\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 26, pp. 1--10.\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-2010/26\hfil Nonlinear eigenvalue problem]
{Existence and uniqueness for a $p$-Laplacian
 nonlinear eigenvalue problem}

\author[G. Franzina, P. D. Lamberti\hfil EJDE-2010/26\hfilneg]
{Giovanni Franzina, Pier Domenico Lamberti}  % in alphabetical order

\address{Giovanni Franzina \newline
Dipartimento di Matematica, University of Trento, Trento, Italy}
\email{g.franzina@email.unitn.it}

\address{Pier Domenico Lamberti \newline
Dipartimento di Matematica Pura e Applicata, 
University of Padova, Padova, Italy}
\email{lamberti@math.unipd.it}

\thanks{Submitted January 14, 2010. Published February 16, 2010.}
\subjclass[2000]{35P30}
\keywords{$p$-laplacian; eigenvalues; existence; uniqueness
results}

\begin{abstract}
 We consider the Dirichlet eigenvalue problem
 $$
 -\mathop{\rm div}(|\nabla u|^{p-2}\nabla u )
  =\lambda \| u\|_q^{p-q}|u|^{q-2}u,
 $$
 where the unknowns $u\in W^{1,p}_0(\Omega )$ (the eigenfunction)
 and $\lambda >0$ (the eigenvalue),  $\Omega $ is an arbitrary
 domain in $\mathbb{R}^N$ with finite measure, $1<p<\infty $,
 $1<q< p^*$, $p^*=Np/(N-p)$ if $1<p<N$ and $p^*=\infty $ if $p\geq N$.
 We study several existence and uniqueness results as well as
 some properties of the solutions. Moreover, we indicate how to
 extend to the general case some proofs known in the classical
 case $p=q$.
\end{abstract}

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

\section{Introduction}

Let $\Omega $ be a domain (i.e., a connected open set) in
$\mathbb{R}^N$ with finite measure, $1<p<\infty$,
$1<q< p^*$ where $p^*=Np/(N-p)$ if $p< N$ and $p^*=\infty $ if $p\geq N$.
It is well-known that the Sobolev space $W^{1,p}_0(\Omega )$
is compactly embedded in $L^q(\Omega )$ and that
\begin{equation}\label{ray}
\lambda_1(p,q):=\inf_{u\in W^{1,p}_0(\Omega ),\, u\ne 0}
\frac{\int_{\Omega }|\nabla u|^pdx}{(\int_{\Omega }|u|^qdx)^{p/ q}}>0 .
\end{equation}
It is also well-known that  the Rayleigh quotient in \eqref{ray}
 admits a minimizer which does not change sign in $\Omega $.
The Euler-Lagrange
equation associated with this minimization problem is
\begin{equation}\label{classiceq}
-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u
)=\lambda \| u \|_{q}^{p-q} |u|^{q-2}u ,
\end{equation}
where $\| u\|_q$ denotes the norm of $u$ in $L^q(\Omega )$.
Usually in the literature, the function $u$ is normalized
in order to get rid of the apparently redundant factor
  $\|u \|_{q}^{p-q}$. However, we prefer to keep it since it allows
to think of this problem as an eigenvalue problem. Indeed,
\eqref{classiceq} is homogeneous; i.e., if $u$ is a solution then
$ku $ is also a solution for all $k\in {\mathbb{R}}$, as one would
expect from an eigenvalue problem.
It turns out that  $\lambda_1(p,q)$ is the smallest eigenvalue
of  \eqref{classiceq} and we refer to it as the first eigenvalue.

The case $p=q$ has been largely investigated by many authors
and it has been often considered as a {\it typical eigenvalue
problem} (cf. e.g., Garc\'{\i}a Azorero and Peral Alonso~\cite{azal});
for extensive references on this subject we refer to
Lindqvist~\cite{lqv}. For the case $p\ne q$ we refer again
to \cite{azal}, \^Otani~\cite{ot88, ot84} and  Dr\'{a}bek, Kufner
and Nicolosi~\cite{drabook} who consider an even more general
 class of {\it nonhomogeneous eigenvalue problems}.

In this paper, we study several results and we indicate how to
adapt to the general case  some proofs known in the case $p=q$.

First of all, we discuss the simplicity of $\lambda_1(p,q)$.
We recall that $\lambda_1(p,q)$ is simple if $q\le p$, as it is
proved in Idogawa and \^{O}tani~\cite{otani95}. If $q>p$ then
$\lambda_1(p,q)$ is not necessarily simple:  for example,
simplicity does not hold if  $\Omega $ is a sufficiently thin annulus,
see Kawohl~\cite{kawohl} and Nazarov~\cite{naza}.
However, if $\Omega $ is a ball then
the simplicity of $\lambda_1(p,q)$ is guaranteed also in the
case $q>p$: here we briefly describe the argument of Erbe and
Tang~\cite{erta}.

By adapting the argument
of Kawohl and Lindqvist~\cite{lqka}, we prove that if $q\le p$ then
the only eigenvalue admitting a non-negative eigenfunction is the
first one.

