\documentclass[11pt]{amsart}

\usepackage{amssymb,amsthm,amsmath,color}
\usepackage[numbers,sort&compress]{natbib}
\usepackage[paper=a4paper, left=2.35cm, right=2.35cm, top=2.5cm, bottom=1.8cm, headheight=5.5pt, footskip=0.8cm, footnotesep=0.8cm, centering, includefoot]{geometry}
%\begin{document}

%\hfill\today\bigskip

\title[Noncoercive resonant $(p,2)$-equations]{Noncoercive resonant $(p,2)$-equations}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author[N.S. Papageorgiou]{Nikolaos S. Papageorgiou}
\address[N.S. Papageorgiou]{National Technical University, Department of Mathematics,
				Zografou Campus, Athens 15780, Greece}
\email{\tt npapg@math.ntua.gr}

\author[V.D. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address[V.D. R\u{a}dulescu]{Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia \& Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13,
          200585 Craiova, Romania}
\email{\tt vicentiu.radulescu@imar.ro}

\keywords{Resonance at zero and at infinity, $C$-condition, critical groups, regularity theory.\\
\phantom{aa} 2010 AMS Subject Classification:
35J20, 35J60, 58E05}

\newtheorem{theorem}{Theorem}
\newtheorem{ex}{Example}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{remark}{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}{Claim}
\newtheorem{step}{Step}
\newtheorem{case}{Case}
\newtheorem{definition}[theorem]{Definition}

%\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\newenvironment{proof-sketch}{\noindent{\bf Sketch of Proof}\hspace*{1em}}{\qed\bigskip}

\newcommand{\eqname}[1]{\tag*{#1}}% Name of equation


%\numberwithin{equation}{section}
\newcommand{\RR}{\mathbb R}
\newcommand{\NN}{\mathbb N}
\newcommand{\PP}{\mathbb P}
\newcommand{\ZZ}{\mathbb Z}
\renewcommand{\le}{\leqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\ge}{\geqslant}
\renewcommand{\geq}{\geqslant}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\baselineskip=16pt plus 1pt minus 1pt

\begin{document}

\begin{abstract}
We consider a nonlinear Dirichlet problem driven by the sum of $p$-Laplacian and a Laplacian (a $(p,2)$-equation) which is resonant at $\pm\infty$ with respect to the principal eigenvalue $\hat{\lambda}_1(p)$ of $(-\Delta_p,W^{1,p}_{0}(\Omega))$ and resonant at zero with respect to any nonprincipal eigenvalue of $(-\Delta,H^1_0(\Omega))$. At $\pm\infty$  the resonance occurs from the right of $\hat{\lambda}_1(p)$ and so the energy functional of the problem is indefinite. Using critical groups, we show that the problem has at least one nontrivial smooth solution. The result complements the recent work of Papageorgiou and R\u adulescu ({\it Appl. Math. Optim.} {\bf 69} (2014), 393-430), where resonant $(p,2)$-equations were examined with the resonance occurring from the left of $\hat{\lambda}_1(p)$ (coercive problem).
\end{abstract}

\maketitle

%\tableofcontents

\section{Introduction}\label{sec1}

Let $\Omega\subseteq\RR^N$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. In this paper, we study the following nonlinear elliptic equation
\begin{equation}\label{eq1}
	-\Delta_pu(z)-\Delta u(z)=f(z,u(z))\ \mbox{in}\ \Omega,\quad u|_{\partial\Omega}=0,\ 2<p.
\end{equation}

In this problem $\Delta_p$ denotes the $p$-Laplacian differential operator defined by
$$\Delta_pu={\rm div}\,(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$

The reaction term $f(z,x)$ is a measurable function which is $C^1$ in the $x$-variable. We assume that $f(z,\cdot)$ is asymptotically as $x\rightarrow\pm\infty$ and $x\rightarrow 0$, resonant (double resonance situation). For this doubly resonant problem, we prove an existence theorem producing a nontrivial smooth solution.

Elliptic equations driven by the sum of a $p$-Laplacian and a Laplacian, are known as $(p,2)$-equations and arise in problems of mathematical physics. We mention the papers of Benci, D'Avenia, Fortunato \& Pisani \cite{2} (quantum physics) and Cherfils \& Ilyasov \cite{4} (plasma physics). Recently there have been some existence and multiplicity results for such equations. We mention the works of Aizicovici, Papageorgiou \& Staicu \cite{1}, Cingolani \& Degiovanni \cite{5}, Mugnai \& Papageorgiou \cite{13}, Papageorgiou \& R\u adulescu \cite{15}, \cite{16}, Sun \cite{17}, Sun, Zhang \& Su \cite{18}.

Our work here is closely related to the paper of Papageorgiou \& R\u adulescu \cite{15} and in fact it complements it. In that paper, the authors examined (${p,2}$)-equations which at $\pm\infty$ are resonant with respect to the principal eigenvalue $\hat{\lambda}_1(p)>0$ of the Dirichlet $p$-Laplacian $(-\Delta_p,W^{1,p}_{0}(\Omega))$. The resonance occurs from the left of $\hat{\lambda}_1(p)$ in the sense that $\hat{\lambda}_1(p)|x|^p-pF(z,x)\rightarrow+\infty$ as $x\rightarrow\pm\infty$ (here $F(z,x)=\int^x_0f(z,s)ds$). Then the corresponding energy functional of the problem is coercive and this permits the use of the direct method of the calculus of variations. So, using this method together with suitable truncation techniques and Morse theory (critical groups), Papageorgiou \& R\u adulescu \cite{15} proved multiplicity theorems producing three or four nontrivial solutions, all with sign information.

It is natural to ask what happens when the resonance at $\pm\infty$ with respect to $\hat{\lambda}_1(p)>0$ occurs from the right, in the sense that $\hat{\lambda}_1|x|^p-pF(z,x)\rightarrow-\infty$ as $x\rightarrow\pm\infty$. In this case, the energy functional is no longer coercive and so the direct method fails and it is not clear if we can have a nontrivial solution.

In this paper we study this case and using Morse theory (critical groups), we show the existence of at least one nontrivial solution.


\section{Mathematical Background}\label{sec2}

Let $X$ be a Banach space and $X^*$ its topological dual. By $\left\langle \cdot,\cdot\right\rangle$ we denote the duality brackets for the pair $(X^*,X)$. We say that $\varphi\in C^1(X,\RR)$ satisfies the ``Cerami condition" (the ``$C$-condition" for short), if the following property holds:
\begin{center}
	``Every sequence $\{u_n\}_{n\geq 1}\subseteq X$ such that $\{\varphi(u_n)\}_{n\geq 1}\subseteq\RR$ is bounded and
	$$(1+||u_n||)\varphi'(u_n)\rightarrow 0\ \mbox{in}\ X^*\ \mbox{as}\ n\rightarrow\infty,$$
	admits a strongly convergent subsequence."
\end{center}

Let us briefly recall some basic definitions and facts concerning critical groups (Morse theory) which we will need in the sequel. So, let $\varphi\in C^1(X,\RR)$ and $c\in\RR$. We introduce the following sets
\begin{itemize}
	\item	$\varphi^c=\{u\in X:\varphi(u)\leq c\}$ (the sublevel set of $\varphi$ at $c$),
	\item $K_{\varphi}=\{u\in X:\varphi'(u)=0\}$ (the critical set of $\varphi$),
	\item $K^c_{\varphi}=\{u\in K_{\varphi}:\varphi(u)=c\}$ (the critical set of $\varphi$ at the level $c$).
\end{itemize}

Suppose $(Y_1,Y_2)$ is a topological pair such that $Y_2\subseteq Y_1\subseteq X$ and $k\in\NN_0$. By $H_k(Y_1,Y_2)$ we denote the $k$th singular homology group for the pair $(Y_1,Y_2)$ with integer coefficients. Suppose that $u\in K^c_{\varphi}$ is isolated. The critical groups of $\varphi$ at $u$ are defined by
$$C_k(\varphi,u)=H_k(\varphi^c\cap U,\varphi^c\cap U\backslash\{u\})\ \mbox{for all}\ k\in\NN_0.$$

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

Suppose that $\varphi$ satisfies the $C$-condition and $-\infty<\inf\varphi(K_{\varphi})$. Then the critical groups of $\varphi$ at infinity, are defined by
$$C_k(\varphi,\infty)=H_k(X,\varphi^c)\ \mbox{for all}\ k\in\NN_0.$$

