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\title[Asymptotically linear elliptic equations with sign changing weight]{Multiple solutions for asymptotically linear elliptic equations with sign changing weight}


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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. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address{Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah, Saudi Arabia}
\email{\tt vicentiu.radulescu@math.cnrs.fr}

\keywords{Indefinite weight, weighted eigenvalue problem, critical groups, maximum principle, mountain pass theorem.\\
\phantom{aa} 2010 AMS Subject Classification:
Primary 35J20, Secondary 35J60, 58E05}

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\begin{abstract}
We consider a semilinear Dirichlet problem driven by the Laplacian and with an indefinite (that is, sign changing) weight and a nonlinearity which is asymptotically linear near $\pm\infty$. Using variational methods together with truncation techniques and Morse theory, we show that the problem has at least three nontrivial solutions, two of which have constant sign (one positive and the other negative).
\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 existence of multiple nontrivial solutions for the following semilinear Dirichlet problem:
\begin{eqnarray}\label{eq1}
-\Delta u(z)=\beta(z)f(u(z))\ \mbox{in}\ \Omega,\quad u|_{\partial\Omega}=0.
\end{eqnarray}

In this problem the weight function $\beta\in L^{\infty}(\Omega)$ is nodal (that is, sign changing) and $f$ is a $C^1$-nonlinearity which exhibits linear growth near $\pm\infty$ and it is superlinear near $0$. Using variational methods coupled with suitable truncation and comparison techniques and Morse theory (critical groups), we show that problem (\ref{eq1}) has at least three nontrivial solutions, two of which have constant sign (one positive and the other negative).

Problems with an indefinite nonlinearity were first investigated by Ouyang \cite{9} on a compact Riemannian manifold using bifurcation theory. Subsequently, Alama and Tarantello \cite{1}, \cite{2} using variational methods extended the results of Ouyang \cite{9}, by considering more general nonlinearities and assuming a thickness condition of the form $\overline{\Omega}_{+}\cap\overline{\Omega}_{-}=\emptyset$, where $\Omega_{+}=\{z\in\Omega:\beta(z)>0\}$ and $\Omega_{-}=\{z\in\Omega:\beta(z)<0\}$. This condition was removed by Berestycki, Capuzzo Dolcetta and Nirenberg \cite{6}, who proved existence of solutions by deriving a priori bounds and using topological methods. However, their condition on the weight $\beta(\cdot)$ is stronger since $\beta\in C^1(\Omega)$ and they assume a nondegeneracy condition of the form $\nabla\beta(z)\neq 0$ when $\beta(z)=0$ (that is, the level set $[\beta=0]$ is a $C^1$-submanifold; so, in this case the zero set is thin). Three solutions theorems for problems with a superlinear $f(\cdot)$ satisfying the Ambrosetti-Rabinowitz condition, without any thickness or thinness condition, were proved for parametric problems and for certain values of the parameter, by Chang and Jiang \cite{7}.

To the best of our knowledge, the first paper involving nonlinear elliptic equations with asymptotically linear terms $f(z,u)$ is due to Amann and Zehnder \cite{amann}, in the context of semilinear equations (that is, if $p=2$). They proved an existence theorem assuming that the limit $\lim_{u\rightarrow \pm\infty}f(z,u)/u=\lambda\in\RR$ exists, does not belong to the spectrum of the Laplace operator in $H^1_0(\Omega)$ (nonresonance at infinity) and there is at least one eigenvalue between $\lambda$ and $\lambda +f'(0)$.
Related multiplicity results for Dirichlet elliptic problems involving asymptotically linear terms near $\pm\infty$ have been studied in some recent papers, see Hu and Papageorgiou \cite{hp11} and Papageorgiou and Smyrlis \cite{ps9}. The hypotheses in the present paper are more general than those imposed in \cite{hp11} and \cite{ps9}. For instance, Hu and Papageorgiou \cite{hp11} establish a {\it three nontrivial solutions} theorem provided that $f$ satisfies a local boundedness assumption and $f_u(z,u)$ has a $q$-polynomial growth for some $0<q<4/(N-2)$.
 Papageorgiou and Smyrlis \cite{ps9} produce five nontrivial smooth solutions, two positive, two negative and one nodal. This is done under several hypotheses, including a behavior of the quotient $\frac{f(z,u)}{|u|^{p-2}u}$ with respect to the first two eigenvalues of the $p$-Laplace operator, the existence of two zeros of the nonlinear term and a certain monotonicity assumption with respect to the mapping $u\longmapsto f(z,u)+|u|^{p-2}u$, see \cite[pp. 3138, 3142]{ps9}. With respect to these works, the main features of the present paper are the following: (i) the presence of a sign changing weight; (ii) the study is performed for  nonlinearities that fulfill general assumptions and whose behavior is described with respect to certain eigenvalues
 of a weighted eigenvalue problem that involves the indefinite potential.

Throughout this paper, for all $x\in\RR$, we denote $x^{\pm}=\max\{\pm x,0\}$.

\section{Mathematical Background}\label{sec2}
In the analysis of problem (\ref{eq1}), we will use the Sobolev space $H_0^1(\Omega)$ and the Banach space $C_0^1(\overline{\Omega})=\{u\in C^1(\overline{\Omega}):u|_{\partial\Omega=0}\}$. We know that $C_0^1(\overline{\Omega})$ is an ordered Banach space with positive cone
$$C_+(\overline{\Omega})=\left\{u\in C_0^1(\overline{\Omega}):u(z)\geq 0\ \mbox{for all}\ z\in\overline{\Omega}\right\}.$$
This cone has a nonempty interior given by
$$\mbox{int}\, C_{+}(\overline{\Omega})=\left\{u\in C_{+}(\overline{\Omega}):u(z)>0\ \mbox{for all}\ z\in\Omega,\ \frac{\partial u}{\partial n}\Big|_{\partial\Omega}<0\right\}$$
with $n(\cdot)$ being the outward unit normal on $\partial\Omega$. Also, prominent in our arguments will be the spectrum of the following weighted linear eigenvalue problem:
\begin{eqnarray}\label{eq2}
-\Delta u(z)=\lambda\beta(z)u(z)\ \mbox{in}\ \Omega,\quad u|_{\partial\Omega}=0.
\end{eqnarray}

