\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 49, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/49\hfil An eigenvalue problem]
{Remarks on an eigenvalue problem associated with
the $p$-Laplace operator}

\author[A. G\u al\u a\c tan, C. Lupu, F. Preda\hfil EJDE-2010/49\hfilneg]
{Alin G\u al\u a\c tan, Cezar Lupu, Felician Preda}  % in alphabetical order



\address{Alin G\u al\u a\c tan \newline
University of Bucharest, Faculty of Mathematics, Str.
Academiei 14, RO--70109, \newline
Bucharest, Romania}
\email{alin9d@yahoo.com}

\address{Cezar Lupu \newline
University of Bucharest, Faculty of Mathematics, Str.
Academiei 14, RO--70109, \newline
Bucharest, Romania}
\email{lupucezar@yahoo.com, lupucezar@gmail.com}

\address{Felician Preda \newline
Institute of Mathematical Statistics and Applied Mathematics
``Gheorghe Mihoc--Caius Iacob'', Calea 13 Septembrie No. 13, Bucharest 5,
RO-050711, Romania}
\email{felician.preda@yahoo.com}

\thanks{Submitted February 10, 2010. Published April 7, 2010.}
\subjclass[2000]{35D05, 35J60, 35J70, 58E05, 15A18, 35P05}
\keywords{$p$-Laplace operator; eigenvalue problem; critical point}

\dedicatory{Dedicated to Professor Gheorghe Moro\c sanu on his 60-th birthday}


\begin{abstract}
 In this article we study eigenvalue problems involving
 $p$-Laplace operator and having a continuous family
 of eigenvalues and at least one isolated eigenvalue.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction and statement of main results}

Eigenvalue problems have been studied in various settings lately.
 The leading example of linear eigenvalue problem
is to find all non-trivial solutions of the equation $\Delta
u+\lambda u=0$ with boundary values zero in a given bounded domain
in $\mathbb{R}^{N}$. This is called a Dirichlet boundary-value
problem.

In this article we study the eigenvalue problem
\begin{equation}\label{e1}
\begin{gathered}
-\Delta_{p} u=\lambda f(x,u), \quad\text{in } \Omega\\
u=0, \quad\text{on } \partial\Omega\,,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^{N}$ is a bounded domain
with smooth boundary, $f:\Omega\times\mathbb{R}\to\mathbb{R}$
is a given function and $\lambda$ is a real number.
The operator
$\Delta_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$
is called $p$-harmonic, and  appears in many contexts in physics
reaction-diffusion problems, non-linear elasticity, etc.
The $p$-harmonic operator is defined as
$$
\Delta_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)
=|\nabla u|^{p-4}\Big(|\nabla u|^{p-2}\Delta u+(p-2)
\sum\frac{\partial u}{\partial x_{i}}
\frac{\partial u}{\partial x_{i}}
\frac{\partial^2 u}{\partial x_{i}\partial x_{j}}\Big),
$$
where $1< p< N$.

\begin{definition} \label{def1.1} \rm
We say that $u\in W_0^{1, p}(\Omega)\setminus\{0\}$ is an eigenfunction
of \eqref{e1}, if
$$
\int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla v\,dx
=\lambda\int_{\Omega}f(x, u)v\,dx,
$$
for all $ v\in W_1^{p}(\Omega)$. The corresponding real number
$\lambda$ is called the eigenvalue of \eqref{e1}.
\end{definition}

The Sobolev space $W_0^{1, p}(\Omega)$ is the completion of
$C_0^{\infty}(\Omega)$ with respect to the norm
$$
\|\varphi\|=\Big(\int_{\Omega}(|\varphi|^{p}+|\operatorname{div}
\varphi|^{p})dx\Big)^{1/p}.
$$
As usual, the space $C_0^{\infty}(\Omega)$ is the class of smooth
functions with compact support in $\Omega$. By standard elliptic
regularity theory an eigenfunction is continuous.
The smallest eigenvalue of \eqref{e1} can be characterized
by the minimum of Rayleigh quotient,
$$
\lambda_1=\inf_{u\in W_{0}^{1, p}(\Omega)\setminus\{0\}}
\frac{\int_{\Omega}|\nabla u|^pdx}{\int_{\Omega}u^pdx}.
$$

