\not \in {\widehat{\sigma }} (q)$.

As we already mentioned earlier, our approach will combine variational arguments (the mountain pass theorem) with critical groups (Morse theory). To do this, we need to know that the energy functional $\varphi (\cdot )$ satisfies the compactness condition (the C-condition). This can be done using the initial (weaker) hypotheses $H_1$(since in $H_2$ we have modified only the behavior of $f(z,\cdot )$ near zero).

Proposition 6

If hypotheses $H_{1}$ hold, then the energy functional $\varphi (\cdot )$ satisfies the C-condition.

Proof

Consider a sequence $\{u_n\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} |\varphi (u_n)|\le c_8~\text { for some } c_8>0, \text { all }n\in \mathbb {N}, \end{aligned}$$

(25)

$$\begin{aligned} (1+\Vert u_n\Vert )\varphi '(u_n)\rightarrow 0~\text { in } W^{-1,p'}_0(\Omega ). \end{aligned}$$

(26)

From (25) we have

$$\begin{aligned} \Vert Du_n\Vert _{p}^{p}+\frac{p}{q}\Vert Du_n\Vert _{q}^{q}- \widehat{\lambda }_1(p) \Vert u_n\Vert _{p}^{p}-\int _{\Omega }pF(z,u_n)dz\le pc_8~~\text {for all } n\in \mathbb {N}. \end{aligned}$$

(27)

From (26), we have

$$\begin{aligned} |\langle \varphi '(u_n),h \rangle |\le \frac{\varepsilon _n\Vert h\Vert }{1+\Vert u_n\Vert }~\text { for all }h\in W^{1,p}_0(\Omega ),~\text { with }\varepsilon _n\rightarrow 0^+. \end{aligned}$$

(28)

In (28), we choose $h=-u^-_n\in W^{1,p}_0(\Omega )$ and obtain

$$\begin{aligned} -\Vert Du^-_n\Vert _{p}^{p}-\Vert Du^-_n\Vert _{q}^{q}+\widehat{\lambda }_1(p)\Vert u^-_n\Vert _{p}^{p}+\int _{\Omega }f(z_1,-u^-_n) (-u^-_n)dz\le \varepsilon _n ~~\text {for all } n\in \mathbb {N}. \end{aligned}$$

(29)

Adding (28) and (29) and using hypothesis $H_1(iv)$ and the fact that $q<p$, we obtain

$$\begin{aligned}&\Vert Du^+_n\Vert _{p}^{p}+\frac{p}{q}\Vert Du^+_n\Vert _{q}^{q}- \widehat{\lambda }_1(p)\Vert u_n^+\Vert _{p}^{p}\nonumber \\ {}&-\int _{\Omega }pF(z,u^+_n)dz\le c_9 ~~\text {for some } c_9>0, \text { all } n\in \mathbb {N}. \end{aligned}$$

(30)

In (28), we use the test function $h=u^+_n\in W^{1,p}_0(\Omega )$ and obtain

$$\begin{aligned} -\Vert Du^+_n\Vert _{p}^{p}-\Vert Du^+_n\Vert _{q}^{q}+\widehat{\lambda }_1 (p)\Vert u^+_n\Vert _{p}^{p}+\int _{\Omega }f(z_1,u^+_n)u^+_ndz\le \varepsilon _n ~~\text {for all } n\in \mathbb {N}. \end{aligned}$$

(31)

We add (30) and(31) and use that $q<p$. We obtain

$$\begin{aligned} \int _{\Omega }[f(z,u^+_n) (u^+_n)-pF(z,u^+_n)]dz\le c_{10}~~\text { for some } c_{10}>0 ,\text { all } n\in \mathbb {N}. \end{aligned}$$

(32)

Using (32), hypothesis $H_1(iii)$ and reasoning as in the “Claim” in the proof of Proposition 4 of Papageorgiou–Rădulescu–Zhang [21], we show that

$$\begin{aligned} \{u_n^+\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0(\Omega )\text { is bounded. } \end{aligned}$$

(33)

For all $n\in \mathbb {N}$, we have

$$\begin{aligned}&\varphi (u_n)=\varphi (u^+_n)+\varphi (-u^-_n)\\ \Rightarrow&\{\varphi (-u^-_n)\}_{n\in \mathbb {N}} \subseteq \mathbb {R}\text { is bounded. } (\text { see } (25), (33)) \end{aligned}$$

But then (33) and Proposition 4 imply that

$$\begin{aligned} \{u_n^-\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0(\Omega )\text { is bounded. } \end{aligned}$$

(34)

From (33) and (34), it follows that

$$\begin{aligned} \{u_n\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0(\Omega )\text { is bounded. } \end{aligned}$$

So, we may assume that

$$\begin{aligned} u_n{\mathop {\longrightarrow }\limits ^{w}} u~\text {in}~W^{1,p}_{0}(\Omega )~\text {and}~u_{n}\rightarrow u~\text {in}~L^{r}(\Omega ). \end{aligned}$$

(35)

In (28), we use the test function $h=u_n-u\in W^{1,p}_{0}(\Omega )$, pass to the limit as $n\rightarrow \infty$ and use (35). We obtain

$$\begin{aligned}&\lim _{n\rightarrow \infty }\langle V(u_n),u_n-u\rangle =0,\\ \Rightarrow&u_n\rightarrow u~\text {in}~W^{1,p}_{0}(\Omega )~ \text { (see Proposition ({1}))},\\ \Rightarrow&\varphi (\cdot ) ~\text { satisfies the C-condition.} \end{aligned}$$

$\square$

We assume that $K_{\varphi }$ is finite or otherwise we already have an infinity of nontrivial solutions of (1) which by the nonlinear regularity theory are smooth(in $C^1_0(\overline{\Omega })$). Next we show the triviality of $C_1(\varphi ,0)$. To do this, we need hypothesis $H_0$ and also hypotheses $H_2$.

Proposition 7

If hypotheses $H_{0},H_{2}$ hold, then $C_1(\varphi ,0)=0.$

Proof

Let $\hat{\theta }\in (\hat{\lambda }_2(q),\infty ){\setminus } \hat{\sigma }(q)$ be as in hypothesis $H_2(iv)$. We consider the $C^1$ function $\psi :W^{1,p}_{0}(\Omega )\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \psi (u)=\frac{1}{q}\Vert Du\Vert _{q}^{q}-\frac{\hat{\theta }}{q}\Vert u\Vert _{q}^{q} ~\text {for all}~u\in W^{1,p}_{0}(\Omega ). \end{aligned}$$

We introduce the homotopy

$$\begin{aligned} h_t(u)=t\varphi (u)+(1-t)\psi (u) ~\text {for all}~t\in [0,1],~\text {all}~u\in W^{1,p}_{0}(\Omega ). \end{aligned}$$

Suppose that we can find $\{t_n\}_{n\in \mathbb {N}}\subseteq [0,1]$ and $\{u_n\}_{n\in \mathbb {N}}\subseteq W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} t_n\rightarrow t~\text {in}~[0,1],u_n\rightarrow 0~\text {in}~W^{1,p}_{0}(\Omega ),(h_{t_n})^{'}(u_n)=0~\text { for all }~n\in \mathbb {N}. \end{aligned}$$

(36)

From the equality in (36), we have

$$\begin{aligned}&t_n\langle A_p(u_n),h\rangle +\langle A_q(u_n),h\rangle \nonumber \\ =&\widehat{\lambda }_1(p)t_n\int _{\Omega }|u_n|^{p-2}u_nhdz+t_n \int _{\Omega }f(z,u_n)hdz+(1-t_n)\hat{\theta }\int _{\Omega }|u_n|^{q-2}u_nhdz \end{aligned}$$

(37)

for all $h\in W^{1,p}_{0}(\Omega )$, all $n\in \mathbb {N}$.

