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\title[On noncoercive elliptic problems]{On noncoercive elliptic problems}


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\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{Department of Mathematics, Faculty of Sciences,
King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia \&
Institute of Mathematics ``Simion Stoilow" of the
Romanian Academy, 014700 Bucharest, Romania}
\email{\tt vicentiu.radulescu@imar.ro}

\keywords{Positive solution, nonlinear regularity, nonlinear maximum principle, critical groups, local linking, nodal solution\\
\phantom{aa} 2010 AMS Subject Classification:
35J20, 35J60, 35J92, Secondary 58E05}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{abstract}
We consider a nonlinear noncoercive elliptic equation driven by the $p$-Laplacian. We show that if the $L^{\infty}$-perturbation has small norm, then the problem admits a positive solution. Moreover, if the $L^{\infty}$-perturbation is nonzero and nonnegative, then we find two positive solutions. Also, we consider a class of semilinear equations with an indefinite and unbounded potential. Using critical groups, we show that there is a nontrivial solution and under a global sign condition, we show that this solutions is nodal. Our results extend and improve a recent work of R\u{a}dulescu \cite{14}.
\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 first (see Section \ref{sec3}), we study the following nonlinear Dirichlet problem
\begin{align}\label{eq:1}
		\left\{\begin{aligned}
			-\Delta_pu(z)=\beta(z)u(z)^{p-1}+f(z,u(z))+g(z)\ &\mbox{in}\ \Omega,\\
			u|_{\partial\Omega}=0,\ u>0,\ 1<p<\infty,\ g\in L^{\infty}(\Omega).&
		\end{aligned}\right\}\eqname{$(1)_g$}
\end{align}

\stepcounter{equation}

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 $\beta\in L^{\infty}(\Omega)$ and $\beta(z)\leq \hat{\lambda}_1(p)$ for a.a. $z\in\Omega$ with strict inequality on a set of positive measure. Here $\hat{\lambda}_1(p)>0$ denotes the principal eigenvalue of $(-\Delta_p,W^{1,p}_{0}(\Omega))$. The perturbation $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in\RR$, $z\longmapsto f(z,x)$ is measurable and for a.a. $z\in\Omega$, $x\longmapsto f(z,x)$ is continuous), which exhibits $(p-1)$-superlinear growth near $+\infty$, but without satisfying the usual  Ambrosetti-Rabinowitz condition ($AR$-condition for short). We show that for $||g||_{\infty}$ sufficiently small, problem \ref{eq:1} admits at least one positive solution. Moreover, we show that if $g$ is nonzero and nonnegative, then a second positive solution can be found. Problem \ref{eq:1} was investigated recently by R\u{a}dulescu \cite{14}, when $p=2$ (semilinear equation) and with a perturbation function $f(z,x)=f(x)$ which is in $C^1(\RR)$ and satisfies the $AR$-condition. Under these conditions, the author shows that the problem has a positive solution (see Theorem 1 of \cite{14}). Our work here generalizes the result of R\u{a}dulescu \cite{14} and provides additional information for problem \ref{eq:1}.

In Section \ref{sec4}, we deal with the following semilinear problem:
\begin{align}\label{eq:2}
	\left\{
		\begin{aligned}
			-\Delta u(z)+\beta(z)u(z)=\lambda u(z)+f(z,u(z))&\ \mbox{in}\ \Omega,\\
			u|_{\partial\Omega}=0.\hspace{5.2cm}&
		\end{aligned}	
	\right\}\eqname{$(2)_{\lambda}$}
\end{align}

\stepcounter{equation}

In this problem, $\beta\in L^\tau (\Omega)$ with $\tau >\frac{N}{2}$ and is general indefinite. Also $\lambda\in\RR$ is a parameter and $f(z,x)$ is a measurable function on $\Omega\times\RR$ which is $C^1$ in the $x\in\RR$ variable and $x\longmapsto f(z,x)$ exhibits $(p-1)$-superlinear growth near $\pm\infty$ again without satisfying the $AR$-condition. We show for all $\lambda\geq \hat{\lambda}_1(2,\beta)$, problem \ref{eq:2} admits a nontrivial solution (by $\hat{\lambda}_1(2,\beta)$ we denote the principal eigenvalue of $(-\Delta+\beta I,\ H^{1}_{0}(\Omega))$). In fact, under a global sign condition on $f(z,\cdot)$, we show that any nontrivial solution of \ref{eq:2} is necessarily nodal (sign changing), that is, the problem has no nontrivial constant sign solutions. Problem \ref{eq:2} was also studied by R\u{a}dulescu \cite{14} under the hypotheses that $\beta\equiv 0$, $f(z,x)=f(x)$ and $f\in C^1(\RR)$ satisfies the $AR$-condition and it is strictly increasing and onto. In fact in \cite{14} it was left as an open problem  whether the strict monotonicity and surjectivity conditions on $f(\cdot)$ can be relaxed. Here we show that the answer to this open problem is affirmative and in fact we go even further establishing the existence of solutions for a broader class of equations with more general perturbations $f(z,x)$.

Our approach is variational based on the critical point theory, coupled with suitable truncation and comparison techniques. In Section \ref{sec4} we also use critical groups. In the next section for the convenience of the reader, we review the main mathematical tools that we will use in this paper.


\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 $(X^*,X)$. Given  $\varphi\in C^1(X)$, we say that it satisfies the Cerami condition (the $C$-condition for short), if the following 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}

This is a compactness-type condition on the functional $\varphi$ which is more general than the more common Palais-Smale condition. The $C$-condition leads to a deformation theorem, from which we can derive the minimax theory for the critical values of $\varphi$. One of the main results in this theory is the so-called ``mountain pass theorem" due to Ambrosetti and Rabinowitz \cite{4}. Here we state this in a slightly more general form (see Gasinski and Papageorgiou \cite{8}).
\begin{theorem}\label{th1}
	Assume that $X$ is a Banach space, $\varphi\in C^1(X)$ satisfies the $C$-condition, $u_0,u_1\in X$ with $||u_1-u_0||>\rho>0$
	 $$\max\{\varphi(u_0),\varphi(u_1)\}<\inf\left[\varphi(u):||u-u_0||=\rho\right]=\eta_{\rho}$$
	and $c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq 1}\varphi(\gamma(t))$ with $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\ \gamma(1)=u_1\}$. Then $c\geq \eta_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

In the analysis of problems \ref{eq:1} and \ref{eq:2}, we will use the Sobolev spaces $W^{1,p}_{0}(\Omega)$ and $H^{1}_{0}(\Omega)$ and the Banach space $C^{1}_{0}(\overline{\Omega})=\{u\in C^1(\overline{\Omega}):u|_{\partial\Omega}=0\}$. The latter is an ordered Banach space with positive cone
$$C_+=\{u\in C^{1}_{0}(\overline{\Omega}):u(z)\geq 0\ \mbox{for all}\ z\in\overline{\Omega}\}.$$

This cone has a nonempty interior
$$\mbox{int}\, C_+=\left\{u\in C_+:u(z)>0\ \mbox{for all}\ z\in\Omega,\ \left.\frac{\partial u}{\partial n}\right|_{\partial\Omega}<0\right\},$$
where $n(\cdot)$ denotes the outward unit normal on $\partial\Omega$.

