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\markboth{ Contributions of Alan C. Lazer}
{ Alfonso Castro }
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Nonlinear Differential Equations, \newline
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 13--19\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
A perspective on the contributions of \\ Alan C. Lazer to critical point theory
%
\thanks{ {\em Mathematics Subject Classifications:} 34B15, 35J65.
\hfil\break\indent
{\em Key words:}
Critical points, boundary-value problems, resonance.
\hfil\break\indent
\copyright 2000 Southwest Texas State University.
\hfil\break\indent Published October 24, 2000.} }
\date{}
\author{ Alfonso Castro }
\maketitle
\begin{abstract}
Over the last thirty five years Professor Alan C.
Lazer has been a leading figure in the development of min-max methods
and critical point theory for applications to partial differential
equations. The author, his former student, summarizes from his own
perspective Professor Lazer's contributions to the subject.
\end{abstract}
\newtheorem{theorem}{Theorem}
Critical point theory has proven to be one of the most
important tools in the study of nonlinear equations.
Among the various critical point techniques, minmax principles
leading to the existence of {\it saddle points}
have played a central role. Professor Lazer has been
one of the main architects of these developments.
Let us begin by
stating the following basic principle proven by Professor Lazer
jointly
with Professor E. M. Landesman and D. R. Meyers
\cite{llm}.
\begin{theorem}
\label{theo1}
Let $H$ be a real Hilbert space and $X, Y$
closed subspaces with $\dim(X) < \infty$ and $H = X \oplus Y$.
Let $f:H \to R$ be a functional of class $C^2$. Let
$\nabla f$ and $D^2f$ denote the gradient and the Hessian of
$f$ respectively. If there exits a positive constant $m$ such that
\begin{equation}
\label{concv}
\langle D^2f(u)x,x\rangle \leq -m\|x\|^2
\quad\mbox{for all } x \in X,\ u \in H
\end{equation}
and
\begin{equation}
\label{convx}
\langle D^2f(u)y,y\rangle \geq m\|y\|^2 \quad\mbox{for all } y \in Y,
\ u \in H,
\end{equation}
then there exists a unique $u_0 \in H$ such that $\nabla f(u_0)=0$.
Moreover
\begin{equation}
\label{maxmin}
f(u_0) = \max_{x \in X} \min_{y \in Y} f(x + y).
\end{equation}
\end{theorem}
The proof of Theorem \ref{theo1} is based on the observation that,
by (\ref{convx}),
the function $f$ is convex on linear manifolds of the form
$\{x + y; y \in Y \}\equiv Y_x$
and $f(x+y)$ tends to $+\infty$ as $\|y\| \to
\infty$. Hence for each $x \in X$ there exists a unique $y = \phi(x)
\in Y$ such that
\begin{equation}
\label{deffi}
f(x+\phi(x)) = \min_{y \in Y}f(x+y).
\end{equation}
Through a clever use of the implicit function theorem Professor
Lazer and his coauthors show that the function $\phi$ is actually
of class $C^1$. Since $f(x) \to -\infty$ as
$\|x\| \to \infty$ (see (\ref{concv})) and $\tilde f(x) \equiv
f(x+\phi(x)) \leq f(x)$, $\tilde f$
attains its maximum value at some point
$x_0$. This implies that $u_0 = x_0 + \phi(x_0)$ satisfies
the claims of Theorem \ref{theo1}.
Noting that the latter argument only uses the convexity of $f$
on the manifold $Y_x$ and that $f(x) \to -\infty$ as
$\|x\| \to \infty$, Professor Lazer with the help of one of his
students, this author, proved in \cite{cl1} the following result.
\begin{theorem}
\label{theo2}
Let $H$ be a real Hilbert space and $X, Y$
closed subspaces with $\dim(X) < \infty$ and $H = X \oplus Y$.
Let $f:H \to R$ be a functional of class $C^2$. Let
$\nabla F$ and $D^2f$ denote the gradient and the Hessian of
$f$ respectively. If there exits a positive constant $m$ such that
\begin{equation}
\label{concv1}
\langle D^2f(u)y,y\rangle \geq m\|y\|^2 \quad\mbox{for all } y \in Y,
\ u \in H
\end{equation}
and
\begin{equation}
\label{-fcoer}
f(x) \to -\infty \quad\mbox{as }\|x\| \to \infty,
\end{equation}
then there exists $u_0 \in H$ such that $\nabla f(u_0)=0$.
Moreover
\begin{equation}
\label{maxmin1}
f(u_0) = \max_{x \in X} \min_{y \in Y} f(x + y).