The second deformation theorem (see, for example, Gasinski \& Papageorgiou \cite[p. 628]{9}), implies that the above definition is independent of the choice of the level $c<\inf\varphi(K_{\varphi})$. We know that if for some $k\in\NN_0$, $C_k(\varphi,\infty)\neq 0$, then we can find $u\in K_{\varphi}$ such that $C_k(\varphi,u)\neq 0$. Also, if $X=H=$ Hilbert space, $\varphi\in C^2(H,\RR)$ and $u\in K_{\varphi}$, then the ``Morse index" of $u$, denoted by $\mu(u)$, is defined to be the supremum of the dimensions of the vector subspaces of $H$ on which $\varphi''(u)$ is negative definite. The ``nullity" of $u$, denoted by $\nu(u)$, is the dimension of ${\rm ker}\,\varphi''(u)$. We say that $u$ is ``nondegenerate", if $\varphi''(u)$ is invertible (that is, $\nu(u)=0$). If $\varphi\in C^2(H,\RR)$ and $u\in K_{\varphi}$ is nondegenerate (hence by the inverse function theorem automatically isolated) with Morse index $\mu(u)=\mu$, then
$$C_k(\varphi,u)=\delta_{k,\mu}\ZZ\ \mbox{for all}\ k\in\NN_0.$$

Hereafter, by $\delta_{k,\mu}$ we denote the Kronecker symbol defined by
$$\delta_{k,\mu}=\left\{
\begin{array}{ll}
	1&\mbox{if}\ k=\mu\\
	0&\mbox{if}\ k\neq\mu .
\end{array}\right.$$

In our analysis of problem (\ref{eq1}) we will use some basic facts about the spectrum of the $p$-Laplacian and the Laplacian.

So, consider the following nonlinear eigenvalue problem
\begin{equation}\label{eq2}
	-\Delta_q u(z)=\hat{\lambda}|u(z))|^{q-2}u(z)\ \mbox{in}\ \Omega,\quad u|_{\partial\Omega}=0\ (1<q<\infty).
\end{equation}

We say that $\hat{\lambda}$ is an eigenvalue of $(-\Delta_q,W^{1,q}_{0}(\Omega))$, if problem (\ref{eq2}) admits a nontrivial solution $\hat{u}\in W^{1,q}_{0}(\Omega)$, which is an eigenfunction corresponding to the eigenvalue $\hat{\lambda}$. There exists a smallest eigenvalue $\hat{\lambda}_1(q)>0$ with the following properties:
\begin{itemize}
	\item $\hat{\lambda}_1(q)$ is isolated (that is, there exists $\epsilon>0$ such that $(\hat{\lambda}_1(q),\hat{\lambda}_1(q)+\epsilon)$ contains no eigenvalues of ($-\Delta_q,W^{1,q}_{0}(\Omega)$));
	\item $\hat{\lambda}_1(q)$ is simple (that is, if $\hat{u},\hat{v}\in W^{1,q}_{0}(\Omega)$ are eigenfunction corresponding to $\hat{\lambda}$, then $\hat{u}=\xi\hat{v}$ with $\xi\in\RR\backslash\{0\}$);
 \begin{equation}\label{eq3}
		\bullet\hspace{0.7cm} \hat{\lambda}_1(q)=\inf\left[\frac{||Du||^q_q}{||u||^q_q}:u\in W^{1,q}_{0}(\Omega),u\neq 0\right]>0.\hspace{4.5cm}
	\end{equation}
\end{itemize}

The infimum in (\ref{eq3}) is realized on the corresponding one dimensional eigenspace. It is clear from (\ref{eq3}) that the elements of this eigenspace do not change sign. By $\hat{u}_1(q)$ we denote the $L^q$-normalized positive eigenfunction corresponding to $\hat{\lambda}_1(q)>0$. From the nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasinski \& Papageorgiou \cite[pp. 737-738]{9}), we have $$\hat{u}_1(q)\in {\rm int}\,  C_+=\left\{u\in C^1_0(\overline{\Omega}):u(z)>0\ \mbox{for all}\ z\in\Omega,\left.\frac{\partial u}{\partial n}\right|_{\partial\Omega}<0\right\}$$ with $n(\cdot)$ being the outward unit normal on $\partial\Omega$ ($C_+$ denotes the positive cone of the ordered Banach space $C^1_0(\overline{\Omega})$, defined by $C_+=\{u\in C^1_0(\overline{\Omega}):u(z)\geq 0\ \mbox{for all}\ z\in\overline{\Omega}\}$ and ${\rm int}\, C_+$ is its interior). Using the Ljusternik-Schnirelmann minimax scheme, we can produce a whole strict
 ly increasing sequence $\{\hat{\lambda}_k(q)\}_{k\geq 1}$ of eigenvalues (known as LS-eigenvalues) such that $\hat{\lambda}_k(q)\rightarrow+\infty$. We can have at least three such sequence of LS-eigenvalues depending on the index used in the minimax scheme. All three coincide in the first two eigenvalues, but we do not know if this is also true for the higher eigenvalues. We only know that their elements are ordered. In general, the spectrum of $(-\Delta_q,W^{1,q}_{0}(\Omega))$ is far from fully described. Note that the isolation of $\hat{\lambda}_1(q)>0$ implies he second eigenvalue $\hat{\lambda}^*_2(q)$ is well-defined by
$$\hat{\lambda}_2^*(q)=\inf[\hat{\lambda}:\hat{\lambda}\ \mbox{is an eigenvalue of (\ref{eq2})},\ \hat{\lambda}>\hat{\lambda}_1(q)].$$

We know that $\hat{\lambda}^*_2(q)=\hat{\lambda}_2(q)$ (that is, the second eigenvalue coincides with the second LS-eigenvalue). So, the Ljusternik-Schnirelmann minimax scheme provides a variational characterization of $\hat{\lambda}_2(q)$. However, for our purposes more convenient is an alternative minimax characterization due to Cuesta, de Figueiredo \& Gossez \cite{7}. Let
\begin{eqnarray*}
	&&\partial B^{L^q}_1=\{u\in L^q(\Omega):||u||_q=1\},\ M=W^{1,q}_{0}(\Omega)\cap\partial B^{L^q}_1,\\
	&&\hat{\Gamma}=\{\hat{\gamma}\in C([-1,1],M):\hat{\gamma}(-1)=-\hat{u}_1(q),\hat{\gamma}(1)=\hat{u}_1(q)\}.
\end{eqnarray*}
\begin{prop}\label{prop1}
We have	$\hat{\lambda}_2(q)=\inf\limits_{\hat{\gamma}\in\hat{\Gamma}}\max\limits_{-1\leq t\leq 1}||D\hat{\gamma}(t)||^q_q$.
\end{prop}

When $q=2$ (linear eigenvalue problem), we have a complete description of the spectrum of $(-\Delta,H^1_0(\Omega))$. This consists of a sequence of distinct eigenvalues $\{\hat{\lambda}_k(2)\}_{k\geq 1}$ such that $\hat{\lambda}_k(2)\rightarrow+\infty$. By $E(\hat{\lambda}_k(2))$ we denote the eigenspace for the eigenvalue $\hat{\lambda}_k(2)$. In this case we have the orthogonal direct sum decomposition
$$H^1_0(\Omega)=\overline{{\underset{\mathrm{k\geq 1}}\oplus}E(\hat{\lambda}_k(2))}.$$

These eigenspaces exhibit the unique continuation property, which says that if $u\in E(\hat{\lambda}_k(2))$ vanishes on a set of positive measure, then $u=0$. Also, each $E(\hat{\lambda}_k(2))$ is finite dimensional and $E(\hat{\lambda}_k(2))\subseteq C^1_0(\overline{\Omega})$ (regularity theory).