Let $\Vert \cdot \Vert _{1,q}$ denote the norm of $W^{1,q}_{0}(\Omega )$($\Vert u\Vert _{1,q}=\Vert Du\Vert _q$ for all $u\in W^{1,q}_{0}(\Omega )$) and recall that $W^{1,p}_{0}(\Omega )\hookrightarrow W^{1,q}_{0}(\Omega ).$ We set $v_n=\frac{u_n}{\Vert u_n\Vert _{1,q}},n\in \mathbb {N}$. Then, $\Vert v_n\Vert _{1,q}=1$ and so we may assume that

$$\begin{aligned} v_n{\mathop {\longrightarrow }\limits ^{w}} v~\text {in}~W^{1,q}_{0}(\Omega )~\text {and}~v_{n}\rightarrow v~\text {in}~L^{q}(\Omega ). \end{aligned}$$

(38)

From (37) we have

$$\begin{aligned}&\Vert u_n\Vert ^{p-q}_{1,q} t_n\langle A_p(v_n),h\rangle +\langle A_q(v_n),h\rangle \nonumber \\ =&\Vert u_n\Vert ^{p-q}_{1,q}\widehat{\lambda }_1(p)\int _{\Omega } |v_n|^{p-2}v_nhdz+t_n\int _{\Omega }\frac{f(z,u_n)}{\Vert u_n\Vert ^{q-1}_{1,q}}hdz +(1-t_n)\hat{\theta }\int _{\Omega }|v_n|^{q-2}v_nhdz\nonumber \\&\quad \quad \quad \quad \quad \quad \quad \text {for all } h\in W^{1,p}_{0}(\Omega ), \text { all } n\in \mathbb {N}. \end{aligned}$$

(39)

Note that $\Vert u_n\Vert _{1,q}\rightarrow 0$ (see (36) and recall that $W^{1,p}_{0}(\Omega )\hookrightarrow W^{1,q}_{0}(\Omega )).$ In (39) we choose the test function $h=v_n-v\in W^{1,p}_{0}(\Omega )$ and exploit the monotonicity of $A_p(\cdot )$. We have

$$\begin{aligned}&\Vert u_n\Vert ^{p-q}_{1,q} t_n\langle A_p(v),v_n-v\rangle + \langle A_q(v_n),v_n-v\rangle \nonumber \\ \le&\Vert u_n\Vert ^{p-q}_{1,q}\widehat{\lambda }_1(p) \int _{\Omega }|v_n|^{p-2}v_n(v_n-v)dz+t_n \int _{\Omega }\frac{f(z,u_n)}{\Vert u_n\Vert ^{q-1}_{1,q}}(v_n-v)dz\nonumber \\&+(1-t_n)\hat{\theta }\int _{\Omega }|v_n|^{q-2}v_n(v_n-v)dz. \end{aligned}$$

(40)

On account of hypothesis $H_0$, we have $p\le q^*$ and so $W^{1,q}_{0}(\Omega )\hookrightarrow L^{p}(\Omega )$ (by the Sobolev embedding theorem). Also, we have

$$\begin{aligned}&|\int _{\Omega }|v_n|^{p-2}v_n(v_n-v)dz| \nonumber \\ \le&\int _{\Omega }|v_n|^{p-1}|v_n-v|dz \nonumber \\ \le&\Vert v_n\Vert ^{p-1}_p\Vert v_n-v\Vert _p\le c_{11} ~ \text {for some}~c_{11}> 0,~\text {all}~n\in \mathbb {N}. \end{aligned}$$

(41)

Let $\langle \cdot ,\cdot \rangle _{1,q}$ denote the duality brackets for the pair $(W^{1,q}_{0}(\Omega ),W^{-1,q{'}}(\Omega ))$ and recall that $W^{-1,q{'}}(\Omega )\hookrightarrow W^{-1,p{'}}(\Omega )$ (see Gasiński–Papageorgiou [9], p.141). If in (40) we pass to the limit as $n\rightarrow \infty$ and use (38) and (41), we obtain

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\langle A_q(v_n),v_n-v\rangle \nonumber \\ =&\limsup _{n\rightarrow \infty }\langle A_q(v_n),v_n-v\rangle _{1,q}\le 0 \nonumber \\ =&v_n\rightarrow v ~\text {in}~W^{1,q}_{0}(\Omega )~\text {and so }\Vert v\Vert _{1,q}=1. \end{aligned}$$

(42)

In (39), we pass to the limit as $n\rightarrow \infty$ and use (42) and hypothesis $H_2(iv)$. We obtain

$$\begin{aligned}&\langle A_q(v),h\rangle _{1,q}=\int _{\Omega }\hat{\theta }|v |^{q-2}vhdz~\text {for all}~h\in W^{1,p}_{0}(\Omega ), \nonumber \\ \Rightarrow&-\Delta _qv=\hat{\theta }|v|^{q-2}v ~\text {in}~\Omega ,v|_{\partial \Omega }=0~(\text {since } W^{1,p}_0(\Omega )\hookrightarrow W^{1,q}_0(\Omega ) \text { densely}). \end{aligned}$$

(43)

But by hypothesis $\hat{\theta }\notin \hat{\sigma }(p)$. So, from (43) we have $v=0,$ which contradicts (42). Therefore, (36) cannot happen and then the homotopy invariance property of critical groups (see Papageorgiou–Rădulescu–Repovš [20], p.509), we have

$$\begin{aligned} C_k(\varphi ,0)=C_k(\psi ,0) \text { for all }k\in \mathbb {N}_0. \end{aligned}$$

(44)

Since $\hat{\theta }>\hat{\lambda }_2(q),\hat{\theta }\notin \hat{\sigma }(p)$, from Theorem 1.1 of Dancer–Perera [7], we have

$$\begin{aligned}&C_1(\psi ,0)=0,\\ \Rightarrow&C_1(\varphi ,0)=0 ~(\text {see}~(44)). \end{aligned}$$

$\square$

Now we have all the necessary tools to produce a second nontrivial solution for problem (1) and have the first multiplicity theorem.

Theorem 8

If hypotheses $H_{0},H_{2}$ hold, then problem (1) has at least two nontrivial solutions $v_{0}\in -\text {int}~C_{+},u_0\in C^1_0(\overline{\Omega }){\setminus }\{0\}$.

Proof

From Proposition 5, we already have a solution $v_{0}\in -\text {int}~C_{+}$ which is a local minimizer of $\varphi (\cdot )$. Recall that without any loss of generality $K_{\varphi }$ is assumed to be finite. Invoking Theorem 5.7.6, p.449, of Papageorgiou–Rădulescu–Repovš [20], we can find $\rho \in (0,1)$ small such that

$$\begin{aligned} \varphi (v_0)<\text {inf}~\bigg [\varphi (u):\Vert u-v_0\Vert =\rho \bigg ]=m_\rho . \end{aligned}$$

(45)

On account of hypothesis $H_2(ii)=H_1(ii)$, if $u\in \text {int}~C_{+}$, then

$$\begin{aligned} \varphi (tu)\rightarrow -\infty ~\text { as }t\in +\infty . \end{aligned}$$

(46)

Moreover, from Proposition 6 we have that

$$\begin{aligned} \varphi (\cdot )~\text { satisfies the C-condition}. \end{aligned}$$

(47)

Then, (45), (46), (47) permit the use of the mountain pass theorem. So, we can find $u_0\in W^{1,p}_{0}(\Omega )$ such that

$$\begin{aligned}&u_0\in K_{\varphi },\varphi (v_0)<m_0\le \varphi (u_0),\\ \Rightarrow&u_0\in C^1_0(\overline{\Omega })~\text { (nonlinear regularity) is a solution of } ({1}),u_0\ne v_0. \end{aligned}$$

From Theorem 6.5.8, p.527, of [20], we have

$$\begin{aligned} C_1(\varphi _1,u_0)\ne 0. \end{aligned}$$

(48)

Then, (48) and Proposition 7 imply $u_0\ne 0$. $\square$

When $q=2$ (a (p, 2)-equation) and if strengthen the regularity of the perturbation $f(z,\cdot )$, we can generate a third nontrivial smooth solution.

The problem under consideration is now the following.

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta _pu(z)-\Delta u(z)= {\hat{\lambda }}_1(p)|u(z)|^{p-2}u(z)+f(z,u(z)) \quad \textrm{in}\ \Omega , \\ u|_{\partial \Omega }=0, \quad 2<p. \end{array}\right\} . \end{aligned}$$

(49)

The new hypotheses on f(z, x) are the following:

$H_{3}$: f: $\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ is a measurable function such that for a.e. $z\in \Omega ~~f(z,\cdot )\in C^1(\mathbb {R})$ and

(i) $|f_x'(z,x)|\le a(z)[1+|x|^{r-2}]$ for a.e.$z\in \Omega$, all $x\in \mathbb {R}$, with $a\in L^{\infty }(\Omega )$, $p<r<p^*$;

(ii), (iii) are the same as the corresponding hypotheses $H_1(ii),(iii)$;

(iv) there exists $m\ge 2$ such that

$$\begin{aligned}&f_x'(z,0)\in [\hat{\lambda }_m(2), \hat{\lambda }_{m+1}(2)] ~\text { for a.e.}z\in \Omega ,\\&f_x'(\cdot ,0)\not \equiv \hat{\lambda }_{m} (2),f_x'(\cdot ,0)\not \equiv \hat{\lambda }_{m+1}(2); \end{aligned}$$

(v) is the same with hypothesis $H_1(v)$.

