% version 2013 \documentclass[11pt]{amsart} \usepackage{graphicx, color} \usepackage{amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{mathrsfs} \textwidth=6in \textheight=9.5in \topmargin=-0.5cm \oddsidemargin=0.5cm \evensidemargin=0.5cm %\usepackage[notref,notcite]{showkeys} \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} %\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} %\hfill\today\bigskip \title[Resonant $(p,2)$--equations with asymmetric reaction]{Resonant $(p,2)$--equations with asymmetric reaction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \author[N.S. Papageorgiou]{Nikolaos S. Papageorgiou} \address{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{Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania \& Department of Mathematics, University of Craiova, Street A.I. Cuza 13, 200585 Craiova, Romania} \email{\tt vicentiu.radulescu@imar.ro} \keywords{$(p,2)$-equation, resonance, nonlinear regularity, critical groups, Picone's identity\\ \phantom{aa} 2010 AMS Subject Classification: 35B34, 35J20, 35J60, 58E05.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a $p$--Laplacian and a Laplacian, $2
2$ be a real number. In this paper we study the following nonlinear nonhomogeneous elliptic equation (($p,2$)-equation): \begin{eqnarray}\label{eq1} -\Delta_pu(z)-\Delta u(z)=f(z,u(z))\quad \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0\,. \end{eqnarray} Here $\Delta_p$ denotes the $p$--Laplacian differential operator defined by $$\Delta_pu=\mbox{div}(||Du||^{p-2}Du)\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$ Also $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory reaction (that is, for all $x\in\RR$, the mapping $z\longmapsto f(z,x)$ is measurable and for a.a. $z\in\Omega$, $ x\longmapsto f(z,x)$ is continuous). The aim of this work is to prove a multiplicity theorem (in particular, a three solutions theorem), when the reaction $f(z,\cdot)$ is $(p-1)$-linear near $\pm\infty$, but exhibits asymmetric behavior at $+\infty$ and at $-\infty$. More precisely, we assume that the quotient $\frac{f(z,x)}{|x|^{p-2}x}$ crosses the principal eigenvalue $\hat{\lambda}_1(p)>0$ of $(-\Delta_p,W^{1,p}_{0}(\Omega))$ as $x\in\RR$ moves from $-\infty$ to $+\infty$. Another interesting feature of our framework is that we allow for resonance to occur at both $+\infty$ and $-\infty$. At $+\infty$ the resonance can occur with respect to the principal eigenvalue $\hat{\lambda}_1(p)>0$, while at $-\infty$ with respect to the second eigenvalue $\hat{\lambda}_2(p)>\hat{\lambda}_1(p)$. Problems with an asymmetric nonlinearity were studied by Chabrowski and Yang \cite{5}, Chang \cite{6}, de Paiva and Massa \cite{10}, de Paiva and Presoto \cite{11}, Motreanu, Motreanu and Papageorgiou \cite{17}, Perera \cite{24} (for semilinear Dirichlet problems), by Motreanu, Motreanu and Papageorgiou \cite{18} (for nonlinear equations driven by the Dirichlet $p$-Laplacian) and by Papageorgiou and R\u{a}dulescu \cite{20} (for semilinear Neumann problems with an indefinite and unbounded potential). None of the aforementioned works permits resonance. We mention that $(p,2)$--equations (that is, equations driven by the sum of a $p$-Laplacian and a Laplacian, with $2
\rho>0$,
$$\max\{\varphi(x_0),\varphi(x_1)\}<\inf \{\varphi(x):||x-x_0||=\rho\}=\eta_{\rho}\,,$$
and $c=\inf\limits_{\gamma\in \Gamma}\max\limits_{0\leq t\leq 1}\ \varphi(\gamma(t))$, where $\Gamma=\{\gamma\in C([0,1],W^{1,p}_{0}(\Omega)):\gamma(0)=x_0,\ \gamma(1)=x_1\}$. Then $c\geq\eta_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}
In the analysis of problem (\ref{eq1}), in addition to the Sobolev space $W^{1,p}_{0}(\Omega)$, we will also use the Banach space $C^{1}_{0}(\bar{\Omega})$ defined by
$$C^{1}_{0}(\bar{\Omega})=\{u\in C^{1}_{0}(\bar{\Omega}):u|_{\partial\Omega}=0\}.$$
This is an ordered Banach space with positive cone
$$C_+=\{u\in C^{1}_{0}(\bar{\Omega}):u(z)\geq 0\ \mbox{for\ all}\ z\in\bar{\Omega}\}.$$
This cone has a nonempty interior given by
$$\mbox{int}\, C_+=\{u\in C_+:u(z)>0\ \mbox{for\ all}\ z\in\Omega,\ \frac{\partial u}{\partial n}(z)<0\ \mbox{for\ all}\ z\in\partial\Omega\}.$$
Here $n(\cdot)$ denote the outward unit normal on $\partial\Omega$.
Suppose that $f_0:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function with subcritical growth in the $x\in\RR$ variable, that is,
$$|f_0(z,x)|\leq a(z)\ (1+|z|^{r-1})\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR,$$
with $a\in L^{\infty}(\Omega)_+,\ 1 0$. We have
\begin{eqnarray}\label{eq37}
u^{+}_{n}(z)\rightarrow+\infty\ \mbox{for\ a.a.}\ z\in\Omega.
\end{eqnarray}
By virtue of hypothesis $H_1(ii)$, given $\xi>0$, we can find $M_6=M_6(\xi)>0$ such that
\begin{eqnarray}\label{eq38}
f(z,x)x-pF(z,x)\geq\xi\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_6.