We consider the following nonlinear eigenvalue problem:
\begin{equation}\label{eq3}
	-\Delta_pu(z)=\hat{\lambda}|u(z)|^{p-2}u(z)\ \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0.
\end{equation}

We say that $\hat{\lambda}\in\RR$ is an eigenvalue of $(-\Delta_p,W^{1,p}_{0}(\Omega))$, if problem (\ref{eq3}) admits a nontrivial solution $\hat{u}\in W^{1,p}_{0}(\Omega)$ known as an eigenfunction corresponding to $\hat{\lambda}$. The nonlinear regularity theory (see, for example, Gasinski and Papageorgiou \cite[pp. 737-738]{8}), implies that $\hat{u}\in C^{1}_{0}(\overline{\Omega})$. We know that $(-\Delta_p,W^{1,p}_{0}(\Omega))$ has a smallest eigenvalue $\hat{\lambda}_1(p)$ such that:

(i)	 $\hat{\lambda}_1(p)>0$ and it is isolated (that is, there exists $\epsilon>0$ such that $\left[\hat{\lambda}_1(p),\hat{\lambda}_1(p)+\epsilon\right)$ does not contain any other eigenvalue of $(-\Delta_p,W^{1,p}_{0}(\Omega))$);

(ii) $\hat{\lambda}_1(p)$ is simple (that is, if $\hat{u},\hat{v}\in C^{1}_{0}(\overline{\Omega})$ are eigenfunctions corresponding to $\hat{\lambda}_1(p)$, then $\hat{u}=\xi\hat{v}$ with $\xi\neq 0$)

\smallskip
\noindent and
	\begin{eqnarray}\label{eq4}
		\hspace{2cm}&&\hat{\lambda}_1(p)=\inf\left[\frac{||Du||^{p}_{p}}{||u||^{p}_{p}}:u\in W^{1,p}_{0}(\Omega),\ u\neq 0\right].
	\end{eqnarray}

The infimum in (\ref{eq4}) is realized on the one-dimensional eigenspace corresponding to $\hat{\lambda}_1(p)>0$. It is clear from (\ref{eq4}) that the elements of this eigenspace do not change sign. Let $\hat{u}_1(p)$ be the $L^p$-normalized (that is, $||\hat{u}_1(p)||_p=1$), positive eigenfunction corresponding to $\hat{\lambda}_1(p)$. The nonlinear maximum principle (see, for example, Gasinski and Papageorgiou \cite[p. 738]{8}), implies that $\hat{u}_1(p)\in \mbox{int}\, C_+$. We mention that $\hat{\lambda}_1(p)$ is the only eigenvalue with eigenfunctions of constant sign. Every eigenvalue $\hat{\lambda}\neq\hat{\lambda}_1(p)$ has nodal eigenfunctions.

As a consequence of the above properties of $\hat{\lambda}_1(p)>0$ and $\hat{u}_1(p)\in \mbox{int}\, C_+$, we have the following lemma (see Papageorgiou and Kyritsi \cite[p. 356]{10}).
\begin{lemma}\label{lem2}
	If $\beta\in L^{\infty}(\Omega)$ and $\beta(z)\leq\hat{\lambda}_1(p)$ a.e. in $\Omega$ with strict inequality on a set of positive measure, then there exists $\xi_0>0$ such that
	$$||Du||^{p}_{p}-\int_{\Omega}\beta(z)|u|^pdz\geq\xi_0||Du||^{p}_{p}\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$
\end{lemma}

To deal with problem \ref{eq:2}, we will use the spectrum of $(-\Delta+\beta I, H^{1}_{0}(\Omega))$. So, we consider the following linear eigenvalue problem
\begin{eqnarray}\label{eq5}
	-\Delta u(z)+\beta(z)u(z)=\lambda u(z)\ \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0.
\end{eqnarray}

Recall that $\beta\in L^\tau (\Omega)$ with $\tau >\frac{N}{2}$ and in general is indefinite (that is, sign-changing). Problem (\ref{eq5}) has a strictly increasing sequence $\{\hat{\lambda}_k(2,\beta)\}_{k\geq 1}\subseteq\RR$ of eigenvalues such that $\hat{\lambda}_k(2,\beta)\rightarrow+\infty$ as $k\rightarrow+\infty$. By $E(\hat{\lambda}_k(2,\beta))$ we denote the eigenspace corresponding to the eigenvalue $\hat{\lambda}_k(2,\beta)$. We have $E(\hat{\lambda}_k(2,\beta))\subseteq C^{1}_{0}(\overline{\Omega})$ and the eigenspace has the so-called unique continuation property (UCP for short), that is, if $\hat{u}\in E(\hat{\lambda}_k(2,\beta))$ and $\hat{u}$ vanishes on a set of positive measure, then $\hat{u}\equiv 0$. We have the following variational characterizations of these eigenvalues:
\begin{eqnarray}\label{eq6}
	 \hat{\lambda}_1(2,\beta)=\inf\left[\frac{||Du||^{2}_{2}+\int_{\Omega}\beta(z)u^2dz}{||u||^{2}_{2}}:u\in H^{1}_{0}(\Omega),\ u\neq 0\right]
\end{eqnarray}
and for $k\geq 2$, we have
\begin{eqnarray}\label{eq7}
	 \hat{\lambda}_k(2,\beta)&&=\sup\left[\frac{||Du||^{2}_{2}+\int_{\Omega}\beta(z)u^2dz}{||u||^{2}_{2}}:u\in \overset{k}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(2,\beta)),u\neq 0\right]\nonumber\\
	&&=\inf\left[\frac{||Du||^{2}_{2}+\int_{\Omega}\beta(z)u^2dz}{||u||^{2}_{2}}:u\in \overline{{\underset{\mathrm{i\geq k}}\oplus}E(\hat{\lambda}_i(2,\beta))},u\neq 0\right].
\end{eqnarray}

In (\ref{eq6}) and (\ref{eq7}), the infimum and the supremum are realized on the corresponding eigenspace $E(\hat{\lambda}_k(2,\beta))$ (see Kyritsi and Papageorgiou \cite{9}).

We have the following orthogonal direct sum decomposition
$$H^{1}_{0}(\Omega)=\overline{H}_k\oplus\hat{H}_k$$
with $\overline{H}_k=\overset{k}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(2,\beta))$ and $\hat{H}_k=\overline{{\underset{\mathrm{i\geq k+1}}\oplus}E(\hat{\lambda}_i(2,\beta))}$.

Next let $X$ be a Banach space and $\varphi\in C^1(X),\ c\in \RR$. We introduce the following sets
$$\varphi^c=\{u\in X:\varphi(u)\leq c\},\ K_{\varphi}=\{u\in X:\varphi'(u)=0\},\ K^{c}_{\varphi}=\{u\in K_{\varphi}:\varphi(u)=c\}.$$

Let $(Y_1,Y_2)$ be a topological pair such that $Y_2\subseteq Y_1\subseteq X$. For every integer $k\geq 0$, by $H_k(Y_1,Y_2)$ we denote the $k^{th}$ relative singular homology group for the pair $(Y_1,Y_2)$ with integer coefficients. Recall that $H_k(Y_1,Y_2)=0$ for all $k<0$. The critical groups of $\varphi$ at an isolated critical point $u\in K^{c}_{\varphi}$ are defined by
$$C_k(\varphi,u)=H_k(\varphi^c\cap U,\ \varphi^c\cap U\backslash\{u\})\ \mbox{for every}\ k\geq 0,$$
where $U$ is a neighborhood of $u$ such that $\varphi^c\cap K_{\varphi}\cap U=\{u\}$. The excision property of singular homology theory, implies that the above definition of critical groups is independent of the choice of the neighborhood $U$.

