Multiple solutions for resonant problems of the Robin $p$-Laplacian plus an indefinite potential

TL;DR

This paper investigates a nonlinear Robin boundary value problem involving the p-Laplacian and an indefinite potential, demonstrating the existence of multiple solutions using advanced variational and topological methods.

Contribution

It establishes the existence of at least three solutions for the problem, including two with fixed signs, under asymptotic resonance conditions.

Findings

At least three nontrivial solutions exist.

Two solutions have fixed signs.

Solutions are obtained via variational, Morse theory, and truncation-perturbation techniques.

Abstract

We study a nonlinear boundary value problem driven by the $p$-Laplacian plus an indefinite potential with Robin boundary condition. The reaction term is a Carath\'eodory function which is asymptotically resonant at $\pm\infty$ with respect to a nonprincipal Ljusternik-Schnirelmann eigenvalue. Using variational methods, together with Morse theory and truncation-perturbation techniques, we show that the problem has at least three nontrivial smooth solutions, two of which have a fixed sign.

Equations

$$\left\{\begin{array}[]{ll}-\Delta_{p}u(z)+\xi(z)|u(z)|^{p-2}u(z)=f(z,u(z))&\mbox{in}\ \Omega,\\ \frac{\partial u}{\partial n_{p}}+\beta(z)|u|^{p-2}u=0&\mbox{on}\ \partial\Omega.\end{array}\right.$$

$${\Delta }_{p}u=\mathrm{div}(|Du{|}^{p-2}Du)\text{mbox}forallu\in W^{1,p}(\Omega ),1<p<\infty .$$

$$\frac{\partial u}{\partial n_p}=|Du{|}^{p-2}(Du,n{)}_{{\mathbb{R}}^N}=|Du{|}^{p-2}\frac{\partial u}{\partial n}\text{mbox}forallu\in W^{1,p}(\Omega ),$$

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

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

$$||u||={\left[||u|{|}_p^p+||Du|{|}_p^p\right]}^{1/p}\text{mbox}forallu\in W^{1,p}(\Omega ).$$

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

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

$$\mathrm{im}{\gamma }_0=W^{\frac{1}{p^{\prime }},p}(\partial \Omega )\left(\frac{1}{p^{\prime }}+\frac{1}{p}=1\right)\text{mbox}and\ker {\gamma }_0=W_0^{1,p}(\Omega ).$$

$$\left\{\begin{array}[]{ll}-\Delta_{p}u(z)+\xi(z)|u(z)|^{p-2}u(z)=\hat{\lambda}|u(z)|^{p-2}u(z)&\mbox{in}\ \Omega,\\ \frac{\partial u}{\partial n_{p}}+\beta(z)|u|^{p-2}u=0&\mbox{on}\ \partial\Omega.\end{array}\right.$$

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

$${\hat{\lambda }}_2=\inf [\hat{\lambda }\in \hat{\sigma }(p):\hat{\lambda }>{\hat{\lambda }}_1].$$

$$K_{\phi }=\{u\in X:{\phi }^{\prime }(u)=0\}(\text{mbox}thecriticalsetof\phi ),$$

$$K_{\phi }^c=\{u\in K_{\phi }:\phi (u)=c\}(\text{mbox}thecriticalsetof\phi \text{mbox}atthelevelc),$$

$${\phi }^c=\{u\in X:\phi (u)⩽c\}(\text{mbox}thesublevelsetof\phi \text{mbox}atc).$$

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

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

$$M(t,u)=\sum _{k⩾0}\mathrm{rank}{C}_k(\phi ,u)\text{mbox}forallt\in \mathbb{R},\text{mbox}allu\in K_{\phi },$$

$$P(t,\infty )=\sum _{k⩾0}\mathrm{rank}{C}_k(\phi ,\infty )\text{mbox}forallt\in \mathbb{R}.$$

$$\sum _{\mathrm{u}\in {\mathrm{K}}_{\phi }}M(t,u)=P(t,\infty )+(1+t)Q(t)\text{mbox}forallt\in \mathbb{R},$$

$$C_k(\phi ,u)={\delta }_{k,0}\mathbb{Z}\text{mbox}forallk\in \mathbb{N},$$

$$\displaystyle\delta_{k,\vartheta}=\left\{\begin{array}[]{ll}1&\mbox{if}\ k=\vartheta\\ 0&\mbox{if}\ k\neq\vartheta\end{array}\right.\ \mbox{for all}\ k,\vartheta\in\mathbb{N}_{0}.$$

$$⟨A(u),h⟩={\int }_{\Omega }|Du{|}^{p-2}(Du,Dh{)}_{{\mathbb{R}}^N}dz\text{mbox}forallu,h\in W^{1,p}(\Omega ).$$

$$‘‘u_n\stackrel{w}{\to }u\text{mbox}in{W}^{1,p}(\Omega ),\underset{n\to \infty }{\mathrm{limsup}}⟨A(u_n),u_n-u⟩⩽0\Rightarrow u_n\to u\text{mbox}in{W}^{1,p}(\Omega )".$$

$$\gamma (u)-{\int }_{\Omega }\vartheta (z)|u{|}^{p}dz⩾c_0||u|{|}^p\text{mbox}forallu\in W^{1,p}(\Omega ).$$

$$u^{\pm }(\cdot )=u(\cdot {)}^{\pm }.$$

$$N_g(u)(\cdot )=g(\cdot ,u(\cdot ))\text{mbox}forallu\in W^{1,p}(\Omega ),$$

$$|f(z,x)|⩽a_{\rho }(z)\text{mbox}foralmostallz\in \Omega ,\text{mbox}all|x|⩽\rho ;$$

$$\underset{x\to \pm \infty }{\lim }\frac{f(z,x)}{|x{|}^{p-2}x}={\hat{\lambda }}_m\text{mbox}uniformlyforalmostallz\in \Omega ;$$

$$\underset{x\to \pm \infty }{\lim }[f(z,x)x-pF(z,x)]=+\infty \text{mbox}uniformlyforalmostallz\in \Omega ;\text{mbox}and$$

$$\vartheta (z)⩽{\hat{\lambda }}_1\text{mbox}foralmostallz\in \Omega ,\vartheta \not\equiv {\hat{\lambda }}_1,$$

$$|f(z,x)|⩽c_1(1+|x{|}^{p-1})\text{mbox}foralmostallz\in \Omega ,\text{mbox}allx\in \mathbb{R},\text{mbox}some{c}_1>0.$$