\end{eqnarray}
For a.a. $z\in\Omega$ and all $x\geq M_6$, we have
\begin{eqnarray}\label{eq39}
\frac{d}{dx}\ \frac{F(z,x)}{x^p}&=&\frac{f(z,x)x^p-pF(z,x)x^{p-1}}{x^{2p}}\nonumber\\
&=&\frac{f(z,x)x-pF(z,x)}{x^{p+1}}\nonumber\\
&\geq&\frac{\xi}{x^{p+1}}\ (\mbox{see\ (\ref{eq38})})\nonumber\\
\Rightarrow&&\frac{F(z,x)}{v^p}-\frac{F(z,x)}{x^p}\geq-\frac{\xi}{p}\ \left[\frac{1}{v^p}-\frac{1}{x^p}\right]\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ v\geq x\geq M_6.
\end{eqnarray}
So, if in (\ref{eq39}) we let $v\rightarrow+\infty$ and use (\ref{eq33}), we obtain
\begin{eqnarray}
&&\frac{\hat{\lambda}_1(p)}{p}-\frac{F(z,x)}{x^p}\geq\frac{\xi}{p}\ \frac{1}{x^p}\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_6\nonumber\\
&\Rightarrow&\frac{\hat{\lambda}_1(p)}{p}\ x^p-F(z,x)\geq\frac{\xi}{p}\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_6\nonumber\\
&\Rightarrow&\lim\limits_{x\rightarrow+\infty}\left[\frac{\hat{\lambda}_1(p)}{p}x^p-F(z,x)\right]
\geq\frac{\xi}{p}\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.\nonumber
\end{eqnarray}
But $\xi>0$ is arbitrary. So, we conclude that
$$\lim\limits_{x\rightarrow+\infty}\left[\frac{\hat{\lambda}_1(p)}{p}x^p-F(z,x)\right]=+\infty\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.$$
Thus, using (\ref{eq37}), we have
\begin{eqnarray}\label{eq40}
&&\frac{\hat{\lambda}_1(p)}{p}u^{+}_{n}(z)^p-F(z,u^{+}_{n}(z))\rightarrow+\infty\ \mbox{for\ a.a.}\ z\in\Omega\nonumber\\
&\Rightarrow&\int_{\Omega}\left[\frac{\hat{\lambda}_1(p)}{p}u^{+}_{n}(z)^p-F_+(z,u^{+}_{n}(z))\right]dz\rightarrow+\infty\ (\mbox{by\ Fatou's\ Lemma}).
\end{eqnarray}
Recall that
\begin{eqnarray}\label{eq41}
&&\frac{1}{p}||Du_n||^{p}_{p}+\frac{1}{2}||Du_n||^{2}_{2}-\int_{\Omega}F_+(z,u^{+}_{n})dz\leq M_4\ \mbox{all}\ n\geq 1\nonumber\\
&&\int_{\Omega}\left[\frac{\hat{\lambda}_1(p)}{p}(u^{+}_{n})^p-F_+(z,u^{+}_{n})\right]dz\leq M_4\ \mbox{for\ all}\ n\geq 1\ (\mbox{see\ (\ref{eq4})}).
\end{eqnarray}
Comparing (\ref{eq40}) and (\ref{eq41}), we have a contradiction which proves that $\varphi_+$ is coercive.
\end{proof}
Now we can produce a first solution for problem (\ref{eq1}) which is positive.
\begin{prop}\label{prop9}
Assume that hypotheses $H_1$ hold. Then problem (\ref{eq1}) admits a positive solution $u_0\in{\rm int}\, C_+$ which is a local minimizer of the energy functional $\varphi$.
\end{prop}
\begin{proof}
From Proposition \ref{prop8}, we know that $\varphi_+$ is coercive. Also, using the Sobolev embedding theorem, we see that $\varphi_+$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $u_0\in W^{1,p}_{0}(\Omega)$ such that
\begin{eqnarray}\label{eq42}
\varphi_+(u_0)=\inf\left\{\varphi_+(u):u\in W^{1,p}_{0}(\Omega)\right\}.
\end{eqnarray}
Hypothesis $H_1(iv)$ implies that given $\epsilon>0$, we can find $\delta=\delta(\epsilon)>0$ such that
\begin{eqnarray}\label{eq43}
F(z,x)\geq\frac{1}{2}(\beta(z)-\epsilon)x^2\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta.
\end{eqnarray}
Since $\hat{u}_1(2)\in\mbox{int}\, C_+$, for $\lambda\in(0,1)$ small we have $\lambda\hat{u}_1(2)(z)\in[0,\delta]$ for all $z\in\bar{\Omega}$. Then
\begin{eqnarray}
\varphi_+(\lambda\hat{u}_1(2))&=&\frac{\lambda^p}{p}||D\hat{u}_1(2)||^{p}_{p}+\frac{\lambda^2}{2}\hat{\lambda}_1(2)-\int_{\Omega}F_+(z,\lambda\hat{u}_1(2))dz\nonumber\\
&\leq&\frac{\lambda^p}{p}||D\hat{u}_1(2)||^{p}_{p}-\frac{\lambda^2}{2}\left[\int_{\Omega}(\beta(z)-\hat{\lambda}_1(2))\hat{u}_1(2)^2dz+\epsilon\hat{\lambda}_1(2)\right]\ \mbox{(see\ (\ref{eq43})).}\nonumber
\end{eqnarray}
The hypothesis on $\beta(\cdot)$ (see $H_1(iv)$) and since $\hat{u}_1(2)\in\mbox{int}\, C_+$, imply that
$$\xi_{\ast}=\int_{\Omega}(\beta(z)-\hat{\lambda}_1(2))\hat{u}_1(2)^2dz>0.$$
So, if we choose $\epsilon\in(0,\frac{\xi_{\ast}}{\hat{\lambda}_1(2)})$, then
\begin{eqnarray}
&&\varphi_+(\lambda\hat{u}_1(2))<0\nonumber\\
\Rightarrow&&\varphi_+(u_0)<0=\varphi_+(0)\ (\mbox{see\ (\ref{eq42})}), \mbox{hence}\ u_0\neq 0.\nonumber
\end{eqnarray}
From (\ref{eq42}) we have
\begin{eqnarray}\label{eq44}
&&\varphi'_{+}(u_0)=0\nonumber\\
\Rightarrow&&A_p(u_n)+A(u_n)=N_f(u_0).
