Robin problems with a general potential and a superlinear reaction

TL;DR

This paper investigates semilinear Robin boundary value problems involving indefinite potentials and superlinear reactions without the Ambrosetti-Rabinowitz condition, establishing existence and multiplicity of solutions using advanced variational methods.

Contribution

It introduces new existence and multiplicity results for Robin problems with indefinite potentials and superlinear reactions, without relying on the Ambrosetti-Rabinowitz condition.

Findings

Proved existence of solutions using variational methods.

Established multiplicity and infinity of solutions.

Applied Morse theory and perturbation techniques.

Abstract

We consider semilinear Robin problems driven by the negative Laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the Ambrosetti-Rabinowitz condition. We prove existence and multiplicity theorems (producing also an infinity of smooth solutions) using variational tools, truncation and perturbation techniques and Morse theory (critical groups).

Equations

$$\left\{\begin{array}[]{ll}-\Delta u(z)+\xi(z)u(z)=f(z,u(z))\ \mbox{in}\ \Omega,\\ \frac{\partial u}{\partial n}+\beta(z)u=0\ \mbox{on}\ \partial\Omega.\end{array}\right\}$$

$$\displaystyle\displaystyle 2^{*}=\left\{\begin{array}[]{ll}\frac{2N}{N-2}&\mbox{if}\ N\geqslant 3\\ +\infty&\mbox{if}\ N=1,2.\end{array}\right.$$

$$C^1(\overline{\Omega })\ni u\mapsto \frac{\partial u}{\partial n}=(Du,n{)}_{{\mathbb{R}}^N},$$

$$(1+||u_n||){\phi }^{\prime }(u_n)\to 0\text{mbox}in{X}^{\ast }\text{mbox}asn\to \infty ,$$

$$\max \{\phi (u_0),\phi (u_1)\}<\inf [\phi (u):||u-u_0||=r]=m_r$$

$$\sup [\phi (u):u\in Y_n\cap \partial B_{{\rho }_n}]<0$$

$$(u,h{)}_{H^1}={\int }_{\Omega }uhdz+{\int }_{\Omega }(Du,Dh{)}_{{\mathbb{R}}^N}dz\text{mbox}forallu,h\in H^1(\Omega ).$$

$$||u||={\left[||u|{|}_2^2+||Du|{|}_2^2\right]}^{{}^1{/}_2}\text{mbox}forallu\in H^1(\Omega ).$$

$$C_{+}=\{u\in C^1(\overline{\Omega }):u(z)⩾0\text{mbox}forallz\in \overline{\Omega }\}.$$

$$D_{+}=\{u\in C_{+}:u(z)>0\text{mbox}forallz\in \overline{\Omega }\}.$$

$${\gamma }_0(u)=u{|}_{\partial \Omega }\text{mbox}forallu\in H^1(\Omega )\cap C(\overline{\Omega }).$$

$$\ker {\gamma }_0=H_0^1(\Omega )\text{mbox}and\mathrm{im}{\gamma }_0=H^{\frac{1}{2},2}(\partial \Omega ).$$

$$\left\{\begin{array}[]{l}-\Delta u(z)+\xi(z)u(z)=\hat{\lambda}u(z)\ \mbox{in}\ \Omega,\\ \frac{\partial u}{\partial n}+\beta(z)u=0\ \mbox{on}\ \partial\Omega.\end{array}\right\}$$

$$\xi \in L^s(\Omega )\text{mbox}withs>N\text{mbox}and\beta \in W^{1,\infty }(\partial \Omega )\text{mbox}with\beta (z)⩾0\text{mbox}forallz\in \partial \Omega .$$

$$\gamma (u)=||Du|{|}_2^2+{\int }_{\Omega }\xi (z)u^{2}dz+{\int }_{\partial \Omega }\beta (z)u^{2}d\sigma \text{mbox}forallu\in H^1(\Omega ).$$

$$\gamma (u)+\mu ||u|{|}_2^2⩾c_0||u|{|}^2\text{mbox}forallu\in H^1(\Omega ),\text{mbox}some{c}_0>0.$$

$${\hat{\lambda }}_1=\inf \left[\frac{\gamma (u)}{||u|{|}_2^2}:u\in H^1(\Omega ),u\ne 0\right].$$

$${\hat{\lambda }}_m$$

$$|f_0(z,x)|⩽a_0(z)(1+|x{|}^{2^{\ast }-1})\text{mbox}foralmostallz\in \Omega ,\text{mbox}allx\in \mathbb{R}.$$

$${\phi }_0(u)=\frac{1}{2}\gamma (u)-{\int }_{\Omega }F(z,u)dz\text{mbox}forallu\in H^1(\Omega ).$$

$${\phi }_0(u_0)⩽{\phi }_0(u_0+h)\text{mbox}forallh\in C^1(\overline{\Omega })\text{mbox}with||h|{|}_{C^1(\overline{\Omega })}⩽{\delta }_1.$$

$${\phi }_0(u_0)⩽{\phi }_0(u_0+h)\text{mbox}forallh\in H^1(\Omega )\text{mbox}with||h||⩽{\delta }_2.$$

$${\phi }^c=\{u\in X:\phi (u)⩽c\},K_{\phi }=\{u\in X:{\phi }^{\prime }(u)=0\},K_{\phi }^c=\{u\in K_{\phi }:\phi (u)=c\}.$$

$$C_k(\phi ,u_0)=H_k({\phi }^c\cap U,{\phi }^c\cap U\backslash \{u_0\})\text{mbox}forallk\in {\mathbb{N}}_0,$$

$$C_k(\phi ,\infty )=H_k(X,{\phi }^c)\text{mbox}forallk\in {\mathbb{N}}_0.$$

$${\phi }^c\text{mbox}isastrongdeformationretractof{\phi }^{\hat{c}}.$$

$$H_k(X,{\phi }^c)=H_k(X,{\phi }^{\hat{c}})\text{mbox}forallk\in {\mathbb{N}}_0.$$

$$M(t,u)$$

$$P(t,\infty )$$

$$\sum _{u\in K_{\phi }}M(t,u)=P(t,\infty )+(1+t)Q(t),$$

$$u=u^{+}-u^{-}\text{mbox},|u|=u^{+}-u^{-}\text{mbox},u^{\pm }\in H^1(\Omega ).$$

Full text

aa 2010 AMS Subject Classification: 35J20, 35J60, Secondary 58E05

1. Introduction

In this paper we study the following semilinear elliptic problem with Robin boundary condition:

[TABLE]

In this problem $\displaystyle\Omega\subseteq\mathbb{R}^{N}$ is a bounded domain with a $\displaystyle C^{2}$-boundary $\displaystyle\partial\Omega$. The potential function $\displaystyle\xi\in L^{s}(\Omega)$ with $\displaystyle s>N$ is in general sign-changing. So, the linear part of (1) is indefinite. The reaction term $\displaystyle f(z,x)$ is a Carathéodory function (that is, for all $\displaystyle x\in\mathbb{R},\ z\mapsto f(z,x)$ is measurable and for almost all $\displaystyle z\in\Omega,\ x\mapsto f(z,x)$ is continuous), which exhibits superlinear growth near $\displaystyle\pm\infty$. However, $\displaystyle f(z,\cdot)$ does not satisfy the (usual in such cases) Ambrosetti-Rabinowitz condition (AR-condition, for short). Instead, we employ a more general condition which incorporates in our framework superlinear functions with “slower” growth near $\displaystyle\pm\infty$, which fail to satisfy the AR-condition. Another nonstandard feature of our work is that $\displaystyle f(z,\cdot)$ does not have subcritical polynomial growth. In our case, the growth of $\displaystyle f(z.\cdot)$ is almost critical in the sense that $\displaystyle\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{|x|^{2^{*}-2}x}=0$ uniformly for almost all $\displaystyle z\in\Omega$, with $\displaystyle 2^{*}$ being the Sobolev critical exponent for 2, defined by

[TABLE]

In the boundary condition, $\displaystyle\frac{\partial u}{\partial n}$ denotes the normal derivative of $\displaystyle u\in H^{1}(\Omega)$ defined by extension of the continuous linear map

[TABLE]

with $\displaystyle n(\cdot)$ being the outward unit normal on $\displaystyle\partial\Omega$. The boundary coefficient is $\displaystyle\beta\in W^{1,\infty}(\partial\Omega)$ and we assume that $\displaystyle\beta(z)\geqslant 0$ for all $\displaystyle z\in\partial\Omega$. When $\displaystyle\beta\equiv 0$, we have the usual Neumann problem.

Our aim in this paper is to prove existence and multiplicity results within this general analytical framework. Recently, there have been such results primarily for Dirichlet problems. We mention the works of Lan and Tang [14] (with $\displaystyle\xi\equiv 0$), Li and Wang [15], Miyagaki and Souto [17] (with $\displaystyle\xi\equiv 0$), Papageorgiou and Papalini [21], Qin, Tang and Zhang [29], Wu and An [34], Zhang-Liu [35]. For Neumann and Robin problems, we mention the works of D’Agui, Marano and Papageorgiou [5], Papageorgiou and Rădulescu [23, 24, 26], Papageorgiou, Rădulescu and Repovš [27], Papageorgiou and Smyrlis [28], Pucci et al. [2, 4], Shi and Li [31]. Superlinear problems were treated by Lan and Tang [14], Li and Wang [15], Miyagaki and Souto [17], who proved only existence results. The superlinear case was not studied in the context of Neumann and Robin problems.

Our approach uses variational methods based on the critical point theory, together with suitable truncation and perturbation techniques and Morse theory (critical groups).

2. Mathematical Background

Let $\displaystyle X$ be a Banach space and $\displaystyle X^{*}$ its topological dual. By $\displaystyle\left\langle\cdot,\cdot\right\rangle$ we denote the duality brackets for the pair $\displaystyle(X^{*},X)$. Given $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$, we say that $\displaystyle\varphi$ satisfies the “Cerami condition” (the “C-condition” for short), if the following property holds:

“Every sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq X$ such that $\displaystyle\{\varphi(u_{n})\}_{n\geqslant 1}\subseteq\mathbb{R}$ is bounded and

[TABLE]

admits a strongly convergent subsequence”.

