A Landesman–Lazer Local Condition for Semilinear Elliptic Problems

Abstract

The objective of this paper is to study the existence, multiplicity and non existence of solutions for semilinear elliptic problems under a local Landesman–Lazer condition. There is no growth restriction at infinity on the nonlinear term and it may change sign. In order to establish the existence of solution we combine the Lyapunov–Schmidt reduction method with truncation and approximation arguments via bootstrap methods. In our applications we also consider the existence of a bifurcation point which may have multiple positive solutions for a fixed value of the parameters.

1 Introduction and Main Results

In this paper we are concerned with the existence, non existence and multiplicity of weak solutions for the following problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u+\mu h(x,u) &{}\text { in } \Omega ,\\ u=0 &{}\text { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(1.1)

where $\Omega$ is a bounded smooth domain of ${\mathbb {R}}^N$, $N\ge 1$; $\lambda >0$; $\mu \ne 0$ is a real parameter and $h:\overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$ is a Carathéodory function.

One of the most celebrated articles in the area of Nonlinear Analysis is the paper due to Landesman et al. (1975) which, via Topological Degree Theory, establishes the existence or non existence of solution for resonant nonlinear elliptic problems. In Landesman et al. (1975), one of the most famous hypotheses for this class of problems was introduced, nowadays called the Landesman–Lazer condition.

We emphasize that, besides providing the basic tools for studying resonant problems, the article Landesman et al. (1975) has allowed researchers to consider the possibility of using different methods to study that class of problems—see (Ahmad et al. 1976; Amann 1979; Arcoya and Gámez 2001; Bartolo and Benci 1983; Rabinowitz 1986; Ruiz 2004; Shaw 1977; Silva 1991) and references therein. In particular we mention the paper by Rabinowitz (1978) where it was demonstrated one of the most important results on the minimax theory: the Saddle Point Theorem.

Supposing that $\lambda =\lambda _1$ and that h is of the form $h(x,s)=f(x)+g(s)$, with $f \in L^2(\Omega )$ and $g:\mathbb {R} \rightarrow \mathbb {R}$ a bounded continuous function with finite limits $g^{\pm }=\lim _{s \rightarrow \pm \infty }g(s)$, the Landesman–Lazer condition for Problem (1.1) may be written as

$$\begin{aligned} \bigg [\int _\Omega (f+g^-)\varphi _1dx\bigg ]\bigg [\int _\Omega (f+g^+)\varphi _1dx\bigg ] <0, \end{aligned}$$

(1.2)

where $\varphi _1$ is a positive eigenfunction associated with $\lambda _1$, the first eigenvalue of the operator $-\Delta$ under Dirichlet boundary conditions.

We observe that the Landesman–Lazer condition (1.2) implies that there exist real numbers $t_1$ and $t_2$, with $t_1<t_2$, such that either

$(H_0^+)$:

$\displaystyle \int _{\Omega }h(x,t_1\varphi _1)\varphi _1dx>0>\int _{\Omega } h(x,t_2\varphi _1)\varphi _1dx,$

or

$(H^-_0)$:

$\displaystyle \int _{\Omega }h(x,t_1\varphi _1)\varphi _1dx<0<\int _{\Omega } h(x,t_2\varphi _1)\varphi _1dx.$

The main objective of this work is to consider the existence, multiplicity and non existence of solutions for Problem (1.1) under the local version of the Landesman–Lazer condition $(H_0^{\pm })$ when the parameters $\mu$ and $\lambda$ are close to zero and $\lambda _1$, respectively. One of the characteristics of our main results is that they do not impose any growth restriction at infinity on the nonlinear term h. It is worthwhile mentioning that in Landesman et al. (1975), and in the vast majority of subsequent results, it is imposed a growth restriction at infinity on the nonlinear term.

In order to establish our first result on the existence of solution for Problem (1.1) we assume h satisfies $(H_0^+)$ and the technical hypothesis:

$(H_1)$:

h is locally $L^\sigma (\Omega )$-bounded, $\sigma >\{N/2,1\}$, i. e., given $S>0$, there is $f_S\in L^\sigma (\Omega )$ such that $|h(x,s)|\le f_S(x)$ for every $|s|\le S$, a. e. in ${\Omega }$.

We observe that our results are proved via variational methods. However, since there is no global growth restriction on the nonlinearity h, the associated functional may not be well defined in $H_{0}^{1}(\Omega )$. We overcome this fact by combining the hypothesis $(H_1)$ with an appropriated truncation of the function h. The hypothesis $(H_1)$ also plays an important role in the approximation method that we use to derive the existence of solution for Problem (1.1).

Now we may state:

Theorem 1.1

Suppose h satisfies $(H_0^+)$ and $(H_1)$. Then there exist positive constants $\mu ^*$ and $\nu ^*$ such that, for every $\mu \in (0,\mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$, Problem (1.1) has a weak solution $u_\mu =t\varphi _1+v$, with $t\in (t_1, t_2)$ and $v\in \langle \varphi _1\rangle ^\perp$.

For proving a corresponding result under the hypothesis $(H_0^-)$, we suppose more regularity on the function h:

$(H_2)$:

h is locally $L^\sigma (\Omega )$-Lipschitz, $\sigma >\{N/2,1\}$, i. e., given $S>0$, there is $\zeta _S\in L^\sigma (\Omega )$ such that $|h(x,s_1)-h(x,s_2)|\le \zeta _S(x)|s_1-s_2|$, for every $|s_1|,~|s_2|\le S$, a. e. in ${\Omega }$.

Theorem 1.2

Suppose h satisfies $(H_0^-)$, $(H_1)$ and $(H_2)$ . Then there exist positive constants $\mu ^*$ and $\nu ^*$ such that, for every $\mu \in (0,\mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$, Problem (1.1) has a weak solution $u_\mu =t\varphi _1+v$, with $t\in (t_1, t_2)$ and $v\in \langle \varphi _1\rangle ^\perp$.

It is important to note that in Theorems 1.1 and 1.2 the projections of the solutions $u_{\mu }$ on the direction of $\varphi _1$ are located between $t_1\varphi _1$ and $t_2\varphi _1$. As a direct consequence of this fact, we may establish multiple solutions for the Problem (1.1) under the following version of $(H_0^{\pm })$:

$(H_0)$:

there exist $k\in \mathbb {N}$ and $t_i\in \mathbb {R}$, $t_i<t_{i+1}$, $i=1,\dots ,k$ such that

$$\begin{aligned} \bigg [\displaystyle \int _\Omega h(x,t_i\varphi _1)\varphi _1dx\bigg ]\bigg [\int _\Omega h(x,t_{i+1}\varphi _1)\varphi _1dx\bigg ]<0. \end{aligned}$$

Theorem 1.3

Suppose h satisfies $(H_0)$, $(H_1)$ and $(H_2)$. Then there exist positive constants $\mu ^*$ and $\nu ^*$ such that, for every $0<|\mu |<\mu ^*$ and $|\lambda -\lambda _1|<|\mu |\nu ^*$, Problem (1.1) has k weak solutions $u_i=\hat{t}_i\varphi _1+v_i$, with $\hat{t}_i\in (t_i,t_{i+1})$ and $v_i\in \langle \varphi _1\rangle ^\perp$, $i=1,\ldots ,k$.

Remark 1.4

Note that the solutions provided by Theorems 1.1–1.3 are of class $C^{1, \gamma }(\overline{\Omega })$ if $N=1$ and of class $C^{0, \gamma }(\overline{\Omega })$ if $N \ge 2$. If we assume $(H_1)$ holds with $\sigma >N$, we may assert that those solutions are actually in $C^{1, \gamma }(\overline{\Omega })$. Using this fact we may verify that the solution $u_{\mu }$ given by Theorems 1.1 and 1.2 is positive or negative in $\Omega$ provided $t_1\ge 0$ or $t_2\le 0$, respectively. Moreover, for $|\mu |>0$ sufficiently small, the solutions of Theorem 1.3 are ordered, see Theorems 2.10 and 2.11 in Sect. 2.

The proofs of Theorems 1.1 and 1.2 are inspired by the Lyapunov–Schmidt Reduction Method, as presented in the articles (Castro and Lazer 1979; Castro 1982; Cossio 2004; Landesman et al. 1975). However, under the hypotheses of those theorems, that method can not be applied directly since we do not impose any global growth restriction on the nonlinear term h. In order to overcome such difficulty we combine the Lyapunov–Schmidt reduction method with a truncation argument and an approximation method based on the bootstrap technique.

Theorems 1.1 and 1.2 imply that the local Landesman–Lazer condition $(H_0^{\pm })$ is sufficient to provide the existence of solution $u_{\mu }$ for Problem (1.1) such that its orthogonal projection on the direction of $\varphi _1$ belongs to the interval $(t_1\varphi _1, t_2\varphi _1)$. It is natural to question if this condition is also necessary. Here we prove such result under the hypotheses (see also Theorem 2.9)

$(H_3)$:

there exists $f\in L^\sigma (\Omega )$, $\sigma > \{N/2,1\}$, such that $|h(x,s)|\le f(x)(1+|s|), \text { for every } s\in \mathbb {R}, \text { a. e. in} ~ {\Omega };$

and

$(H_4)$:

there exist real numbers $t_1$ and $t_2$, with $t_1<t_2$, such that $\displaystyle \int _{\Omega }h(x,t\varphi _1)\varphi _1dx\ne 0,~~\text { for every } t\in [t_1,t_2].$

Theorem 1.5

Suppose h satisfies $(H_3)$ and $(H_4)$. Then there exist positive constants $\mu ^*$ and $\nu ^*$ such that, for every $0<|\mu |<\mu ^*$ and $|\lambda -\lambda _1|<|\mu |\nu ^*$, Problem (1.1) has no weak solution $u_\mu =t\varphi _1+v$, with $t\in [t_1, t_2]$ and $v\in \langle \varphi _1\rangle ^\perp$.

It is important to note that, under our hypotheses, h may change sign in $\Omega$. This characterizes the Problem (1.1) as indefinite. This class of problems has been object of an intense research in the last three decades since the articles by Alama and Tarantello (1993), Berestycki et al. (1994) and Ouyang (1991) (see Alama and Del Pino 1996; Alama and Tarantello 1996; Chang and Jiang 2004; Costa and Tehrani 2001; De Figueiredo et al. 2003, 2006; Medeiros et al. 2014; Silva and Silva 2013 and references there in). Here we present, as a consequence of our main theorems, results on the existence of solutions for this class of problems, relating the hypotheses assumed for such problems with the Landesman–Lazer condition. More specifically, we consider the existence of a solution for the following problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u+\beta b_1(x)u^{q}+ b_2(x) u^{p} &{}\text { in } \Omega ,\\ u=0 &{}\text { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(1.3)

where $\Omega$ is a bounded smooth domain of $\mathbb {R}^N$, $N\ge 1$; $\lambda , \beta >0$; $p>q$, with $p\ne 1$, and $b_1$, $b_2\in L^\sigma (\Omega )$, with $\sigma > N$.