Moreover,  by exploiting our point of view, we also give an alternative
proof of a uniqueness result of Dr\'{a}bek~\cite[Thm.~1.1]{dra} for
the equation $-\Delta_pu=|u|^{q-2}u$, see Theorem~\ref{drabrev}.

Finally, in the general case $1<q<p^*$, we observe that the point
spectrum $\sigma (p,q)$ is closed as in the case $p=q$ considered
in \cite{lqv} and we indicate how to apply the
Ljusternik-Schnirelman min-max procedure in order to define a
divergent sequence of eigenvalues $\lambda_n(p,q)$, $n\in
{\mathbb{N}}$. Note that the existence of infinitely many
solutions to equation \eqref{classiceq} is also proved in
\cite{azal} where the cases $q<p $ and $q>p$ are treated
separately; instead, here we adopt a  unified approach.


We point out that in this paper we do not assume that $\Omega $
is bounded as largely done in the literature, but only that its
measure is finite.

\section{The eigenvalue problem}

Let $\Omega $ be a domain  in $\mathbb{R}^N$ with finite measure
and $1<p<\infty $. By $W^{1,p}(\Omega )$ we denote the Sobolev space
of those functions in $L^{p}(\Omega )$ with first order weak
derivatives in $L^p(\Omega )$ endowed with its usual norm.
By $W^{1,p}_0(\Omega )$ we denote the closure in $W^{1,p}(\Omega )$
of the $C^{\infty }$-functions with compact support in $\Omega$.

 It is well-known that the Poincar\'{e} inequality holds. Namely,
for every $1<q<p^*$ there exists $C>0$ depending only on $N,p,q$
such that
\begin{equation} \label{poi}
\|u\|_{L^q(\Omega )} \le C |\Omega
|^{\frac{1}{q}-\frac{1}{p}+\frac{1}{N}}\| \nabla u\|_{L^p(\Omega)},
\end{equation}
for all $u\in W^{1,p}_0(\Omega )$. In particular it follows that
$\lambda_1(p,q)$ defined in \eqref{ray} is positive and satisfies
the inequality
\begin{equation}
\label{poibis}
\lambda_1(p,q)> \frac{1}{C^p|\Omega|^{p(\frac{1}{q}-\frac{1}{p}
+\frac{1}{N} )}}\,.
\end{equation}
Moreover, since the measure of $\Omega $ is finite, the
embedding $W^{1,p}_0(\Omega )\subset L^q(\Omega )$ is compact:
this combined with the reflexivity of $W^{1,p}_0(\Omega )$ guarantees
the existence of a minimizer in \eqref{ray}. As we mentioned in the
introduction, equation \eqref{classiceq} is the Euler-Lagrange
equation corresponding to the minimization problem \eqref{ray}.
It is then natural to give the following definition where,
as usual, equation \eqref{classiceq} is  interpreted in the weak sense.

\begin{definition}\label{weakdef} \rm
Let $\Omega$ be a domain in $\mathbb{R}^N$ with finite measure,
$1<p<\infty $ and $1<q< p^*$. We say that $\lambda >0 $
is an eigenvalue of equation \eqref{classiceq} if there exists
$u\in W^{1,p}_0(\Omega )\setminus\{0\}$ such that
\begin{equation} \label{weakeq}
\int_{\Omega }|\nabla u |^{p-2}\nabla u\nabla \varphi dx
=\lambda \| u \|_{q}^{p-q} \int_{\Omega }|u|^{q-2}u\varphi dx,
\end{equation}
for all $\varphi \in W^{1,p}_0(\Omega )$. The eigenfunctions
corresponding to $\lambda $ are the solutions $u$ to  \eqref{weakeq}.
\end{definition}

It is clear that all eigenvalues are positive and that
$\lambda_1(p,q)$ is the least eigenvalue. Moreover, the eigenfunctions
corresponding to $\lambda_1(p,q)$ are exactly the minimizers
in \eqref{ray}. We recall the following known result.

\begin{theorem}\label{bound}
Let $\Omega$ be a domain in $\mathbb{R}^N$ with finite measure,
$1<p<\infty $ and $1<q< p^*$. Let $\lambda >0 $ be an eigenvalue
of equation \eqref{weakeq} and $u\in W^{1,p}_0(\Omega )$ be a
corresponding eigenfunction. Then $u$ is bounded and its first
derivatives are locally H\"{o}lder continuous.
Moreover, if $u\geq 0$ in $\Omega $ then $u>0$ in $\Omega$.
\end{theorem}

As done in \cite[Lemma~5.2]{lqv} for the case $q=p$, the boundedness
of $u$ can be proved by using the method of \cite[Lemma~5.1]{ladur}.
The H\"older regularity of the first order derivatives  follows
by Tolksdorf~\cite{tol}.
We note that the argument in \cite{lqv} allows to give a quantitative
bound for $u$. Namely, by a slight modification of \cite[Lemma~5.2]{lqv}
one can prove that  there exists a constant $M>0$, depending only
on $p,q,N$, such that
\begin{equation}\label{qtbound}
 \|u\|_{L^\infty(\Omega)} \le M \lambda^{\frac{1}{\delta p}}
\|u\|_{L^1(\Omega)},
\end{equation}
where $\delta = 1/N$ if $q\le p$ and $\delta = 1/q-1/p+1/N$ if $q>p$.
We refer to Franzina~\cite{fra} for details.
Finally, the fact that a non-negative eigenfunction does not vanish
in $\Omega$ can be deduced by the strong maximum principle in
Garc\'{i}a-Meli\`{a}n and Sabina de Lis~\cite[Theorem~1]{garmel}.