Now we have variational characterizations for all the eigenvalues, namely
\begin{eqnarray}
	\hat{\lambda}_1(2)&=&\inf\left[\frac{||Du||^2_2}{||u||^2_2}:u\in H^1_0(\Omega),u\neq 0\right]\ (\mbox{see (\ref{eq3}) with}\ q=2)\label{eq4}\\
	\hat{\lambda}_k(2)&=&\inf\left[\frac{||Du||^2_2}{||u||^2_2}:u\in\overline{{\underset{\mathrm{i\geq k}}\oplus}E(\hat{\lambda}_i(2))},\ u\neq 0\right]\nonumber\\
	&=&\sup\left[\frac{||Du||^2_2}{||u||^2_2}:u\in\overset{k}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(2)),\ u\neq 0\right]\ \mbox{for all}\ k\geq 2.\label{eq5}
\end{eqnarray}

In (\ref{eq4}) and (\ref{eq5}) the infima and suprema are realized on the corresponding eigenspaces.

We mention that for all $1<q<\infty$ and all $k\geq 2$, the eigenfunctions corresponding to $\hat{\lambda}_k(q)$ are nodal (that is, sign changing).

We introduce the following linear subspace of $W^{1,q}_{0}(\Omega)\ (1<q<\infty)$:
$$V_q=\{u\in W^{1,q}_{0}(\Omega):\int_{\Omega}\hat{u}_1(q)^{q-1}udz=0\}.$$

We have the following direct sum decomposition
$$W^{1,q}_{0}(\Omega)=\RR\hat{u}_1(q)\oplus V_q.$$

We define
\begin{equation}\label{eq6}
	\tilde{\lambda}(q)=\inf\left[\frac{||Du||^q_q}{||u||^q_q}:u\in V_q,\ u\neq 0\right].
\end{equation}
\begin{prop}\label{prop2}
We have	$\hat{\lambda}_1(q)<\tilde{\lambda}(q)\leq\hat{\lambda}_2(q).$
\end{prop}
\begin{proof}
	From (\ref{eq3}) and (\ref{eq6}), we have
	$$\hat{\lambda}_1(q)\leq \tilde{\lambda}(q).$$
	
	Suppose that equality holds. So, we can find $\{u_n\}_{n\geq 1}\subseteq V_q$ such that
	\begin{equation}\label{eq7}
		||u_n||_q=1\ \mbox{for all}\ n\in\NN\ \mbox{and}\ ||Du||^q_q\rightarrow\hat{\lambda}_1(q)=\tilde{\lambda}(q)\ \mbox{as}\ n\rightarrow\infty .
	\end{equation}
	
	From (\ref{eq3}) it is clear that $\{u_n\}_{n\geq 1}\subseteq W^{1,q}_{0}(\Omega)$ is bounded and so we may assume that
	\begin{equation}\label{eq8}
		u_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ W^{1,q}_{0}(\Omega)\ \mbox{and}\ u_n\rightarrow u\ \mbox{in}\ L^q(\Omega).
	\end{equation}
	
	Then $u\in V_q$ and $||u||_q=1$. Also, from (\ref{eq6}) and (\ref{eq8}), we have
	\begin{eqnarray*}
		 &&\hat{\lambda}_1(q)\leq||Du||^q_q\leq\liminf_{n\rightarrow\infty}||Du_n||^q_q=\tilde{\lambda}(q)=\hat{\lambda}_1(q),\\
		&\Rightarrow&\hat{\lambda}_1(q)=||Du||^q_q\ \mbox{with}\ ||u||_q=1,\\
		&\Rightarrow&u=\pm\hat{u}_1(q),
	\end{eqnarray*}
	a contradiction to the fact that $u\in V_q$. Therefore, we have
	$$\hat{\lambda}_1(q)<\tilde{\lambda}(q).$$
	
	Next we show that
	$$\tilde{\lambda}(q)\leq\hat{\lambda}_2(q).$$
	
	Again we argue by contradiction. So, suppose that $\hat{\lambda}_2(q)<\tilde{\lambda}(q)$. From Proposition \ref{prop1} we know that we can find $\hat{\gamma}\in\hat{\Gamma}$ such that
	\begin{equation}\label{eq9}
		||D\hat{\gamma}(t)||^q_q<\tilde{\lambda}(q)\ \mbox{for all}\ t\in[-1,1].
	\end{equation}
	
	From the definition of $\hat{\Gamma}$, we have
	$$\hat{\gamma}(-1)=-\hat{u}_1(q)\in {\rm int}\, C_+\ \mbox{and}\ \hat{\gamma}(1)=\hat{u}_1(q)\in {\rm int}\, C_+.$$
	
	Consider the function $\xi:[-1,1]\rightarrow\RR$ defined by
	$$\xi(t)=\int_{\Omega}\hat{\gamma}(t)(z)\hat{u}_1(q)(z)^{q-1}dz.$$
	
	Then $\xi(-1)=-||\hat{u}_1(q)||^q_q=-1<0<1=||\hat{u}_1(q)||^q_q=\xi(1)$. Since $\xi(\cdot)$ is continuous, it follows  from Bolzano's theorem that we can find $t_0\in(-1,1)$ such that
	\begin{eqnarray*}
		&&\xi(t_0)=0,\\
		&\Rightarrow&\int_{\Omega}\hat{\gamma}(t_0)\hat{u}_1(q)^{q-1}dz=0,\\
		&\Rightarrow&\hat{\gamma}(t_0)\in V_q,\\
		&\Rightarrow&\tilde{\lambda}(q)\leq||D\hat{\gamma}(t_0)||^q_q\ (\mbox{see (\ref{eq6})})
	\end{eqnarray*}
	and this contradicts (\ref{eq9}). Therefore we conclude that $\tilde{\lambda}(q)\leq\hat{\lambda}_2(q)$.
	\end{proof}
\begin{remark}
	In general we do not know if the inequality $\tilde{\lambda}(q)\leq\hat{\lambda}_2(q)$ can be strict. Note that if $q=2$, then $\tilde{\lambda}(2)=\hat{\lambda}_2(2)$.
\end{remark}

In the sequel for $1<q<\infty$, by $A_q:W^{1,q}_{0}(\Omega)\rightarrow W^{-1,q'}(\Omega)=W^{1,q}_{0}(\Omega)^*$ $\left(\frac{1}{q}+\frac{1}{q'}=1\right)$, we denote the map defined by
\begin{equation}\label{eq10}
	\left\langle A_q(u),v\right\rangle=\int_{\Omega}|Du|^{q-2}(Du,Dv)_{\RR^N}dz\ \mbox{for all}\ u,v\in W^{1,q}_{0}(\Omega).
\end{equation}

>From Gasinski \& Papageorgiou \cite[p. 746]{9}, we have:
\begin{prop}\label{prop3}
	If $1<q<\infty$ and $A_q:W^{1,q}_{0}(\Omega)\rightarrow W^{-1,q'}(\Omega)$ is defined by (\ref{eq10}), then $A_q$ is monotone, continuous (hence maximal monotone), bounded (that is, maps bounded sets to bounded sets) and of type $(S)_+$, that is, if $u_n\stackrel{w}{\rightarrow}u$ in $W^{1,q}_{0}(\Omega)$ and
	$$\limsup\limits_{n\rightarrow\infty}\left\langle A_q(u_n),u_n-u\right\rangle\leq 0,$$
	then $u_n\rightarrow u$ in $W^{1,q}_{0}(\Omega)$.
\end{prop}

If $q=2$, we write $A=A_2\in\mathcal{L}(H^1_0(\Omega),H^{-1}(\Omega))$.