Suppose that $\varphi\in C^1(X)$ satisfies the $C$-condition and $\inf\varphi(K_{\varphi})>-\infty$. Let $c<\inf\varphi(K_{\varphi})$. The critical groups of $\varphi$ at infinity are defined by
$$C_k(\varphi,\infty)=H_k(X,\varphi^c)\ \mbox{for all}\ k\geq 0.$$

The second deformation theorem (see, for example, Gasinski and Papageorgiou \cite[p. 628]{8}), implies that this definition is independent of the choice of level $c<\inf\varphi(K_{\varphi})$. If for some $k\geq 0$, $C_k(\varphi,0)\neq 0$, $C_k(\varphi,\infty)=0$, then $\varphi$ admits a nontrivial critical point.

We conclude this section by fixing our notation. By $||\cdot||$ we denote the norm of the Sobolev space $W^{1,p}_{0}(\Omega)$. By virtue of the Poincar\'e inequality, we have
$$||u||=||Du||_p\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$

For every $x\in\RR$, we set $x^{\pm}=\max\{\pm x,0\}$. Then given $u\in W^{1,p}_{0}(\Omega)$, we define $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We have
$$u^{\pm}\in W^{1,p}_{0}(\Omega),\ u=u^+-u^-,\ |u|=u^++u^-.$$

By $|\cdot|_N$ we denote the Lebesgue measure on $\RR^N$. Finally, if $h:\Omega\times\RR\rightarrow\RR$ is a measurable function (for example, a Carath\'eodory function), then we define
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega)$$
(the Nemytskii operator corresponding to function $h(\cdot,\cdot)$). Note that $z\longmapsto N_f(u)(z)$ is measurable.


\section{Solutions for Problem \ref{eq:1}}\label{sec3}

In this section, we show that for $||g||_{\infty}$ small, problem \ref{eq:1} has at least one positive solution and for nonzero and nonnegative $g$, it has two positive solutions. The hypotheses on the perturbation $f(z,x)$, are the following:

\smallskip
$H_1:$ $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
	\item[(i)] $|f(z,x)|\leq a(z)(1+x^{r-1})$ for a.a $z\in\Omega$, all $x\geq 0$, with $a\in L^{\infty}(\Omega)_+$, $p<r<p^*=
	\left\{	\begin{array} {ll}
			\frac{Np}{N-p}&\mbox{if}\ p<N\\
			+\infty&\mbox{if}\ p\leq N
		\end{array}\right.$;
	\item[(ii)] if $F(z,x)=\int^{x}_{0}f(z,s)ds$, then
	$$\lim\limits_{x\rightarrow+\infty}\frac{F(z,x)}{x^p}=+\infty\ \mbox{uniformly for a.a}\ z\in\Omega;$$
	\item[(iii)] there exists $\eta_0>0$ and $\tau\in\left(\max\{1,(r-p)\frac{N}{p}\},p^*\right)$ such that
	$$0<\eta_0\leq\liminf\limits_{x\rightarrow+\infty}\frac{f(z,x)x-pF(z,x)}{x^{\tau}}\ \mbox{uniformly for a.a.}\ z\in\Omega;$$
	\item[(iv)] $\displaystyle\lim\limits_{x\rightarrow 0^+}\frac{f(z,x)}{x^{p-1}}=0$ uniformly for a.a. $z\in\Omega$;
	\item[(v)] for every $\rho>0$, there exists $\xi_{\rho}>0$ such that for a.a. $z\in\Omega$, the application $x\longmapsto f(z,x)+\xi_{\rho}x^{p-1}$ is nondecreasing on $[0,\rho]$.
\end{itemize}

\begin{remark}
	Since we are interested on positive solutions and the above hypotheses concern the positive semiaxis $\RR_+=\left[0,+\infty\right)$, without any loss of generality, we may assume that for a.a. $z\in\Omega$ $f(z,x)=0$ for all $x\leq 0$. Hypotheses $H_1(ii),(iii)$ imply that $f(z,\cdot)$ is $(p-1)$-superlinear near $+\infty$, more precisely we have
	$$\lim\limits_{x\rightarrow+\infty}\frac{f(z,x)}{x^{p-1}}=+\infty\ \mbox{uniformly for a.a.}\ z\in\Omega.$$
\end{remark}

{\it Note that we do not employ the usual in such cases $AR$-condition (see \cite{4}). Instead we use a weaker condition (see hypothesis $H_1(iii)$) which incorporates in our framework $(p-1)$-superlinear perturbations with ``slower" growth near $+\infty$. For example, the function
$$f(x)=x^{p-1}\left[\ln x+\frac{1}{p}\right]\ \mbox{for all}\ x\geq 0$$
(for the sake of simplicity we have dropped the $z$-dependence), satisfies hypotheses $H_1$ but fails to satisfy the $AR$-condition. So, Theorem 1 of \cite{14} does not apply to this function. If $f(z,\cdot)\in C^1(\RR)$ and $f'_x(z,\cdot)$ is bounded on bounded sets, then hypothesis $H_1(v)$ is satisfied.}

First we show that we cannot have a positive solution for problem \ref{eq:1} for every $g\in L^{\infty}(\Omega)$. To this end, let $u\in W^{1,p}_{0}(\Omega)$ be a positive solution for problem \ref{eq:1}. The nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasinski and Papageorgiou \cite[pp. 737-738]{8}), imply that $u\in \mbox{int}\, C_+$. Recall that $\hat{u}_1(p)\in \mbox{int}\, C_+$. So, invoking Lemma 3.3 of Filippakis, Kristaly and Papageorgiou \cite{6}, we can find $c_1,c_2>0$ such that
\begin{eqnarray}\label{eq8}
	&&c_1u\leq \hat{u}_1(p)\leq c_2u\nonumber\\
	\Rightarrow&&c_1\leq\frac{\hat{u}_1(p)}{u}\leq c_2\ \mbox{in}\ \Omega.
\end{eqnarray}

Let $R(\hat{u}_1(p),u)(z)=|D\hat{u}_1(p)(z)|^p-|Du(z)|^{p-2}$ $\left(Du(z),D\left(\frac{\hat{u}_1(p)^p}{u^{p-1}}\right)(z)\right)_{\RR^N}$. From the nonlinear Picone identity of Allegretto and Huang \cite{3}, we have
\begin{eqnarray}\label{eq9}
	0&\leq&\int_{\Omega}R(\hat{u}_1(p),u)dz\nonumber\\
	 &=&||D\hat{u}_1(p)||^{p}_{p}-\int_{\Omega}(-\Delta_pu)\frac{\hat{u}_1(p)^p}{u^{p-1}}dz\nonumber\\
	&&\hspace{0.1cm}(\mbox{using the nonlinear Green's identity, see Gasinski and Papageorgiou \cite[p. 211]{8}}\nonumber\\
	 &=&\hat{\lambda}_1(p)||\hat{u}_1(p)||^{p}_{p}-\int_{\Omega}\left[\beta(z)u^{p-1}+f(z,u)+g(z)\right]\frac{\hat{u}_1(p)^p}{u^{p-1}}dz\nonumber\\
	&=&\int_{\Omega}\left[\hat{\lambda}_1(p)-\beta(z)\right]\hat{u}_1(p)^pdz-
\int_{\Omega}f(z,u)\frac{\hat{u}_1(p)^p}{u^{p-1}}dz-\int_{\Omega}g(z)\frac{\hat{u}_1(p)^p}{u^{p-1}}dz.
\end{eqnarray}

We know that
\begin{eqnarray}\label{eq10}
	\vartheta_0=\int_{\Omega}\left[\hat{\lambda}_1(p)-\beta(z)\right]\hat{u}_1(p)^pdz>0.
\end{eqnarray}

As we already observed, hypotheses $H_1(ii),(iii)$ imply that
\begin{eqnarray}\label{eq11}
	\lim\limits_{x\rightarrow+\infty}\frac{f(z,x)}{x^{p-1}}=+\infty\ \mbox{uniformly for a.a}\ z\in\Omega.
\end{eqnarray}

