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\markboth{\hfil Multiple solutions \hfil EJDE--1995/01}%
{EJDE--1995/01\hfil Steve B. Robinson\hfil}
\begin{document}
\ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations}\newline
Vol. {\bf 1995}(1995), No. 01, pp. 1-14. Published January 26, 1995.\newline
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113 }
\vspace{\bigskipamount} \\
Multiple Solutions For Semilinear Elliptic Boundary Value
Problems At Resonance
\thanks{ {\em 1991 Mathematics Subject Classification:}
35J60.\newline\indent
{\em Key words and phrases:} Landesman-Lazer condition, Leray-Schauder
degree,\newline\indent
Palais-Smale condition, coercivity, mountain pass, Morse index,
multiple solutions.
\newline\indent
\copyright 1995 Southwest Texas State University and University of
North Texas.\newline\indent
Submitted: May 20, 1994.} }
\date{}
\author{Steve B. Robinson}
\maketitle
\begin{abstract}
In recent years several nonlinear techniques have been very successful
in proving the existence of weak solutions for semilinear elliptic boundary
value problems at resonance. One technique involves a variational approach
where solutions are characterized as saddle points for a related functional.
This argument requires that the Palais-Smale condition and some coercivity
conditions are satisfied so that the saddle point theorem of Ambrossetti
and Rabinowitz can be applied. A second technique has been to apply the
topological ideas of Leray-Schauder degree. This argument typically creates
a homotopy with a uniquely solvable linear problem at one end and the
nonlinear problem at the other, and then an a priori bound is established so
that the homotopy invariance of Leray-Schauder degree can be applied.
In this paper we prove that both techniques are applicable in a wide variety
of cases, and that having both techniques at our disposal gives more detailed
information about solution sets, which leads to improved existence results
such as the existence of multiple solutions.
\end{abstract}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{claim}{Claim}
\section{Introduction}
The fundamental question that we address in this paper is: Under what
conditions are both topological and variational existence theorems applicable
to the problem
\begin{equation}
\begin{array}{c}
\Delta u+\lambda_{k}u+g(u)+h=0\,,x \in \Omega\,, \\ \\
u|_{\partial\Omega}=0\, ,
\end{array}
\label{bvp}
\end{equation}
where $\Omega$ is a smooth bounded domain in $\Bbb R^{n}$, $\lambda_{k}$ is
an eigenvalue of $-\Delta$, $g:{\Bbb R}\rightarrow{\Bbb R}$ is a continuous
function, and $h\in L^{2}(\Omega)$? Of particular interest are double
resonance problems where the term $(\lambda_k+g(u)/u)$ ranges
between consecutive eigenvalues of $-\Delta$ for large $|u|$. One-sided
resonance is similar except that the given term is strictly bounded away
from one of the eigenvalues.
As an application we prove the existence of multiple nontrivial
solutions for a class of problems where $h=0$ and $g$ is a $C^{1}$ function
such that $g(0)=0$ and $g'(0)$ is known. Problems such as this often arise
in applications such as Population Biology, where $u$ represents a
steady-state population density, $(\lambda_{k}+g(u)/u)$ represents
a population dependent growth rate, and $(\lambda_{k}+g'(0))$ represents a
growth rate in the absence of certain environmental restrictions such as
crowding.
Our results improve upon previous work in the following ways:
Theorem 1 improves upon the basic existence result in \cite{ro} by including
a saddle point characterization of at least one solution. This characterization
is an important part of the subsequent multiplicity result. This improvement
comes at the price of a somewhat less general solvability condition than in
\cite{ro}. Theorems 2 and 3 improve upon the multiplicity results of
\cite{ah} and \cite{rurola}. In \cite{ah} it is assumed that $g$ is bounded,
and the argument relies on a standard Landesman-Lazer condition. In
\cite{rurola} the results of \cite{ah} are extended by allowing $g$ to have
linear growth, and by using a generalized Landesman-Lazer condition. However,
the variational argument in \cite{rurola} assumes one-sided resonance at the
principal eigenvalue. In this paper $g$ is allowed linear growth and we
provide a variational argument that is valid for double resonance problems
between arbitrary consecutive eigenvalues. Moreover, we rely on a
Landesman-Lazer type condition that is more general than that in
\cite{rurola}.
For purposes of clarity we consider only boundary value
problems for the Laplace operator and with Dirichlet boundary conditions.
However, it will be clear that our variational arguments apply to boundary
value problems with more general elliptic operators, more general boundary
conditions, and with nonlinear terms of the form $g(x,u)$ where
$g$ is Caratheodory.
The discussion begins in Section 2, where we state Theorem 1 along with some
clarifying comments. Sections 3 and 4 provide the elements of a variational
existence proof using an Ambrossetti-Rabinowitz type saddle point
argument. Theorem 1 then follows as a consequence of these arguments combined
with the basic degree-theoretic result of \cite{ro}, and so we get a theorem
that provides a better description of the solution set than either the
variational or topological arguments do separately. In Section 5 we use the
combined topological and variational characteristics of the solution set to
prove the existence of multiple solutions for a certain class of problems.