Finally let us fix our notation. By $|\cdot|_N$ we denote the Lebesgue measure on $\RR^N$. Also, if $h:\Omega\times\RR\rightarrow\RR$ is a measurable function, then we set
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))\ \mbox{for all}\ W^{1,q}_{0}(\Omega),$$
the Nemytskii operator corresponding to $h$. By $||\cdot||$ we denote the norm for the Sobolev space $W^{1,p}_{0}(\Omega)$. The Poincar\'e inequality implies that
$$||u||=||Du||_p\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$

Let $p^*$ denote the critical Sobolev exponent, that is, $$p^*=\left\{\begin{array}{ll}
	\displaystyle\frac{Np}{N-p}&\mbox{if}\ p<N\\
\displaystyle	+\infty&\mbox{if}\ N\leq p.
\end{array}\right.$$


\section{Existence of Nontrivial Solutions}

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

$H:$ $f:\Omega\times\RR\rightarrow\RR$ is a measurable function such that for almost all $z\in\Omega$, $f(z,0)=0,\ f(z,\cdot)\in C^1(\RR)$ and
\begin{itemize}
	\item[(i)] $|f'_x(z,x)|\leq a(z)(1+|x|^{r-1})$ for almost all $z\in\Omega$, all $x\in\RR$, with $a\in L^{\infty}(\Omega)_+,\ p<r<p^*$;
	\item[(ii)] $\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{|x|^{p-2}x}=\hat{\lambda}_1(p)$ uniformly for almost all $z\in\Omega$;
	\item[(iii)] if $F(z,x)=\int^x_0f(z,s)ds$, then there exist $\tau>2$ and $\beta_0>0$ such that
	$$\limsup\limits_{x\rightarrow\pm\infty}\frac{f(z,x)x-pF(z,x)}{|x|^{\tau}}\leq-\beta_0\ \mbox{uniformly for almost all}\ z\in\Omega;$$
	\item[(iv)]  there exist $l\in\NN,\ l\geq 2$, $\delta>0$ and $\eta\in L^\infty(\Omega)$ such that $\hat{\lambda}_l(2)\leq \eta(z)$ for almost all $z\in\Omega$, $\eta\not\equiv \hat{\lambda}_l(2)$,
	$$\eta (z)x^2\leq f(z,x)x\leq \hat{\lambda}_{l+1}(2)x^2\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ 0<|x|\leq\delta,$$ and the second inequality is strict on a set of positive measure.
\end{itemize}


\begin{remark}
	Hypothesis $H(ii)$ implies that asymptotically as $x\rightarrow\pm\infty$, we have resonance with respect to the principal eigenvalue $\hat{\lambda}_1(p)>0$ of $(-\Delta_p,W^{1,p}_{0}(\Omega))$. Hypothesis $H(iii)$ implies that the resonance occurs from the right of $\hat{\lambda}_1(p)$. Indeed, as will see in the proof of Proposition \ref{prop5}, this hypothesis implies that
		$$\hat{\lambda}_1(p)|x|^p-F(z,x)\rightarrow-\infty\ \mbox{as}\ x\rightarrow\pm\infty .$$
		
		Hypothesis $H(iv)$ implies that we have resonance also at zero (double resonance).
   Resonance is possible with respect to $\hat{\lambda}_{l+1}(2)$ as $x\rightarrow 0$, since we can have
 $$\lim_{x\rightarrow 0}\frac{f(z,x)}{x}=\hat{\lambda}_{l+1}(2)\quad\mbox{uniformly for almost all $z\in\Omega$.}$$
 Symmetrically, we may assume that
 $$ \hat{\lambda}_{l}(2)x^2\leq f(z,x)x\leq\hat\eta (z)x^2\quad\mbox{for almost all $z\in\Omega$, all $0<|x|\leq\delta$}$$  with $\hat\eta\in L^\infty (\Omega)$, $\hat\eta (z)\leq\hat{\lambda}_{l+1}(2)$ for almost all $z\in\Omega$, $\hat\eta\not\equiv\hat\lambda_{l+1}(2)$ and the first inequality is strict on a set of positive measure.
 Note that hypothesis $H(iv)$ is in contrast to the situation in Papageorgiou \& R\u adulescu \cite{15}, where the hypotheses on the reaction term do not permit resonance at zero.
\end{remark}


Let $\varphi:W^{1,p}_{0}(\Omega)\rightarrow\RR$ be the energy functional for problem (\ref{eq1}) defined by
$$\varphi(u)=\frac{1}{p}||Du||^p_p+\frac{1}{2}||Du||^2_2-\int_{\Omega}F(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$

Evidently $\varphi\in C^2(W^{1,p}_{0}(\Omega))$.
\begin{prop}\label{prop4}
	If hypotheses $H(i),(ii),(iii)$ hold, then the functional $\varphi$ satisfies the C-condition.
\end{prop}
\begin{proof}
	Let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ be a sequence such that
	\begin{eqnarray}
		&&|\varphi(u_n)|\leq M_1\ \mbox{for some}\ M_1>0,\ \mbox{all}\ n\in\NN,\label{eq11}\\
		&&(1+||u_n||)\varphi'(u_n)\rightarrow 0\ \mbox{in}\ W^{-1,p'}(\Omega)\ \mbox{as}\ n\rightarrow\infty .\label{eq12}
	\end{eqnarray}
	
	From (\ref{eq11}) we have
	\begin{eqnarray}\label{eq13}
		&&\left|\left\langle A_p(u_n),h\right\rangle+\left\langle A(u_n),h\right\rangle-\int_{\Omega}f(z,u_n)hdz\right|\leq\frac{\epsilon_n||h||}{1+||u_n||}\\
		&&\hspace{1cm}\mbox{for all}\ h\in W^{1,p}_{0}(\Omega),\ \mbox{with}\ \epsilon_n\rightarrow 0^+.\nonumber
	\end{eqnarray}
	
	We claim that $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ is bounded. We argue indirectly. So, suppose that the sequence is unbounded. By passing to a subsequence if necessary, we may assume that
	\begin{equation}\label{eq14}
		||u_n||\rightarrow\infty\ \mbox{as}\ n\rightarrow\infty .
	\end{equation}
	
	Let $y_n=\frac{u_n}{||u_n||}$, $ n\in\NN$. Then $||y_n||=1$ for all $n\in\NN$ and so we may assume that
	\begin{equation}\label{eq15}
		y_n\stackrel{w}{\rightarrow}y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^p(\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{equation}
	
	From (\ref{eq13}) we have
	\begin{equation}\label{eq16}
		\left|\left\langle A_p(y_n),h\right\rangle+\frac{1}{||u_n||^{p-2}}\left\langle A(y_n),h\right\rangle-\int_{\Omega}\frac{N_f(u_n)}{||u_n||^{p-1}}hdz\right|\leq\frac{\epsilon_n||h||}{(1+||u_n||)||u_n||^{p-1}}\ \mbox{for all}\ n\in\NN\,.
	\end{equation}
	
	Hypotheses $H(i),(ii)$ imply that
	\begin{equation}\label{eq17}
		|f(z,x)|\leq c_1(1+|x|^{p-1})\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR,\ \mbox{some}\ c_1>0.
	\end{equation}
	
	From (\ref{eq17}) it follows that
	\begin{equation}\label{eq18}
		\left\{\frac{N_f(u_n)}{||u_n||^{p-1}}\right\}_{n\geq 1}\subseteq L^{p'}(\Omega)\ \mbox{is bounded}.
	\end{equation}
	
	So, by passing to a subsequence if necessary and using hypothesis $H(ii)$, we have
	\begin{equation}\label{eq19}
		\frac{N_f(u_n)}{||u_n||^{p-1}}\stackrel{w}{\rightarrow}\hat{\lambda}_1(p)|y|^{p-2}y\ \mbox{in}\ L^{p'}(\Omega)
	\end{equation}
(see Filippakis \& Papageorgiou \cite{8}, proof of Proposition 4.4). In (\ref{eq16}) we choose $h=y_n-y\in W^{1,p}_{0}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq14}), (\ref{eq15}), (\ref{eq18}) and the fact that $2<p$. Then

\begin{eqnarray}\label{eq20}
	&&\lim\limits_{n\rightarrow\infty}\left\langle A_p(y_n),y_n-y\right\rangle=0,\nonumber\\
	&\Rightarrow&y_n\rightarrow y\ \mbox{in}\ W^{1,p}_{0}(\Omega),\ \mbox{hence}\ ||y||=1.
\end{eqnarray}

So, if in (\ref{eq16}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq14}), (\ref{eq19}), (\ref{eq20}) and the fact that $2<p$, then
\begin{eqnarray*}
	&&\left\langle A_p(y),h\right\rangle=\hat{\lambda}_1(p)\int_{\Omega}|y|^{p-2}yhdz\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega),\\
	&\Rightarrow&-\Delta_py(z)=\hat{\lambda}_1(p)|y(z)|^{p-2}y(z)\ \mbox{for almost all}\ z\in\Omega,\ y|_{\partial\Omega}=0,\\
	&\Rightarrow&y=\xi\hat{u}_1(p)\ \mbox{with}\ \xi\neq 0\ (\mbox{see (\ref{eq20})}).
\end{eqnarray*}