Full text

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

1. Introduction

Let $\displaystyle\Omega\subseteq\mathbb{R}^{N}$ be a bounded domain with a $\displaystyle C^{2}$-boundary $\displaystyle\partial\Omega$. In this paper we study the following nonlinear Robin boundary value problem

[TABLE]

Here $\displaystyle\Delta_{p}$ denotes the $\displaystyle p$-Laplacian differential operator defined by

[TABLE]

Also, $\displaystyle\xi\in L^{\infty}(\Omega)$ is an indefinite (that is, sign changing) potential function. The reaction term $\displaystyle f(z,x)$ is a Carathéodory function (that is, for all $\displaystyle x\in\mathbb{R}$ the mapping $\displaystyle z\mapsto f(z,x)$ is measurable and for almost all $\displaystyle z\in\Omega$ the function $\displaystyle x\mapsto f(z,x)$ is continuous) which is $\displaystyle p$-sublinear in $\displaystyle x\in\mathbb{R}$ and asymptotically interacts as $\displaystyle x\rightarrow\pm\infty$ with the nonprincipal part of the spectrum of $\displaystyle W^{1,p}(\Omega)\ni u\mapsto-\Delta_{p}u+\xi(z)|u|^{p-2}u$ with Robin boundary condition. In the boundary condition, $\displaystyle\frac{\partial u}{\partial n_{p}}$ denotes the generalized normal derivative on $\displaystyle\partial\Omega$, defined by

[TABLE]

with $\displaystyle n(\cdot)$ being the outward unit normal on $\displaystyle\partial\Omega$. The boundary coefficient function $\displaystyle\beta(\cdot)$ satisfies $\displaystyle\beta\in C^{0,\alpha}(\partial\Omega)$ with $\displaystyle 0<\alpha<1$ and $\displaystyle\beta(z)\geqslant 0$ for all $\displaystyle z\in\partial\Omega$. When $\displaystyle\beta\equiv 0$, we get the Neumann problem. In this resonant setting, we prove a multiplicity theorem, producing three nontrivial smooth solutions.

Previous three solutions theorems for equations driven by the $\displaystyle p$-Laplacian with zero potential function (that is, $\displaystyle\xi\equiv 0$), were proved by Gasinski & Papageorgiou [9], Guo & Liu [10], Liu & Liu [13], Motreanu, Motreanu & Papageorgiou [14], Papageorgiou & Papageorgiou [18], Papageorgiou & Rădulescu [19, 21] (Dirichlet problems) and Gasinski & Papageorgiou [8], Motreanu, Motreanu & Papageorgiou [15] (Neumann problems). Recently, Mugnai & Papageorgiou [17] considered Neumann problems driven by the $\displaystyle p$-Laplacian plus an indefinite potential, while Papageorgiou & Rădulescu [20, 22] studied Robin problems driven by the $\displaystyle p$-Laplacian with no potential function. Of the aforementioned works, resonant problems are treated only in [13], [14], [17], and [20]. In all these works resonance occurs with respect to the principal eigenvalue. The reason for this is that due to the nonlinearity of the differential operator, the eigenspaces are not linear spaces and consequently the underlying Sobolev space does not have a direct sum decomposition in terms of the eigenspaces. In addition, we have only partial knowledge of the spectrum of the operator. These drawbacks make the study of problems which are resonant with respect to the nonprincipal part of the spectrum, rather difficult and require different tools to prove existence and multiplicity theorems. Our approach uses variational methods based on the critical point theory for linking sets over cones (since the eigenspaces are cones), together with Morse theory (critical groups). With these tools we prove the existence of at least three nontrivial smooth solutions.

2. Mathematical Background

In this section, for the convenience of the reader, we briefly recall the main mathematical tools which will be used in the sequel.

So, let $\displaystyle X$ be a Banach space with $\displaystyle X^{*}$ 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”.

Since the underlying Banach space is in general not locally compact (in most applications $\displaystyle X$ is infinite dimensional), the necessary compactness condition needed to have a coherent theory is passed to the functional $\displaystyle\varphi$ by introducing the aforementioned C-condition. This is analogous to what happens in infinite dimensional degree theory (the Leray-Schauder degree theory). The C-condition leads to a deformation theorem describing the changes in the topological structure of the sublevel sets $\displaystyle\varphi$. From this, one can derive the minimax theory of the critical values of $\displaystyle\varphi$. Prominent in that theory, is the so-called “mountain pass theorem” due to Ambrosetti and Rabinowitz [2].

Theorem 1**.**

Suppose that $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ satisfies the C-condition, $\displaystyle u_{0},u_{1}\in X$, $\displaystyle||u_{1}-u_{0}||>\rho>0$

[TABLE]

and $\displaystyle c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leqslant t\leqslant 1}\varphi(\gamma(t))$, where $\displaystyle\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_{0},\gamma(1)=u_{1}\}$. Then $\displaystyle c\geqslant m_{\rho}$ and $\displaystyle c$ is a critical value of $\displaystyle\varphi$.

For our problem, the underlying space is the Sobolev space $\displaystyle W^{1,p}(\Omega)$ with the norm

[TABLE]

In addition, we will also use the ordered Banach space $\displaystyle C^{1}(\overline{\Omega})$, with positive (order) cone given by

[TABLE]

In $\displaystyle C_{+}$ we consider the nonempty open set $\displaystyle D_{+}$ given by

[TABLE]

To deal with the Robin boundary condition, we will need the “boundary” Lebesgue spaces $\displaystyle L^{q}(\partial\Omega)\ (1\leqslant q\leqslant\infty)$. So, on $\displaystyle\partial\Omega$ we consider the $\displaystyle(N-1)$-dimensional Hausdorff (surface) measure denoted by $\displaystyle\sigma(\cdot)$. Using this measure, we can define the Lebesgue spaces $\displaystyle L^{q}(\partial\Omega)$. From the theory of Sobolev spaces, we know there exists a unique linear continuous map $\displaystyle\gamma_{0}:W^{1,p}(\Omega)\rightarrow L^{q}(\partial\Omega)$, known as the trace map, such that $\displaystyle\gamma_{0}(u)=u|_{\partial\Omega}$ for all $\displaystyle u\in W^{1,p}(\Omega)\cap C(\overline{\Omega})$. So, we can understand the trace map as representing the boundary values of $\displaystyle u\in W^{1,p}(\Omega)$. We know that $\displaystyle\gamma_{0}$ is a compact mapping into $\displaystyle L^{q}(\partial\Omega)$ for all $\displaystyle q\in\left[1,\frac{p(N-1)}{N-p}\right)$ and

[TABLE]

In the sequel, for the sake of notational simplicity, we shall drop the use of the map $\displaystyle\gamma_{0}$. It is understood that all restrictions of the Sobolev functions $\displaystyle u\in W^{1,p}(\Omega)$ on $\displaystyle\partial\Omega$, are defined in the sense of traces.