The proofs in Section 5 are similar to those in \cite{ah} and \cite{rurola}.
It is well known that the Landesman-Lazer condition implies
coerciveness statements and the Palais-Smale condition in a natural way, see
\cite{ah} for details. One consequence of the work in this paper is that
generalized Landesman-Lazer conditions imply a similar structure. However,
there are some interesting differences. For example, although the functional
related to problem (\ref{bvp}) will be coercive over one subspace and
anticoercive over its orthogonal complement, its growth in either direction
might be relatively slow. This possibility of slower growth makes it more
difficult to establish a compactness condition. In fact, we will not prove
the usual Palais-Smale condition in Section 4, but rather a less restrictive
version often credited to G. Cerami. For a detailed discussion of this
compactness condition and for additional references see \cite{babefo}.
Before continuing it is helpful to establish the notation that will be used
throughout the paper.
\begin{description}
\item{} $H_{0}^{1}(\Omega)$ is the completion of $C_{0}^{\infty}(\Omega)$
in $L^{2}(\Omega)$ with respect to the norm
$\|u\|=\left(\int_{\Omega}|\nabla u|^{2}\right )^{1/2}$.
\item{} $\lambda_{j}:= j^{th}$ distinct eigenvalue of
$-\Delta$, where $0<\lambda_{1}<\lambda_{2}<\cdots$.
\item{} $V^{j}:= \mbox{Ker}(\Delta+\lambda_{j})$,
$V^{-}=\bigoplus_{jk+1}V^{j}$.
\item{} Given $u\in H_{0}^{1}(\Omega)$, then $u^{-}$, $u^{k}$, $u^{k+1}$,
and $u^{+}$ are its orthogonal components in
$V^{-}$, $V^{k}$, $V^{k+1}$, and $V^{+}$, respectively.
\item{} $G(x,u) :=\int_{0}^{u}(g(s)+h(x))ds$.
\item{} $\tilde{g}(u):=(\lambda_{k+1}-\lambda_{k})u-g(u)$.
\item{} $\tilde{G}(x,u) :=\int_{0}^{u}(\tilde{g}(s)-h)ds=
\left ( \frac{\lambda_{k+1}-\lambda_{k}}{2}\right )u^{2}-G(x,u)$.
\item{} $f(u):=\frac{1}{2}\int_{\Omega}|\nabla u|^{2}-
\frac12 \lambda_{k}\int_{\Omega}u^{2}
-\int_{\Omega}G(x,u)$, for $u\in H_{0}^{1}(\Omega)$.
\end{description}
Notice that under reasonable conditions on $g$, $f$ is a functional on
$H_{0}^{1}(\Omega)$ that is twice Frechet-differentiable with
\begin{eqnarray*}
f'(u)v&=&\int_{\Omega}\nabla u\cdot \nabla v-\lambda_{k}\int_{\Omega}uv
-\int_{\Omega}(g(u)+h)v\,,\mbox{ and}\\
f''(u)(v,w)&=&\int_{\Omega}\nabla v\cdot\nabla w-\lambda_{k}\int_{\Omega}vw-
\int_{\Omega}g'(u)vw\; .
\end{eqnarray*}
It is a standard fact that solutions of (\ref{bvp}) correspond to critical
points of $f$, and that $f'$ has the form Identity-Compact, see \cite{ra},
so that Leray-Schauder techniques are applicable. We will use the notation
$\mbox{deg}_{LS}(f',U,0)$ for the Leray-Schauder degree of $f'$ with
respect to the set $U$ and the value $0$.
\section{A General Existence Theorem For Double Resonance Problems}
\begin{theorem}
If $g$ satisfies
\begin{description}
\item{(g1): } $\displaystyle 0\leq
\liminf_{|s|\rightarrow\infty}\frac{g(s)}{s}\leq
\limsup_{|s|\rightarrow\infty}\frac{g(s)}{s}\leq
\lambda_{k+1}-\lambda_{k}$,
\item{(g2):} If $ \|u_{n}\|\rightarrow\infty$ such that
$\|u_{n}^{k}\|/\|u_{n}\|\rightarrow 1$, then
$\exists N,\delta > 0$ such that \newline
$\left < g(u_{n})+h,u_{n}^{k}\right >_{L^{2}} \geq\delta$
for all $ n>N$, and
\item{(g3):} If $\|u_{n}\|\rightarrow\infty$ such that
$\|u_n^{k+1}\|/\|u_{n}\|\rightarrow 1$,
then $\exists N,\delta > 0$ such that \newline
$\left < \tilde{g}(u_{n})-h,u_{n}^{k+1}\right >_{L^{2}} \geq \delta$
for all $ n>N$,
\end{description}
then problem (\ref{bvp}) has a nonempty
solution set, $S$, and there is an $R>0$ such that $S\subset B_{R}(0)$ and
${\rm deg}_{LS}(f',B_{R}(0),0)=(-1)^{m}$, where $m$ represents the dimension of
$V^{-}\bigoplus V^{k}$. Moreover, we have the following
saddle point result: There are real constants $\beta > \alpha$ and a
bounded neighborhood $D$ of $0$ in $V^{-}\bigoplus V^{k}$ such that
$f|_{\partial D}\leq\alpha$ and $f|_{V^{k+1}\bigoplus V^{+}}\geq\beta$, and
there is a critical value $c\geq\beta$ such that
\[ c=\inf_{h\in\Gamma}\max_{u\in\overline{D}}f(h(u))\; ,\]
where
\[ \Gamma=\{ h\in C(\overline{D},H^{1}_{0}(\Omega))|h=id \mbox{ on }
\partial D\}.\]
\end{theorem}
The proof of the the degree computation in this theorem is given in \cite{ro}
by establishing an a priori bound for the solution set of the family of
equations
\begin{eqnarray}
&\Delta u+\frac12(\lambda_k+\lambda_{k+1})u+
t\left( g(u)+h-\frac12 (\lambda_{k+1}-\lambda_k)u\right)=0, \;
x \in \Omega,\; t\in [0,1]& \nonumber\\
&u|_{\partial\Omega}=0\,.&
\end{eqnarray}
The homotopy invariance of Leray-Schauder degree is then applicable,
and it is straight forward to compute the degree for the linear problem
at $t=0$.
