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\title{\sc Sublinear eigenvalue problems associated to
the Laplace operator revisited\thanks{Correspondence address: Vicen\c{t}iu R\u{a}dulescu, Department of
Mathematics, University of Craiova,  200585 Craiova, Romania. E-mail:
{\tt
vicentiu.radulescu@imar.ro}}}
\author{\sc Mihai Mih\u ailescu$\,^{a,b}$ and Vicen\c{t}iu R\u{a}dulescu$\,^{a,c,d}$\\
\small
$^a\,$Department of Mathematics, University of Craiova,  200585 Craiova,
Romania\\
\small $^b\,$Department of Mathematics, Central European
University,  1051 Budapest,
Hungary\\
\small $^c\,$Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,\\
\small P.O. Box 1-764, 014700 Bucharest, Romania\\
\small $^d\,$Institute of Mathematics, Physics and Mechanics, P.O.B. 2964, Ljubljana, Slovenia 1001\\
\small
E-mail addresses: {\tt mmihailes@yahoo.com}\qquad {\tt
vicentiu.radulescu@imar.ro}}
\begin{document}
\baselineskip16pt \maketitle \noindent{\small{\sc Abstract}. Eigenvalue problems involving the Laplace operator on
bounded domains lead to a discrete or a continuous set of eigenvalues. In this paper we highlight the case of an eigenvalue
problem involving the Laplace operator which possesses, on the one hand, a continuous family of eigenvalues and, on
the other hand, at least one more eigenvalue which is isolated in the set of eigenvalues of that problem.\\
\small{\bf 2000 Mathematics Subject Classification:}  35D05, 35J60, 35J70, 58E05, 15A18, 35P05. \\
\small{\bf Key words:}  Laplace operator, eigenvalue problem, critical point.}

\section{Introduction and the main result}
Throughout this paper we assume that $\Omega\subset\RR^N$ is a bounded domain with
smooth boundary. By an eigenvalue problem involving the Laplace operator
we understand a problem of the type
\begin{equation}\label{zero}
\left\{\begin{array}{lll}
-\Delta u=\lambda f(x,u), &\mbox{in}& \Omega\\
u=0, &\mbox{on}& \partial\Omega\,,
\end{array}\right.
\end{equation}
where $f:\Omega\times\RR\rightarrow\RR$ is a given function and $\lambda\in\RR$
is a real number. We  say that $\lambda$ is an {\it eigenvalue} of
problem \eq{zero} if there exists $u\in H_0^1(\Omega)\setminus\{0\}$ such
that for any $v\in H_0^1(\Omega)$,
$$\int_\Omega\nabla u\nabla v\;dx-\lambda\int_\Omega f(x,u)v\;dx=0\,.$$
 Moreover,  if $\lambda$ is an eigenvalue of problem \eq{zero}
then  $u\in H_0^{1}(\Omega)\setminus\{0\}$ given in the above definition is called the {\it eigenfunction} corresponding
to the eigenvalue $\lambda$. In this paper are interested in finding
positive eigenvalues for problems of type \eq{zero}.


The study of eigenvalue problems involving the Laplace operator
guides our mind back to a basic result in the elementary theory of
partial differential equations which asserts that the problem
(which represents a particular case of problem \eq{zero}, obtained
when $f(x,u)=u$)
\begin{equation}\label{DEVP}
\left\{\begin{array}{lll}
-\Delta u=\lambda u, &\mbox{in}& \Omega\\
u=0, &\mbox{on}& \partial\Omega\,,
\end{array}\right.
\end{equation}
 possesses an unbounded sequence of eigenvalues
$0<\lambda_{1}<\lambda_{2}\leq
...\leq\lambda_{n}\leq ...$. This celebrated result goes
back to the Riesz-Fredholm theory of self-adjoint and compact
operators on Hilbert spaces.