The following function satisfies these hypotheses:

$$\begin{aligned} f(x)=\left\{ \begin{array}{lll} \theta x&{} \text {if }~ x<1 \\ \theta x^{r-1}+(r-2)\theta \ln x &{} \text {if }~ 1\le x \end{array}\right. \end{aligned}$$

with $\theta \in (\hat{\lambda }_m(2),\hat{\lambda }_{m+1}(2))$.

Now, the energy function $\varphi :W^{1,p}_{0}(\Omega )\rightarrow \mathbb {R}$ has the following form

$$\begin{aligned} \varphi (u)=\frac{1}{p}\Vert Du\Vert _{p}^{p}+\frac{1}{2}\Vert Du\Vert _{2}^{2}- \frac{\hat{\lambda }_{1}(p)}{p}\Vert u\Vert _{p}^{p}-\int _{\Omega }F(z,u)dz ~\text {for all}~u\in W^{1,p}_{0}(\Omega ). \end{aligned}$$

In this case $\varphi \in C^2(W^{1,p}_{0}(\Omega ))$(recall $p>2$). The asymmetric behavior of the reaction as $x\rightarrow \pm \infty$ leads to the following result due to Papageorgiou–Winkert [22](Proposition 4.8).

Proposition 9

If hypotheses $H_3$ hold, then $C_k(\varphi ,\infty )=0$ for all $k\in \mathbb {N}_0$.

Using Morse theoretic tools (critical groups), we can generate a third nontrivial smooth solution and have the second multiplicity theorem.

Theorem 10

If hypotheses $H_0,H_3$ hold, then problem (49) has at least three nontrivial solutions

$$\begin{aligned} v_0\in \text {-int}~C_+,u_0,w_0\in C^1_0(\overline{\Omega })\setminus \{0\}. \end{aligned}$$

Proof

From Theorem 8, we already have two nontrivial solutions.

$$\begin{aligned} v_0\in ~\text {-int}~C_+,u_0\in C^1_0(\overline{\Omega })\setminus \{0\}. \end{aligned}$$

Recall that $v_0$ is a local minimizer of $\varphi (\cdot )$(see Proposition 5). So, we have

$$\begin{aligned} C_k(\varphi ,v_0)=\delta _{k,0}\mathbb {Z}~\text {for all}~k\in \mathbb {N}_0. \end{aligned}$$

(50)

Also, we know that

$$\begin{aligned} C_1(\varphi ,u_0)\ne 0~(\text {see}~(48)). \end{aligned}$$

Since $\varphi \in C^2(W^{1,p}_{0}(\Omega ))$, from Claim 3 in the proof of Proposition 3.5 of Papageorgiou–Rădulescu [19], we have

$$\begin{aligned} C_k(\varphi ,v_0)=\delta _{k,1}\mathbb {Z}~\text {for all}~k\in \mathbb {N}_0. \end{aligned}$$

(51)

Consider the function $\hat{\psi }\in C^2(H^1_0(\Omega ))$ defined by

$$\begin{aligned} \hat{\psi }(u)=\frac{1}{2}\Vert Du\Vert _{2}^{2}-\frac{1}{2}\int _{\Omega }f_x'(z,0)u^2dz ~\text {for all}~u\in H^{1}_{0}(\Omega ). \end{aligned}$$

On account of hypothesis $H_3(iv)$ and of the unique continuation property of the eigenspaces of $(-\Delta , H^{1}_{0}(\Omega ))$, we have that $u=0$ is a nondegenerate critical point of $\hat{\psi }(\cdot )$ with Morse index $d_m=\text {dim}~\overline{H}_m=\text {dim}~ \bigoplus ^m_{k=1}E(\widehat{\lambda }_k(2))$. Therefore,

$$\begin{aligned} C_k(\hat{\psi },0)=\delta _{k,d_m}\mathbb {Z}~\text {for all}~k\in \mathbb {N}_0~\text {(see [{20}], Proposition 6.2.6, p.479).} \end{aligned}$$

(52)

Let $\psi =\hat{\psi }|_{W^{1,p}_{0}(\Omega )}$(recall that $2<p$). Then, Theorem 6.6.26, p.545, of [20] implies that

$$\begin{aligned}&C_k(\psi ,0)=C_k(\hat{\psi },0)~\text {for all}~k\in \mathbb {N}_0,\nonumber \\ \Rightarrow&C_k(\psi ,0)=\delta _{k,d_m}\mathbb {Z}~\text {for all}~k\in \mathbb {N}_0~~ (\text {see } ({52})). \end{aligned}$$

(53)

A homotopy invariance argument as in the proof of Proposition 7, shows that

$$\begin{aligned}&C_k(\varphi ,0)=C_k(\psi ,0)~\text {for all}~k\in \mathbb {N}_0,\nonumber \\ \Rightarrow&C_k(\varphi ,0)=\delta _{k,d_m}\mathbb {Z}~\text {for all}~k\in \mathbb {N}_0. \end{aligned}$$

(54)

From Proposition 9, we know that

$$\begin{aligned} C_k(\varphi ,\infty )=0~\text {for all}~k\in \mathbb {N}_0. \end{aligned}$$

(55)

Suppose $K_{\varphi }=\{v_0,u_0,0\}$. From (50),(51),(54),(55) and the Morse identity with $t=-1$(see (8)), we have

$$\begin{aligned} (-1)^0+(-1)^1+(-1)^{dm}=0, \end{aligned}$$

a contradiction. Hence, there exists $w_0\in K_{\varphi }\subseteq C^1_0(\overline{\Omega })$ such that $w_0\notin \{v_0,u_0,0\}$. Therefore, $w_0$ is the third nontrivial smooth solution of (49). $\square$

4 Semilinear Equations

In this section, we deal with the special case of semilinear equations driven by the Dirichlet Laplacian. In what follows $\hat{\lambda }_k=\hat{\lambda }_k(2)$ for all $k\in \mathbb {N}$ and $\hat{u}_1=\hat{u}_1(2)\in ~\text {int}~C_+$.

The equation under consideration is the following

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta u(z)={\hat{\lambda }}_1 u(z) +f(z,u(z)) \quad \textrm{in}\ \Omega , \\ u|_{\partial \Omega }=0. \end{array}\right\} . \end{aligned}$$

(56)

The hypotheses on the perturbation f(z, x) are the following:

$H_{4}:f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ is a Carathéodory function such that for $a.e.~z\in \Omega ,~f(z,0)=0$ and

(i) $|f(z,x)|\le a(z)[1+|x|^{r-1}]$ for $a.e.~z\in \Omega$, all $x\in \mathbb {R}$, with $a\in L^{\infty }(\Omega )$, $2< r<2^{*};$

(ii) if $F(z,x)=\int ^x_0f(z,s)ds$, then $\lim _{x\rightarrow +\infty }\frac{F(z,x)}{x^2}=+\infty$ uniformly for $a.e.~z\in \Omega$ and there exists $\mu \in ((r-2)\frac{N}{2},2^*)$ such that

$$\begin{aligned} 0<\beta _0\le \liminf _{x\rightarrow +\infty }\frac{f(z,x) x-2F(z,x)}{x^\mu }~\text {uniformly for a.e.}~z\in \Omega ; \end{aligned}$$

(iii) $\lim _{x\rightarrow -\infty }\frac{f(z,x)}{x}=0, \lim _{x\rightarrow -\infty }f(z,x)=-\infty$ uniformly for a.e.$z\in \Omega$ and $\lim _{x\rightarrow -\infty }[f(z,x)x-2F(z,x)]=+\infty$ for a.e. $x\in \Omega$;

(iv) there exist $\theta \in L^{\infty }(\Omega )$ and $\hat{t},\hat{\eta }>0$ such that

$$\begin{aligned}&\theta (z)\le 0~\text {for a.e.}~z\in \Omega , \theta \not \equiv 0, \limsup _{x\rightarrow 0}\frac{f(z,x)}{x}\le \theta (z)~ \text {uniformly for a.e.}~z\in \Omega ,\\&\int _{\Omega }F(z,-\hat{t}\hat{u}_1)dz>0,~e(z,x)=f(z,x)x-2F(z,x)\ge -\hat{\eta }~\text { for a.e.}~z\in \Omega ,~\text {all}~x\le 0; \end{aligned}$$

(v) for every $\rho >0$, there exists $\hat{\xi }_{\rho }>0$ such that for a.e. $z\in \Omega$, the function $x\rightarrow f(z,x)+\hat{\xi }_{\rho }x$ is nondecreasing on $[-\rho ,\rho ]$.

