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\title[Resonant Neumann problems with indefinite and unbounded potential]{Resonant Neumann problems with indefinite and unbounded potential}


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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 \& Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,
          014700 Bucharest, Romania}
\email{\tt vicentiu.radulescu@math.cnrs.fr}

\keywords{Indefinite and unbounded potential, reduction method, resonance, unique continuation property, regularity, critical groups.\\
\phantom{aa} 2010 AMS Subject Classification:
35J20, 35J60, 58E05}

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\begin{abstract}
In this paper, we report on some recent results obtained in our joint paper \cite{npvrtams}.
We establish multiplicity properties for a class of semilinear Neumann problems driven by the Laplacian plus on unbounded and indefinite potential. The reaction is a Carath\'eodory function which exhibits linear growth near $\pm\infty$. We allow for resonance to occur with respect to a nonprincipal nonnegative eigenvalue. The approach combines critical point theory, Morse theory and the Lyapunov-Schmidt method.
\end{abstract}

\maketitle

\section{Introduction}\label{sec1}

Let $\Omega\subseteq \RR^{N}$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. Consider the following semilinear Neumann problem:
\begin{eqnarray}\label{eq1}
-\Delta u(z)+\beta(z)u(z)=f(z,u(z))\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}

Here $n(\cdot)$ denotes the outward unit normal on $\partial\Omega$. The potential function $\beta(\cdot)$ is in general unbounded and sign changing.  More  precisely, we assume that $\beta\in L^{s}(\Omega)$ with $s>N$. Also, the reaction $f(z,x)$ is a Carath\'eodory function that exhibits linear growth near $\pm\infty$. We allow for resonance to occur with respect to any nonnegative nonprincipal eigenvalue of $(-\Delta+\beta(\cdot),\ H^1(\Omega))$. So, we assume that asymptotically at $\pm\infty$ the quotient $\frac{f(z,x)}{x}$ is located in the spectral interval $[\hat{\lambda}_m,\hat{\lambda}_{m+1}]$ with $m\geq \max\{m_0,2\}$, where $\hat{\lambda}_{m_0}$ is the first nonnegative eigenvalue of $(-\Delta+\beta(\cdot),\ H^1(\Omega))$. Hence, if $\beta\equiv 0$, then $m_0=2$ and so $m\geq 2$. We allow resonance with respect to the left end $\hat{\lambda}_m$ and nonuniform nonresonance with respect to the right end $\hat{\lambda}_{m+1}$. Problems with double resonance (that is
 , possible resonance at both ends of the spectral interval), were studied by O'Regan, Papageorgiou \& Smyrlis \cite{25}, with $\beta\equiv 0$ (see also Hu \& Papageorgiou \cite{16} for Dirichlet problems with $\beta\neq 0$).

The following linear eigenvalue problem has a central role in the analysis of problem (\ref{eq1}):
\begin{eqnarray}\label{eq2}
-\Delta u(z)+\beta(z)u(z)=\lambda u(z)\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}
This eigenvalue problem was studied by Papageorgiou \& Smyrlis \cite{28}. So, suppose that $\beta\in L^{N/2}(\Omega)$ if $N\geq 3$, $\beta\in L^r(\Omega)$ with $r>1$ if $N=2$ and $\beta\in L^1(\Omega)$ if $N=1$. Let $\tau:H^1(\Omega)\rightarrow \RR$ be the energy functional defined by
\begin{eqnarray*}
\tau(u)=\|Du\|^2_2+\int_{\Omega}\beta(z)u(z)^2dz\ \mbox{for\ all}\ u\in H^1(\Omega).
\end{eqnarray*}
Then the eigenvalue problem (\ref{eq2}) has a smallest eigenvalue $\hat{\lambda}_1>-\infty$ given by
\begin{eqnarray}\label{eq3}
\hat{\lambda}_1=\inf\left[\frac{\tau(u)}{\|u\|^2_2}:\ u\in H^1(\Omega),\ u\neq 0\right].
\end{eqnarray}
>From (\ref{eq3}) it follows that we can find $\xi_0>\max\{-\hat{\lambda}_1,0\}$ such that
\begin{eqnarray}\label{eq4}
\tau(u)+\xi_0\|u\|^2_2\geq c_1\|u\|^2\ \mbox{for\ all}\ u\in H^1(\Omega)\ \mbox{and\ some}\ c_1>0.
\end{eqnarray}
Using (\ref{eq4}) and the spectral theorem for compact self-adjoint operators (see, for example, Gasinski \& Papageorgiou \cite[p. 297]{11}), we obtain a sequence $\{\hat{\lambda}_k\}_{k\geq 1}$ consisting of all the eigenvalues of (\ref{eq2}) such that  $\hat{\lambda}_k\rightarrow +\infty$ when $k\rightarrow\infty$. To these eigenvalues corresponds a sequence $\{\hat{u}_n\}_{n\geq 1}\subseteq H^1(\Omega)$ of eigenfunctions which form an orthonormal basis of $L^2(\Omega)$ and an orthogonal basis of $H^1(\Omega)$. Moreover, if $\beta\in L^s(\Omega)$ with $s>N$, then the regularity results of Wang \cite{34}, imply that $\{\hat{u}_n\}_{n\geq 1}\subseteq C^1(\overline{\Omega})$. These eigenvalues admit variational characterizations in terms of the Rayleigh quotient $\frac{\tau(u)}{\|u\|^2_2}$ for all $u\in H^1(\Omega)\setminus\{0\}$. In what follows, by $E(\hat{\lambda}_k)$, we denote the eigenspace corresponding to the eigenvalue $\hat{\lambda}_k,\ k\geq 1$.