Considering the nomenclature for elliptic problems used in the literature, Problem (1.3) is superlinear or sublinear at infinity if $p>1$ or $0<p<1$ and it is superlinear, linear or sublinear at the origin if $q>1$, $q=1$ or $0<q<1$, respectively.

Setting

$$\begin{aligned} r_1:=\int _\Omega b_1\varphi _1^{q+1}dx ~~\text { and }~~ r_2:=\int _\Omega b_2\varphi _1^{p+1}dx, \end{aligned}$$

for the problem linear or superlinear at the origin and superlinear at infinity we establish the following result:

Proposition 1.6

Suppose $p>q\ge 1$ and $r_1r_2<0$. Then there exist positive constants $\beta ^*$ and $\nu ^*$ such that Problem (1.3) has a positive weak solution for every $\beta \in (0,\beta ^*)$ and $|\lambda -\lambda _1|<\beta ^\frac{p-1}{p-q}\nu ^*$.

For the problem sublinear at the origin and superlinear or sublinear at infinity we may state:

Proposition 1.7

Suppose $r_1>0>r_2$. Then

(i):

if $0<q<1<p$, there exist positive constants $\beta _1^*$ and $\nu ^*_1$ such that Problem (1.3) has a positive weak solution for every $\beta \in (0,\beta ^*_1)$ and $|\lambda -\lambda _1|<\beta ^\frac{p-1}{p-q}\nu ^*_1$.

(ii):

if $0<q<p<1$, there exist positive constants $\beta _2^*$ and $\nu ^*_2$ such that Problem (1.3) has a positive weak solution for every $\beta \in (\beta ^*_2,\infty )$ and $|\lambda -\lambda _1|<\beta ^\frac{p-1}{p-q}\nu ^*_2$.

We emphasize that in Proposition 1.6 and in item (i) of Proposition 1.7, we do not assume the restriction $p<{(N+2)}/{(N-2)}$ to derive the existence of a solution for (1.3). Furthermore, in Proposition 1.6 it is worthwhile mentioning that for the subcritical problem with a nonlinear term superlinear at the origin and infinity $(1<q<p<(N+2)/(N-2))$, we are not able to apply a minimax theorem Rabinowitz (1986) to find a positive solution for Problem (1.3) since we do not know if the associated functional satisfies any version of the Palais–Smale compactness condition. Note that, under our hypothesis, the nonlinear term $f(x,s)=\beta b_1(x)s^q+b_2(x)s^p$ does not satisfy $\lim _{s \rightarrow \infty } |f(x,s)-b_2(x)s^p|/s=0$ and we do not suppose that Problem (1.3) has a thick zero set (see e.g. Alama and Tarantello 1993; Medeiros et al. 2014).

For the problem sublinear at the origin we refer the reader to the papers by Ambrosetti et al. (1994) and De Figueiredo et al. (2003).

We also note that, as an application of Theorem 1.3, we establish the existence of multiple solutions for Problem (1.1) with an indefinite nonlinear term h(x, s) that is a polynomial function with respect to the second variable (see Proposition 3.1).

Inspired by the work of Landesman et al. (1975), we consider applications of our main results for Problem (1.1) with h of the form $h(x)=f(x)+g(s)$. More specifically, we are interested in the existence of solution for the problem:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u+\mu (f(x)+g(u)) &{}\text { in } \Omega ,\\ u=0 &{}\text { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(1.4)

where $\Omega$ is a bounded smooth domain of $\mathbb {R}^N$, $N\ge 1$; $\lambda , \mu >0$ and $f\in L^{\sigma }(\Omega )$, with $\sigma > \{N/2, 1\}$.

In our first result on existence of a solution for Problem (1.4) we suppose

$(G_1)$:

$g:\mathbb {R}\rightarrow \mathbb {R}$ a continuous function and there exists $M>0$ such that

$$\begin{aligned} g(s)\ge -M \text { if } s\le 0 \text { and } g(s)\le M \text { if } s\ge 0. \end{aligned}$$

Denoting by $g_i^-:=\liminf _{s\rightarrow -\infty }g(s)$ and $g_s^+:=\limsup _{s\rightarrow +\infty }g(s)$, we assume:

$$\begin{aligned} (LL^+) \qquad \qquad \qquad \qquad \quad \quad \int _\Omega (f+g^-_i)\varphi _1dx>0>\int _\Omega (f+g^+_s)\varphi _1dx. \end{aligned}$$

Proposition 1.8

Suppose g satisfies $(G_1)$ and $(LL^+)$. Then there exist positive constants $\mu ^*$ and $\nu ^*$ such that, for every $\mu \in (0,\mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$, Problem (1.4) has a weak solution.

Next, supposing

$(\hat{G}_1)$:

$g:\mathbb {R}\longrightarrow \mathbb {R}$ is a locally Lipschitz function and there exists $M>0$ such that

$$\begin{aligned} g(s)\le M \text { if } s\le 0 \text { and } g(s)\ge -M \text { if } s\ge 0 \end{aligned}$$

and

$$\begin{aligned} (LL^-) \qquad \qquad \qquad \qquad \quad \quad \int _\Omega (f+g^-_s)\varphi _1dx<0<\int _\Omega (f+g^+_i)\varphi _1dx, \end{aligned}$$

where $g_s^-:=\limsup _{s\rightarrow -\infty }g(s)$ and $g_i^+:=\liminf _{s\rightarrow +\infty }g(s)$, we obtain

Proposition 1.9

Suppose g satisfies $(\hat{G}_1)$ and $(LL^-)$. Then there exist positive constants $\mu ^*$ and $\nu ^*$ such that, for every $\mu \in (0,\mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$, Problem (1.4) has a weak solution.

We observe that the above results allow us to consider g such that $g_i^-=+\infty$ and $g_s^+=-\infty$ or $g_s^-=-\infty$ and $g_i^+=+\infty$, respectively. Moreover g may have unbounded oscillatory behavior.

In our last application we consider the existence of a bifurcation point for the solutions of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u+ f(\alpha , x, u) + b(x)u^p &{}\text { in } \Omega ;\\ u=0 &{}\text { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(1.5)

where $\Omega$ is a bounded smooth domain of $\mathbb {R}^N$, $N\ge 1$, $\lambda \in \mathbb {R}$, $p>1$, $\alpha \in \mathbb {R}^m$, $m\ge 0$, and $f:\mathbb {R}^m\times \Omega \times \mathbb {R}\rightarrow \mathbb {R}$ is given by

$$\begin{aligned} f(\alpha , x, s)=\sum _{i=1}^m \alpha _ib_i(x)s^{p_i}, \text {for every}\ \alpha =(\alpha _1, \ldots , \alpha _m) \in \mathbb {R}^m, s \in \mathbb {R}, ~\text {a. e. in} ~\Omega \nonumber \\ \end{aligned}$$

(1.6)

with $1<p_i<p, p_i\ne p_j, i \ne j \in \{1, \ldots , m\}$. For $m=0$, we take $\mathbb {R}^m=\{0\}$ and $f\equiv 0$.

We also suppose that $b_1, \ldots , b_m, b \in L^{\sigma }(\Omega ), \sigma >N$, and that b satisfies

$$\begin{aligned} r=\int _\Omega b(x)\varphi _1^{p+1} dx \ne 0. \end{aligned}$$

(1.7)

Denoting by $\mathbb {H}$ the Hilbert space $\mathbb {R}\times \mathbb {R}^m\times H_0^1(\Omega )$, a solution of Problem (1.5) is a point $(\lambda , \alpha , u) \in \mathbb {H}$ such that u solves (1.5) for the parameters $(\lambda , \alpha )$. The elements of the subspace of $\mathbb {H}$ formed by the solutions $\{(\lambda , \alpha , 0)/ (\lambda , \alpha ) \in \mathbb {R}\times \mathbb {R}^m\}$ will be referred to as the trivial solutions of Problem (1.5). We recall that $(\lambda _0, \alpha _0, 0)$ is a bifurcation point for the solutions of (1.5) if $(\lambda _0, \alpha _0, 0)$ is in the closure of the set of nontrivial solutions of (1.5) or, equivalently, if every neighborhood of $(\lambda _0, \alpha _0, 0)$ in $\mathbb {H}$ contains a nontrivial solution of (1.5).

Our main objective here is to find hypotheses on f that guarantee that $(\lambda _1, 0, 0)$ is a bifurcation point such that close to it we may find multiple nontrivial positive solutions associated with the same parameters $(\lambda , \alpha )$.

We say that $(\lambda _0, \alpha _0, 0) \in \mathbb {H}$ is a bifurcation point of multiplicity k for the positive solutions of (1.5) if every neighborhood of $(\lambda _0, \alpha _0, 0)$ in $\mathbb {H}$ possesses k distinct nontrivial solutions $(\lambda , \alpha , u_1), \ldots , (\lambda , \alpha , u_k)$ with $u_1, \ldots , u_k$ positive in $\Omega$.

Considering $m\ge 1$, we define $r_i=\int _\Omega b_i(x)\varphi _1^{p_{i+1}} dx, 1 \le i \le m$, and we set

$$\begin{aligned} J:=\{i \in \{1, \ldots , m\}; r_i\ne 0\}. \end{aligned}$$

We denote by $k_J$ the number of elements of J. We also set $k_J=0$ if $m=0$.

Proposition 1.10

Suppose b satisfies (1.7). Then $(\lambda _1, 0, 0)$ is a bifurcation point of multiplicity $k_J+1$ for the positive solutions of (1.5).

We note that for the particular case in which $f\equiv 0$ (corresponding to $m=0$) we have the following nonlinear eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u+ b(x)u^p, &{} \text { in } \Omega ;\\ u=0 &{}\text { on } \partial \Omega . \end{array} \right. \end{aligned}$$

(1.8)

As a consequence of Proposition 1.10 and its proof, we may assert that if r, given by (1.7), is positive then there exists $\underline{\lambda } <\lambda _1$ such that Problem (1.8) has a positive solution for every $\underline{\lambda }<\lambda <\lambda _1$. On the other hand, if r is negative then there exists $\overline{\lambda }>\lambda _1$ such that Problem (1.8) has a positive solution for every $\lambda _1<\lambda <\overline{\lambda }$.

