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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 41, 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-2010/41\hfil Multiplicity results]
{Multiplicity results for {\lowercase {\em p}}-sublinear
{\lowercase {\em p}}-Laplacian problems involving indefinite
eigenvalue problems via Morse theory}

\author[K. Perera, R. P. Agarwal, D. O'Regan\hfil EJDE-2010/41\hfilneg]
{Kanishka Perera, Ravi P. Agarwal, Donal O'Regan}

\address{Kanishka Perera \newline
Department of Mathematical Sciences\\
Florida Institute of Technology\\
Melbourne, FL 32901, USA}
\email{kperera@fit.edu}

\address{Ravi P. Agarwal \newline
Department of Mathematical Sciences\\
Florida Institute of Technology\\
Melbourne, FL 32901, USA}
\email{agarwal@fit.edu}

\address{Donal O'Regan \newline
Department of Mathematics\\
National University of Ireland\\
Galway, Ireland}
\email{donal.oregan@nuigalway.ie}

\thanks{Submitted December 2, 2009. Published March 19, 2010.}
\subjclass[2000]{35J20, 47J10, 58E05}
\keywords{$p$-Laplacian problems;
 $p$-sublinear; multiplicity results; \hfill\break\indent
 indefinite eigenvalue problems;
   Morse theory}

\begin{abstract}
 We establish some multiplicity results for a class of
 $p$-sublinear $p$-Laplacian problems involving indefinite
 eigenvalue problems using Morse theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

The purpose of this note is to establish some multiplicity
results for a class of $p$-sublinear $p$-Laplacian problems
involving indefinite eigenvalue problems using Morse theory.

As motivation, we begin by recalling a well-known result
for the semilinear elliptic boundary value problem
\begin{equation} \label{1.1}
\begin{gathered}
- \Delta u  = f(x,u) \quad \text{in } \Omega\\
u  = 0 \quad \text{on } \partial\Omega
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n \ge 1$,
$f$ is a Carath\'{e}odory function on $\Omega \times \mathbb{R}$
satisfying the sublinear growth condition
\begin{equation} \label{1.2}
|f(x,t)| \le C\, (|t|^{r-1} + 1)
\end{equation}
for some $r \in (1,2)$, and $C$ denotes a generic positive constant. Weak solutions of \eqref{1.1} coincide with the critical points of the $C^1$-functional
\[
\Phi(u) = \int_\Omega \frac{1}{2}\, |\nabla u|^2 - F(x,u), \quad
u \in H^1_0(\Omega)
\]
where $F(x,t) = \int_0^s f(x,s)\, ds$ is the primitive of $f$.
By \eqref{1.2}, $\Phi$ is bounded from below and satisfies
the (PS) condition.

Assume that
\begin{equation} \label{1.3}
\lim_{t \to 0}\, \frac{f(x,t)}{t} = \lambda, \quad \text{uniformly a.e.},
\end{equation}
which implies $f(x,0) = 0$ a.e.\ and hence \eqref{1.1}
has the trivial solution $u(x) \equiv 0$.
Let $\lambda_1 < \lambda_2 \le \dots$ denote the Dirichlet
eigenvalues of the negative Laplacian on $\Omega$.
If $\lambda > \lambda_1$ and is not an eigenvalue, then  \eqref{1.1}
has at least two nontrivial solutions. Indeed,
if $\lambda_k < \lambda < \lambda_{k+1}$, then the (cohomological)
critical groups of $\Phi$ at zero are given by
\[
C^q(\Phi,0) \approx \delta_{qk}\, \mathcal{G}
\]
where $\mathcal{G}$ is the coefficient group and $\delta_{\cdot,\cdot}$
denotes the Kronecker delta (see, e.g., Chang \cite{MR94e:58023}
or Mawhin and Willem \cite{MR90e:58016}), so $\Phi$ has
two nontrivial critical points by the following
``three critical points theorem'' of Chang \cite{MR622618}
and Liu and Li \cite{MR802575}.