Throughout this paper, our hypotheses on the potential function $\beta(\cdot)$ are the following:

\medskip
$H_0$: $\beta\in L^{s}(\Omega)$ with $s>N$ and $\beta^{+}\in L^{\infty}(\Omega).$

\section{Existence of multiple solutions}\label{sec3}

We assume that the resonance occurs at $\pm\infty$ with respect to any nonnegative nonprincipal eigenvalue of $(-\Delta-\beta, H^1(\Omega))$. So, in what follows, $\hat{\lambda}_{m_0}$ denotes the first nonnegative eigenvalue of this operator.

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

$H_1:\ f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that  $f(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
\item [(i)] there exist an integer $m\geq \max\{m_0,2\}$ and a function $\eta\in L^{\infty}(\Omega)_{+}$ such that
\begin{eqnarray*}
&&\eta(z)\leq \hat{\lambda}_{m+1}\ \mbox{a.e.\ in}\ \Omega,\ \eta\neq\hat{\lambda}_{m+1}\\
&&(f(z,x)-f(z,y))(x-y)\leq\eta(z)(x-y)^2\ \mbox{for\ a.a}\ z\in\Omega,\ \mbox{all}\ x,y \in\RR;
\end{eqnarray*}
\item [(ii)] $\hat{\lambda}_{m}\leq \liminf\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{x}$ uniformly for a.a. $z\in\Omega$;
\item [(iii)] if $F(z,x)=\int\limits_0^x f(z,s)ds$, then we have
$$\lim\limits_{x\rightarrow\pm\infty}[f(z,x)x-2F(z,x)]=-\infty\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;$$
\item [(iv)] there exists a function $\vartheta\in L^{\infty}(\Omega)$ such that
\begin{eqnarray*}
&&\vartheta(z)\leq \hat{\lambda}_{1}\ \mbox{a.e.\ in}\ \Omega,\ \vartheta\neq\hat{\lambda}_{1}\\
&&\limsup\limits_{x\rightarrow 0}\frac{2F(z,x)}{x^2}\leq\vartheta(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;
\end{eqnarray*}
\item [(v)] for every $\varrho>0$, there exists $\xi_{\varrho}>0$ such that
$$f(z,x)x+\xi_{\varrho} x^2\geq 0\ \mbox{for\ a.a}\ z\in\Omega,\ \mbox{all}\ |x|\leq \varrho.$$
\end{itemize}

\smallskip
We observe that
hypotheses $H_1 (i), (ii)$ imply that asymptotically at $\pm\infty$, the quotient $\frac{f(z,x)}{x}$ is in the spectral interval $[\hat{\lambda}_{m}, \hat{\lambda}_{m+1}]$ with possible resonance with respect to $ \hat{\lambda}_{m}$ (see $H_1(ii)$), while at the other end we have nonuniform nonresonance (see $H_1(i)$).


The following function satisfies hypotheses $H_1$ above. For the sake of simplicity, we drop the $z$-dependence:
\begin{eqnarray*}
f(x)=\left\{  \begin{array}{ll}
  \vartheta x+\xi |x|^{p-2}x&\mbox{if}\ |x|\leq 1 \\
    \lambda x+\frac{c}{x} &\mbox{if}\ 1<|x|,
    \end{array}\right.
\end{eqnarray*}
with $\vartheta<\hat{\lambda}_{1}$, $p>2$, $\xi=\lambda+c-\vartheta$, $\lambda\in [\hat{\lambda}_{m},\hat{\lambda}_{m+1})$ for some integer $m\geq \max\{m_0,2\}$, $c>0$, $2c<\lambda$.