\end{equation}
\end{theorem}
In \cite{c}, condition (\ref{concv1})
is further relaxed by using the variational characterization
in (\ref{deffi}) to prove that $\phi$ is continuous and
$\tilde f$ of class $C^1$ when $f$ is of class $C^1$.
These observations bypass the usage
of the implicit function theorem.
The latter results were motivated by the Dirichlet boundary
value problem
\begin{equation}
\label{diri}
\gathered
\Delta u(x) + g(x,u(x)) = 0 \quad\mbox{ for } x\in\Omega \\
u(x) =0 \quad\mbox{for } x\in \partial \Omega,
\endgathered
\end{equation}
where $\Omega$ is a smooth bounded region in $R^n$, and
$g:\Omega \times R \to R$ is a sufficiently regular
function satisfying an adequate {\it growth condition.}
Indeed if we let $H$ denote the Sobolev space of square integrable
functions in $\Omega$ whose first order partial derivatives are
also square integrable in $\Omega$ and which vanish on $\partial
\Omega$ (see \cite{adams}), $G(x,t) = \int_0^t g(x,s) ds$,
and $J:H \to R$ is the functional
given by
\begin{equation}
\label{defJ}
J(u) = \int_{\Omega} \{\|\nabla(u(x))\|^2/2 - G(x,u(x))\}\,dx,
\end{equation}
then the critical points of $J$ are the solutions to
(\ref{diri}). Moreover, $J$ obeys the
assumptions of Theorem \ref{theo1} when
there exits an integer $N$ and real numbers $a, b$
such that
\begin{equation}
\label{nores}
\lambda_N < a \leq \frac{\partial g}{\partial t}(x,t) \leq
b < \lambda_{N+1} \ \ \mbox{for all }
\ \ (x,t) \in \Omega \times R,
\end{equation}
where $\lambda_N, \lambda_{N+1}$ are consecutive eigenvalues of
\begin{equation}
\label{eigen}
\gathered
\Delta u(x) + \lambda_k u(x) = 0 \quad\mbox{for } x\in\Omega \\
u(x) = 0 \quad\mbox{for } x\in \partial \Omega\,.
\endgathered
\end{equation}
Thus under this hypothesis the problem
(\ref{diri}) has a unique solution. On the other hand if
(\ref{nores}) is replaced by the weaker condition
\begin{equation}
\label{nores1}
\frac{\partial g}{\partial t}(x,t) \leq b, \quad
\mbox{and}\quad G(x,t) \geq \frac{at^2}{2} + C,
\end{equation}
where $a,b$ are as in (\ref{nores}) and $C$ is an arbitrary real number
then $J$ satisfies the assumptions of Theorem \ref{theo2}.
Hence (\ref{diri})
has a solution which need not be unique (see
\cite{cl1}, \cite{chen}, and Theorem \ref{theo6} below).
Conditions (\ref{nores}) and (\ref{nores1}), also known
as {\it non-resonance conditions}, open the issue of what
happens when the range of $\frac{\partial g}{\partial t}$ includes an eigenvalue
$\lambda_k$. Professor Lazer, in collaboration with
Professor E. M. Landesman, provided in \cite{ll} a breakthrough in this
problem by considering the case in
which
\begin{equation}
\label{reshyp}
\gathered
g(x,u) = h(u) + \lambda_k u - p(x),\\
h(-\infty) \equiv \lim_{t \to -\infty}h(t)
< h(x) < h(+\infty) \equiv \lim_{t \to \infty}h(t)
\quad\mbox{for all } x \in {\mathbb R},\\
p \in L^2(\Omega), \ \mbox{and } \\
\lambda_k \mbox{ is a simple eigenvalue.}
\endgathered
\end{equation}
They proved that if $w$ is an eigenfunction corresponding to
the eigenvalue $\lambda_k$ then (\ref{diri}) has
a solution if and only if
\begin{eqnarray}
\lefteqn{ h(-\infty)\int_{\Omega^+}|w| dx - h(\infty)\int_{\Omega^-}|w| \,dx }
\nonumber \\
&<& \int_{\Omega}p \cdot w dx \label{rescond} \\
&<& h(\infty)\int_{\Omega^+}|w| dx - h(-\infty)\int_{\Omega^-}|w| \,dx,\nonumber
\end{eqnarray}
here
$\Omega^+ = \{x; w(x) \geq 0\}$, and
$\Omega^- = \{x; w(x) \leq 0\}$.