Next, we recall some basic facts about the spectrum of the differential operator $\displaystyle u\mapsto-\Delta_{p}u+\xi(z)|u|^{p-2}u$ with Robin boundary condition. For details we refer to Mugnai & Papageorgiou [17] and Papageorgiou & Rădulescu [20].

We consider the following nonlinear eigenvalue problem:

[TABLE]

We say that $\displaystyle\hat{\lambda}\in\mathbb{R}$ is an eigenvalue of the differential operator, if problem (2) admits a nontrivial solution $\displaystyle\hat{u}\in W^{1,p}(\Omega)$, known as an eigenfunction corresponding to the eigenvalue $\displaystyle\hat{\lambda}$. We know that there exists a smallest eigenvalue $\displaystyle\hat{\lambda}_{1}\in\mathbb{R}$ which has the following properties:

• •

$\displaystyle\hat{\lambda}_{1}$ is isolated in the spectrum $\displaystyle\hat{\sigma}(p)$ of the differential operator;

• •

$\displaystyle\hat{\lambda}_{1}$ is simple (that is, if $\displaystyle\hat{u}_{1},\hat{u}_{2}$ are eigenfunctions corresponding to $\displaystyle\hat{\lambda}_{1}$, then $\displaystyle\hat{u}_{1}=\xi\hat{u}_{2}$ for some $\displaystyle\xi\neq 0$); and

• •

$\displaystyle\hat{\lambda}_{1}=\inf\left[\frac{\gamma(u)}{||u||^{p}_{p}}:u\in W^{1,p}(\Omega),u\neq 0\right]$, where $\displaystyle\gamma:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ is the $\displaystyle C^{1}$-functional defined by

[TABLE]

The infimum in (3) is realized on the corresponding one dimensional eigenspace. Evidently, the elements of this eigenspace do not change sign. In what follows, by $\displaystyle\hat{u}_{1}$ we denote the positive $\displaystyle L^{p}$-normalized (that is, $\displaystyle||\hat{u}_{1}||_{p}=1$) eigenfunction corresponding to the eigenvalue $\displaystyle\hat{\lambda}_{1}$. The nonlinear regularity theory (see Lieberman [12, Theorem 2]), implies that $\displaystyle\hat{u}_{1}\in C_{+}\backslash\{0\}$. Moreover, using the nonlinear strong maximum principle (see, for example, Gasinski & Papageorgiou [7, p. 738]), we show that $\displaystyle\hat{u}_{1}\in D_{+}$.

It is easy to show that the spectrum $\displaystyle\hat{\sigma}(p)$ is closed and so given the isolation of $\displaystyle\hat{\lambda}_{1}$, the second eigenvalue of (2) is well-defined by

[TABLE]

Using the Ljusternik-Schnirelmann minimax scheme with the Fadell-Rabinowitz cohomological index ind$\displaystyle\,(\cdot)$ (see [6]), we produce a whole sequence $\displaystyle\{\hat{\lambda}_{k}\}_{k\geqslant 1}$ of distinct eigenvalues such that $\displaystyle\hat{\lambda}_{k}\rightarrow+\infty$ (see Cingolani & Degiovanni [4]). However, we do not know if this sequence exhausts $\displaystyle\hat{\sigma}(p)$. This is the case if $\displaystyle p=2$ (linear eigenvalue problem) or $\displaystyle N=1$ (ordinary differential equation). All eigenvalues $\displaystyle\hat{\lambda}\neq\hat{\lambda}_{1}$ have nodal (that is, sign changing) eigenfunctions.

As we already mentioned in the introduction, our approach involves also the use of critical groups (Morse theory). So, let us briefly recall some basic definitions and facts from that theory.

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 every $\displaystyle k\in\mathbb{N}_{0}$, by $\displaystyle H_{k}(Y_{1},Y_{2})$ we denote the $\displaystyle k$th-relative singular homology group with integer coefficients for the pair $\displaystyle(Y_{1},Y_{2})$. Recall that $\displaystyle H_{k}(Y_{1},Y_{2})=0$ for all $\displaystyle k\in-\mathbb{N}$. If $\displaystyle u\in K^{c}_{\varphi}$ is isolated, then the critical groups of $\displaystyle\varphi$ at $\displaystyle u$ are defined by

[TABLE]

Here, $\displaystyle U$ is a neighborhood of $\displaystyle u$ such that $\displaystyle K_{\varphi}\cap\varphi^{c}\cap U=\{u\}$. The excision property of the singular homology implies that the above definition of critical groups is independent of the particular choice of the neighborhood $\displaystyle U$.

If $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ satisfies the C-condition and $\displaystyle\inf\varphi(K_{\varphi})>-\infty$, then the critical groups of $\displaystyle\varphi$ at infinity are defined by

[TABLE]

with $\displaystyle c<\inf\varphi(K_{\varphi})$. The second deformation theorem (see Gasinski & Papageorgiou [7, p. 628]), implies that this definition is independent of the choice of the level $\displaystyle c<\inf\varphi(K_{\varphi})$.

Assuming that $\displaystyle K_{\varphi}$ is infinite, we define

[TABLE]

The Morse relation says that

[TABLE]

where $\displaystyle Q(t)={\underset{\mathrm{k\leqslant 0}}{\sum}}\beta_{k}t^{k}$ is a formal series in $\displaystyle t\in\mathbb{R}$ with nonnegative integer coefficients $\displaystyle\beta_{k}$.