\end{eqnarray}
On (\ref{eq44}) we act with $-u^{-}_{n}\in W^{1,p}_{0}(\Omega)$. We obtain
$$||Du^{-}_{n}||^{p}_{p}+||Du^{-}_{n}||^{2}_{2}=0,\ \mbox{hence}\ u_0\geq 0,\ u_0\neq 0.$$
Then from (\ref{eq44}) we have
$$-\Delta_pu_0(z)-\Delta u_0(z)=f(z,u_0(z))\ \mbox{a.e.\ in}\ \Omega,\ u_0|_{\partial\Omega}=0.$$
From Ladyzhenskaya and Uraltseva \cite[p. 286]{14}, we know that $u_0\in L^{\infty}(\Omega)$. Then Theorem \ref{th1} of Lieberman \cite{15} implies that $u_0\in C_+\backslash\{0\}$.
Evidently hypotheses $H_1(i),\ (iv)$ imply that for every $\rho>0$, we can find $\hat{\xi}_{\rho}>0$ such that $f(z,x)x+\xi_{\rho}|x|^p\geq 0$ for a.a. $z\in\Omega$, all $|x|\leq\rho$. Let $\rho=||u_0||_{\infty}$ and let $\hat{\xi}_{\rho}>0$ as just mentioned. We have
\begin{eqnarray}\label{eq45}
&&-\Delta_pu_0(z)-\Delta u_0(z)+\hat{\xi}_{\rho}u_0(z)^{p-1}\nonumber\\
&&=f(z,u_0(z))+\hat{\xi}_{\rho}u_0(z)^{p-1}\geq 0\ \mbox{for\ a.a.}\ z\in\Omega\nonumber\\
\Rightarrow&&\Delta_pu_0(z)+\Delta u_0(z)\leq\xi_{\rho}u_0(z)^{p-1}\ \mbox{for\ a.a.}\ z\in\Omega.
\end{eqnarray}
Let $a:\RR^N\rightarrow\RR^N$ be the $C^1$-map defined by
$$a(y)=||y||^{p-2}y+y\ (\mbox{recall}\ p>2).$$
We have $\mbox{div}\ a(Du)=\Delta_pu+\Delta u$ for all $u\in W^{1,p}_{0}(\Omega)$ and
\begin{eqnarray}
\nabla a(y)&=&||y||^{p-2}\left(I+(p-2)\frac{y\otimes y}{||y||^2}\right)+I\ \mbox{for\ all}\ y\in\RR^N\nonumber\\
&\Rightarrow&(\nabla a(y)\xi,\xi)_{\RR^N}\geq||\xi||^2\ \mbox{for\ all}\ y,\xi\in\RR^N.\nonumber
\end{eqnarray}
Then the tangency principle of Pucci and Serrin \cite[p. 35]{25} implies that
$$u_0(z)>0\ \mbox{for\ all}\ z\in\Omega.$$
So, from (\ref{eq45}) and the boundary point theorem of Pucci and Serrin \cite[p. 120]{25}, we conclude that $u_0\in\mbox{int}\, C_+$.
Note that $\varphi|_{C_+}=\varphi_+|_{C_+}$. So, $u_0\in\mbox{int}\ C_+$ is a local $C^{1}_{0}(\bar{\Omega})$-minimizer of $\varphi$. Invoking Proposition \ref{prop2}, we conclude that $u_0$ is a local $W^{1,p}_{0}(\Omega)$-minimizer of $\varphi$.
\end{proof}
To produce a second nontrivial solution, we need to restrict the behavior of $f(z,\cdot)$ near zero. More precisely, the new hypotheses on the reaction $f(z,x)$ are the following:
\smallskip
$\underline{H_2}:\ f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$, hypotheses $H_2(i),\ (ii),\ (iii),\ (v)$ are the same as the corresponding hypotheses $H_1(i),\ (ii),\ (iii),\ (v)$ and
\begin{itemize}
\item[(iv)]there exist an integer $m\geq 2$ and functions $\beta,\hat{\beta}\in L^{\infty}(\Omega)_+$ such that
\begin{eqnarray}
&&\hat{\lambda}_m(2)\leq\beta(z)\leq\hat{\beta}(z)\leq\hat{\lambda}_{m+1}(2)\ \mbox{a.e.\ in}\ \Omega,\ \hat{\lambda}_m(2)\neq\beta,\ \hat{\lambda}_{m+1}(2)\neq\hat{\beta}\ \mbox{and}\nonumber\\
&&\beta(z)\leq\liminf\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\limsup\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\hat{\beta}(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.\nonumber
\end{eqnarray}
\end{itemize}
\begin{theorem}\label{th10}
Assume that hypotheses $H_2$ hold. Then problem (\ref{eq1}) admits at least two nontrivial solutions
$$u_0\in{\rm int}\, C_+\ \mbox{and}\ \hat{u}\in C^{1}_{0}(\bar{\Omega}),\ u_0\neq\hat{u}.$$
\end{theorem}
\begin{proof}
From Proposition \ref{prop9} we already have one nontrivial solution $u_0\in\mbox{int}\, C_+$, which is a local minimizer of $\varphi$. Hence as in Aizicovici, Papageorgiou and Staicu \cite{1} (see the proof of Proposition 29), we can find $\rho\in(0,1)$ small such that
\begin{eqnarray}\label{eq46}
\varphi(u_0)<\inf \left\{\varphi(u):||u-u_0||=\rho\right\}=m_{\rho}\,.