This is a compactness-type condition on $\displaystyle\varphi$, which compensates for the fact that the ambient space $\displaystyle X$ is in general not locally compact. It leads to a deformation theorem from which one can derive the minimax theory of the critical values of $\displaystyle\varphi$. A fundamental result of this theory is the so-called “mountain pass theorem”, which we state here in a slightly more general form (see Gasinski and Papageorgiou [9, p. 648]). We also point out that Theorem 1 is a direct consequence of Ekeland [7, Corollaries 4 and 9].

Theorem 1**.**

Let $\displaystyle X$ be a Banach space. Assume that $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ satisfies the C-condition and for some $\displaystyle u_{0},u_{1}\in X$ with $\displaystyle||u_{1}-u_{0}||>r>0$ we have

[TABLE]

and $\displaystyle c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leqslant t\leqslant 1}\varphi(\gamma(t))\ \mbox{with}\ \Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_{0},\gamma(1)=u_{1}\}$. Then $\displaystyle c\geqslant m_{r}$ and $\displaystyle c$ is a critical value of $\displaystyle\varphi$ (that is, there exists $\displaystyle u_{0}\in X$ such that $\displaystyle\varphi^{\prime}(u_{0})=0$ and $\displaystyle\varphi(u_{0})=c$).

It is well known that when the functional $\displaystyle\varphi$ has symmetry properties, then we can have an infinity of critical points. In this direction, we mention two such results which we will use in the sequel. The first is the so-called “symmetric mountain pass theorem” due to Rabinowitz [30, Theorem 9.12, p. 55] (see also Gasinski and Papageorgiou [9, Corollary 5.4.35, p. 688]).

Theorem 2**.**

Let $\displaystyle X$ be an infinite dimensional Banach space such that $\displaystyle X=Y\oplus V$ with $\displaystyle Y$ finite dimensional. Assume that $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ satisfies the C-condition and that

• (i)

there exist $\displaystyle\vartheta,\rho>0$ such that $\displaystyle\varphi|_{\partial B_{\rho}\cap V}\geqslant\vartheta>0$ (here $\displaystyle\partial B_{\rho}=\{x\in X:||x||=\rho\}$);

• (ii)

for every finite dimensional subspace $\displaystyle E\subseteq X$, we can find $\displaystyle R=R(E)$ such that $\displaystyle\varphi|_{X\backslash B_{R}}\leqslant 0$ (here $\displaystyle B_{R}=\{u\in X:||u||<R\}$).

Then $\displaystyle\varphi$ has an unbounded sequence of critical points.

The second such abstract multiplicity result that we will need, is a variant of a classical result of Clark [3], due to Heinz [11] and Kajikiya [13].

Theorem 3**.**

If $\displaystyle X$ is a Banach space, $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ satisfies the C-condition, is even and bounded below, $\displaystyle\varphi(0)=0$ and for every $\displaystyle n\in\mathbb{N}$ there exist an $\displaystyle n$-dimensional subspace $\displaystyle Y_{n}$ of $\displaystyle X$ and $\displaystyle\rho_{n}>0$ such that

[TABLE]

then there exists a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq X$ of critical points of $\displaystyle\varphi$ such that $\displaystyle u_{n}\rightarrow 0$ in $\displaystyle X$.

In the analysis of problem (1), we will use the following three spaces:

• $\displaystyle\bullet$

the Sobolev space $\displaystyle H^{1}(\Omega)$;

• $\displaystyle\bullet$

the Banach space $\displaystyle C^{1}(\overline{\Omega})$;

• $\displaystyle\bullet$

the “boundary” Lebesgue spaces $\displaystyle L^{q}(\partial\Omega)$ with $\displaystyle 1\leqslant q\leqslant\infty$.

The Sobolev space $\displaystyle H^{1}(\Omega)$ is a Hilbert space with inner product given by

[TABLE]

By $\displaystyle||\cdot||$ we denote the corresponding norm defined by

[TABLE]

The Banach space $\displaystyle C^{1}(\overline{\Omega})$ is an ordered Banach space with positive (order) cone given by

[TABLE]

This cone has a nonempty interior given by

[TABLE]

On $\displaystyle\partial\Omega$ we consider the $\displaystyle(N-1)$-dimensional Hausdorff (surface) measure $\displaystyle\sigma(\cdot)$. Using this measure, we can define in the usual way the “boundary” Lebesgue space $\displaystyle L^{q}(\partial\Omega),1\leqslant q\leqslant\infty$. From the theory of Sobolev spaces, we know that there exists a unique continuous linear map $\displaystyle\gamma_{0}:H^{1}(\Omega)\rightarrow L^{2}(\partial\Omega)$, known as the “trace map”, such that

[TABLE]

So, the trace map extends the notion of boundary values to every Sobolev function. We know that the map $\displaystyle\gamma_{0}$ is compact into $\displaystyle L^{q}(\partial\Omega)$ for all $\displaystyle q\in\left[1,\frac{2(N-1)}{N-2}\right)$ if $\displaystyle N\geqslant 3$ and into $\displaystyle L^{q}(\partial\Omega)$ for all $\displaystyle q\geqslant 1$ if $\displaystyle N=1,2$. Moreover, we have

[TABLE]

In the sequel, for the sake of notational simplicity, we drop the use of the trace map. All restrictions of Sobolev functions on $\displaystyle\partial\Omega$ are understood in the sense of traces.

We will need some facts about the spectrum of the differential operator $\displaystyle u\mapsto-\Delta u+\xi(z)u$ with Robin boundary condition. So, we consider the following linear eigenvalue problem:

[TABLE]

We assume that

[TABLE]

Let $\displaystyle\gamma:H^{1}(\Omega)\rightarrow\mathbb{R}$ be the $\displaystyle C^{1}$-functional defined by

[TABLE]

From D’Agui, Marano and Papageorgiou [5], we know that we can find $\displaystyle\mu>0$ such that

[TABLE]

Using (4) and the spectral theorem for compact self-adjoint operators on a Hilbert space, we show that the spectrum $\displaystyle\hat{\sigma}$(3) of (3) consists of a sequence $\displaystyle\{\hat{\lambda}_{k}\}_{k\in\mathbb{N}}$ of eigenvalues such that $\displaystyle\hat{\lambda}_{k}\rightarrow+\infty$. By $\displaystyle E(\hat{\lambda}_{k})$ $\displaystyle(k\in\mathbb{N})$ we denote the corresponding eigenspace. These items have the following properties:

• $\displaystyle\bullet$

$\displaystyle\hat{\lambda}_{1}$ is simple (that is, $\displaystyle{\rm dim}\,E(\hat{\lambda}_{1})=1$) and

[TABLE]

• $\displaystyle\bullet$

For every $\displaystyle m\geqslant 2$ we have

[TABLE]

• $\displaystyle\bullet$

For every $\displaystyle k\in\mathbb{N},\ E(\hat{\lambda}_{k})$ is finite dimensional, $\displaystyle E(\hat{\lambda}_{k})\subseteq C^{1}(\overline{\Omega})$ and it has the “unique continuation property” (UCP for short), that is, if $\displaystyle u\in E(\hat{\lambda}_{k})$ and vanishes on a set of positive measure, then $\displaystyle u\equiv 0$.

Note that in (5) the infimum is realized on $\displaystyle E(\hat{\lambda}_{1})$ and in (6) both the infimum and the supremum, are realized on $\displaystyle E(\hat{\lambda}_{m})$. The above properties, imply that the nontrivial elements of $\displaystyle E(\hat{\lambda}_{1})$ have constant sign, while the nontrivial elements of $\displaystyle E(\hat{\lambda}_{m})$ (for $\displaystyle m\geqslant 2$) are all nodal (that is, sign changing) functions. By $\displaystyle\hat{u}_{1}$ we denote the $\displaystyle L^{2}$-normalized (that is, $\displaystyle||\hat{u}_{1}||_{2}=1$) positive eigenfunction. We know that $\displaystyle\hat{u}_{1}\in C_{+}$ and by the Harnack inequality (see, for example, Motreanu, Motreanu and Papageorgiou [18, p. 211]), we have $\displaystyle\hat{u}_{1}(z)>0$ for all $\displaystyle z\in\Omega$. Moreover, assuming that $\displaystyle\xi^{+}\in L^{\infty}(\Omega)$ and using the strong maximum principle, we have $\displaystyle\hat{u}_{1}\in D_{+}$.

Now let $\displaystyle f_{0}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a Carathéodory function satisfying

[TABLE]

We set $\displaystyle F_{0}(z,x)=\int^{x}_{0}f_{0}(z,s)ds$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\varphi_{0}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

The next result is a special case of a more general result of Papageorgiou and Rădulescu [22, 25].

Proposition 4**.**

Assume that $\displaystyle u_{0}\in H^{1}(\Omega)$ is a local $\displaystyle C^{1}(\overline{\Omega})$-minimizer of $\displaystyle\varphi_{0}$, that is, there exists $\displaystyle\delta_{1}>0$ such that

[TABLE]

Then $\displaystyle u_{0}\in C^{1}(\overline{\Omega})$ and $\displaystyle u_{0}$ is also a local $\displaystyle H^{1}(\Omega)$-minimizer of $\displaystyle\varphi_{0}$, that is, there exists $\displaystyle\delta_{2}>0$ such that

[TABLE]

Next let us recall a few basic definitions and facts from Morse theory, which we will need in the sequel. So, let $\displaystyle X$ be a Banach space, $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ and $\displaystyle c\in\mathbb{R}$. We introduce the following sets:

[TABLE]

Let $\displaystyle(Y_{1},Y_{2})$ be a topological pair such that $\displaystyle Y_{2}\subseteq Y_{1}\subseteq X$. For $\displaystyle k\in\mathbb{N}_{0}$, let $\displaystyle H_{k}(Y_{1},Y_{2})$ denote the $\displaystyle k$th relative singular homology group for the pair $\displaystyle(Y_{1},Y_{2})$ with integer coefficients (for $\displaystyle k\in-\mathbb{N}$, we have $\displaystyle H_{k}(Y_{1},Y_{2})=0$). Given $\displaystyle u_{0}\in K^{c}_{\varphi}$ isolated, the critical groups of $\displaystyle\varphi$ at $\displaystyle u_{0}$ are defined by

[TABLE]

with $\displaystyle U$ being a neighbourhood of $\displaystyle u_{0}$ satisfying $\displaystyle\varphi^{c}\cap K_{\varphi}\cap U=\{u_{0}\}$. The excision property of singular homology implies that this definition of critical groups is independent of the choice of the neighbourhood $\displaystyle U$.