In what concerns $\lambda_{1}$, the lowest eigenvalue of
problem \eq{DEVP}, we remember that it can be characterized from a
variational point of view as the minimum of the Rayleigh quotient,
that is,
\begin{equation}\label{vpunu}\lambda_{1}=\inf_{u\in
H_0^1(\Omega)\setminus\{0\}}\frac{\di\int_{\Omega}|\nabla
u|^2\;dx} {\di\int_{\Omega}u^2\;dx}\,.
\end{equation}
Moreover, it is known that $\lambda_{1}$ is simple, that is,
all the associated eigenfunctions  are merely multiples of each
other (see, e.g. Gilbarg and Trudinger \cite{GT}). Furthermore,
the corresponding eigenfunctions of $\lambda_{1}$ never
change signs in $\Omega$.


Going further, another type of eigenvalue problems involving the
Laplace operator (obtained in the case when we take in \eq{zero}, $f(x,u)=|u|^{p(x)-2}u$)
is given by the nonlinear model equation
\begin{equation}\label{DEVP1}
\left\{\begin{array}{lll}
-\Delta u=\lambda|u|^{p(x)-2}u, &\mbox{in}& \Omega\\
u=0, &\mbox{on}& \partial\Omega\,,
\end{array}\right.
\end{equation}
where $p(x):\overline\Omega\rightarrow(1,2^*)$ is a given continuous function and $2^*$ denotes the critical Sobolev exponent, that is,
$$2^*=
\left\{
\begin{array}{lll}
&\di\frac{2N}{N-2} &\qquad\mbox{if $N\geq 3$}\\
&\di +\infty &\qquad\mbox{if $N\in\{1,2\}$.}
\end{array}\right.
$$
Obviously,  the case when $p$ is a constant function on
$\overline\Omega$ is allowed but we avoid the case when $p\equiv
2$ since this case is the object of problem \eq{DEVP}, discussed
above. For this problem the growth rate of the function $p$
is essential in the description of the set of eigenvalues.
First, assuming that $\min_{\overline\Omega}p>2$ it can be proved
(by using a mountain-pass argument) that any $\lambda>0$ is an
eigenvalue of problem \eq{DEVP1}. Next, in the case when
$\min_{\overline\Omega}p<2$ it can be proved (by using  Ekeland's
variational principle) that the problem has a continuous family of
eigenvalues which lies in a neighborhood of the origin (see, e.g.,
Mih\u ailescu and R\u adulescu \cite{pams} or Fan \cite{fan} for
some extensions). Finally, we point out that the above result can
be completed in the particular case when
$\max_{\overline\Omega}p<2$. More exactly, in this situation it
can be proved that  the energy functional associated to problem
\eq{DEVP1} has a nontrivial (global) minimum point for any
positive $\lambda$ large enough. In other words, if
$\max_{\overline\Omega}p<2$ then there exist two positive
constants $\mu_1$ and $\mu_2$ such that any
$\lambda\in(0,\mu_1)\cup(\mu_2,\infty)$ is an eigenvalue of
problem \eq{DEVP1}.

We notice that in all the situations presented above on \eq{DEVP1} the set of
eigenvalues is not completely described, excepting the case when $\min_{\overline\Omega}p>2$. However,
in all the cases the set of eigenvalues possesses a continuous subfamily.

In what concerns the eigenvalue problems involving quasilinear
operators we remember, in the case of homogeneous elliptic
operators, the contributions of Anane \cite{anane}, de Th\'elin
\cite{the2}, Lindqvist \cite{L} and Filippucci-Pucci-R\u
adulescu \cite{FPR}, while in the case of nonhomogeneous elliptic
operators we point out the recent advances of Fan-Zhang-Zhao
\cite{FZZ}, Mih\u ailescu-R\u adulescu
\cite{pams,manuscripta,blms}, Mih\u ailescu-Pucci-R\u adulescu
\cite{mprcras,mpr} and Fan \cite{fan}.

Motivated by the above results on problems \eq{DEVP} and \eq{DEVP1}
which show that the eigenvalue problems involving the Laplace
operator lead to a discrete spectrum (see the case of problem
\eq{DEVP}) or a continuous spectrum (see the case of problem
\eq{DEVP1} in the different forms  pointed out above) we consider
important to supplement the above situations by studying a new
eigenvalue  problem involving the Laplace operator which possesses,
on the one hand, a continuous family of eigenvalues and, on the
other hand, at least one more eigenvalue which is isolated in the
set of eigenvalues of that problem.