From (\ref{eq11}) and hypothesis $H_1(i)$, we see that given $\xi>\vartheta_0$ we can find $c_3=c_3(\xi)>0$ such that
\begin{eqnarray}\label{eq12}
	f(z,x)\geq \xi x^{p-1}-c_3\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
\end{eqnarray}

Returning to (\ref{eq9}) and using (\ref{eq10}) and (\ref{eq12}), we have
$$\int_{\Omega}(g(z)-c_3)\left(\frac{\hat{u}_1(p)}{u}\right)^{p-1}\hat{u}_1(p)dz\leq\vartheta_0-
\xi||\hat{u}_1(p)||^{p}_{p}<0$$
(recall that $\xi>\vartheta_0$  and $||\hat{u}_1(p)||_p=1$).
Since $\frac{\hat{u}_1(p)}{u}\in L^{\infty}(\Omega)_+$ (see (\ref{eq8})), if $g(z)>c_3$ for almost all $z\in\Omega$, we have a contradiction. This suggests
that in order to guarantee a positive solution of \ref{eq:1} we need to restrict  $||g||_\infty$.

Let $g\in L^{\infty}(\Omega)$ and let $e_g:\Omega\times\RR\rightarrow\RR$ be the Carath\'eodory function defined by
\begin{eqnarray}\label{eq13}
	e_g(z,x)=\left\{
		\begin{array}{ll}
			g(z)&\mbox{if}\ x\leq 0\\
			\beta(z)x^{p-1}+f(z,x)+g(z)&\mbox{if}\ x>0.
		\end{array}\right.
\end{eqnarray}

We set $E_g(z,x)=\int^{x}_{0}e_g(z,s)ds$ and consider the $C^1$-functional $\varphi_g:W^{1,p}_{0}(\Omega)\rightarrow\RR$ defined by
$$\varphi_g(u)=\frac{1}{p}||Du||^{p}_{p}-\int_{\Omega}E_g(z,u)dz\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$

From Papageorgiou and Smyrlis \cite{13}, we have:
\begin{prop}\label{prop3}
	If hypotheses $H_1$ hold then for every $g\in L^{\infty}(\Omega)$ the functional $\varphi_g$ satisfies the $C$-condition.
\end{prop}

The next result is an immediate consequence of hypothesis, $H_1(ii)$ and (\ref{eq13}).
\begin{prop}\label{prop4}
	If hypotheses $H_1$ hold, $u\in \mbox{int}\, C_+$ and $g\in L^{\infty}(\Omega)$, then $\varphi_g(tu)\rightarrow-\infty$ as $t\rightarrow+\infty$.
\end{prop}

The next proposition shows that the mountain pass geometry (see Theorem \ref{th1}) is satisfied by the functional $\varphi_g$ for $||g||_{\infty}$ small.
\begin{prop}\label{prop5}
	If hypotheses $H_1$ hold, then there exist $\delta_0>0$ and $\rho_0=\rho_0(\delta_0)>0$ such that
	$$||g||_{\infty}<\delta_0\Rightarrow\varphi_g(u)\geq m_0>0\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega)\ \mbox{with}\ ||u||=\rho_0.$$
\end{prop}
\begin{proof}
	Hypotheses $H_1(i)$ and $(iv)$ imply that given $\epsilon>0$, we can find $c_5=c_5(\epsilon)>0$ such that
	\begin{eqnarray}\label{eq14}
		F(z,x)\leq \frac{\epsilon}{p}x^p+c_5 x^r\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
	\end{eqnarray}
	
	Then for every $u\in W^{1,p}_{0}(\Omega)$, we have
	\begin{eqnarray*}
		\varphi_g(u)&=&\frac{1}{p}||Du||^{p}_{p}-\int_{\Omega}E_g(z,u)dz\\
		 &\geq&\frac{1}{p}||Du||^{p}_{p}-\frac{1}{p}\int_{\Omega}\beta(z)|u|^pdz-\int_{\Omega}F(z,u)dz-c_6||g||_{\infty}||u||\\
		&&\hspace{4cm}\mbox{for some}\ c_6>0\ \mbox{(see (\ref{eq13}))}\\
		 &\geq&\frac{1}{p}\left[\xi_0-\frac{\epsilon}{\hat{\lambda}_1(p)}\right]||u||^p-c_7||u||^r-c_6||g||_{\infty}||u||\ \mbox{for some}\ c_7>0\\
		&&\hspace{4cm}(\mbox{see Lemma \ref{lem2} and (\ref{eq4})})
	\end{eqnarray*}
	
	Choosing $\epsilon\in(0,\hat{\lambda}_1(p)\xi_0)$, we obtain
	\begin{eqnarray}\label{eq15}
		\varphi_g(u)&\geq&c_8||u||^p-c_7||u||^r-c_6||g||_{\infty}||u||\ \mbox{with}\ c_8=\frac{\hat{\lambda}_1(p)\xi_0-\epsilon}{p\hat{\lambda}_1(p)}>0\nonumber\\
		&=&\left[c_8-\left(c_7||u||^{r-p}+c_6||g||_{\infty}||u||^{1-p}\right)\right]||u||^p.
	\end{eqnarray}
	
	Let $\gamma(t)=c_7 t^{r-p}+c_6||g||_{\infty}t^{1-p}$ for all $t\geq 0$. Evidently $\gamma\in C^1(0,\infty)$ and since $1<p<r$, we have
	$$\gamma(t)\rightarrow+\infty\ \mbox{as}\ t\rightarrow 0^+\ \mbox{and}\ t\rightarrow+\infty.$$
	
	So, we can find $t_0\in(0,+\infty)$ such that
	\begin{eqnarray*}
		&&\gamma(t_0)=\inf\limits_{t\geq 0}\gamma\\
		&\Rightarrow& \gamma'(t_0)=0\\
		&\Rightarrow&(r-p)c_7 t^{r-p-1}_{0}=(p-1)c_6||g||_{\infty}t_0^{-p}\\
		&\Rightarrow&t_0=\left[\frac{(p-1)c_6||g||_{\infty}}{(r-p)c_7}\right]^{\frac{1}{r-1}}.
	\end{eqnarray*}
	
	Then $\gamma(t_0)\rightarrow 0^+$ as $||g||_{\infty}\rightarrow 0^+$. So, we can find $\delta_0>0$ such that
	\begin{eqnarray*}
		&&||g||_{\infty}<\delta_0\Rightarrow\gamma(t_0)<c_8\\
		&\Rightarrow&\varphi_g(u)\geq m_0>0=\varphi_g(0)\ \mbox{for all}\ ||u||=t_0=\rho_0.
	\end{eqnarray*}
This completes the proof.
\end{proof}

These propositions lead to the following existence theorem for problem \ref{eq:1} when $||g||_{\infty}$ is small.
\begin{theorem}\label{th6}
	If hypotheses $H_1$ hold, then there exists $\delta_1\in\left(0,\delta_0\right]$ such that  if $||g||_{\infty}<\delta_1$, then problem \ref{eq:1} has at least one positive solution $u_0\in \mbox{int}\, C_+$.
\end{theorem}
\begin{proof}
	Proposition \ref{prop3}, \ref{prop4} and \ref{prop5} imply that when $||g||_{\infty}<\delta_0$, then the functional $\varphi_g$ satisfies the mountain pass geometry and the $C$-condition. So, we can apply Theorem \ref{th1} (the mountain pass theorem) and find $u_0\in W^{1,p}_{0}(\Omega)$ such that
	\begin{eqnarray*}
		&&\varphi'_g(u_0)=0\ \mbox{and}\ \varphi_g(0)=0<m_0\leq \varphi_g(u_0)\\
		&\Rightarrow&u_0\neq 0.
	\end{eqnarray*}
	