\end{eqnarray}
Hypothesis $H_2(iii)$ implies that
\begin{eqnarray}\label{eq47}
\varphi(t\hat{u}_1(p))\rightarrow-\infty\ \mbox{as}\ t\rightarrow-\infty.
\end{eqnarray}
Recall that $\varphi$ satisfies the $C$-condition (see Proposition \ref{prop7}). This fact together with (\ref{eq46}) and (\ref{eq47}) permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $\hat{u}\in W^{1,p}_{0}(\Omega)$ such that
\begin{eqnarray}\label{eq48}
\hat{u}\in K_{\varphi}\ \mbox{and}\ \varphi(u_0)0$ such that
$$h_t(x)\leq\gamma_0\Rightarrow(1+||x||_X)||(h_t)'(x)||_{X^{\ast}}\geq\delta_0(||x||^{q}_{X}+||x||^{p}_{X})\ \mbox{for\ all}\ t\in[0,1].$$
Then $C_k(h_0,\infty)=C_k(h_1,\infty)$ for all $k\geq 0$.
\end{prop}
\begin{proof}
Since $h\in C^1([0,1]\times X)$, we know that it admits a pseudo-gradient vector field $\hat{v}_t(x)$ (see, for example, Gasinski and Papageorgiou \cite[p. 616]{12}). From the construction of the pseudo-gradient vector field we deduce that
$$\hat{v}_t(x)=(\partial_t h_t(x),\ v_t(x)),$$
with $(t,x)\longmapsto v_t(x)$ locally Lipschitz and for all $t\in[0,1],\ v_t(\cdot)$ is the pseudo-gradient vector field corresponding to $h_t(\cdot)$. So, for all $t\in[0,1]$ and all $x\in X$, we have
\begin{eqnarray}\label{eq6}
||(h_t)'(x)||^{2}_{X^{\ast}}\leq\left\langle (h_t)'(x),v_t(x)\right\rangle\ \mbox{and}\ ||v_t(x)||_{X}\leq 2||(h_t)'(x)||_{X^{\ast}}\,.
\end{eqnarray}
Given $t\in[0,1]$, we consider the map $w_t:X\rightarrow X$ defined by
$$w_t(x)=-\frac{|\partial_t h_t(x)|}{||(h_t)'(x)||^{2}_{X^{\ast}}}v_t(x)\ \mbox{for\ all}\ x\in X.$$
Evidently, $[t,x]\longmapsto w_t(x)$ is well-defined and locally Lipschitz. Let $\gamma\leq \gamma_0$ be such that
$${h_0}^\gamma\neq\O\ \mbox{or}\ {h_1}^\gamma\neq\O\,.$$
If we can not find such a $\gamma\leq\gamma_0$, then $C_k(h,\infty)=C_k(h,\infty)=\delta_{k,0}\ZZ\ \mbox{for\ all}\ k\geq 0$.
Assume that $h^{\gamma}_{0}\neq\O$ and let $y\in h^{\gamma}_{0}$. We consider the following abstract Cauchy problem
\begin{eqnarray}\label{eq7}
\frac{d\sigma}{dt}=w_t(\sigma)\ \mbox{on}\ [0,1],\ \sigma(0)=y.
\end{eqnarray}
Problem (\ref{eq7}) admits a local flow $\sigma(t,y)$ (see, for example, Gasinski and Papageorgiou \cite[p. 618]{12}). In what follows, for notational simplicity, we drop $y$ from the description of $\sigma$. Using the chain rule, we have
$$ \begin{array}{ll}
\displaystyle \frac{d}{dt}\ h_t(\sigma)&=\displaystyle\left\langle (h_t)'(\sigma),\ \frac{d\sigma}{dt}\right\rangle+\partial_t h_t(\sigma)\\
&\displaystyle=\left\langle (h_t)'(\sigma),\ \frac{-|\partial_t h_t(\sigma)|}{||(h_t)'(\sigma)||^{2}_{X^{\ast}}}\ v_t(\sigma)\right\rangle+\partial_t h_t(\sigma)\\
&\displaystyle\leq -|\partial_t h_t(\sigma)|+\partial_t h_t(\sigma)\ (\mbox{see}\ (\ref{eq6}))\\
&\displaystyle\leq 0\\
\Rightarrow&\displaystyle t\longmapsto h_t(\sigma)\ \mbox{is\ nonincreasing}.
\end{array}$$
Hence for $t\geq 0$ small, we have
\begin{eqnarray}\label{eq8}
&&h_t(\sigma(t))\leq h_0(\sigma(0))=h_0(y)\leq\gamma\leq\gamma_0\nonumber\\
\Rightarrow&&(1+||\sigma(t)||_X)\ ||(h_t)'(\sigma(t))||_{X^{\ast}}\ \geq\ \delta_0(||\sigma(t)||^{q}_{X}+||\sigma(t)||^{p}_{X}).