Suppose that $\displaystyle\varphi$ satisfies the C-condition and that $\displaystyle\inf\varphi(K_{\varphi})>-\infty$. Let $\displaystyle c<\inf\varphi(K_{\varphi})$. The critical groups of $\displaystyle\varphi$ at infinity, are defined by

[TABLE]

This definition is independent of the choice of $\displaystyle c<\inf\varphi(K_{\varphi})$. Indeed, let $\displaystyle c<\hat{c}<\inf\varphi(K_{\varphi})$. Then from a corollary of the second deformation theorem (see Motreanu, Motreanu and Papageorgiou [18, Corollary 5.35, p. 115]) we have that

[TABLE]

Therefore, we have

[TABLE]

We assume that $\displaystyle K_{\varphi}$ is finite and introduce the following quantities:

[TABLE]

Then the Morse relation says that

[TABLE]

where $\displaystyle Q(t)=\sum_{k\geqslant 0}\hat{\beta}_{k}t^{k}$ is a formal series in $\displaystyle t\in\mathbb{R}$, with nonnegative integer coefficients $\displaystyle\hat{\beta}_{k}$.

Finally we fix our notation. So, for $\displaystyle x\in\mathbb{R}$, we set $\displaystyle x^{\pm}=\max\{\pm x,0\}$. Then for $\displaystyle u\in H^{1}(\Omega)$ we define $\displaystyle u^{\pm}(\cdot)=u(\cdot)^{\pm}$ and we have

[TABLE]

Given a measurable function $\displaystyle g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ (for example, a Carathéodory function), by $\displaystyle N_{g}$ we denote the Nemytskii map corresponding to $\displaystyle g$, that is,

[TABLE]

Evidently, $\displaystyle z\mapsto N_{g}(u)(z)$ is measurable on $\displaystyle\Omega$. By $\displaystyle|\cdot|_{N}$ we denote the Lebesgue measure on $\displaystyle\mathbb{R}^{N}$. We set

[TABLE]

Then we have the following orthogonal direct sum decomposition of the Sobolev space $\displaystyle H^{1}(\Omega)$:

[TABLE]

with $\displaystyle H_{-}=\underset{\text{k=1}}{\overset{\text{m}_{-}}{\oplus}}E(\hat{\lambda}_{k})$, $\displaystyle H_{+}=\overline{\mathop{\oplus}\limits_{k\geqslant m_{+}}E(\hat{\lambda}_{k})}$. So, every $\displaystyle u\in H^{1}(\Omega)$ admits a unique sum decomposition

[TABLE]

• If $\displaystyle 0\notin\hat{\sigma}(2)=\{\hat{\lambda}_{k}\}_{k\in\mathbb{N}}$, then $\displaystyle E(0)={0}$ and $\displaystyle m_{-}=m_{+}-1$.
• If $\displaystyle\xi\geqslant 0$ and $\displaystyle\xi\not\equiv 0$ or $\displaystyle\beta\not\equiv 0$, then $\displaystyle\hat{\lambda}_{1}>0$ and so $\displaystyle m_{+}=1$ and $\displaystyle m_{-}=0$.
• If $\displaystyle\xi\equiv 0$, $\displaystyle\beta\equiv 0$ (Neumann problem with zero potential), then

[TABLE]

• If $\displaystyle u,v\in H^{1}(\Omega)$, and $\displaystyle v\leqslant u$, then by $\displaystyle[v,u]$ we denote the order interval defined by

[TABLE]

3. Existence Theorems

In this section we prove two existence theorems. The two existence results differ on the geometry near the origin of the energy (Euler) functional.

For the first existence theorem, we assume that $\displaystyle f(z,\cdot)$ is strictly sublinear near the origin. More precisely, our hypotheses on the data of problem (1) are the following:

• $\displaystyle H(\xi):$

$\displaystyle\xi\in L^{s}(\Omega)$ with $\displaystyle s>\mathbb{N}$.

• $\displaystyle H(\beta):$

$\displaystyle\beta\in W^{1,\infty}(\partial\Omega)$ with $\displaystyle\beta(z)\geqslant 0$ for all $\displaystyle z\in\partial\Omega$.

Remark 1**.**

When $\displaystyle\beta\equiv 0$, we have the usual Neumann problem.

• $\displaystyle H(f)_{1}:$

$\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that

• $\displaystyle(i)$

for every $\displaystyle\rho>0$, there exists $\displaystyle a_{\rho}\in L^{\infty}(\Omega)_{+}$ such that

[TABLE]

[TABLE]

• $\displaystyle(ii)$

if $\displaystyle F(z,x)=\int_{0}^{x}f(z,s)ds$ and $\displaystyle\tau(z,x)=f(z,x)x-2F(z,x)$, then

$\displaystyle\lim_{x\rightarrow\pm\infty}\frac{F(z,x)}{x^{2}}=+\infty$ uniformly for almost all $\displaystyle z\in\Omega$ and there exists $\displaystyle e\in L^{1}(\Omega)$ such that

[TABLE]

• $\displaystyle(iii)$

$\displaystyle\lim_{x\rightarrow 0}\frac{f(z,x)}{x}=0$ uniformly for almost all $\displaystyle z\in\Omega$ and there exists $\displaystyle\delta>0$ such that

[TABLE]

Remark 2**.**

Hypothesis $\displaystyle H(f)_{1}(i)$ is more general than the usual subcritical polynomial growth which says that

[TABLE]

with $\displaystyle c_{1}>0$ and $\displaystyle 1\leqslant r<2^{*}$. Here the growth of $\displaystyle f(z,\cdot)$ is almost critical and this means we face the difficulty that the embedding of $\displaystyle H^{1}(\Omega)$ into $\displaystyle L^{2^{*}}(\Omega)$ is not compact. We overcome this difficulty without use of the concentration-compactness principle. Instead we use Vitali’s theorem. Hypothesis $\displaystyle H(f)_{1}(ii)$ implies that

[TABLE]

Therefore $\displaystyle f(z,\cdot)$ is superlinear near $\displaystyle\pm\infty$. Usually such problems are studied using the so-called Ambrosetti-Rabinowitz condition (the AR-condition for short). We recall that the AR-condition says that there exist $\displaystyle q>2$ and $\displaystyle M>0$ such that

[TABLE]

Integrating (7a) and using (7b), we obtain the following weaker condition

[TABLE]

From (7a) and (9), we see that $\displaystyle f(z,\cdot)$ has at least ($\displaystyle q-1$)-polynomial growth. This restriction removes from consideration superlinear functions with ”slower” growth near $\displaystyle\pm\infty$. For example, consider a function $\displaystyle f(x)$ which satisfies:

[TABLE]

In this case the primitive is $\displaystyle F(x)=\frac{1}{2}x^{2}\ln|x|$ for all $\displaystyle|x|\geqslant M$ and so (4) fails. In particular, then the AR-condition (see (7a) and (7b)) does not hold. In contrast $\displaystyle f(\cdot)$ satisfies our hypothesis $\displaystyle H(f)_{1}(ii)$. This condition is a slightly more general form of a condition used by Li and Yang [15]. It is satisfied, if there exists $\displaystyle M>0$ such that

[TABLE]

Hypothesis $\displaystyle H(f)_{1}(iii)$ implies that $\displaystyle f(z,\cdot)$ is sublinear near zero.

Examples: The following functions satisfy hypotheses $\displaystyle H(f)_{1}$. For the sake of simplicity, we drop the $\displaystyle z$-dependence:

[TABLE]

and

[TABLE]

with $\displaystyle r>2$ and $\displaystyle c=1-\frac{1}{\ln{2}}\left[1-\frac{1}{2^{*}}\frac{1}{2\ln{2}}\right]$.

Note that we have

[TABLE]

[TABLE]

Observe that $\displaystyle f_{1}(\cdot)$ although superlinear, fails to satisfy the AR-condition, while $\displaystyle f_{2}$ has almost critical growth.

Let $\displaystyle\varphi:H^{1}(\Omega)\rightarrow\mathbb{R}$ be the energy (Euler) functional for problem (1) defined by

[TABLE]

Evidently, $\displaystyle\varphi\in C^{1}(H^{1}(\Omega))$. First we show that the functional $\displaystyle\varphi$ satisfies the C-condition.

Proposition 5**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)_{1}(i),(ii)$ hold, then the functional $\displaystyle\varphi$ satisfies the $\displaystyle C$-condition.

Proof.

We consider a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ such that

[TABLE]

From (11) we have

[TABLE]

for all $\displaystyle h\in H^{1}(\Omega)$, with $\displaystyle\varepsilon_{n}\rightarrow 0^{+}$.

In (12) we choose $\displaystyle h=u_{n}\in H^{1}(\Omega)$ and obtain

[TABLE]

On the other hand, by (10), we have

[TABLE]

We add (13) and (14) and obtain

[TABLE]

Claim 1**.**

$\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$* is bounded*

We argue by contradiction. So, we assume that the Claim is not true. Then by passing to a subsequence if necessary, we have

[TABLE]

Let $\displaystyle y_{n}=\frac{u_{n}}{||u_{n}||}$ for all $\displaystyle n\in\mathbb{N}$. Then $\displaystyle||y_{n}||=1$ and so we may assume that

[TABLE]

(note that since $\displaystyle s>N$ (see hypothesis $\displaystyle H(\xi)$), we have $\displaystyle\frac{2s}{s-1}<2^{*}$).

First, we assume that $\displaystyle y\neq 0$. Let $\displaystyle\Omega^{*}=\{z\in\Omega:y(z)\neq 0\}$. Then $\displaystyle|\Omega^{*}|_{N}>0$ and we have

[TABLE]

Using hypothesis $\displaystyle H(f)_{1}(ii)$ we have

[TABLE]

Using Fatou’s lemma we can say that

[TABLE]

Hypothesis $\displaystyle H(f)(ii)$ implies that we can find $\displaystyle M_{3}>0$ such that

[TABLE]

From (16) we see that we may assume that

[TABLE]

Then we have

[TABLE]

[TABLE]

By (10) we have

[TABLE]

(see (20)).