\begin{corollary}\label{signcor}
Let $\Omega$ be a domain in $\mathbb{R}^N$ with finite measure,
$1<p<\infty $ and $1<q< p^*$. Let $u\in W^{1,p}_0(\Omega )\setminus \{0\}$
 be an eigenfunction corresponding to $\lambda_1(p,q)$.
Then either $u>0$ or $u<0$ in $\Omega $.
\end{corollary}

\begin{proof}
 Clearly $u$ is a minimizer in \eqref{ray}.
Then also $|u|$ is a minimizer, hence a first eigenfunction.
Thus by Theorem~\ref{bound} $|u|$ cannot vanish in $\Omega $.
\end{proof}

\section{On the simplicity of $\lambda_1(p,q)$}


It is known that if $q\le p$ then $\lambda_1(p,q)$ is simple.
In fact we have the following theorem by Idogawa and
\^{O}tani~\cite[Theorem~4]{otani95} the proof of which works word
by  word  also when $\Omega$ is not bounded.

\begin{theorem}\label{uniray}
Let $\Omega$ be a domain in $\mathbb{R}^N$ with finite measure
and $1<q\le p<\infty $. Then $\lambda_1(p,q)$ is simple; i.e.,
the eigenfunctions corresponding to $\lambda_1(p,q)$ define
a linear space of dimension one.
\end{theorem}

We refer to Kawohl, Lucia and Prashanth~\cite{kaw} for a recent
 generalization of the previous result to some indefinite
quasilinear problems. For the case $p=q$ we refer to
Lindqvist~\cite{lqv0}.

In general, Theorem~\ref{uniray} does not hold if $q>p$;
 see \cite{kawohl} and \cite{naza} where the case of a sufficiently
thin annulus is considered. However, as one may expect,
if $\Omega$ is a ball then $\lambda_1(p,q)$ is simple. Basically,
this depends on the following theorem,  cf. \cite{kawohl}.

\begin{theorem}\label{rad}
 Let $\Omega $ be a ball in ${\mathbb{R}}^{N}$ centered at zero,
 $1<p<\infty $ and  $1<q<p^*$. Then the eigenfunctions corresponding
to $\lambda_1(p,q)$ are radial functions.
\end{theorem}

Theorem~\ref{rad} allows to pass to spherical coordinates and to
reduce our problem to an ODE as follows.
If $\Omega $ is a ball centered at zero and  $u$ is a radial
function, $u(x) =\phi (|x |)$,  then
$$
-\Delta_pu= -(p-1)|\phi '(r)|^{p-2}\phi ''(r)
-\frac{N-1}{r}|\phi '(r)|^{p-2}\phi'(r),
$$
which is well-defined for all $r> 0$ such that $\phi $ is twice
differentiable in $r$. Recall that by standard regularity theory
an eigenfunction $u$ is twice differentiable on the set
$\{x\in \Omega : \nabla u(x)\ne 0 \} $.
By writing  \eqref{classiceq} in spherical coordinates and
multiplying both sides by $r^{N-1}$ it follows that if
$u=\phi (|x|)$ is a radial eigenfunction corresponding
to the eigenvalue $\lambda$ and $\|u\|_{L^q(\Omega )}=1$ then
\begin{equation}\label{radeq}
-( r^{N-1}|\phi '|^{p-2}\phi ' )'=\lambda r^{N-1} |\phi |^{q-2}\phi .
\end{equation}
If in addition $u$ is a first eigenfunction then $u$ does not change
sign in $\Omega $; thus, by integrating equation \eqref{radeq},
one can easily prove that $\phi '$ vanishes only at $r=0$.
Hence $\phi $ is twice differentiable for all $r>0$ and  \eqref{radeq}
is satisfied in the classical sense for all $r>0$.

To prove the simplicity of $\lambda_1(p,q)$ we use the following Lemma.
The proof is more or less standard (further details can be
found in Franzina~\cite{fra}).