Recall that $\hat{u}_1(p)\in {\rm int}\, C_+$, hence $|y(z)|>0$ for all $z\in\Omega$ and so
$$|u_n(z)|\rightarrow+\infty\ \mbox{for all}\ z\in\Omega,\ \mbox{as}\ n\rightarrow\infty.$$

Hypothesis $H(iii)$ implies that
\begin{equation}\label{eq21}
	\limsup\limits_{n\rightarrow\infty}\frac{f(z,u_n(z))u_n(z)-pF(z,u_n(z))}{|u_n(z)|^{\tau}}\leq-\beta_0<0\ \mbox{for almost all}\ z\in\Omega.
\end{equation}

 By hypothesis $H(iii)$ we see that we can find $\beta_1\in (0,\beta_0)$ and $M>0$ such that
$$\frac{f(z,x)x-pF(z,x)}{|x|^\tau}\leq -\beta_1\quad\mbox{for almost all $z\in\Omega$, all $|x|\geq M$.}$$ Then we have
$$\begin{array}{ll} &\displaystyle \frac{1}{||u_n||^\tau}\int_\Omega [f(z,u_n)u_n-pF(z,u_n)]dz\\
&\displaystyle =\int_{\{|u_n|\geq M\}}\frac{f(z,u_n)u_n-pF(z,u_n)}{|u_n|^\tau}\,|y_n|^\tau dz+
\frac{1}{||u_n||^\tau}\int_{\{|u_n|< M\}}[f(z,u_n)u_n-pF(z,u_n)]dz.\end{array}$$
Recalling that $||u_n||\rightarrow\infty$ (see \eqref{eq14}), we see that $$\frac{1}{||u_n||^\tau}\int_{\{|u_n|< M\}}[f(z,u_n)u_n-pF(z,u_n)]dz\rightarrow 0.$$
On the other hand from \eqref{eq21}, Fatou's lemma and because $y\not=0$ (see \eqref{eq20}), we infer that \begin{eqnarray}
&& \limsup_{n\rightarrow\infty} \int_{\{|u_n|\geq M\}}\frac{f(z,u_n)u_n-pF(z,u_n)}{|u_n|^\tau}\,|y_n|^\tau dz<0,\nonumber\\
&\Rightarrow& \limsup\limits_{n\rightarrow\infty}\frac{1}{||u_n||^{\tau}}\int_{\Omega}[f(z,u_n)u_n-pF(z,u_n)]dz<0.
\label{eq22}\end{eqnarray}



On the other hand from (\ref{eq11}) we have
\begin{equation}\label{eq23}
	-pM_1\leq||Du_n||^p_p+\frac{p}{2}||Du_n||^2_2-\int_{\Omega}pF(z,u_n)dz\ \mbox{for all}\ n\in\NN.
\end{equation}

Also in (\ref{eq13}) we choose $h=u_n\in W^{1,p}_{0}(\Omega)$. We obtain
\begin{equation}\label{eq24}
	-\epsilon_n\leq-||Du_n||^p_p-||Du_n||^2_2+\int_{\Omega}f(z,u_n)u_ndz\ \mbox{for all}\ n\in\NN.
\end{equation}

Adding (\ref{eq23}) and (\ref{eq24}) we have for all $n\in\NN$
\begin{eqnarray}\label{eq25}
	&&-M_2\leq\left(\frac{p}{2}-1\right)||Du_n||^2_2+\int_{\Omega}[f(z,u_n)u_n-pF(z,u_n)]dz,\nonumber\\
	 &\Rightarrow&-\frac{M_2}{||u_n||^{\tau}}\leq\left(\frac{p}{2}-1\right)\frac{||Dy_n||^2_2}{||u_n||^{\tau-2}}+\frac{1}{||u_n||^{\tau}}\int_{\Omega}[f(z,u_n)u_n-pF(z,u_n)]dz,\nonumber\\
	&\Rightarrow&-\frac{M_2}{||u_n||^{\tau}}\leq\frac{c_2}{||u_n||^{\tau-2}}+
\frac{1}{||u_n||^{\tau}}\int_{\Omega}[f(z,u_n)u_n-pf(z,u_n)]dz
\end{eqnarray}
for some $c_2>0$, all $n\in\NN$ (since $\{Dy_n\}_{n\geq 1}\subseteq L^2(\Omega,\RR^N)$ is bounded and $2<p$).

Passing to the limit as $n\rightarrow\infty$ in (\ref{eq25}), using (\ref{eq14}) and recalling that $\tau>2$ (see hypothesis $H(iii)$), we have
\begin{equation}\label{eq26}
	0\leq\liminf\limits_{n\rightarrow\infty}\frac{1}{||u_n||^{\tau}}\int_{\Omega}[f(z,u_n)u_n-pF(z,u_n)]dz.
\end{equation}

Comparing (\ref{eq22}) and (\ref{eq26}), we reach a contradiction. This proves that
$$\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is bounded}.$$

So, we may assume that
\begin{equation}\label{eq27}
	u_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ u_n\rightarrow u\ \mbox{in}\ L^p(\Omega)\ \mbox{as}\ n\rightarrow\infty.
\end{equation}

>From (\ref{eq17}) and (\ref{eq27}) it is clear that
\begin{equation}\label{eq28}
	\{N_f(u_n)\}_{n\geq 1}\subseteq L^{p'}(\Omega)\ \mbox{is bounded}.
\end{equation}

If in (\ref{eq13}) we choose $h=u_n-u\in W^{1,p}_{0}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq27}), (\ref{eq28}), then
\begin{eqnarray*}
	&&\lim\limits_{n\rightarrow\infty}\left[\left\langle A_p(u_n),u_n-u\right\rangle+\left\langle A(u_n),u_n-u\right\rangle\right]=0,\\
	&\Rightarrow&\limsup\limits_{n\rightarrow\infty}\left[\left\langle A_p(u_n),u_n-u\right\rangle+\left\langle A(u),u_n-u\right\rangle\right]\leq 0\\
	&&\hspace{1cm}(\mbox{recall $A$ is monotone, see Proposition \ref{prop3}}),\\
	&\Rightarrow&\limsup\limits_{n\rightarrow\infty}\left\langle A_p(u_n),u_n-u\right\rangle\leq 0,\\
	&\Rightarrow&u_n\rightarrow u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ (\mbox{see Proposition \ref{prop3}}),\\
	&\Rightarrow&\varphi\ \mbox{satisfies the $C$-condition}.
\end{eqnarray*}
The proof is now complete.
\end{proof}	

 In the sequel we assume that $K_\varphi$ is finite or otherwise we already have an infinity of solutions for problem \eqref{eq1}, which belong to $C^1_0(\overline\Omega )$ (see the proof of Theorem \ref{th7}). So, we are done. 

\begin{prop}\label{prop5}
	If hypotheses $H(i),(ii),(iii)$ hold, then $C_1(\varphi,\infty)\neq 0$.
\end{prop}
\begin{proof}
	Recall that
	$$W^{1,p}_{0}(\Omega)=\RR\hat{u}_1(p)\oplus V_p.$$
	\begin{claim}\label{cl1}
		$\varphi|_{\RR\hat{u}_1(p)}$ is anticoercive (that is, $\varphi(t\hat{u}_1(p))\rightarrow-\infty$ as $t\rightarrow\pm\infty$).
	\end{claim}
	
	It is clear from hypothesis $H(iii)$ that we can always assume that $\tau<p$.
	