Comparing (21) and (22), we get a contradiction.

Now suppose that $\displaystyle y=0$. Let $\displaystyle\eta>0$ and set $\displaystyle v_{n}=(2\eta)^{{}^{1}/_{2}}y_{n}\in H^{1}(\Omega)$ for all $\displaystyle n\in\mathbb{N}$. Then from (17) and since $\displaystyle y=0$, we have

[TABLE]

Let $\displaystyle c_{4}=\sup\limits_{n\geqslant 1}||v_{n}||^{2^{*}}_{2^{*}}$ (see (23)). From hypothesis $\displaystyle H(f)_{1}(i)$ we see that given $\displaystyle\varepsilon>0$, we can find $\displaystyle c_{5}=c_{5}(\varepsilon)>0$ such that

[TABLE]

Let $\displaystyle E\subseteq\Omega$ be a measurable set such that $\displaystyle|E|_{N}\leqslant\frac{\varepsilon}{2c_{5}}$. Then we have

[TABLE]

[TABLE]

Also note that by (23) and by passing to a subsequence if necessary, we have

[TABLE]

So, invoking Vitali’s theorem (the extended dominated convergence theorem) we have

[TABLE]

From (16), we see that we can find $\displaystyle n_{0}\in\mathbb{N}$ such that

[TABLE]

We choose $\displaystyle t_{n}\in[0,1]$ such that

[TABLE]

From (26) and (27) we have

[TABLE]

Recall that $\displaystyle y=0$. So, from (17), (25) and (3), we see that we can find $\displaystyle n_{1}\in\mathbb{N},\ n_{1}\geqslant n_{0}$ such that

[TABLE]

But $\displaystyle\eta>0$ was arbitrary. Therefore it follows that

[TABLE]

We have

[TABLE]

From (29) and (30) we see that we can find $\displaystyle n_{2}\in\mathbb{N}$ such that

[TABLE]

From (27) and (31), we have

[TABLE]

From hypothesis $\displaystyle H(f)_{1}(ii)$ and (31), we have

[TABLE]

We return to (32) and use (3). Then

[TABLE]

Comparing (29) and (34) we get a contradiction. This proves the Claim.

On account of the Claim, we may assume that

[TABLE]

In (12) we choose $\displaystyle h=u_{n}-u\in H^{1}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (35). Then

[TABLE]

This proves that $\displaystyle\varphi$ satisfies the $\displaystyle C$-condition. ∎

We assume that $\displaystyle K_{\varphi}$ is finite (otherwise we already have an infinity of nontrivial solutions for problem (1)). Then the finiteness of $\displaystyle K_{\varphi}$ and Proposition 5 permit the computation of the critical groups of $\displaystyle\varphi$ at infinity.

Proposition 6**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)_{1},(i),(ii)$ hold, then $\displaystyle C_{k}(\varphi,\infty)=0$ for all $\displaystyle k\in\mathbb{N}$.

Proof.

Hypotheses $\displaystyle H(f)_{1}(i),(ii)$ imply that given any $\displaystyle\eta>0$, we can find $\displaystyle c_{7}=c_{7}(\eta)>0$ such that

[TABLE]

Let $\displaystyle\partial B_{1}=\{u\in H^{1}(\Omega):||u||=1\}$. Then for $\displaystyle u\in\partial B_{1}$ and $\displaystyle t>0$, we have

[TABLE]

(see hypotheses $\displaystyle H(\xi),H(\beta)$).

Recall that $\displaystyle\eta>0$ is arbitrary. So, we can choose $\displaystyle\eta>\frac{c_{8}}{||u||_{2}^{2}}$. Then it follows from (37) that

[TABLE]

For $\displaystyle u\in\partial B_{1}$ and $\displaystyle t>0$, we have

[TABLE]

From (38) and (39) we infer that

[TABLE]

Invoking the implicit function theorem, we can find $\displaystyle\vartheta\in C(\partial B_{1})$ such that

[TABLE]

We extend $\displaystyle\vartheta$ on $\displaystyle H^{1}(\Omega)\backslash\{0\}$ by defining

[TABLE]

We have that $\displaystyle\hat{\vartheta}\in C(H^{1}(\Omega)\backslash\{0\})$ and $\displaystyle\varphi(\hat{\vartheta}(u)u)=\rho_{0}$. Also

[TABLE]

Therefore, if we define

[TABLE]

then $\displaystyle\vartheta_{0}\in C^{1}(H^{1}(\Omega)\backslash\{0\})\ \mbox{(see (\ref{eq40}))}$.

Consider the deformation $\displaystyle h:[0,1]\times(H^{1}(\Omega)\backslash\{0\})\rightarrow H^{1}(\Omega)\backslash\{0\}$ defined by

[TABLE]

We have

[TABLE]

These properties imply that

[TABLE]

Consider the map $\displaystyle r_{0}:H^{1}(\Omega)\backslash\{0\}\rightarrow\partial B_{1}$ defined by

[TABLE]

We see that

[TABLE]

Also, if we consider the deformation $\displaystyle h_{0}:[0,1]\times(H^{1}(\Omega)\backslash\{0\})\rightarrow H^{1}(\Omega)\backslash\{0\}$ defined by

[TABLE]

then we see that

[TABLE]

From (44), (45) and Theorem 6.5 of Dugundji [6, p. 325] it follows that

[TABLE]

From (43) and (46) we infer that

[TABLE]

(see Motreanu, Motreanu and Papageorgiou [18, p. 143]).

Since $\displaystyle\partial B_{1}$ is the unit sphere of the infinite dimensional Hilbert space $\displaystyle H^{1}(\Omega)$ then it is contractible (see Gasinski and Papageorgiou [10, Problems 4.154 and 4.159]). Therefore

[TABLE]

(see Motreanu, Motreanu and Papageorgiou [18, p. 147]).

From (47) and (48) it follows that

[TABLE]

Taking $\displaystyle\rho_{0}$ more negative if necessary (see (40)), we have

[TABLE]

∎

We also compute the critical groups of $\displaystyle\varphi$ at the origin. Recall that we have the orthogonal direct sum decomposition

[TABLE]

with $\displaystyle H_{-}=\underset{\text{k=1}}{\overset{\text{m\_}}{\oplus}}E(\hat{\lambda}_{k}),H_{+}=\overline{\mathop{\oplus}\limits_{k\geqslant m_{+}}E(\hat{\lambda}_{k})}$ (see Section 2). We set

[TABLE]

• $\displaystyle\bullet$

$\displaystyle d_{-}=0$ if $\displaystyle\hat{\lambda}_{k}\geqslant 0$ for all $\displaystyle k\in\mathbb{N}_{0}$ (that is, $\displaystyle H_{-}=\{0\}$).

• $\displaystyle\bullet$

$\displaystyle d^{0}_{-}=0$ if $\displaystyle\hat{\lambda}_{k}>0$ for all $\displaystyle k\in\mathbb{N}_{0}$ (that is, $\displaystyle H_{-}\oplus E(0)=\{0\}$).

Proposition 7**.**

If hypotheses $\displaystyle H(\xi)$, $\displaystyle H(\beta)$, $\displaystyle H(f)_{1}$ hold, then $\displaystyle C_{d_{-}}(\varphi,0)\neq 0$ or $\displaystyle C_{d_{-}^{0}}(\varphi,0)\neq 0$.

Proof.

First we assume that hypothesis $\displaystyle H(f)_{1}(iii)$ [a] holds.

Hypotheses $\displaystyle H(f)_{1}$ imply that given $\displaystyle\varepsilon>0$, we can find $\displaystyle c_{9}=c_{9}(\varepsilon)>0$ such that

[TABLE]

(if $\displaystyle N=1,2$, then we replace $\displaystyle 2^{*}$ by $\displaystyle r>2$). For $\displaystyle u\in\ H_{-}$ we have

[TABLE]

Recall that $\displaystyle\hat{\lambda}_{m_{-}}<0$. So, if we choose $\displaystyle\varepsilon\in(0,-\hat{\lambda}_{m_{-}})$ and exploit the fact that since $\displaystyle H_{-}$ is finite dimensional all norms are equivalent, then from (51) we have

[TABLE]

Since $\displaystyle 2<2^{*}$, we see from (52) that we can find $\displaystyle\rho_{1}\in(0,1)$ small such that

[TABLE]

Recall that $\displaystyle E(0)$ is finite dimensional. So, all norms are equivalent and we can find $\displaystyle\rho_{0}>0$ such that

[TABLE]

Here, $\displaystyle\delta>0$ is as postulated by hypothesis $\displaystyle H(f)_{1}(iii)$.

Let $\displaystyle u\in E(0)\oplus H_{+}$. Then $\displaystyle u$ admits a unique sum decomposition

[TABLE]

Note that

[TABLE]

since $\displaystyle u^{0}$ is the orthogonal projection of $\displaystyle u$ on $\displaystyle E(0)$ and the orthogonal projection operator has operator norm equal to 1.