From (4) it follows easily that, if for some $\displaystyle m\in\mathbb{N}$ we have $\displaystyle C_{m}(\varphi,\infty)\neq 0$, then there exists $\displaystyle u\in K_{\varphi}$ such that $\displaystyle C_{m}(\varphi,u)\neq 0$. Moreover, if $\displaystyle u\in X$ is a local minimizer of $\displaystyle\varphi$, then

[TABLE]

with $\displaystyle\delta_{k,0}$ being the Kronecker symbol, that is,

[TABLE]

Let $\displaystyle A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^{*}$ be the nonlinear map defined by

[TABLE]

From Motreanu, Motreanu & Papageorgiou [16, p. 40], we have the following property.

Proposition 2**.**

The map $\displaystyle A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^{*}$ is bounded (maps bounded sets into bounded sets), demicontinuous, maximal monotone and of type $\displaystyle(S)_{+}$, that is,

[TABLE]

Also, as a consequence of the properties of $\displaystyle\hat{\lambda}_{1}$ and $\displaystyle\hat{u}_{1}$, we have (see Papageorgiou & Rădulescu [20]).

Proposition 3**.**

If $\displaystyle\vartheta\in L^{\infty}(\Omega),\ \vartheta(z)\leqslant\hat{\lambda}_{1}$ for almost everywhere $\displaystyle z$ in $\displaystyle\Omega$, $\displaystyle\vartheta\not\equiv\hat{\lambda}_{1}$, then there exists $\displaystyle c_{0}>0$ such that

[TABLE]

We now fix our notation. Given $\displaystyle x\in\mathbb{R}$, we set $\displaystyle x^{\pm}=\max\{\pm x,0\}$. Then for $\displaystyle u\in W^{1,p}(\Omega)$ we define

[TABLE]

We have $\displaystyle u^{\pm}\in W^{1,p}(\Omega),\ u=u^{+}-u^{-},\ |u|=u^{+}+u^{-}$. Given a measurable function $\displaystyle g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ (for example, a Carathéodory function), we define

[TABLE]

which is the Nemytskii (superposition) operator corresponding to $\displaystyle g$.

Finally, we introduce the hypotheses on the data of problem (1):

$\displaystyle H(\xi):\ \xi\in L^{\infty}(\Omega)$.

$\displaystyle H(\beta):\ \beta\in C^{0,\alpha}(\partial\Omega)$ with $\displaystyle\alpha\in(0,1)$ and $\displaystyle\beta(z)\geqslant 0$ for all $\displaystyle z\in\partial\Omega$.

Remark 1**.**

If $\displaystyle\beta\equiv 0$, then we obtain the Neumann problem. So, our framework here includes the Neumann problem as a special case.

$\displaystyle H(f):\ f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that $\displaystyle f(z,0)=0$ for almost all $\displaystyle z\in\Omega$ and

• (i)

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

[TABLE]

• (ii)

there exists an integer $\displaystyle m\geqslant 2$ such that

[TABLE]

• (iii)

if $\displaystyle F(z,x)=\int^{x}_{0}f(z,s)ds$, then

[TABLE]

• (iv)

there exist functions $\displaystyle\hat{\vartheta},\vartheta\in L^{\infty}(\Omega)$ such that

[TABLE]

$\displaystyle\hat{\vartheta}(z)\leqslant\liminf\limits_{x\rightarrow 0}\frac{f(z,x)}{|x|^{p-2}x}\leqslant\limsup\limits_{x\rightarrow 0}\frac{f(z,x)}{|x|^{p-2}x}\leqslant\vartheta(z)$ uniformly for almost all $\displaystyle z\in\Omega$.

Remark 2**.**

Hypothesis $\displaystyle H(ii)$ implies that asymptotically at $\displaystyle\pm\infty$, we have resonance with respect to any nonprincipal Ljusternik-Schnirelmann eigenvalue of the differential operator. Hypotheses $\displaystyle H(f)(i),(ii)$ imply that

[TABLE]

Hypothesis $\displaystyle H(f)(iv)$ says that we have nonuniform nonresonance at zero with respect to the principal eigenvalue $\displaystyle\hat{\lambda}_{1}$ from the left.

3. Constant Sign Solutions

In this section, using minimax methods, we establish the existence of at least two nontrivial smooth solutions of constant sign (one positive and one negative).

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

[TABLE]

Evidently, $\displaystyle\varphi\in C^{1}(W^{1,p}(\Omega))$. Let $\displaystyle\mu>||\xi||_{\infty}$ (see hypothesis $\displaystyle H(\xi)$) and consider the following truncations-perturbations of the reaction term $\displaystyle f(z,\cdot)$:

[TABLE]

Both are Carathéodory functions. We define

[TABLE]

and consider the $\displaystyle C^{1}$-functionals $\displaystyle\hat{\varphi}_{\pm}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Proposition 4**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)(i),(ii),(iii)$ hold, then the functionals $\displaystyle\hat{\varphi}_{\pm}$ satisfy the C-condition.

Proof.

We shall present the proof for the functional $\displaystyle\hat{\varphi}_{+}$, the proof for $\displaystyle\hat{\varphi}_{-}$ is similar.

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

[TABLE]

From (13) we have

[TABLE]

In (3) we choose $\displaystyle h=-u_{n}^{-}\in W^{1,p}(\Omega)$. Then

[TABLE]

From (3), (15) and (8), we have

[TABLE]

Claim 1**.**

$\displaystyle\{u^{+}_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$* is bounded.*

We argue indirectly. So, suppose that the claim is not true. By passing to a subsequence if necessary, we may assume that $\displaystyle||u^{+}_{n}||\rightarrow\infty$. We set $\displaystyle y_{n}=\frac{u^{+}_{n}}{||u^{+}_{n}||}$, $\displaystyle n\in\mathbb{N}$. Then

[TABLE]

So, we may assume that

[TABLE]

From (16) we have

[TABLE]

From (5) it is clear that

[TABLE]

In (18) we choose $\displaystyle h=y_{n}-y\in W^{1,p}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (17) and (19). Then

[TABLE]

Because (19) and hypothesis $\displaystyle H(f)(ii)$, at least for a subsequence we have

[TABLE]

(see Aizicovici, Papageorgiou & Staicu [1], proof of Proposition 16). Therefore, if in (18) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (20), (21), then

[TABLE]

From (3) and since $\displaystyle y\neq 0$ (see (20)) and $\displaystyle m\geqslant 2$ (see hypothesis $\displaystyle H(f)(ii)$), it follows that $\displaystyle y$ must be nodal (that is, sign changing), which contradicts (20). This proves the claim.