 The study of eigenvalues involving Laplace and $p$-Laplace
operators starts with the following basic problem, which represents
a particular case of \eqref{e1},
\begin{equation}\label{e2}
\begin{gathered}
-\Delta u=\lambda u, \quad\text{in } \Omega\\
u=0, \quad\text{on } \partial\Omega\,.
\end{gathered}
\end{equation}
As  mentioned in \cite{Mihailescu-Radulescu2} problem
\eqref{e2} goes back to the Riesz-Fredholm theory for compact operators
on Hilbert spaces, where it is proved that it has an unbounded
sequence of eigenvalues $
0<\lambda_1<\lambda_{2}\leq\dots\leq\lambda_{n}\dots$. Also,
in \cite{Mihailescu-Radulescu2} other eigenvalue problems are
mentioned; for example we have problems involving $p(x)$-Laplace operator
in the case when $ f(x, u)=|u|^{p(x)-2}u$ where we obtain the
nonlinear model equation
\begin{equation}\label{e3}
\begin{gathered}
-\Delta_{p(x)} u=\lambda |u|^{p(x)-2}u, \quad\text{in } \Omega\\
u=0, \quad\text{on } \partial\Omega\,,
\end{gathered}
\end{equation}
where $p(\cdot):\overline{\Omega}\to (1, 2^{*})$ is a given
continuous function and $2^{*}$ denotes the critical Sobolev exponent,
$$
2^*= \begin{cases}
\frac{2N}{N-2} & \text{if }N\geq 3\\
+\infty &\text{if }N\in\{1,2\}.
\end{cases}
$$
By specific methods of nonlinear analysis (Ekeland variational
principle, mountain pass theorem, etc) many properties are
established about problem \eqref{e3}. For further discussions of this
problem as well as generalizations and extensions we refer to
\cite{Fan,Mihailescu-Radulescu1,Mihailescu-Radulescu2}.
In the particular case, when
$ f(x,u)=|u|^{p-2}u$ we obtain the  eigenvalue problem
\begin{equation}\label{e4}
\begin{gathered}
-\Delta_{p} u=\lambda |u|^{p-2}u, \quad\text{in } \Omega\\
u=0, \quad\text{on } \partial\Omega\,,
\end{gathered}
\end{equation}
which was introduced by Lieb \cite{Lieb} in 1983 and then
studied by Lindqvist in \cite{Lindqvist}, \cite{Kawohl-Lindqvist}
and a modified eigenvalue problem \eqref{e4} involving the
weight function $V(\cdot)$ which changes sign and has nontrivial
positive part by Cuesta in \cite{Cuesta}. Inspired by the work
of Mih\u ailescu and R\u adulescu from \cite{Mihailescu-Radulescu2},
we study \eqref{e1} in the case when
\begin{equation}\label{e5}
f(x, t)=\begin{cases}
 h(x, t) &\text{if }t\geq 0 \\
 t &\text{if }t<0,
\end{cases}
\end{equation}
where $h:\Omega\times [0, \infty)\to\mathbb{R}$ is a
Carath\'eodory function satisfying the following properties
\begin{itemize}
\item[(P1)] there exists a positive constant $k\in (0,1)$ such
that $|h(x, t)|\leq k\cdot t^{p-1}$,  for all $t\geq 0$ and
a.e. $x\in\Omega$;

\item[(P2)] there exists $t_{0}>0$ such that $ H(x,
t_{0})=\int_0^{t_{0}}h(x, s)ds>0$ for a.e. $x\in\Omega$;

\item[(P3)] $\lim_{t\to\infty}\frac{h(x, t)}{t^{p-1}}=0$,
uniformly in $x$.
\end{itemize}

These assumptions are related to those used by Diaz and
Saa \cite{diazsaa}, to deduce an existence and uniqueness
result for a quasilinear problem with Dirichlet boundary condition
(see Brezis and Oswald \cite{breosw} for the semilinear case).

Examples of functions satisfying properties (P1), (P2) and (P3)
are  mentioned in \cite{Mihailescu-Radulescu2}.
Regarding \eqref{e1}, we also point out the recent
work of Pucci and R\u adulescu \cite{Pucci-Radulescu} in which
they study the problem for polyharmonic operator provided that $f$
satisfies the same conditions as those in \cite{Mihailescu-Radulescu2}.