We define $\displaystyle\Omega_{\delta}=\left\{z\in\Omega:|\hat{u}(z)|\leqslant\frac{\delta}{2}\right\}.$ Then for $\displaystyle u\in E(0)\oplus H_{+}$ with $\displaystyle||u||\leqslant\rho_{0}$, we have

[TABLE]

Also, for $\displaystyle u\in E(0)\oplus H_{+}$ with $\displaystyle||u||\leqslant\rho_{0}$, we have

[TABLE]

So, for $\displaystyle u\in E(0)\oplus H_{+}$ with $\displaystyle||u||\leqslant\rho_{0}$, exploiting the orthogonality of the component spaces, we have

[TABLE]

Since $\displaystyle 2<2^{*}$, choosing $\displaystyle\rho_{2}\in(0,\rho_{0}]$ small, from (3) we have

[TABLE]

Then (53) and (59) imply that

[TABLE]

Then by Motreanu, Motreanu and Papageorgiou [18, pp. 169, 171], we have

[TABLE]

Now assume that hypothesis $\displaystyle H(f)_{1}(iii)$ [b] holds. We consider the following orthogonal direct sum decomposition

[TABLE]

Then for $\displaystyle u\in H_{+}$ we have

[TABLE]

Choosing $\displaystyle\varepsilon>0$ small, we have

[TABLE]

Since $\displaystyle 2<2^{*}$, we can find $\displaystyle\rho_{1}\in(0,1)$ small such that

[TABLE]

Now suppose that $\displaystyle u\in Z=H_{-}\oplus E(0)$. The space Z is finite dimensional and so all norms are equivalent. Therefore we can find $\displaystyle\rho_{2}>0$ such that

[TABLE]

Here, $\displaystyle\delta>0$ is as postulated in hypothesis $\displaystyle H(f)(iii)$. Every $\displaystyle u\in Z$ can be written in a unique way as

[TABLE]

Exploiting the orthogonality of the component spaces, for $\displaystyle u\in Z$ with $\displaystyle||u||\leqslant\rho_{2}$, we have

[TABLE]

Since $\displaystyle\hat{\lambda}_{m_{-}}<0$, it follows that

[TABLE]

Then (60) and (62) imply that

[TABLE]

As before, by Motreanu, Motreanu and Papageorgiou [18, pp. 169, 171], we have

[TABLE]

∎

Now we are ready for our first existence theorem.

Theorem 8**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)_{1}$ hold, then problem (1) admits a nontrivial solution $\displaystyle\tilde{u}\in C^{1}(\overline{\Omega})$.

Proof.

From Proposition 6, we have that

[TABLE]

Also, Proposition 7 says that

[TABLE]

Then (64), (65) and Corollary 6.92 of Motreanu, Motreanu and Papageorgiou [18, p. 173] imply that we can find $\displaystyle\tilde{u}\in K_{\varphi}\backslash\{0\}$. From Papageorgiou and Rădulescu [22], we have

[TABLE]

We define the following functions

[TABLE]

Hypotheses $\displaystyle H(f)_{1}$, imply that given $\displaystyle\varepsilon>0$, we can find $\displaystyle c_{16}=c_{16}(\varepsilon)>0$ such that

[TABLE]

Then from (67), (68) and the Sobolev embedding theorem, we have

[TABLE]

Also, it is clear from (66) and hypothesis $\displaystyle H(f)_{1}(i)$ that

[TABLE]

We rewrite (66) as follows

[TABLE]

Invoking Lemma 5.1 of Wang [33], we have

[TABLE]

Then hypotheses $\displaystyle H(\xi),H(f)_{1}(i)$ imply that

[TABLE]

Invoking Lemma 5.2 of Wang [33] (the Calderon-Zygmund estimates), we have

[TABLE]

∎

In Theorem 8, the reaction term $\displaystyle f(z,\cdot)$ is strictly sublinear near zero (see hypothesis $\displaystyle H(f)_{1}(iii)$). In the next existence theorem, we change the geometry near zero and assume that $\displaystyle f(z,\cdot)$ is linear near zero. In fact we permit double resonance with respect to any nonprincipal spectral interval.

The new hypotheses on the reaction term $\displaystyle f(z,x)$ are the following:

$\displaystyle H(f)_{2}:f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that hypotheses $\displaystyle H(f)_{2}(i),(ii)$ are the same as the corresponding hypotheses $\displaystyle H(f)_{1}(i),(ii)$ and

$\displaystyle(iii)$ there exist $\displaystyle m\in N,m\geqslant 2$ and $\displaystyle\delta>0$ such that

[TABLE]

Remark 3**.**

The behaviour of $\displaystyle f(z,\cdot)$ near $\displaystyle\pm\infty$ remains the same. However, near zero, the growth of $\displaystyle f(z,\cdot)$ has changed. In fact the new condition for $\displaystyle f(z,\cdot)$ near zero implies linear growth. Also, permits resonance with respect to both endpoints of the nonprincipal spectral interval $\displaystyle[\hat{\lambda}_{m},\hat{\lambda}_{m+1}]$, $\displaystyle m\geqslant 2$ (double resonance). This means that the computation of the critical groups of $\displaystyle\varphi$ at the origin changes.

Proposition 9**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)_{2}$ hold, then $\displaystyle C_{k}(\varphi,0)=\delta_{k,d_{m}}\mathbb{Z}$ for all $\displaystyle k\in\mathbb{N}_{0}$ with $\displaystyle d_{m}=dim\underset{\text{i=1}}{\overset{\text{m}}{\oplus}}E(\hat{\lambda}_{i})$.

Proof.

Let $\displaystyle\vartheta\in(\hat{\lambda}_{m},\hat{\lambda}_{m+1})$ and consider the $\displaystyle C^{2}$-functional $\displaystyle\Psi:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

In this case we consider the following orthogonal direct sum decomposition of the Hilbert space $\displaystyle H^{1}(\Omega)$:

[TABLE]

The choice of $\displaystyle\vartheta$ and (6) imply that

[TABLE]

Then Proposition 2.3 of Su [32] implies that

[TABLE]

Consider the homotopy $\displaystyle h_{*}:[0,1]\times H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

As in the proof of Theorem 8, using the regularity theorem of Wang [33], we have that

[TABLE]

Let $\displaystyle t>0$ and suppose that $\displaystyle u\in C^{1}(\overline{\Omega})$ satisfies $\displaystyle 0<||u||_{C^{1}(\overline{\Omega})}\leqslant\delta$, with $\displaystyle\delta>0$ as postulated by hypothesis $\displaystyle H(f)_{2}(iii)$. Then

[TABLE]

Using the orthogonal direct sum decomposition (69), we can write $\displaystyle u$ in a unique way as

[TABLE]

Exploiting the orthogonality of the component spaces, we have

[TABLE]

Recall the choice of $\displaystyle u\in C^{1}(\overline{\Omega})$ and use hypothesis $\displaystyle H(f)_{2}(iii)$. Then

[TABLE]

Therefore

[TABLE]

(see (6)).

Also, again via the orthogonality of the component spaces, we have

[TABLE]

for some $\displaystyle c_{17}>0$ (recall that $\displaystyle\partial\in(\hat{\lambda}_{m},\hat{\lambda}_{m+1})$).

Returning to (71) and using $\displaystyle v=\hat{u}-\overline{u}\in H^{1}(\Omega)$ and relations (72), (73), we obtain

[TABLE]

For $\displaystyle t=0$, we have $\displaystyle h_{*}(0,\cdot)=\varphi(\cdot)$ and $\displaystyle 0\in K_{\varphi}$ is isolated (recall that we assumed that $\displaystyle K_{\varphi}$ is finite or otherwise we already have an infinity of nontrivial solutions). Therefore from the homotopy invariance property of critical groups (see Gasinski and Papageorgiou [10, Theorem 5.125, p. 836]), we have

[TABLE]

∎

Using this Proposition and reasoning as in the proof of Theorem 8, we obtain the following existence theorem.

Theorem 10**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f_{2})$ hold, then problem (1) admits a nontrivial smooth solution

[TABLE]

4. Equations with Concave Terms

In this section we examine what happens when the reaction function exhibits a concave term (that is, a term which is strictly superlinear near zero). So, the geometry is different from both cases considered in Section 3. To deal with this new problem, we introduce a parameter $\displaystyle\lambda>0$ in the concave term and for all $\displaystyle\lambda>0$ small we prove multiplicity results for the equation.

So, now the problem under consideration, is:

[TABLE]

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

$\displaystyle H(f)_{3}:$ $\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that hypotheses $\displaystyle H(f)_{3}(i),(ii)$ are the some as the corresponding hypotheses $\displaystyle H(f)_{1}(i),(ii)$ and

• (iii)

there exist functions $\displaystyle\eta,\eta_{0}\in L^{\infty}(\Omega)$ such that

[TABLE]

For every $\displaystyle\lambda>0$, let $\displaystyle\varphi_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ be the energy (Euler) functional for problem ($\displaystyle P_{\lambda}$) defined by

[TABLE]

Evidently, $\displaystyle\varphi_{\lambda}\in C^{1}(H^{1}(\Omega))$.

Also, we consider the following truncations-perturbations of the reaction in problem ($\displaystyle P_{\lambda}$):

[TABLE]

Here $\displaystyle\mu>0$ is as in (4). Both $\displaystyle\hat{f}^{+}_{\lambda}$ and $\displaystyle\hat{f}^{-}_{\lambda}$ are Carathéodory functions. We set $\displaystyle\hat{F}^{\pm}_{\lambda}(z,x)=\int^{x}_{0}\hat{f}^{\pm}_{\lambda}(z,s)ds$ and consider the $\displaystyle C^{1}$-functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Since hypotheses $\displaystyle H(f)_{3}(i),(ii)$ are the same as $\displaystyle H(f)_{1}(i),(ii)$ and the concave term $\displaystyle\lambda|x|^{q-2}x$ does not affect the behavior of the reaction near $\displaystyle\pm\infty$, from Proposition 5, we have:

Proposition 11**.**

If hypotheses $\displaystyle H(\xi),\,H(\beta),\,H(f)_{3}$ hold and $\displaystyle\lambda>0$, then the functional $\displaystyle\varphi_{\lambda}$ satisfies the C-condition.

Using similar arguments, we can also prove the following result:

Proposition 12**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)_{3}$ hold and $\displaystyle\lambda>0$, then the functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda}$ satisfy the C-condition.

Proof.

As we already indicated, the proof is basically the same as that of Proposition 5. So, we only present the first part of the proof, which differs a little due to the unilateral nature of the functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda}$ (see (76) and (79)).

So, let $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ be a sequence such that

[TABLE]

From (81) we have

[TABLE]

In (82) we choose $\displaystyle h=-u^{-}_{n}\in H^{1}(\Omega)$. Then

[TABLE]

Using (76) and reasoning as in the proof of Proposition 5, we show that

[TABLE]

From (83) and (84) it follows that

[TABLE]

From this via the Kadec-Klee property, as in the proof of Proposition 5, we establish that

[TABLE]

In a similar fashion, using this time (79), we show that $\displaystyle\hat{\varphi}^{-}_{\lambda}$ satisfies the C-condition. ∎

Hypothesis $\displaystyle H(f)_{3}(ii)$ (the superlinearity condition) implies that the functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda}$ are unbounded below.