We  study problem \eq{zero} in the case when
\begin{equation}\label{definitief}
f(x,t)=\left\{\begin{array}{lll}
h(x,t), &\mbox{if}& t\geq 0\\
t, &\mbox{if}& t< 0\,,
\end{array}\right.
\end{equation}
where $h:\Omega\times[0,\infty)\rightarrow\RR$ is a Carath\'eodory function satisfying the following hypotheses
\smallskip

\noindent (H1) there exists a positive constant $C\in(0,1)$ such that $|h(x,t)|\leq Ct$ for any $t\geq 0$ and a.e. $x\in\Omega$;
\smallskip

\noindent (H2) there exists $t_0>0$ such that $H(x,t_0):=\int_0^{t_0}h(x,s)\;ds>0$, for a.e. $x\in\Omega$;
\smallskip

\noindent (H3) $\lim_{t\rightarrow\infty}\frac{h(x,t)}{t}=0$, uniformly in $x$.
\smallskip

\noindent{\bf Examples.} We point out certain examples of functions $h$ which satisfies the hypotheses (H1)-(H3):

1. $h(x,t)=\sin\,(t/2)$, for any $t\geq 0$ and any $x\in\Omega$;
\smallskip

2. $h(x,t)=k\log(1+t)$, for any $t\geq 0$ and any $x\in\Omega$, where $k\in(0,1)$ is a constant;
\smallskip

3. $h(x,t)=g(x)(t^{q(x)-1}-t^{p(x)-1})$, for any $t\geq 0$ and any
$x\in\Omega$, where $p$, $q:\overline\Omega\rightarrow(1,2)$
are continuous functions satisfying
$\max_{\overline\Omega}p<\min_{\overline\Omega}q$, and $g\in
L^\infty(\Omega)$ satisfies $0<\inf_\Omega g\leq\sup_\Omega g<1$.

\medskip
The main result of the present paper establishes a striking property of the eigenvalue problem \eq{zero}, provided that $f$ is defined as in \eq{definitief} and satisfies the above assumptions. More precisely, we prove that the first eigenvalue of the Laplace operator in $H^1_0(\Omega)$ is an {\it isolated} eigenvalue of \eq{zero} and, moreover, {\it any} $\lambda$ sufficiently large is an eigenvalue, while the interval $(0,\lambda_1)$ does {\it not} contain any eigenvalue. This shows that problem \eq{zero} has both isolated eigenvalues and a continuous spectrum in a neighbourhood of $+\infty$.

\begin{teo}\label{t1}
Assume that $f$ is given by relation \eq{definitief} and
conditions (H1), (H2) and (H3) are fulfilled.  Then  $\lambda_1$ defined in \eq{vpunu} is
an isolated eigenvalue of problem \eq{zero} and the corresponding set of eigenvectors
 is a cone. Moreover, any
$\lambda\in(0,\lambda_1)$ is not an eigenvalue of problem \eq{zero}
but there exists $\mu_1>\lambda_1$ such that any
$\lambda\in(\mu_1,\infty)$ is  an eigenvalue of problem \eq{zero}.
\end{teo}

 Finally, we notice that similar results as those given by Theorem \ref{t1} can be formulated  for
equations of type \eq{probl2} but replacing the Laplace operator $\Delta u$ by the $p$-Laplace operator, that is
$\Delta_p u:={\rm div}(|\nabla u|^{p-2}\nabla u)$, with $1<p<\infty$. Certainly, in that case hypotheses (H1)-(H3) should
be modified accordingly with the new situation. This statement is supported by the fact that the first eigenvalue of
the $p$-Laplace operator on bounded domains satisfies similar properties as the one obtained in the case of the Laplace operator
(see, e.g., \cite{anane}) combined with the remark that the results on problem \eq{P} can be easily  extended  to the case of the
$p$-Laplace operator.