Remarks: The asymptotic conditions as $x\rightarrow \pm \infty$ (see $H_4(ii),(iii)$) remain similar as before when we examined (p, q)-equations. Again we have a “resonant-superlinear” problem, similar to the one studied by Domingos da Silva-Ribeiro [8]. However, our conditions on the perturbation f(z, x) are less restrictive. So, our multiplicity theorem (see Theorem 13 ) extends Theorem 1.2 and Corollary 1.1 of Domingos da Silva-Ribeiro [8].

The following function satisfies hypotheses $H_4$. As before for the sake of simplicity, we drop the z-dependence

$$\begin{aligned} f(x)=\left\{ \begin{array}{lll} -\ln |x|+\theta &{} \text {if }~ x<-1 \\ cx-\sin x &{} \text {if }~ -1\le x\le 1\\ \theta x^{r-1} &{} \text {if }~ 1 < x \end{array}\right. \end{aligned}$$

with $\sin 1<c<1$ and $\theta =c-\sin 1>0$.

We introduce the $C^1-$functional $\zeta _{\pm }:H^1_0(\Omega )\rightarrow \mathbb {R}$ defined by

$$\begin{aligned}&\zeta _+(u)=\frac{1}{2}\Vert Du\Vert ^2_2-\frac{\hat{\lambda }_1}{2} \Vert u^+\Vert ^2_2-\int _{\Omega }F(z,u^+)dz,\\&\zeta _-(u)=\frac{1}{2}\Vert Du\Vert ^2_2-\frac{\hat{\lambda }_1}{2} \Vert u^-\Vert ^2_2-\int _{\Omega }F(z,-u^-)dz,~\text {for all}~u\in H^1_0(\Omega ). \end{aligned}$$

Reasoning as in Proposition 4, we have

Proposition 11

If hypotheses $H_{4}$ hold, then $\zeta _-(\cdot )$ is coercive.

Next we determine what kind of critical point for $\zeta _{\pm }(\cdot )$ is the origin($u=0$).

Proposition 12

If hypotheses $H_{4}$ hold, then $u=0$ is a local minimizer for the functionals $\zeta _{\pm }(\cdot )$.

Proof

On account of hypothesis $H_4(iv)$, given $\varepsilon >0$, we can find $\delta =\delta (\varepsilon )>0$ such that

$$\begin{aligned} F(z,x)\le \frac{1}{2}[\theta (z)+\varepsilon ]x^2 ~\text {for a.e.}~z\in \Omega ,~\text {all}~|x|\le \delta . \end{aligned}$$

(57)

Let $u\in C^1_0(\overline{\Omega })$ with $\Vert u\Vert _{C^1_0(\overline{\Omega })}\le \delta$. We have

$$\begin{aligned} \zeta _{-}(u)&=\frac{1}{2}\Vert Du\Vert ^2_2-\frac{\hat{\lambda }_1}{2} \Vert u^-\Vert ^2_2-\int _{\Omega }F(z,-u^-)dz \nonumber \\&\ge \frac{1}{2}\Vert Du\Vert ^2_2-\frac{\hat{\lambda }_1}{2}\Vert u^- \Vert ^2_2-\frac{1}{2}\big [\int _{\Omega }\theta (z) (u^-)^2dz+ \frac{\varepsilon }{2}\Vert u^-\Vert ^2_2\big ]\nonumber \\&\ge \frac{1}{2}\bigg [\Vert Du\Vert ^2_2- \int _{\Omega } [\hat{\lambda }_1+\theta (z)](u^-)^2 dz- \frac{\varepsilon }{2\hat{\lambda }_1} \Vert Du^-\Vert ^2_2 \bigg ] ~~(\text {see}~(52)). \end{aligned}$$

(58)

Note that

$$\begin{aligned} \hat{\lambda }_1+\theta (z)\le \hat{\lambda }_1~\text {for a.e.}~z\in \Omega ,\hat{\lambda }_1+\theta (\cdot )\not \equiv \hat{\lambda }_1. \end{aligned}$$

So, from Proposition 2 we have

$$\begin{aligned} \Vert Du^-\Vert ^2_2- \int _{\Omega }[\hat{\lambda }_1+\theta (z)](u^-)^2dz\ge c_{12} \Vert Du^-\Vert ^2_2 ~\text {for some } c_{12}>0. \end{aligned}$$

Returning to (58) we have

$$\begin{aligned} \zeta _-(u)\ge \frac{1}{2}\bigg [\Vert Du^+\Vert ^2_2+(c_{10}- \frac{\varepsilon }{\hat{\lambda }_1})\Vert Du^-\Vert ^2_2\bigg ]. \end{aligned}$$

Choosing $\varepsilon \in (0,\hat{\lambda }_1 c_{10})$, we obtain

$$\begin{aligned} \zeta _-(u)\ge c_{13}\Vert u\Vert ^2\ge 0=\zeta _-(0) ~\text {for some}~c_{13}>0,~\text {all}~u\in C^1_0(\overline{\Omega }),\Vert u\Vert _{C^1_0(\overline{\Omega })}\le \delta . \end{aligned}$$

This means that

$u=0$ is a local $C^1_0(\overline{\Omega })-$minimizer of $\zeta _-(\cdot )$,

$\Rightarrow u=0$ is a local $H^1_0(\overline{\Omega })$-minimizer of $\zeta _-(\cdot )$,(see Brezis-Nirenberg [3] and [11]).

Similarly we show that $u=0$ is a local minimizer for $\zeta _+(\cdot )$ too. $\square$

Now we can have our multiplicity theorem for problem (56).

Theorem 13

If hypotheses $H_{4}$ hold, then problem (56) has at least three nontrivial solutions

$$\begin{aligned} v_0,\hat{v}\in -\text {int}~C_+,\hat{u}\in \text {int}~C_+. \end{aligned}$$

Proof

From Proposition 11 we know that $\zeta _-(\cdot )$ is coercive. Also using the Sobolev embedding theorem, we see that $\zeta _-(\cdot )$ is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find $v_0\in H^1_0(\Omega )$ such that

$$\begin{aligned} \zeta _-(v_0)=\text {inf}~\big [\zeta _-(u):u\in H^1_0(\Omega )\big ]. \end{aligned}$$

(59)

Using hypothesis $H_4(iv)$ we have

$$\begin{aligned}&\zeta _-(-\hat{t}\hat{u}_1)=-\int _{\Omega }F(z,-\hat{t}\hat{u}_1)dz<0,\\ \Rightarrow&\zeta _-(v_0)<0=\zeta _-(0) (\text {see}(59)),\\ \Rightarrow&v_0\ne 0. \end{aligned}$$

From (59), we have

$$\begin{aligned}&\zeta '_-(v_0)=0 ~\text {in}~H^{-1}(\Omega ), \nonumber \\ \Rightarrow&\langle \zeta '_-(v_0),h \rangle =0~\text { for all }~h\in H^1_0(\Omega ). \end{aligned}$$

(60)

In (60) we choose $h=v^+_0\in H^1_0(\Omega ).$ We obtain

$$\begin{aligned} \Vert Dv^+_0\Vert ^2_2=0,~~ \Rightarrow ~~ v_0\le 0,v_0\ne 0. \end{aligned}$$

We have

$$\begin{aligned} -\Delta v_0=\hat{\lambda }_1v_0+f(z,v_0) ~\text {in}~\Omega ,v_0|_{\partial \Omega }=0. \end{aligned}$$

Then, the classical regularity theory (see Gilbarg–Trudinger [13]) implies $v_0\in (-C_+)\setminus \{0\}$. Let $\rho =\Vert v_0\Vert _{\infty }$ and let $\hat{\xi }_{\rho }>0$ be as postulated by hypothesis $H_4(v)$. We have

$$\begin{aligned}&-\Delta v_0+\hat{\xi }_{\rho }v_0=\hat{\lambda }_1 v_0+f(z,v_0)+\hat{\xi }_{\rho }v_0\ge 0\\ \Rightarrow&\Delta (-v_0)\le \hat{\xi }_{\rho }(-v_0),\\ \Rightarrow&v_0\in -\text {int}~C_+~\text { (by the Hopf maximum principle)}. \end{aligned}$$