Problem (1.8) for dimensions $N\ge 3$ with critical exponent, $p= (N+2)/(N-2$), and $b\equiv 1$ has been considered by Brézis and Nirenberg (1983). Applying variational methods, those authors proved that Problem (1.8) has a positive has for every $\lambda \in (0,\lambda _1)$ if $N\ge 4$. They also proved that if $\Omega$ is a ball and $N=3$, Problem (1.8) has a positive solution if and only if $\lambda \in (\lambda _1/4, \lambda _1)$. We also observe that, assuming more regularity on b, as a consequence of the results derived by Rabinowitz (1971) via bifurcation theory, we may assert that Problem (1.8) possesses a component ${\mathcal {C}}$ of positive solutions which meets $(\lambda _1,0)$ and it is unbounded in $\mathbb {R}\times H_0^1(\Omega )$. In view of the above observation we may expect that the projection of ${\mathcal {C}}$ on the $\lambda$-axis contains the interval $(\underline{\lambda },\lambda _1)$ or $(\lambda _1,\overline{\lambda })$ provided r is either positive or negative, respectively.

We emphasize that Proposition 1.10 presents us the possibility of having a bifurcation for which we may find multiple positive solutions associated with the same value of the parameters. We also note that, considering $m\ge 1$ and taking appropriated functions $\alpha (\lambda )$, $\lambda \in \mathbb {R}$, in the expression of the function f given by (1.6), we may find nonlinear eigenvalue problems for which $(\lambda _1,0)$ is a bifurcation point of multiplicity greater than one for the positive solutions of the problem (see Proposition 3.3).

As mentioned previously we do not suppose that h satisfies any global growth restriction, which prevents a direct application of variational methods since the associated functional is not well defined. To overcome such difficulty in our proof of Theorem 1.1, firstly we establish a version of this theorem assuming that |h| is bounded by a function in $L^{\sigma }(\Omega )$. In this case we use a minimization argument to find a solution of class $C^{0, \gamma }(\overline{\Omega })$$u_{\mu }=t\varphi _1+v, t \in (t_1, t_2)$ and $v=v(\mu )\in \langle \varphi _1\rangle ^\perp$. Next, considering an appropriated truncation of h, we apply this version of Theorem 1.1 and a bootstrap argument to verify that $\Vert v\Vert _{\infty } \rightarrow 0$ as $\mu \rightarrow 0^+$. This allows us to show that the solution we have found is actually a solution of Theorem 1.1 whenever $\mu >0$ is sufficiently small.

In order to prove Theorem 1.2, we first establish a version of it by supposing that |h| is bounded by a function in $L^{\sigma }(\Omega )$ and that h is Lipschitz with respect to the second variable, with the constant of Lipschitz given by a function in $L^{\sigma }(\Omega )$. We note that the proof of this version is based on the Liapunov–Schmidt Reduction Method. Next, as in the proof of Theorem 1.1, we complete the proof of Theorem 1.2 by truncating h and by using an approximation method.

This paper is organized as follows: in the Sect. 2 we present the proofs of the main results: we reserve the Sects. 2.1, 2.2 and 2.3 for the proofs of Theorems 1.1, 1.2–1.3 and 1.5, respectively. In Sect. 2.4 we establish the results mentioned in Remark 1.4 on positivity, negativity and ordering of the solutions of Problem (1.1). In Sect. 3 we proof Propositions 1.6–1.9 and 1.10. There, as an application of Theorem 1.3, we establish a result on the existence of multiple solutions when h(x, s) is a polynomial function with respect to the variable s. Moreover we also present a result, for a version of Problem (1.8), that states that $(\lambda _1, 0)$ is a bifurcation point with multiple positive solutions associated with the same value of the parameter $\lambda$.

Throughout this work, we denote by

$$\begin{aligned} \Vert u\Vert= & {} \bigg (\int _\Omega |\nabla u|^2dx\bigg )^\frac{1}{2}, ~~ \Vert u\Vert _{k,q}=\bigg (\sum _{0\le |\alpha |\le k}\Vert D^\alpha u\Vert ^q_q\bigg )^{\frac{1}{q}}, ~~\text { and }~~\\ \Vert u\Vert _q= & {} \bigg (\int _\Omega |u|^qdx\bigg )^\frac{1}{q}, \end{aligned}$$

the norms of the spaces $H^1_0(\Omega )$, $W^{k,q}(\Omega )$ and $L^q(\Omega )$, with $q\in [1,\infty )$, respectively. Moreover, we consider $C^{1,\gamma }(\overline{\Omega })$ equipped with its usual norm $\Vert \cdot \Vert _{1,\gamma }$. The symbols A, $d_q$, K and $K_i$, $i=1,2,\ldots$, represent positive constants, reserving $d_q$ for the imbedding constant of $H^1_0(\Omega )$ in $L^q(\Omega )$, $q\in [1,2^*]$.

2 Proofs of the Main Results

We reserve this section for our proofs of Theorems 1.1–1.3 and 1.5. Here we also verify the assertion made in Remark 1.4 on the positivity/negativity of the solution provided by Theorems 1.1 and 1.2 and on the ordering of the solutions given by Theorem 1.3.

Henceforth in this section we suppose without loss of generality that there exists $\bar{\lambda }>0$ such that $0<\lambda <\bar{\lambda }$, with $\bar{\lambda } < \lambda _2$, the second eigenvalue of the operator $-\Delta$ under the Dirichlet boundary conditions.

2.1 Proof of Theorem 1.1

We begin by proving a version of Theorem 1.1 with |h| bounded by a function in $L^\sigma (\Omega )$. More specifically we suppose that

($H_1^*$):

there exists $f\in L^\sigma (\Omega ), \sigma > \{N/2, 1\},$ such that $|h(x,s)|\le f(x), \text { for every} s\in \mathbb {R}, ~\text {a. e. in } {\Omega }.$

Theorem 2.1

Suppose h satisfies $(H_0^+)$ and ($H_1^*$). Then there exist positive constants $\mu ^*$ and $\nu ^*$ such that, for every $\mu \in (0,\mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$, Problem (1.1) has a weak solution $u_\mu =t\varphi _1+v$, with $t\in (t_1, t_2)$ and $v\in \langle \varphi _1\rangle ^\perp$.

We note that, under the hypothesis ($H_1^*$), the associated functional $I_\mu :H^1_0(\Omega )\rightarrow \mathbb {R}$ given by

$$\begin{aligned} I_\mu (u)=\frac{1}{2}\Vert u\Vert ^2-\frac{\lambda }{2}\Vert u\Vert _2^2-\mu \int _{\Omega }H(x,u)dx, \end{aligned}$$

(2.1)

where $H(x,t)= \int _0^t h(x,s)ds$, is well defined and it is of class $C^1$. In addition, the critical points of $I_\mu$ are weak solutions of Problem (1.1).

As mentioned in the introduction, our proof of Theorem 2.1 is based on a minimization argument. More specifically, we shall verify that for $\mu >0$ sufficiently small the infimum of $I_\mu$ on the set $C:=\{u=\tau \varphi _1+v; \tau \in [t_1, t_2], v \in \langle \varphi _1\rangle ^\perp \}$ is attained on an interior point of C. On this direction, we prove:

Lemma 2.2

Suppose h satisfies ($H_1^*$). Then, for every $\mu >0$, the functional $I_\mu$ is bounded from below and coercive on C.

Proof

From $\lambda<\overline{\lambda }<\lambda _2$, ($H_1^*$), the Hölder inequality and the Sobolev Imbedding Theorem, for every $u=\tau \varphi _1+v, v\in \langle \varphi _1\rangle ^\perp$, we get

$$\begin{aligned} I_\mu (u)\ge \frac{-(\overline{\lambda }-\lambda _1)}{2\lambda _1}{\tau }^2 + \frac{(\lambda _2-\overline{\lambda })}{2\lambda _2}\Vert v\Vert ^2 -\mu d_{\sigma ^\prime }\Vert f\Vert _\sigma \Vert v\Vert . \end{aligned}$$

Since $\tau \in [t_1, t_2]$ whenever $u \in C$, the above inequality implies that $I_\mu$ is bounded from below and coercive on C. The lemma is proved. $\square$

Now we define

$$\begin{aligned} m_C:=\inf \{I_\mu (u); u\in C\} \end{aligned}$$

(2.2)

and, for every $\tau \in [t_1, t_2]$,

$$\begin{aligned} m_\tau :=\inf \{I_\mu (u); u=\tau \varphi _1+v, v\in \langle \varphi _1\rangle ^\perp \}. \end{aligned}$$

(2.3)

It follows frow Lemma 2.2 that $m_\tau \ge m_C>-\infty$ for every $\tau \in [t_1, t_2]$. Actually, we may also obtain:

Corollary 2.3

For every $\mu >0$, there exists $u_\mu \in C$ such that $I_\mu (u_\mu )=m_C$. Moreover, for every $\mu >0$ and $\tau \in [t_1, t_2]$, there exists $v_\mu \in \langle \varphi _1\rangle ^\perp$ such that $I_\mu (\tau \varphi _1+v_\mu )=m_\tau$.

Proof

Noting that ($H_1^*$) implies that $I_\mu$ is weakly lower semicontinuous, we may invoke Lemma 2.2 and the fact that C is a closed convex set to conclude that, for every $\mu >0$, there exists $u_\mu \in C$ such that $I_\mu (u_\mu )=m_C$. An analogous argument implies that for every $\mu >0$ and $\tau \in [t_1, t_2]$, we may find $v_\mu \in \langle \varphi _1\rangle ^\perp$ such that $I_\mu (\tau \varphi _1+v_\mu )=m_\tau$. The proof of the corollary is complete. $\square$

For every $\tau \in [t_1, t_2]$ and $\mu >0$, we set

$$\begin{aligned} S_\tau :=\{v\in \langle \varphi _1\rangle ^\perp ; I_\mu (\tau \varphi _1+v)=m_\tau \}. \end{aligned}$$

In view of Corollary 2.3, $S_\tau \ne \emptyset$ for every $\mu >0$ and $\tau \in [t_1, t_2]$. Considering the extreme points of the interval $[t_1, t_2]$, we obtain:

Lemma 2.4

Suppose h satisfies ($H_1^*$). Then, given $\delta >0$, there exists $\mu _1>0$ such that $\Vert v\Vert <\delta , \text { for every } v\in S_{t_1}\cup S_{t_2}.$

Proof

Without loss of generality we suppose that $v\in S_{t_1}$. Using Corollary 2.3, ($H_1^*$), the Hölder Inequality and the Sobolev Imbedding Theorem, we obtain

$$\begin{aligned} \Vert v\Vert ^2=\lambda \Vert v\Vert ^2_2+\mu \int _\Omega h(x,t_1\varphi _1+v)v dx \le \overline{\lambda }\Vert v\Vert ^2/{\lambda _2}+\mu d_{\sigma ^\prime }\Vert f\Vert _\sigma \Vert v\Vert . \end{aligned}$$

Hence $\Vert v\Vert \le {\lambda _2}\mu d_{\sigma ^\prime }\Vert f\Vert _\sigma /(\lambda _2-\overline{\lambda }).$ Taking $0<\mu _2<\min \{\mu _1,{\delta (\lambda _2-\overline{\lambda })}/{\lambda _2d_{\sigma ^\prime }\Vert f\Vert _\sigma }\}$, we complete the proof of Lemma 2.4. $\square$

Now we may present:

Proof of Theorem 2.1

By Corollary 2.3, for every $\mu >0$, there exists $u_\mu =t\varphi _1+v \in C$ such that $I(u_{\mu })=m_C$, $m_C$ given by (2.2). It is clear that $m_C=m_t$ and, consequently, $v \in S_t$. In order to prove Theorem 2.1, it suffices to find $\mu ^*>0$ and $\nu ^*>0$ such that $u_{\mu } \in \text {int}(C)$ for every $\mu \in (0, \mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$.