\begin{proposition} \label{prop1.1}
Let $\Phi$ be a $C^1$-functional defined on a Banach space.
If $\Phi$ is bounded from below, satisfies {\em (PS)}, and
$C^k(\Phi,0) \ne 0$ for some $k \ge 1$, then $\Phi$ has
two nontrivial critical points.
\end{proposition}

\begin{remark} \rm
Li and Willem \cite{MR96a:58045} used a local linking to
obtain a similar result when $\lambda$ is an eigenvalue
and $f$ satisfies a suitable sign condition near zero.
\end{remark}

The above result can be extended to the corresponding
$p$-sublinear  $p$-Laplacian problem
\begin{equation} \label{1.7}
\begin{gathered}
- \Delta_p\, u  = f(x,u) \quad \text{in } \Omega\\
u  = 0 \quad \text{on } \partial\Omega
\end{gathered}
\end{equation}
where $\Delta_p\, u = \mathop{\rm div} \big(|\nabla u|^{p-2}\, \nabla u\big)$
is the $p$-Laplacian of $u$, $p \in (1,\infty)$, and $f$
now satisfies \eqref{1.2} with $r \in (1,p)$.
Then the associated variational functional
\[
\Phi(u) = \int_\Omega \frac{1}{p}\, |\nabla u|^p - F(x,u), \quad
u \in W^{1,p}_0(\Omega)
\]
is bounded from below and satisfies (PS).
Assume that
\begin{equation} \label{1.8}
\lim_{t \to 0}\, \frac{f(x,t)}{|t|^{p-2}\, t} = \lambda, \quad
\text{uniformly a.e.}
\end{equation}
The associated quasilinear eigenvalue problem
\begin{gather*}
- \Delta_p\, u  = \lambda\, |u|^{p-2}\, u \quad \text{in } \Omega\\
u  = 0 \quad \text{on } \partial\Omega
\end{gather*}
is far more complicated. It is known that the first eigenvalue
$\lambda_1$ is positive, simple, and has an associated
eigenfunction $\varphi_1$ that is positive in $\Omega$
(see Anane \cite{MR89e:35124} and Lindqvist
\cite{MR90h:35088, MR1139483}). Moreover, $\lambda_1$
is isolated in the spectrum $\sigma(- \Delta_p)$,
so the second eigenvalue
$\lambda_2 = \inf\, \sigma(- \Delta_p) \cap (\lambda_1,\infty)$
is well-defined. In the ODE case $n = 1$, where $\Omega$ is
an interval, the spectrum consists of a sequence of simple
eigenvalues $\lambda_k \nearrow \infty$, and the eigenfunction
$\varphi_k$ associated with $\lambda_k$ has exactly $k-1$ interior
zeroes (see, e.g., Dr{\'a}bek \cite{MR94e:47084}).
In the PDE case $n \ge 2$, an increasing and unbounded sequence
of eigenvalues can be constructed using a standard minimax
scheme involving the Krasnoselskii's genus, but it is not known
whether this gives a complete list of the eigenvalues.

Perera \cite{MR1998432} used a minimax scheme involving the
$\mathbb{Z}_2$-cohomological index of Fadell and Rabinowitz \cite{MR57:17677}
to construct a new sequence of eigenvalues
 $\lambda_k \nearrow \infty$ such that if
$\lambda_k < \lambda < \lambda_{k+1}$ in \eqref{1.8}, then
\[
C^k(\Phi,0) \ne 0
\]
and hence $\Phi$ has two nontrivial critical points by Proposition
\ref{prop1.1}. Thus, problem \eqref{1.7} has at least two nontrivial
solutions when $\lambda > \lambda_1$ is not an eigenvalue from this
particular sequence.