From (15) and the claim it follows that $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ is bounded and so we may assume that

[TABLE]

Evidently,

[TABLE]

(see (5), (8), (23)). So, if in (3) we choose $\displaystyle h=u_{n}-u\in W^{1,p}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (23), (24), then

[TABLE]

Similarly for the functional $\displaystyle\hat{\varphi}_{-}$, using this time (11). ∎

Proposition 5**.**

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

Proof.

Let $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ be a sequence such that

[TABLE]

From (26) we have

[TABLE]

In (3) we choose $\displaystyle h=u_{n}\in W^{1,p}(\Omega)$ and obtain

[TABLE]

On the other hand, from (25) we have

[TABLE]

We add (28) and (29). Then

[TABLE]

Claim 2**.**

$\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$* is bounded.*

We argue again by contradiction. So, suppose that $\displaystyle||u_{n}||\rightarrow\infty$. We set $\displaystyle y_{n}=\frac{u_{n}}{||u_{n}||}$ for all $\displaystyle n\in\mathbb{N}$. Then $\displaystyle||y_{n}||=1$ and so we may assume that

[TABLE]

From (3) we have

[TABLE]

From (5) it is clear that

[TABLE]

So, if in (3) we choose $\displaystyle h=y_{n}-y\in W^{1,p}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (31), (33), then

[TABLE]

Since $\displaystyle y\neq 0$ (see (34)), for $\displaystyle D=\{z\in\Omega:y(z)\neq 0\}$ we have $\displaystyle|D|_{N}>0$, with $\displaystyle|\cdot|_{N}$ being the Lebesgue measure on $\displaystyle\mathbb{R}^{N}$ and

[TABLE]

Then hypothesis $\displaystyle H(f)(iii)$, Fatou’s lemma and (35) imply that

[TABLE]

Also, hypotheses $\displaystyle H(f)(i),(iii)$ imply that we can find $\displaystyle c_{3}>0$ such that

[TABLE]

Then

[TABLE]

Comparing (30) and (3), we reach a contradiction. This proves the claim.

Because of the claim, we may assume that

[TABLE]

From (5) and (39) we see that

[TABLE]

So, if in (3) we choose $\displaystyle h=u_{n}-u\in W^{1,p}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (39), (40), then

[TABLE]

∎

Proposition 6**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)(iv)$ and (5) hold, then $\displaystyle u=0$ is a local minimizer of the functionals $\displaystyle\hat{\varphi}_{\pm}$ and $\displaystyle\varphi$.

Proof.

We give the proof for the functional $\displaystyle\hat{\varphi}_{+}$, the proofs for $\displaystyle\hat{\varphi}_{-}$ and $\displaystyle\varphi$ are similar.

Hypothesis $\displaystyle H(f)(iv)$ implies that given $\displaystyle\epsilon>0$, we can find $\displaystyle\delta=\delta(\epsilon)>0$ such that

[TABLE]

If $\displaystyle r>p$, then because of (5) we can find $\displaystyle c_{4}=c_{4}(\epsilon,r)>0$ such that

[TABLE]

From (41) and (42) it follows that we can find $\displaystyle c_{5}>0$ such that

[TABLE]

(recall that $\displaystyle\vartheta\in L^{\infty}(\Omega)$, see hypothesis $\displaystyle H(f)(iv)$).

So, for all $\displaystyle u\in W^{1,p}(\Omega)$, we have

[TABLE]

Choosing $\displaystyle\epsilon\in(0,c_{0})$, from (44) we have

[TABLE]

Since $\displaystyle r>p$, from (45) we see that we can find $\displaystyle\rho\in(0,1)$ small such that

[TABLE]

Similarly for the functionals $\displaystyle\hat{\varphi}_{-}$ and $\displaystyle\varphi$. ∎

We are now ready to produce two nontrivial smooth solutions of constant sign.

Proposition 7**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)$ hold, then problem (1) has at least two nontrivial constant sign smooth solutions

[TABLE]

Proof.

Using (8) and (11) and the nonlinear regularity theory (see Lieberman [12, Theorem 2]), we can easily check that

[TABLE]

So, we may assume that $\displaystyle u=0$ is an isolated critical point of $\displaystyle\hat{\varphi}_{\pm}$ or otherwise we already have whole sequence of distinct smooth positive and negative solutions of (1) which as we will see in the last part of this proof, using the nonlinear strong maximum principle (see, for example, Gasinski & Papageorgiou [7, p. 738]), belong in $\displaystyle D_{+}$ and in $\displaystyle-D_{+}$, respectively. Thus we are done.

Because $\displaystyle u=0$ is an isolated critical point of $\displaystyle\hat{\varphi}_{+}$ and a local minimizer of $\displaystyle\hat{\varphi}_{+}$ (see Proposition 6), we can find $\displaystyle\rho\in(0,1)$ small such that

[TABLE]

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

Hypothesis $\displaystyle H(f)(ii)$ implies that

[TABLE]

From (47), (48) and Proposition 4 we see that we can apply Theorem 1 (the mountain pass theorem) and find $\displaystyle u_{0}\in W^{1,p}(\Omega)$ such that

[TABLE]

From Papageorgiou & Rădulescu [22] we know that $\displaystyle u_{0}\in L^{\infty}(\Omega)$. Let $\displaystyle\rho_{0}=||u_{0}||_{\infty}$. Hypotheses $\displaystyle H(f)(i),(iv)$ imply that we can find $\displaystyle\hat{\xi}_{\rho_{0}}>0$ such that

[TABLE]

So, we have (see Papageorgiou & Rădulescu [20])

[TABLE]

In a similar fashion, working this time with the functional $\displaystyle\hat{\varphi}_{-}$, we produce a negative smooth solution $\displaystyle v_{0}\in-D_{+}$ for problem (1). ∎

4. Three Solutions Theorem

In this section, using Morse theoretic tools (critical groups), we establish the existence of a third nontrivial smooth and thus prove the three solutions theorem for problem (1) under conditions of resonance.

We start by examining the critical groups of $\displaystyle\varphi$ at infinity.

Proposition 8**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)$ hold and $\displaystyle K_{\varphi}$ is finite, then $\displaystyle C_{m}(\varphi,\infty)\neq 0$.