From the continuity of the functional $J:H_0^1(\Omega )\rightarrow \mathbb {R}$, given by

$$\begin{aligned} J(u)=\displaystyle \int _{\Omega } h(x, u)\varphi _1 dx, ~~~~ \text {for every} ~u \in H_0^1(\Omega ), \end{aligned}$$

and the hypothesis $(H_0^+)$, we find $a>0$ and $\delta >0$ such that, for every $u \in H_0^1(\Omega )$, with $\Vert u\Vert <\delta$, we have

$$\begin{aligned} \int _\Omega h(x,t_1\varphi _1+u)\varphi _1dx>a>0>-a>\int _\Omega h(x,t_2\varphi _1+u)\varphi _1dx. \end{aligned}$$

(2.4)

Considering the above value of $\delta$, we take $\mu ^{*}=\mu _1$, $\mu _1$ given by Lemma 2.4. Taking $\mu \in (0, \mu ^*)$, by Lemma 2.4 and (2.4), for every $v_1 \in S_{t_1}$ and $v_2 \in S_{t_2}$, we obtain

$$\begin{aligned} \langle I^{'}_\mu (t_1\varphi _1+v_1),\varphi _1\rangle <-\mu \left[ \frac{(\lambda -\lambda _1)t_1}{\lambda _1\mu }\Vert \varphi _1\Vert ^2+a\right] \end{aligned}$$

and

$$\begin{aligned} \langle I^{'}_\mu (t_2\varphi _1+v_2),\varphi _1\rangle >-\mu \left[ \frac{(\lambda -\lambda _1)t_2}{\lambda _1\mu }\Vert \varphi _1\Vert ^2-a\right] . \end{aligned}$$

Hence, taking $0<\nu ^*<({\lambda _1a})/{(1+|t_1|+|t_2|)\Vert \varphi _1\Vert ^2}$, for every $0<\mu <\mu ^*$ and $|\lambda -\lambda _1|<\mu \nu ^*$, we get

$$\begin{aligned} \langle I^{'}_\mu (t_1\varphi _1+v_1),\varphi _1\rangle<0<\langle I^{'}_\mu (t_2\varphi _1+v_2),\varphi _1\rangle , \end{aligned}$$

(2.5)

whenever $v_1 \in S_{t_1}$ and $v_2 \in S_{t_2}$. Supposing, for example, that $u_{\mu }=t_1\varphi _1+v$, it follows from (2.5) that there is $\tau \in (t_1, t_2)$ such that $m_C\le m_{\tau } \le I_{\mu }(\tau \varphi _1+v) < I_{\mu }(t_1\varphi _1+v)=I_{\mu }(u_{\mu })=m_C$. This contradiction implies that we may not have $t=t_1$. A similar argument also implies that we may not have $t=t_2$. Consequently $u_{\mu } \in \text {int}(C)$. The proof of Theorem 2.1 is complete. $\square$

Before proving Theorem 1.1 we state the following estimate:

Lemma 2.5

Suppose h satisfies ($H_1^*$). Consider $\mu ^*$ and $\nu ^*$ the positive constants given by Theorem 2.1. Then there exists $b_\sigma >0$ such that $\Vert v\Vert _{2,\sigma }\le b_\sigma \mu$ for every $\mu \in (0,\mu ^*)$, $|\lambda -\lambda _1|<\mu \nu ^*$ and $u_\mu =t\varphi _1+v$ a weak solution of Problem (1.1) with $t\in (t_1, t_2)$ and $v\in \langle \varphi _1\rangle ^\perp$.

Proof of Lemma 2.5

We present a proof of the lemma for $N\ge 3$ since if $N=1$ or $N=2$ the proof may be easily adapted by using the fact that $H^1_0(\Omega )$ is continuously imbedded in $L^p(\Omega )$, for every $p\ge 1$.

First of all we claim that there exists $\hat{b}_{\sigma }>0$ such that, for every $\mu \in (0, \mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$,

$$\begin{aligned} \Vert v\Vert _{\sigma }\le \hat{b}_{\sigma }\mu . \end{aligned}$$

(2.6)

As $u=t\varphi _1+v$ is a weak solution of Problem (1.1) we obtain that v is weak solution of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta v = (\lambda -\lambda _1)t\varphi _1+ \lambda v+\mu h(x,t\varphi _1+v) &{}\text { in } \Omega ,\\ v=0 &{}\text { on } \partial \Omega . \end{array} \right. \end{aligned}$$

Hence, using $0<\lambda<\overline{\lambda }<\lambda _2$, ($H_1^*$), the Hölder Inequality and the Sobolev Imbedding Theorem, we obtain

$$\begin{aligned} \Vert v\Vert \le \frac{\lambda _2}{\lambda _2-\overline{\lambda }}d_{\sigma ^\prime }\Vert f\Vert _\sigma \mu . \end{aligned}$$

(2.7)

Taking $p_0=2^*=2N/(N-2)$, by (2.7) and the continuity of the Sobolev Imbedding $H_0^1(\Omega )\hookrightarrow L^{2^*}(\Omega )$, we find $K_0>0$ such that

$$\begin{aligned} \Vert v\Vert _{p_0}\le K_0\mu . \end{aligned}$$

(2.8)

Next, we define $\hat{h}(x)=(\lambda -\lambda _1)t\varphi _1(x)+\lambda v(x)+\mu h(x,t\varphi _1(x)+v(x))$, for every $x\in \Omega$, and we consider $\hat{p_0}=\min \{\sigma ,p_0\}$. From ($H_1^*$), $|\lambda -\lambda _1|<\mu \nu ^*$, (2.8) and the fact that $L^\sigma (\Omega )$ and $L^{p_0}(\Omega )$ are continuously imbedded in $L^{\hat{p_0}}(\Omega )$, we have that there exists $\hat{K_0}>0$ such that

$$\begin{aligned} \Vert \hat{h}\Vert _{p_0}\le \hat{K_0} \mu . \end{aligned}$$

(2.9)

Thus, by Agmon–Douglis–Nirenberg Theorem (Agmon et al. 1959), we find $\hat{K}>0$ such that

$$\begin{aligned} \Vert v\Vert _{2,\hat{p_0}} \le \hat{K}\hat{K_0}\mu . \end{aligned}$$

(2.10)

If $p_0\ge N/2$, we have that $\hat{p_0}\ge N/2$. In this case the estimate (2.6) is a direct consequence of (2.10) and the continuity of the Sobolev Imbedding $W^{2, N/2}(\Omega )\hookrightarrow L^{\sigma }(\Omega )$. On the other hand, if $p_0<N/2$, we have that $\hat{p_0}=p_0$. In this case, from (2.10) and the fact that $W^{2, p_0}(\Omega )$ is continuously imbedded in $L^{p_1}(\Omega )$, $p_1=Np_0/(N-p_0)$, we find $K_1>0$ such that $\Vert v\Vert _{p_1} \le {K_1}\mu .$

Arguing as above we have that either (2.6) holds or there exist $0<p_0<p_1< \cdots<p_m<N/2$ and $K_1, K_2, \ldots , K_m>0$ such that $p_i=Np_{i-1}/(N-2p_{i-1}),~i=1, \ldots , m$, and $\Vert v\Vert _{p_i} \le {K_i}\mu$, $i=0, 1, \ldots , m$. Noting that those relations imply that $\lim _{m\rightarrow \infty }p_m=\infty$, we conclude that (2.6) must hold. The claim is proved.

From the above claim and the arguments used to derive (2.9) and (2.10), we find $b_\sigma >0$ such that $\Vert v\Vert _{2, \sigma }\le b_\sigma \mu$ for every $\mu \in (0, \mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$. The proof of Lemma 2.5 is complete. $\square$

Now we are in condition of presenting a

Proof of Theorem 1.1

We consider the truncated function $h_R$ defined by

$$\begin{aligned} h_R(x,s)=\chi (s) h(x,s), \text { for every } x\in \overline{\Omega }, ~s\in \mathbb {R}, \end{aligned}$$

(2.11)

where $R>\max \{ |t_1|,|t_2|\}\Vert \varphi _1\Vert _\infty >0$ and $\chi \in C^\infty (\mathbb {R},[0,1])$ is a function satisfiying $\chi (s)\equiv 1$, if $|s|\le R+1$, and $\chi (s)\equiv 0$, if $|s|\ge R+2$. Associated with $h_R$, we consider the truncaded problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u+\mu h_R(x,u) &{}\text { in } \Omega ,\\ u=0 &{}\text { on } \partial \Omega . \end{array} \right. \end{aligned}$$

(2.12)

As $\Vert t_i\varphi _1\Vert _\infty <R$, $i=1,2$, it follows from (2.11) and $(H_0^+)$ that $h_R$ satisfies the hypothesis $(H_0^+)$. Moreover, from $(H_1)$ we may assert that $h_R$ satisfies ($H_1^*$) with $f=f_{R+2}$. Applying Theorem 2.1 we find $\hat{\mu }$ and $\nu ^*$ such that, for every $\mu \in (0,\hat{\mu })$ and $|\lambda -\lambda _1|<\mu \nu ^*$, Problem (2.12) has a weak solution $u_\mu =t_\mu \varphi _1+v_\mu$, with $t_\mu \in (t_1, t_2)$ and $v_\mu \in \langle \varphi _1\rangle ^\perp$. Since $\sigma >N/2$, we may apply the Sobolev Imbedding Theorem and Lemma 2.5 to conclude that $\Vert v_\mu \Vert _\infty \rightarrow 0 \text { as } \mu \rightarrow 0^+.$ Consequently there exists $\mu ^*\in (0,\hat{\mu })$ such that $\Vert v_\mu \Vert _\infty < 1$ for every $\mu \in (0,\mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$. Thus $\Vert u_\mu \Vert _\infty \le |t_\mu |\Vert \varphi _1\Vert _\infty +\Vert v_\mu \Vert _\infty < R+1.$ Hence $\chi (u_\mu )=1$ and $h_R(x,u_\mu )=h(x,u_\mu )$, almost everywhere in $\Omega$. It follows that $u_\mu$ is actually a solution of Problem (1.1). This concludes the proof of Theorem 1.1. $\square$