Note that \eqref{1.8} implies $tf(x,t) > 0$ for $t \ne 0$ near zero
when $\lambda > 0$. Naturally we may ask whether these results hold
without such a sign condition. More specifically, can we
replace \eqref{1.8} with
\begin{equation} \label{1.91}
\lim_{t \to 0}\, \frac{f(x,t)}{|t|^{p-2}\, t} = \lambda\, V(x),
\quad \text{uniformly a.e.}
\end{equation}
and let $V$ change sign?

This leads us to the indefinite eigenvalue problem
\begin{equation} \label{1.92}
\begin{gathered}
- \Delta_p\, u  = \lambda\, V(x)\, |u|^{p-2}\, u \quad \text{in } \Omega\\
u  = 0 \quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}
We assume that the weight function $V \in L^s(\Omega)$ for some
\begin{equation} \label{1.94}
s\, \begin{cases}
> n/p, & p \le n\\[5pt]
= 1, & p > n.
\end{cases}
\end{equation}
Then the smallest positive and largest negative eigenvalues
of \eqref{1.92} are given by
\[
\lambda^+_1 = \inf_{\substack{u \in W^{1,p}_0(\Omega)\\[1pt]
\int_\Omega V(x)\, |u|^p > 0}}\,
\frac{\int_\Omega |\nabla u|^p}{\int_\Omega V(x)\, |u|^p}, \quad
 \lambda^-_1 = \sup_{\substack{u \in W^{1,p}_0(\Omega)\\[1pt]
\int_\Omega V(x)\, |u|^p < 0}}\,
\frac{\int_\Omega |\nabla u|^p}{\int_\Omega V(x)\, |u|^p},
\]
respectively. Noting that \eqref{1.91} implies
\begin{equation} \label{1.93}
F(x,t) = \frac{\lambda}{p}\, V(x)\, |t|^p + o(|t|^p)
\quad\text{as $t \to 0$, uniformly a.e.},
\end{equation}
we shall prove the following result.

\begin{theorem} \label{thm1.2}
Assume \eqref{1.2} with $r \in (1,p)$, $V \in L^s(\Omega)$
with $s$ satisfying \eqref{1.94}, and \eqref{1.93}.
If $\lambda \notin (\lambda^-_1,\lambda^+_1)$ and is not an eigenvalue
 of \eqref{1.92}, then problem \eqref{1.7} has at least two nontrivial
solutions.
\end{theorem}

Since $\lambda^-_1 = - \infty$ when $V \ge 0$ a.e.\ and
$\lambda^+_1 = + \infty$ when $V \le 0$ a.e., this theorem applies
in all possible cases:
\begin{itemize}
\item[(i)] $V$ changes sign:
$\dots < \lambda^-_1 < 0 < \lambda^+_1 < \dots$,
\item[(ii)] $V \ge 0$ a.e. and $\not\equiv 0$:
$- \infty = \lambda^-_1 < 0 < \lambda^+_1 < \dots$,
\item[(iii)] $V \le 0$ a.e. and $\not\equiv 0$:
$\dots < \lambda^-_1 < 0 < \lambda^+_1 = + \infty$,
\item[(iv)] $V \equiv 0$:
$- \infty = \lambda^-_1 < \lambda^+_1 = + \infty$
(in this case the theorem is vacuously true).
\end{itemize}

Our proof will be based on an abstract framework for indefinite
eigenvalue problems introduced in Perera, Agarwal,
and O'Regan \cite{PeAgO'R}, which we will recall in the next section.

\section{Preliminaries}

In this section we recall an abstract framework for indefinite
eigenvalue problems introduced in Perera, Agarwal, and
O'Regan \cite{PeAgO'R}.