The saddle point characterization will follow from the arguments
in the next two sections below. For later reference we remark that
if $\Gamma$ is simply a collection of curves with fixed endpoints,
then we refer to the corresponding solution as a solution of {\it mountain pass
type}. This would occur, for example, if we had $\lambda_{k}=\lambda_{1}$
so that $V^{-}\equiv 0$ and $V^{1}$ is one dimensional.
The existence of at least one solution is true in a much more general
setting. For example, in \cite{ro} a theorem of this type is proved for a
class of boundary value problems over unbounded domains. Also, in \cite{rola}
it is shown that $\delta$ can be replaced by $0$ in $(g2)$ and $(g3)$,
although the boundedness of the solution set is lost, and so no compactness
condition of Palais-Smale type is possible.
It has also been shown that many well-known solvability conditions are
special cases of $(g2)$ and $(g3)$. For example the standard Landesman-Lazer
condition (see \cite{lala}), the solvability conditions used by Fucik,
Krbec, and Hess (see \cite{kefu} and \cite{he}), the double resonance
conditions used by Berestycki and DeFigueredo (see \cite{befi}), and some
cases of the density conditions at infinity (see \cite{dego}). For
comparisons of solvability conditions see \cite{ru}, \cite{ro}, and
\cite{rola}. A notable exception is the sign condition used in \cite{iank}
and many other recent papers. The sign condition is a special case of the
more general theorem where we replace $\delta$ by $0$ in $(g2)$ and $(g3)$,
see \cite{rola}, but will not be included
in the results of this paper. A second interesting exception is the
well-known solvability condition of Ahmad, Lazer, and Paul, see
\cite{ahlapa}, for problems with bounded nonlinear terms. It can be shown
that the ALP condition is not a consequence of the most general form of
{\it (g2)} and {\it (g3)} and vice versa.
\section{Coercivity}
In this section we prove
\begin{lemma}
Assume that $g$ is a continuous function satisfying (g1)-(g3).
Then $f$ is coercive on $V^{k+1}\bigoplus V^{+}$ and is anticoercive on
$V^{-}\bigoplus V^{k}$.
\end{lemma}
The proof of this lemma requires several technical preliminary results. We will
concentrate on proving that $f$ is coercive on $V^{k+1}\bigoplus V^{+}$ and
remark that the anticoercive statement follows by a similar argument. In fact
the second argument can be simplified by using the fact that $V^{-}\bigoplus
V^{k}$ is finite dimensional.
The interested reader can verify that
$f|_{V^{+}}$ and $f|_{V^{k+1}}$ are coercive as a direct consequence of
conditions {\it (g1)} and {\it (g3)}, respectively. The technical difficulty
arises when we study the functional over the combined space.
\begin{claim}
Given any $r>0$, $f$ is coercive on the solid cone
\[ C_{r}:=\{ u\in V^{k+1}\bigoplus V^{+}:\|u^{+}\|\geq r\|u\|\}.\]
\label{lem1}
\end{claim}
\paragraph{Proof:} Rewrite $f$ as
\begin{equation}
f(u)=\frac{1}{2}\int_{\Omega}|\nabla
u|^{2}-\frac{\lambda_{k+1}}{2}\int_{\Omega}u^{2}+\int_{\Omega}
\tilde{G}(x,u)\,,\label{eq:blob}
\end{equation}
and $f'$ as
\[
f'(u)v=\int_{\Omega}\nabla u\cdot\nabla v-
\lambda_{k+1}\int_{\Omega}uv+\int_{\Omega}(\tilde{g}(u)-h)v\,.