We assume that $K_{\zeta }$ is finite. Otherwise we already have an infinity of negative smooth solutions and so we are done. From Proposition 12 we know that $u=0$ is a local minimizer of $\zeta _-(\cdot )$. Using Theorem 5.7.6, p.449, of [20], we can find $\rho \in (0,1)$ small such that

$$\begin{aligned} \zeta _-(v_0)<\zeta _-(0)<\text {inf} \big [\zeta _-(u):\Vert u\Vert =\rho \big ]=m_-,\quad \rho <\Vert v_0\Vert . \end{aligned}$$

(61)

Since $\zeta _-(\cdot )$ is coercive (see Proposition 11), we have that

$$\begin{aligned} \zeta _-(\cdot ) ~\text { satisfies the C-condition\quad (see [{20}]), Proposition 5.1.15, p.369).} \end{aligned}$$

(62)

Then (61), (62) and the mountain pass theorem, imply that we can find $\hat{v}\in H^1_0(\Omega )$ such that

$$\begin{aligned}&\hat{v}\in K_{\zeta _-}\subseteq -C_+,m_-\le \zeta _-(\hat{v}),\\ \Rightarrow&\hat{v}\in -\text {int}~C_+~\text { (by the Hopf maximum principle)}. \end{aligned}$$

From Proposition 12 we know that $u=0$ is also a local minimizer for $\zeta _+(\cdot ).$ By the regularity theory $K_{\zeta _+}\subseteq C_+$ and again without any loss of generality, we assume that $K_{\zeta _+}$ is finite. So, as before we can find $\rho \in (0,1)$ small such that

$$\begin{aligned} \zeta _+(0)=0<\text {inf}\big [\zeta _+(u):\Vert u\Vert =\rho \big ]=m_+. \end{aligned}$$

(63)

From Papageorgiou–Rădulescu–Zhang [21] (see the “Claim” in the proof of Proposition 4), we have that

$$\begin{aligned} \zeta _+(\cdot ) ~\text { satisfies the C-condition}. \end{aligned}$$

(64)

Finally on account of hypothesis $H_4(ii)$, if $u\in \text {int}~C_+$, then we have

$$\begin{aligned} \zeta _+(tu)\rightarrow -\infty \text { as }t\rightarrow \infty . \end{aligned}$$

(65)

Then, (63), (64), (65) permit the use of the mountain pass theorem. So, we can find ${\hat{u}}\in H^1_0(\Omega )$ such that

$$\begin{aligned}&\hat{u}\in K_{\zeta _+}\subseteq C_+,\zeta _+(0)=0<m_+\le \zeta _+(\hat{u}),\\ \Rightarrow&\hat{u}\in \text {int}~C_+~\text { is a third solution of } ~(56). \end{aligned}$$

So, we have produced three nontrivial smooth solutions and provided sign information for all of them. $\square$

Next for problem (56) we consider the case where the perturbation $f(z,\cdot )$ is sublinear “resonant-sublinear” equation). To the best of our knowledge, this case was not considered in the past.

The hypotheses on f(z, x) are the following:

$H_{5}:f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ is a measurable function such that for $a.e.~z\in \Omega ,~f(z,0)=0,f(z,\cdot )\in C^1(\mathbb {R})$ and

(i) $|f'_x(z,x)|\le a(z)[1+|x|^{r-2}]$ for $a.e.~z\in \Omega$, all $x\in \mathbb {R}$, with $a\in L^{\infty }(\Omega )$, $2< r<2^{*};$

(ii) there exist $m\in \mathbb {N}$ and functions $\theta ,\hat{\theta }\in L^{\infty }(\Omega )$ such that

$$\begin{aligned}&\hat{\lambda }_m-\hat{\lambda }_1\le \theta (z)\le \hat{\theta }(z)\le \hat{\lambda }_{m+1}-\hat{\lambda }_1~\text { for a.e.}z\in \Omega ,\\&\theta \not \equiv \hat{\lambda }_m-\hat{\lambda }_1,\hat{\theta } \not \equiv \hat{\lambda }_{m+1}-\hat{\lambda }_1,\\&\theta (z)\le \liminf _{x\rightarrow +\infty }\frac{f(z,x)}{x} \le \limsup _{x\rightarrow +\infty } \frac{f(z,x)}{x} \le \hat{\theta }(z)~\text { uniformly for a.e.}z\in \Omega ; \end{aligned}$$

(iii) $\lim _{x\rightarrow -\infty }\frac{f(z,x)}{x}=0, \lim _{x\rightarrow -\infty }f(z,x)=-\infty$ uniformly for a.e.$z\in \Omega$;

$$\begin{aligned}&f(z,x)x-2F(z,x)\rightarrow +\infty ~\text { for a.e.} z\in \Omega ,~\text {as}~x\rightarrow -\infty , \\&-\hat{\eta }\le f(z,x)x-2F(z,x)~ \text { for a.e.}z\in \Omega ~\text {with}~\hat{\eta }>0; \end{aligned}$$

(iv) there exists $l \in \mathbb {N}$ such that

$$\begin{aligned}&f'_x(z,0)\in [\hat{\lambda }_{l},\hat{\lambda }_{l+1}]~ \text { for a.e.}z\in \Omega ,\\&f'_x(\cdot ,0)\not \equiv \hat{\lambda }_{l}, f'_x(\cdot ,0)\not \equiv \hat{\lambda }_{l+1}. \end{aligned}$$

(v) for every $\rho >0$, there exists $\hat{\xi }_{\rho }>0$ such that for a.e.$z\in \Omega$, the function $x\rightarrow f(z,x)+\hat{\xi }_{\rho }x$ is nondecreasing on $[-\rho ,\rho ]$.

In addition to the functionals $\zeta _{\pm }$, let $\zeta :H^1_0(\Omega )\rightarrow \mathbb {R}$ be the energy functional for problem (56) defined by

$$\begin{aligned} \zeta (u)=\frac{1}{2}\Vert Du\Vert ^2_2- \frac{\hat{\lambda }_1}{2}\Vert u\Vert ^2_2-\int _{\Omega }F(z,u)dz~\text { for all }u\in H^1_0(\Omega ). \end{aligned}$$

Note that $\zeta \in C^2(H^1_0(\Omega ))$. Next we show that $\zeta (\cdot )$ satisfies the compactness condition (the C-condition).

Proposition 14

If hypotheses $H_5$ hold, then the functional $\zeta (\cdot )$ satisfies the C-condition.

Proof

We consider a sequence $\{u_n\}_{n\in \mathbb {N}}\subseteq {H^1_0(\Omega )}$ such that

$$\begin{aligned}&|\zeta (u_n)|\le c_{14} ~\text { for some }~ c_{14}>0,~\text { all }n\in \mathbb {N},\end{aligned}$$

(66)

$$\begin{aligned}&(1+\Vert u_n\Vert )\zeta '(u_n)\rightarrow 0~\text {in}~H^{-1} (\Omega )~\text {as}~n\rightarrow \infty . \end{aligned}$$

(67)

From (67), we have

$$\begin{aligned} |\langle A(u_n),h \rangle -\hat{\lambda }_1\int _{\Omega } u_nhdz-\int _{\Omega }f(z,u_n)hdz|\le \frac{\varepsilon _n\Vert h\Vert }{1+\Vert u_n\Vert }\\ ~~\text {for all } h\in H^1_0(\Omega ), \text { with } \varepsilon _n\rightarrow 0^+.\nonumber \end{aligned}$$

(68)

In (68), we use the test function $h=-u^-_n\in H^1_0(\Omega )$. We obtain

$$\begin{aligned} \Vert Du^-_n\Vert ^2_2-\hat{\lambda }_1\Vert u^-_n\Vert ^2_2-\int _{\Omega }f(z,-u^-_n) (-u^-_n)dz\le \varepsilon _n~~\text {for all } n\in \mathbb {N}. \end{aligned}$$

(69)

Suppose that $\Vert u^-_n\Vert \rightarrow \infty$ and let $v_n=\frac{v^-_n}{\Vert v^-_n\Vert },n\in \mathbb {N}$. Then $\Vert v_n\Vert =1,v_n\ge 0$ for all $n\in \mathbb {N}$. So, we may assume that

$$\begin{aligned} v_n{\mathop {\longrightarrow }\limits ^{w}} v~\text {in}~H^{1}_{0}(\Omega ),v_{n}\rightarrow v~\text {in}~L^{2}(\Omega ). \end{aligned}$$

(70)