Proposition 13**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)_{3}$ hold, $\displaystyle\lambda>0$ and $\displaystyle u\in D_{+}$ then $\displaystyle\hat{\varphi}^{\pm}_{\lambda}(tu)\rightarrow-\infty$ as $\displaystyle t\rightarrow\pm\infty$.

The next result will help us to verify the mountain pass geometry (see Theorem 1) for the functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda}$ when $\displaystyle\lambda>0$ is small.

Proposition 14**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)_{3}$ hold, then we can find $\displaystyle\lambda^{\pm}_{*}>0$ such that for every $\displaystyle\lambda\in(0,\lambda^{\pm}_{*})$ we can find $\displaystyle\rho^{\pm}_{\lambda}>0$ for which we have

[TABLE]

Proof.

Hypotheses $\displaystyle H_{f}(3)(i),(iii)$ imply that given $\displaystyle\epsilon>0$, we can find $\displaystyle c_{\epsilon}>0$ such that

[TABLE]

For every $\displaystyle u\in H^{1}(\Omega)$, we have

[TABLE]

for some $\displaystyle c_{19}>0$ (see Lemma 4.11 of Mugnai and Papageorgiou [19] and (4)).

So, choosing $\displaystyle\epsilon\in(0,c_{19})$, we have

[TABLE]

Consider the function

[TABLE]

Since $\displaystyle q<2<2^{*}$, we see that

[TABLE]

So, we can find $\displaystyle t_{0}=t_{0}(\lambda)\in(0,+\infty)$ such that

[TABLE]

We have

[TABLE]

Hence it follows that

[TABLE]

So, we can find $\displaystyle\lambda^{+}_{*}>0$ such that

[TABLE]

Returning to (86) and using this fact we obtain

[TABLE]

Similarly for the functional $\displaystyle\hat{\varphi}^{-}_{\lambda}$. ∎

To produce multiple solutions of constant sign, we need to strengthen the condition on the potential function. The new hypothesis on $\displaystyle\xi(\cdot)$ is:

$\displaystyle H(\xi)^{\prime}:$ $\displaystyle\xi\in L^{s}(\Omega)$ with $\displaystyle s>N$ and $\displaystyle\xi^{+}\in L^{\infty}(\Omega)$.

Proposition 15**.**

If hypotheses $\displaystyle H(\xi)^{\prime},H(\beta),H(f)_{3}$ hold, then

• (a)

for every $\displaystyle\lambda\in(0,\lambda^{+}_{*})$ problem ($\displaystyle P_{\lambda}$) has two positive solutions

[TABLE]

• (b)

for every $\displaystyle\lambda\in(0,\lambda^{-}_{*})$ problem ($\displaystyle P_{\lambda}$) has two negative solutions

[TABLE]

• (c)

for every $\displaystyle\lambda\in(0,\lambda_{*})$ (here $\displaystyle\lambda_{*}=\min\{\lambda^{+}_{*},\lambda^{-}_{*}\}$) problem ($\displaystyle P_{\lambda}$) has at least four nontrivial solutions of constant sign

[TABLE]

Proof.

(a) Let $\displaystyle\lambda\in(0,\lambda^{+}_{*})$ and let $\displaystyle\rho^{+}_{\lambda}>0$ be as postulated by Proposition 14. We consider the set

[TABLE]

This set is weakly compact. Also, using the Sobolev embedding theorem and the compactness of the trace map, we see that $\displaystyle\hat{\varphi}^{+}_{\lambda}$ is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find $\displaystyle u_{0}\in H^{1}(\Omega)$ such that

[TABLE]

Since $\displaystyle q<2$, for $\displaystyle t\in(0,1)$ small we have

[TABLE]

This fact and (87) imply that

[TABLE]

In (4) we choose $\displaystyle h=-u^{-}_{0}\in H^{1}(\Omega)$. Using (76) we obtain

[TABLE]

Then because of (76), equation (4) becomes

[TABLE]

As in the proof of Theorem 8, from the regularity theory of Wang [33], we have

[TABLE]

Hypotheses $\displaystyle H(f)_{3}(i),(iii)$, imply that

[TABLE]

Then from (4) and hypothesis $\displaystyle H(\xi)^{\prime}$, we have

[TABLE]

From (76) it is clear that

[TABLE]

So, from (90) it follows that

[TABLE]

Next we will produce a second positive smooth solution.

We know that

[TABLE]

Also, from Propositions 12 and 13, we have

[TABLE]

Then (91), (92) and (93) permit the use of Theorem 1 (the mountain pass theorem). So, we can find $\displaystyle\hat{u}\in H^{1}(\Omega)$ such that

[TABLE]

From (91) and (94) we have that

[TABLE]

(b) Reasoning in a similar fashion, this time using the functional $\displaystyle\hat{\varphi}^{-}_{\lambda}$ and (79), we produce two negative smooth solutions

[TABLE]

with $\displaystyle v_{0}$ being a local minimizer of $\displaystyle\varphi_{\lambda}$.

(c) Follows from (a) and (b). ∎

In fact we can show that problem ($\displaystyle P_{\lambda}$) has extremal constant sign solutions. So, for all $\displaystyle\lambda\in(0,\lambda^{+}_{*})$ there exists a smallest positive solution $\displaystyle u^{*}_{\lambda}\in D_{+}$ and for all $\displaystyle\lambda\in(0,\lambda^{-}_{*})$ there is a biggest negative solution $\displaystyle v^{*}_{\lambda}\in-D_{+}$.

Let $\displaystyle S^{+}_{\lambda}$ (respectively $\displaystyle S^{-}_{\lambda}$) be the set of positive (respectively negative) solutions of problem. From Proposition 15 we know that

[TABLE]

Moreover, from Filippakis and Papageorgiou [8] (see Lemmata 4.1 and 4.2), we have that

• •

$\displaystyle S^{+}_{\lambda}$ is downward directed (that is, if $\displaystyle u_{1},u_{2}\in S^{+}_{\lambda}$, then there exists $\displaystyle u\in S^{+}_{\lambda}$ such that $\displaystyle u\leqslant u_{1}$ and $\displaystyle u\leqslant u_{2}$).

• •

$\displaystyle S^{-}_{\lambda}$ is upward directed (that is, if $\displaystyle v_{1},v_{2}\in S^{-}_{\lambda}$, then there exists $\displaystyle v\in S^{-}_{\lambda}$ such that $\displaystyle v_{1}\leqslant v,v_{2}\leqslant v$).

Note that hypotheses $\displaystyle H(f)_{3}(i),(iii)$ imply that

[TABLE]

We may always assume that $\displaystyle c_{22}\geqslant\mu$ (see (4)). This unilateral growth condition on $\displaystyle f(z,\cdot)$, leads to the following auxiliary Robin problem:

[TABLE]

Proposition 16**.**

If hypotheses $\displaystyle H(\xi)^{\prime},H(\beta),H(f)_{3}$ hold and $\displaystyle\lambda>0$, then problem ($\displaystyle Au_{\lambda}$) has a unique positive solution

[TABLE]

and because problem ($\displaystyle Au_{\lambda}$) is odd it follows that

[TABLE]

is the unique negative solution of ($\displaystyle Au_{\lambda}$).

Proof.

First, we show the existence of a positive solution.

To this end, let $\displaystyle\psi^{+}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ be the $\displaystyle C^{1}$-functional defined by

[TABLE]

Also, $\displaystyle\psi^{+}_{\lambda}$ is sequentially weakly lower semicontinuous.

So, we can find $\displaystyle\tilde{u}_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

Since $\displaystyle q<2<2^{*}$, for $\displaystyle t\in(0,1)$ small we have

[TABLE]

From (96) we have

[TABLE]

In (98) we choose $\displaystyle h=-\tilde{u}^{-}_{\lambda}\in H^{1}(\Omega)$. Then

[TABLE]

Therefore equation (98) becomes

[TABLE]

(as before using the regularity theory of Wang [33]).

From (4) we have

[TABLE]

Next we show the uniqueness of this positive solution. So, suppose that $\displaystyle\tilde{u}_{\lambda},\bar{u}_{\lambda}\in D_{+}$ are two solutions of ($\displaystyle Au_{\lambda}$). The solution set of ($\displaystyle Au_{\lambda}$) is downward directed (see [8]). So, we may assume that $\displaystyle\bar{u}_{\lambda}\leqslant\tilde{u}_{\lambda}$,

[TABLE]

We subtract (101) from (100) and obtain

[TABLE]

This proves the uniqueness of the positive solution $\displaystyle\tilde{u}_{\lambda}\in D_{+}$ of ($\displaystyle Au_{\lambda}$).

Problem ($\displaystyle Au_{\lambda}$) is odd. So, it follows that

[TABLE]

is the unique negative solution of ($\displaystyle Au_{\lambda}$). ∎

Remark 4**.**

We present an alternative way of proving the uniqueness of the positive solution $\displaystyle\tilde{u}_{\lambda}\in D_{+}$ of ($\displaystyle Au_{\lambda}$). So, let $\displaystyle\tilde{u}_{\lambda},\bar{u}_{\lambda}\in D_{+}$ be two positive solutions of ($\displaystyle Au_{\lambda}$). Let $\displaystyle t>0$ be the biggest positive real such that

[TABLE]

(see Filippakis and Papageorgiou [8, Lemma 3.6]). Suppose that $\displaystyle t\in(0,1)$. Since $\displaystyle q<2<2^{*}$, we can find $\displaystyle\hat{\vartheta}>0$ such that the function

[TABLE]

is increasing on $\displaystyle[0,\max\{||\tilde{u}_{\lambda}||_{\infty},||\bar{u}_{\lambda}||_{\infty}\}]$.

We have

[TABLE]

Note that $\displaystyle\tilde{u}_{\lambda}\not\equiv t\bar{u}_{\lambda}$. Indeed, if $\displaystyle\tilde{u}_{\lambda}\equiv t\bar{u}_{\lambda}$, then

[TABLE]

a contradiction.

Then from (4) and the strong maximum principle, we have

[TABLE]

which contradicts the maximality of $\displaystyle t>0$. Hence $\displaystyle t\geqslant 1$ and we have

[TABLE]

Interchanging the roles of $\displaystyle\tilde{u}_{\lambda}$ and $\displaystyle\bar{u}_{\lambda}$ in the above argument, we also have

[TABLE]

This proves the uniqueness of the positive solution of ($\displaystyle Au_{\lambda}$).