Proof.

Let $\displaystyle\lambda\in(\hat{\lambda}_{m},\hat{\lambda}_{m+1})\backslash\hat{\sigma}(p)$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\psi:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Consider the homotopy $\displaystyle h(t,u)=h_{t}(u)$ defined by

[TABLE]

Claim 3**.**

There exist $\displaystyle\eta\in\mathbb{R}$ and $\displaystyle\delta_{0}>0$ such that

[TABLE]

We argue indirectly. So, suppose that the claim is not true. Because the function $\displaystyle(t,u)\mapsto h_{t}(u)$ maps bounded sets into bounded sets, we can find $\displaystyle\{t_{n}\}_{n\geqslant 1}\subseteq[0,1]$ and $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ such that

[TABLE]

From the last convergence in (51) we have

[TABLE]

We set $\displaystyle y_{n}=\frac{u_{n}}{||u_{n}||}\ n\in\mathbb{N}$. Then $\displaystyle||y_{n}||=1$ for all $\displaystyle n\in\mathbb{N}$ and so by passing to a suitable subsequence if necessary, we may assume that

[TABLE]

From (4) we have

[TABLE]

From (5) we see that

[TABLE]

From hypothesis $\displaystyle H(f)(ii)$, (55) and by passing to a subsequence if necessary, we have

[TABLE]

(see Aizicovici, Papageorgiou & Staicu [1], proof of Proposition 30).

In (4) we choose $\displaystyle h=y_{n}-y\in W^{1,p}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (53), (56). We obtain

[TABLE]

Therefore, if in (4) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (56) and (57), then

[TABLE]

with $\displaystyle\lambda_{t}=(1-t)\hat{\lambda}_{m}+t\lambda$ (see Papageorgiou & Rădulescu [20]). Note that $\displaystyle\lambda_{t}\in\left[\hat{\lambda}_{m},\hat{\lambda}_{m+1}\right)$.

If $\displaystyle\lambda_{t}\notin\hat{\sigma}(p)$, then from (4) we infer that $\displaystyle y=0$, a contradiction to (57).

If $\displaystyle\lambda_{t}\in\hat{\sigma}(p)$, then from (57) we see that for $\displaystyle D=\{z\in\Omega:y(z)\neq 0\}$ we have $\displaystyle|D|_{N}>0$. Also, we can say that

[TABLE]

Hypotheses $\displaystyle H(f)(i),(ii)$ imply that we can find $\displaystyle c_{9}>0$ such that

[TABLE]

We have

[TABLE]

On the other hand, from the third convergence in (51), we see that we can find $\displaystyle n_{0}\in\mathbb{N}$ such that

[TABLE]

Also, if in (4) we choose $\displaystyle h=u_{n}\in W^{1,p}(\Omega)$, then

[TABLE]

By taking $\displaystyle n_{0}\in\mathbb{N}$ even bigger if necessary, we may assume that $\displaystyle\epsilon_{n}\in(0,1)$ for all $\displaystyle n\geqslant n_{0}$ (recall that $\displaystyle\epsilon_{n}\rightarrow 0^{+}$). Adding (62) and (63), we obtain

[TABLE]

Note that we can assume that $\displaystyle t_{n}\in\left[0,1\right)$ for all $\displaystyle n\geqslant n_{0}$. Indeed, if there is a subsequence $\displaystyle\{t_{n_{k}}\}_{k\geqslant 1}$ of $\displaystyle\{t_{n}\}_{n\geqslant 1}$ such that $\displaystyle t_{n_{k}}=1$ for all $\displaystyle k\in\mathbb{N}$, then $\displaystyle t=1$ (see (51)) and from the previous argument we have

[TABLE]

which contradicts (57).

So, $\displaystyle t_{n}\in\left[0,1\right)$ for all $\displaystyle n\geqslant n_{0}$ and from (64) we have

[TABLE]

Comparing (61) and (65) we reach a contradiction. This proves the claim.

A careful reading of the proof of the claim, reveals that the same argument with minor changes, shows that for every $\displaystyle t\in[0,1],h_{t}(\cdot)$ satisfies the C-condition. So, we can apply Theorem 5.1.21 of Chang [3, p. 334] (with $\displaystyle a=\eta,b\equiv+\infty$, see also Liang & Su [11, Proposition 3.2]) and infer that

[TABLE]

We introduce the following two sets

[TABLE]

Both are symmetric sets and $\displaystyle C_{r}\cap E=\emptyset$. The set $\displaystyle\partial B_{r}=\{u\in W^{1,p}(\Omega):||u||=r\}$ is a Banach $\displaystyle C^{1}$-manifold and so it is locally contractible. The set $\displaystyle C_{r}$ is an open subset of $\displaystyle\partial B_{r}$, so it is locally contractible, too. Evidently, the open set $\displaystyle W^{1,p}(\Omega)\backslash E$ is locally contractible. Since $\displaystyle\lambda\in(\hat{\lambda}_{m},\hat{\lambda}_{m+1})\backslash\hat{\sigma}(p)$, we have

[TABLE]

and recall that ind$\displaystyle\,(\cdot)$ denotes the Fadell-Rabinowitz cohomological index, see [6]. Moreover, from Theorem 3.6 of Cingolani & Degiovanni [4], we know that the sets $\displaystyle C_{r}$ and $\displaystyle E$ homologically link in dimension $\displaystyle m$. So, we can apply Theorem 3.2 of Cingolani & Degiovanni [4] and infer that

[TABLE]

But since $\displaystyle\lambda\in(\hat{\lambda}_{m},\hat{\lambda}_{m+1})\backslash\hat{\sigma}(p)$, we have

[TABLE]

∎

In what follows we outline an alternative approach to showing that $\displaystyle C_{m}(\varphi,\infty)\neq 0$.

Note that the $\displaystyle p$-homogeneity of $\displaystyle\psi$ implies that

[TABLE]

(just use the radial contraction). In a similar way, we can see that

[TABLE]

Here, $\displaystyle\partial B^{L^{p}}_{1}=\{u\in L^{p}(\Omega):||u||_{p}=1\}$. Let $\displaystyle\ast\in\psi^{0}\backslash\{0\}$ and consider the following triple of sets

[TABLE]

For this triple we consider the corresponding “reduced” long exact sequence of singular homology groups (see Motreanu, Motreanu & Papageorgiou [16], Proposition 6.21, p. 146).