From (69) we have

$$\begin{aligned} \Vert Dv^-_n\Vert ^2_2-\hat{\lambda }_1\Vert v_n\Vert ^2_2-\int _{\Omega } \frac{f(z,-u^-_n)}{\Vert u^-_n\Vert }v_ndz\le \frac{\varepsilon _n}{\Vert u^-_n\Vert ^2}~~\text {for all } n\in \mathbb {N}. \end{aligned}$$

(71)

Note that $\{\frac{f(\cdot ,-u^-_n(\cdot ))}{\Vert u^-_n\Vert }\}_{n\in \mathbb {N}}\subseteq L^2(\Omega )$ is bounded and so on account of hypothesis $H_5(iii)$, we have at least for a subsequence we have that

$$\begin{aligned}&\quad \quad \quad \frac{f(\cdot ,-u^-_n(\cdot ))}{\Vert u^-_n\Vert }\ {\mathop {\longrightarrow }\limits ^{w}} 0~\text {in}~L^{2}(\Omega )\\&\text {(see Aizicovici--Papageorgiou--Staicu [1], proof of Proposition 16).} \end{aligned}$$

Therefore, if we pass to the limit as $n\rightarrow \infty$ in (71) and use (70) we obtain

$$\begin{aligned}&\Vert Dv\Vert ^2_2\le \hat{\lambda }_1\Vert v\Vert ^2_2,\\ \Rightarrow&\Vert Dv\Vert ^2_2=\hat{\lambda }_1\Vert v\Vert ^2_2 ~~~(\text {see} ({5})),\\ \Rightarrow&v=\mu \hat{u}_1~\text { with }\mu \ge 0~~~(\text {recall }v\ge 0). \end{aligned}$$

If $\mu =0$, then $v=0$ and so from (71) we have

$$\begin{aligned} \Vert Dv_n\Vert _2\rightarrow 0,~~\Rightarrow v_n\rightarrow 0 ~\text {in}~H^{1}_{0}(\Omega ) \end{aligned}$$

a contradiction to the fact that $\Vert v_n\Vert =1$ for all $n\in \mathbb {N}$.

If $\mu >0$, then $v=\mu \hat{u}_1\in ~\text {int}~C_+$ and so we have

$$\begin{aligned} u^-_n(z)\rightarrow +\infty ~\text { for a.e.}z\in \Omega . \end{aligned}$$

(72)

From (66) we have

$$\begin{aligned}&\Vert Du^+_n\Vert ^2_2+\Vert Du^-_n\Vert ^2_2-\Vert u^+_n\Vert ^2_2-\Vert u^-_n\Vert ^2_2\nonumber \\&-\int _{\Omega } 2F(z,u^+_n)dz-\int _{\Omega } 2F(z,-u^-_n)dz\le 2c_{14}~\text {for all}~n\in \mathbb {N}. \end{aligned}$$

(73)

From (68) with $h=u^+_n\in H^1_0(\Omega )$, we have

$$\begin{aligned}&-\Vert Du^+_n\Vert ^2_2+\hat{\lambda }_1\Vert u^+_n\Vert ^2_2+ \int _{\Omega } f(z,u^+_n)u^+_n dz\le \varepsilon _n, \nonumber \\ \Rightarrow&-\Vert Du^+_n\Vert ^2_2+\hat{\lambda }_1\Vert u^+_n\Vert ^2_2 +\int _{\Omega } 2F(z,u^+_n)dz\le c_{15}\\&\text {for some } c_{15}>0, \text { all } n\in \mathbb {N}~~~(\text {see hypothesis } H_5(iii)).\nonumber \end{aligned}$$

(74)

Adding (73) and (74), we obtain

$$\begin{aligned} \Vert Du^-_n\Vert ^2_2-\hat{\lambda }_1\Vert u^-_n\Vert ^2_2- \int _{\Omega }2F(z,-u^-_n) dz\le c_{16}~~\text {for some } c_{16}>0, \text { all } n\in \mathbb {N}. \end{aligned}$$

(75)

In (68), we use the test function $h=-u^-_n\in H^1_0(\Omega )$ and obtain

$$\begin{aligned} -\Vert Du^-_n\Vert ^2_2+\hat{\lambda }_1\Vert u^-_n\Vert ^2_2+ \int _{\Omega }f(z,-u^-_n) (-u^-_n)dz\le \varepsilon _n~~\text {for all } n\in \mathbb {N}. \end{aligned}$$

(76)

We add (75) and (76) and have

$$\begin{aligned} \int _{\Omega }[f(z,-u^-_n) (-u^-_n)-2F(z,-u^-_n)]dz\le c_{17}~~\text {for some } c_{17}>0, \text { all } n\in \mathbb {N}. \end{aligned}$$

Using hypothesis $H_5(iii)$,(72) and Fatou’s lemma, we have a contradiction. This proves that

$$\begin{aligned} \{u^-_n\}_{n\in \mathbb {N}}\subseteq H^{1}_{0}(\Omega ) ~\text { is bounded}. \end{aligned}$$

(77)

Now suppose that $\Vert v^+_n\Vert \rightarrow \infty$. Let $y_n=\frac{u^+_n}{\Vert u^+_n\Vert },n\in \mathbb {N}$. Then $\Vert y_n\Vert =1,y_n\ge 0$ for all $n\in \mathbb {N}$. So, we may assume that

$$\begin{aligned} y_n{\mathop {\longrightarrow }\limits ^{w}} y~\text {in}~H^{1}_{0}(\Omega ),y_{n}\rightarrow y~\text {in}~L^{2}(\Omega ). \end{aligned}$$

(78)

From (67) and (77), we have

$$\begin{aligned}&\langle A(u^+_n),h \rangle -\hat{\lambda }_1\int _{\Omega }u^+_nhdz-\int _{\Omega }f(z,u^+_n)hdz\le c_{18}\Vert h\Vert \nonumber \\&\quad \quad \quad \quad \quad \quad \quad \text {for some } c_{18}>0, \text { all } h\in H^{1}_{0}(\Omega ), \text { all } n\in \mathbb {N},\nonumber \\&\Rightarrow \langle A(y_n),h \rangle -\hat{\lambda }_1\int _{\Omega }y_nhdz-\int _{\Omega }\frac{f(z,u^+_n)}{\Vert u^+_n\Vert }hdz\le \varepsilon '_n~~\text {with } \varepsilon '_n\rightarrow 0^+ \text { as } n\rightarrow \infty . \end{aligned}$$

(79)

In (79), we use the test function $h=y_n-y\in H^{1}_{0}(\Omega )$ and we note that $\{\frac{f(\cdot ,u^+_n(\cdot ))}{\Vert u^+_n\Vert }\}_{n\in \mathbb {N}}\subseteq L^2(\Omega )$ is bounded (see hypotheses $H_5(i), (ii)$). So, if we pass to the limit as $n\rightarrow \infty$, we have

$$\begin{aligned}&\lim _{n\rightarrow \infty }\langle A(y_n),y_n-y\rangle =0,\nonumber \\ \Rightarrow&\Vert Dy_n\Vert _2\rightarrow \Vert Dy\Vert _2,\nonumber \\ \Rightarrow&y_n\rightarrow y ~\text {in}~H^{1}_{0}(\Omega ) ~\text { and so } \Vert y\Vert =1,y\ge 0\\&~~\text {(by the Kadec--Klee property of Hilbert spaces).}\nonumber \end{aligned}$$

(80)

Recall that $\{\frac{f(\cdot ,u^+_n(\cdot ))}{\Vert u^+_n\Vert }\}_{n\in \mathbb {N}}\subseteq L^2(\Omega )$ is bounded. So, we may assume that

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{f(\cdot ,u^+_n(\cdot ))}{\Vert u^+_n\Vert } {\mathop {\longrightarrow }\limits ^{w}}\eta \quad \textrm{in}\ L^2(\Omega ), \\ \eta =\widetilde{\theta }(\cdot )y, \quad \text { with }\theta (z) \le \widetilde{\theta }(z)~\text { for a.e.}z\in \Omega . \end{array}\right. \end{aligned}$$

(81)

(see hypothesis $H_5(ii)$ and see [1], proof of Proposition 16). So, if in (79) we pass to the limit as $n\rightarrow \infty$ and use (80) and (81), we obtain

$$\begin{aligned}&\langle A(y),h \rangle =\int _{\Omega }[\hat{\lambda }_1+\widetilde{\theta }(z)]yhdz ~\text {for all}~h\in H^{1}_{0}(\Omega ),\nonumber \\ \Rightarrow&-A(y) (z)=[\hat{\lambda }_1+\widetilde{\theta }(z)] y(z)~\text {in}~\Omega ,y|_{\partial \Omega }=0. \end{aligned}$$