\]
Applying the right hand side of the inequality {\it (g1)}, we can say that
given any $\epsilon>0$ there is a constant $\rho>0$ such that
$\tilde{g}(s)s\geq-\epsilon s^{2}$ for every $|s|>\rho$. Thus there is a
constant, $a$, depending on $g$ and $\rho$, such that
\begin{eqnarray*}
\int_{\Omega}\tilde{g}(u)u&\geq & -\epsilon\int_{\Omega}u^{2}-a \,,\\
&\geq&-\frac1{\lambda_{1}} \epsilon \|u\|^{2}-a\,,
\end{eqnarray*}
where we have applied Poincare's Inequality. Further,
\begin{eqnarray*}
\int_{\Omega}|\nabla u|^{2}-\lambda_{k+1}\int_{\Omega}u^{2}
&=& \int_{\Omega}|\nabla u^{+}|^{2}-
\lambda_{k+1}\int_{\Omega}(u^{+})^{2} \mbox{ for } u\in
V^{k+1}\bigoplus V^{+} \\
&\geq& (1-\frac{\lambda_{k+1}}{\lambda_{k+2}})
\|u^{+}\|^{2}\mbox{ for } u\in
V^{k+1}\bigoplus V^{+} \\
&\geq& r^{2}(1-\frac{\lambda_{k+1}}{\lambda_{k+2}})
\|u\|^{2}\mbox{ for } u\in C_{r}\,.
\end{eqnarray*}
The previous inequalities imply
\[
f'(tu)u\geq t(b-\frac{\epsilon}{\lambda_{1}})\|u\|^{2}-t\|h\|\,\|u\|-a,
\; t\geq 0,
\]
where $b=r^{2}(1-\frac{\lambda_{k+1}}{\lambda_{k+2}})>0$.
Therefore
\[
f(u)=f(0)+\int_{0}^{1}f'(tu)u\,dt \geq f(0)+ \frac{1}{2}
\left [ (b-\frac{\epsilon}{\lambda_{1}})\|u\|^{2}-\|h\|\,\|u\|\right]-a\,,
\]
and so an appropriate choice of $\epsilon$ finishes the proof.
\begin{claim}
Given any $r>0\;f$ achieves a minimum on the cylinder
\[ K_{r}:=\{u \in V^{k+1}\bigoplus V^{+}:\|u^{k+1}\|=r\}\; .\]
\end{claim}
\paragraph{Proof:}
Applying Claim 1, it is easy to see that $f$ is coercive when restricted to
$K_{r}$ . Thus if $\{ u_{n}\}$ is a sequence in $H_{0}^{1}(\Omega)$ such
that $\{f(u_{n})\}$ is bounded, then $\{u_{n}\}$ must be
bounded as well. If we also know that $f'(u_{n})\rightarrow 0$, then,
using the fact that $f'$ is of the form Identity-Compact, we can show that
$\{u_{n}\}$ must have a converging subsequence.
In other words $f|_{K_{r}}$ satisfies the Palais-Smale condition.
It is then an easy exercise to show that $f|_{K_{r}}$ achieves a minimum.
\paragraph{Proof of Lemma 1:}
We prove that $f$ is coercive by examining its behavior on a sequence of
``minimizers," as described in Claim 2. Let
$\{u_{n}\}\subset V^{k+1}\bigoplus V^{+}$
such that $\|u_{n}\|\rightarrow\infty$, and such that
\begin{equation}
f(u_{n})\leq f(u) \mbox{ for all } u\in K_{\|u_{n}^{k+1}\|}\;
.\label{min}\end{equation}
We will show that $f(u_{n})\rightarrow\infty$ for some
subsequence of $\{u_{n}\}$. It will follow that no sequence of minimizers
is bounded above, and hence that
\[\lim_{\|u\|\rightarrow\infty}f(u)=\infty \mbox{ for }
u\in V^{k+1}\bigoplus V^{+}\; .\]
If $\{u_{n}\}$ is contained in any solid cone, $C_{r}$, as in Claim 1,
then $f(u_{n})\rightarrow\infty$, so we need only consider the case where
$\|u_{n}^{k+1}\|/\|u_{n}\|\rightarrow 1$, which brings condition
{\it (g3)} into play. Since this implies
$\|u_{n}^{k+1}\|\rightarrow\infty$,
we may also assume that $\|u_{n+1}^{k+1}\|>\|u_{n}^{k+1}\|$ for all $n$.
For any $n$ it is clear that
$(\|u_{n}^{k+1}\|/\|u_{n+1}^{k+1}\|) u_{n+1}^{k+1}+u_{n+1}^{+}\in
K_{\|u_{n}^{k+1}\|}$, so property (\ref{min}) implies
\[f(u_{n+1})-f(u_{n})\geq f(u_{n+1})-
f\left (\frac{\|u_{n}^{k+1}\|}{\|u_{n+1}^{k+1}\|}u_{n+1}^{k+1}+
u_{n+1}^{+}\right )\; .