2.2 Proofs of Theorems 1.2 and 1.3

As in Sect. 2.1, we firstly prove a version of Theorem 1.2 with h satisfying ($H_1^*$) and a hypothesis stronger than $(H_2)$:

$(H_2^*)$:

there exists $\zeta \in L^\sigma (\Omega )$, $\sigma >\{N/2, 1\},$ such that

$|h(x,s_2)-h(x,s_1)|\le \zeta (x)|s_2-s_1|, \text { for every } s_1,s_2\in \mathbb {R}, ~\text {a. e. in } {\Omega }.$

Our initial goal is to prove

Theorem 2.6

Suppose h satisfies $(H_0^-)$, ($H_1^*$) and $(H_2^*)$. Then there exist positive constants $\mu ^*$ and $\nu ^*$ such that, for every $\mu \in (0,\mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$, Problem (1.1) has a solution $u_\mu =t\varphi _1+v$, with $t\in (t_1, t_2)$ and $v\in \langle \varphi _1\rangle ^\perp$.

As in Sect. 2.1, we prove Theorem 2.6 by finding a critical point of the functional $I_\mu$, given by (2.1). In order to establish the existence of such critical point we use the Lyapunov-Schmidt Reduction Method. For the sake of completeness we announce the version of that method that we will be using in our proof of Theorem 1.2 (see Castro 1982).

Theorem 2.7

Let Y and Z be closed subspaces of a real Hilbert X such that $X=Y\oplus Z$. Let $\Phi :X\rightarrow \mathbb {R}$ be a functional of class $C^1$. If there exists an increasing function $\phi :(0,\infty )\rightarrow (0,\infty )$ such that $\phi (s)\rightarrow \infty$ as $s\rightarrow \infty$ and

$$\begin{aligned}&\langle \Phi ^{'}(y+z_1)-\Phi ^{'}(y+z_2),z_1-z_2 \rangle \\&\quad \ge \Vert z_1-z_2\Vert \phi (\Vert z_1-z_2\Vert ), \text { for every } y\in Y, z_1, z_2\in Z. \end{aligned}$$

Then,

(i):

There exists a continuous function $\psi : Y\rightarrow Z$ such that $\Phi (y+\psi (y))=\min _{z\in Z}\Phi (y+z)$. Moreover, $\psi (y)$ is the only element of Z such that $\langle \Phi ^{'}(y+\psi (y)),z\rangle =0$, for every $z\in Z$.

(ii):

The function  $\hat{\Phi }: Y\rightarrow \mathbb {R}$  defined by  $\hat{\Phi }(y)=\Phi (y+\psi (y))$  is of class $C^1$ and

$$\begin{aligned} \langle \hat{\Phi }^{'}(y_1),y_2\rangle =\langle \Phi ^{'}(y_1+\psi (y_1)),y_2 \rangle , \text { for every } ~y_1, y_2\in Y. \end{aligned}$$

(iii):

$y\in Y$ is a critical point of $\hat{\Phi }$ if and only if $y+\psi (y)$ is a critical point of $\Phi$.

We are ready to prove Theorem 2.6.

Proof of Theorem 2.6

From ($H_1^*$), $(H_2^*)$, the Hölder Inequality and the Sobolev Imbedding Theorem, for every $\tau \in \mathbb {R}$ and $v_1$, $v_2\in \langle \varphi _1\rangle ^\perp$, we have that

$$\begin{aligned}&\langle I^{'}_\mu (\tau \varphi _1+v_1)- I^{'}_\mu (\tau \varphi _1+v_2),v_1-v_2 \rangle \\&\quad \ge \Vert v_1-v_2\Vert ^2-\frac{\overline{\lambda }}{\lambda _2}\Vert v_1-v_2\Vert ^2-\mu \int _\Omega [~h(x,\tau \varphi _1+v_1)\\&\qquad -\,h(x,\tau \varphi _1+v_2)~](v_1-v_2)dx\\&\quad \ge \frac{\lambda _2-\overline{\lambda }}{\lambda _2}\Vert v_1-v_2\Vert ^2-\mu \Vert \zeta \Vert _\sigma \Vert v_1-v_2\Vert _{2\sigma ^\prime }^2\\&\quad \ge \bigg [\frac{\lambda _2-\overline{\lambda }}{\lambda _2}-\mu \Vert \zeta \Vert _\sigma d_{2\sigma ^\prime }^2\bigg ]\Vert v_1-v_2\Vert ^2. \end{aligned}$$

Therefore, taking $0<\mu _1<{(\lambda _2-\overline{\lambda })/2\lambda _2\Vert \zeta \Vert _\sigma d_{2\sigma ^\prime }^2}$ we obtain that I satisfies the hypothesis of Theorem 2.7, with $Y=\langle \varphi _1\rangle ,~ Z=\langle \varphi _1\rangle ^\perp$ and $\phi (s)=(\lambda _2-\overline{\lambda })s/2\lambda _2$, for every $s \in (0,\infty )$. Hence, by Theorem 2.7-(i), there exists a continuous function $\psi :\langle \varphi _1\rangle \rightarrow \langle \varphi _1\rangle ^\perp$ such that $\psi (\tau \varphi _1)$, with $\tau \in \mathbb {R}$, is the only element of the space $\langle \varphi _1\rangle ^\perp$ that satisfies

$$\begin{aligned} \langle I^{'}_\mu (\tau \varphi _1+\psi (\tau \varphi _1)),v\rangle =0, \text { for every } \tau \in \mathbb {R},~ v\in \langle \varphi _1\rangle ^\perp . \end{aligned}$$

(2.13)

Moreover, from Theorem 2.7-(ii), the functional $\hat{I}:\langle \varphi _1\rangle \rightarrow \mathbb {R}$, given by $\hat{I}_\mu (\tau \varphi _1)=I_\mu (\tau \varphi _1+\psi (\tau \varphi _1))$, is of class $C^1$ and

$$\begin{aligned} \langle \hat{I}^{'}_\mu (\tau \varphi _1),\varphi _1\rangle= & {} -\mu \bigg [\frac{\lambda -\lambda _1}{\lambda _1\mu }\Vert \varphi _1\Vert ^2\tau +\int _{\Omega }h(x,\tau \varphi _1+\psi (\tau \varphi _1))\varphi _1dx\bigg ]. \end{aligned}$$

(2.14)

On the other hand, from ($H_1^*$) and $(H_0^-)$ we may work as in (2.4) to conclude that there exists $\delta >0$ such that, for every $u\in H_0^1(\Omega )$, with $\Vert u\Vert <\delta$, we have

$$\begin{aligned} \int _{\Omega }h(x,t_1\varphi _1+u)\varphi _1dx< & {} \frac{1}{2}\int _\Omega h(x,t_1\varphi _1)\varphi _1dx\nonumber \\< & {} 0<\frac{1}{2}\int _\Omega h(x,t_2\varphi _1)\varphi _{1}dx< \int _{\Omega }h(x,t_2\varphi _1+u)\varphi _1dx.\nonumber \\ \end{aligned}$$

(2.15)

Taking $v=\psi (\tau \varphi _1)$ in (2.13), it follows from ($H_1^*$), the Hölder Inequality and the Sobolev Imbedding Theorem that

$$\begin{aligned} \Vert \psi (\tau \varphi _1)\Vert ^2 \le \frac{\overline{\lambda }}{\lambda _2}\Vert \psi (\tau \varphi _1)\Vert ^2+\mu d_{\sigma ^\prime }\Vert f\Vert _\sigma \Vert \psi (\tau \varphi _1)\Vert . \end{aligned}$$

Hence $\Vert \psi (\tau \varphi _1)\Vert \le \lambda _2/(\lambda _2-\overline{\lambda })\mu d_{\sigma ^\prime }\Vert f\Vert _\sigma , \text { for every }\mu \in (0,\mu _1),~\tau \in \mathbb {R}.$ Consequently, taking $\mu ^*<\min \{\mu _1,{(\lambda _2-\overline{\lambda })\delta }/{\lambda _2d_{\sigma ^\prime }\Vert f\Vert _\sigma }\}$, where $\delta >0$ is given in (2.15), we obtain that $\Vert \psi (\tau \varphi _1)\Vert <\delta$, for every $\tau \in \mathbb {R}$ and $\mu \in (0,\mu ^*)$. Consequently, from (2.14) and (2.15),

$$\begin{aligned} \langle \hat{I}^{'}_\mu (t_1\varphi _1),\varphi _1\rangle >-\mu \bigg [\frac{|\lambda -\lambda _1|}{\lambda _1\mu }\Vert \varphi _1\Vert ^2 |t_1|+\frac{1}{2}\int _\Omega h(x,t_1\varphi _1)\varphi _1dx\bigg ] \end{aligned}$$

and

$$\begin{aligned} \langle \hat{I}^{'}_\mu (t_2\varphi _1),\varphi _1\rangle <-\mu \bigg [-\frac{|\lambda -\lambda _1|}{\lambda _1\mu }\Vert \varphi _1\Vert ^2 |t_2|+\frac{1}{2}\int _\Omega h(x,t_2\varphi _1)\varphi _1dx\bigg ]. \end{aligned}$$

Therefore, there exists $\nu ^*>0$ such that $\langle \hat{I}^{'}_\mu (t_1\varphi _1),\varphi _1\rangle >0$ and $\langle \hat{I}^{'}_\mu (t_2\varphi _1),\varphi _1 \rangle <0$, for every $\mu \in (0,\mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$. By the Intermediate Value Theorem, there exists  $t\in (t_1,t_2)$  such that  $\langle \hat{I}^{'}_\mu (t\varphi _1),\varphi _1 \rangle =0$. Thus, by Theorem 2.7-(iii),  $u_\mu =t\varphi _1+\psi (t\varphi _1)$  is a critical point of the functional $I_\mu$. This concludes the proof of Theorem 2.6. $\square$

Next we present the proofs of Theorems 1.2 and 1.3:

Proof of Theorem 1.2

We consider $h_R$ defined by (2.11) and the associated truncated problem (2.12). As in the proof of Theorem 1.1, we have that $h_R$ satisfies the hypothesis $(H_0^-)$ and ($H_1^*$) with $f=f_{R+2}$. Furthermore, from the definition of h, $(H_1)$ and $(H_2)$, we obtain that $h_R$ satisfies $(H_2^*)$ with $\zeta =\Vert \chi '\Vert _\infty f_{R+2}+\zeta _{R+2} \in L^\sigma (\Omega ), ~\sigma > \{N/2, 1\}$, where $f_{R+2}$ and $\zeta _{R+2}$ are given by $(H_1)$ and $(H_2)$, respectively. These facts allow us to apply Theorem 2.6, finding positive constants $\hat{\mu }$ and $\nu ^*$ such that, for every $\mu \in (0, \hat{\mu })$ and $|\lambda -\lambda _1|<\mu \nu ^*$, Problem (2.12) has a weak solution $u_\mu =t\varphi _1+v$, with $t \in (t_1, t_2)$ and $v\in \langle \varphi _1\rangle ^\perp$. Next, using Lemma 2.5 and arguing as in the proof of Theorem 1.1, we obtain $\mu ^* \in (0, \hat{\mu })$ such that $u_\mu$ is a solution of Problem (1.1) for every $\mu \in (0, \mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$. The proof of Theorem 1.2 is complete. $\square$

Proof of Theorem 1.3

Taking $-h$, whenever $\mu <0$, we may apply Theorems 1.1 or 1.2, for every $i \in \{1,\dots ,k\}$, to find positive constants $\mu _i$ and $\nu _i$ such that, for every $0<|\mu |<\mu _i$ and $|\lambda -\lambda _1|<|\mu |\nu _i$, Problem (1.1) has a weak solution $u_i=\hat{t}_i\varphi _1+v_i$, with $\hat{t}_i\in (t_i,t_{i+1})$ and $v_i\in \langle \varphi _1\rangle ^\perp$. The proof of Theorem 1.3 is completed by taking $0<\mu ^*<\min \{\mu _i;i=1,\dots ,k\}$ and $0<\nu ^*<\min \{\nu _i;i=1,\dots ,k\}$. $\square$

Remark 2.8

We observe that, by Lemma 2.5 and the proofs of Theorems 1.1 and 1.2, we may assert that there exists $b_\sigma >0$ such that $\Vert v\Vert _{2, \sigma }\le b_\sigma \mu$ for the solutions $u_\mu =t\varphi _1+v$, given by these theorems. In particular, since $\sigma >N/2$, we obtain that $v_\mu \rightarrow 0$ in $C(\overline{\Omega })$ as $\mu \rightarrow 0^+$. We also note that a similar remark may be made for the k solutions obtained in Theorem 1.3.

2.3 Proof of Theorem 1.5

Here we present a proof of Theorem 1.5. We also establish a version of this theorem for solutions uniformly bounded in $H^1_0(\Omega )$ under the hypothesis that h has subcritical growth.

Proof of Theorem 1.5

Arguing by contradiction, we suppose there exist $(\mu _k)\subset \mathbb {R}\backslash \{0\}$, $(\hat{\lambda }_k)\subset \mathbb {R}$ and $(u_k)\subset H_0^1(\Omega )$ such that $|\mu _k|\rightarrow 0$, $|\hat{\lambda }_k-\lambda _1|<|\mu _k|/k$ and $u_k=\tau _k\varphi _1+v_k$, with $\tau _k \in [t_1, t_2]$ and $v_k \in \langle \varphi _1\rangle ^\perp$, a weak solution of Problem (1.1), with parameters $\mu =\mu _k$ and $\lambda =\hat{\lambda }_k.$ Then, using $(H_3)$, the Hölder inequality and the Sobolev Imbedding Theorem, we find $K>0$ such that

$$\begin{aligned} \Vert v_k\Vert ^2= & {} \hat{\lambda }_k\Vert v_k\Vert _2^2+\mu _k\int _\Omega h(x, u_k)v_k dx\\\le & {} \left( \lambda _1+\frac{|\mu _k|}{K}\right) \frac{\Vert v_k\Vert ^2}{\lambda _2}+|\mu _k|K(\Vert f\Vert _{\sigma } d_{\sigma '}\Vert v_k\Vert +\Vert f\Vert _\sigma d_{2\sigma '}^2\Vert v_k\Vert ^2). \end{aligned}$$

Consequently,

$$\begin{aligned} \left[ 1-\left( \frac{|\mu _k|}{K\lambda _2}+\frac{\lambda _1}{\lambda _2}\right) -|\mu _k|K\Vert f\Vert _\sigma d_{2\sigma '}^2\right] \Vert v_k\Vert \le |\mu _k|K\Vert f\Vert _\sigma d_{\sigma '}. \end{aligned}$$

Since $\mu _k\rightarrow 0$, we conclude that $\Vert v_k\Vert \rightarrow 0$. Hence, from the compactness of the Sobolev Imbedding $H_0^1(\Omega )\hookrightarrow L^{2\sigma '}(\Omega )$, we may suppose that $u_k\rightarrow \tau _0\varphi _1$, a. e. in $\Omega$, and that there exists $g \in L^{2\sigma '}(\Omega )$ such that, for every $k \in \mathbb {N}$, $|u_k|\le g$, a. e. in $\Omega$.

Next, invoking $(H_3)$, $(H_4)$ and the Lebesgue Dominated Convergence Theorem, we obtain

$$\begin{aligned} \int _\Omega h(x, u_k)\varphi _1 dx \rightarrow \int _\Omega h(x, \tau _0\varphi _1)\varphi _1 dx \ne 0. \end{aligned}$$

Consequently, from $|\hat{\lambda }_k-\lambda _1| |\tau _k|/|\mu _k| \rightarrow 0$, we get

$$\begin{aligned} \langle I'_{\mu _k}(u_k), \varphi _1\rangle =-\mu _k\left[ \frac{\hat{\lambda }_k-\lambda _1}{\mu _k\lambda _1}\Vert \varphi _1\Vert ^2\tau _k+\int _\Omega h(x, u_k)\varphi _1 dx\right] \ne 0, \end{aligned}$$

for every $k \in \mathbb {N}$ sufficiently large. However this contradicts the fact that $u_k$ is a solution of Problem (1.1). The proof of Theorem 1.5 is complete. $\square$

Supposing

$(\hat{H}_3)$:

there exist $a>0$ and $1<p<\infty ~(p<(N+2)(N-2) ~\text {if}~ N \ge 3$), such that $|h(x,s)|\le a(1+|s|^p), \text { for every } s\in \mathbb {R}, \text { a. e. in} ~ {\Omega },$

we may state a version of Theorem 1.5 for solutions of Problem (1.1) which are uniformly bounded in $H_0^1(\Omega )$:

Theorem 2.9

Suppose h satisfies $(\hat{H}_3)$ and $(H_4)$. Then, given $M>0$, there exist positive constants $\mu ^*$ and $\nu ^*$ such that for every $0<|\mu |<\mu ^*$ and $|\lambda -\lambda _1|<|\mu |\nu ^*$, Problem (1.1) has no weak solution $u_\mu =t\varphi _1+v$, with $t \in [t_1, t_2]$ and $v \in \langle \varphi _1\rangle ^\perp$, with $\Vert u_\mu \Vert \le M$.

Proof

Arguing by contradiction, we suppose that there exist $M>0,~(\mu _k) \subset \mathbb {R}\backslash \{0\},~(\hat{\lambda }_k) \subset \mathbb {R}$ and $(u_k)\subset H_0^1(\Omega )$ such that $|\mu _k|\rightarrow 0$, $|\hat{\lambda }_k-\lambda _1|<|\mu _k|/k$ and $u_k=\tau _k\varphi _1+v_k$, with $\tau _k \in [t_1, t_2]$, $v_k \in \langle \varphi _1\rangle ^\perp$ and $\Vert u_k\Vert \le M$, a weak solution of Problem (1.1), with parameters $\mu =\mu _k$ and $\lambda =\hat{\lambda }_k$.

Invoking $(\hat{H}_3)$, we may write

$$\begin{aligned} |h(x, u_k(x)|\le a(1+|u_k(x)|^{p-1})(1+|u_k(x)|),~\text {a.e. in}~\Omega . \end{aligned}$$

Next, using the above inequality, $\Vert u_k\Vert \le M$ and the Sobolev Imbedding Theorem, we find $f \in L^\sigma (\Omega )$, $\sigma >\{1, N/2\}$, such that

$$\begin{aligned} |h(x, u_k(x)|\le f(x)(1+|u_k(x)|),~\text {a.e. in}~\Omega . \end{aligned}$$

Now, following the argument used in the proof of Theorem 1.5, we derive a contradiction with the fact that $u_k$ is a solution of Problem (1.1). The theorem is proved. $\square$

2.4 Positivity, Negativity and Ordering of the Solutions

Here we verify the assertions made in Remark 1.4. We denote by $(\hat{H}_1)$ the hypothesis $(H_1)$ with $\sigma >N$.

Theorem 2.10

Suppose h satisfies $(H_0^+)$ and $(\hat{H}_1)$ or $(H_0^-)$, $(\hat{H}_1)$ and $(H_2)$. Then there exist positive constants $\mu ^{*}$ and $\nu ^{*}$ such that, for every $\mu \in (0,\mu ^{*})$ and $|\lambda -\lambda _1|<\mu \nu ^{*}$, Problem (1.1) has a positive or negative solution $u_\mu =t\varphi _1+v$, with $t\in (t_1, t_2)$ and $v\in \langle \varphi _1\rangle ^\perp$, provided $t_1\ge 0$ or $t_2\le 0$, respectively.

Proof

Since the proves of the other cases are similar we just present a proof of Theorem 2.10 for h satisfying $(H_0^+)$ and $(\hat{H}_1)$ with $t_1\ge 0$. Using the continuity of the function $\Phi (t)=\int _{\Omega }h(x,t\varphi _1)\varphi _1dx, ~ t\in \mathbb {R},$ we may assume without loss of generality that $t_1>0$. Then, applying Theorem 1.1, we find $\mu _1>0$ and $\nu ^*>0$ such that, for every $0<\mu <\mu _1$ and $|\lambda -\lambda _1|<\mu \nu ^{*}$, Problem (1.1) has a solution $u_\mu =t_\mu \varphi _1+v_\mu$, with $t_\mu \in (t_1, t_2)$ and $v_\mu \in \langle \varphi _1\rangle ^\perp$. Furthermore, from $\sigma > N$, Lemma 2.5 and the Sobolev Imbedding Theorem, we obtain that $v_\mu \rightarrow 0$ in $C^1(\overline{\Omega })$ as $\mu \rightarrow 0^+$.