(82)

From (81) and hypothesis $H_5(ii)$, we have

$$\begin{aligned} \hat{\lambda }_m\le \hat{\lambda }_1+\widetilde{\theta }(z) ~\text { for a.e.}z\in \Omega , \hat{\lambda }_1+\widetilde{\theta }(\cdot )\not \equiv \hat{\lambda }_m. \end{aligned}$$

Invoking Proposition 3 we have

$$\begin{aligned} \widetilde{\lambda }_1(\hat{\lambda }_1+\widetilde{\theta }(\cdot )) <\widetilde{\lambda }_1(\hat{\lambda }_m)\le \widetilde{\lambda }_1(\hat{\lambda }_1)=1,\\ \Rightarrow y \text { must be nodal, a contradiction } (\text { see } ({82}),({80})). \end{aligned}$$

This proves that

$$\begin{aligned}&\{u^+_n\}_{n\in \mathbb {N}}\subseteq H^{1}_{0}(\Omega ) ~\text { is bounded}.\\ \Rightarrow&\{u_n\}_{n\in \mathbb {N}}\subseteq H^{1}_{0} (\Omega ) ~\text { is bounded } (\text {see } ({77})). \end{aligned}$$

We may assume that

$$\begin{aligned} u_n{\mathop {\longrightarrow }\limits ^{w}} u~\text {in}~H^{1}_{0} (\Omega ),u_{n}\rightarrow u~\text {in}~L^{2}(\Omega ). \end{aligned}$$

(83)

In (68) we choose the test function $h=u_n-u\in H^{1}_{0}(\Omega )$, pass to the limit as $n\rightarrow \infty$ and use (83). We obtain

$$\begin{aligned}&\lim _{n\rightarrow \infty }\langle A(u_n),u_n-u\rangle =0,\nonumber \\ \Rightarrow&\Vert Du_n\Vert \rightarrow \Vert Du\Vert _2,\nonumber \\ \Rightarrow&u_n\rightarrow u ~\text {in}~H^{1}_{0}(\Omega ) ~\text { (Kadec--Klee property)}. \end{aligned}$$

This proves that $\zeta (\cdot )$ satisfies the C-condition. $\square$

Proposition 14 allows us to compute the critical groups of $\zeta (\cdot )$ at infinity. Recall that as before without any loss of generality, we assume that $K_{\zeta }$ is finite.

Proposition 15

If hypotheses $H_5$ hold, then $C_k(\zeta ,\infty )=0$ for all $k\in \mathbb {N}_0$.

Proof

Let $\beta \in L^{\infty }(\Omega )$ such that $\beta (z)>0$ for a.e.$z\in \Omega$ and $\vartheta _0\in (\hat{\lambda }_m- \hat{\lambda }_1,\hat{\lambda }_{m+1}-\hat{\lambda }_1)$. We consider the homotopy $(t,u)\rightarrow \hat{h}_t(u)$ defined by

$$\begin{aligned} \hat{h}_t(u)=\frac{1}{2}\Vert Du\Vert ^2_2-\frac{\hat{\lambda }_1}{2} \Vert u\Vert ^2_2-\frac{(1-t)\theta _0}{2}\Vert u^+\Vert ^2_2-t\int _{\Omega } F(z,u)dz-(1-t)\int _{\Omega }\beta (z)udz\\ \text {for all } t\in [0,1], \text { all } u\in H^{1}_{0}(\Omega ). \end{aligned}$$

Note that

$$\begin{aligned}&\hat{h}_0(u)=\gamma (u)=\frac{1}{2}\Vert Du\Vert ^2_2-\frac{\hat{\lambda }_1}{2} \Vert u\Vert ^2_2-\frac{\theta _0}{2}\Vert u^+\Vert ^2_2-\int _{\Omega }\beta (z)udz,\\&\hat{h}_1(u)=\zeta (u)~\text { for all }u\in H^{1}_{0}(\Omega ). \end{aligned}$$

Suppose we can find $\{t_n\}_{n\in \mathbb {N}}\subseteq [0,1]$ and$\{u_n\}_{n\in \mathbb {N}}\subseteq H^{1}_{0}(\Omega )$ such that

$$\begin{aligned} \hat{h}_{t_n}(u_n)\rightarrow -\infty ~\text { and }~(1+\Vert u_n\Vert ) (\hat{h}_{t_n})'(u_n)\rightarrow 0~\text {in}~H^{-1}(\Omega ). \end{aligned}$$

(84)

From the second convergence in (84), we have

$$\begin{aligned}&|\langle A(u_n),h \rangle -\hat{\lambda }_1\int _{\Omega }u_nhdz-(1-t) \theta _0\int _{\Omega }u^+_nhdz-t_n\int _{\Omega }f(z,u_n)hdz \nonumber \\&-(1-t_n)\int _{\Omega }\beta (z)hdz|\le \frac{\varepsilon _n\Vert h\Vert }{1+\Vert u_n\Vert } ~\text { for all }h\in H^{1}_{0}(\Omega ),~\text {with}~\varepsilon _n\rightarrow 0^+. \end{aligned}$$

(85)

Assume that $\Vert u_n\Vert \rightarrow \infty$ and set $v_n=\frac{u_n}{\Vert u_n\Vert },n\in \mathbb {N}.$ Then $\Vert v_n\Vert =1$ for all $n\in \mathbb {N}$ and we may assume that

$$\begin{aligned} v_n{\mathop {\longrightarrow }\limits ^{w}} v~\text {in}~H^{1}_{0}(\Omega ),v_{n}\rightarrow v~\text {in}~L^{2}(\Omega ). \end{aligned}$$

(86)

From (85) we have

$$\begin{aligned}&|\langle A(v_n),h \rangle -\hat{\lambda }_1\int _{\Omega }v_nhdz-(1-t)\theta _0 \int _{\Omega }v^+_nhdz-t_n\int _{\Omega }\frac{f(z,u_n)}{\Vert u_n\Vert }hdz \nonumber \\&-(1-t_n)\int _{\Omega }\frac{\beta (z)}{\Vert u_n\Vert }hdz|\le \varepsilon '_n~\text { with }\varepsilon '_n\rightarrow 0~\text {as}~n\rightarrow \infty . \end{aligned}$$

(87)

In (87) we use the test function $h=v_n-v\in H^{1}_{0}(\Omega )$, pass to the limit as $n\rightarrow \infty$ and use (86). Then,

$$\begin{aligned}&\lim _{n\rightarrow \infty }\langle A(v_n),v_n-v\rangle =0,\nonumber \\ \Rightarrow&\Vert Dv_n\Vert _2\rightarrow \Vert Dv\Vert _2,\nonumber \\ \Rightarrow&v_n\rightarrow v ~\text {in}~H^{1}_{0}(\Omega ) ~ \text { (Kadec--Klee property)},\Vert v\Vert =1. \end{aligned}$$

(88)

Hypotheses $H_5(i),(ii),(iii)$ imply that

$$\begin{aligned} \frac{f(\cdot ,u_n(\cdot ))}{\Vert u_n\Vert }{\mathop {\longrightarrow }\limits ^{w}}\theta ^*(\cdot )v^+ \quad \textrm{in}\ L^2(\Omega ), \end{aligned}$$

(89)

with $\theta ^*\in L^{\infty }(\Omega )$ such that $\theta (z)\le \theta ^*(z)\le \hat{\theta }(z)$ for a.e.$z\in \Omega$ (see [1]). Therefore, if in (87) we let $n\rightarrow \infty$ and use (88),(89), we obtain

$$\begin{aligned} \langle A(v_n),h \rangle =\hat{\lambda }_1\int _{\Omega }vhdz+\int _{\Omega }[(1-t) \theta _0+t\theta ^*(z)]v^+hdz~\text {for all}~h\in H^{1}_{0}(\Omega ). \end{aligned}$$

(90)

Suppose $v^-\ne 0$ and in (90) choose the test function $h=-v^-\in H^{1}_{0}(\Omega )$. We have

$$\begin{aligned} \Vert Dv^-\Vert ^2_2=\hat{\lambda }_1\Vert v^-\Vert ^2_2, \quad \Rightarrow v=\mu \hat{u}_1~\text { with }\mu <0. \end{aligned}$$

Then, $v(z)<0$ for all $z\in \Omega$ and so

$$\begin{aligned} u_n(z)\rightarrow -\infty ~\text { for a.e.}z\in \Omega . \end{aligned}$$