	In particular, let $g\equiv 0$ and let $\bar{u}_0$ be the critical point of $\varphi_0$ obtained above. We have
	\begin{eqnarray}\label{eq16}
		A(\bar{u}_0)=N_{e_0}(\bar{u}_0).
	\end{eqnarray}
	
	On (\ref{eq16}) we act with $-\bar{u}^-_0\in W^{1,p}_{0}(\Omega)$. Then
	\begin{eqnarray*}
		&&||D\bar{u}_0||^{p}_{p}=0\ (\mbox{see (\ref{eq13}) with}\ g\equiv 0)\\
		&\Rightarrow& \bar{u}_0\geq 0,\ \bar{u}_0\neq 0.
	\end{eqnarray*}
	
	So, $\bar{u}_0$ is a positive solution of problem $(1)_0$ (with $g\equiv 0$). Nonlinear regularity theory, implies that $\bar{u}_0\in C_+\backslash\{0\}$. Let $\rho=||\bar{u}_0||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H_1(v)$. We have
	\begin{eqnarray*}
		&&-\Delta_p\bar{u}_0(z)+\xi_{\rho}\bar{u}_0(z)^{p-1}\\
		&=&\beta(z)u_0(z)^{p-1}+f(z,u_0(z))+\xi_{\rho}\bar{u}_0(z)^{p-1}\geq 0\ \mbox{a.e. in}\ \Omega\\
		\Rightarrow&&\Delta_p\bar{u}_0(z)\leq \xi_{\rho}\bar{u}_0(z)^{p-1}\ \mbox{a.e. in}\ \Omega\\
		\Rightarrow&&\bar{u}_0\in \mbox{int}\, C_+\ (\mbox{by the nonlinear maximum principle, see \cite[p. 738]{8}}).
	\end{eqnarray*}
	
	So, every positive solution of $(1)_0$ (with $g\equiv 0$), belongs to $\mbox{int}\, C_+$.
	
	Now, let $\{g_n\}_{n\geq 1}\subseteq L^{\infty}(\Omega)$ with $||g_n||_{\infty}<\delta_0$ for all $n\geq 1$ and assume that $g_n\rightarrow 0$ in $L^{\infty}(\Omega)$. Let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ be the corresponding critical points of $\varphi_{g_n}$ obtained in the beginning of the proof via the mountain pass theorem (see Theorem \ref{th1}). We have
	$$-\Delta_p u_n(z)=e_{g_n}(z,u_n(z))\ \mbox{a.e. in}\ \Omega,\ u_n|_{\partial\Omega}=0,\ n\geq 1.$$
	
	From Gasinski and Papageorgiou \cite[p. 737]{8}, we can find $c_9>0$ such that
	$$||u_n||_{\infty}\leq c_9\ \mbox{for all}\ n\geq 1.$$
	
	So, there exist $\alpha\in (0,1)$ and $c_{10}>0$ such that
	$$u_n\in C^{1,\alpha}_{0}(\overline{\Omega})\ \mbox{and}\ ||u_n||_{C^{1,\alpha}_{0}(\overline{\Omega})}\leq c_{10}\ \mbox{for all}\ n\geq 1$$
	(see Gasinski and Papageorgiou \cite[p. 738]{8}). Exploiting the compact embedding of $C^{1,\alpha}_{0}(\overline{\Omega})$ into $C^1(\overline{\Omega})$, we may assume that
	\begin{eqnarray}\label{eq17}
		u_n\rightarrow\tilde{u}\ \mbox{in}\ C^{1}_{0}(\overline{\Omega}),\ \mbox{with}\ \tilde{u}\ \mbox{solution of}\ (1)_0.
	\end{eqnarray}
	
	Recall that for all $n\geq 1$ we have
	\begin{eqnarray*}
		&&\varphi_{g_n}(u_n)\geq m_0>0=\varphi_{g_n}(0)\\ 
&&(\mbox{note that, by Proposition \ref{prop5}, since $||g_n||_\infty <\delta_0$ for all $n\in\NN$, $m_0$ does not depend on $n$})\\
		&\Rightarrow& \varphi_0(\tilde{u})\geq m_0>0=\varphi_0(0)\ (\mbox{see (\ref{eq17}) and (\ref{eq13})})\\
		&\Rightarrow&\tilde{u}\neq 0,\ \mbox{hence}\ \tilde{u}\in \mbox{int}\, C_+\ \mbox{as established earlier}.
	\end{eqnarray*}
	
	From (\ref{eq17}) it follows that
	$$u_n\in \mbox{int}\, C_+\ \mbox{for all}\ n\geq n_0.$$
	
	Therefore, we can find $\delta_1\in\left(0,\delta_0\right]$ such that  for $||g||_{\infty}<\delta_1$ problem \ref{eq:1} has at least one positive solution $u_0\in \mbox{int}\, C_+$.
\end{proof}

We can improve the conclusion of the above theorem and produce a second positive solution, provided $g$ is nonzero and nonnegative and as before has small $L^{\infty}(\Omega)$-norm.
\begin{theorem}\label{th7}
	If hypotheses $H_1$ hold then there exists $\delta_1\in\left(0,\delta_0\right]$ such that  if $0<||g||_{\infty}<\delta_1$ and $g\geq 0$, then problem \ref{eq:1} has at least two positive solutions
	$$u_0,\hat{u}\in \mbox{int}\, C_+,\ u_0\leq\hat{u},\ u_0\neq \hat{u}.$$
\end{theorem}
\begin{proof}
	From Theorem \ref{th6} we know that there exists $\delta_1\in\left(0,\delta_0\right]$ such that  if $||g||_{\infty}<\delta_1$, then problem \ref{eq:1} has at least one positive solution $u\in \mbox{int}\, C_+$.
	
	Now we assume that $0<||g||_{\infty}<\delta_1$ and $g\geq 0$. Let $\eta\in(0,\delta_1-||g||_{\infty})$ and let $g^*=g+\eta$. Evidently $||g^*||_{\infty}<\delta_1$ and so problem $(1)_{g^*}$ has a positive solution $u^*\in \mbox{int}\, C_+$.
	
	\begin{claim}\label{claim1}
		We can find a positive solution $u_0\in \mbox{int}\, C_+$ of \ref{eq:1} such that  $u_0\leq u^*$.
	\end{claim}
	
	We have
	\begin{eqnarray}\label{eq18}
		&&A(u^*)=\beta(z)(u^*)^{p-1}+N_f(u^*)+g^*\geq \beta(z)(u^*)^{p-1}+N_f(u^*)+g\\
		&&\hspace{4cm}\mbox{in}\ W^{-1,p'}(\Omega)=W^{1,p}_{0}(\Omega)^*\left(\frac{1}{p}+\frac{1}{p'}=1\right).\nonumber
	\end{eqnarray}
	
	We consider the following Carath\'eodory function
	\begin{eqnarray}\label{eq19}
		\gamma_g(z,x)=\left\{
			\begin{array}{ll}
				g(z)&\mbox{if}\ x<0\\
				\beta(z)x^{p-1}+f(z,x)+g(z)&\mbox{if}\ 0\leq x\leq u^*(z)\\
				\beta(z)u^*(z)^{p-1}+f(z,u^*(z))+g(z)&\mbox{if}\ u^*(z)<x.
			\end{array}\right.
	\end{eqnarray}
	