Next we show that $\displaystyle\tilde{u}_{\lambda}\in D_{+}$ (respectively $\displaystyle\tilde{v}_{\lambda}\in-D_{+}$) is a lower bound (upper bound) for the elements of $\displaystyle S^{+}_{\lambda}$ (respectively $\displaystyle S^{-}_{\lambda}$).

Proposition 17**.**

If hypotheses $\displaystyle H(\xi)^{\prime},H(\beta),H(f)_{3}$ hold, then

• (a)

$\displaystyle\tilde{u}_{\lambda}\leqslant u$* for all $\displaystyle u\in S^{+}_{\lambda},\ \lambda\in(0,\lambda^{+}_{*})$;*

• (b)

$\displaystyle v\leqslant\tilde{v}_{\lambda}$* for all $\displaystyle v\in S^{-}_{\lambda},\ \lambda\in(0,\lambda^{-}_{*})$.*

Proof.

(a) Let $\displaystyle u\in S^{+}_{\lambda}$ and let $\displaystyle g^{+}_{\lambda}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be the Carathéodory function defined by

[TABLE]

We set $\displaystyle G^{+}_{\lambda}(z,x)=\int^{x}_{0}g^{+}_{\lambda}(z,s)ds$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\hat{\psi}^{+}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

From (4) and (107) it is clear that $\displaystyle\hat{\psi}^{+}_{\lambda}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\displaystyle\bar{u}_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

As before, since $\displaystyle q<2<2^{*}$, we have

[TABLE]

From (108), we have

[TABLE]

In (4) first we choose $\displaystyle h=-\bar{u}^{-}_{\lambda}\in H^{1}(\Omega)$. Then

[TABLE]

Also in (4) we choose $\displaystyle h=(\bar{u}_{\lambda}-u)^{+}\in H^{1}(\Omega)$. Then

[TABLE]

So, we have proved that

[TABLE]

On account of (107) and (111), equation (4) becomes

[TABLE]

From (111) we conclude that

[TABLE]

(b) In a similar fashion we show that

[TABLE]

∎

Now we are ready to produce extremal constant sign solutions for problem ($\displaystyle P_{\lambda}$), that is, a smallest element for the set $\displaystyle S^{+}_{\lambda}$ $\displaystyle(\lambda\in(0,\lambda^{+}_{*}))$ and a biggest element for the set $\displaystyle S^{-}_{\lambda}$ $\displaystyle(\lambda\in(0,\lambda^{-}_{*}))$.

Theorem 18**.**

If hypotheses $\displaystyle H(\xi)^{\prime},H(\beta),H(f)_{3}$ hold, then

• (a)

for every $\displaystyle\lambda\in(0,\lambda^{+}_{*})$ problem $\displaystyle S^{+}_{\lambda}$ has a smallest element $\displaystyle u^{*}_{\lambda}\in D_{+}$;

• (b)

for every $\displaystyle\lambda\in(0,\lambda^{-}_{*})$ problem $\displaystyle S^{-}_{\lambda}$ has a biggest element $\displaystyle v^{*}_{\lambda}\in-D_{+}.$

Proof.

(a) Recall that $\displaystyle S^{+}_{\lambda}$ $\displaystyle(\lambda\in(0,\lambda^{+}_{*}))$ is downward directed. So, invoking Lemma 3.10 of Hu and Papageorgiou [12, p. 178], we can find a decreasing sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq S^{+}_{\lambda}$ such that

[TABLE]

Evidently, $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ is bounded. So, we may assume that

[TABLE]

We have

[TABLE]

In (113) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (112). We obtain

[TABLE]

Also we have

[TABLE]

Then from (4) and (115) we infer that

[TABLE]

(b) Similarly we produce $\displaystyle v^{*}_{\lambda}\in-D_{+}$ the biggest element of $\displaystyle S^{-}_{\lambda}$. ∎

Using these extremal constant sign solutions of ($\displaystyle P_{\lambda}$), we can generate nodal (that is, sign changing) solutions.

Proposition 19**.**

If hypotheses $\displaystyle H(\xi)^{\prime},H(\beta),H(f)_{3}$ hold and $\displaystyle\lambda\in(0,\lambda_{*})$ (recall $\displaystyle\lambda_{*}=\min\{\lambda^{+}_{\lambda},\lambda^{-}_{*}\}$), then problem ($\displaystyle P_{\lambda}$) admits a nodal solution $\displaystyle\hat{y}\in C^{1}(\overline{\Omega})\backslash\{0\}$.

Proof.

Let $\displaystyle u^{*}_{\lambda}\in D_{+}$ and $\displaystyle v^{*}_{\lambda}\in-D_{+}$ be the two extremal constant sign solutions of ($\displaystyle P_{\lambda}$) produced in Theorem 18. Using them we introduce the following Carathéodory function

[TABLE]

Let $\displaystyle K_{\lambda}(z,x)=\int^{x}_{0}k_{\lambda}(z,s)ds$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\eta_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Also we consider the positive and negative truncations of $\displaystyle k_{\lambda}(z,\cdot)$ (that is, $\displaystyle k^{\pm}_{\lambda}(z,x)=k_{\lambda}(z,\pm x^{\pm})$), set $\displaystyle K^{\pm}_{\lambda}(z,x)=\int^{x}_{0}k^{\pm}_{\lambda}(z,s)ds$ and consider the corresponding $\displaystyle C^{1}$-functionals $\displaystyle\eta^{\pm}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Claim 2**.**

$\displaystyle K_{\eta_{\lambda}}\subseteq[v^{*}_{\lambda},u^{*}_{\lambda}]\cap C^{1}(\overline{\Omega}),\ K_{\eta^{+}_{\lambda}}=\{0,u^{*}_{\lambda}\},\ K_{\eta^{-}_{\lambda}}=\{0,v^{*}_{\lambda}\}$.

Let $\displaystyle u\in K_{\eta_{\lambda}}$. We have

[TABLE]

In (120) we choose $\displaystyle h=(u-u^{*}_{\lambda})^{+}\in H^{1}(\Omega)$. Then

[TABLE]

Similarly, choosing $\displaystyle h=(v^{*}_{\lambda}-u)^{+}\in H^{1}(\Omega)$ in (120), we show that

[TABLE]

So, from the above and the regularity theory of Wang [33], we have

[TABLE]

In a similar fashion, we show that

[TABLE]

The extremality of $\displaystyle u^{*}_{\lambda}\in D_{+}$ and of $\displaystyle v^{*}_{\lambda}\in-D_{+}$, implies that

[TABLE]

This proves Claim 2.

Claim 3**.**

$\displaystyle u^{*}_{\lambda}\in D_{+}$* and $\displaystyle v^{*}_{\lambda}\in-D_{+}$ are local minimizers of $\displaystyle\eta_{\lambda}$.*

Evidently, $\displaystyle\eta^{+}_{\lambda}$ is coercive (see (119)) and sequentially weakly lower semicontinuous. So, we can find $\displaystyle\tilde{u}^{*}_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

As before, since $\displaystyle q<2<2^{*}$, we have

[TABLE]

From (121) we have that

[TABLE]

From (122), (123) and Claim 2 it follows that $\displaystyle\tilde{u}^{*}_{\lambda}=u^{*}_{\lambda}\in D_{+}$. Note that

[TABLE]

Similarly for $\displaystyle v^{*}_{\lambda}\in-D_{+}$, using this time the functional $\displaystyle\eta^{-}_{\lambda}$. This proves Claim 3.

Without any loss of generality, we may assume that

[TABLE]

The reasoning is similar if the opposite inequality holds. Also, we assume that $\displaystyle K_{\eta_{\lambda}}$ if finite or on account of Claim 2 we already have an infinity of smooth nodal solutions and so we are done. Then Claim 3 implies that we can find $\displaystyle\rho\in(0,1)$ small such that

[TABLE]

(see Aizicovici, Papageorgiou and Staicu [1], proof of Proposition 29).

From (4) and (119) it is clear that $\displaystyle\eta_{\lambda}$ is coercive. Hence

[TABLE]

Then (124) and (125) permit the use of Theorem 1 (the mountain pass theorem). So, we can find $\displaystyle\hat{y}\in H^{1}(\Omega)$ such that

[TABLE]

Claim 2 together with (124) and (126) imply that

[TABLE]

Since $\displaystyle\hat{y}$ is a critical point of $\displaystyle\eta_{\lambda}$ of mountain pass type, we have

[TABLE]

(see Motreanu, Motreanu and Papageorgiou [18, Corollary 6.81, p. 168]).

On the other hand, the presence of the concave term $\displaystyle\lambda|x|^{q-2}x\ (q<2)$ in the reaction function, hypothesis $\displaystyle H(f)_{3}(iii)$ and Lemma 3.4 of D’Agui, Marano and Papageorgiou [5] imply that

[TABLE]

Comparing (127) and (128), we conclude that

[TABLE]

∎

So, we can state the following multiplicity theorem for problem ($\displaystyle P_{\lambda}$).

Theorem 20**.**

If hypotheses $\displaystyle H(\xi)^{\prime},H(\beta),H(f)_{3}$ hold, then we can find a parameter value $\displaystyle\lambda_{*}>0$ such that for every $\displaystyle\lambda\in(0,\lambda_{*})$ problem ($\displaystyle P_{\lambda}$) has at least five nontrivial smooth solutions

[TABLE]

If we strengthen the regularity of $\displaystyle f(z,\cdot)$, we can improve Theorem 20 and produce a sixth nontrivial smooth solution. However, we do not provide any sign information for this sixth solution.