\]
By substituting into expression (\ref{eq:blob}), we get
\[f(u_{n+1})-f(u_{n})\geq\int_{\Omega}\left [\tilde{G}(x,u_{n+1})-
\tilde{G}\left (x,\frac{\|u_{n}^{k+1}\|}{\|u_{n+1}^{k+1}\|}u_{n+1}^{k+1}
+u_{n+1}^{+}\right )\right ]\; .\]
Let \[\gamma_{n}(t)=t+(1-t)\frac{\|u_{n}^{k+1}\|}{\|u_{n+1}^{k+1}\|}\; ,\]
and let
\[ v_{n}(t)=\gamma_{n}(t)u_{n+1}^{k+1}+u_{n+1}^{+},\]
so $v_{n}(0)=(\|u_{n}^{k+1}\|/\|u_{n+1}^{k+1}\|)u_{n+1}^{k+1}
+u_{n+1}^{+}$ and $v_{n}(1)=u_{n+1}$. Next let
\[F_{n}(t)=\int_{\Omega}\tilde{G}(x,v_{n}(t))\; , \]
so
\begin{eqnarray*}
f(u_{n+1})-f(u_{n}) &\geq& F_{n}(1)-F_{n}(0) \\
&=& \int_{0}^{1}F_{n}'(t)\,dt \\
&=& \int_{0}^{1}\left <\tilde{g}(v_{n}(t)),
\gamma_{n}'(t)u_{n+1}^{k+1}\right >_{L^{2}}\, dt \\
&=&\int_{0}^{1}\left <\tilde{g}(v_{n}(t)),
\gamma_{n}(t)u_{n+1}^{k+1}\right >_{L^{2}}
(\frac{\gamma_{n}'(t)}{\gamma_{n}(t)})\,dt\, .
\end{eqnarray*}
We claim that, without loss of generality, there is a $\delta >0$ such
that
\[
\left<\tilde{g}(v_{n}(t)),\gamma_{n}(t)u_{n+1}^{k+1}\right >_{L^{2}}
\geq\delta \]
for every $n$ and every $t\in [0,1]$.
If not there would be a subsequence $\{u_{n}\}$ and a corresponding sequence
$\{t_{n}\}\subset [0,1]$, such that
\[ \limsup_{n\rightarrow\infty}
\left <\tilde{g}(v_{n}(t_{n})),\gamma_{n}(t_{n})u_{n+1}^{k+1}\right >_{L^{2}}
\leq 0 \; .\]
Observe that
$\|v_{n}(t_{n})^{+}\|=\|u_{n+1}^{+}\|$, and
$\|v_{n}(t_{n})^{k+1}\|=\gamma_{n}(t_{n})\|u_{n+1}^{k+1}\|$, so
$$\|u_{n}^{k+1}\|\leq \|v_{n}(t_{n})^{k+1}\|\leq \|u_{n+1}^{k+1}\|\,.$$
It follows that
$\lim_{n\rightarrow\infty}\|v_{n}(t_{n})^{k+1}\|/\|v_{n}(t_{n})\|=1$,
but this contradicts {\it (g3)}.
Applying this result we have
\begin{eqnarray*}
f(u_{n+1})-f(u_{n})
&\geq& \delta\int_{0}^{1}\frac{\gamma_{n}'(t)}{\gamma_{n}(t)}\,dt
\;\; \forall\; n \\
&=&\delta\left [\ln (\gamma_{n}(1))-\ln (\gamma_{n}(0))\right ] \\
&=&\delta\left [\ln (1)-\ln (\frac{\|u_{n}^{k+1}\|}{\|u_{n+1}^{k+1}\|})
\right ]\\
&=& \delta\left [\ln (\|u_{n+1}^{k+1}\|)-\ln (\|u_{n}^{k+1}\|) \right ]\,.
\end{eqnarray*}
Hence
\begin{eqnarray*}
f(u_{n+1}) &=&f(u_{1})+\sum_{j=1}^{n}\left [ f(u_{j+1})-f(u_{j})\right ]\\
&\geq& f(u_{1})+\delta\sum_{j=1}^{n}\left [ \ln (\|u_{j+1}^{k+1}\|)-
\ln (\|u_{j}^{k+1}\|)\right ]\\
&=&f(u_{1})-\delta\ln (\|u_{1}^{k+1}\|)+\delta\ln (\|u_{n+1}^{k+1}\|)\; ,
\end{eqnarray*}
therefore $f(u_{n})\rightarrow\infty$ ,
and coerciveness is proved.
In the next section we will give more thought to the fact that $f$ might
grow only as fast as a logarithm.
\section{A Compactness Condition}
In this section we show that $f$ satisfies a less restrictive form of
Palais-Smale condition that is still sufficient to imply mountain pass and
saddle point theorems.
It is a simple task to prove the Palais-Smale condition as a consequence of
the Landesman-Lazer condition, or as a consequence of the following
generalized conditions:
{\it
\begin{description}
\item{$(g2)'$:} If $ \|u_{n}\|\rightarrow\infty$ such that
$\|u_{n}^{k}\|/\|u_{n}\|\rightarrow 1$, then
$\exists N,\delta > 0$ such that \newline
$\left < g(u_{n})+h,u_{n}^{k}/\|u_{n}^{k}\|\right >_{L^{2}}
\geq\delta$ for all $ n>N$, and
\item{$(g3)'$:} If $\|u_{n}\|\rightarrow\infty$ such that
$\|u_{n}^{k+1}\|/\|u_{n}\|\rightarrow 1$,
then $\exists N,\delta > 0$ such that \newline
$\left < \tilde{g}(u_{n})-h,u_{n}^{k+1}/\|u_{n}^{k+1}\|
\right >_{L^{2}} \geq \delta$ for all $ n>N$,
\end{description} }
We remark that $(g2)'$ and $(g3)'$ were satisfied in both \cite{ah} and
\cite{rurola}, and we refer to these
papers for the simple Palais-Smale argument.