When the weight $\beta\in L^{\infty}(\Omega)$ is nodal, it is well-known (see, for example, Gasinski and Papageorgiou \cite[p. 714]{8}), that problem (\ref{eq2}) has a double sequence of distinct eigenvalues
$$\ldots<\hat{\lambda}_k^{-}(\beta)<\ldots <\hat{\lambda}_2^{-}(\beta)<\hat{\lambda}_1^{-}(\beta)<0<\hat{\lambda}_1^{+}(\beta)<\hat{\lambda}_2^{+}(\beta)<\ldots <\hat{\lambda}_k^{+}(\beta)<\ldots$$

Let $|\cdot|_N$ denote the Lebesgue measure on $\RR^N$. If $\Omega_{+}=\{z\in\Omega:\beta(z)>0\}$ and $|\Omega_{+}|_N>0$, then $\hat{\lambda}_n^{+}(\beta)\to +\infty$ as $n\to\infty$. Similarly, if $\Omega_{-}=\{z\in\Omega:\beta(z)<0\}$ and $|\Omega_{-}|_N>0$, then $\hat{\lambda}_n^{-}(\beta)\to -\infty$. On the other hand, if $|\Omega_{+}|_N=0$, then $\hat{\lambda}_n^{+}(\beta)=0$ for all $n\geq 1$, while if $|\Omega_{-}|_N=0$, then $\hat{\lambda}_n^{-}(\beta)=0$ for all $n\geq 1$.

Suppose that $\Omega_{+}$ is an open connected set with a $C^2$-boundary $\partial\Omega_{+}$ and consider the following weighted linear eigenvalue problem:
\begin{eqnarray}\label{eq3}
-\Delta u(z)=\lambda\beta^{+}(z)u(z)\ \mbox{in}\ \Omega_+,\quad u|_{\partial\Omega_+}=0.
\end{eqnarray}

According to our previous discussion, problem (\ref{eq3}) has only positive eigenvalues, namely $\{\hat{\lambda}_k^{\Omega_{+}}(\beta^{+})\}_{k\geq 1}$, $\hat{\lambda}_k^{\Omega_{+}}(\beta^{+})\to+\infty$ as $k\to\infty$ and $\hat{\lambda}_1^{\Omega_{+}}(\beta^{+})>0$ and it is simple. We have the following variational characterization of $\hat{\lambda}_1^{\Omega_{+}}(\beta^{+})$:
\begin{eqnarray}\label{eq4}
\hat{\lambda}_1^{\Omega_{+}}(\beta^{+})=\inf\left[\frac{\|Du\|_{L^2(\Omega_{+},\RR^N)}^2}{\int\limits_{\Omega_{+}}\beta^{+}(z)u^2dz}:u\in H_0^1(\Omega_{+}),u\neq 0\right].
\end{eqnarray}

The infimum in relation (\ref{eq4}) is realized on the one-dimensional eigenspace corresponding to $\hat{\lambda}_1^{\Omega_{+}}(\beta^{+})>0$. Standard regularity theory implies that the elements of this eigenspace belong in $C_0^1(\overline{\Omega_+})$. In fact from (\ref{eq4}) we can see that they do not change sign. In what follows by $\tilde{u}_1(\Omega_{+})$ we denote the $L^2$-normalized (that is, $\|\tilde{u}_1(\Omega_{+})\|_{L^2(\Omega_{+})}=1$) positive eigenfunction. Using the maximum principle (see, for example, Gasinski and Papageorgiou \cite[p. 738]{8}) we have $\tilde{u}_1(\Omega_{+})\in \mbox{int}\, C_{+}(\Omega_{+})$.

Our variational approach, will be based on the well-known mountain pass theorem of Ambrosetti and Rabinowitz \cite{4}, formulated here in a slightly more general form using the $C$-compactness condition on the functional instead of the more common $PS$-condition (Palais-Smale condition).

So, let $X$ be a Banach and $X^*$ its topological dual. By $\langle\cdot,\cdot\rangle$ we denote the duality brackets for the pair $(X^*,X)$. Given $\varphi\in C^1(X)$, we say that $\varphi$ satisfies the $C$-condition, if the following is true:

``Every sequence $\{u_n\}_{n\geq 1}\subseteq X$ such that  $\{\varphi(u_n)\}_{n\geq 1}\subseteq\RR$ is bounded and
$$(1+\|u_n\|)\varphi'(u_n)\to 0\ \mbox{in}\ X^*\ \mbox{as}\ n\to\infty,$$
admits a strongly convergent subsequence."

This compactness-type condition on $\varphi$, leads to a deformation theorem, which in turn produces a minimax theory for the critical values of $\varphi$. A major result in that theory, is the mountain pass theorem.
\begin{theorem}\label{th1}
Assume that $\varphi\in C^1(X)$ satisfies the $C$-condition, $u_0,u_1\in X$, $\rho>0$, $\|u_1-u_0\|>\rho$,
$$\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):\|u-u_0\|=\rho]=m_{\rho}$$
and $c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq 1}\varphi(\gamma(t))$ where $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\gamma(1)=u_1\}$. Then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

Also, we will use some tools from Morse theory (critical groups), which for the benefit of the reader, we briefly review below. So, for $\varphi\in C^1(X)$ and $c\in\RR$, we introduce the following sets:
$$\varphi^c=\{u\in X:\varphi(u)\leq c\},\ K_{\varphi}=\{u\in X:\varphi'(u)=0\}\ \mbox{and}\ K_{\varphi}^c=\{u\in  K_{\varphi}:\varphi(u)=c\}.$$