The new conditions on the perturbation term $\displaystyle f(z,x)$ are the following:

$\displaystyle H(f)_{4}:$ $\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a measurable function such that for almost all $\displaystyle z\in\Omega,\ f(z,\cdot)\in C^{1}(\mathbb{R})$ and

• (i)

for every $\displaystyle\rho>0$, there exists $\displaystyle a_{\rho}\in L^{\infty}(\Omega)$ such that

[TABLE]

• (ii)

$\displaystyle\lim\limits_{x\rightarrow\pm\infty}\frac{F(z,x)}{x^{2}}=+\infty$ uniformly for almost all $\displaystyle z\in\Omega$ and there exists $\displaystyle e\in L^{1}(\Omega)$ such that

[TABLE]

(recall that $\displaystyle F(z,x)=\int^{x}_{0}f(z,s)ds$ and $\displaystyle\tau(z,x)=f(z,x)x-2F(z,x)$);

• (iii)

$\displaystyle f(z,0)=0$ for almost all $\displaystyle z\in\Omega$, $\displaystyle f^{\prime}_{x}(z,0)=\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}$ uniformly for almost all $\displaystyle z\in\Omega$ and

[TABLE]

Under the above hypotheses, we have $\displaystyle\varphi_{\lambda}\in C^{2}(H^{1}(\Omega)\backslash\{0\})$.

Proposition 21**.**

If hypotheses $\displaystyle H(\xi)^{\prime},H(\beta),H(f)_{4}$ hold and $\displaystyle\lambda\in(0,\lambda_{*})$, then problem ($\displaystyle P_{\lambda}$) admits a sixth nontrivial smooth solution

[TABLE]

Proof.

From Proposition 15 we know that

[TABLE]

Also, recall (see the proof of Proposition 15), that $\displaystyle\hat{u}\in D_{+}$ is a critical point of mountain pass type for $\displaystyle\hat{\varphi}^{+}_{\lambda}$ and $\displaystyle\hat{v}\in-D_{+}$ is a critical point of mountain pass type for $\displaystyle\hat{\varphi}^{-}_{\lambda}$. Note that

[TABLE]

Also, as we already pointed out in the proof of Proposition 19 (see (128) and recall $\displaystyle\left.\eta_{\lambda}\right|_{[v^{*}_{\lambda},u^{*}_{\lambda}]}=\left.\varphi_{\lambda}\right|_{[v^{*}_{\lambda},u^{*}_{\lambda}]}$), we have

[TABLE]

From Proposition 6, we have

[TABLE]

Let $\displaystyle\hat{m}=\max\{||u_{0}||_{\infty},||\hat{u}||_{\infty},||v_{0}||_{\infty},||\hat{v}||_{\infty}\}$. Hypotheses $\displaystyle H(f)_{4}(i),(iii)$ imply that we can find $\displaystyle\hat{\xi}>0$ such that for almost all $\displaystyle z\in\Omega$, the function

[TABLE]

is nondecreasing on $\displaystyle[-\hat{m},\hat{m}]$. With $\displaystyle\hat{y}\in C^{1}(\overline{\Omega})\backslash\{0\}$ being the nodal solution we have

[TABLE]

Similarly we show that

[TABLE]

So, finally we have

[TABLE]

Recall that

[TABLE]

Suppose that $\displaystyle K_{\varphi_{\lambda}}=\{0,u_{0},v_{0},\hat{u},\hat{v},\hat{y}\}$. Then from (129), (4), (131), (132) and (133) and using the Morse relation with $\displaystyle t=-1$ (see (7)), we have

[TABLE]

a contradiction. So, there exists $\displaystyle\tilde{y}\in H^{1}(\Omega)$ such that

[TABLE]

This is the sixth nontrivial smooth solution of problem ($\displaystyle P_{\lambda}$). ∎

So, we can state the following new multiplicity theorem for problem ($\displaystyle P_{\lambda}$).

Theorem 22**.**

If hypotheses $\displaystyle H(\xi)^{\prime},H(\beta),H(f)_{4}$ hold, then there exists a parameter value $\displaystyle\lambda_{*}>0$ such that for every $\displaystyle\lambda\in(0,\lambda_{*})$ problem ($\displaystyle P_{\lambda}$) has at least six nontrivial smooth solutions

[TABLE]

5. Infinitely Many Solutions

In this section, we generate an infinity of nontrivial smooth solutions by introducing symmetry on the reaction term. We prove two such results. The first concerns problem ($\displaystyle P_{\lambda}$) and the solutions we produce are nodal. The second result deals with problem (1) and produces an infinity of nontrivial smooth solutions but without any sign information.

For the first theorem, the hypotheses on the perturbation term $\displaystyle f(z,x)$ are the following:

$\displaystyle H(f)_{5}:$ $\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function which satisfies hypotheses $\displaystyle H(f)_{3}$ and in addition for almost all $\displaystyle z\in\Omega,\ f(z,\cdot)$ is odd.

Theorem 23**.**

If hypotheses $\displaystyle H(\xi)^{\prime},H(\beta),H(f)_{5}$ hold, then we can find a parameter value $\displaystyle\lambda_{*}>0$ such that for every $\displaystyle\lambda\in(0,\lambda_{*})$ problem ($\displaystyle P_{\lambda}$) admits a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq C^{1}(\overline{\Omega})$ of distinct nodal solutions such that

[TABLE]

Proof.

Let $\displaystyle\lambda_{*}=\min\{\lambda^{+}_{*},\lambda^{-}_{*}\}$ be as in Proposition 15(c) and consider the $\displaystyle C^{1}$-functional $\displaystyle\eta_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ introduced in the proof of Proposition 19. We have the following properties:

• •

$\displaystyle\eta_{\lambda}$ is even;

• •

$\displaystyle\eta_{\lambda}$ is coercive (hence it is bounded below and it satisfies the C-condition);

• •

$\displaystyle\eta_{\lambda}(0)=0$.

Let $\displaystyle V$ be a finite dimensional subspace of $\displaystyle H^{1}(\Omega)$. So, all norms on $\displaystyle V$ are equivalent. Also, from (119) and hypotheses $\displaystyle H(f)_{5}(i),(iii)=H(f)_{3}(i),(iii)$ we see that we can find $\displaystyle c_{24}>0$ such that

[TABLE]

So, for every $\displaystyle u\in V$ and recalling that all norms are equivalent, we have

[TABLE]

Since $\displaystyle q<2$, we can find $\displaystyle\rho_{\lambda}\in(0,1)$ small such that

[TABLE]

Therefore we can apply Theorem 3 and find $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq K_{\eta_{\lambda}}\subseteq[v^{*}_{\lambda},u^{*}_{\lambda}]\cap C^{1}(\overline{\Omega})$ (see Claim 2 in the proof of Proposition 19) such that

[TABLE]

From the regularity theory of Wang [33] we know that

[TABLE]

Exploiting the compact embedding of $\displaystyle C^{1,\alpha}(\overline{\Omega})$ into $\displaystyle C^{1}(\overline{\Omega})$, from (136) and (137) we infer that

[TABLE]

Moreover, since $\displaystyle u_{n}\in[v^{*}_{\lambda},u^{*}_{\lambda}]$ for all $\displaystyle n\in\mathbb{N}$, $\displaystyle u_{n}\in C^{1}(\overline{\Omega})$ is nodal. ∎

The second result of this section is about problem (1). For this result the hypotheses on the reaction term $\displaystyle f(z,x)$ are the following:

$\displaystyle H(f)_{6}:$ $\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that for almost all $\displaystyle z\in\Omega$ $\displaystyle f(z,x)$ is odd and hypotheses $\displaystyle H(f)_{6}(i),(ii)$ are the same as the corresponding hypotheses $\displaystyle H(f)_{5}(i),(ii)$.

Remark 5**.**

We point out that in the above hypotheses there are no conditions on $\displaystyle f(z,\cdot)$ near zero.

Theorem 24**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)_{6}$ hold, then problem (1) admits an unbounded sequence of solutions $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq C^{1}(\overline{\Omega})$.

Proof.

From Proposition 5 we know that the energy (Euler) functional $\displaystyle\varphi$ satisfies the C-condition and $\displaystyle\varphi(0)=0$.

We consider the following orthogonal direct sum decomposition of $\displaystyle H^{1}(\Omega)$

[TABLE]

with $\displaystyle H_{-}=\overset{m_{-}}{\underset{\mathrm{k=1}}{\oplus}}E(\hat{\lambda}_{k}),\ H_{+}=\overline{{\underset{\mathrm{k\geqslant m_{+}}}{\oplus}}E(\hat{\lambda}_{k})}$ (see Section 2). Then every $\displaystyle u\in H^{1}(\Omega)$ admits a unique sum decomposition of the term

[TABLE]

Hypothesis $\displaystyle H(f)_{6}(i)$ implies that given $\displaystyle\epsilon>0$, we can find $\displaystyle c_{28}=c_{28}(\epsilon)>0$ such that

[TABLE]

Let $\displaystyle u\in H_{+}$. We have

[TABLE]

If $\displaystyle\hat{u}\in H_{+}$ is such that

[TABLE]

then we have

[TABLE]

Also, for $\displaystyle l\geqslant m_{+}$ and $\displaystyle u\in\overline{{\underset{\mathrm{k\geqslant l}}{\oplus}}E(\hat{\lambda}_{k})}$, we have

[TABLE]

Therefore

[TABLE]

So, if $\displaystyle u\in V_{l}$ with $\displaystyle l\geqslant m_{+}$ big and with $\displaystyle||u||=\hat{\rho}$, we have

[TABLE]

Returning to (139), we see that

[TABLE]

Next let $\displaystyle Z\subseteq H^{1}(\Omega)$ be a finite dimensional subspace. From hypotheses $\displaystyle H(f)_{6}(i),(ii)$, we know that given any $\displaystyle\eta>0$, we can find $\displaystyle c_{32}=c_{32}(\eta)>0$ such that

[TABLE]

For $\displaystyle u\in Z$ we have

[TABLE]

(since $\displaystyle Z$ is finite dimensional all norms are equivalent).

But $\displaystyle\eta>0$ is arbitrary. So, we choose $\displaystyle\eta>\frac{c_{33}}{c_{34}}>0$ and from (142) we infer that

[TABLE]

Then (140) and (143) permit the use of Theorem 2 (the symmetric mountain pass theorem). So, we can find $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ such that

[TABLE]

Hence $\displaystyle u_{n}$ is a solution of (1) and $\displaystyle u_{n}\in C^{1}(\overline{\Omega})$ with $\displaystyle||u_{n}||_{C^{1}(\overline{\Omega})}\rightarrow+\infty$. ∎

Acknowledgments. This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025.