\begin{lemma}\label{uniqlem}
Let $1<p<\infty $, $1<q<p^*$ and $\lambda , c>0$. Then the Cauchy problem
\begin{equation}\label{cauchy}
\begin{gathered}
-( r^{N-1}|\phi '|^{p-2}\phi ' )'=\lambda r^{N-1} |\phi |^{q-2}\phi , \quad
 r\in (0,R),\\
\phi (0)=c,\quad  \phi'(0)=0,
\end{gathered}
\end{equation}
has at most one positive solution  $\phi$ in $C^1[0,R]\cap C^2(0,R)$.
\end{lemma}

\begin{proof}
 We consider the operator $T$ of $C[0,R]$ to $C[0,R]$ defined by
\begin{equation}
 T (\phi) (r) = c - \int_0^r g^{-1}\Big( \frac{\lambda}{t^{N-1}}
 \int_0^t s^{N-1}|\phi|^{q-2}\phi\,ds\Big)\,dt, \quad r\in [0,R],
\end{equation}
for all $\phi\in C[0,R]$, where $g(t)=|t|^{p-2}t$ if $t\ne 0$
and $g(0)=0$ and $g^{-1}$ denotes the inverse function of $g$.
It's easily seen that every positive solution to the Cauchy
problem \eqref{cauchy} is a fixed point of the operator $T$ of class
$C^1[0,R]\cap C^2(0,R)$.

Now let  $\phi_1,\phi_2\in C^1[0,R]\cap C^2(0,R)$ be two positive
solutions to  problem \eqref{cauchy}.
One can prove that there exists  $\epsilon_1>0$ such that
\[
 \|T(\phi_1)-T(\phi_2)\|_{C[0,\varepsilon]}
 \leqslant C_1(\varepsilon) \|\phi_1 - \phi_2 \|_{C[0,\varepsilon]},
\]
 for all $\varepsilon \in [0,\epsilon_1] $ where $C_1(\varepsilon)<1$.
It follows that $\phi_1=\phi_2$ in a neighborhood of zero.
Furthermore, let
$R_0 = \sup\{\varepsilon>0 : \phi_1=\phi_2\text{ on }[0,\varepsilon]\}$.
Arguing by contradiction, assume that
$R_0<R$. Then one can prove that there exists  $0<\epsilon_2<R-R_0$
such that
\[
    \|T(\phi_1)-T(\phi_2)\|_{C[R_0,R_0+\varepsilon]}
\leqslant C_2(\varepsilon) \|\phi_1 - \phi_2 \|_{C[R_0,R_0+\varepsilon]},
\]
for all $\varepsilon\in[0,\epsilon_2]$, where $C_2(\varepsilon)<1$.
This implies that $\phi_1=\phi_2 $  in a neighborhood of $R_0$,
a contradiction.
\end{proof}


We point out that Lemma \ref{uniqlem} does not immediately
imply that $\lambda_1(p,q)$ is simple in a ball; if $N>1$ further
technical work is required and the main step is the following
Lemma for which we refer  to Erbe and Tang~\cite[Lemma~3.1]{erta}.

\begin{lemma}\label{erbetanglem}
Let $N>1$, $1<p<\infty $,  $1<q<p^*$ and   $c_1,c_2>0$.
Let $\phi_1,\phi_2\in C^1[0,R]\cap C^2(0,R)$ be two
positive solutions to the Cauchy problem \eqref{cauchy}
with $c=c_1, c_2$ respectively.
If $c_1\le c_2 $ then $\phi_1\le \phi_2 $.
\end{lemma}

By using Lemmas~\ref{uniqlem} and \ref{erbetanglem} we can
deduce the validity of the following result.

\begin{theorem}
 Let $\Omega $ be a ball in ${\mathbb{R}}^{N}$, $1<p<\infty $
and $1<q<p^*$. Then $\lambda_1(p,q)$ is simple.
\end{theorem}

\begin{proof}
 For the case $N=1$ we refer to \cite[Theorem~I]{ot84}. Assume
now that $N>1$ and that $\Omega $ is a ball of radius $R$ centered
at zero. Let $u_1, u_2$ be two nonzero eigenfunctions corresponding
to the first eigenvalue $\lambda_1(p,q)$. We have to prove that $u_1$
and $u_2$ are proportional. To do so we can directly assume that
$\| u_1\|_q=\| u_2\|_q=1$. Moreover, by Corollary \ref{signcor} we
can assume without loss of generality that $u_1,u_2>0$ on $\Omega$.
 By Theorem~\ref{rad} $u_1,u_2$ are radial functions hence they can
be written as $u_1=\phi_1(|x|)$,
$u_2=\phi_2(|x|)$ for suitable positive functions
$\phi_1,\phi_2\in C^1[0,R]\cap C^2(0,R)$ satisfying condition
$\phi_1'(0)=\phi_2'(0)=0$ and equation \eqref{radeq} with
$\lambda =\lambda_1(p,q)$.  If $\phi_1(0)\ne \phi_2(0)$, say
$\phi_1(0)<\phi_2(0)$, then by Lemma~\ref{erbetanglem}
$\phi_1\le \phi _2 $ in $\Omega $ hence $\| u_1\|_q< \| u_2\|_q$,
since by continuity $u_1<u_2$ in a neighborhood of zero.
A contradiction.
Thus  $\phi _1(0)=\phi _2(0)$, hence by Lemma~\ref{uniqlem}
$\phi_1=\phi_2$ or equivalently $u_1=u_2$.
\end{proof}