	Hypothesis $H(iii)$ implies that we can find $\beta_1\in (0,\beta_0)$ and $M_3>0$ such that
	\begin{equation}\label{eq29}
		f(z,x)x-pF(z,x)\leq-\beta_1|x|^{\tau }\quad \mbox{for almost all $z\in\Omega$, all $|x|\geq M_3$.}.
	\end{equation}
	
	We have
	\begin{eqnarray}\label{eq30}
		\frac{d}{dx}\ \frac{F(z,x)}{|x|^p}&=&\frac{f(z,x)|x|^p-p|x|^{p-2}xF(z,x)}{|x|^{2p}}\nonumber\\
		&=&\frac{f(z,x)x-pF(z,x)}{|x|^{p}x}\nonumber\\
		&\leq&-\beta_1\frac{1}{|x|^{p-\tau +1}}\ \mbox{if $x\geq M_3$ and}\ \geq-\beta_1\frac{1}{|x|^{p-\tau }x}\ \mbox{if $x\leq M_3$},\nonumber\\
		 &\Rightarrow&\frac{F(z,x)}{|x|^p}-\frac{F(z,y)}{|y|^p}\leq\frac{\beta_1}{p-\tau}\left[\frac{1}{|x|^{p-\tau}}-\frac{1}{|y|^{p-\tau}}\right]\\
		&&\hspace{3cm}\mbox{for all}\ z\in\Omega,\ \mbox{all}\ |x|\geq|y|\geq M_3.\nonumber
	\end{eqnarray}
	
	Note that hypothesis $H(ii)$ implies that
	\begin{equation}\label{eq31}
		\lim\limits_{x\rightarrow\pm\infty}\frac{pF(z,x)}{|x|^p}=\hat{\lambda}_1(p)\ \mbox{uniformly for almost all}\ z\in\Omega\,.
	\end{equation}
	
	So, if in (\ref{eq30}) we let $|x|\rightarrow+\infty$ and use (\ref{eq31}), then
	\begin{eqnarray}\label{eq32}
		&&\frac{1}{p}\hat{\lambda}_1(p)-\frac{F(z,y)}{|y|^p}\leq-\frac{\beta_1}{p-\tau}\ \frac{1}{|y|^{p-\tau}}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |y|\geq M_3\nonumber\\
		&&\hspace{8cm}(\mbox{recall}\ \tau<p),\nonumber\\
		&\Rightarrow&\frac{\hat{\lambda}_1(p)}{p}|y|^p-F(z,y)\leq-\frac{\beta_1}{p-\tau}|y|^{\tau}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |y|\geq M_3.
	\end{eqnarray}
	
	Hypothesis $H(i)$ and (\ref{eq32}) imply that we can find $c_3>0$ such that
	\begin{equation}\label{eq33}
		\frac{\hat{\lambda}_1(p)}{p}|y|^p-F(z,y)\leq-\frac{\beta_1}{p-\tau}|y|^{\tau}+c_3\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ y\in\RR\,.
	\end{equation}
	
	Then we have
	\begin{eqnarray}\label{eq34}
		 \varphi(t\hat{u}_1(p))&=&\int_{\Omega}\left[\frac{\hat{\lambda}_1(p)}{p}|t\hat{u}_1(p)|^p-F(z,t\hat{u}_1(p))\right]dz+\frac{t^2}{2}||D\hat{u}_1(p)||^2_2\nonumber\\
		 &\leq&-\frac{\beta_1|t|^{\tau}}{p-\tau}||\hat{u}_1(p)||^{\tau}_{\tau}+\frac{t^2}{2}||D\hat{u}_1||^2_2+c_3|\Omega|_N\ \mbox{(see (\ref{eq33}))}.
	\end{eqnarray}
	
	Since $\tau>2$ (see hypothesis $H(iii)$), from (\ref{eq34}) we infer that
	\begin{eqnarray*}
		&&\varphi(t\hat{u}_1(p))\rightarrow-\infty\ \mbox{as}\ t\rightarrow\pm\infty,\\
		&\Rightarrow&\varphi|_{\RR\hat{u}_1(p)}\ \mbox{is anticoercive}.
	\end{eqnarray*}
	
	This proves Claim \ref{cl1}.
	\begin{claim}\label{cl2}
		$\varphi|_{V_p}$ is bounded below.
	\end{claim}
	
	From (\ref{eq31}) and hypothesis $H(i)$, we see that given $\epsilon>0$, we can find $c_4=c_4(\epsilon)>0$ such that
	\begin{equation}\label{eq35}
		F(z,x)\leq\frac{\hat{\lambda}_1(p)+\epsilon}{p}|x|^p+c_4\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
	\end{equation}
	
	For $u\in V_p$, we have
	\begin{equation}\label{eq36}
		\varphi(u)\geq\frac{1}{p}\left[\hat{\lambda}(p)-(\hat{\lambda}_1(p)+\epsilon)\right]||u||^p_p-c_4|\Omega|_N\ \mbox{(see (\ref{eq6}) and (\ref{eq35}))}.
	\end{equation}
	
	Choosing $\epsilon\in(0,\hat{\lambda}(p)-\hat{\lambda}_1(p))$ (see Proposition \ref{prop1}), from (\ref{eq36}) we see that
	$$\varphi|_{V_p}\ \mbox{is bounded below.}$$
	
	This proves Claim \ref{cl2}.
	
	We assume that $K_{\varphi}$ is finite or otherwise we already have infinitely many nontrivial solutions for problem (\ref{eq1}) and so we are done. Then Claims \ref{cl1},\ref{cl2} and Proposition \ref{prop4}, permit the use of Proposition 6.63, p. 160, of Motreanu, Motreanu \& Papageorgiou \cite{12} and have $C_1(\varphi,\infty)\neq 0$.
\end{proof}

Next we compute the critical group of $\varphi$ at the origin.
\begin{prop}\label{prop6}
	If hypotheses $H(i),(iv)$ hold, then $C_k(\varphi,0)=\delta_{k,d_l}\ZZ$ for all $k\in\NN_0$ with $d_l=\mbox{dim}\,\overset{l}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(2))$.
\end{prop}
\begin{proof}
	Consider the $C^2$-functional $\hat{\psi}:H^1_0(\Omega)\rightarrow\RR$ defined by
	$$\hat{\psi}(u)=\frac{1}{2}||Du||^2_2-\int_{\Omega}F(z,u)dz\ \mbox{for all}\ u\in H^1_0(\Omega).$$
	
	Let $\psi=\hat{\psi}|_{W^{1,p}_{0}(\Omega)}$ (recall $2<p$).
	
	\begin{claim}\label{cl3}
We have		$C_k(\psi,0)=\delta_{k,d_l}\ZZ$ for all $k\in\NN_0$.
	\end{claim}
	
	Let $\eta_0\in(\hat{\lambda}_l(2),\hat{\lambda}_{l+1}(2))$ and consider the $C^2$-functional $\sigma:H^1_0(\Omega)\rightarrow\RR$ defined by
	$$\sigma(u)=\frac{1}{2}||Du||^2_2-\frac{\eta_0}{2}||u||^2_2\ \mbox{for all}\ u\in H^1_0(\Omega).$$
	
	Evidently $u=0$ is a nondegenerate critical point of $\sigma$ with Morse index $d_l$. Therefore
	\begin{equation}\label{eq37}
		C_k(\sigma,0)=\delta_{k,d_l}\ZZ\ \mbox{for all}\ k\in\NN_0.
	\end{equation}
	
	We consider the homotopy $h(t,u)$ defined by
	$$h(t,u)=(1-t)\hat{\psi}(u)+t\sigma(u)\ \mbox{for all}\ (t,u)\in[0,1]\times H^1_0(\Omega).$$
	
	Let $t\in\left(0,1\right]$ and $u\in C^1_0(\overline{\Omega})$ with $||u||_{C^1_0(\overline{\Omega})}\leq\delta$ (here $\delta>0$ is as postulated in hypothesis $H(iv)$). Let $\left\langle \cdot,\cdot\right\rangle_0$ denote the duality brackets for the pair $(H^{-1}(\Omega),H^1_0(\Omega))$. We have
	\begin{equation}\label{eq38}
		\left\langle h'_u(t,u),y\right\rangle_0=(1-t)\left\langle \hat{\psi}'(u),y\right\rangle_0+t\left\langle \sigma'(u),y\right\rangle_0\ \mbox{for all}\ y\in H^1_0(\Omega).
	\end{equation}
	
	We consider the following orthogonal direct sum decomposition of the Hilbert space $H^1_0(\Omega)$:
	$$H^1_0(\Omega)=\bar{H}_l\oplus\hat{H}_l\ \mbox{with}\ \bar{H}_l=\overset{l}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(2)),\ \hat{H}_l=\overline{{\underset{\mathrm{i\geq l+1}}\oplus}E(\hat{\lambda}_i(2))}\,.$$
	
	Then every $v\in H^1_0(\Omega)$ admits a unique sum decomposition as
	$$v=\bar{v}+\hat{v}\ \mbox{with}\ \bar{v}\in\bar{H}_l\ \mbox{and}\ \hat{v}\in\hat{H}_l.$$
	