The main result of this article establishes a property of
the \eqref{e1} provided that $f$ is defined as above and
satisfies (P1), (P2) and (P3). It and shows that \eqref{e1} has
both isolated eigenvalues and a continuous spectrum in a neighborhood
of the origin.

\begin{theorem} \label{thm1.2}
Assume that $f$ is defined by the relation \eqref{e5} and satisfies
properties {\rm (P1), (P2), (P3)}. Then the  eigenvalue
$\lambda_1$ defined by the Rayleigh quotient is isolated,
and the corresponding set of eigenvectors form a cone.
Moreover, there is no eigenvalue
$\lambda\in (0, \lambda_1)$, but there exists
$\mu_1>\lambda_1$ such that any $\lambda>\mu_1$
is an eigenvalue of \eqref{e1}.
\end{theorem}

\section{Proof of the main result}

We shall use the method of Stamppachia and for any
$u\in W_0^{1,p}(\Omega)$ we denote
$ u_{\pm}=\max\{\pm u(x), 0\}$, for all
$x\in\Omega$. Then $ u_{+}, u_{-}\in W_{0}^{1, p}(\Omega)$ and
$$
\nabla u_+=\begin{cases}
0, &\text{if } u\leq 0\\
\nabla u, &\text{if } u>0\,,
\end{cases}
\quad
\nabla u_-=\begin{cases}
0, &\text{if } u\geq 0\\
\nabla u, &\text{if } u<0\,,
\end{cases}
$$
It follows that, with $f$ given by \eqref{e5},
\eqref{e1} becomes
\begin{equation}\label{e6}
\begin{gathered}
-\Delta_{p} u=\lambda[h(x,u_+)-u_-], \quad\text{in } \Omega\\
u=0, \quad\text{on } \partial\Omega\,,
\end{gathered}
\end{equation}
and $\lambda>0$ is an eigenvalue of  \eqref{e6}
 if there exists $u\in W_0^{1, p}(\Omega)\setminus\{0\}$ such that
\begin{equation}\label{e7}
\int_\Omega |\nabla u_+|^{p-2}\nabla u_+\nabla v\,dx
-\int_\Omega |\nabla u_-|^{p-2}\nabla u_-\nabla v\,dx
-\lambda\int_\Omega[h(x,u_+)-u_-]v\,dx=0\,,
\end{equation}
for all $v\in W_0^{1, p}(\Omega)$.

To prove the main result, Theorem \ref{thm1.2}, we shall begin with the
following lemmata.

\begin{lemma} \label{lem2.1}
There are no eigenvalues of \eqref{e6} in the interval
$(0, \lambda_1)$.
\end{lemma}

\begin{proof}
 Assume that $\lambda>0$ is an eigenvalue of
 \eqref{e6} and $u$ is its corresponding eigenfunction.
We put $v=u_{+}$ and $ v=u_{-}$ in \eqref{e7} and we infer that
\begin{equation}\label{e8}
\int_{\Omega}|\nabla u_{+}|^p\,dx
=\lambda\int_{\Omega}h(x, u_{+})u_{+}dx
\end{equation}
and
\begin{equation}\label{e9}
\int_{\Omega}|\nabla u_{-}|^pdx
=\lambda\int_{\Omega}u_{-}^pdx.
\end{equation}
By property (P1) and relations \eqref{e8} and \eqref{e9}, we obtain
$$
\lambda_1\int_{\Omega}u_{+}^pdx
\leq\int_{\Omega}|\nabla u_{+}|^pdx
=\lambda\int_{\Omega}h(x, u_{+})u_{+}dx\leq\lambda\int_{\Omega}u_{+}^pdx
$$
and
$$
\lambda_1\int_{\Omega}u_{-}^pdx\leq\int_{\Omega}|\nabla u_{-}|^pdx
=\lambda\int_{\Omega}u_{-}^pdx.
$$
If $\lambda$ is an eigenvalue of problem \eqref{e6}, then the
corresponding eigenvector $u$ is not null and thus, at least one
of the eigenfunctions $u_{+}$ and $u_{-}$ is not the zero function.
 This means that $\lambda$ is an eigenvalue of \eqref{e6},
and by the definition of the Rayleigh quotient,
$\lambda\geq\lambda_1$.
\end{proof}

\begin{lemma} \label{lem2.2}
$\lambda_1$ is an eigenvalue of \eqref{e6}, and is  isolated.
Moreover, the set of eigenvectors corresponding to $\lambda_1$
form a cone.
\end{lemma}