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 with integer coefficients for the pair $(Y_1,Y_2)$. The critical groups of $\varphi$ at an isolated $u\in K_{\varphi}^c$, are defined by
$$C_k(\varphi,u)=H_k(\varphi^c\cap U, \varphi^c\cap U\setminus\{u\})\ \mbox{for all}\ k\geq 0,$$
where $U$ is a neighborhood of $u$, such that $K_{\varphi}\cap\varphi^c\cap U=\{u\}$. The excision property of singular homology theory, implies that this definition is independent of the particular 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
$$C_k(\varphi,\infty)=H_k(X,\varphi^c)\ \mbox{for all}\ k\geq 0.$$
Using the second deformation theorem (see, for example, Gasinski and Papageorgiou \cite[p. 628]{8}), we see that this definition is independent of the particular choice of the level $c<\inf\varphi(K_{\varphi})$. We know that if $C_k(\varphi,\infty)\neq 0$, then there exists $u\in K_{\varphi}$ such that  $C_k(\varphi,u)\neq 0$ (a consequence of the so-called Morse relation, see, for example, Ambrosetti and Malchiodi \cite[p. 222]{3}).

Recently the authors proved the following result which is useful in the computation of critical groups at infinity (see Papageorgiou and R\u adulescu \cite{11}).

\begin{prop}\label{prop2}
 Assume that $(t,u)\longmapsto h_t(u)$ belongs to $C^1([0,1]\times X)$, maps bounded sets to bounded sets, the maps $u\to (h_t)'(u)$ and $t\longmapsto\partial_t h_t(u)$ are both locally Lipschitz, $h_0$ and $h_1$ satisfy the $C$-condition,
$$|\partial_t h_t(u)|\leq c_1\|u\|^p\ \mbox{for all}\ u\in X,$$
with $c_1>0$, $1<p<\infty$ and there exist $\xi_0\in\RR$ and $\delta_0>0$ such that
$$h_t(u)\leq\xi_0\Rightarrow (1+\|u\|)\|(h_t)'(u)\|_{*}\geq\delta_0\|u\|^p\ \mbox{for all}\ t\in [0,1].$$
Then $C_k(h_0,\infty)=C_k(h_1,\infty)$ for all $k\geq 0$.
\end{prop}
We conclude this section by fixing our notation. In what follows by $\|\cdot\|$ we denote the norm of the Sobolev space $H_0^1(\Omega)$. By virtue of the Poincar\'e inequality, we have
$$\|u\|=\|Du\|_{L^2(\Omega,\RR^N)}\ \mbox{for all}\ u\in H_0^1(\Omega).$$

 For all $u\in H_0^1(\Omega)$, we define $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u^{\pm}\in H_0^1(\Omega),\ u=u^{+}-u^{-},\ |u|=u^{+}+u^{-}.$$
Also, by $N_f$ we denote the Nemitsky (superposition) operator corresponding to $f$, that is,
$$N_f(u)(\cdot)=f(u(\cdot))\ \mbox{for all}\ u\in H_0^1(\Omega).$$
Finally by $A\in\mathcal{L}(H_0^1(\Omega),H^{-1}(\Omega)=H_0^1(\Omega)^{*})$ we denote the bounded linear operator defined by
$$\langle A(u),y\rangle=\int\limits_{\Omega}(Du,Dy)_{\RR^N}dz\ \mbox{for all}\ u,y\in H_0^1(\Omega).$$

\section{Multiplicity Theorem}\label{sec3}
Our hypotheses on the data of problem (\ref{eq1}), are the following:

\smallskip
$H(\beta)$: $\beta\in L^{\infty}(\Omega)$, $\beta^{+},\ \beta^{-}\neq 0$ and $\Omega_{+}=\{z\in\Omega:\beta(z)>0\}$ is a connected open set with a $C^2$-boundary $\partial\Omega_{+}$.

\smallskip
$H(f)$: $f\in C^1(\RR)$, $f(0)=0$ and
\begin{itemize}
\item [(i)] $|f'(x)|\leq a(1+|x|^{r-1})$ for all $x\in\RR$ with $a>0$ and $2\leq r<2^*=\left\{  \begin{array}{ll}
   \frac{2N}{N-2} &\mbox{if}\ N\geq 3 \\
    +\infty &\mbox{if}\ N=1,2;
    \end{array}\right.$
\item [(ii)] there exists an integer $m\geq 2$ such that
\begin{eqnarray*}
&&\hat{\lambda}_1^{\Omega_{+}}(\beta^{+})<\hat{\lambda}_m^{+}(\beta)\\
&&\hat{\lambda}_m^{+}(\beta)<\liminf\limits_{x\to\pm\infty}\frac{f(x)}{x}\leq \limsup\limits_{x\to\pm\infty}\frac{f(x)}{x}<\hat{\lambda}_{m+1}^{+}(\beta);
\end{eqnarray*}
\item [(iii)] $f'(x)=\lim\limits_{x\to 0}\frac{f(x)}{x}=0$.
\end{itemize}

First we will produce two constant sign solutions, one positive and the other negative. To this end, we introduce the positive and negative truncations of $f(\cdot)$, namely the $C^1$-functions (see hypothesis $H(f)$(iii))
$$f_{+}(x)=f(x^{+})\ \mbox{and}\ f_{-}(x)=f(-x^{-})\ \mbox{for all}\ x\in\RR.$$
Let $F_{\pm}(x)=\int\limits_0^x f_{\pm}(s)ds$ and consider the $C^2$-functionals $\varphi_{\pm}:H_0^1(\Omega)\to\RR$ defined by
$$\varphi_{\pm}(u)=\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}\beta(z)F_{\pm}(u(z))dz\ \mbox{for all}\ u\in H_0^1(\Omega).$$
Also, let $\varphi:H_0^1(\Omega)\to\RR$ be the energy functional for problem (\ref{eq1}) defined by
$$\varphi(u)=\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}\beta(z)F(u(z))dz\ \mbox{for all}\ u\in H_0^1(\Omega),$$
where $F(x)=\int\limits_0^x f(s)ds$. We know that $\varphi\in C^2(H_0^1(\Omega)).$
\begin{prop}\label{prop3}
If hypotheses $H(\beta)$ and $H(f)$ hold, then the functionals $\varphi_{\pm}$ satisfy the $C$-condition.
\end{prop}
\begin{proof}
We do the proof for the functional $\varphi_{+}$, the proof for $\varphi_{-}$ being similar.