Then, reasoning as in the proof of Proposition 4 (see the part of the proof after (15)), we reach a contradiction. This means that $v\ge 0$ and from (90) we have

$$\begin{aligned} -\Delta v(z)=[\hat{\lambda }_1+\hat{\theta }_t(z)]v(z)~\text {in}~\Omega ,v|_{\partial \Omega }=0, \end{aligned}$$

(91)

with $\hat{\theta }_t(z)=(1-t)\theta _0+t\theta ^*(z),\hat{\theta }_t\in L^{\infty }(\Omega ),0\le t\le 1$. From the choice of $\theta _0$ and (89) we see that

$$\begin{aligned} \left\{ \begin{array}{lll} \hat{\lambda }_m\le \hat{\lambda }_1+\hat{\theta }_t(z)\le \hat{\lambda }_{m+1} ~\text { for a.e.}z\in \Omega ,\\ \hat{\lambda }_m\not \equiv \hat{\lambda }_1+\hat{\theta }_t(\cdot ),\hat{\lambda }_{m+1}\not \equiv \hat{\lambda }_1+\hat{\theta }_t(\cdot ). \end{array}\right\} . \end{aligned}$$

(92)

Using Proposition 3, we have

$$\begin{aligned} \widetilde{\lambda }_1(\hat{\lambda }_1+\hat{\theta }_t)< \widetilde{\lambda }_1(\hat{\lambda }_m)\le \widetilde{\lambda }_1(\hat{\lambda }_1)=1, \end{aligned}$$

$\Rightarrow v$ must be nodal (see (91)), a contradiction.

Therefore $\{u_n\}_{n\in \mathbb {N}}\subseteq H^{1}_{0}(\Omega )$ is bounded and this implies that $\{h_{t_n}(u_n)\}_{n\in \mathbb {N}}\subseteq \mathbb {R}$ is bounded, contradicting (84). So, (84) cannot happen and then using Proposition 3.2 of Liang-Su [15] (see also Chang [4], Theorem 5.1.21, p.334), we have

$$\begin{aligned}&C_k(h_0,\infty )=C_k(h_1,\infty )~~\text { for all }k\in \mathbb {N}_0, \nonumber \\ \Rightarrow&C_k(\gamma ,\infty )= C_k(\zeta ,\infty )~\text { for all }k\in \mathbb {N}_0. \end{aligned}$$

(93)

Let $u\in K_\gamma$ We have

$$\begin{aligned} -\Delta u(z)=\hat{\lambda }_1u(z)+\theta _0u^+(z)+\beta (z)~\text {in}~\Omega ,u|_{\partial \Omega }=0. \end{aligned}$$

(94)

Suppose $u^-\ne 0$ and act on (94) with $-u^-\in H^{1}_{0}(\Omega )$. Then,

$$\begin{aligned} 0\le \Vert Du^-\Vert ^2_2 -\hat{\lambda }_1\Vert u^-\Vert ^2_2-\int _{\Omega }\beta (z)u^-dz<0 \end{aligned}$$

(recall $\beta (z)>0$ for a.e.$z\in \Omega$), a contradiction. Hence, $u\ge 0,u\ne 0$(since $\beta \ne 0$). From (94), the regularity theory (see Gilbarg-Trudinger [13]) and the Hopf maximum principle we infer that $u\in ~\text {int}~C_+$ (note that since $\beta \ne 0$, then $u\ne 0$). Let $y\in ~\text {int}~C_+$. Using Picone’s identity (see Motreanu–Motreanu–Papageorgiou [18], p.255), we have

$$\begin{aligned} 0&\le \Vert Dy\Vert ^2_2-\int _{\Omega }(Du,D(\frac{y^2}{u}))_{\mathbb {R}^N}dz\\&= \Vert Dy\Vert ^2_2-\int _{\Omega }(-\Delta u)\frac{y^2}{u}dz~ \text { (by Green's identity)} \\&= \Vert Dy\Vert ^2_2-\int _{\Omega }[\hat{\lambda }_1+\theta _0] y^2dz-\int _{\Omega }\beta (z)\frac{y^2}{u}dz~\text { (see (94))}\\&\le \Vert Dy\Vert ^2_2-\int _{\Omega }[\hat{\lambda }_1+\theta _0]y^2dz \end{aligned}$$

Let $y=\hat{u}_1\in ~\text {int}~C_+$. We have

$$\begin{aligned} 0\le -\theta _0\int _\Omega \hat{u}_1^2\,dz<0~~~\text { see (5) } \end{aligned}$$

a contradiction. Therefore, $K_{\gamma }=\emptyset$ and the Proposition 6.2.28, p.491, in [20], implies that

$$\begin{aligned}&C_k(\gamma ,\infty )=0 ~\text { for all }k\in \mathbb {N}_0, \\ \Rightarrow&C_k(\zeta ,\infty )=0 ~\text { for all }k \in \mathbb {N}_0~\text { (see (93)) }. \end{aligned}$$

$\square$

Remark: If $m=2$ (see hypothesis $H_5(ii)$), then we can have an alternative proof that $K_\gamma =\emptyset$. We outline this alternative proof. Let $u\in K_\gamma .$ We have

$$\begin{aligned} -\Delta u=\hat{\lambda }_1u+\theta _0u^++\beta (z)~\text {in}~\Omega ,u|_{\partial \Omega }=0. \end{aligned}$$

As before acting with $-u^-\in H^{1}_{0}(\Omega )$, we infer that $u\ge 0,u\ne 0$. On the other hand choosing $\theta _0$ close to $\hat{\lambda }_2-\hat{\lambda }_1$ and invoking the antimaximum principle (see Motreanu–Motreanu–Papageorgiou [18], p.263), we infer that $u\in -\text {int}~C_+$, a contradiction.

Note that hypothesis $H_5(iv)$ implies that $u=0$ is a nondegenerate critical point of $\zeta (\cdot )$ with Morse index $d_l=\text {dim}~\overline{H}_l=\text {dim}~\bigoplus ^l_{k=1}E(\widehat{\lambda }_k)$. Then using Proposition 6.2.6, p.479, of [20], we have:

Proposition 16

If hypotheses $H_5$ hold, then $C_k(\zeta ,0)=\delta _{k,d_l}\mathbb {Z}$ for all $k\in \mathbb {N}_0$.

We are ready for the multiplicity theorem of the “resonant-sublinear” case.

Theorem 17

If hypotheses $H_5$ hold, then problem (56) has at least three nontrivial solutions $v_0\in -\text {int}~C_+,u_0,\hat{u}\in C^1_0(\overline{\Omega })$.

Proof

As before using the functional $\zeta _-(\cdot )$ which is coercive via the Weierstrass–Tonelli theorem, we produce $v_0\in -\text {int}~C_+$ a solution of (56) which is a local minimizer of $\zeta (\cdot )$. Hence,

$$\begin{aligned} C_k(\zeta ,v_0)=\delta _{k,0}\mathbb {Z} ~\text { for all }k\in \mathbb {N}_0. \end{aligned}$$

(95)

Using $v_0$ and Proposition 14, as in the proof of Theorem 8, using the mountain pass theorem, we generate a second nontrivial solution $u_0\in C^1_0(\overline{\Omega })$ (regularity theory). For this solution, we have

$$\begin{aligned} C_k(\zeta ,u_0)=\delta _{k,1}\mathbb {Z} ~\text { for all }k\in \mathbb {N}_0.~\text {(see [{20}], p.529)}. \end{aligned}$$

(96)

From Proposition 15 and 16, we have

$$\begin{aligned} C_k(\zeta ,\infty )=0 ~\text { for all }k\in \mathbb {N}_0, C_k(\zeta ,0)=\delta _{k,d_l}\mathbb {Z}~\text { for all }k\in \mathbb {N}_0. \end{aligned}$$

(97)

Suppose $K_{\zeta }=\{v_0,u_0,0\}$. Then from (95),(96),(97) and the Morse identity with $t=-1$, we have

$$\begin{aligned}&(-1)^0+(-1)^1+(-1)^{d_l}=0,\\ \Rightarrow&(-1)^{d_l}=0, ~\text { a contradiction.} \end{aligned}$$

Therefore, there exists $\hat{u}\in K_{\zeta },\hat{u}\notin \{v_0,u_0,0\}$. Hence $\hat{u}\in C^1_0(\overline{\Omega })$(regularity theory) is the third nontrivial smooth solution of (56). $\square$

Remark: It is interesting to know if the above result for the “resonant-sublinear” case remains valid if we consider (p, q)-equations.