\section{Proof of the main result}
For any $u\in H_0^1(\Omega)$ we denote
$$u_\pm(x)=\max\{\pm u(x),0\},\;\;\;\forall\;x\in\Omega\,.$$
Then $u_+$, $u_-\in H_0^1(\Omega)$ and
$$
\nabla u_+=\left\{\begin{array}{lll}
0, &\mbox{if}& [u\leq 0]\\
\nabla u, &\mbox{if}& [u>0]\,,
\end{array}\right.\;\;\;\;\;
\nabla u_-=\left\{\begin{array}{lll}
0, &\mbox{if}& [u\geq 0]\\
\nabla u, &\mbox{if}& [u<0]\,,
\end{array}\right.
$$
(see, e.g. \cite[Theorem 7.6]{GT}). Thus, problem \eq{zero} with $f$ given by relation
\eq{definitief} becomes
\begin{equation}\label{probl2}
\left\{\begin{array}{lll}
-\Delta u=\lambda[h(x,u_+)-u_-], &\mbox{in}& \Omega\\
u=0, &\mbox{on}& \partial\Omega\,,
\end{array}\right.
\end{equation}
and $\lambda>0$ is an eigenvalue of problem \eq{probl2} if there
exists $u\in H_0^1(\Omega)\setminus\{0\}$ such that
\begin{equation}\label{defvp}\int_\Omega\nabla u_+\nabla v\;dx-\int_\Omega\nabla u_-\nabla v\;dx-\lambda\int_\Omega[h(x,u_+)-u_-]v\;dx=0\,,\end{equation}
for any $v\in H_0^1(\Omega)$.

\begin{lemma}\label{l0}
Any $\lambda\in(0,\lambda_1)$ is not an eigenvalue of problem \eq{probl2}.
\end{lemma}
\proof Assume that $\lambda>0$ is an eigenvalue of problem \eq{probl2} with the corresponding eigenfunction $u$.
Letting $v=u_+$ and $v=u_-$ in the definition of the eigenvalue $\lambda$ we find that  the following two relations hold true
\begin{equation}\label{x4}
\int_\Omega|\nabla u_+|^2\;dx=\lambda\int_\Omega h(x,u_+)u_+\;dx
\end{equation}
and
\begin{equation}\label{x5}
\int_\Omega|\nabla u_-|^2\;dx=\lambda\int_\Omega u_-^2\;dx\,.
\end{equation}
In this context, hypothesis (H1) and relations \eq{vpunu}, \eq{x4} and \eq{x5} imply
$$\lambda_1\int_\Omega u_+^2\;dx\leq\int_\Omega|\nabla u_+|^2\;dx=\lambda\int_\Omega h(x,u_+)u_+\;dx
\leq\lambda \int_\Omega u_+^2\;dx$$
and
$$\lambda_1\int_\Omega u_-^2\;dx\leq\int_\Omega|\nabla u_-|^2\;dx=\lambda\int_\Omega u_-^2\;dx\,.$$
If $\lambda$ is an eigenvalue of problem \eq{probl2} then $u\neq 0$ and thus, at least one of the functions
$u_+$ and $u_-$ is not the zero function. Thus, the last two inequalities show that $\lambda$ is an
eigenvalue of problem \eq{probl2} only if $\lambda\geq\lambda_1$.\endproof

\begin{lemma}\label{163276}
$\lambda_1$ is an eigenvalue of problem \eq{probl2}. Moreover, the set of eigenvectors corresponding
to $\lambda_1$ is a cone.
\end{lemma}
\proof
Indeed, as we already pointed out, $\lambda_1$ is the lowest eigenvalue of problem \eq{DEVP}, it
is simple, that is, all the associated eigenfunctions  are merely multiples of each other (see, e.g.,
Gilbarg and Trudinger \cite{GT}) and the corresponding eigenfunctions of $\lambda_{1}$ never
change signs in $\Omega$. In other words, there exists $e_1\in H_0^1(\Omega)\setminus\{0\}$,
with $e_1(x)<0$ for any $x\in\Omega$ such that
$$\int_\Omega\nabla e_1\nabla v\;dx-\lambda_1\int_\Omega e_1v\;dx=0\,,$$
for any $v\in H_0^1(\Omega)$. Thus,  we have $(e_1)_+=0$ and $(e_1)=-e_1$ and we deduce that
relation \eq{defvp} holds true with $u=e_1\in H_0^1(\Omega)\setminus\{0\}$ and $\lambda=\lambda_1$.
In other words, $\lambda_1$ is an eigenvalue of problem \eq{probl2} and undoubtedly, the set of its corresponding
eigenvectors lies in a cone of $H_0^1(\Omega)$. The proof of Lemma \ref{163276} is
complete.\endproof