So, let $\{u_n\}_{n\geq 1}\subseteq H_0^1(\Omega)$ be a sequence such that  $\{\varphi_{+}(u_n)\}_{n\geq 1}\subseteq\RR$ is bounded and
\begin{eqnarray}\label{eq5}
(1+\|u_n\|)\varphi_{+}'(u_n)\to 0\ \mbox{in}\ H^{-1}(\Omega)=H_0^1(\Omega)^*.
\end{eqnarray}
From (\ref{eq5}), we have
\begin{eqnarray}\label{eq6}
\left|\langle A(u_n),h\rangle-\int\limits_{\Omega}\beta(z)f_{+}(u_n)h dz\right|\leq\frac{\varepsilon_n\|h\|}{1+\|u_n\|}\ \mbox{for all}\ h\in H_0^1(\Omega),\ \mbox{with}\ \varepsilon_n\to 0^{+}.
\end{eqnarray}
In (\ref{eq6}) we choose $h=-u_n^{-}\in H_0^1(\Omega)$. Then
\begin{eqnarray}\label{eq7}
&&\|Du_n^{-}\|_{L^2(\Omega,\RR^N)}^2\leq\varepsilon_n\ \mbox{for all}\ n\geq 1\nonumber\\
&\Rightarrow& u_n^{-}\to 0\ \mbox{in}\ H_0^1(\Omega).
\end{eqnarray}
Suppose that $\|u_n^{+}\|\to\infty$. We set $y_n=\frac{u_n^{+}}{\|u_n^{+}\|}$, $n\geq 1$. Then $\|y_n\|=1$, $y_n\geq 0$ for all $n\geq 1$. So, by passing to a suitable subsequence if necessary, we may assume that
\begin{eqnarray}\label{eq8}
y_n\overset{w}{\rightarrow}y\ \mbox{in}\ H_0^1(\Omega)\ \mbox{and}\ y_n\to y\ \mbox{in}\ L^2(\Omega)\ \mbox{as}\ n\to\infty.
\end{eqnarray}
From (\ref{eq5}) and (\ref{eq6}) it follows that
\begin{eqnarray}\label{eq9}
&&\left|\langle A(y_n),h\rangle-\int\limits_{\Omega}\beta(z)\frac{f(u_n^{+})}{\|u_n^{+}\|}h dz\right|\leq \varepsilon_n'\|h\|\ \mbox{for all}\ h\in H_0^1(\Omega),\ \mbox{with}\ \varepsilon_n'\to 0^{+}\ \mbox{as}\ n\to\infty.
\end{eqnarray}
From hypotheses $H(\beta)$ and $H(f)$(i),(ii) we deduce that $\left\{\beta\frac{N_f(u_n^{+})}{\|u_n^{+}\|}\right\}_{n\geq 1}\subseteq L^2(\Omega)$ is bounded. So, if in (\ref{eq9}) we choose $h=y_n-y\in H_0^1(\Omega)$ and pass to the limit as $n\to\infty$, using (\ref{eq8}) we obtain
\begin{eqnarray}
&&\lim\limits_{n\to\infty}\langle A(y_n),y_n-y\rangle=0,\nonumber\\
&\Rightarrow& \|Dy_n\|_{L^2(\Omega,\RR^N)}^2\to \|Dy\|_{L^2(\Omega,\RR^N)}^2\nonumber\\
&\Rightarrow& y_n\to y\ \mbox{in}\ H_0^1(\Omega)\ (\mbox{by the Kadec-Klee property of Hilbert spaces; see}\ (\ref{eq8}))\label{eq10}\\
&\Rightarrow& \|y\|=1,\ y\geq 0.\label{eq11}
\end{eqnarray}
Since $\left\{\beta\frac{N_f(u_n^{+})}{\|u_n^{+}\|}\right\}_{n\geq 1}\subseteq L^2(\Omega)$ is bounded, by passing to a suitable subsequence if necessary and using hypothesis $H(f)$(ii), we obtain
\begin{eqnarray}\label{eq12}
\beta\frac{N_f(u_n^{+})}{\|u_n^{+}\|}\overset{w}{\rightarrow}\beta\eta y\ \mbox{in}\ L^2(\Omega)\ \mbox{with}\ \eta\in (\hat{\lambda}_m^{+}(\beta),\hat{\lambda}_{m+1}^{+}(\beta)).
\end{eqnarray}
So, if in (\ref{eq9}) we pass to the limit as $n\to\infty$ and use (\ref{eq10}) and (\ref{eq12}), then
\begin{eqnarray}\label{eq13}
&&\langle A(y),h\rangle=\int\limits_{\Omega}\beta(z)\eta y h dz\ \mbox{for all}\ h\in H_0^1(\Omega),\nonumber\\
&\Rightarrow& A(y)=\beta(z)\eta y,\nonumber\\
&\Rightarrow& -\Delta y(z)=\eta\beta(z) y(z)\ \mbox{a.e. in}\ \Omega,\ y|_{\partial\Omega}=0.
\end{eqnarray}

From (\ref{eq12}) and (\ref{eq13}) it follows that $y=0$, which contradicts (\ref{eq11}). This implies that $\{u_n^{+}\}_{n\geq 1}\subseteq H_0^1(\Omega)$ is bounded. This in conjunction with (\ref{eq7}), imply that $\{u_n\}_{n\geq 1}\subseteq H_0^1(\Omega)$ is bounded. So, we may assume that
\begin{eqnarray}\label{eq14}
u_n\overset{w}{\rightarrow}u\ \mbox{in}\ H_0^1(\Omega)\ \mbox{and}\ u_n\to u\ \mbox{in}\ L^2(\Omega).
\end{eqnarray}
In (\ref{eq6}) we choose $h=u_n-u\in H_0^1(\Omega)$, pass to the limit as $n\to\infty$ and use (\ref{eq6}).