A similarly easy argument based upon the conditions {\it (g2)} and
{\it (g3)} does not appear to be available. In order to understand why this new
situation is more delicate it is worthwhile considering a simple but
instructive example. Notice that in the previous section the estimates
revealed that, over the eigenspaces $V^{k}$ and $V^{k+1}$, the functional
might have only logarithmic growth. Thus we consider the following
situation: Let $z:{\Bbb R}^{2}\rightarrow{\Bbb R}:z(x,y)=\log (1+x^{2})
-\log (1+y^{2})$. This function is coercive over the $x$-axis, anticoercive
over the $y$-axis and satisfies conditions {\it (g2)} and
{\it (g3)}, since, for example,
$\left < \nabla z,(x,0)\right > = 2x^{2}/(1+x^{2})\rightarrow 2$ as
$x\rightarrow\infty$. However, $z$ does not satisfy the Palais-Smale
condition, because $z=0$ on the level set $|x|=|y|$, but
$\nabla z=(2x/(1+x^2),-2y/(1+y^2))\rightarrow (0,0)$ as
$x,y\rightarrow\infty$.
It turns out that the functional $f$, as well as the example above,
satisfies the following compactness condition, which is a special case of
the condition used in \cite{babefo}.
\begin{lemma}
If $\{u_{n}\}\subset H_{0}^{1}(\Omega)$ such that
$(1+\|u_{n}\|)\|f'(u_{n})\|\rightarrow 0$, then $\{u_{n}\}$ contains a
converging subsequence.
\end{lemma}
\paragraph{Proof:}
Suppose $\{u_{n}\}$ is such a sequence. Once again, $f'$ is of the form
Identity-Compact, so it suffices to show that $\{u_{n}\}$
is bounded. We argue by contradiction, so suppose
that $\|u_{n}\|\rightarrow\infty$. Since bounded sets in $H_{0}^{1}(\Omega)$
are weakly compact, and since $H^{1}_{0}(\Omega)$ embeds compactly in
$L^{2}(\Omega)$, without loss of generality we may assume
that there is a unit vector $w\in H_{0}^{1}(\Omega)$ and a bounded measurable
$\gamma (x)$ such that
\begin{eqnarray*}
\frac{u_{n}}{\|u_{n}\|}&\rightharpoonup& w \mbox{ in } H_{0}^{1}(\Omega)\,,
\\
\frac{u_{n}}{\|u_{n}\|}&\rightarrow& w \mbox{ in } L_{2}(\Omega)\,,\mbox{ and}
\\
\frac{g(u_{n})}{\|u_{n}\|}&\rightharpoonup& \gamma w \mbox{ in }
L_{2}(\Omega)\,,
\end{eqnarray*}
where $0\leq\gamma\leq\lambda_{k+1}-\lambda_{k}$. The last statement follows
directly from writing $g(u_{n})/\|u_{n}\|=
(g(u_n)/u_n)(u_n/\|u_{n}\|)$ for $u_{n}(x)\neq 0$ and
then applying condition {\it (g1)}.
Notice that for any $v\in H_{0}^{1}(\Omega)$
\[
\frac{f'(u_n)v}{\|u_{n}\|}=\int_{\Omega}\left ( \frac{\nabla
u_{n}}{\|u_{n}\|}\right )\cdot \nabla v
-\lambda_{k}\int_{\Omega}\left ( \frac{u_{n}}{\|u_{n}\|}\right ) v-
\int_{\Omega}\left ( \frac{g(u_{n}) }{\|u_{n}\|}\right ) v-
\frac{1}{\|u_{n}\|} \int_{\Omega}hv\; .
\]
Allowing $n\rightarrow\infty$ we get
\[
0=\int_{\Omega}\nabla w\cdot\nabla v -\lambda_{k}\int_{\Omega}wv-
\int_{\Omega}\gamma wv\;\;\; \forall v\in H^{1}_{0}(\Omega)\,.
\]
Thus $w$ is a nontrivial weak solution of the boundary value problem
\[
\begin{array}{c}
\Delta w +(\lambda_{k}+\gamma)w=0,\;\; x\in\Omega\,, \\ \\
u|_{\partial\Omega}=0\,.
\end{array}
\]
Since $\lambda_{k}\leq \lambda_{k}+\gamma \leq \lambda_{k+1}$, a standard
argument involving the maximum principle and the unique continuation
property implies that either $\lambda_{k}+\gamma \equiv \lambda_{k}$ a.e.
and $w\equiv w^{k}$, or $\lambda_{k}+\gamma \equiv \lambda_{k+1}$ a.e.
and $w\equiv w^{k+1}$. (The details of this argument are
available in many papers, see \cite{ro} or \cite{iank2}, for example.)