	We set $\Gamma_g(z,x)=\int^{x}_{0}\gamma_g(z,s)ds$ and consider the $C^1$-functional $\tau_g:W^{1,p}_{0}(\Omega)\rightarrow \RR$ defined by
	$$\tau_g(u)=\frac{1}{p}||Du||^{p}_{p}-\int_{\Omega}\Gamma_g(z,u)dz\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$
	
	It is clear from (\ref{eq19}) that $\tau_g$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $u_0\in W^{1,p}_{0}(\Omega)$ such that
	\begin{eqnarray}\label{eq20}
		&&\tau_g(u_0)=\inf[\tau_g(u):u\in W^{1,p}_{0}(\Omega)]\nonumber\\
		&\Rightarrow&\tau'_g(u_0)=0\nonumber\\
		&\Rightarrow&A(u_0)=N_{\gamma_g}(u_0).
	\end{eqnarray}
	
	In (\ref{eq20}) first we act with $-u_0^-\in W^{1,p}_{0}(\Omega)$. Then
	\begin{eqnarray*}
		&&||Du_0^-||^{p}_{p}=\int_{\Omega}g(z)(-u_0^-)dz\leq 0\ (\mbox{see (\ref{eq19}) and recall}\ g\geq 0)\\
		&\Rightarrow&u_0\geq 0.
	\end{eqnarray*}
	
	Also, on (\ref{eq20}) we act with $(u_0-u^*)^+\in W^{1,p}_{0}(\Omega)$. Then
	\begin{eqnarray*}
		\left\langle A(u_0),(u_0-u^*)^+\right\rangle&=&\int_{\Omega}\gamma_g(z,u_0)(u_0-u^*)^+dz\\
		&=&\int_{\Omega}\left[\beta(z)(u^*)^{p-1}+f(z,u^*)+g\right](u_0-u^*)^+dz\ (\mbox{see (\ref{eq19})})\\
		&\leq&\left\langle A(u^*),(u_0-u^*)^+\right\rangle\ (\mbox{see (\ref{eq18})}),
	\end{eqnarray*}
	\begin{eqnarray*}
		 &\Rightarrow&\int_{\{u_0>u^*\}}\left(|Du_0|^{p-2}Du_0-|Du^*|^{p-2}Du^*,Du_0-Du^*\right)_{\RR^N}dz\leq 0\\
		&\Rightarrow&\left|\{u_0>u^*\}\right|_N=0,\ \mbox{hence}\ u_0\leq u^*.
	\end{eqnarray*}
	
	So, we have proved that
	$$u_0\in[0,u^*]=\{u\in W^{1,p}_{0}(\Omega):0\leq u(z)\leq u^*(z)\ \mbox{a.e. in}\ \Omega\}.$$
	
	Then from (\ref{eq19}) and (\ref{eq20}) it follows that $u_0$ is a solution of \ref{eq:1} and since $g\neq 0,\ u_0\neq 0$. The nonlinear regularity theory and the nonlinear maximum principle imply that $u_0\in \mbox{int}\, C_+$.
	
	Using $u_0\in \mbox{int}\, C_+$ we introduce the following truncation of the reaction of problem \ref{eq:1}:
	\begin{eqnarray}\label{eq21}
		k_g(z,x)=\left\{
			\begin{array}{ll}
				\beta(z)u_0(z)^{p-1}+f(z,u_0(z))+g(z)&\mbox{if}\ x<u_0(z)\\
				\beta(z)x^{p-1}+f(z,x)+g(z)&\mbox{if}\ u_0(z)\leq x.
			\end{array}
		\right.
	\end{eqnarray}
	
	We set $K_g(z,x)=\int^{x}_{0}k_g(z,s)ds$ and consider the $C^1$-functional $\psi_g:W^{1,p}_{0}(\Omega)\rightarrow\RR$ defined by
	$$\psi_g(u)=\frac{1}{p}||Du||^{p}_{p}-\int_{\Omega}K_g(z,u)dz\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$
	
	If $[u_0)=\{u\in W_0^{1,p}(\Omega): u_0(z)\leq u(z)\ \mbox{for almost all $z\in\Omega$}\}$, then from (\ref{eq13})) we see that
	\begin{eqnarray}\label{eq22}
		\psi_g|_{[u_0)}=\varphi|_{[u_0)}+\xi^*\ \mbox{for some}\ \xi^*\in\RR\,.
	\end{eqnarray}
	
	From (\ref{eq22}) and Proposition \ref{prop3} it follows that
	\begin{eqnarray}\label{eq23}
		\psi_g\ \mbox{satisfies the}\ C-\mbox{condition}.
	\end{eqnarray}
	
	Moreover, Proposition \ref{prop4} implies that for any $u\in \mbox{int}\, C_+$, we have
	\begin{eqnarray}\label{eq24}
		\psi_g(tu)\rightarrow-\infty\ \mbox{as}\ t\rightarrow+\infty.
	\end{eqnarray}
	
	\begin{claim}\label{claim2}
	We have	$K_{\psi_g}\subseteq\left[u_0\right)=\{u\in W^{1,p}_{0}(\Omega):u_0(z)\leq u(z)\ \mbox{a.e. in}\ \Omega\}$
	\end{claim}
	
		Indeed, let $u\in K_{\psi_g}$. Then
		\begin{eqnarray}\label{eq25}
			A(u)=N_{k_g}(u).
		\end{eqnarray}
		
		On (\ref{eq25}) we act with $(u_0-u)^+\in W^{1,p}_{0}(\Omega)$. We have
		\begin{eqnarray*}
			\left\langle A(u),(u_0-u)^+\right\rangle&=&\int_{\Omega}k_g(z,u)(u_0-u)^+dz\\
			&=&\int_{\Omega}\left[\beta(z)u_0^{p-1}+f(z,u_0)+g(z)\right](u_0-u)^+dz\\
			&=&\left\langle A(u_0),(u_0-u)^+\right\rangle\ (\mbox{since}\ u_0\ \mbox{is a solution of}\ (1)_g)
		\end{eqnarray*}
		\begin{eqnarray*}
			 &\Rightarrow&\int_{\{u_0>u\}}\left(|Du_0|^{p-2}Du_0-|Du|^{p-2}Du,Du_0-Du\right)_{\RR^N}dz=0\\
			&\Rightarrow&\left|\{u_0>u\}\right|_N=0,\ \mbox{hence}\ u_0\leq u.
		\end{eqnarray*}
		
This proves Claim \ref{claim2}.

By virtue of Claim \ref{claim2} every element of $K_{\psi_g}$ is a positive solution of \ref{eq:1}. Arguing by contradiction, suppose $K_{\psi_g}=\{u_0\}$ (see (\ref{eq21})).
\begin{claim}\label{claim3}
	$u_0\in \mbox{int}\, C_+$ is a local minimizer of the functional $\psi_g$.
\end{claim}

Recall that $0\leq u_0\leq u^*$ and consider the following truncation of $k_g(z,\cdot):$
\begin{eqnarray}\label{eq26}
	\hat{k}_g(z,x)=\left\{
		\begin{array}{ll}
			k_g(z,x)&\mbox{if}\ x<u^*(z)\\
			k_g(z,u^*(z))&\mbox{if}\ u^*(z)\leq x.
		\end{array}
	\right.
\end{eqnarray}

This is a Carath\'eodory function. We set $\hat{K}_g(z,x)=\int^{x}_{0}\hat{k}_g(z,s)ds$ and consider the $C^1$-functional $\hat{\psi}_g:W^{1,p}_{0}(\Omega)\rightarrow\RR$ defined by
$$\hat{\psi}_g(u)=\frac{1}{p}||Du||^{p}_{p}-\int_{\Omega}\hat{K}_g(z,u)dz\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$