We have
\begin{eqnarray*}
&&\lim\limits_{n\to\infty}\langle A(u_n),u_n-u\rangle=0,\\
&\Rightarrow& u_n\to u\ \mbox{in}\ H_0^1(\Omega)\ (\mbox{as before via the Kadec-Klee property}),\\
&\Rightarrow& \varphi_{+}\ \mbox{satisfies the C-condition}.
\end{eqnarray*}
In a similar fashion, we show that $\varphi_{-}$ satisfies the $C$-condition.
\end{proof}

Minor changes in the above proof, lead also to the following result.
\begin{prop}\label{prop4}
If hypotheses $H(\beta)$ and $H(f)$ hold, then the energy functional $\varphi$ satisfies the $C$-condition.
\end{prop}
Next we show that the functional $\varphi_{\pm}$ satisfy the mountain pass geometry.
\begin{prop}\label{prop5}
If hypotheses $H(\beta)$ and $H(f)$ hold, then there exist $\rho_{\pm}>0$ such that  $\inf[\varphi_{\pm}(u):\|u\|=\rho_{\pm}]=\hat{m}_{\pm}>0$.
\end{prop}
\begin{proof}
We do the proof for the functional $\varphi_{+}$, the proof for $\varphi_{-}$ being similar.

Hypotheses $H(f)$ imply that given $\varepsilon>0$ and $\vartheta\in (2,2^*)$, we can find $c_1=c_1(\varepsilon,\vartheta)>0$ such that
\begin{eqnarray}\label{eq15}
&&|f(x)|\leq\varepsilon|x|+c_1|x|^{\vartheta -1}\ \mbox{for all}\ x\in\RR,\nonumber\\
&\Rightarrow& |F(x)|\leq \frac{\varepsilon}{2}x^2+c_2|x|^{\vartheta}\ \mbox{for all}\ x\in\RR\ \mbox{and with}\ c_2=\frac{c_1}{\vartheta}>0.
\end{eqnarray}
Then for all $u\in H^1(\Omega)$, we have
\begin{eqnarray*}
\varphi_{+}(u)&=&\frac{1}{2}\|Du\|_{L^2(\Omega,\RR^N)}^2-\int\limits_{\Omega}\beta(z)F(u^{+})dz\\
&\geq&\frac{1}{2}\|u\|^2-\frac{\varepsilon}{2}\|\beta\|_{L^{\infty}(\Omega)}\|u^{+}\|_{L^2(\Omega)}^2-c_2\|u^{+}\|_{L^{\vartheta}(\Omega)}^{\vartheta}\\
&\geq&\frac{1}{2}\left(1-\frac{\varepsilon\|\beta\|_{L^{\infty}(\Omega)}}{\hat{\lambda}_1^{+}(\beta)}\right)\|u\|^2-c_3\|u\|^{\vartheta}\ \mbox{for some}\ c_3>0\ (\mbox{see}\ (\ref{eq4})\ \mbox{and}\ (\ref{eq15})).
\end{eqnarray*}
Choosing $\varepsilon\in\left(0,\frac{\hat{\lambda}_1^{+}(\beta)}{\|\beta\|_{L^{\infty}(\Omega)}}\right)$, we obtain
\begin{eqnarray}\label{eq16}
\varphi_{+}(u)\geq\frac{c_4}{2}\|u\|^2-c_3\|u\|^{\vartheta}\ \mbox{with}\ c_4=c_4(\varepsilon)>0.
\end{eqnarray}
Because $\vartheta>2$, from (\ref{eq16}) we see that for $\rho_{+}\in (0,1)$ small, we have
$$\varphi_{+}(u)\geq\hat{m}_{+}>0\ \mbox{for all}\ u\in H_0^1(\Omega),\ \mbox{with}\ \|u\|=\rho_{+}.$$
Similarly for the functional $\varphi_{-}$.
\end{proof}

Recall that $\tilde{u}_1(\Omega_{+})\in \mbox{int}\, C_{+}(\Omega_{+})$ is the principal eigenfunction of $-\Delta$ in $H_0^1(\Omega_{+})$ with weight $\beta^{+}\in L^{\infty}(\Omega)_{+}$. We extend $\tilde{u}_1$ to all of $\overline{\Omega}$ by setting $\tilde{u}_1$ to be equal to $0$ on $\overline{\Omega}\setminus\Omega_{+}$. We denote this extension by $\overline{u}_1$. Evidently $\overline{u}_1\in C_{+}(\overline{\Omega})\cap H_0^1(\Omega)$.