Let $(W,\|\cdot\|)$ be a real reflexive Banach space with the
dual $(W^\ast,\|\cdot\|)$ and the duality pairing
$(\cdot,\cdot)$. We consider the nonlinear eigenvalue problem
\begin{equation} \label{4.0.1}
A_p\, u = \lambda\, B_p\, u
\end{equation}
in $W^\ast$, where $A_p \in C(W,W^\ast)$ is
\begin{itemize}
\item[(A1)] $(p-1)$-homogeneous and odd for some $p \in (1,\infty)$:
    \[
    A_p(\alpha u) = |\alpha|^{p-2}\, \alpha\, A_p\, u \quad
\forall u \in W,\, \alpha \in \mathbb{R},
    \]
\item[(A2)] uniformly positive: $\exists\, c_0 > 0$ such that
    \[
    (A_p\, u,u) \ge c_0\, \|u\|^p \quad \forall u \in W,
    \]
\item[(A3)] a potential operator: there is a functional
$I_p \in C^1(W,\mathbb{R})$, called a potential for $A_p$, such that
    \[
    I_p'(u) = A_p\, u \quad \forall u \in W,
    \]
\item[(A4)] of type $(S)$: for any sequence $\{u_j\} \subset W$,
    \[
    u_j \rightharpoonup u, \quad (A_p\, u_j,u_j - u)
 \to 0 \implies u_j \to u,
    \]
\end{itemize}
and $B_p \in C(W,W^\ast)$ is
\begin{itemize}
\item[(B1)] $(p-1)$-homogeneous and odd,
\item[(B2)] a compact potential operator.
\end{itemize}

The following proposition is often useful for verifying (A4).

\begin{proposition}[{\cite[Proposition 1.0.3]{PeAgO'R}}]
\label{prop2.1}
If $W$ is uniformly convex and
\[
(A_p\, u,v) \le r\, \|u\|^{p-1}\, \|v\|, \quad (A_p\, u,u)
= r\, \|u\|^p \quad \forall u, v \in W
\]
for some $r > 0$, then (A4) holds.
\end{proposition}

By \cite[Proposition 1.0.2]{PeAgO'R}, the potentials $I_p$ and
$J_p$ of $A_p$ and $B_p$ satisfying $I_p(0) = 0 = J_p(0)$ are given by
\[
I_p(u) = \frac{1}{p}\, (A_p\, u,u), \quad
J_p(u) = \frac{1}{p}\, (B_p\, u,u),
\]
respectively, and are $p$-homogeneous and even. Let
\[
\mathcal{M} = \{u \in W : I_p(u) = 1\}, \quad
\mathcal{M}^\pm = \{u \in \mathcal{M} : J_p(u) \gtrless 0\}.
\]
Then $\mathcal{M} \subset W \setminus \{0\}$ is a bounded complete
symmetric $C^1$-Finsler manifold radially homeomorphic to the
unit sphere in $W$, $\mathcal{M}^\pm$ are symmetric open submanifolds
of $\mathcal{M}$, and the positive (resp. negative) eigenvalues
of \eqref{4.0.1} coincide with the critical values of the
even $C^1$-functionals
\[
\Psi^\pm(u) = \frac{1}{J_p(u)}, \quad u \in \mathcal{M}^\pm
\]
(see \cite[Sections 9.1 and 9.2]{PeAgO'R}).

Denote by $\mathcal{F}^\pm$ the classes of symmetric subsets of
$\mathcal{M}^\pm$ and by $i(M)$ the Fadell-Rabinowitz cohomological
index of $M \in \mathcal{F}^\pm$. Then
\begin{gather*}
\lambda^+_k  := \inf_{\substack{M \in \mathcal{F}^+\\[1.5pt]
i(M) \ge k}}\, \sup_{u \in M}\, \Psi^+(u), \quad
 1 \le k \le i(\mathcal{M}^+),\\
\lambda^-_k  := \sup_{\substack{M \in \mathcal{F}^-\\
i(M) \ge k}}\, \inf_{u \in M}\, \Psi^-(u), \quad 1 \le k \le i(\mathcal{M}^-)
\end{gather*}
define nondecreasing (resp. nonincreasing) sequences of positive
(resp. negative) eigenvalues of \eqref{4.0.1} that are unbounded
when $i(\mathcal{M}^\pm) = \infty$
(see \cite[Theorems 9.1.2 and 9.2.1]{PeAgO'R}).
When $\mathcal{M}^\pm = \emptyset$, we set $\lambda^\pm_1 = \pm \infty$
for convenience.