[TABLE]

where $\displaystyle j_{*}$ is the homomorphism induced by the inclusion $\displaystyle(\psi^{0},\ast){\overset{j}{\hookrightarrow}}(\psi^{0},\psi^{0}\backslash\{0\})$ and $\displaystyle\hat{\partial}_{*}$ is the composed boundary homomorphism (see Motreanu, Motreanu & Papageorgiou [16], Proposition 6.14, p. 143). From (68) we have

[TABLE]

(see Motreanu, Motreanu & Papageorgiou [16], Proposition 6.24, p. 147). Also, from the exactness of (70) we have

[TABLE]

Here, $\displaystyle\tilde{H}_{k-1}(\psi^{0}\cap\partial B^{L^{p}}_{1})$ denotes the reduced homology group.

Since $\displaystyle\psi^{0}\cap\partial B^{L^{p}}_{1}=\{u\in W^{1,p}(\Omega):\gamma(u)\leqslant\lambda||u||^{p}_{p},||u||_{p}=1\}$ and $\displaystyle\lambda\in(\hat{\lambda}_{m},\hat{\lambda}_{m+1})\backslash\hat{\sigma}(p)$ as in Perera [24], we can show that

[TABLE]

We also compute the critical groups at infinity for the functionals $\displaystyle\hat{\varphi}_{\pm}$.

Proposition 9**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H$ hold and $\displaystyle K_{\hat{\varphi}_{\pm}}$ are finite, then $\displaystyle C_{k}(\hat{\varphi}_{\pm},\infty)=0$ for all $\displaystyle k\in\mathbb{N}_{0}$.

Proof.

We give the proof for the functional $\displaystyle\hat{\varphi}_{+}$, the proof for $\displaystyle\hat{\varphi}_{-}$ is similar.

As before, let $\displaystyle\lambda\in(\hat{\lambda}_{m},\hat{\lambda}_{m+1})\backslash\hat{\sigma}(p)$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\hat{\psi}_{+}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

with $\displaystyle\mu>||\xi||_{\infty}$ (see hypothesis $\displaystyle H(\xi)$). We introduce the homotopy $\displaystyle\hat{h}^{+}(t,u)=\hat{h}^{+}_{t}(u)$ defined by

[TABLE]

Claim 4**.**

There exist $\displaystyle\hat{\eta}\in\mathbb{R}$ and $\displaystyle\hat{\delta}_{0}>0$ such that

[TABLE]

We again argue by contradiction. So, suppose that the claim is not true. Then since $\displaystyle(t,u)\mapsto\hat{h}^{+}_{t}(u)$ maps bounded sets to bounded sets, we can find sequences $\displaystyle\{t_{n}\}_{n\geqslant 1}\subseteq[0,1]$ and $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ such that

[TABLE]

From the last convergence established in (73), we have

[TABLE]

In (4) we choose $\displaystyle h=-u^{-}_{n}\in W^{1,p}(\Omega)$. Then

[TABLE]

From (73) we know that $\displaystyle||u_{n}||\rightarrow\infty$. Because of (75) it follows that

[TABLE]

Let $\displaystyle y_{n}=\frac{u^{+}_{n}}{||u^{+}_{n}||}$, $\displaystyle n\in\mathbb{N}$. Then $\displaystyle||y_{n}||=1$ for all $\displaystyle n\in\mathbb{N}$. So, by passing to a subsequence if necessary, we may assume that

[TABLE]

From (4) and (75) we have

[TABLE]

It is clear from (5) that

[TABLE]

Using (78) and hypothesis $\displaystyle H(f)(ii)$, for at least a subsequence, we have

[TABLE]

In (4) we choose $\displaystyle h=y_{n}-y\in W^{1,p}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (76) and (79). Then

[TABLE]

Passing to the limit as $\displaystyle n\rightarrow\infty$ in (4) and using (79) and (80), we obtain

[TABLE]

with $\displaystyle\lambda_{t}=(1-t)\hat{\lambda}_{m}+t\lambda$.

We see that $\displaystyle\lambda_{t}\in\left[\hat{\lambda}_{m},\hat{\lambda}_{m+1}\right)$ and since $\displaystyle m\geqslant 2$, if $\displaystyle\lambda_{t}\in\hat{\sigma}(p)$, then it must be a nonprincipal eigenvalue and so from (81) it follows that

[TABLE]

both contradicting (80). If $\displaystyle\lambda_{t}\notin\hat{\sigma}(p)$, then $\displaystyle y=0$ (see (81)), again a contradiction.

This proves Claim 4.

Claim 4 says that we can find $\displaystyle\hat{\eta}_{0}\leqslant\hat{\eta}$ such that

[TABLE]

The above argument with minor changes also shows that

for all $\displaystyle t\in[0,1],\ \hat{h}^{+}_{t}$ satisfies the C-condition.

We apply Theorem 5.1.21 of Chang [3, p. 334] (with $\displaystyle a\equiv\hat{\eta}_{0}$ and $\displaystyle b\equiv+\infty$; see also Liang & Su [11, Proposition 3.2]) and obtain

[TABLE]

Next, we consider the homotopy $\displaystyle\tilde{h}^{+}(t,u)=\tilde{h}^{+}_{t}(u)$ defined by

[TABLE]

Claim 5**.**

$\displaystyle(\tilde{h}^{+}_{t})^{\prime}(u)=0$* for all $\displaystyle t\in[0,1]$, all $\displaystyle u\in W^{1,p}(\Omega),\ u\neq 0$.*

We proceed by contradiction. So, suppose that we can find $\displaystyle t\in[0,1]$ and $\displaystyle u\in W^{1,p}(\Omega),\ u\neq 0$ such that

[TABLE]

In (84) we choose $\displaystyle h=-u^{-}\in W^{1,p}(\Omega)$. Then

[TABLE]

Then equation (84) becomes

[TABLE]

As before, from (4) and the nonlinear regularity theory (see [12]), we have

[TABLE]

Also, we have

[TABLE]

Let $\displaystyle v\in D_{+}$ and consider the function

[TABLE]

Using the nonlinear Picone’s identity (see, for example, Motreanu, Motreanu & Papageorgiou [16, p. 255]), we have

[TABLE]

Let $\displaystyle v=\hat{u}_{1}\in D_{+}$. Then from (86) we have

[TABLE]

a contradiction.