	In (\ref{eq38}) we choose $y=\hat{u}-\bar{u}\in H^1_0(\Omega)$. Exploiting the orthogonality of the component spaces in the above decomposition, we have
	\begin{equation}\label{eq39}
		\left\langle \hat{\psi}'(u),\hat{u}-\bar{u}\right\rangle_0=||D\hat{u}||^2_2-||D\bar{u}||^2_2-\int_{\Omega}f(z,u)(\hat{u}-\bar{u})dz.
	\end{equation}
	
	Note that hypothesis $H(iv)$ implies that
$$\eta(z)\leq\frac{f(z,x)}{x}\leq\hat\lambda_{l+1}(2)\quad\mbox{for almost all $z\in\Omega$, all $0<|x|\leq\delta$.}$$
Let $y=\hat u-\bar{u}\in H^1_0(\Omega)$. Then
$$\begin{array}{ll}
f(z,u)(\hat u-\bar{u})=f(z,u)y&\displaystyle =\frac{f(z,u)}{u}uy\\

&\displaystyle \leq \left\{\begin{array}{lll} &\hat\lambda_{l+1}(2)(\hat u^2-\bar{u}^2)&\quad\mbox{if $uy>0$}\\
& \eta (z)(\hat u^2-\bar{u}^2)&\quad\mbox{if $uy<0$}\end{array}\right. \\
&\displaystyle \leq\hat\lambda_{l+1}(2)\hat u^2-\eta (z)\bar{u}^2.\end{array} $$ So, we have proved that
	\begin{equation}\label{eq40}
		f(z,u(z))(\hat{u}-\bar{u})(z)\leq\hat{\lambda}_{l+1}(2)\hat{u}(z)^2-\eta(z)\bar{u}(z)^2\ \mbox{for almost all}\ z\in\Omega\,.
	\end{equation}    
	
	Using (\ref{eq40}) in (\ref{eq39}), we obtain
	\begin{eqnarray}\label{eq41}
		\left\langle \hat{\psi}'(u),\hat{u}-\bar{u}\right\rangle_0&=&||D\hat{u}||^2_2-\hat{\lambda}_{l+1}(2)||\hat{u}||^2_2-\left[||D\bar{u}||^2_2-
\hat{\lambda}_l(2)||\bar{u}||^2_2\right]\nonumber\\
		&\geq&0\ \mbox{(see (\ref{eq5}))}.
	\end{eqnarray}
	
	Also we have
	\begin{eqnarray}\label{eq42}
		\left\langle \sigma'(u),\hat{u}-\bar{u}\right\rangle_0&=&||D\hat{u}||^2_2-\eta_0||\hat{u}||^2_2-\left[||D\bar{u}||^2_2-\eta_0||\bar{u}||^2_2\right]\nonumber\\
		&\geq&c_5||Du||^2\ \mbox{for some}\ c_5>0\ (\mbox{recall that}\ \eta_0\in(\hat{\lambda}_l(2),\hat{\lambda}_{l+1}(2))).
	\end{eqnarray}
	
	Returning to (\ref{eq39}) and using (\ref{eq41}) and (\ref{eq42}), we have
	\begin{equation}\label{eq43}
		\left\langle h'_u(t,u),\hat{u}-\bar{u}\right\rangle_0\geq tc_5||Du||^2>0\ (\mbox{recall that}\ t\in\left(0,1\right]).
	\end{equation}
	
	We claim that $0\in K_{\hat{\psi}}$ is isolated. Arguing by contradiction, suppose that the claim is false. Then we can find $\{u_n\}_{n\geq 1}\subseteq H^1_0(\Omega)$ such that
	\begin{equation}\label{eq44}
		u_n\rightarrow 0\ \mbox{in}\ H^1_0(\Omega)\ \mbox{and}\ \hat{\psi}'(u_n)=0\ \mbox{for all}\ n\in\NN\,.
	\end{equation}
	
	From the equality in (\ref{eq44}), we have
	\begin{equation}\label{eq45}
		-\Delta u_n(z)=f(z,u_n(z))\ \mbox{for almost all}\ z\in\Omega,\ u_n|_{\partial\Omega}=0\ \mbox{for all}\ n\in\NN
	\end{equation}
	
	Standard regularity theory (the Calderon-Zygmund estimates), implies that we can find $\alpha\in(0,1)$ and $M_4>0$ such that
	\begin{equation}\label{eq46}
		u_n\in C^{1,\alpha}_0(\overline{\Omega})\ \mbox{and}\ ||u_n||_{C^{1,\alpha}_0(\overline{\Omega})}\leq M_4\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	Exploiting the compact embedding of $C^{1,\alpha}_0(\overline{\Omega})$ into $C^1_0(\overline{\Omega})$, from (\ref{eq44}) and (\ref{eq46}) we have
	\begin{eqnarray}\label{eq47}
		&&u_n\rightarrow 0\ \mbox{in}\ C^1_0(\overline{\Omega})\ \mbox{as}\ n\rightarrow\infty,\nonumber\\
		&\Rightarrow&\eta(z)u_n(z)^2\leq f(z,u_n(z))u_n(z)\leq\hat{\lambda}_{l+1}(2)u_n(z)^2\\
		&&\hspace{1cm}\mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ n\geq n_0\ (\mbox{see hypothesis}\ H(iv)).\nonumber
	\end{eqnarray}
 Relation \eqref{eq47} implies that $$f(z,u_n(z))(\hat{u}_n(z)-\bar{u}_n(z))\leq\hat\lambda_{l+1}(2)\hat{u}_n(z)^2-\eta(z)\bar{u}_n(z)^2\ \mbox{for almost all $z\in\Omega$, all $n\geq n_0$.}$$
	
	Multiplying (\ref{eq45}) with $(\hat{u}_n-\bar{u}_n)(z)$, integrating over $\Omega$ and using Green's identity, the orthogonality of the component spaces and (\ref{eq47}), we have
	\begin{eqnarray*}
		&&||D\hat{u}_n||^2_2-||D\bar{u}_n||^2_2\\
		&&=\int_{\Omega}(Du_n,D\hat{u}_n-D\bar{u}_n)_{\RR^N}dz\ (\mbox{since $u_n=\hat{u}_n+\bar{u}_n$})\\
&&=\int_\Omega f(z,u_n)(\hat{u}_n-\bar{u}_n)dz\ (\mbox{see \eqref{eq45}})\\
		&&\leq\int_{\Omega}[\hat{\lambda}_{l+1}(2)\hat{u}^2_n-\eta(z)\bar{u}^2_n]dz\\
&&\Rightarrow 0\leq ||D\hat{u}_n||^2_2-\hat\lambda_{l+1}(2)||\hat u_n||_2^2\leq ||D\bar{u}_n||^2_2-\int_\Omega \eta (z)\bar u_n^2dz\leq-\hat\xi_0||\bar u_n||^2\\
&&\qquad \mbox{for some $\hat\xi_0>0$ (see Papageorgiou, R\u adulescu and Repov\v{s} \cite{prramo})},\\
		&\Rightarrow& \bar{u}_n=0\ \mbox{and}\ u_n=\hat{u}_n\in E(\hat\lambda_{l+1}(2))\ \mbox{for all $n\geq n_0$}.
	\end{eqnarray*}
	The unique continuation property of the eigenspaces implies that
$$\hat{u}_n(z)\not=0\quad\mbox{for almost all $z\in\Omega$.}$$
Then
$$\hat\lambda_{l+1}(2)||u_n||^2_2=||Du_n||^2_2=\int_\Omega f(z,u_n)u_ndz<\hat\lambda_{l+1}(2)||u_n||^2_2\quad\mbox{for all $n\geq n_0$}, $$
since $u_n(z)\not=0$ for almost all $z\in\Omega$, see hypothesis $H(iv)$. This contradiction proves that $0\in K_{\hat{\psi}}$ is isolated.