\begin{proof}
Indeed, as we already pointed out, $\lambda_1$ is the smallest
eigenvalue of  \eqref{e2}, it is simple, that is, all the associated
eigenfunctions  are merely multiples of each other (see, e.g.,
Gilbarg and Trudinger \cite{Gilbarg-Trudinger}) and the corresponding
eigenfunctions of $\lambda_1$ never
change signs in $\Omega$. In other words, there exists
$e_1\in W_0^{1, p}(\Omega)\setminus\{0\}$,
with $e_1(x)<0$ for any $x\in\Omega$ such that
$$
\int_\Omega |\nabla e_1|^{p-2}\nabla e_1\nabla v\,dx
-\lambda_1\int_\Omega e_1v\,dx=0\,,
$$
for any $v\in W_0^{1, p}(\Omega)$. Thus,  we have $(e_1)_+=0$ and
$(e_1)=-e_1$ and we deduce that
relation \eqref{e7} holds  with
$u=e_1\in W_0^{1, p}(\Omega)\setminus\{0\}$ and $\lambda=\lambda_1$.
In other words, $\lambda_1$ is an eigenvalue of \eqref{e1} and the
set of its corresponding eigenvectors lies in a cone of
$W_{0}^{1, p}(\Omega)$. Now, we prove that $\lambda_1$ isolated
in the set of eigenvalues of problem \eqref{e6}. Indeed, by the
Lemma \ref{lem2.1} we have that there does not exist an eigenvalue of
\eqref{e6} in the interval $(0, \lambda_1)$.
On the other hand it is clear that if $\lambda$ is also an
eigenvalue of  \eqref{e6} for which $u_{+}$ is not
identically zero, then we have
$$
\lambda_1\int_\Omega u_+^p\,dx\leq\int_\Omega|\nabla u_+|^p\,dx
=\lambda\int_\Omega h(x,u_+)u_+\,dx
\leq\lambda k \int_\Omega u_+^p\,dx\,,
$$
and thus since $k\in (0,1)$ we have
$\lambda\geq\frac{\lambda_1}{k}>\lambda_1$.
This means that for any eigenvalue $\lambda\in (0, \lambda_1/k)$
of  \eqref{e6} we must have $u_{+}=0$. It follows that $\lambda$
is an eigenvalue of  \eqref{e2} with the corresponding
eigenfunction negative in $\Omega$. As it has been already
noticed, the set of eigenvalues of  \eqref{e2} is discrete and
$\lambda_1<\lambda_{2}$. Now, let us consider
$\epsilon=\min\{\lambda_1/k, \lambda_{2}\}$ and we have that
$\epsilon>\lambda_1$ and any $\lambda\in (\lambda_1,
\epsilon)$ cannot be an eigenvalue of  \eqref{e2}
and \eqref{e6} and
thus $\lambda_1$ is isolated in the set of eigenvalues of
 \eqref{e6}.
\end{proof}

Next, we show that there exists $\mu_1>0$ such that any
$\lambda\in(\mu_1,\infty)$ is an eigenvalue
of  \eqref{e6}. With that end in view, we consider the
eigenvalue problem
\begin{equation}\label{e10}
\begin{gathered}
-\Delta_{p} u=\lambda h(x,u_+), \quad\text{in } \Omega\\
u=0, \quad\text{on } \partial\Omega\,,
\end{gathered}
\end{equation}
We say that $\lambda$ is an eigenvalue of \eqref{e10}
 if there exists $u\in W_0^{1, p}(\Omega)\setminus\{0\}$ such that
$$
\int_\Omega |\nabla u|^{p-2}\nabla u\nabla v\,dx
-\lambda\int_\Omega h(x,u_+)v\,dx=0\,,
$$
for any $v\in W_0^{1, p}(\Omega)$.

We notice that if $\lambda$ is an eigenvalue for \eqref{e10}
 with the corresponding eigenfunction $u$, then taking
$v=u_-$ in the above relation we deduce that $u_-=0$, and thus, we find $u\geq 0$. In other words, the eigenvalues
of \eqref{e10} possesses nonnegative corresponding eigenfunctions.
Moreover, the above discussion show that
an eigenvalue of \eqref{e10} is an eigenvalue of  \eqref{e6}.