Thus we have that
either $\|u_{n}^{k}\|/\|u_{n}\|\rightarrow 1$ or
$\|u_{n}^{k+1}\|/\|u_{n}\|\rightarrow 1$, so either condition
{\it (g2)} or {\it (g3)} is applicable. Suppose $w\equiv w^{k}$
(The argument for $w\equiv w^{k+1}$ is similar). Then by {\it (g2)}, and by
passing to a subsequence if necessary, we can assume there is a
$\delta >0$ such that
\[
f'(u_{n})u_{n}^{k}=-\int_{\Omega}(g(u_{n})+h)u_{n}^{k}\leq -\delta\;\;
\forall n\,.
\]
Therefore
\[
\|f'(u_{n})\|\, \|u_{n}^{k}\|\geq \delta\;\;
\forall n\,,
\]
which contradicts $(1+\|u_{n}\|)\|f'(u_{n})\|\rightarrow 0$, and the
proof is done.
As a consequence of Lemma 2 we know that the functional $f$ satisfies a
variant of the Palais-Smale condition discussed in \cite{babefo}. We
refer to this paper for proofs of a deformation lemma as well as
the standard mountain pass and saddle point theorems.
Hence Lemmas 1 and 2 imply the variational characterization in
Theorem 1.
\section{ Existence of Multiple Solutions }
In this section we consider the following restricted version of problem
(\ref{bvp}).
\begin{equation}
\begin{array}{c}
\Delta u+\lambda_{k}u+g(u)=0,\;\; x\in \Omega\; , \\ \\
u|_{\partial\Omega}=0\; ,
\end{array}
\label{bvp2}
\end{equation}
where $g$ is a $C^{1}$ function such that $g(0)=0$. Thus it is given that
there is a trivial solution to the problem and we are interested in proving
the existence of nontrivial solutions. Observe that weak solutions are
classical solutions in this case.
Most of the work for proving the following theorems has already been
accomplished in the preceding sections. The general outline of the arguments
is similar to that used in \cite{ah} and \cite{rurola}, and we will make use
of the results in \cite{am} and \cite{ho}. Ambrossetti's result in \cite{am}
states that if $f$ has a nondegenerate critical point of mountain pass type,
then the Morse index of $f$ at this point is $1$.
We will use the notation $\mbox{ind}_{M}(f,u)$ for the Morse index of
$f$ at a nondegenerate critical point $u$. Recall that this quantity is
the dimension of the subspace where $f''(u)$ is negative definite.
Hofer's result in \cite{ho} states that if $f$ has an isolated critical point
of mountain pass type, then the Leray-Schauder index of $f'$ at this point is
$-1$. We will use the
notation $\mbox{ind}_{LS}(f',u)$ for the Leray-Schauder index of $f'$
at a critical point $u$. Recall that this quantity is defined as
$\lim_{r\rightarrow 0}\mbox{deg}_{LS}(f',B_{r}(u),0)$, if this limit
exists, where $B_{r}(u)$ is the $r$-ball centered at $u$.
The computation of $\mbox{ind}_{LS}(f',0)$ in the following proofs is
standard, but for more detail see Theorem 2.8.1 in \cite{ni}.
Both theorems of this section generalize the results in \cite{rurola} by
allowing double resonance rather than just one-sided resonance, and
by allowing a more general Landesman-Lazer type solvability condition.
Further, the first theorem of this section allows double resonance between
any two consecutive eigenvalues rather than just between the first two
eigenvalues.
A particular example of a nonlinear term that does not satisfy the conditions
in \cite{ah} or \cite{rurola} would be $g(u)=u/(1+u^2)$. Since
$\lim_{|u|\rightarrow\infty}g(u)=0$ it follows that neither the standard
Landesman-Lazer condition used in \cite{ah} nor the generalization used in
\cite{rurola} can be satisfied. However it was shown in \cite{rola}, using a
simple dominated convergence argument, that this nonlinear term does satisfy
{\it (g2)} and {\it (g3)}. A number of modifications can
be made to this example without changing its basic characteristics, e.g.
adding certain types of terms with linear growth. For a detailed discussion
with examples see \cite{rola}. Finally, it is easy to see that this example
can be modified for $u$ within any specified interval $[-r,r]$ so that
$g'(0)$ satisfies the hypotheses of the following theorems. For further
remarks and examples on how generalized Landesman-Lazer conditions compare to
solvability conditions used in other variational arguments, such as in
\cite{cool}, see \cite{rurola}.
\begin{theorem}
Suppose $g$ satisfies (g1)-(g3) and $\lambda_{k}+g'(0)<\lambda_{1}$.
Then problem (\ref{bvp2}) has at least two nontrivial solutions.