\end{eqnarray}
Then
$$\begin{array}{ll}
\displaystyle |w_t(\sigma(t))|&\displaystyle\leq\ \frac{|\partial_t h_t(\sigma(t))|}{||(h_t)'(\sigma(t))||^{2}_{X^{\ast}}}\ ||v_t(\sigma(t))||_{X}\\
&\displaystyle\leq\ \frac{C_1(||\sigma(t)||^{q}_{X}+||\sigma(t)||^{p}_{X})}{||(h_t)'(\sigma(t))||^{2}_{X^{\ast}}}\ 2||(h_t)'(x)||_{X^{\tau}}\ (\mbox{see}\ (\ref{eq6}))\\
&\displaystyle\leq\ \frac{C_1(||\sigma(t)||^{q}_{X}+||\sigma(t)||^{p}_{X})}{\delta_0\left(||\sigma(t)||^{q}_{X}+||\sigma(t)||^{p}_{X}\right)}\ (1+||\sigma(t)||_X)\ (\mbox{see}\ (\ref{eq8}))\\
&\displaystyle=\ \frac{C_1}{\delta_0}(1+||\sigma(t)||_X)\ \mbox{for\ all}\ t\in[0,1]\ \mbox{small}.
\end{array}$$
This means that the flow in (\ref{eq7}) is global on $[0,1]$.
Then $\sigma(t,\cdot)$ is a homeomorphism of $h^{\gamma}_{0}$ onto a subset $D_0$ of $h^{\gamma}_{1}$. Also, reversing the time $t\rightarrow 1-t$ and using the corresponding global flow $\sigma_{\ast}(\cdot,v)$ (here $v\in h^{\gamma}_{1}$), we deduce that $h^{\gamma}_{1}$ is homeomorphic to a subset $D_1$ of $h^{\gamma}_{0}$.
Let
$$\eta(t,y)=\sigma_{\ast}(t,\sigma(1,y))\ \mbox{for\ all}\ (t,y)\in[0,1]\times h^{\gamma}_{0}.$$
Then we have
\begin{eqnarray}\label{eq9}
\eta(0,\cdot)\ \mbox{is\ homotopy \ equivalent\ to\ id}|_{D_0}(\cdot)\ \mbox{and}\ \eta(1,\cdot)=(\sigma_{\ast})_1\circ\sigma_1(\cdot).
\end{eqnarray}
Similarly, if
$$\eta_{\ast}(t,v)=\sigma(t,\sigma_{\ast}(1,v))\ \mbox{for\ all}\ (t,v)\in[0,1]\times h^{\gamma}_{1},$$
then
\begin{eqnarray}\label{eq10}
\eta_{\ast}(0,\cdot)\ \mbox{is\ homotopy\ equivalent\ to\ id}|_{D_1}(\cdot)\ \mbox{and}\ \eta_{\ast}(1,\cdot)=\sigma_1\circ(\sigma_{\ast})_1(\cdot)\,.
\end{eqnarray}
Recall that $D_0$ and $H^{\gamma}_{0}$ are homeomorphic. Similarly $D_1$ and $h^{\gamma}_{1}$ are homeomorphic. Combining these facts with (\ref{eq9}) and (\ref{eq10}), we infer that the level sets $h^{\gamma}_{0}$ and $h^{\gamma}_{1}$ are homotopy equivalent. Therefore
\begin{eqnarray}
&&H_k(X,h^{\gamma}_{0})=H_k(X,h^{\gamma}_{1})\ \mbox{for\ all}\ k\geq 0\ (\mbox{see\ Granas\ and\ Dugundji\ \cite[p. 387]{13}})\nonumber\\
\Rightarrow&&C_k(h_0,\infty)=C_k(h,\infty)\ \mbox{for\ all}\ k\geq 0.\nonumber
\end{eqnarray}
This completes the proof.
\end{proof}
\section{Two Nontrivial Solutions}
In this section we establish the existence of two nontrivial solutions for problem (\ref{eq1}) without imposing any differentiability condition on $f(z,\cdot)$.
First we produce a positive solution. To this end, we impose the following conditions on the reaction $f(z,x)$:
$\underline{H_1}:\ f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
\item[(i)]$|f(z,x)|\leq a(z)(1+|x|^{p-1})$ for a.a. $z\in\Omega$, all $x\in\RR$ with $a\in L^{\infty}(\Omega)_+$;
\item[(ii)]$\limsup\limits_{x\rightarrow+\infty}\frac{f(z,x)}{x^{p-1}}\leq\hat{\lambda}_1(p)$ uniformly for a.a. $z\in\Omega$ and if $F(z,x)=\int^{x}_{0}f(z,s)ds$, then
$$\lim\limits_{x\rightarrow+\infty}[f(z,x)x-pF(z,x)]=+\infty\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;$$
\item[(iii)]there exists a function $\eta\in L^{\infty}(\Omega)_+$ such that
\begin{eqnarray}
&&\hat{\lambda}_1(p)\leq\eta(z)\ \mbox{for\ a.a.}\ z\in\Omega,\ \hat{\lambda}_1(p)\neq\eta\ \mbox{and}\nonumber\\
&&\eta(z)\leq\liminf\limits_{x\rightarrow-\infty}\frac{f(z,x)}{|x|^{p-2}x}\leq\limsup\limits_{x\rightarrow-\infty}\frac{f(z,x)}{|x|^{p-2}x}\leq\hat{\lambda}_2(p)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;\nonumber
\end{eqnarray}
\item[(iv)]there exist functions $\beta,\hat{\beta}\in L^{\infty}(\Omega)_+$ such that
\begin{eqnarray}
&&\hat{\lambda}_1(2)\leq\beta(z)\ \mbox{for\ a.a.}\ z\in\Omega,\ \hat{\lambda}_1(2)\neq\beta\ \mbox{and}\nonumber\\
&&\beta(z)\leq\liminf\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\limsup\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\hat{\beta}(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;\nonumber
\end{eqnarray}
\item[(v)]$f(z,x)x-pF(z,x)\geq 0$ for a.a. $z\in\Omega$, all $x\leq 0$, and $f(z,\cdot)$ is lower locally Lipschitz on $[0,+\infty)$.
