N. \end{gathered} \end{equation} As usual $V^{\pm} (x) =\max\{\pm V(x),0\}$. We are interested in positive eigengenvalues. The goal of this paper is the study of the main properties (isolation, simplicity) of the {\it least positive eigenvalue}, \begin{equation}\label{e:lambda1} \lambda_1 \eqdef \inf \Big\{\int_{\Omega}|\nabla u|^p \, dx \, : \, u\in W_0^{1,p}(\Omega) \hbox{ and } \int_{\Omega}V \,| u|^p \, dx =1\Big\}. \end{equation} We prove that $\lambda_1$ is associated to a $C_{\text loc}^{\alpha} (\Omega)$ eigenfunction which is positive in $\Omega$ and unique (up to a multiplicative constant). Moreover $\lambda_1$ is the unique positive eigenvalue associated to a nonnegative eigenfunction. These properties are well known in the case of bounded weights (see \cite{An} for indefinite weigths and \cite{Lin} for the case $V\equiv 1$). For non-negative weights satisfying \eqref{e:condV} see \cite{Ag-Pe, Dr}, and for indefinite weights with different integrability conditions see \cite{Al-Hu, Sz-Wi}. Of course the main difficulty to prove the different properties of $\lambda_1$ is the lack of regularity of the eigenfunctions. The results (as far as we know) concerning the regularity of weak solutions to degenerate elliptic equations are proved by \cite{Gu-Ve, La-Ur, Se}. These authors prove, for a class of degenerate quasilinear problems more general that the one considered here, that solutions of \eqref{e:EPV} are essentially bounded in $\Omega$ and at least of class $C_{\text loc}^{\alpha} (\Omega)$ for some $0<\alpha <1$. In the case of a bounded weight one can prove better results. In fact the results of \cite{To1},\cite{To2} and \cite{Di} imply that the solutions of problem \eqref{e:EPV} for $V$ bounded are at least of class $C_{\text loc}^{1,\alpha}$ (see Remark~\ref{r:regu}). This lack of regularity can be a handicap if one wants to use for instance ``Diaz-Saa's inequality'', which is a classical tool to prove the simplicity, or the ``strong maximum principle'' of Vazquez, a property which is used repeatedly in this context. We will show in this paper how to deal with this lack of regularity by using for instance ``Picone's identity'' instead of Diaz-Saa's inequality and ``Harnack's inequality'' instead of Vazquez's results. This paper is organized as follows. In section 2 we recall some results about the existence of sequences of eigenvalues for problem \eqref{e:EPV}. We also recall some regularity results that we will use later. In section 3 we give some basic properties of $\lambda_1$ and we study the sign of the eigenfunctions. In section 4 we study simplicity, isolation and monotonicity properties of $\lambda_1$. We conclude this work in section 5 where we comment on some new results from \cite{Ar-Ca-Cu-Go1, Dr-Ro} and \cite{An-Ts} on the second positive eigenvalue. This work is mainly motivated by the study of asymmetric elliptic problems with weights done in \cite{Ar-Ca-Cu-Go1}. Some of the results proved here were announced in that paper. \section{Preliminaries } Throughout this paper $\Omega$ will be a bounded domain of $\mathbb{R}^N$ and we will always assume that condition~\eqref{e:condV} is satisfied. $W_0^{1,p}(\Omega)$ will denote the usual Sobolev space with norm $||u||=(\int_{\Omega}|\nabla u|^p dx )^{1/p}$. We will write $||\cdot ||_p$ for the $L^p-$norm. $\langle\cdot, \cdot \rangle$ will denote the duality product between $W_0^{1,p} (\Omega) $ and its dual $W^{-1,p'} (\Omega) $. We will write $Y=L^{s'p}(\Omega)$ if $1