\end{theorem}
\paragraph{Proof:}
Notice that for any $v\in H_{0}^{1}(\Omega)$
\[f''(0)(v,v)=\int_{\Omega}|\nabla v|^{2}-\lambda_{k}\int_{\Omega}v^{2}-
g'(0)\int_{\Omega}v^{2}\; .\]
If $\lambda_{k}+g'(0)\geq 0$, then by Poincare's inequality
\[
f''(0)(v,v)\geq \left (1-\frac{\lambda_{k}+g'(0)}{\lambda_{1}}\right )
\|v\|^{2},
\]
and if
$\lambda_{k}+g'(0)\leq 0$, then \[ f''(0)(v,v)\geq \|v\|^{2}.\]
It follows that $0$ is a nondegenerate critical
point of $f$ with $\mbox{ind}_{M}(f,0)=0$ and \linebreak
$\mbox{ind}_{LS}(f',0)=1$. Moreover, $0$ is a strict local minimum of
$f$, so there is an $r>0$ and an $\alpha >0$ such that
$f|_{\partial B_{r}(0)}\geq \alpha$. Since $f|_{V^{-}\bigoplus V^{k}}$
is anticoercive, we can find a $w\in \overline{B_{r}(0)}^{\;c}$ such that
$f(w)\leq 0$. Now let
$\Gamma=\{ h\in C([0,1], H^{1}_{0}(\Omega))|h(0)=0,h(1)=w \}$.
The compactness condition of Lemma 2 justifies a standard deformation
argument to show that
\[ c=\inf_{h\in\Gamma}\max_{0\leq t\leq 1}f(h(t))\; \]
is a critical value for $f$. Thus there is at least one critical point,
$u_{0}$, of mountain pass type, and, since $f(u_{0})\geq\alpha >0$,
it is clear that $u_{0}$ is nontrivial.
Suppose $\{0,u_{0}\}$ is the entire solution set of (\ref{bvp2}).
Then $u_{0}$ is an isolated critical point of mountain pass type, and it is
justified to apply the result in \cite{ho} to get
$\mbox{ind}_{LS}(f',u_{0})=-1$. The addition property of
Leray-Schauder degree implies that for $R$ as in Theorem 1
\[ \mbox{deg}_{LS}(f',B_{R}(0),0)=
\mbox{ind}_{LS}(f',0)+\mbox{ind}_{LS}(f',u_{0}) = 0,\] but this
contradicts the conclusion of Theorem 1. Hence there
must be at least one more nontrivial solution.
\begin{theorem}
Suppose $g$ satisfies (g1)-(g3), $k=1$, and there is an $m\geq 2$ such that
\newline $\lambda_{m}<\lambda_{1}+g'(0)<\lambda_{m+1}$.
Then problem (\ref{bvp2}) has at least two nontrivial solutions.
\end{theorem}
\paragraph{Proof:}
Observe that since $k=1$ we have
\[f''(0)(v,v)=\int_{\Omega}|\nabla
v|^{2}-\lambda_{1}\int_{\Omega}v^{2}-g'(0)\int_{\Omega}v^{2},\]
so for $v\in \bigoplus_{j\leq m}V^{j}$ we have
\[f''(0)(v,v)\leq (\lambda_{m}-\lambda_{1}-g'(0))\int_{\Omega}v^{2} ,\]
and, since the $L_{2}$ and $H^{1}_{0}$ norms are equivalent on the finite
dimensional space $\bigoplus_{j\leq m}V^{j}$, there is a constant $c>0$
such that
\[f''(0)(v,v)\leq -c\|v\|^{2}.\]
For $v\in \bigoplus_{j\geq m+1}V^{j}$ we have
\[
f''(0)(v,v)\geq\left ( 1-\frac{\lambda_{1}+g'(0)}{\lambda_{m+1}}\right )
\|v\|^{2} .
\]
Therefore $0$ is a nondegenerate critical point of $f$ with
$\mbox{ind}_{M}(f,0)=d$ and $\mbox{ind}_{LS}(f',0)=(-1)^{d}$,
where $d$ is the dimension of $\bigoplus_{j\leq m}V^{j}$.
Clearly $d\geq 2$ so, by the result in \cite{am}, $0$ cannot
be a critical point of mountain pass type, else the Morse index would be $1$.
But Theorem 1 for the case $k=1$ states that $f$ must have at least one
critical point of mountain pass type, call it $u_{0}$, and so there is at
least one nontrivial solution.
Suppose $\{0,u_{0}\}$ is the entire set of solutions. As in the proof of the
previous theorem, apply Hofer's result and the additive property of
Leray-Schauder degree to get
\[ \mbox{deg}_{LS}(f',B_{R}(0),0)=
\mbox{ind}_{LS}(f',0)+\mbox{ind}_{LS}(f',u_{0}) = (-1)^{d}-1.\]
Observe that the right hand side of this equality is even, which contradicts
the conclusion of Theorem 1. Hence there must be at least one more nontrivial
solution.
\paragraph{Acknowledgements.}
We would like to thank the referee for helpful comments and suggestions in
the preparation of this paper, and in particular for pointing out the reference
\cite{hi}, where some similar multiplicity results are proved for problems with
non-differentiable $g$. As the referee mentioned, the results in \cite{hi}
require that the range of $g(u)/u$ includes only a single eigenvalue
of simple multiplicity, which is not the case in our theorems.
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{\sc Steve B. Robinson\newline
Department of Mathematics and Computer Science \newline
Wake Forest University\newline
Winston-Salem, NC 27109\newline}
E-mail address: robinson@mthcsc.wfu.edu
\end{document}