\begin{lemma}\label{17715}
$\lambda_1$  is isolated in the set of eigenvalues of problem \eq{probl2}.
\end{lemma}
\proof By Lemma \ref{l0} we know that in the interval
$(0,\lambda_1)$ there is no eigenvalue of problem \eq{probl2}. On
the other hand, hypothesis (H1) and relations \eq{vpunu}  and
\eq{x4} show that if $\lambda$ is an eigenvalue of problem
\eq{probl2} for which the positive part of its corresponding
eigenfunction, that is $u_+$, is not identically zero then
$$\lambda_1\int_\Omega u_+^2\;dx\leq\int_\Omega|\nabla u_+|^2\;dx=\lambda\int_\Omega h(x,u_+)u_+\;dx
\leq\lambda C \int_\Omega u_+^2\;dx\,,$$
and thus, since $C\in(0,1)$ we infer $\lambda\geq\frac{\lambda_1}{C}>\lambda_1\,.$
We deduce that for any eigenvalue $\lambda\in(0,\lambda_1/C)$ of problem \eq{probl2} we must have $u_+=0$. It
follows that if $\lambda\in(0,\lambda_1/C)$ is an eigenvalue of problem \eq{probl2} then it is actually
an eigenvalue of problem \eq{DEVP} with the corresponding eigenfunction negative in $\Omega$. But, we already noticed that
the set of eigenvalues of problem \eq{DEVP} is discrete and $\lambda_1<\lambda_2$. In other words, taking
$\delta=\min\{\lambda_1/C,\lambda_2\}$ we find that $\delta>\lambda_1$ and any $\lambda\in(\lambda_1,\delta)$
can not be an eigenvalue of problem \eq{DEVP} and, consequently, any $\lambda\in(\lambda_1,\delta)$ is not an eigenvalue of
problem \eq{probl2}. We conclude that $\lambda_1$ is isolated in the set of eigenvalues of problem \eq{probl2}.
The proof of Lemma \ref{17715} is complete.  \endproof


Next, we show that there exists $\mu_1>0$ such that any $\lambda\in(\mu_1,\infty)$ is an eigenvalue
of problem \eq{probl2}. With that end in view, we consider the eigenvalue problem
\begin{equation}\label{P}
\left\{\begin{array}{lll}
-\Delta u=\lambda h(x,u_+), &\mbox{in}& \Omega\\
u=0, &\mbox{on}& \partial\Omega\,,
\end{array}\right.
\end{equation}
We say that $\lambda$ is an eigenvalue of problem \eq{P} if there exists $u\in H_0^1(\Omega)\setminus\{0\}$ such that
$$\int_\Omega\nabla u\nabla v\;dx-\lambda\int_\Omega h(x,u_+)v\;dx=0\,,$$
for any $v\in H_0^1(\Omega)$.

We notice that if $\lambda$ is an eigenvalue for \eq{P} 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 problem \eq{P} possesses nonnegative corresponding eigenfunctions. Moreover, the above discussion show that
an eigenvalue of problem \eq{P} is an eigenvalue of problem \eq{probl2}.