\end{itemize}
\smallskip
Let $\varphi:W^{1,p}_{0}(\Omega)\rightarrow\RR$ be the energy functional for problem (\ref{eq1}) defined by
$$\varphi(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}F(z,u(z))dz\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
\begin{prop}\label{prop7}
Assume that hypotheses $H_1$ 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 such that
\begin{eqnarray}
&&|\varphi(u_n)|\leq M_1\ \mbox{for\ all}\ n\geq 1,\ \mbox{some}\ M_1>0\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{eq12}) we have
\begin{eqnarray}\label{eq13}
&&\left|\left\langle A_p(u_n),h\right\rangle-\int_{\Omega}f(z,u_n)h\ dz\right|\leq\frac{\epsilon_n||h||}{1+||u_n||}\ \mbox{for\ all}\ h\in W^{1,p}_{0}(\Omega)\ \mbox{with}\ \epsilon_n\rightarrow 0^+.
\end{eqnarray}
In (\ref{eq13}) we choose $h=u^{+}_{n}\in W^{1,p}_{0}(\Omega)$. Then
\begin{eqnarray}\label{eq14}
&&\left|||Du^{+}_{n}||^{p}_{p}+||Du^{+}_{n}||^{2}_{2}-\int_{\Omega}f(z,u^{+}_{n})u^{+}_{n}dz\right|\leq\epsilon_n\ \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
Using (\ref{eq14}) we will show that the sequence $\{u^{+}_{n}\}_{n\geq 1}\subset W^{1,p}_{0}(\Omega)$ is bounded. Arguing indirectly, suppose that the sequence is not bounded in $W^{1,p}_{0}(\Omega)$. Then by passing to a subsequence if necessary, we may assume that $||u^{+}_{n}||\rightarrow\infty$. Let $y_n=\frac{u^{+}_{n}}{||u^{+}_{n}||}\ n\geq 1$. Then $||y_n||=1$ and $y_n\geq 0$ for all $n\geq 1$. We may assume that
\begin{eqnarray}\label{eq15}
y_n\stackrel{w}{\longrightarrow}y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^{p}(\Omega).
\end{eqnarray}
From (\ref{eq14}), we have
\begin{eqnarray}\label{eq16}
||Dy_n||^{p}_{p}\leq\frac{\epsilon_n}{||u^{+}_{n}||^p}+\int_{\Omega}\frac{f(z,u^{+}_{n})}{||u^{+}_{n}||^{p-1}}y_ndz\ \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
From hypothesis $H_1(i)$ it is clear that
$$\left\{\frac{N_f(u^{+}_{n})}{||u^{+}_{n}||^{p-1}}\right\}_{n\geq 1}\subseteq L^{p'}(\Omega)\ \mbox{is\ bounded.}$$
So, we may assume that
\begin{eqnarray}\label{eq17}
\frac{N_f(u^{+}_{n})}{||u^{+}_{n}||^{p-1}}\ \stackrel{w}{\longrightarrow}g\ \mbox{in}\ L^{p'}(\Omega)\ \mbox{as}\ n\rightarrow\infty.
\end{eqnarray}
Using hypothesis $H_1(ii)$, as in Aizicovici, Papageorgiou and Staicu \cite{1}, we show that
\begin{eqnarray}\label{eq18}
g(z)=\vartheta(z)y(z)^{p-1}\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{with}\ \vartheta\in L^{\infty}(\Omega),\ \vartheta(z)\leq\hat{\lambda}_1(p)\ \mbox{a.e.\ in}\ \Omega.
\end{eqnarray}
Hence, if in (\ref{eq16}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq15}), (\ref{eq17}),\ (\ref{eq18}) we obtain
\begin{eqnarray}\label{eq19}
||Dy||-\int_{\Omega}\vartheta(z)y^p dz\leq 0.
\end{eqnarray}
If $\vartheta\neq\hat{\lambda}_1(p)$, then from (\ref{eq19}) and Lemma \ref{lem5}, we have
$$c_0||y||^p\leq 0,\ \mbox{hence}\ y=0.$$
Then from (\ref{eq16}) it follows that $Dy_n\rightarrow 0$ in $L^p(\Omega,\RR^N)$ and so $y_n\rightarrow 0$ in $W^{1,p}_{0}(\Omega)$, a contradiction to the fact that $||y_n||=1$ for all $n\geq 1$.
Next suppose that $\vartheta(z)=\hat{\lambda}_1(p)$ a.e. in $\Omega$. Then from (\ref{eq19}) and (\ref{eq4}) we have
\begin{eqnarray}
&&||Dy||^{p}_{p}=\hat{\lambda}_1(p)||y||^{p}_{p}\nonumber\\
\Rightarrow&&y=\xi\hat{u}_1(p)\ \mbox{for\ some}\ \xi>0\ \ (\mbox{see\ \eqref{eq4}}).\nonumber
\end{eqnarray}
Since $y\in\mbox{int}\, C_+$, we have $u^{+}_{n}(z)\rightarrow+\infty$ for a.a. $z\in\Omega$ and so by virtue of hypothesis $H_1(ii)$ we have
\begin{eqnarray}\label{eq20}
&&f(z,u^{+}_{n}(z))u^{+}_{n}(z)-pF(z,u^{+}_{n}(z))\rightarrow\infty\ \mbox{for\ a.a.}\ z\in\Omega\nonumber\\
\Rightarrow&&\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}-pF(z,u^{+}_{n})\right]dz\rightarrow+\infty\ (\mbox{by\ Fatou's\ lemma}).