Now we consider the operator equation
\begin{equation} \label{2.6}
A_p\, u = F'(u)
\end{equation}
where $F \in C^1(W,\mathbb{R})$ with $F'$ compact, whose solutions coincide with the critical points of the functional
\[
\Phi(u) = I_p(u) - F(u), \quad u \in W.
\]
The following proposition is useful for verifying
the (PS) condition for $\Phi$.

\begin{proposition}[{\cite[Lemma 3.1.3]{PeAgO'R}}] \label{prop2.2}
Every bounded {\em (PS)} sequence of $\Phi$ has a convergent subsequence.
\end{proposition}

Suppose that $u = 0$ is a solution of \eqref{2.6} and the asymptotic
behavior of $F$ near zero is given by
\begin{equation} \label{2.21}
F(u) = \lambda\, J_p(u) + o(\|u\|^p) \quad\text{as } u \to 0.
\end{equation}

\begin{proposition}[{\cite[Proposition 9.4.1]{PeAgO'R}}] \label{prop2.3}
Assume (A1) - (A4), (B1), (B2), and \eqref{2.21} hold, $F'$ is compact,
and zero is an isolated critical point of $\Phi$.
\begin{itemize}
\item[(i)] If $\lambda^-_1 < \lambda < \lambda^+_1$,
then $C^q(\Phi,0) \approx \delta_{q0}\, \mathbb{Z}_2$.
\item[(ii)] If $\lambda^-_{k+1} < \lambda < \lambda^-_k$ or
$\lambda^+_k < \lambda < \lambda^+_{k+1}$, then $C^k(\Phi,0) \ne 0$.
\end{itemize}
\end{proposition}

\section{Proof of Theorem \ref{thm1.2}}

First let us verify that our problem fits into the abstract
framework of the previous section. Let $W = W^{1,p}_0(\Omega)$,
\[
(A_p\, u,v) = \int_\Omega |\nabla u|^{p-2}\,
\nabla u \cdot \nabla v, \quad
(B_p\, u,v) = \int_\Omega V(x)\, |u|^{p-2}\, uv,
\]
and
\[
F(u) = \int_\Omega F(x,u).
\]
Then (A1) and (B1) are clear, $(A_p\, u,u) = \|u\|^p$
in (A2), and (A3) and (B2) hold with
\[
I_p(u) = \frac{1}{p} \int_\Omega |\nabla u|^p, \quad
J_p(u) = \frac{1}{p} \int_\Omega V(x)\, |u|^p,
\]
respectively. By the H\"{o}lder inequality,
\[
(A_p\, u,v) \le \Big(\int_\Omega |\nabla u|^p\Big)^{1-\tfrac{1}{p}}
 \Big(\int_\Omega |\nabla v|^p\Big)^{1/p} = \|u\|^{p-1}\, \|v\|,
\]
so (A4) follows from Proposition \ref{prop2.1}.
By \eqref{1.2} and \eqref{1.93}, \eqref{2.21} also holds.

Since $\lambda \notin (\lambda^-_1,\lambda^+_1)$ and is not an
eigenvalue of \eqref{1.92}, it now follows from
Proposition \ref{prop2.3} that $C^k(\Phi,0) \ne 0$ for
some $k \ge 1$. By \eqref{1.2},
\[
|F(x,t)| \le C\, (|t|^r + 1),
\]
so by the Sobolev imbedding,
\[
\Phi(u) \ge \frac{1}{p}\, \|u\|^p - C\, (\|u\|^r + 1) \quad
\forall u \in W^{1,p}_0(\Omega).
\]
Since $p > r$, it follows that $\Phi$ is bounded from below
 and coercive. Then every (PS) sequence of $\Phi$ is bounded
and hence $\Phi$ satisfies the (PS) condition by
Proposition \ref{prop2.2}. Thus, $\Phi$ has two nontrivial
critical points by Proposition  \ref{prop1.1}.

\begin{remark} \rm
Note that it suffices to assume
$\lambda \notin (\lambda^-_1,\lambda^+_1)$ is not an eigenvalue
from the particular sequences $(\lambda^\pm_k)$.
\end{remark}

\begin{thebibliography}{00}

\bibitem{MR89e:35124}
Aomar Anane.
\newblock Simplicit\'e et isolation de la premi\`ere valeur propre du
  {$p$}-laplacien avec poids.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I Math.}, 305(16):725--728, 1987.