\section{Further uniqueness results}

By Corollary~\ref{signcor} the first eigenvalue admits a non-negative
eigenfunction. It is well-known that no other eigenvalues enjoy
this property when $p=q$. This can be proved also in the case $q\le p$.
The proof of the following theorem  exploits an argument used by
\^{O}tani and Teshima~\cite{otsh} in the case of bounded smooth
open sets  combined with an argument  of Lindqvist and Kawhol~\cite{lqka}
which allows to deal with rough boundaries.

\begin{theorem}\label{unipos}
Let $\Omega$ be a domain in $\mathbb{R}^N$ with finite measure
and $1<q\le p$. If $\lambda $ is an eigenvalue of  \eqref{classiceq}
admitting a positive eigenfunction then $\lambda =\lambda_1(p,q)$.
\end{theorem}

\begin{proof}
We argue by contradiction and assume that $\lambda_1(p,q)< \lambda $.
Let $u_1\in W^{1,p}_0(\Omega )\setminus\{ 0\}$ be a positive
eigenfunction corresponding to $\lambda_1=\lambda_1(p,q)$
and  $u$ be a positive eigenfunction corresponding to
$\lambda $. We directly  assume that $
u_1  \le u $ in $\Omega$, otherwise one can use the approximation
argument of \cite{lqka} (which works also in the case of unbounded
domains). Since $q\le p$ it follows that for all nonnegative
test functions $\varphi$,
\begin{equation}\label{mainineq}
\begin{aligned}
\int_{\Omega}|\nabla u_1|^{p-2}\nabla u_1 \cdot \nabla\varphi dx
&= \lambda \| (\lambda_1/\lambda )^{\frac{1}{p-1}} u_1 \|^{p-q}_{q}
 \int_{\Omega }\left(\left(\lambda_1/\lambda  \right)^{\frac{1}{p-1}}u_1
\right)^{q-1}\varphi dx  \\
&\leq \lambda \| \eta u \|^{p-q}_{q} \int_{\Omega }
\left(\eta u\right)^{q-1}\varphi dx \\
&= \int_{\Omega}|\nabla\eta  u|^{p-2}\nabla\eta  u \cdot \nabla\varphi dx,
\end{aligned}
\end{equation}
where $\eta = (\lambda_1/\lambda )^{\frac{1}{p-1}}$.
By choosing $\varphi =\max \{u_1-\eta u,0\}$ in \eqref{mainineq} and
using the argument  in the proof of \cite[Lemma~3]{otsh}
(see also \cite{lqka}) we deduce that $u_1\le \eta u$ and
by iteration $u_1\le \eta^n u$ for all $n\in {\mathbb{N}}$.
Since $0<\eta<1$, by passing to the limit as $n\to \infty$ we
obtain $u_1=0$, a contradiction.
\end{proof}

By Theorem~\ref{unipos} we deduce the validity of the following
corollary which is well-known in the case of bounded smooth
domains (cf. e.g. Huang~\cite{huang}).

\begin{corollary}
Let $\Omega$ be a domain in $\mathbb{R}^N$ with finite measure
and $1<q< p $. The equation
\begin{equation} \label{eqnorm}
-\Delta_pv= |v|^{q-2}v
\end{equation}
has a unique positive solution in $W^{1,p}_0(\Omega )\setminus \{0 \}$.
\end{corollary}

\begin{proof}
Existence follows immediately by observing that if $u$ is a nonzero
eigenfunction of  \eqref{classiceq} then
$$
v=\frac{u}{\lambda^{\frac{1}{p-q}} \| u\| _q}
$$
is a solution to  \eqref{eqnorm}, hence the first eigenfunction
provides a positive solution to \eqref{eqnorm}.
We now prove uniqueness. Observe that if $v\ne 0 $ is a solution to
 \eqref{eqnorm} then $v$ is an eigenfunction
corresponding to the eigenvalue
\begin{equation} \label{jbis}
\lambda =\frac{1}{\| v\|_q^{p-q}} .
\end{equation}
Accordingly, by Theorem~\ref{unipos} two positive solutions
$v_1,v_2$ of \eqref{eqnorm} would be eigenfunctions corresponding to
$\lambda_1(p,q)$. Thus such solutions would be proportional by
Theorem~\ref{uniray}. Since $p\ne q$ proportionality implies
coincidence and then $v_1=v_2$.
\end{proof}

We recall that the solutions to \eqref{eqnorm} are exactly the
critical points of the functional
$$
J(v):=\frac{1}{p}\int_{\Omega }|\nabla v|^pdx
-\frac{1}{q}\int_{\Omega }|v|^qdx ,
$$
defined for all $v\in W^{1,p}_0(\Omega )$. The functional $J$ can
be used to give a condition equivalent to the simplicity of
$\lambda_1(p,q)$. In fact, we have the following

\begin{lemma}\label{equiv}
Let $\Omega$ be a domain in $\mathbb{R}^N$ with finite measure,
$1<p<\infty $ and $1<q<p^*$ with $q\ne p$. Let $S_{pq} $ be the set of all nontrivial 
solutions to  \eqref{eqnorm}. If $w\in S_{pq}$ is a point of
minimum for the restriction of $J$ to $S_{pq}$ then $w$ is an
 eigenfunction corresponding to $\lambda_1(p,q)$.
In particular, $\lambda_{1}(p,q)$ is simple if and only if
the restriction of $J$ to $S_{pq}$ has a unique (up to the sign)
point of minimum.
\end{lemma}