For each $\lambda>0$ we define the energy functional associated to problem \eq{P} by $I_\lambda:H_0^1(\Omega)\rightarrow\RR$,
$$I_\lambda(u)=\frac{1}{2}\int_\Omega|\nabla u|^2\;dx-\lambda\int_\Omega H(x,u_+)\;dx\,,$$
where  $H(x,t)=\int_0^th(x,s)\;ds$. Standard arguments show that $I_\lambda\in C^1(H_0^1(\Omega),\RR)$ with the derivative
given by
$$\langle I_\lambda^{'}(u),v\rangle=\int_\Omega\nabla u\nabla v\;dx-\lambda\int_\Omega h(x,u_+)v\;dx\,,$$
for any $u$, $v\in H_0^1(\Omega)$. Thus, $\lambda>0$ is an eigenvalue of problem \eq{P} if and only if there
exists a critical nontrivial point of functional $I_\lambda$.

\begin{lemma}\label{l1}
The functional $I_\lambda$ is bounded from below and coercive.
\end{lemma}
\proof
By hypothesis (H3) we deduce that
$$\lim_{t\rightarrow\infty}\frac{H(x,t)}{t^2}=0,\;\;\;{\rm uniformly}\;{\rm in}\;\Omega\,.$$
Then for a given $\lambda>0$ there exists a positive constant $C_\lambda>0$ such that
$$\lambda H(x,t)\leq\frac{\lambda_1}{4}t^2+C_\lambda,\;\;\;\forall\;t\geq 0,\;{\rm a.e.}\;x\in\Omega\,,$$
where $\lambda_1$ is given by relation \eq{vpunu}.

Thus, we find that for any $u\in H_0^1(\Omega)$ it holds true
$$I_\lambda(u)\geq\frac{1}{2}\int_\Omega|\nabla u|^2\;dx-\frac{\lambda_1}{4}\int_\Omega u^2\;dx-C_\lambda|\Omega|\geq\frac{1}{4}\|u\|^2-C_\lambda|\Omega|\,,$$
where by $\|\cdot\|$ is denoted the norm on $H_0^1(\Omega)$, that is $\|u\|=(\int_\Omega |\nabla u|^2\;dx)^{1/2}$.
This shows that $I_\lambda$ is bounded from below and coercive.  The proof of Lemma \ref{l1} is complete. \endproof

\begin{lemma}\label{l2}
There exists $\lambda^\star>0$ such that assuming that $\lambda\geq\lambda^\star$ we have $\inf_{H_0^1(\Omega)}I_\lambda<0$.
\end{lemma}
\proof
Hypothesis (H2) implies that there exists $t_0>0$ such that
$$H(x,t_0)>0\,\;\;\;{\rm a.e.}\;x\in\overline\Omega\,.$$
Let $\Omega_1\subset\Omega$ be a compact subset, sufficiently large, and $u_0\in C_0^1(\Omega)\subset H_0^1(\Omega)$
such that $u_0(x)=t_0$ for any $x\in\Omega_1$ and $0\leq u_0(x)\leq t_0$ for any $x\in\Omega\setminus\Omega_1$.

Thus, by hypothesis (H1) we have
\begin{eqnarray*}
\int_\Omega H(x,u_0)\;dx&\geq&
\int_{\Omega_1} H(x,t_0)\;dx-\int_{\Omega\setminus\Omega_1}Cu_0^2\;dx\\
&\geq&\int_{\Omega_1} H(x,t_0)\;dx-Ct_0^2|\Omega\setminus\Omega_1|>0\,.
\end{eqnarray*}
We conclude that $I_\lambda(u_0)<0$ for $\lambda>0$ sufficiently large, and thus, $\inf_{H_0^1(\Omega)}I_\lambda<0$.
The proof of Lemma \ref{l2} is complete.  \endproof

Lemmas \ref{l1} and \ref{l2} show that for any $\lambda>0$ large
enough, the functional $I_\lambda$ possesses a negative global
minimum (see, \cite[Theorem 1.2]{S}), and thus, any $\lambda>0$
large enough is an eigenvalue of problem \eq{P} and consequently of
problem \eq{probl2}. Combining that fact with the results of Lemmas
\ref{l0}, \ref{163276} and \ref{17715} we conclude that Theorem
\ref{t1} holds true.


\medskip

{\bf Acknowledgments.} The authors have been supported by Grant
CNCSIS 79/2007 ``Degenerate and Singular Nonlinear Processes".

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\end{document} 