This surprising result is, without doubt, a corner stone in the
study of nonlinear boundary value problems and, hence, one
of the most cited papers in this area. Actually, as pointed
out by Professor Landesman to the author in a personal
communication, previously Professor Lazer had considered
a somewhat more complicated case with his student D. E. Leach in \cite{lleach}.
By 1974, when this author had the privilege of meeting Professor Lazer, he
was interested in providing a variational proof of his result
in \cite{ll}. He concluded this successfully
in \cite{alp}, where he and his coauthors established in a
{\it semivariational way} the following result.
\begin{theorem}
\label{theo3}
Let $Y$ denote a subspace solutions to the problem
$\Delta u + \lambda_k u = 0$ in $\Omega$ with
$u= 0$ on $\partial \Omega$.
Assume $g(x,t) - \lambda_k t$ is a continuous bounded function.
If for $w \in Y$
\begin{equation}
\label{reson1}
\int_{\Omega}(G(x,w(x)) - \frac{\lambda_k w^2(x)}{2}) dx \to
\pm \infty \ \ \mbox{as } \ \ \|w\| \to \infty ,
\end{equation}
then (\ref{diri}) has a solution.
\end{theorem}
Motivated by this result, Professor P. H. Rabinowitz (see \cite{rab})
provided a variational proof of Theorem \ref{theo3}. Rabinowitz's
variational proof includes the following general {\it saddle point
principle}.
\begin{theorem}
\label{theo4}
Let $H$ be a real Banach space and $X, Y$
closed subspaces with $\dim(X) < \infty$ and $H = X \oplus Y$.
Let $f:H \to {\mathbb R}$ be a functional of class $C^1$ that
satisfies de Palais-Smale condition (i.e., every sequence
$\{u_n\}$ for which $\{f(u_n)\}$ is bounded and
$\{f'(u_n)\}$ converges to zero, has a convergent subsequence.)
Suppose that
\begin{equation}
\label{fbddinY}
\inf_{y \in Y} f(y) = d > -\infty
\end{equation}
and
\begin{equation}
\label{-fcoer1}
f(x) \to -\infty \quad \mbox{as} \quad \|x\| \to \infty.
\end{equation}
Let $D = \{x \in X; \|x\| \leq r\}$ with $r$ big enough so that
$\|x\| = r$ implies $F(x) > d$. If $\Gamma$ denotes the set
of continuous mappings $p:D \to H$ such that $G(x) = x$
if $\|x\|=r$ then
\begin{equation}
\label{defc}
c \equiv \inf_{p \in \Gamma} \max_{x \in D} f(p(x)) > -\infty
\end{equation}
and there exists $u_0 \in H$ such that
$f(u_0) = c$ and $f'(u_0) = 0$.
\end{theorem}
For further details on the
proof and applicability of the latter result the reader is
referred to \cite{rabmon}.
If one defines the Morse index of a critical point $u$
as the number of negative eigenvalues of $D^2f(u)$ from the
hypotheses of Theorem \ref{theo1} one sees that the Morse
index of $u_0$ is $\dim X$. Given the similarities between
the assumptions of Theorem \ref{theo1} and those of
Theorem \ref{theo4} one would be tempted to conjecture
that the Morse index of the critical point arising in
Theorem \ref{theo4} is $\dim X$.
This, in general, is not true (see \cite{lsol}).
Professor Lazer, in collaboration with Professor S. Solimini,
in \cite{lsol} provides a detailed account of this problem.
They prove the following results in \cite{lsol}.
\begin{theorem}
\label{theo5}
If $f$ satisfies the hypotheses of Theorem \ref{theo4}
and has only a finite number of critical points, all of
which are nondegenerate, then there exists a critical point
of $f$ with Morse index equal to $\dim X$.
\end{theorem}
The reader is invited to consult \cite{Ghoussoub}
for extensive ellaborations on the results of
Professors Lazer and Solimini
in \cite{lsol}.
Professor Lazer has masterfully utilized the
variational characterizations provided
in (\ref{maxmin1}) and (\ref{defc}) to establish the
existence of
multiple solutions for problems like (\ref{diri}). For example, in
a paper with this author (see Theorem A of \cite{cl2}) he proved the
following multiplicity result.
\begin{theorem}
\label{theo6}
If $a, b$ are as in (\ref{nores1}) $g(x,0) =0$, and
$\frac{\partial g}{\partial t}(x,0) < \lambda_N$ then
the problem (\ref{diri}) has at least two solutions.
Moreover, if $\frac{\partial g}{\partial t}(x,0) \not =
\lambda_k$ for any $k$ then (\ref{diri}) has at
least three solutions.