N$. The Lebesgue norm of $Y$ (or the infinity norm in the case $Y=C (\Omega)$ ) will be denoted by $||\cdot ||_Y$. Notice that hypothesis (1.2) on $s$ implies that the Sobolev imbedding $i:W_0^{1,p}(\Omega)\hookrightarrow Y$ is compact. We will also denote $p'=\frac{p}{p-1}$ the H\"{o}lder conjugate exponent of $p$ and $p^*$ the critical exponent, that is $p^* = \infty $ if $p\geq N$ and $p^* =\frac{Np}{N-p}$ if $1

1$ it is proved for instance in \cite{Cu2} that $\lambda_k =\mu_k$ for all $k\geq 1$ but this last equality remains an open question when $N>1$. \end{rem} \vspace{4mm} We conclude this section recalling some results about the regularity and boundedness of the eigenfunctions of \eqref{e:EPV}. The first part of the next proposition is proved in \cite[Propositions 1.2 and 1.3]{Gu-Ve} (see also in \cite[Th\'eor\`emes 7.1-7.2, pg.262]{La-Ur}) The second part can be found in \cite[Theorem 8]{Se}. \begin{prop}\cite{Gu-Ve, Se} \label{p:regu} Let $u\in W_0^{1,p}(\Omega)\setminus\{0\}$ be an eigenfunction associated to $\lambda$. Then {\bf (i)} $u\in L^{\infty}(\Omega)$ and {\bf (ii)} $u$ is locally H\"{o}lder continuous, that is, there exists $\alpha =\alpha (p,N, ||\lambda V||_s ) \in \, ]0,1[$ s.t. for any subdomain $\Omega' \subset \Omega$ there exist $C=C(p,N,||\lambda V||_s , dist (\Omega', \partial\Omega))$ such that $$|u(x)-u(y)|\leq C ||u||_{\infty}|x-y|^{\alpha}, \; \; \forall \,x,y\in \Omega' .$$ \end{prop} \vspace{2mm} \begin{rem}\rm\label{r:regu} Under the hypothesis (1.2) on $V$ we can not assure that the solutions of (1.1) are of class $C_{\text loc}^{1,\alpha}$. The $C_{\text loc}^{1,\alpha}$ regularity proved by \cite{To1, To2, Di} could be applied here provided the weight $V$ is either bounded or belongs to some $ L^{r}(\Omega)$ with $r>Np'$. \end{rem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Sign properties of the eigenvalues} \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{prop}\label{p:lambda1} The infimum $\lambda_1$ in \eqref{e:lambda1} is achieved at some $u\in {\mathcal M}$, $\lambda_1 >0$ and $\lambda_1$ is the least positive eigenvalue of problem \eqref{e:EPV}. Moreover $\lambda_1 =\Phi(u)$ for some $u\in {\mathcal M}$ if and only if $u$ is an eigenfunction associated to $\lambda_1$. \end{prop} {\it Proof.} The proof is an straight application of Theorem 1.2 of \cite{St} and the Lagrange's multiplier rule. \qed \vspace{2mm} The following ``strong maximum principle'' holds : \begin{prop} \label{p:SMP} If $u\in W_0^{1,p}(\Omega)$ is a non-negative weak solution of \eqref{e:EPV} then either $u\equiv 0$ or $u(x)>0$ for all $x\in \Omega$. \end{prop} {\it Proof.} The result is a direct consequence of the following Harnack's inequality for nonnegative solutions of \eqref{e:EPV}. We referee here to \cite[Theorems 5,6 and 9, pg.264-270]{Se}. ``{\em Let $u\in W^{1,p}(\Omega) $ be a non-negative weak solution of \eqref{e:EPV} and assume that $B(x_0 ,3r)\subset \Omega$ for some $r>0$ and $x_0 \in \Omega$. Then for some $C=C(p,N,r,\lambda V,\Omega )$, $$\max_{\overline{B}(x_0 , r)} u \leq C \min_{\overline{B}(x_0 ,r)} u. '' $$ } \qed \vspace{2mm} We also have the following result : \begin{prop}\label{p:positive} The eigenfunctions associated to $\lambda_1$ are either positive or negative in $\Omega$. \end{prop} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} Let $u\in {\mathcal M} $ be an eigenfunction associated to $\lambda_1$. Then $u$ achieves the infimum in \eqref{e:lambda1}. Since $||\nabla |u|~||_p = ||\nabla u ||_p $ and $|u|\in {\mathcal M}$ it follows that $|u|$ achieves also the infimum in \eqref{e:lambda1} and therefore, from Proposition \ref{p:lambda1}, $|u|$ is an eigenfunction for $\lambda_1$. By Proposition~\ref{p:SMP} we conclude that $|u(x)|>0 \; \; \forall x\in \Omega$ and consequently $u$ is either positive or negative in $\Omega.