\end{eqnarray}
On the other hand from (\ref{eq11}) we have
\begin{eqnarray}\label{eq21}
||Du_n||^{p}_{p}+\frac{p}{2}||Du_n||^{2}_{2}-\int_{\Omega}pF(z,u_n)dz\leq p\ M_1\ \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
Also from (\ref{eq13}) with $h=u_n\in W^{1,p}_{0}(\Omega)$, we obtain
\begin{eqnarray}\label{eq22}
-||Du_n||^{p}_{p}-||Du_n||^{2}_{2}+\int_{\Omega}f(z,u_n)u_ndz\leq\epsilon_n\ \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
Adding (\ref{eq21}) and (\ref{eq22}) we have
\begin{eqnarray}\label{eq23}
&&\left(\frac{p}{2}-1\right)||Du_n||^{2}_{2}+\int_{\Omega}\left[f(z,u_n)u_n-pF(z,u_n)\right]dz\leq M_2\ \mbox{for\ some}\ M_2>0\nonumber\\
&&\hspace{10cm}\mbox{all}\ n\geq 1\nonumber\\
&\Rightarrow&\int_{\Omega}\left[f(z,u_n)u_n-pF(z,u_n)\right]dz\leq M_2\ \mbox{for\ all}\ n\geq 1\ (\mbox{recall}\ p>2)\nonumber\\
&\Rightarrow&\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}-pF(z,u^{+}_{n})\right]dz+
\int_{\Omega}\left[f(z,-u^{-}_{n})(-u^{-}_{n})-pF(z,-u^{-}_{n})\right]dz\leq M_2\nonumber\\
&&\hspace{10cm}\mbox{for\ all}\ n\geq 1\nonumber\\
&\Rightarrow&\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}-pF(z,u^{+}_{n})\right]dz\leq M_2\ \mbox{for\ all}\ n\geq 1\ (\mbox{see}\ H_1(v)).
\end{eqnarray}
Comparing (\ref{eq20}) and (\ref{eq23}), we reach a contradiction which proves that
\begin{eqnarray}\label{eq24}
\{u^{+}_{n}\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is\ bounded}.
\end{eqnarray}
Next we show that $\{u^{-}_{n}\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ is bounded. Again we argue by contradiction. So, assume that $||u^{-}_{n}||\rightarrow\infty$ and let $v_n=\frac{u^{-}_{n}}{||u^{-}_{n}||}\ n\geq 1$. Then $||v_n||=1,\ v_n\geq 0$ for all $n\geq 1$ and so we may assume that
\begin{eqnarray}\label{eq25}
v_n\stackrel{w}{\longrightarrow} v\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ v_n\rightarrow v\ \mbox{in}\ L^{p}(\Omega),\ v\geq 0.
\end{eqnarray}
From (\ref{eq13}) and (\ref{eq24}), we have
\begin{eqnarray}\label{eq26}
&&\left|\left\langle A_p(-u^{-}_{n}),h\right\rangle+\left\langle A(-u^{-}_{n}),h\right\rangle-\int_{\Omega}f(z,-u^{-}_{n})h\ dz\right|\leq M_3||h||\nonumber\\
&&\hspace{8cm}\mbox{for\ some}\ M_3>0,\ \mbox{all}\ n\geq 1\nonumber\\
&\Rightarrow&\left|\left\langle A_p(-v_{n}),h\right\rangle+\frac{1}{||u^{-}_{n}||^{p-2}}\left\langle A(-v_n),h\right\rangle-\int_{\Omega}\frac{f(z,-u^{-}_{n})}{||u^{-}_{n}||^{p-1}}h\ dz\right|\leq\frac{M_3||h||}{||u^{-}_{n}||^{p-1}}\\
&&\hspace{10.5cm}\mbox{for\ all}\ n\geq 1.\nonumber
\end{eqnarray}
Hypothesis $H_1(i)$ implies that
$$\left\{\frac{N_f(-u^{-}_{n})}{||u^{-}_{n}||^{p-1}}\right\}_{n\geq 1}\subseteq L^{p'}(\Omega)\ \mbox{is\ bounded}.$$
Passing to a subsequence if necessary and using hypothesis $H_1(iii)$ we have
\begin{eqnarray}\label{eq27}
\frac{N_f(-u^{-}_{n})}{||u^{-}_{n}||^{p-1}}\ \stackrel{w}{\longrightarrow}\ -\hat{\eta}v^{p-1}\ \mbox{in}\ L^{p'}(\Omega)\ \mbox{with}\ \eta(z)\leq\hat{\eta}(z)\leq\hat{\lambda}_2(p)\ \mbox{for\ a.a.}\ z\in\Omega.
\end{eqnarray}
In (\ref{eq26}) we choose $h=v-v_n\in W^{1,p}_{0}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq25}) and (\ref{eq27}). Since $p>2$, we obtain
\begin{eqnarray}\label{eq28}
&&\lim\limits_{n\rightarrow\infty}\left\langle A_p(-v_n),\ v-v_n\right\rangle=0\nonumber\\
&\Rightarrow&v_n\rightarrow v\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ (\mbox{see\ Proposition\ \ref{prop3}}),\ \mbox{hence}\ ||v||=1,\ v\geq 0.
\end{eqnarray}
Therefore, if in (\ref{eq26}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq27}) and (\ref{eq28}) and the fact that $p>2$, we deduce that
\begin{eqnarray}\label{eq29}
\left\langle A_p(-v),h\right\rangle&=&\int_{\Omega}-\hat{\eta}v^{p-1}h\ dz\ \mbox{for\ all}\ h\in W^{1,p}_{0}(\Omega)\nonumber\\
&\Rightarrow&A_p(v)=\hat{\eta}v^{p-1}\nonumber\\
&\Rightarrow&-\Delta_p v(z)=\hat{\eta}(z)v(z)^{p-1}\ \mbox{a.e.\ in}\ \Omega,\ v|_{\partial\Omega}=0.