\end{theorem}
The key ingredient in the proof of Theorem \ref{theo6}
is the fact that the solution obtained via the characterization
(\ref{maxmin1}), if isolated,
gives a critical point of Morse index $N$ while
zero is a critical point of Morse index less than $N$.
The third solution in Theorem \ref{theo6} comes from
the combining the fact that the degree of $\nabla J$ in a large ball is
$N$. Actually the argument extends to the case when
the first part of the hypothesis in (\ref{nores1})
is replaced by
\begin{equation}
\label{nores2}
G(x,t) \leq \frac{bt^2}{2} + C\,.
\end{equation}
In this case
one may use theorems \ref{theo4} and \ref{theo5} to show that
either $c$ or a {\it dual} value $\bar c$ is a critical value
containing a critical point whose Morse index is not that of
$0$. For other results where multiple solutions for (\ref{diri})
are studied using the above devices the reader is referred
to \cite{cc}, \cite{ccn}, \cite{ccnejde}, and references therein.
Another line of research motivated by Professor Lazer's work in
\cite{llm} is the continuous dependence of the saddle point
arising in (\ref{maxmin}) and its applications.
Extensive developments in this direction are due to Professor
H. Amann and his coworkers (see \cite{amann}, \cite{az}).
The
following generalization of Theorem
\ref{theo1} can be found in \cite{amann}.
\begin{theorem}
\label{theo7}
Let $H$ be a real Hilbert space and $X, Y, Z$
closed subspaces with $H = X \oplus Y \oplus Z$.
Let $f:H \to {\mathbb R}$ be a functional of class $C^2$. Let
$\nabla f$ and $D^2f$ denote the gradient $f$.
If there exits a positive constant $m$ such that
\begin{equation}
\label{concv2} \gathered
\langle f(x+y+z) - f(x_1+y + z), x - x_1\rangle
\leq -m\|x- x_1\|^2 \\
\mbox{for all}\quad x, x_1 \in X,\quad y \in Y,\quad z \in Z\,,
\endgathered
\end{equation}
and
\begin{equation}
\label{convx1} \gathered
\langle f(x+y+z) - f(x+y_1 + z), y - y_1\rangle \geq m\|y- y_1\|^2 \\
\mbox{for all } x \in X,\quad y, y_1 \in Y, \quad z \in Z,\,
\endgathered
\end{equation}
then there exists a continuous function $\phi: Z \to X \oplus Y$
such that $<\nabla f(z + \phi(z)), x + y>=0$ for all
$(x,y,z) \in X \times Y \times Z$.
Moreover
\begin{equation}
\label{minmax1}
\hat f(z) \equiv f(z + \phi(z)) = \max_{x \in X} \min_{y \in Y} f(x + y + z)
\end{equation}
is of Class $C^1$, and $z_0$ is a critical point of $\hat f$ if and only
if $z_0 + \phi(z_0)$ is a critical point of $f$
\end{theorem}
This latter theorem is particularly useful for the study of
equations of the form $L(u) + N(u) = 0$ where $L$ is a self-adjoint
linear operator having infinitely many eigenvalues both positive and
negative. This is the case where $L$ comes from a wave operator
(see \cite{ccai}) or a Hammerstein integral operator (see \cite{c}).
The techniques involved in the construction of the function
$\phi$ in (\ref{deffi}) have been extended in many directions
and Professor Lazer has utilized them exquisitely. For example, in
\cite{lmck}, he and Professor P. J. McKenna use it to
study the existence of multiple solutions for (\ref{diri})
in the case of {\it jumping nonlinearities}. They establish
sufficient conditions on $g$
for (\ref{diri}) to have three solutions when
$$
g(x,t) = h(t) - s
\theta (x) + p(x), \ \lim_{t \to -\infty}h'(t), \lim_{t \to \infty}h'(t)
\in (\lambda_{N-1}, \lambda_{N+1}),
$$
where $p$ is orthogonal to $\theta$, and
$\theta $ is and eigenfunction correponding to the
eigenvalue $\lambda_1$ of (\ref{eigen}).
\paragraph{Conclusion.} Over thirty five years Professor Lazer has provided
the Nonlinear Analysis community with fundamental new ideas and has
left for others to ellaborate. AL, please keep on giving
us exciting food for thought.
\begin{thebibliography}{99} {\frenchspacing
\bibitem{adams} R. Adams,
{\it Sobolev Spaces,}
New York: Academic Press (1975).
\bibitem{alp} S. Ahmad, A. C. Lazer, and J. Paul,
{\it Elementary critical point theory
and perturbations of elliptic boundary
value problems at resonance,}
Indiana Univ. Math. J., Vol. 25 (1976), pp. 933-944.