$\qed \vspace{2mm} In what follows we will use the so-called ``Picone's identity'' proved in \cite{Al-Hu}. We recall it here for completness. \begin{thm}\label{t:picone}\cite{Al-Hu} Let $v>0, u\geq 0$ be two continuous functions in $\Omega$ differentiable a.e. Denote $$\begin{array}{l} L(u,v)=|\nabla u|^p +(p-1)\frac{u^p}{v^p}|\nabla v|^p - p\frac{u^{p-1}}{v^{p-1}}|\nabla v|^{p-2}\nabla v \nabla u\, ,\\ \\ R(u,v)=|\nabla u|^p -|\nabla v|^{p-2} \nabla (\frac{u^p}{v^{p-1}})\nabla v . \end{array} $$ Then {\bf (i)} $L(u,v)=R(u,v)$, {\bf (ii)} $L(u,v)\geq 0$ a.e. and {\bf (iii)} $L(u,v)=0$ a.e. in $\Omega$ if and only if $u=kv$ for some $k\in \mathbb{R}$. \end{thm} \vspace{2mm} In the next theorem we give an estimate of the measure of the nodal domains of an eigenfunction $u$. We recall that a {\it nodal domain of $u$} is a connected component of $\Omega \setminus\{x\in \Omega \colon u(x)=0\}$. The same result for positive weights can be found in \cite{Ag-Pe}. Our exponent $\gamma$ is slightly different. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\vspace{2mm} \begin{thm} \label{t:nodal} Any eigenfunction $v$ associated to a positive eigenvalue $0<\lambda \not=\lambda_1$ changes sign. Moreover if ${\mathcal N}$ is a nodal domain of $v$ then \begin{equation}\label{e:omegapm} |{\mathcal N} |\geq (C\lambda ||V||_s )^{-\gamma} \end{equation} where $\gamma= \frac{sN}{sp-N} $ and C is some constant depending only on $N$ and $p$ if $p\not=N$ and on $N$ and $s'$ if $p=N$. \end{thm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} Assume by contradiction that $v\geq 0$, the case $v\leq 0$ being completely analogous. By Proposition~\ref{p:SMP} it follows that $v(x)>0$ for all $x\in \Omega$. Let $\varphi >0$ be an eigenfunction associated to $\lambda_1$. For any $\epsilon>0$ we apply Picone's identity to the pair $\varphi ,v+\epsilon $. We have \begin{equation}\label{e:aa} \begin{array}{ll} 0\leq \int_{\Omega}L(\varphi, v+\epsilon)\, dx = \int_{\Omega}R(\varphi, v+\epsilon)\, dx =& \\ &\\ \lambda_1 \int_{\Omega} V \, \varphi ^p \, dx -\int_{\Omega} |\nabla v|^{p-2}\nabla (\frac{\varphi^p }{( v+\epsilon)^{p-1}})\nabla v\, dx .& \end{array} \end{equation} Notice that $\frac{\varphi^p }{( v+\epsilon)^{p-1}}$ belongs to $W_0^{1,p} (\Omega)$ and then it is admissible in the weak formulation of $-\Delta_p v=\lambda V |v|^{p-2}v$. Then if follows from \eqref{e:aa} that $$0\leq \int_{\Omega} V \, \varphi ^p (\lambda_1-\lambda \frac{v^{p-1} }{( v+\epsilon)^{p-1}})\, dx .$$ Letting $\epsilon \rightarrow 0$ it comes that $0\leq \int_{\Omega} V \, \varphi ^p (\lambda_1-\lambda )\, dx $ which is imposible because $\lambda >\lambda_1$ and $\int_{\Omega}V \, \varphi ^p \, dx >0$. Hence we have proved that $v$ must change sign. Next we prove estimate \eqref{e:omegapm}. Assume that $v>0$ in ${\mathcal N}$, the case $v<0$ being completely analogous. We observe that because $v\in W_0^{1,p}(\Omega) \cap C(\Omega)$ then $v_{|_{\mathcal N}} \in W_0^{1,p} ({\mathcal N})$. Hence the function $w$ defined as $w(x)=v(x)$ if $x\in {\mathcal N}$ and $w(x)=0$ if $x\in \Omega\setminus{\mathcal N}$ belongs to $W_0^{1,p}(\Omega)$. Let us start with the case $1

N$ we have on the one hand
$$\int_{\mathcal{N}} |\nabla v|^p \, dx \leq
\lambda ||V||_1 ||v||_{\infty ,{\mathcal N}}^p \, ,$$
and on the other hand, from Morrey's lemma,
$$C||v||_{\infty, {\mathcal N}} \leq |{\mathcal N} |^{-1/p + 1/N }
|| \nabla v||_{p, {\mathcal N}} $$ for some $C=C(N,p)$.