We claim that

$$\begin{aligned} \lim _{\mu \rightarrow 0^+} \frac{|v_\mu (x)|}{d(x, \partial \Omega )}=0, ~\text {uniformly for}~x \in \Omega . \end{aligned}$$

Supposing the claim, we use that $t\ge t_1>0$ and the fact that there exists $K>0$ such that $\varphi _1(x)\ge K d(x, \partial \Omega )$, for every $x \in \Omega$, to get $\mu ^* \in (0, \mu _1)$ such that, for every $\mu \in (0, \mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^{*}$, $u_\mu (x)\ge \frac{t_1 K}{2} d(x, \partial \Omega )>0$, for every $x \in \Omega$.

In order to conclude the proof of the theorem it remains to prove the claim. Arguing by contradiction, we suppose there exists $\epsilon >0$, $(x_n) \subset \Omega$ and $(\mu _n)\subset \mathbb {R}^+\backslash \{0\}$ such that $|v_{\mu _n}(x_n)|\ge \epsilon d(x_n, \partial \Omega )$, for every $n \in \mathbb {N}$ and $\mu _n\rightarrow 0$ as $n\rightarrow \infty$. Taking $(y_n)\subset \partial \Omega$ such that $d(x_n, \partial \Omega )=|x_n-y_n|$, from $v_{\mu _n}(y_n)=0$ and the Mean Value Theorem, for every $n \in \mathbb {N}$, we find $\theta _n \in (0, 1)$ such that

$$\begin{aligned} \epsilon \le \frac{|v_{\mu _n}(x_{n})|}{d(x_{n},\partial \Omega )}= & {} \frac{|v_{\mu _n}(x_{n})-v_{\mu _n}(y_{n})|}{|y_n-x_n|}\\= & {} |\langle \nabla v_{\mu _{n}}(x_{n}+\theta _n(y_{n}-x_{n}),\frac{y_n-x_n}{|y_n-x_n|}\rangle | \le \Vert \nabla v_{\mu _{n}}\Vert _\infty . \end{aligned}$$

However this contradicts $v_{\mu _n}\rightarrow 0$ in $C^1(\overline{\Omega })$ as $n\rightarrow \infty$. The proof of Theorem 2.10 is complete. $\square$

Arguing as in the proof of Theorem 2.10, we may establish the ordering of the solutions given by Theorem 1.3.

Theorem 2.11

Suppose h satisfies $(H_0)$, $(\hat{H}_1)$ and $({H}_2)$ with $k\ge 2$. Then there exist positive constants $\mu ^{*}$ and $\nu ^{*}$ such that, for every $0<|\mu |<\mu ^{*}$ and $|\lambda -\lambda _1|<|\mu |\nu ^{*}$, Problem (1.1) has k solutions $u_i=\hat{t}_i\varphi _1+v_i$, with $\hat{t}_i\in (t_i,t_{i+1})$ and $v_i\in \langle \varphi _1\rangle ^\perp$, such that $u_i<u_{i+1}$, $i=1,\ldots ,k-1$.

Proof

Arguing as in the proof of Theorem 2.10, for every $i \in \{1, \ldots , k\}$ we find $\tilde{t}_i \in (t_i, t_{i+1})$ such that

$$\begin{aligned} \left[ \int _\Omega h(x, t_i\varphi _1)\varphi _1 dx \right] \left[ \int _\Omega h(x, \tilde{t}_i\varphi _1)\varphi _1 dx \right] <0,~i=1, \ldots , k. \end{aligned}$$

Hence, applying Theorems 1.1 and 1.2, we find $\mu _1>0$ and $\nu ^*>0$ such that, for every $0<|\mu |<\mu _1$ and $|\lambda -\lambda _1|<|\mu |\nu ^{*}$, Problem (1.1) has k solutions $u_i=\hat{t}_i\varphi _1+v_i$, with $\hat{t}_i\in (t_i,\tilde{t}_i)\subset (t_i,t_{i+1})$ and $v_i\in \langle \varphi _1\rangle ^\perp$, $i=1, \ldots , k$. Furthermore, $v_i\rightarrow 0$ in $C^1(\overline{\Omega })$ as $\mu \rightarrow 0$. Noting that $u_{i+1}-u_i=(\hat{t}_{i+1}-\hat{t}_i)\varphi _1+(v_{i+1}-v_i)$, with $\hat{t}_{i+1}-\hat{t}_i\ge t_{i+1}-\tilde{t}_i\ge d>0$, for some $d>0$, we may argue as in the proof of Theorem 2.10 to verify that there exists $\mu ^{*}>0$ such that the thesis of Theorem 2.11 holds for every $i=1, \ldots , k-1$. The proof of the theorem is complete. $\square$

3 Applications of the Main Results

The goal of this section is to present the proofs of Propositions 1.6–1.9. We also present an application of Theorem 1.3, deriving the existence of multiple solutions for Problem (1.1) when h(x, s) is a polynomial function with respect to the second variable.

Proofs of Propositions 1.6 and 1.7

Since the argument used in the proof of Proposition 1.7 is analogous, we shall only present the proof of Proposition 1.6. Considering the rescaling $u=\beta ^\frac{1}{p-q}w, ~\beta >0,$ we obtain that u is a positive solution of Problem 1.3 if, and only if, w is a positive solution of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta w = \lambda w+\mu (b_1 w^{q}+ b_2 w^{p})&{}\text { in } \Omega ,\\ w = 0 &{}\text { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(3.1)

with $\mu =\beta ^\frac{p-1}{p-q}$. Defining $h:{\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$ by $h(x, s)=0$, for $s \le 0$, and $h(x,s)=b_1(x)s^{q}+b_2(x)s^{p}$, for $s \ge 0$, we have that h is a Carathéodory function. Furthermore, since $\sigma >N$ and $p, q \ge 1$, h satisfies $(\hat{H}_1)$ and $(H_2)$. Considering the function $\Phi :\mathbb {R}\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \Phi (t)=\int _\Omega h(x,t\varphi _1)\varphi _1dx, ~ \text {for every} ~ t\ge 0, \end{aligned}$$

(3.2)

we obtain that $\Phi (t)=r_1t^q+r_2t^p$, for every $t \ge 0$. Hence, using that $p>q>0$ and $r_1 r_2<0$, we may find $0<t_1<t_2$ such that h satisfies either $(H_0^+)$ or $(H_0^-)$. Next, applying Theorem 2.10, we find $\mu ^*, \nu ^*>0$ such that Problem (3.1) possesses a positive solution for every $\mu \in (0, \mu ^*)$ and $|\lambda -\lambda _1|<\mu \nu ^*$. The proof of Proposition 1.6 follows directly from this result by taking $\beta ^*=(\mu ^*)^{\frac{p-q}{p-1}}$. $\square$

Proofs of Propositions 1.8 and 1.9

We present only the proof of Proposition 1.8 since the argument used here may be easily adapted to prove Proposition 1.9. Since $h(x, s):= f(x)+g(s)$ satisfies $(H_1)$, in order to derive the existence of a solution for Problem (1.4) via Theorem 1.1 we just need to verify that h satisfies $(H_0^+)$: since, by $(G_1)$, $g(s)+M\ge 0$, for every $s\le 0$, we may apply Fatou’s Lemma and $(LL^+)$ to get

$$\begin{aligned}&\liminf _{t\rightarrow -\infty }\int _\Omega h(x,t\varphi _1)\varphi _1dx =\int _\Omega f\varphi _1dx \\&\quad +\liminf _{t\rightarrow -\infty }\int _\Omega g(t\varphi _1)\varphi _1dx \ge \int _\Omega (f+g^-_i)\varphi _1dx>0. \end{aligned}$$

Analogously, from $M-g(s)\ge 0$, for every $s\ge 0$, Fatou’s Lemma and $(LL^+)$, we obtain

$$\begin{aligned}&\limsup _{t\rightarrow \infty }\int _\Omega h(x,t\varphi _1)\varphi _1dx =\int _\Omega f\varphi _1dx\\&\quad +\limsup _{t\rightarrow \infty }\int _\Omega g(t\varphi _1)\varphi _1dx \le \int _\Omega (f+g^+_s)\varphi _1dx<0. \end{aligned}$$

From the above relations, we find real numbers $t_1<0<t_2$ such that the condition $(H_0^+)$ is valid for these values. Proposition 1.8 is proved. $\square$

We conclude this section by presenting an application of Theorem 1.3 when h(x, s) is a polynomial function in the variable s, i.e., when h is given by

$$\begin{aligned} h(x,s)=\sum _{i=0}^{m}\alpha _i(x)s^i, \text {for every} ~s \in \mathbb {R} ~\text {a. e. in} ~\Omega , \end{aligned}$$

(3.3)

with $\alpha _i\in L^\sigma (\Omega ), ~ \sigma > \{N/2, 1\}.$ In this case, the associated function $\Phi$, given by (3.2), is also a polynomial function in the variable t. More specifically, we have that

$$\begin{aligned} \Phi (t)=\sum _{i=0}^m d_it^i, ~\text {with} ~ d_i=\displaystyle \int _\Omega \alpha _i(x)\varphi _1^{i+1}dx, ~i=1, \ldots , m. \end{aligned}$$

As a consequence of Theorem 1.3, we establish the existence of multiple solutions for Problem (1.1) in function of the number of roots of odd multiplicity of $\Phi$:

Proposition 3.1

Suppose h(x, s) is a polynomial function in the variable s. If the function $\Phi$ has k roots of odd multiplicity, $\tau _1,\dots ,\tau _k$, then there exist positive constants $\mu ^*$ and $\nu ^*$ such that, for every $0<|\mu |<\mu ^*$ and $|\lambda -\lambda _1|<|\mu |\nu ^*$, Problem (1.1) has k weak solutions $u_1,\dots ,u_k$. Moreover, if $|\lambda -\lambda _1|/\mu \rightarrow 0$, as $\mu \rightarrow 0$, the solution $u_i$ converges in $C(\overline{\Omega })$ to $\tau _i\varphi _1$ as $\mu \rightarrow 0$, $i=1,\dots ,k$.