\begin{prop}\label{prop6}
If hypotheses $H(\beta)$ and $H(f)$ hold, then $\limsup\limits_{t\to\pm\infty}\varphi_{\pm}(t\overline{u}_1)<0$.
\end{prop}

\begin{proof}
For $t>0$, we have
\begin{eqnarray*}
\varphi_{+}(t\overline{u}_1)&=&\frac{t^2}{2}\|D\overline{u}_1\|_{L^2(\Omega,\RR^N)}^2-\int\limits_{\Omega}\beta(z)F_{+}(t\overline{u}_1)dz\\
&=&\frac{t^2}{2}\|\overline{u}_1\|^2-\int\limits_{\Omega}\beta^{+}(z)F(t\overline{u}_1)dz\ (\mbox{note that}\ \overline{u}_1\in C_{+}(\overline{\Omega})),\\
\Rightarrow &&\frac{\varphi_{+}(t\overline{u}_1)}{t^2}=\frac{1}{2}\|\overline{u}_1\|^2-\int\limits_{\Omega}\beta^{+}(z)\frac{F(t\overline{u}_1)}{t^2}dz,
\end{eqnarray*}
\begin{eqnarray*}
\Rightarrow \limsup\limits_{t\to+\infty}\frac{\varphi_{+}(t\overline{u}_1)}{t^2}&\leq&\frac{1}{2}\|\overline{u}_1\|^2-\int\limits_{\Omega}\beta^{+}(z) \liminf\limits_{t\to+\infty}\frac{F(t\overline{u}_1)}{t^2}dz\ (\mbox{by Fatou's lemma})\\
&\leq&\frac{1}{2}\|\overline{u}_1\|^2-\frac{\eta}{2}\int\limits_{\Omega}\beta^{+}(z)\overline{u}_1^2dz\ \mbox{with}\ \eta\in (\hat{\lambda}_m^{+}(\beta),\hat{\lambda}_{m+1}^{+}(\beta))\\
& & (\mbox{see hypothesis}\ H(f)(ii))\\
&=&\frac{1}{2}\|D\tilde{u}_1\|_{L^2(\Omega_{+},\RR^N)}^2-\frac{\eta}{2}\int\limits_{\Omega_{+}}\beta^{+}(z)\tilde{u}_1^2dz\ (\mbox{recall the definition of}\ \overline{u}_1)\\
&\leq&\frac{1}{2}\left[1-\frac{\eta}{\hat{\lambda}_1^{\Omega_{+}}(\beta^{+})}\right]\|\tilde{u}_1\|_{H_0^1(\Omega_{+})}^2\ (\mbox{see}\ (\ref{eq4}))\\
&<& 0\ (\mbox{see}\ H(f)(ii)).
\end{eqnarray*}
Similarly for the functional $\varphi_{-}$.
\end{proof}

Now we are ready to produce constant sign solutions.
\begin{prop}\label{prop7}
If hypotheses $H(\beta)$ and $H(f)$ hold, then problem (\ref{eq1}) has at least two nontrivial constant sign solutions
$$u_0\in\mbox{int}\, C_{+}(\overline{\Omega})\ \mbox{and}\ v_0\in -\mbox{int}\, C_{+}(\overline{\Omega}).$$
\end{prop}
\begin{proof}
By virtue of Proposition \ref{prop3}, \ref{prop5} and \ref{prop6}, we can apply Theorem \ref{th1} (the mountain pass theorem) and obtain $u_0\in H_0^1(\Omega)$ such that
\begin{eqnarray}\label{eq17}
\varphi_{+}'(u_0)=0\ \mbox{and}\ \varphi_{+}(0)=0<\hat{m}_{+}\leq\varphi_{+}(u_0).
\end{eqnarray}
From (\ref{eq17}) it is clear that $u_0\neq 0$. We have
\begin{eqnarray}\label{eq18}
A(u_0)=\beta(z)N_{f_{+}}(u_0).
\end{eqnarray}
On (\ref{eq18}) we act with $-u_0^{-}\in H_0^1(\Omega)$ and obtain
$$\|u_0^{-}\|^2=0,\ \mbox{hence}\ u_0\geq 0,\ u_0\neq 0.$$
Then (\ref{eq18}) becomes
\begin{eqnarray*}
&&A(u_0)=\beta(z)N_{f}(u_0)\\
&\Rightarrow& -\Delta u_0(z)=\beta(z)u_0(z)\ \mbox{a.e. in}\ \Omega,\quad u_0|_{\partial\Omega}=0.
\end{eqnarray*}
Standard regularity theory implies that $u_0\in C_{+}(\overline{\Omega})\setminus\{0\}$. Let $\rho=\|u_0\|_{\infty}$. Hypotheses $H(\beta)$ and $H(f)$(i),(iii) imply that we can find $\xi_{\rho}>0$ such that
$$\beta(z)f(x)+\xi_{\rho}x\geq 0\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ 0\leq x\leq \rho.$$
So, we have
\begin{eqnarray*}
&&-\Delta u_0(z)+\xi_{\rho}u_0(z)=\beta(z)f(u_0(z))+\xi_{\rho}u_0(z)\geq 0\ \mbox{a.e. in}\ \Omega,\\
&\Rightarrow& \Delta u_0(z)\leq \xi_{\rho}u_0(z)\ \mbox{a.e. in}\ \Omega,\\
&\Rightarrow& u_0\in \mbox{int}\, C_{+}(\overline{\Omega})\ (\mbox{by the maximum principle, see \cite[p. 738]{8}}).
\end{eqnarray*}
Similarly, working this time with the functional $\varphi_{-}$, we obtain another nontrivial constant sign solution $ v_0\in -\mbox{int}\, C_{+}(\overline{\Omega})$.
\end{proof}

Next we will produce a third nontrivial solution for problem (\ref{eq1}). To do this, we use tools from Morse theory (critical groups). So,  we first compute the critical groups of the energy functional $\varphi$ at infinity. This particular computation will be based on Proposition \ref{prop2}.
\begin{prop}\label{prop8}
If hypotheses $H(\beta)$ and $H(f)$ hold, then
$C_k(\tau,\infty)=\delta_{k,d_m}\ZZ$ for all $k\geq 0$ with some $d_m\geq 2$.
\end{prop}

\begin{proof}
Let $\hat{\eta}\in (\hat{\lambda}_m^{+}(\beta),\hat{\lambda}_{m+1}^{+}(\beta))$ and consider the $C^2$-functional $\tau:H_0^1(\Omega)\to\RR$ defined by
$$\tau(u)=\frac{1}{2}\|Du\|_2^2-\frac{\hat{\eta}}{2}\int\limits_{\Omega}\beta(z)u^2 dz\ \mbox{for all}\ u\in H_0^1(\Omega).$$
We consider the homotopy $h(t,u)=h_t(u)$ defined by
$$h_t(u)=(1-t)\varphi(u)+t\tau(u)\ \mbox{for all}\ (t,u)\in [0,1]\times H_0^1(\Omega).$$