\bibitem{amann} H. Amann,
{\it Saddle points and multiple solutions of differential
equations,}
Math. Z., Vol. 196 (1979), pp. 127-166.
\bibitem{az} H. Amann and E. Zehnder,
{\it Nontrivial solutions for a class of nonresonance problems
and applications to nonlinear partial
differential
equations,}
Ann. Scuola Normal Sup. Pisa Cl. Sci. (4), Vol. 7 (1980), pp.
539-603.
\bibitem{c} A. Castro, {\it Hammerstein integral equations
with indefinite kernel,} Internat. J.
Math. and Math. Sci., Vol 1 (1978), pp. 187-201.
\bibitem{ccai} A. Castro and J. F. Caicedo, {\it A semilinear wave
equation
with derivative of nonlinearity containing multiple
eigenvalues of infinite multiplicity}, Contemporary Mathematics,
Vol. 208 (1997), pp. 111-132.
\bibitem{cc} A. Castro and J. Cossio,
{\it Multiple solutions for a semilinear Dirichlet problem,}
SIAM J. Math. Anal., Vol. 25 (1994), pp. 1554-1561.
\bibitem{ccn} A. Castro, J. Cossio and J. M. Neuberger,
{\it A sign-changing solution for a superlinear Dirichlet problem,}
Rocky Mountain Journal of Mathematics, Vol. 27, No. 4 (1997), pp. 1041-1053.
\bibitem{ccnejde} A. Castro, J. Cossio and J. M.
Neuberger,
{\it A minmax principle, index of the critical point, and
existence of sign changing solutions to elliptic boundary value problems,}
Electronic Journal of Differential Equations, http://ejde.math.swt.edu,
Vol. 1998 (1998), No. 2, pp. 1-18.
\bibitem{cl1} A. Castro and A. C. Lazer,
{\it Applications of a max-min principle,}
Rev. Colombiana Mat., Vol. X (1976), pp. 141-149.
\bibitem{cl2} A. Castro and A. C. Lazer,
{\it Critical point theory and the number of
solutions of a nonlinear Dirichlet problem,}
Annali di Mat. Pura ed Appl., Vol. CXX (1979), pp. 141-149.
\bibitem{chen} C. Chang,
{\it On an application of the saddle-point theorem,}
Dynamics Systems and Applications,
Vol. 7 (1998), pp. 377-388.
\bibitem{Ghoussoub}
N. Ghoussoub,
{\it Duality and perturbation methods in critical point theory},
Cambridge Tracts in Mathematics, Cambridge University Press (1993).
\bibitem{ll} E. M. Landesman, and A. C. Lazer,
{\it Nonlinear perturbations of linear elliptic boundary
value problems at resonance},
J. Math. and Mechanics, Vol. 19 (1970), pp. 609-623.
\bibitem{lleach} A. C. Lazer, and D. Leach,
{\it On a nonlinear two-point boundary value problem,}
J. Math. Anal. Appl., Vol. 26 (1969), pp. 20-27.
\bibitem{llm} A. C. Lazer, E. M. Landesman, and D. Meyers,
{\it On saddle point problems in the calculus of variations,
the Ritz algorith, and monotone convergence,}
J. Math. Anal. Appl., Vol. 53 (1975), pp. 594-614.
\bibitem{lmck} A. C. Lazer and P. J. McKenna, {\it Critical point
theory and boundary value problems with nonlinearities
crossing multiple eigenvalues,}
Comm. Partial Diff. Eqns., Vol. 10 (1985), 107-150.
\bibitem{lsol} A. C. Lazer, and S. Solimini,
{\it Nontrivial solutions of operator equations
and Morse indices of critical points of min-max type,}
Nonlinear Analysis T. M. A., Vol. 12 (1988),
pp. 761-775.
\bibitem{rab} P. H. Rabinowitz,
{\it Some minmax theorems and applications to
nonlinear partial differential equations,}
Nonlinear Analysis: A collection of papers in honor
of Erich Rothe, Academic Press, New York, (1978), pp. 161-177.
\bibitem{rabmon} P. H. Rabinowitz,
{\it Minmax methods in critical point theory
with applications to differential equations,}
C.B.M.S., Amer. Math. Soc., No. 65 (1986).
}\end{thebibliography}
\medskip
\noindent{\sc Alfonso Castro}\\
Division of Mathematics and Statistics\\
The University of Texas at San Antonio\\
San Antonio TX 78249-0664\\
e-mail: castro@math.utsa.edu
\end{document}