Our first multiplicity property is the following ``three solutions" theorem.

\begin{theorem}\label{teor1}
Assume that hypotheses $H_0$ and $H_1$ hold. Then problem (\ref{eq1}) has at least three nontrivial solutions
$$u_0\in \mbox{int}\, C_{+},\  v_0\in -\mbox{int}\, C_{+}\ \mbox{and}\ y_0\in C^1(\overline{\Omega}).$$
\end{theorem}

{\it Sketch of the proof.} We set $\hat{F}_{\pm}(z,x)=\int\limits_0^x \hat{f}_{\pm}(z,s)ds$ and introduce the $C^1$- functionals $\varphi,\hat{\varphi}_{\pm}:H^1(\Omega)\rightarrow\RR$ defined by
\begin{eqnarray*}
&&\varphi(u)=\frac{1}{2}\tau(u)-\int_{\Omega} F(z,u(z))dz\\
&&\hat{\varphi}_{\pm}(u)=\frac{1}{2}\tau(u)+\frac{\xi_0}{2}\|u\|_2^2-\int_{\Omega} \hat{F}_{\pm}(z,u(z))dz\ \mbox{for\ all}\ u\in H^1(\Omega).
\end{eqnarray*}

The main steps of the proof are the following:

(i)
If hypotheses $H_0$ and $H_1$ hold, then the functionals $\hat{\varphi}_{\pm}$ satisfy the Cerami compactness condition.


(ii)
If hypotheses $H_0$ and $H_1$ hold, then the functional $\varphi$ satisfies the Cerami compactness condition.


(iii)
If hypotheses $H_0$ and $H_1$ hold, then $u=0$ is a minimizer for the functionals $\hat{\varphi}_{\pm}$ and $\varphi$.


(iv)
If hypotheses $H_0$ and $H_1$ hold, then $\hat{\varphi}_{\pm}(t\hat{u}_1)\rightarrow -\infty$ as $t\rightarrow \pm\infty$.


(v)
If hypotheses $H_0$ and $H_1$ hold, then problem (\ref{eq1}) admits at least two nontrivial constant sign solutions
$$u_0\in \mbox{int}\, C_{+}\ \mbox{and}\ v_0\in -\mbox{int}\, C_{+}.$$



To produce a third nontrivial solution, we will employ the so-called Lyapunov-Schmidt reduction technique as this was formulated for elliptic equations by Amann \cite{2}, Castro \& Lazer \cite{7}, and Thews \cite{32}.
To this end, we introduce the following subspaces of $H^1(\Omega)$:
$$Y=\oplus_{i=1}^{m}E(\hat{\lambda}_i)\ \mbox{and}\ \hat{H}=Y^{\bot}=\overline{\oplus_{i\geq m+1}E(\hat{\lambda}_i)}.$$
We have the following orthogonal direct sum decomposition:
$H^1(\Omega)=Y\oplus \hat{H}.$

(vi)
If hypotheses $H_0$ and $H_1$ hold, then there exists a continuous map $\gamma_0:Y\rightarrow \hat{H}$ such that
$$\varphi(y+\gamma_0(y))=\inf \left[\varphi(y+\hat{u}): \hat{u}\in \hat{H}\right]\ \mbox{for\ all}\ y\in Y.$$


(vii) Let $\psi(y)=\varphi(y+\gamma_0(y))$ for all $y\in Y$.
If hypotheses $H_0$ and $H_1$ hold, then $\psi$ is anticoercive (that is, if $\|y\|\rightarrow\infty$, then $\psi(y)\rightarrow -\infty$).

We refer to Papageorgiou \& R\u adulescu \cite{npvrtams} for detailed arguments of the proof.

By strengthening the regularity  of the reaction $f(z,x)$ we can improve Theorem \ref{teor1} and produce four distinct solutions. In what follows we assume that $f$ satisfies the following hypotheses:

\smallskip
$H_2:\ f:\Omega\times\RR\rightarrow\RR$ 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)] there exist an integer $m\geq \max\{m_0,2\}$ and a function $\eta\in L^{\infty}(\Omega)_{+}$ such that
\begin{eqnarray*}
&&\eta(z)\leq \hat{\lambda}_{m+1}\ \mbox{a.e.\ in}\ \Omega,\ \eta\neq \hat{\lambda}_{m+1},\\
&&|f_x'(z,x)|\leq \eta(z)\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR;
\end{eqnarray*}
\item [(ii)] $\hat{\lambda}_{m}\leq \liminf\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{x}\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega$;
\item [(iii)] $\lim\limits_{x\rightarrow\pm\infty}[f(z,x)x-2F(z,x)]=-\infty\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega$;
\item [(iv)] there exists a function $\vartheta\in L^{\infty}(\Omega)_{+}$ such that
\begin{eqnarray*}
&&\vartheta(z)\leq \hat{\lambda}_1\ \mbox{a.e.\ in}\ \Omega,\ \vartheta\neq \hat{\lambda}_1,\\
&&\limsup\limits_{x\rightarrow 0}\frac{2F(z,x)}{x^2}\leq \vartheta(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.
\end{eqnarray*}
\end{itemize}

\smallskip
Note that hypothesis $H_2(i)$ and the mean value theorem imply that
$$|f(z,x)|\leq\eta(z)|x|\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR.$$
Also, for every $\varrho>0$, there exists $\xi_{\varrho}>0$ such that  for a.a. $z\in\Omega$, the mapping $x\mapsto f(z,x)+\xi_{\varrho}x$ is nondecreasing on $[-\varrho,\varrho]$.

\smallskip
A straightforward computation shows that the following function satisfies hypotheses $H_2$ (for the sake of simplicity we drop the $z$-dependence):
\begin{eqnarray*}
f(x)=\left\{  \begin{array}{ll}
 \displaystyle  \vartheta x+\frac{\xi x}{1+|x|} &\mbox{if}\ |x|\leq 1 \\
   \displaystyle  \lambda x+\frac{c}{x} &\mbox{if}\ 1<|x|,
    \end{array}\right.
\end{eqnarray*}
with $\vartheta<\hat{\lambda}_1,\ \lambda\in (\hat{\lambda}_m,\hat{\lambda}_{m+1})$ for some integer $m\geq \max\{m_0,2\},\ \xi,\ c>0,\ \vartheta+\xi<\hat{\lambda}_{m+1}$.

Our second multiplicity result is the following.

\begin{theorem}\label{teor2}
Assume that hypotheses $H_0$ and $H_2$ hold. Then problem (\ref{eq1}) admits at least four nontrivial solutions
$$u_0\in \mbox{int}\, C_{+},\ v_0\in -\mbox{int}\, C_{+}\ \mbox{and}\ y_0,\hat{y}\in C^1(\overline{\Omega}).$$
\end{theorem}


We refer to \cite{npvrtams} for the proof of Theorem \ref{teor2}, as well as for related results on coercive or anticoercive problems, or for the case when the Lyapunov-Schmidt reduction method is implemented on an infinite dimensional subspace of $H^1(\Omega)$.

\medskip
{\bf Acknowledgments}.  V. R\u{a}dulescu acknowledges the support through Grant
CNCS PCE-47/2011.

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\bibitem{2} {\sc H. Amann}, {\em Saddle points and multiple solutions of differential equations}, Math. Z. \textbf{169} (1979), 127-166.

\bibitem{7} {\sc A. Castro, A. Lazer}, {\em Critical point theory and the number of solutions of a nonlinear Dirichlet problem}, Annali. Mat. Pura Appl. \textbf{120} (1979), 113-137.

\bibitem{11} {\sc L. Gasinski, N.S. Papageorgiou}, {\em Nonlinear Analysis}, Chapman \& Hall/CRC, Boca Raton, 2006.

\bibitem{16} {\sc S. Hu, N.S. Papageorgiou}, {\em Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities}, Comm. Pure Appl. Anal. {\bf 11} (2012), 2005-2021.

\bibitem{25} {\sc N.S. Papageorgiou, S.T. Kyritsi}, {\em Handbook of Applied Analysis}, Springer, New York, 2009.

\bibitem{npvrtams} {\sc N.S. Papageorgiou, V. R\u adulescu}, {\em Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential}, Transactions Amer. Math. Soc., in press.

\bibitem{28} {\sc N.S. Papageorgiou, G. Smyrlis}, {\em On a class of parametric Neumann problem with indefinite and unbounded potential}, Forum Math. DOI: \textbf{10.1515}/forum-2012-0042.

\bibitem{32} {\sc K. Thews}, {\em Nontrivial solutions of elliptic equations at resonance}, Proc. Royal Soc. Edinburgh \textbf{85A} (1980), 119-129.

\bibitem{34} {\sc X. Wang}, {\em Neumann problems of semilinear elliptic equations involving critical Sobolev exponents}, J.~Differential Equations \textbf{93} (1991), 283-310.

\end{thebibliography}


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