Now, for each $\lambda>0$ we define the energy functional associated
to \eqref{e10} by $I_\lambda:W_0^{1, p}(\Omega)\rightarrow\mathbb{R}$,
$$
I_\lambda(u)=\frac{1}{p}\int_\Omega|\nabla u|^p\,dx
-\lambda\int_\Omega H(x,u_+)\,dx\,,
$$
where  $H(x,t)=\int_0^th(x,s)\,ds$. Standard arguments show
that $I_\lambda\in C^1(W_0^{1, p}(\Omega),\mathbb{R})$ with the
derivative given by
$$
\langle I_\lambda^{\prime}(u),v\rangle
=\int_\Omega |\nabla u|^{p-2}\nabla u\nabla v\,dx
-\lambda\int_\Omega h(x,u_+)v\,dx\,,
$$
for any $u$, $v\in W_0^{1, p}(\Omega)$. Thus, $\lambda>0$ is an
eigenvalue of \eqref{e10} if and only if there
exists a critical nontrivial point of functional $I_\lambda$.

\begin{lemma} \label{lem2.3}
The functional $ I_{\lambda}$ defined as above is bounded from
below and coercive. Moreover, there exists $\lambda^{\star}>0$
such that assuming that $\lambda\geq\lambda^{\star}$ we have
$\inf_{W_{0}^{1, p}(\Omega)}I_{\lambda}<0$.
\end{lemma}

\begin{proof}
By  (P3) we deduce that
$$
\lim_{t\rightarrow\infty}\frac{H(x,t)}{t^p}=0,\quad
\text{uniformly  in }\Omega\,.
$$
Then for a given $\lambda>0$ and $\lambda_1$ defined as the Rayleigh
quotient, there exists a positive constant $C_\lambda>0$ such that
$$
\lambda H(x,t)\leq\frac{\lambda_1}{2p}t^p+C_\lambda,\quad
\forall\;t\geq 0,\text{ a.e. } x\in\Omega\,.
$$
Thus, for any $u\in W_0^{1, p}(\Omega)$,
$$
I_\lambda(u)\geq\frac{1}{p}\int_\Omega|\nabla u|^p\,dx
-\frac{\lambda_1}{2p}\int_\Omega u^p\,dx-C_\lambda|\Omega|
\geq\frac{1}{2p}\|u\|^p-C_\lambda|\Omega|\,,
$$
where by $\|\cdot\|_{p}$ is denoted the norm on
$W_0^{1, p}(\Omega)$, that is
$\|u\|_{p}=(\int_\Omega |\nabla u|^p\,dx)^{1/p}$.
This shows that $I_\lambda$ is bounded from
below and coercive. Now, we prove the second part of the lemma. We
employ the property (P2) which states that there exists $t_{0}>0$
such that $H(x, t_{0})>0$ a.e. for all $x\in\overline{\Omega}$.
Let us consider $\Omega_1\subset\Omega$ be a sufficiently large
compact subset and
$u_{0}\in C_{0}^{1}(\Omega)\subset W_{0}^{1, p}(\Omega)$
such that $u_{0}(x)=t_{0}$ for $x\in\Omega_1$ and
$0\leq u_{0}(x)\leq t_{0}$ for any $x\in\Omega-\Omega_1$. By
(P1) we have
$$
\int_{\Omega}H(x, u_{0})dx\geq\int_{\Omega_1}H(x, t_{0})dx
-\int_{\Omega-\Omega_1}ku_{0}^pdx
\geq\int_{\Omega}H(x, t_{0})dx-kt_{0}^p|\Omega-\Omega_1|>0.
$$
This means that $ I_{\lambda}(u_{0})<0$ for sufficiently large
$\lambda>0$ and thus, we obtain $\inf_{W_0^{1,p}(\Omega)}I_{\lambda}<0$.
\end{proof}

By Lemma \ref{lem2.3}, the functional $I_{\lambda}$ has a negative global
minimum for $\lambda>0$ sufficiently large and any large $\lambda>0$
is an eigenvalue of \eqref{e1} and thus is an eigenvalue of  \eqref{e6}.
 By Lemma \ref{lem2.1}, the statement of
Theorem \ref{thm1.2} holds.

\subsection*{Acknowledgments}
The second author is grateful to Department of Mathematics of
the Central European University, Budapest for hospitality during
February--May 2010 when this research has been completed.
C.L. thanks ``Dinu Patriciu" Foundation for support.

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\end{document}