\begin{proof}
Note that if $v\in S_{pq}$ then
$\int_{\Omega }|\nabla v|^pdx=\int_{\Omega }|v|^qdx$.  Thus
$$
J(v)=\Big(\frac{1}{p}-\frac{1}{q}\Big)\int_{\Omega }|v|^qdx,
$$
for all $v\in S_{pq}$. Moreover, $v\in S_{pq}$ if and only if $v$
is an eigenfunction of equation \eqref{classiceq} corresponding
to an eigenvalue $\lambda$ satisfying \eqref{jbis}.
It follows that a function $w\in S_{pq}$ minimizes  the
restriction $J_{|S_{pq}}$ of $J$ to $S_{pq}$ if and only
if $w$ minimizes  the functional defined on $S_{pq}$ by \eqref{jbis}.
In particular, if $w\in S_{pq}$ minimizes $J_{|S_{pq}}$  and
$\lambda_1(p,q) $ is simple then $w=ku$, $k\in {\mathbb{R}}$ where
$u$  is the eigenfunction corresponding to $\lambda_1(p,q)$ uniquely
determined by the conditions $u>0$ in $\Omega$ and $\| u\|_q=1$;
moreover, since $ku$ satisfies equation \eqref{eqnorm} then
$k=\pm \lambda_{1}(p,q)^{\frac{1}{q-p}}$, hence $w$ is uniquely
determined up to the sign.  To conclude the proof it suffices to observe that if $v$ is an eigenfunction 
corresponding to $\lambda_1(p,q)$ and satisfies (\ref{jbis}) with $\lambda =\lambda_1(p,q)$ then $v$ minimizes 
$J_{|S_{pq}}$.
\end{proof}

By Lemma \ref{equiv} and Theorem~\ref{uniray} we deduce the
following result of  Dr\'{a}bek~\cite[Thm.~1.1]{dra}
for  the case $1<q<p$.

\begin{theorem}
\label{drabrev}
Let $\Omega$ be a domain in $\mathbb{R}^N$ with finite measure
and $1<q<p$. Equation \eqref{eqnorm} has a unique (up to the sign) nontrivial 
solution $w\in W^{1,p}_0(\Omega )$
with the following property:  $ J(w)\le J(v) $ if $v\in  W^{1,p}_0(\Omega )$ is a nontrivial solution to equation \eqref{eqnorm}.
Moreover, $w$  is an eigenfunction corresponding to $\lambda_1(p,q)$.
\end{theorem}

\begin{proof}
The proof  follows immediately by Lemma~\ref{equiv} and by
observing  that by Theorem~\ref{uniray} $\lambda_1(p,q)$ is
simple since $q<p$.
\end{proof}

\section{On the spectrum $\sigma (p,q)$}

We denote by $\sigma (p,q)$ the set of all the eigenvalues of
 \eqref{classiceq}. We  refer to $\sigma (p,q)$ as the spectrum
of the $p$-Laplacian.

The following result is well-known in the case $p=q$:
the proof given in  \cite[Theorem~5.1]{lqv}  does not require
any significant modification.

\begin{theorem}
Let $\Omega $ be a domain in $\mathbb{R}^N$ with finite measure,
$1<p<\infty $  and $1<q<p^*$. Then $\sigma (p,q) $ is a closed set.
\end{theorem}

As in the case $p=q$, it is possible to produce an infinite sequence
of eigenvalues by means of a min-max procedure which generalizes
the well-known min-max Courant principle.
Namely, for all $n\in {\mathbb{N}}$ we set
\begin{equation}
\lambda_n(p,q)=
\inf_{{\mathcal {M}}\in{\mathfrak M}(p,q )}
\sup_{u\in {\mathcal {M}}}
\frac{\int_{\Omega}|\nabla u|^p{ d}x}
{\big(\int_{\Omega}|u|^q{ d}x\big)^{p/q}},
\end{equation}
where ${\mathfrak{M}}(p,q)$ is the family of those conic
subsets ${\mathcal {M}}$ of
$W^{1,p}_0(\Omega)\setminus\{0\}$,
whose intersection with the unit sphere of $L^q(\Omega )$ is compact
in $W^{1,p}_0(\Omega )$ and whose Krasnoselskii's genus
$\gamma ({\mathcal {M}})$ is greater
than or equal to $n$. Recall that
\begin{equation}
\label{genus}
\gamma ( {\mathcal {M}})=\min\left\{k\in\mathbb{N}:
\exists F\in C\left(\mathcal{M},\mathbb{R}^k\setminus\{0\}\right),
F(f)=-F(-f)\, \forall f\in {\mathcal {M}}\right\},
\end{equation}
where
$C({\mathcal {M}},\mathbb{R}^k\setminus\{0\})$
denotes the space of all continuous functions of ${\mathcal {M}}$
to $\mathbb{R}^k\setminus\{0\}$. It is understood that
$\gamma ({\mathcal {M}})=\infty$ if the set in the right-hand side
of \eqref{genus} is empty.