Note that $\hat{\psi}_g$ is coercive (see (\ref{eq26})) and sequentially weakly lower semicontinuous. So, we can find $\tilde{u}\in W^{1,p}_{0}(\Omega)$ such that
\begin{eqnarray}\label{eq27}
	&&\hat{\psi}_g(\tilde{u})=\inf\left[\hat{\psi}_g(u):u\in W^{1,p}_{0}(\Omega)\right]\nonumber\\
	&\Rightarrow&\hat{\psi}_g'(\tilde{u})=0\nonumber\\
	&\Rightarrow&A(\tilde{u})=N_{\hat{k}_g}(\tilde{u}).
\end{eqnarray}

On (\ref{eq27}), first we can act with $(u_0-\tilde{u})^+\in W^{1,p}_{0}(\Omega)$ and as before using (\ref{eq21}) and (\ref{eq26}), we obtain $u_0\leq\tilde{u}$. Then on  (\ref{eq27}) we act with $(\tilde{u}-u^*)^+\in W^{1,p}_{0}(\Omega)$ and using (\ref{eq18}), (\ref{eq21}), (\ref{eq26}), we show that $\tilde{u}\leq u^*$. Therefore
\begin{eqnarray*}
	&&\tilde{u}\in[u_0,u^*]=\{u\in W^{1,p}_{0}(\Omega):u_0(z)\leq u(z)\leq u^*(z)\ \mbox{a.e. in}\ \Omega\}\\
	&\Rightarrow&\tilde{u}=u_0\ (\mbox{see (\ref{eq21}), (\ref{eq26}) and recall}\ K_{\psi_g}=\{u_0\}).
\end{eqnarray*}

Let $\rho=||u_0||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H_1(v)$. Then
\begin{eqnarray*}
	&&-\Delta_pu_0(z)+\xi_{\rho}u_0(z)^{p-1}\\
	&=&\beta(z)u_0(z)^{p-1}+f(z,u_0(z))+g(z)+\xi_{\rho}u_0(z)^{p-1}\\
	&\leq&\beta(z)u^*(z)^{p-1}+f(z,u^*(z))+g^*(z)+\xi_{\rho}u^*(z)^{p-1}\\
	&&\hspace{3cm}(\mbox{see}\ H_1(v)\ \mbox{and recall that}\ u_0\leq u^*,\ g\leq g^*)\\
	&=&-\Delta_pu^*(z)+\xi_{\rho}u^*(z)^{p-1}\ \mbox{a.e. in}\ \Omega\\
	&\Rightarrow&u^*-u_0\in \mbox{int}\, C_+\ (\mbox{see Arcoya and Ruiz \cite[Proposition 2.6]{5}}).
\end{eqnarray*}

Also, recall that $u_0\in \mbox{int}\, C_+$. Since $\psi_g|_{[0,u^*]}=\hat{\psi}_g|_{[0,u^*]}$ (see (\ref{eq21}) and (\ref{eq26})) it follows that $u_0\in \mbox{int}\, C_+$ is a local $C^{1}_{0}(\overline{\Omega})$-minimizer of $\psi_g$. Then from Garcia Azorero, Manfredi and Peral Alonso \cite[Theorem 1.1]{7}, it follows that $u_0\in \mbox{int}\, C_+$ is a local $W^{1,p}_{0}(\Omega)$-minimizer of $\psi_g$. This proves Claim \ref{claim3}.

By virtue of Claim \ref{claim3}, we can find $\rho\in(0,1)$ small such that
\begin{eqnarray}\label{eq28}
	\psi_g(u_0)<\inf\left[\psi_g(u):||u-u_0||=\rho\right]=\eta_{\rho}
\end{eqnarray}
(see Aizicovici, Papageorgiou and Staicu \cite{1} (proof of Proposition 29)). From (\ref{eq23}), (\ref{eq24}) and (\ref{eq28}), we see that we can apply Theorem \ref{th1} (the mountain pass theorem). So, there exists $\hat{u}\in W^{1,p}_{0}(\Omega)$ such that
\begin{eqnarray}\label{eq29}
	\hat{u}\in K_{\psi_g}\ \mbox{and}\ \eta_{\rho}\leq\psi_g(\hat{u}).
\end{eqnarray}

From Claim \ref{claim2}, (\ref{eq28}) and (\ref{eq29}) it follows that
$$u_0\leq\hat{u},\ \hat{u}\neq u_0\ \mbox{and}\ \hat{u}\in \mbox{int}\, C_+\ \mbox{solves problem}\ \eqref{eq:1}\ (\mbox{see (\ref{eq21})}).$$ This completes the proof.
\end{proof}
\begin{remark}
	The results of this section can be extended to problems driven by a nonhomogeneous differential operator $\mbox{div}\, a(Du)$ with $a:\RR^N\rightarrow\RR^N$ as in Papageorgiou and R\u{a}dulescu \cite{12} (see also Papageorgiou and R\u adulescu \cite{11}). For the sake of simplicity in the presentation, we have chosen to work with the $p$-Laplacian.
\end{remark}

\section{Solutions for Problem \ref{eq:2}}\label{sec4}

In this section we deal with problem \ref{eq:2}.

The hypotheses on the data of problem \ref{eq:2} are the following:

\smallskip
$H(\beta):$ $\beta\in L^\tau (\Omega)$ with $\tau >\frac{N}{2}$.

$H_2:$ $f:\Omega\times\RR\rightarrow\RR$ is a measurable function such that  for a.a. $z\in\Omega$ $f(z,0)=0,\ f(z,\cdot)\in C^1(\RR)$ and
\begin{itemize}
	\item[(i)] $|f_x'(z,x)|\leq a(z)(1+|x|^{r-2})$ for a.a. $z\in\Omega$, all $x\in\RR$, with $a\in L^{\infty}(\Omega)_+$, $2<r<2^*$;
	\item[(ii)] if $F(z,x)=\int^{x}_{0}f(z,s)ds$, then
	$$\lim\limits_{x\rightarrow\pm\infty}\frac{F(z,x)}{x^2}=+\infty\ \mbox{uniformly for a.a.}\ z\in\Omega;$$
	\item[(iii)] there exist $\eta_0>0$ and $\vartheta\in\left(\max\left\{1,(r-2)\frac{N}{2}\right\},2^*\right)$ such that
	 $$0<\eta_0\leq\liminf\limits_{x\rightarrow\pm\infty}\frac{f(z,x)x-2F(z,x)}{|x|^{\vartheta}}\ \mbox{uniformly for a.a.}\ z\in\Omega;$$
	\item[(iv)] $0=f_x'(z,0)=\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}$ uniformly for a.a. $z\in\Omega$;
	\item[(v)] there exists $\delta>0$ such that  $f(z,x)x\geq 0$ for a.a. $z\in\Omega$, all $|x|\leq\delta$.
\end{itemize}
\begin{theorem}\label{th8}
	If hypotheses $H_2$ hold and $\lambda\geq\hat{\lambda}_1(2)$, then problem \ref{eq:2} admits at least one nontrivial solution $u_0\in C^{1}_{0}(\overline{\Omega})$.
\end{theorem}
\begin{proof}
	Let $k\geq 1$ such that  $\lambda\in\left[\hat{\lambda}_k(2),\hat{\lambda}_{k+1}(2)\right)$. We set
	$$\bar{H}_k=\overset{k}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(2))\ \mbox{and}\ \hat{H}_k=\bar{H}_k^\perp=\overline{{\underset{{i\geq k+1}}\oplus}E(\hat{\lambda}_i(2))}.$$
	
	We have the following orthogonal direct sum decomposition
	$$H^{1}_{0}(\Omega)=\bar{H}_k\oplus\hat{H}_k.$$
	