{\bf Claim.} {\it
There exist $\mu\in\RR$ and $\delta>0$ such that
$$h_t(u)\leq\mu\Rightarrow (1+\|u\|)\|(h_t)'(u)\|_{*}^2\geq\delta\|u\|^2\ \mbox{for all}\ t\in [0,1].$$
}

We argue by contradiction. So, suppose that the Claim is not true. Since the homotopy $(t,u)\longmapsto h_t(u)$ maps bounded sets to bounded sets, we can find $\{t_n\}_{n\geq 1}\subseteq [0,1]$ and $\{u_n\}_{n\geq 1}\subseteq H_0^1(\Omega)$ such that
\begin{eqnarray}\label{eq19}
\left\{  \begin{array}{ll}
   t_n\to t,\ \|u_n\|\to\infty,\ h_{t_n}(u_n)\to-\infty\ \mbox{as}\ n\to\infty\ \mbox{and} \\
  \displaystyle  |\langle (h_{t_n})'(u_n),v\rangle|\leq\frac{\|v\|}{n(1+\|u_n\|)}\|u_n\|^2\ \mbox{for all}\ v\in H_0^1(\Omega),\ \mbox{all}\ n\geq 1.
    \end{array}\right.
\end{eqnarray}
From (\ref{eq19}), we have
\begin{eqnarray}\label{eq20}
&&\left|\langle A(u_n),v\rangle-(1-t_n)\int\limits_{\Omega}\beta(z)f(u_n)v dz-t_n\hat{\eta}\int\limits_{\Omega}\beta(z)u_n v dz\right|\leq\frac{\|v\|}{n(1+\|u_n\|)}\|u_n\|^2\\
&& \mbox{for all}\ n\geq 1.\nonumber
\end{eqnarray}

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{eq21}
&&y_n\overset{w}{\to}y\ \mbox{in}\ H_0^1(\Omega)\ \mbox{and}\ y_n\to y\ \mbox{in}\ L^2(\Omega).
\end{eqnarray}
From (\ref{eq20}) we obtain
\begin{eqnarray}\label{eq22}
&&\left|\langle A(y_n),v\rangle-(1-t_n)\int\limits_{\Omega}\beta(z)\frac{f(u_n)}{\|u_n\|}v dz-t_n\hat{\eta}\int\limits_{\Omega}\beta(z)y_n v dz\right|\leq\frac{\|v\|}{n}\ \mbox{for all}\ n\geq 1.
\end{eqnarray}
Recall that $\left\{\beta\frac{N_f(u_n)}{\|u_n\|}\right\}_{n\geq 1}\subseteq L^2(\Omega)$ is bounded. So, if in (\ref{eq22}) we choose $v=y_n-y\in H_0^1(\Omega)$ and pass to the limit as $n\to\infty$,  we obtain
\begin{eqnarray}
&&\lim\limits_{n\to\infty}\langle A(y_n),y_n-y\rangle=0,\nonumber\\
&\Rightarrow& y_n\to y\ \mbox{in}\  H_0^1(\Omega)\ (\mbox{as before via the Kadec-Klee property}),\label{eq23}\\
&\Rightarrow& \|y\|=1.\label{eq24}
\end{eqnarray}
Moreover, we know that, at least for a subsequence, we have
\begin{eqnarray}\label{eq25}
&&\beta\frac{N_f(u_n)}{\|u_n\|}\overset{w}{\rightarrow}\eta\beta y\ \mbox{in}\  L^2(\Omega)\ \mbox{with}\ \eta\in(\hat{\lambda}_m^{+}(\beta),\hat{\lambda}_{m+1}^{+}(\beta))\ (\mbox{see hypothesis}\ H(f)(ii)).
\end{eqnarray}
So, if in (\ref{eq22}) we pass to the limit as $n\to\infty$ and use (\ref{eq23}) and (\ref{eq25}), then
\begin{eqnarray}\label{eq26}
&&\langle A(y),v\rangle=\eta_t\int\limits_{\Omega}\beta(z)y v dz\ \mbox{for all}\ v\in H_0^1(\Omega)\ \mbox{with}\ \eta_t=(1-t)\eta+t\hat{\eta}.\nonumber\\
&\Rightarrow& A(y)=\eta_t\beta(z)y,\nonumber\\
&\Rightarrow& -\Delta y(z)=\eta_t\beta(z)y(z)\ \mbox{a.e. in}\ \Omega,\quad y|_{\partial\Omega}=0.
\end{eqnarray}
Note that $\eta_t\in (\hat{\lambda}_m^{+}(\beta),\hat{\lambda}_{m+1}^{+}(\beta))$. Then from (\ref{eq26}) it follows that $y=0$, which contradicts (\ref{eq24}). This proves the Claim.

From Proposition \ref{prop4} we know that $\varphi$ satisfies the $C$-condition. Similarly, since $\hat{\eta}\in (\hat{\lambda}_m^{+}(\beta),\hat{\lambda}_{m+1}^{+}(\beta))$, it is easily seen that $\tau$ also satisfies the $C$-condition.