The following theorem is proved by applying the abstract result
of Szulkin~\cite[Cor.~4.1, p.~132 ]{szu} as done in
Cuesta~\cite[Prop.~4.5, p.~85]{cue} where  one can find a
detailed proof for  the case $p=q$.

\begin{theorem}
Let $\Omega $ be domain in $\mathbb{R}^N$ with finite measure,
$1<p<\infty $ and $1<q<p^*$. Then
$\lambda_n(p,q)\in \sigma (p,q)$ and
$\lim_{n\to\infty }\lambda_n(p,q)=\infty$.
\end{theorem}

\begin{proof}
Let $I$ and $E$ be the functions of $W^{1,p}_0(\Omega )$ to
${\mathbb{R}}$ defined by
$$
I(u)=\Big(\int_{\Omega }|u|^q dx\Big)^{p/q}, \quad
E(u)=\int_{\Omega }|\nabla u|^pdx,
$$
for all $u\in W^{1,p}_0(\Omega ) $ and let
$M=\{u\in W^{1,p}_0(\Omega ):\ I(u)=1 \}$.  Note that $I$ and $E$
are of class $C^1$ and that $M$ is a closed submanifold  of
$W^{1,p}_0(\Omega )$ of codimension one whose tangent space at a
point $u$ is given by $T_uM=\ker d_uI$.
It is clear that the eigenvalues $\lambda$ of  \eqref{classiceq}
are exactly the critical levels of $E$ restricted to $M$; i.e.,
are those real numbers $\lambda$ for which there exists $u\in M$
such that  $E(u)=\lambda $ and $T_uM\subset \ker d_uE$.
It is not difficult to adapt the argument in Cuesta~\cite{cue}
 to prove that $E$ satisfies the well-known Palais-Smale condition
on $M$. Thus, by applying \cite[Cor.~4.1, p.~132 ]{szu} to
the functions $I$, $E$ it follows that the numbers $\lambda_n(p,q)$
are critical levels  of  $E$ restricted to $M$.

It remains to prove that $\lim_{n\to \infty}\lambda_n(p,q)=\infty $.
To do so we use an argument in Zeidler~\cite[Ch.~44]{zei}.
First of all, we recall that by \cite[Lemma~44.32]{zei} for
every $n\in {\mathbb{N}}$ there exist a finite-dimensional
subspace $X_n$ of $W^{1,p}_0(\Omega )$ and an odd continuous operator
$P_n$ of $W^{1,p}_0(\Omega )$ to $X_n$ such that for every
$u\in W^{1,p}_0(\Omega )$ we have that
$\| P_nu\|_{W^{1,p}_0(\Omega )}\le \| u\|_{W^{1,p}_0(\Omega )} $
and  $P_nu_n$ converges weakly in $W^{1,p}_0(\Omega )$ to $u$ for
all sequences $u_n$, $n\in {\mathbb{N}}$ weakly convergent in
$W^{1,p}_0(\Omega )$ to $u$.
Clearly, it  suffices to prove that for any fixed $L>0$ there
exists $n\in {\mathbb{N}}$ such that $\sup_{u\in {\mathcal{A}}}E(u) >L$
for all symmetric subsets ${\mathcal{A}}$ of $M$ such that
${\mathcal{A}}$ is compact in $W^{1,p}_0(\Omega )$ and
$\gamma ({\mathcal{A}})\geq n$. Assume to the contrary that there
exists $L>0$ such that this is not the case and set
$B_L=\{u\in M:\ E(u)\le L \}$. By means of a simple contradiction
argument one can prove that there exists $n_L\in {\mathbb{N}}$
such that $\inf_{u\in B_L}\| P_{n_L}u\|_{W^{1,p}_0(\Omega )}>0 $
hence $P_{n_L}u\ne 0$ for all $u\in B_L$.
Let $k_L=\dim X_{n_L}+1$. By assumption there exists a symmetric
subset ${\mathcal{A}}$ of $M$ such that ${\mathcal{A}}$ is compact,
 $\gamma ({\mathcal{A}})\geq k_L$ and $\sup_{u\in {\mathcal{A}}}E(u)\le L$. Since ${\mathcal{A}}\subset B_L$ then $P_{n_L}u\ne 0$ for all $u\in {\mathcal{A}}$
hence $\gamma (P_{n_L}({\mathcal{A}}))\geq k_L$; on the other
hand $P_{n_L}({\mathcal{A}})\subset X_{n_L} $ hence
$\gamma (P_{n_L}({\mathcal{A}}))\le \dim X_{n_L}= k_L -1$,
a contradiction.
\end{proof}

We remark that, despite the  results of
Binding and Rynne~\cite{bin} who have recently  provided examples
of nonlinear eigenvalue problems for which not all eigenvalues are
variational, it is not clear yet  whether for our problem the
variational eigenvalues exhaust the spectrum if $N>1$, not even in
the classical case $p=q$. However, a complete description of
$\sigma (p,q)$ is available for $N=1$, see \^Otani~\cite{ot84}
and Dr\'{a}bek and Man\'{a}sevich~\cite{draman}.