Proof

Without loss of generality, we may suppose that $\tau _1<\tau _2< \cdots <\tau _k$. From the hypothesis of Proposition 3.1, we may write

$$\begin{aligned} \Phi (t)=(t-\tau _1)^{2n_1-1}\ldots (t-\tau _k)^{2n_k-1}(t-c_1)^{2z_1}\ldots (t-c_l)^{2z_l}p(t), \end{aligned}$$

with $n_1, \ldots , n_k, z_1, \ldots , z_l \in \mathbb {N}$ and p(t) a product of irreducible quadratic polynomials. As a direct consequence of above expression, we may find $t_1, \ldots , t_{k+1}$ such that $t_1\in (-\infty ,\tau _1)$, $t_{k+1}\in (\tau _k,\infty )$ and $t_i\in (\tau _{i-1},\tau _i)$, $i=2,\ldots ,k$, $c_j\notin (t_i,t_{i+1})$, for every $i=1,\dots ,k$ and $j=1,\dots ,l$; and

$$\begin{aligned} \Phi (t_i)\Phi (t_{i+1})=\left[ \int _\Omega h(x, t_i\varphi _1)\varphi _1 dx\right] \left[ \int _\Omega h(x, t_{i+1}\varphi _1)\varphi _1 dx\right] <0, ~i=1, \ldots , k. \end{aligned}$$

Hence h satisfies $(H_0)$. Noting that h also satisfies $(H_1)$ and $(H_2)$, we may apply Theorem 1.3 to find positive constants $\mu ^*$ and $\nu ^*$ such that, for every $0<|\mu |<\mu ^*$ and $|\lambda -\lambda _1|<|\mu |\nu ^*$, Problem (1.1) has k solutions $u_i=\hat{t}_i\varphi _1+v_i$, with $\hat{t}_i\in (t_i,t_{i+1})$ and $v_i\in \langle \varphi _1\rangle ^\perp$, $i=1,\ldots ,k$.

Next we fix $i \in \{1, \ldots , k\}$. By Remark 2.8, $v_i=v_i(\mu )\rightarrow 0$ in $C(\overline{\Omega })$ as $\mu \rightarrow 0$. Hence, considering sequences $(\mu _m)\subset \mathbb {R}\backslash \{0\}$ and $(\lambda _m)\subset \mathbb {R}$ such that $\mu _m\rightarrow 0$ and $|\lambda _m-\lambda _1|/\mu _m\rightarrow 0$ as $m\rightarrow \infty$, and taking a subsequence, if necessary, we may suppose that $u_{i,m}=u_i(\mu _m)=\hat{t}_i(\mu _m)\varphi _1+v_i(\mu _m)\rightarrow \hat{t}_0\varphi _1$ in $C(\overline{\Omega })$ as $m\rightarrow \infty$. Consequently, from

$$\begin{aligned} 0= & {} -\frac{1}{\mu _m}\langle I^{'}(\hat{t}_i(\mu _m)\varphi _1+v_i(\mu _m)),\varphi _1\rangle \\= & {} \frac{\lambda _m-\lambda _1}{\mu _m}\Vert \varphi _1\Vert ^2_2\hat{t}_i(\mu _m)+\int _{\Omega }h(x,\hat{t}_i(\mu _m)\varphi _1+v_i(\mu _m))\varphi _1dx, \end{aligned}$$

and the Lebesgue Dominated Convergence Theorem, we obtain that $\Phi (\hat{t}_0)=\int _\Omega h(x,\hat{t}_0\varphi _1)\varphi _1dx=0.$ As, by construction, $\tau _i$ is the only root of $\Phi$ in the interval $[t_i,t_{i+1}]$, we obtain that $u_i(\mu _m)\rightarrow \tau _i\varphi _1$ in $C(\overline{\Omega })$ as $m\rightarrow \infty$. The proof of Proposition 3.1 is complete. $\square$

Remark 3.2

Assuming that $\alpha _i\in L^\sigma (\Omega )$, with $\sigma > N$, by Theorem 2.11 we may assert that the solutions $u_1, \ldots , u_k$ provided by Proposition 3.1, are ordered and of class $C^{1,\gamma }(\overline{\Omega })$.

Proof of Proposition 1.10

For simplicity of notation we set $k=k_J$. Moreover, reordering the terms in the expression of f we may suppose that $J=\{1, \ldots , k\}$. Since the argument for k even or zero is similar we shall prove Proposition 1.10 for k odd. First we consider that $r>0$.

Given $a_1, \ldots , a_k, a \in \mathbb {R}$, we define the function $g:\mathbb {R}^+\rightarrow \mathbb {R}$ by

$$\begin{aligned} g(t)=\frac{t}{\lambda _1}+a_1t^{p_1}+\cdots +a_kt^{p_k}+at^p, ~\text {for every}~t\ge 0. \end{aligned}$$

Since $1<p_i<p$ and $p_i\ne p_j$, for $i \ne j \in \{1, \ldots , k\}$, we may find $a_1, \ldots , a_k \in \mathbb {R}\backslash \{0\}, a>0$ and $0<t_1<t_2<\cdots <t_{k+2}$ such that

$$\begin{aligned} g(t_1)>0, \ \ \ \ g(t_l)g(t_{l+1})<0, \text{ for } \text{ every } 1\le l\le k+1. \end{aligned}$$

(3.4)

Next, considering $\lambda >\lambda _1$, we take $\mu = \lambda -\lambda _1 >0$, and $\alpha =(\alpha _1, \ldots , \alpha _m) \in \mathbb {R}^m$ defined by

$$\begin{aligned} \left\{ \begin{array}{rcl} \alpha _i &{} = &{}\frac{a_i}{r_i} \left( \frac{r}{a}\right) ^{\frac{p_i-1}{p-1}}\mu ^{\frac{p-p_i}{p-1}}, \text{ for } \text{ every } 1\le i\le k;\\ \alpha _i &{} = &{}\left( \frac{r}{a}\right) ^{\frac{p_i-1}{p-1}} \mu ^{\frac{p-p_i}{p-1}}, \quad \,\, \text{ for } \text{ every } k+1\le i\le m. \end{array} \right. \end{aligned}$$

(3.5)

We may assert that $(\lambda , \alpha , u)$, u positive in $\Omega$, is a solution of Problem (1.5), with $\lambda >\lambda _1$ if, and only if, $w = u/\beta (\lambda )$, $\beta (\lambda ) = \left( a\mu /r\right) ^{\frac{1}{p-1}}$, is a positive solution of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta w = \lambda _1 w+ \mu h( x, w), &{}\text { in } \Omega ;\\ w=0 &{}\text { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(3.6)

where, for every $(x,s)\in \Omega \times \mathbb {R}$,

$$\begin{aligned} h(x,s) = s + \sum _{i=1}^k \frac{a_i}{r_i}b_i(x)s^{p_i} + \sum _{i=k+1}^m b_i(x)s^{p_i} +\frac{a}{r} b(x)s^p. \end{aligned}$$

Considering the function $\Phi$ defined by (3.2) and the above expression of h, we obtain that $\Phi (t) = g(t)$, for every $t\ge 0$. Hence, by (3.4), Theorem 1.3 and Remark 1.4, we find $\mu ^*>0$ such that, for every $0<\mu <\mu ^*$, Problem (3.6) possesses $k+1$ positive solutions, $w_1, \ldots ,w_{k+1}\in H^1_0(\Omega )$. Moreover, there exists $M>0$ such that $\Vert w_i\Vert \le M$, for every $1\le i\le k+1$. Consequently, taking $\overline{\lambda } = \lambda _1 + \mu ^*$, we obtain that for every $\lambda \in (\lambda _1, \overline{\lambda })$ and $\alpha (\lambda )$, given by (3.5), Problem (1.5) has $k+1$ solutions $\{(\lambda , \alpha (\lambda ), u_i); u_i = \beta (\lambda )w_i, \ w_i \text{ positive } \text{ in } \Omega ,\ 1\le i\le k+1\}$. Since $\Vert w_i\Vert \le M$, $1\le i\le k+1$, and $\alpha (\lambda )\rightarrow 0$ and $\beta (\lambda )\rightarrow 0$ as $\lambda \rightarrow \lambda _1^+$, we have proved that $(\lambda _1, 0,0)$ is a point of bifurcation of multiplicity $k+1$ for the positive solutions of Problem (1.5).

If $r<0$, we may argue as above to find $\underline{\lambda }<\lambda _1$, $\alpha \in \mathbb {R}^m$, and $u_i\in H^1_0(\Omega )$, $u_i$ positive in $\Omega$, $1\le i\le k+1$, such that, for every $\lambda \in (\underline{\lambda }, \lambda _1)$, $(\lambda , \alpha (\lambda ), u_i)$ is a solution of (1.5), $1\le i\le k+1$. Furthermore $\alpha (\lambda )\rightarrow 0$ in $\mathbb {R}^m$, $u_i(\lambda )\rightarrow 0$ in $H^1_0(\Omega )$, $1\le i \le k+1$, as $\lambda \rightarrow \lambda _1^-$. We conclude that $(\lambda _1, 0,0)$ is a point of bifurcation of multiplicity $k+1$ for the positive solutions of Problem (1.5). The proof of Proposition 1.10 is complete. $\square$

Fixed $a_1,\ldots ,a_m \in \mathbb {R}$, we may consider $\hat{f}(\lambda , x, s)= f(\alpha (\lambda ),x,s)$, with f given by (1.6), and $\alpha (\lambda )= (\alpha _1,\ldots , \alpha _m)\in \mathbb {R}$ defined by

$$\begin{aligned} \alpha _i = a_i |\lambda -\lambda _1|^{\frac{p-p_i}{p-1}}, \ 1\le i\le m. \end{aligned}$$

(3.7)

Next we consider the nonlinear eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u+ \hat{f}(\lambda , x, u) + b(x)u^p &{}\text { in } \Omega ;\\ u=0 &{}\text { on } \partial \Omega . \end{array} \right. \end{aligned}$$

(3.8)

Associated with this nonlinear eigenvalue problem we have the functions

$$\begin{aligned} \Phi _{\pm }(t) = \pm \frac{t}{\lambda _1} + \sum _{i=1}^m a_i r_i t^{p_i} + r t^p, \text{ for } \text{ every } t\ge 0. \end{aligned}$$

Denoting by $k_+$ and $k_-$ the numbers of times $\Phi _+$ and $\Phi _-$ change sign on $(0,\infty )$, respectively, as a consequence of our argument (see the proof of Proposition 1.10) we may state:

Proposition 3.3

Suppose $\hat{f}(\lambda ,x,s)= f(\alpha (\lambda ),x, s)$ with f and $\alpha (\lambda )$ given by (1.6) and (3.7), respectively. Then $(\lambda _1,0)$ is a bifurcation point of multiplicity $k_+$ and $k_-$ for the positive solutions of the Problem (3.8) on the interval $(\lambda _1,\infty )$ and $(0,\lambda _1)$, respectively.