This proves Claim 5.

The homotopy invariance property of the singular homology groups implies that for small $\displaystyle r>0$, we have

[TABLE]

Claim 5 implies that

[TABLE]

Also, by definition we have

[TABLE]

From (87), (88), (89) we infer that

[TABLE]

Since $\displaystyle\lambda\in(\hat{\lambda}_{m},\hat{\lambda}_{m+1})\backslash\hat{\sigma}(p)$, we have

[TABLE]

Similarly for the functional $\displaystyle\hat{\varphi}_{-}$, using this time the $\displaystyle C^{1}$-functional $\displaystyle\hat{\psi}_{-}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

∎

This proposition permits the exact computation of the critical groups of $\displaystyle\varphi$ at the two nontrivial constant sign smooth solutions $\displaystyle u_{0}\in D_{1}$ and $\displaystyle v_{0}\in-D_{+}$ (see Proposition 7).

Proposition 10**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)$ hold, $\displaystyle u_{0}\in D_{+}$ and $\displaystyle v_{0}\in-D_{+}$ from Proposition 7 are the only nontrivial constant sign smooth solutions of (1) and $\displaystyle K_{\varphi}$ is finite, then $\displaystyle C_{k}(\varphi,u_{0})=C_{k}(\varphi,v_{0})=\delta_{k,1}\mathbb{Z}$ for all $\displaystyle k\in\mathbb{N}_{0}$.

Proof.

We give the proof for the solution $\displaystyle u_{0}\in D_{+}$, the proof for $\displaystyle v_{0}\in-D_{+}$ is similar.

From (46) and the hypothesis of the proposition, we have

[TABLE]

Let $\displaystyle\eta<0<a<\hat{m}^{+}_{\rho}$ (see (47)). We consider the following triple of sets

[TABLE]

For this triple, we consider the corresponding long exact sequence of singular homology groups (see Motreanu, Motreanu & Papageorgiou [16, Proposition 6.14, p. 143]). We have

[TABLE]

Here, $\displaystyle j_{*}$ is the group homomorphism induced by the inclusion

[TABLE]

and $\displaystyle\hat{\partial}_{*}$ is the composed boundary homomorphism (see [16]). By the rank theorem, we have

[TABLE]

From the choice of $\displaystyle\eta$ and since $\displaystyle K_{\hat{\varphi}_{+}}=\{0,u_{0}\}$, we have

[TABLE]

Also because $\displaystyle a\in(0,\hat{m}^{+}_{\rho})$ and since $\displaystyle K_{\hat{\varphi}_{+}}=\{0,u_{0}\}$, we have

[TABLE]

Finally, because $\displaystyle\eta<0<a<\hat{m}^{+}_{\rho}$ and since $\displaystyle K=\{0,u_{0}\}$, we have

[TABLE]

Returning to (4) and using (93), (94), (95), we see that

[TABLE]

On the other hand, $\displaystyle u_{0}\in D_{+}$ is a critical point of mountain pass type (see the proof of Proposition 7). So, we have

[TABLE]

(see Corollary 6.81, p. 168 of Motreanu, Motreanu & Papageorgiou [16]). From (96), (97) and since in (91) the part $\displaystyle k\geqslant 2$ is trivial (see (93), (94)), we infer that

[TABLE]

We consider the homotopy $\displaystyle h^{+}(t,u)=h^{+}_{t}(u)$ defined by

[TABLE]

Suppose we could find $\displaystyle\{t_{n}\}_{n\geqslant 1}\subseteq[0,1]$ and $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ such that

[TABLE]

From the equality in (99), we have

[TABLE]

From (4) and Papageorgiou & Rădulescu [22], we know that we can find $\displaystyle M_{4}>0$ such that

[TABLE]

Using Theorem 2 of Lieberman [12], we can find $\displaystyle\alpha\in(0,1)$ and $\displaystyle M_{5}>0$ such that

[TABLE]

Exploiting the compact embedding of $\displaystyle C^{1,\alpha}(\overline{\Omega})$ into $\displaystyle C^{1}(\overline{\Omega})$, we infer from (99) and (101) that

[TABLE]

Since $\displaystyle D_{+}\subseteq C^{1}(\overline{\Omega})$ is open, we can find $\displaystyle n_{0}\in\mathbb{N}$ such that

[TABLE]

But note that $\displaystyle\varphi|_{C_{+}}=\hat{\varphi}_{+}|_{C_{+}}$ (see (8)). So, we can see that $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq K_{\varphi}$ and this contradicts our hypothesis that the critical set $\displaystyle K_{\varphi}$ is finite. Therefore (99) cannot happen and so we can use Theorem 5.2 of Corvellec & Hantoute [5] (homotopy invariance of critical groups) and have

[TABLE]

Similarly for the negative solution $\displaystyle v_{0}\in-D_{+}$ using this time the functional $\displaystyle\hat{\varphi}_{-}$ and (11). ∎

We are now ready for the complete multiplicity theorem (three solutions theorem) for problem (1).

Theorem 11**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)$ hold, then problem (1) has at least three nontrivial smooth solutions

[TABLE]

Proof.

From Proposition 7, we already have two nontrivial constant sign smooth solutions

[TABLE]

Suppose that $\displaystyle K_{\varphi}$ is finite. Otherwise we already have an infinity of nontrivial solutions in addition to $\displaystyle u_{0},v_{0}$ which belong in $\displaystyle C^{1}(\overline{\Omega})$ (by the nonlinear regularity theory, see [12]) and so we are done.

From Proposition 10, we have

[TABLE]

From Proposition 6, we have

[TABLE]

According to Proposition 8, $\displaystyle C_{m}(\varphi,\infty)\neq 0$. So, there exists $\displaystyle y_{0}\in W^{1,p}(\Omega)$ such that

[TABLE]

Since $\displaystyle m\geqslant 2$, we infer from (102), (103), (104) that

[TABLE]

Therefore $\displaystyle y_{0}$ is the third nontrivial solution of (1) (see (104)) and by the nonlinear regularity theory (see [12]), we have $\displaystyle y_{0}\in C^{1}(\overline{\Omega})$. ∎

Acknowledgments. This research was supported in part by the Slovenian Research Agency grants P1-0292, J1-7025, and J1-6721. V. Rădulescu was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130.