\end{eqnarray}
From Proposition \ref{prop4}, we have
$$\hat{\lambda}_1(p,\hat{\eta})<\hat{\lambda}_1(p,\hat{\lambda}_1(p))=1.$$
So, from (\ref{eq29}) it follows that $v$ must be nodal, which contradicts (\ref{eq28}). This proves that
\begin{eqnarray}
&&\{u^{-}_{n}\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is\ bounded}\nonumber\\
&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is\ bounded\ (see\ (\ref{eq24})).}\nonumber
\end{eqnarray}
Hence, we may assume that
\begin{eqnarray}\label{eq30}
u_n\stackrel{w}{\longrightarrow} u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ u_n\rightarrow u\ \mbox{in}\ L^{p}(\Omega).
\end{eqnarray}
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{eq30}). 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\nonumber\\
&\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\ (\mbox{since}\ A\ \mbox{is\ monotone})\nonumber\\
&\Rightarrow&\limsup\limits_{n\rightarrow\infty}\left\langle A_p(u_n),u_n-u\right\rangle\leq 0\nonumber\\
&\Rightarrow&u_n\rightarrow u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ (\mbox{see\ Proposition\ \ref{prop3}}).\nonumber
\end{eqnarray}
This proves that $\varphi$ satisfies the $C$-condition.
\end{proof}
We consider the positive truncation of $f(z,\cdot)$ defined by
$$f_+(z,x)=f(z,x^+).$$
This is a Carath\'eodory function. We set $F_+(z,x)=\int^{x}_{0}f_+(z,s)ds$ and consider the $C^1$-functional $\varphi_+:W^{1,p}_{0}(\Omega)\rightarrow\RR$ defined by
$$\varphi_+(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}F_+(z,u(z))dz\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
\begin{prop}\label{prop8}
Assume that hypotheses $H_1$ hold. Then the functional $\varphi_+$ is coercive.
\end{prop}
\begin{proof}
We argue indirectly. So, suppose that $\varphi_+$ is not coercive. Then we can find $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ and $M_4>0$ such that
$$||u_n||\rightarrow\infty\ \mbox{as}\ n\rightarrow\infty\ \mbox{and}\ \varphi_+(u_n)\leq M_4\ \mbox{for\ all}\ n\geq 1.$$
Let $y_n=\frac{u_n}{||u_n||},\ n\geq 1$. Then $||y_n||=1$ for all $n\geq 1$ and so we may assume that
\begin{eqnarray}\label{eq31}
y_n\stackrel{w}{\longrightarrow} y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^{p}(\Omega).
\end{eqnarray}
We have
\begin{eqnarray}\label{eq32}
&&\frac{1}{p}||Du_n||^{p}_{p}+\frac{1}{2}||Du_n||^{2}_{2}-\int_{\Omega}F_+(z,u_n)dz\leq M_4\ \mbox{for\ all}\ n\geq 1\nonumber\\
&\Rightarrow&\frac{1}{p}||Dy_n||^{p}_{p}+\frac{1}{2||u_n||^{p-2}}||Dy_n||^{2}_{2}-\int_{\Omega}\frac{F_+(z,u_n)}{||u_n||^{p}}dz\leq\frac{M_4}{||u_n||^p}\ \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
Hypothesis $H_1(ii)$ implies that given $\epsilon>0$, we can find $M_5=M_5(\epsilon)>0$ such that
\begin{eqnarray}
&&f(z,x)\leq(\hat{\lambda}_1(p)+\epsilon)x^{p-1}\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_5\nonumber\\
&\Rightarrow&F(z,x)\leq\frac{1}{p}(\hat{\lambda}_1(p)+\epsilon)x^p\ \mbox{for\ a.a.}\ x\in\Omega,\ \mbox{all}\ x\geq M_5\nonumber\\
&\Rightarrow&\frac{pF(z,x)}{x^p}\leq\hat{\lambda}_1(p)+\epsilon\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_5\nonumber\\
&\Rightarrow&\limsup\limits_{x\rightarrow+\infty}\frac{pF(z,x)}{x^p}\leq\hat{\lambda}_1(p)+\epsilon\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.\nonumber
\end{eqnarray}
Since $\epsilon>0$, is arbitrary, we let $\epsilon\downarrow 0$ to conclude that
\begin{eqnarray}\label{eq33}
\limsup\limits_{x\rightarrow+\infty}\ \frac{pF_+(z,x)}{x^p}\leq\hat{\lambda}_1(p)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.
\end{eqnarray}
Hypothesis $H_1(i)$ implies that
$$\Rightarrow\left\{\frac{F_+(\cdot,u_n(\cdot))}{||u_n||^p}\right\}_{n\geq 1}\subseteq L^1(\Omega)\ \mbox{uniformly\ integrable.}$$
Then from the Dunford-Pettis theorem and using (\ref{eq33}), at least for a subsequence, we have
\begin{eqnarray}\label{eq34}
&&\frac{F_+(\cdot,u_n(\cdot))}{||u_n||^p}\ \stackrel{w}{\longrightarrow}\ \frac{1}{p}\vartheta(y^+)^p\ \mbox{in}\ L^1(\Omega)\ \mbox{with}\ \vartheta\in L^{\infty}(\Omega),\ \vartheta(z)\leq\hat{\lambda}_1(p)\ \mbox{a.a.\ in}\ \Omega.
\end{eqnarray}
We return to (\ref{eq32}), pass to the limit as $n\rightarrow\infty$ and use (\ref{eq31}) and (\ref{eq34}). Since $2