	By virtue of hypotheses $H_2(iv),(v)$, given $\epsilon>0$, we can find $\delta_1\in\left(0,\delta\right]$ such that
	\begin{eqnarray}\label{eq30}
		0\leq F(z,x)\leq\frac{\epsilon}{2}x^2\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta_1.
	\end{eqnarray}
	
	Since $\bar{H}_k$ is finite dimensional, all norms are equivalent and so we can find $\rho_0>0$ such that
	\begin{eqnarray}\label{eq31}
		||u||\leq\rho_0\Rightarrow||u||_{\infty}\leq\delta_1\ \mbox{for all}\ u\in\bar{H}_k.
	\end{eqnarray}
	
	Let $\varphi_{\lambda}:H^{1}_{0}(\Omega)\rightarrow\RR$ be the energy functional for problem \ref{eq:2} defined by
	 $$\varphi_{\lambda}(u)=\frac{1}{2}\tau(u)-\frac{\lambda}{2}||u||^{2}_{2}-\int_{\Omega}F(z,u)dz\ \mbox{for all}\ u\in H^{1}_{0}(\Omega)$$
	with $\tau(u)=||Du||^{2}_{2}+\int_{\Omega}\beta(z)u^2dz$ for all $u\in H^{1}_{0}(\Omega)$. Evidently $\varphi_{\lambda}\in C^2(H^{1}_{0}(\Omega))$.
	
	For $u\in\bar{H}_k$ with $||u||\leq\rho_0$, we have
	\begin{eqnarray*}
		\varphi_{\lambda}(u)&\leq&\frac{1}{2}\tau(u)-\frac{\lambda}{2}||u||^{2}_{2}\ (\mbox{see (\ref{eq31})})\\
		&\leq&0\ (\mbox{see (\ref{eq7}) and recall that}\ \lambda\leq\hat{\lambda}_k(2)).
	\end{eqnarray*}
	
	From (\ref{eq30}) and hypothesis $H_2(i)$, we have
	\begin{eqnarray}\label{eq32}
		F(z,x)\leq\frac{\epsilon}{2}x^2+c_{11}|x|^r\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR,\ \mbox{some}\ c_{11}=c_{11}(\epsilon)>0.
	\end{eqnarray}
	
	For $u\in\hat{H}_k$, we have
	 $$\varphi_{\lambda}(u)\geq\frac{1}{2}\tau(u)-\frac{\lambda+\epsilon}{2}||u||^{2}_{2}-c_{11}||u||^{r}_{r}\ (\mbox{see (\ref{eq32})}).$$
	
	Choose $\epsilon>0$ small such that  $\lambda+\epsilon<\hat{\lambda}_{k+1}(2)$ (recall $\lambda\in\left[\hat{\lambda}_k(2),\hat{\lambda}_{k+1}(2)\right)$). Then we have
	\begin{eqnarray}\label{eq33}
		\varphi_{\lambda}(u)\geq c_{12}||u||^2-c_{13}||u||^r\ \mbox{for some}\ c_{12},c_{13}>0\ (\mbox{see (\ref{eq7})}).
	\end{eqnarray}
	
	Since $r>2$, from (\ref{eq33}) it follows that we can find $\rho\in\left(0,\rho_0\right]$ small such that
	$$\varphi_{\lambda}(u)\geq 0\ \mbox{for all}\ u\in\hat{H}_k\ \mbox{with}\ ||u||\leq\rho.$$
	
	So, we have proved that $\varphi_{\lambda}$ has a local linking at the origin with respect to the orthogonal direct sum decomposition $H^{1}_{0}(\Omega)=\overline{H}_k\oplus\hat{H}_k$. Since $\varphi_{\lambda}\in C^2(H^{1}_{0}(\Omega))$, from Su \cite[Proposition 2.3]{16}, we have
	\begin{eqnarray}\label{eq34}
		C_k(\varphi_{\lambda},0)=\delta_{k,d_k}\ZZ\ \mbox{with}\ d_k=\mbox{dim}\,\overline{H}_k.
	\end{eqnarray}
	
	On the other hand, from Aizicovici, Papageorgiou and Staicu \cite{2}, we have
	\begin{eqnarray}\label{eq35}
		C_k(\varphi_{\lambda},\infty)=0\ \mbox{for all}\ k\geq 0.
	\end{eqnarray}
	
	From (\ref{eq34}) and (\ref{eq35}) it follows that we can find $u_0\in K_{\varphi_{\lambda}}\backslash\{0\}$. Then $u_0$ solves problem \ref{eq:2} and from the regularity theory (see Struwe \cite[p. 218]{15}), we have that $u_0\in H^{1}_{0}(\overline{\Omega})$.
\end{proof}

If we strengthen hypothesis $H_2(v)$, we can improve the conclusion of Theorem \ref{th8} and provide more information about the solution $u_0$.

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

\smallskip
$H_3:$ $f:\Omega\times\RR\rightarrow\RR$ is a measurable function such that  for a.a. $z\in\Omega$, $f(z,0)=0,\ f(z,\cdot)\in C^1(\RR)$, hypotheses $H_3(i)\rightarrow(iv)$ are the same as the corresponding hypotheses $H_2(i)\rightarrow(iv)$ and\\ $(v)$ $f(z,x)x\geq 0$ for a.a. $z\in\Omega$, all $x\in\RR$ and the inequality is strict for all $(z,x)\in\Omega_0\times\RR$ with $|\Omega_0|_N>0$ and $x\neq 0$.
\begin{theorem}\label{th9}
	If hypotheses $H_3$ hold and $\lambda\geq\hat{\lambda}_1(2)$, then problem \ref{eq:2} admits a nodal solution $u_0\in C^{1}_{0}(\overline{\Omega})$.
\end{theorem}
\begin{proof}
	From Theorem \ref{th8} we know that problem \ref{eq:2} has a nontrivial solution $u_0\in H^{1}_{0}(\overline{\Omega})$. Suppose that $u_0$ has constant sign and to fix things assume that $u_0\geq 0$. We have
	\begin{eqnarray}\label{eq36}
		A(u)+\beta(z)u=\lambda u+N_f(u).
	\end{eqnarray}
	
	On (\ref{eq36}) we act with $\hat{u}_1(2,\beta)\in \mbox{int}\, C_+$. Then
	\begin{eqnarray*}
		&&\left\langle A(u)+\beta u,\hat{u}_1(2,\beta)\right\rangle=\lambda\int_{\Omega}u\hat{u}_1(2,\beta)dz+\int_{\Omega}f(z,u)\hat{u}_1(2,\beta)dz\\
		 &\Rightarrow&(\hat{\lambda}_1(2,\beta)-\lambda)\int_{\Omega}u\hat{u}_1(2,\beta)dz=\int_{\Omega}f(z,u)\hat{u}_1(2,\beta)dz.
	\end{eqnarray*}
	
	Note that $(\hat{\lambda}_1(2,\beta)-\lambda)\int_{\Omega}u\hat{u}_1(2,\beta)dz\leq 0$, while $\int_{\Omega}f(z,u)\hat{u}_1(2,\beta)dz>0$ (see $H_3(v)$ and recall that we have assumed that $u\geq 0$). So, we have a contradiction and this proves that $u_0$ is nodal.
\end{proof}
\begin{remark}
	Our results here answer the question posed in R\u adulescu \cite{14} and show that hypothesis (\ref{eq8}) in \cite{14} is not necessary. Finally we stress that our approach here differs from that of \cite{8}.
\end{remark}

\medskip
{\bf Acknowledgement}. The authors wish to thank two knowledgeable referees for their comments and remarks that helped to improve the paper.


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