The following theorem is a restatement of
\cite[Theorems~3.1, 4.1]{draman}. We include a detailed proof
of \eqref{onedimeq} for the convenience of the reader.
Recall that the function  defined by
$$
\arcsin_{pq}(t)= \frac{q}{2}\int ^{\frac{2t}{q}}_0
\frac{ds}{(1-s^q)^{\frac{1}{p}}}\, ,
$$
for all $t\in [0,q/2] $, is a strictly increasing
function of $[0,q/2]$ onto $[0, \pi_{pq}/2]$ where
$\pi_{pq}= 2\arcsin _{pq} (q/2)=B(1/q, 1-1/p)$ and $B$
denotes  the Euler Beta function. The inverse function
of $\arcsin _{pq} $, which is denoted  by $\sin_{pq}$,
is extended to $[-\pi_{pq},  \pi_{pq}]$ by setting
$\sin_{pq}(\theta )=\sin_{pq}(\pi_{pq}-\theta )$ for all
$\theta \in ]\pi_{pq}/2, \pi_{pq}] $,
$\sin_{pq}(\theta )=-\sin_{pq}(-\theta ) $ for all
$\theta \in [-\pi_{pq},0[$,  and then it is extended by periodicity
to the whole of ${\mathbb{R}}$.

\begin{theorem}
If $N=1$ and $\Omega =(0, a)$ with $a>0$ then
\begin{equation}\label{onedimeq}
\lambda_1(p,q)=q^{\frac{p(1-q)}{q}}
\Big(\frac{2\pi_{pq}}{a^{\frac{1}{q}-\frac{1}{p}+1}}\Big)  ^p
\Big(1-\frac{1}{p}\Big)\Big(\frac{1}{q}-\frac{1}{p}+1 \Big)^{\frac{p-q}{q}},
\end{equation}
and  $\lambda _n(p,q)= n^p\lambda_1(p,q)$ for all $n\in {\mathbb{N}}$.
Moreover, $\sigma (p,q)=\{\lambda_n(p,q):\ n\in {\mathbb{N}}\}$,
$\lambda_n(p,q)  $ is simple for all $n\in {\mathbb{N}}$ and the
corresponding eigenspace is spanned by the function
$u_n(x)=\sin_{pq}\left(\frac{n\pi_{pq}}{a}x \right)$, $x\in (0,a)$.
\end{theorem}

\begin{proof}
Let $u$ be an eigenfunction corresponding to $\lambda_1(p,q)$ with
$u>0$ on $(0,a)$ and $\|u\|_{L^q(0,a)}=1$.
By \cite[Lemma~2.5]{ot84} it follows that
\begin{equation}\label{ident}
\frac{p-1}{p}|u'(x)|^p+\frac{\lambda_1(p,q) }{q}|u(x)|^q=\frac{p-1}{p}|u'(0)|^p,
\end{equation}
for all $x\in (0,a)$.
Recall that $u'(x) >0 $ for all $x\in (0,a/2)$ and $u'(a/2)=0$.  Thus, by setting $y=u(x)/u(a/2)$ and by means of a change if variables in integrals it follows by \eqref{ident}  that
\begin{equation}\label{ident1}
\begin{aligned}
a&=2\int_0^{a/2}dx\\
&= 2\Big(\frac{q(p-1)}{\lambda_1(p,q)p}
\Big)^{1/q}|u'(0)|^{p/q-1}\int_0^1(1-y^q)^{-1/p}dy \\
& = \frac{2}{q}\Big(\frac{q(p-1)}{\lambda_1(p,q)p}
 \Big)^{1/q}|u'(0)|^{p/q-1}\pi_{pq}.
\end{aligned}
\end{equation}
By integrating \eqref{ident} and recalling that
$\lambda_1(p,q)=\| u'\|^p_{L^p(0,a)} $ it follows that
\begin{equation}
\label{ident2}
\Big( \frac{p-1}{p}+\frac{1}{q} \Big)\lambda_1(p,q)
=\frac{p-1}{p}|u'(0)|^pa.
\end{equation}
By combining \eqref{ident1} and \eqref{ident2} we deduce \eqref{onedimeq}.

By \cite[Proposition~4.2, Theorem~II]{ot84}, one can easily deduce
that $\sigma (p,q)=\{  n^p\lambda_1(p,q):\ n\in {\mathbb{N}} \}$
which, combined with the argument used in
\cite[Proposition~4.6]{cue} for the case $p=q$, allows to conclude
that $ \lambda _n(p,q)= n^p\lambda_1(p,q) $. For the proof of the
last part of the statement we refer to \cite{draman}.
\end{proof}


\subsection*{Acknowledgements}
The authors are thankful to Professors B. Kawohl and
P. Lindqvist for useful discussions and references.
Detailed proofs of most of the results presented in this paper
were discussed by G. Franzina in a part of his dissertation~\cite{fra}
under the guidance of P.D. Lamberti.


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\end{document}