So, we can apply Proposition \ref{prop2} and infer that
\begin{eqnarray}\label{eq27}
C_k(\varphi,\infty)=C_k(\tau,\infty)\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
The fact that $\hat{\eta}\in (\hat{\lambda}_m^{+}(\beta),\hat{\lambda}_{m+1}^{+}(\beta))$, implies that $K_{\tau}=\{0\}$. Therefore
\begin{eqnarray}\label{eq28}
C_k(\tau,\infty)=C_k(\tau,0)\  \mbox{for all}\ k\geq 0.
\end{eqnarray}
Moreover, $u=0$ is a nondegenerate critical point of $\tau$ (that is, $\tau''(0)$ is invertible) and from the minimax characterization of the eigenvalues $\{\hat{\lambda}_k^{+}(\beta)\}_{k\geq 1}$ (see Gasinski and Papageorgiou \cite[p. 714]{8}), we see that the Morse index of $\tau$ at $u=0$ is $d_m\geq 2$. So, we have
\begin{eqnarray*}
&&C_k(\tau,0)=\delta_{k,d_m}\ZZ\ \mbox{for all}\ k\geq 0,\\
&\Rightarrow& C_k(\tau,\infty)=\delta_{k,d_m}\ZZ\ \mbox{for all}\ k\geq 0\ \mbox{with}\ d_m\geq 2\ (\mbox{see}\ (\ref{eq27}),\ (\ref{eq28})).
\end{eqnarray*}
This completes the proof.
\end{proof}

Now we can produce the third nontrivial solution.
\begin{prop}\label{prop9}
If hypotheses $H(\beta)$ and $H(f)$ hold then problem (\ref{eq1}) has a third nontrivial solution $y_0\in C_0^1(\overline{\Omega})$.
\end{prop}
\begin{proof}
From Proposition \ref{prop7} we already have two nontrivial solutions of constant sign
$$u_0\in\mbox{int}\, C_{+}(\overline{\Omega})\ \mbox{and}\ v_0\in -\mbox{int}\, C_{+}(\overline{\Omega}).$$
From the proof of that Proposition, we know that

(i) $u_0$ is a critical point of $\varphi_{+}$ of mountain pass type;

 (ii) $v_0$ is a critical point of $\varphi_{-}$ of mountain pass type.
\\
Hence we have
\begin{eqnarray}\label{eq29}
C_1(\varphi_{+},u_0)\neq 0\ \mbox{and}\ C_1(\varphi_{-},v_0)\neq 0.
\end{eqnarray}
Note that $\varphi_{+}|_{C_{+}}=\varphi|_{C_{+}}$ and $\varphi_{-}|_{-C_{+}}=\varphi|_{-C_{+}}$. Since $u_0\in\mbox{int}\, C_{+}(\overline{\Omega})$ and $v_0\in -\mbox{int}\, C_{+}(\overline{\Omega})$ it follows that
\begin{eqnarray}\label{eq30}
\left\{  \begin{array}{ll}
   C_k\left(\varphi_{+}|_{C_0^1(\overline{\Omega})},u_0\right)=C_k\left(\varphi|_{C_0^1(\overline{\Omega})},u_0\right)\ \mbox{for all}\ k\geq 0 \\
    C_k\left(\varphi_{-}|_{C_0^1(\overline{\Omega})},v_0\right)=C_k\left(\varphi|_{C_0^1(\overline{\Omega})},v_0\right)\ \mbox{for all}\ k\geq 0.
    \end{array}\right.
\end{eqnarray}
Since $C_0^1(\overline{\Omega})$ is dense in $H_0^1(\overline{\Omega})$, from Palais \cite{10} and relation (\ref{eq30}), we have
\begin{eqnarray}\label{eq31}
&&C_k(\varphi_{+},u_0)=C_k(\varphi,u_0)\ \mbox{and}\ C_k(\varphi_{-},v_0)=C_k(\varphi,v_0)\ \mbox{for all}\ k\geq 0,\nonumber\\
&\Rightarrow& C_1(\varphi,u_0)\neq 0\ \mbox{and}\ C_1(\varphi,v_0)\neq 0\ (\mbox{see}\ (\ref{eq29})),\nonumber\\
&\Rightarrow& C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\ZZ\ \mbox{for all}\ k\geq 0\ (\mbox{see Bartsch \cite[Proposition 2.5]{5}}).
\end{eqnarray}
Also, using (\ref{eq15}) we have
\begin{eqnarray}\label{eq32}
&&\varphi(u)\geq c_5\|u\|^2-c_6\|u\|^{\vartheta}\ \mbox{for all}\ u\in H_0^1(\Omega)\ \mbox{and some}\ c_5,c_6>0\ (\mbox{recall}\ \vartheta>2),\nonumber\\
&\Rightarrow& u=0\ \mbox{is a local minimizer of}\ \varphi,\nonumber\\
&\Rightarrow& C_k(\varphi,0)=\delta_{k,0}\ZZ\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
From Proposition \ref{prop8} we know that $C_k(\varphi,\infty)=\delta_{k,d_m}\ZZ$ for all $k\geq 0$. This means that there exists $y_0\in K_{\varphi}$ such that
\begin{eqnarray}\label{eq33}
C_{d_m}(\varphi,y_0)\neq 0.
\end{eqnarray}
Since $d_m\geq 2$, comparing (\ref{eq32}) with (\ref{eq30}) and (\ref{eq31}), we infer
\begin{eqnarray*}
&&y_0\notin\{0,u_0,v_0\},\\
&\Rightarrow& y_0\ \mbox{is a third nontrivial solution of}\ (\ref{eq1})\ (\mbox{since}\ y_0\in K_{\varphi}).
\end{eqnarray*}
Standard regularity theory implies that $y_0\in C_0^1(\overline{\Omega})$.
\end{proof}

So, summarizing the situation, we can state the following multiplicity theorem for problem (\ref{eq1}).
\begin{theorem}\label{th10}
If hypotheses $H(\beta)$ and $H(f)$ hold, then problem (\ref{eq1}) has at least three nontrivial solutions
$$u_0\in\mbox{int}\, C_{+}(\overline{\Omega}),\ v_0\in -\mbox{int}\, C_{+}(\overline{\Omega})\ \mbox{and}\ y_0\in C_0^1(\overline{\Omega}).$$
\end{theorem}

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\end{document}