	
	
	
	Since $h(0,\cdot)=\hat{\psi}(\cdot)$, using (\ref{eq43}) we see that $u=0$ is an isolated critical  point of $h(t,\cdot)$ for all $t\in[0,1]$. Invoking Theorem 5.2 of Corvellec \& Hantoute \cite{6}, we have
	\begin{eqnarray}\label{eq48}
		&&C_k(\hat{\psi},0)=C_k(\sigma,0)\ \mbox{for all}\ k\in\NN_0,\nonumber\\
		&\Rightarrow&C_k(\hat{\psi},0)=\delta_{k,d_l}\ZZ\ \mbox{for all}\ k\in\NN_0\ (\mbox{see (\ref{eq37})}).
	\end{eqnarray}
	
	Because $W^{1,p}_{0}(\Omega)$ is dense in $H^1_0(\Omega)$, it follows that
	\begin{eqnarray}\label{eq49}
		&&C_k(\hat{\psi},0)=C_k(\psi,0)\ \mbox{for all}\ k\in\NN_0\ (\mbox{see Chang \cite[p. 14]{3}  and Palais \cite{14}}),\nonumber\\
		&\Rightarrow&C_k(\psi,0)=\delta_{k,d_l}\ZZ\ \mbox{for all}\ k\in\NN_0\ (\mbox{see (\ref{eq48})}).
	\end{eqnarray}
	
	This proves the Claim.
	
	We have
	\begin{eqnarray}
		&&|\varphi(u)-\psi(u)|\leq\frac{1}{p}||u||^p\label{eq50}\\
		&&|\left\langle \varphi'(u)-\psi'(u),h\right\rangle|\leq c_6||u||^{p-1}||h||\ \mbox{for some}\ c_6>0,\nonumber\\
		&\Rightarrow&||\varphi'(u)-\psi'(u)||_*\leq c_6||u||^{p-1}.\label{eq51}
	\end{eqnarray}
	
	From (\ref{eq50}), (\ref{eq51}) and the continuity of critical groups in the $C^1$-topology (see Corvellec \& Hantoute \cite[Theorem 5.1]{6}), we have
	\begin{eqnarray*}
		&&C_k(\varphi,0)=C_k(\psi,0)\ \mbox{for all}\ k\in\NN_0,\\
		&\Rightarrow&C_k(\varphi,0)=\delta_{k,d_l}\ZZ\ \mbox{for all}\ k\in\NN_0\ (\mbox{see (\ref{eq49})}).
	\end{eqnarray*}
\end{proof}
	
	Now we are ready for the existence theorem.
	\begin{theorem}\label{th7}
		Assume that hypotheses $H$ hold. Then problem (\ref{eq1}) admits a nontrivial solution $u_0\in C^1_0(\overline{\Omega})$.
	\end{theorem}
\begin{proof}
>From Proposition \ref{prop5} we know that
$$C_1(\varphi,\infty)\neq 0.$$

So, we can find $u_0\in W^{1,p}_{0}(\Omega)$ such that
\begin{equation}\label{eq52}
	u_0\in K_{\varphi}\ \mbox{and}\ C_1(\varphi,u_0)\neq 0.
\end{equation}

Since $\varphi\in C^2(W^{1,p}_{0}(\Omega))$ from (\ref{eq52}) and Papageorgiou \& R\u adulescu \cite{15}, it follows that
\begin{equation}\label{eq53}
	C_k(\varphi,u_0)=\delta_{k,1}\ZZ\ \mbox{for all}\ k\in\NN_0.
\end{equation}

>From Proposition \ref{prop6}, we have
\begin{equation}\label{eq54}
	C_k(\varphi,0)=\delta_{k,d_l}\ZZ\ \mbox{for all}\ k\in\NN_0.
\end{equation}

Comparing (\ref{eq53}) and (\ref{eq54}) we see that $u_0\neq 0$ (recall $l\geq 2$). Since $u_0\in K_{\varphi}$ (see (\ref{eq52})) it follows that $u_0$  is a nontrivial solution of (\ref{eq1}). Moreover, from Ladyzhenskaya \& Uraltseva \cite[Theorem 7.1, p. 286]{10}, we have $u_0\in L^{\infty}(\Omega)$ and so we can apply Theorem 1 of Lieberman \cite{11} and conclude that $u_0\in C^1_0(\overline{\Omega})$.
\end{proof}

\medskip
{\bf Acknowledgement.} The authors wish to thank a knowledgeable referee for his/her corrections and remarks that improved the paper considerably. V. R\u adulescu acknowledges the support through the grant UEFISCDI-PCCA-23/2014 funded by the Romanian Research Authority.



\begin{thebibliography}{99}

\bibitem{1} {\sc S. Aizicovici, N.S. Papageorgiou, V. Staicu}, {\em Nodal solutions for $(p,2)$-equations}, Trans. Amer. Math. Soc. \textbf{367} (2015), 7343-7372.

\bibitem{2} {\sc V. Benci, P. D'Avenia, D. Fortunato, L. Pisani}, {\em Solutions in several space dimensions: Derrick's problem and infinitely many solutions}, Arch. Rational Mech Anal. \textbf{154} (2000), 297-324.

\bibitem{3} {\sc K.C. Chang}, {\em Infinite Dimensional Morse Theory and Multiple Solutions Problems}, Birkh\"{a}user, Boston (1993).

\bibitem{4} {\sc L. Cherfils, Y. Ilyasov}, {\em On the stationary solutions of generalized reaction diffusion equations with p\&q-Laplacian},  Commun. Pure Appl. Anal. \textbf{4} (2005), 9-22.

\bibitem{5} {\sc S. Cingolani, M. Degiovanni}, {\em Nontrivial solutions for p-Laplace equations with right hand side having p-linear growth at infinity}, Commun. Partial Diff. Equations \textbf{30} (2005), 1191-1203.

\bibitem{6} {\sc J.N. Corvellec, A. Hantoute}, {\em Homotopical stability of isolated critical points of continuous functionals}, Set-Valued Anal. \textbf{10} (2002), 143-164.

\bibitem{7} {\sc M. Cuesta, D. de Figueiredo, J.P. Gossez}, {\em The begining of the Fu\v{c}ik spectrum for the p-Laplacian}, J. Differential Equations \textbf{159} (1999), 212-238.

\bibitem{8} {\sc M. Filippakis, N.S. Papageorgiou}, {\em Multiple constant sign and nodal solutions for nonlinear elliptic equations with the $p$-Laplacian}, J. Differential Equations \textbf{245} (2008), 1883-1922.

\bibitem{9} {\sc L. Gasinski, N.S. Papageorgiou}, {\em Nonlinear Analysis}, Chapman \& Hall / CRC Press, Boca Raton, Fl (2006).

\bibitem{10} {\sc O. Ladyzhenskaya, N. Uraltseva}, {\em Linear and Quasilinear Elliptic Equations}, Academic Press, New York (1968).

\bibitem{11} {\sc G. Lieberman}, {\em Boundary regularity for solutions of degenerate elliptic equations}, Nonlinear Anal. \textbf{12} (1988), 1203-1219.

\bibitem{12} {\sc D. Motreanu, V. Motreanu, N.S. Papageorgiou}, {\em Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems}, Springer, New York (2014).

\bibitem{13} {\sc D. Mugnai, N.S. Papageorgiou}, {\em Wang's multiplicity result for superlinear $(p,q)$-equations without the Ambrosetti-Rabinowitz condition},  Trans. Amer. Math. Soc. \textbf{366} (2014), 4919-4937.

\bibitem{14} {\sc R. Palais}, {\em Homotopy theory of infinite dimensional manifolds}, Topology \textbf{5} (1966), 1-16.

\bibitem{15} {\sc N.S. Papageorgiou, V.D. R\u adulescu}, {\em Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance}, Appl. Math. Optim. \textbf{69} (2014), 393-430.

\bibitem{16} {\sc N.S. Papageorgiou, V.D. R\u adulescu}, {\em Resonant $(p,2)$-equations with asymmetric reaction}, Anal. Appl. \textbf{13} (2015), 481-506.

\bibitem{prramo} {\sc N.S. Papageorgiou, V.D. R\u adulescu, D.D. Repov\v{s}}, {\em On a class of parametric $(p,2)$-equations}, Appl. Math. Optim., in press (DOI: 10.1007/s00245-016-9330-z).

\bibitem{17} {\sc M. Sun}, {\em Multiplicity of solutions for a class of quasilinear elliptic equations at resonance}, J. Math. Anal. Appl. \textbf{386} (2012), 661-668.

\bibitem{18} {\sc M. Sun, M. Zhang, J. Su}, {\em Critical groups at zero and multiple solutions for a quasilinear elliptic equation}, J. Math. Anal. Appl. \textbf{428} (2015), 696-712.


\end{thebibliography}


\end{document}