Then inequality \eqref{e:omegapm} holds.\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{2mm}
\begin{coro}
Each eigenfunction has a finite number of nodal domains.
\end{coro}
{\it Proof.} Let ${\mathcal N}_j $ be a nodal domain of an eigenfunction
associated to some positive eigenvalue $\lambda$. It follows from (3.1) that
%$$|\mathcal{N}_j |\geq (C\lambda ||V||_s )^{-\gamma}$$ and
$$|\Omega |\geq \sum_j |{\mathcal N}_j |\geq (C\lambda ||V||_s )^{-\gamma}\sum_j 1$$
and the claim follows.\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{On the first eigenvalue}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We have proved in the previous section that the eigenfunctions associated to $\lambda_1$ have definite sign in $\Omega$.
We are now going to prove that
this property implies, through Picone's identity, that $\lambda_1$ is simple.
We will also prove that $\lambda_1$ is isolated in the spectrum of \eqref{e:EPV}
as a consequence of Theorem~\ref{t:nodal}.
Finally we give a result on the strict monotoniticy of $\lambda_1$ with respect to both the domain and
the weight.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}\label{p:simple}
$\lambda_1$ is simple in the sense that the
eigenfunctions associated to it
are merely a constant multiple of each other.
\end{prop}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¸
{\it Proof.} We proceed as in the first part of
the proof of Theorem~\ref{t:nodal}. Let $u,v$ be two eigenfunctions associated to $\lambda_1$. We can assume without
restriction that
$u$ and $v$ are positive in $\Omega$. Let $\epsilon >0$. From Picone's identity we have
$$\begin{array}{ll}
0\leq \int_{\Omega}L(u, v+\epsilon)\, dx =
\int_{\Omega}R(u, v+\epsilon)\, dx =&\\
&\\
\lambda_1 \int_{\Omega} V \, u ^p
\, dx -\int_{\Omega} |\nabla v|^{p-2}\nabla (\frac{u^p }{( v+\epsilon)^{p-1}}) \nabla v\, dx .&
\end{array}
$$
The function $\frac{u^p }{( v+\epsilon)^{p-1}}$ belongs to
$W_0^{1,p} (\Omega)$ and then it is admissible
for the weak formulation of $-\Delta_p v=\lambda_1 V |v|^{p-2}v$. It follows then from the previous equation
that
$$0\leq \int_{\Omega}L(u,v+\epsilon)\, dx =\lambda_1 \int_{\Omega} V \, u^p (1-
\frac{v^{p-1} }{( v+\epsilon)^{p-1}})\, dx .$$
Letting $\epsilon \rightarrow 0$ it follows that
$L(u,v)=0$. Then, by Theorem~\ref{t:picone}, there exists $k\in \mathbb{R}$ such that $u=kv$.\qed
\vspace{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}\label{p:isolation}
$\lambda_1$ is isolated, that is, there exists $\delta >0$ such
that in the interval $(\lambda_1 ,\lambda_1 +\delta)$ there are no other
eigenvalues of \eqref{e:EPV}.
\end{prop}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¸
{\it Proof.} The result follows easily from the estimate
\eqref{e:omegapm}. Assume by contradiction that
there exists a sequence of eigenvalues of \eqref{e:EPV}
$\lambda_n $ with
$0<\lambda_n \searrow \lambda_1$. Let $u_n$ be an eigenfunction associated
to $\lambda_n$.