\bibitem{MR622618}
Kung~Ching Chang.
\newblock Solutions of asymptotically linear operator equations via {M}orse
  theory.
\newblock {\em Comm. Pure Appl. Math.}, 34(5):693--712, 1981.

\bibitem{MR94e:58023}
Kung-ching Chang.
\newblock {\em Infinite-dimensional {M}orse theory and multiple solution
  problems}, volume~6 of {\em Progress in Nonlinear Differential Equations and
  their Applications}.
\newblock Birkh\"auser Boston Inc., Boston, MA, 1993.

\bibitem{MR94e:47084}
P.~Dr{\'a}bek.
\newblock {\em Solvability and bifurcations of nonlinear equations}, volume 264
  of {\em Pitman Research Notes in Mathematics Series}.
\newblock Longman Scientific \& Technical, Harlow, 1992.

\bibitem{MR57:17677}
Edward~R. Fadell and Paul~H. Rabinowitz.
\newblock Generalized cohomological index theories for {L}ie group actions with
  an application to bifurcation questions for {H}amiltonian systems.
\newblock {\em Invent. Math.}, 45(2):139--174, 1978.

\bibitem{MR96a:58045}
Shu~Jie Li and Michel Willem.
\newblock Applications of local linking to critical point theory.
\newblock {\em J. Math. Anal. Appl.}, 189(1):6--32, 1995.

\bibitem{MR90h:35088}
Peter Lindqvist.
\newblock On the equation {${\rm div}\,(\vert \nabla u\vert \sp {p-2}\nabla
  u)+\lambda\vert u\vert \sp {p-2}u=0$}.
\newblock {\em Proc. Amer. Math. Soc.}, 109(1):157--164, 1990.

\bibitem{MR1139483}
Peter Lindqvist.
\newblock Addendum: ``{O}n the equation {${\rm div}(\vert \nabla u\vert \sp
  {p-2}\nabla u)+\lambda\vert u\vert \sp {p-2}u=0$}'' [{P}roc.\ {A}mer.\
  {M}ath.\ {S}oc.\ 109 (1990), no.\ 1, 157--164; {MR} 90h:35088].
\newblock {\em Proc. Amer. Math. Soc.}, 116(2):583--584, 1992.

\bibitem{MR802575}
Jia~Quan Liu and Shu~Jie Li.
\newblock An existence theorem for multiple critical points and its
  application.
\newblock {\em Kexue Tongbao (Chinese)}, 29(17):1025--1027, 1984.

\bibitem{MR90e:58016}
Jean Mawhin and Michel Willem.
\newblock {\em Critical point theory and {H}amiltonian systems}, volume~74 of
  {\em Applied Mathematical Sciences}.
\newblock Springer-Verlag, New York, 1989.

\bibitem{MR1998432}
Kanishka Perera.
\newblock Nontrivial critical groups in {$p$}-{L}aplacian problems via the
  {Y}ang index.
\newblock {\em Topol. Methods Nonlinear Anal.}, 21(2):301--309, 2003.

\bibitem{PeAgO'R}
Kanishka Perera, Ravi~P. Agarwal, and Donal O'Regan.
\newblock {\em Morse Theoretic Aspects of {$p$}-{L}aplacian Type Operators},
  volume 161 of {\em Mathematical Surveys and Monographs}.
\newblock American Mathematical Society, Providence, RI, 2010.

\end{thebibliography}

\end{document}