Since $0<\int_\Omega |\nabla u_n |^p \, dx =\lambda_n
\int_\Omega V\, |u_n|^p \, dx$ we can define
$$v_n := \frac{u_n }{(\int_\Omega V\, |u_n|^p \, dx)^{1/p}}.$$
$v_n $ is bounded in $W_0^{1,p}(\Omega)$
so there exist a subsequence (still denoted $v_n $) and
$v\in W_0^{1,p}(\Omega)$ such that $v_n \rightarrow v$ in $Y$
and weakly in $W_0^{1,p}(\Omega)$.
Moreover $\int_\Omega V\, |v|^p \, dx =1$.
On the other hand
$$\int_\Omega |\nabla v|^p \, dx \leq
\displaystyle{\liminf_{n\rightarrow \infty}
\int_\Omega |\nabla v_n |^p \, dx=\lambda_1}$$
and then
$\int_\Omega |\nabla v|^p \, dx=\lambda_1 $ by \eqref{e:lambda1}.
Using Proposition~\ref{p:lambda1} we conclude that
$v$ is an eigenfunction associated to
$\lambda_1$. It follows then from Proposition~\ref{p:positive}
that either $v>0$ or $v<0$. In the case $v>0$ (the other case is analogous) we conclude from the convergence in measure
of the sequence $v_n$ towards $v$ that
\begin{equation}
\label{e:nega}
|\Omega_n^-|\rightarrow 0
\end{equation}
where $\Omega_n^-$ denotes the negative
set of $u_n$. But \eqref{e:nega} clearly
contradicts estimate \eqref{e:omegapm}.
\qed
\vspace{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the sequel we will denote the least positive eigenvalue
of \eqref{e:EPV} by $\lambda_1 (V)$ or
$\lambda_1 (\Omega)$ when comparing $\lambda_1$ for different weights or domains.
We will always assume that condition \eqref{e:condV} is satisfied for
the weights appearing in the claims.
%\vspace{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
Let $V_1 ,V_2$ be two weights and assume that
$V_1 \leq V_2 \; a.e. $ and $|\{x\in \Omega\, :\,
V_1 (x) \lambda_{1}\}.
$$
There exist several variational characterizations of
$\underline{\lambda}_2 $ through mimimax formulae. For instance in \cite{An-Ts} it is proved
that $\underline{\lambda}_2=\lambda_2$ when $V\in L^{\infty}(\Omega)$
and in
\cite{Dr-Ro} that
$\underline{\lambda}_2=\mu_2 $ when $V\equiv 1$.
A further variational characterization has been given by
\cite{Cu-De-Go} in the case $V\equiv 1$. This last characterization has been generalized recently by
\cite{Ar-Ca-Cu-Go1} to weights as those considered here. The following result
was obtained as a consequence of the construction of the first
curve of the Fu\v cik spectrum in
\cite{Ar-Ca-Cu-Go1}.
%
\begin{thm}\cite{Ar-Ca-Cu-Go1}\label{t:lambda2}
Assume that $V$ satisfies \eqref{e:condV}. Then
$$\underline{\lambda}_2 =
\inf_{h\in {\mathcal F}}\max_{u\in h([-1,1]}\int_{\Omega}
|\nabla u|^p \, dx
$$where
${\mathcal F} \eqdef \{ \gamma \in C ([-1, +1],{\mathcal M}) \,:\, \gamma(\pm 1)=\pm\varphi_{1}
\}
$
and $\varphi_1 \in {\mathcal M}$ is the positive eigenfunction associated
to $\lambda_1$.
\end{thm}
\vspace{2mm}
\begin{rem}\rm
Notice that ${\mathcal F} \subset \Delta_2 \subset \Gamma_2$. The variational characterization of Theorem~\ref{t:lambda2} is slighty better than the one of \cite{An-Ts} and \cite{Dr-Ro} because it suffices to minimize along a smaller family of subsets of ${\mathcal M}$
to get the same value.
\end{rem}
\vspace{2mm}
A straight consequence of this result is the following :
%\vspace{2mm}
\begin{coro}\label{c:lambda2}
$\displaystyle{\underline{\lambda}_2=\lambda_2 =\mu_2 =\inf_{h\in {\mathcal F}}\max_{u\in h([-1,1]}\int_{\Omega}
|\nabla u|^p \, dx
}$.
\end{coro}
%%%%%%%%%%%%%%%¸
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\end{document}