\documentclass[twoside]{article}
\usepackage{amsfonts} % used for R in Real numbers
\pagestyle{myheadings}
\markboth{Bounds for Nonlinear eigenvalue problem}
{R. D. Benguria \& M. C. Depassier}
\begin{document}
\setcounter{page}{23}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 23--27\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} \\
%
Bounds for nonlinear eigenvalue problems
% 
\thanks{ {\em Mathematics Subject Classifications:}  35B05, 35B32.
 \hfil\break\indent 
{\em Key words:}  nonlinear elliptic boundary-value problems, bifurcations.
 \hfil\break\indent
\copyright 2001 Southwest Texas State University. 
\hfil\break\indent Published January 8, 2001. } } 

\date{}
\author{Rafael D. Benguria \& M. Cristina Depassier}


\maketitle


\begin{abstract} 
We develop a technique for obtaining bounds on bifurcation curves of 
nonlinear boundary-value problems defined through nonlinear elliptic
partial differential equations. 

\end{abstract}

\newtheorem{theorem}{Theorem}[section]

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\catcode`@=11
\@addtoreset{equation}{section}
\catcode`@=12

\section {Introduction} 

Recently, we have obtained a variational characterization for the principal 
solution of a two point boundary-value problem in the line \cite{BD98}. 
In particular, we proved the following result.

\begin{theorem}\label{thm:t1}
Let the pair $(\lambda,u)$ be the principal solution (i.e., with $u(x) \ge 0$)
of the two point boundary-value problem
\begin{equation}
\frac{d^2u}{dx^2}+\lambda \, u = N(u)
\label{eq:i1}
\end{equation}
subject to $u'(0)=u(1)=0$. Let $u_m=u(0)$, the sup-norm of the solution. 
Here $N(u)$ is a general nonlinear term, which is continuous in $(0,u_m)$. 
Then,
\begin{equation}
\lambda[u_m]=\max_{g \in  D}
\left( \int_0^{u_m} N(u) g(u) \, du + \frac{1}{2} \left(\int_0^{u_m} 
g'(u)^{1/3} \, du\right)^3  \right)/\int_0^{u_m} u g(u) \, du,
\label{eq:i2}
\end{equation}
where $ D= \{g \bigm| g \in C^1(0,u_m), g'>0, g(0)=0\}$.
Moreover, the maximum is attained at some $\hat g \in  D$, which is 
unique up to a  multiplicative constant.
\end{theorem}
 
This theorem cannot be extended, as such, to higher dimensional 
boundary-value problems since the methods used in the proof depend 
heavily on the one dimensional character of (\ref{eq:i1}). Nevertheless, 
at least for one particular three dimensional boundary-value problem (namely, 
the Thomas--Fermi equation) we were able to obtain a, suitably modified, 
variational characterization of the principal solution \cite{BY98}. Thus, 
there are hopes  that at least for some boundary-value problems defined 
through partial differential  equations one can obtain a variational 
characterization of the principal solution. The  purpose of this article 
is to illustrate how the methods used in \cite{BD98,BY98} 
can be extended to find bounds for the principal solution of boundary value 
problems defined through elliptic partial differential equations. 
Unfortunately, in general we fall short of obtaining a variational 
characterization. We will proceed through a well known example, since 
we believe the methods are well illustrated by it, and it is clear how to 
extend them to more general situations. 


\section{Nonlinear eigenvalue problem defined through a semilinear elliptic equation} 

Consider the boundary-value problem
\begin{equation}
-\Delta u + u^3 = \lambda u  \quad \mbox{in $\Omega \subset \mathbb{R}^3$,}
\label{eq:e1}
\end{equation}
with 
\begin{equation}
u=0 \quad \mbox{in $\partial \Omega$}.
\label{eq:e2}
\end{equation}
Here $\Omega$ is a bounded, smooth, domain in $\mathbb{R}^3$. Then we have,

\begin{theorem}\label{thm:t2}
Let the pair $(\lambda,u)$ be the principal solution (i.e., with $u(x) 
\ge 0$) of the boundary-value problem (\ref{eq:e1}), (\ref{eq:e2}), then
\begin{equation}
4 \pi u(x) \le
\frac{2}{3\sqrt{3}} \lim_{\epsilon \to 0}
\int_{\Omega \setminus B_{\epsilon}(x)}
\frac{(\Delta g + \lambda g)^{3/2}}{g^{1/2}} \, dy,
\label{eq:e3}
\end{equation}
where $B_{\epsilon}(x) \subset \Omega$ is a ball of radius $\epsilon$
centered at $x$. Here the function $g$ satisfies,
\begin{equation}
g \in C^2(\Omega \setminus B_{\epsilon}(x)) \cap 
C^0(\overline{\Omega \setminus B_{\epsilon}(x)}),
\label{eq:e4}
\end{equation}
\begin{equation}
g=0 \quad \mbox{in $\partial \Omega$,}
\label{eq:e5}
\end{equation}
\begin{equation}
g(y) \approx \frac{1}{\vert x -y \vert} \quad 
\mbox{in the neighborhood of $x$,}
\label{eq:e6}
\end{equation}
(i.e., $g(y)$ behaves like the fundamental solution around $x$), 
\begin{equation}
g(y) > 0  \quad \mbox{and}  \quad \Delta g + \lambda g >0 \quad 
\mbox{in $\Omega \setminus B_{\epsilon}(x)$,}
\label{eq:e7}
\end{equation}
but is otherwise arbitrary.
\end{theorem}
\noindent
{\it Proof:}
Pick any function $g$ satisfying (\ref{eq:e4}), (\ref{eq:e5}), 
(\ref{eq:e6}) and (\ref{eq:e7}). If we multiply (\ref{eq:e1}) by $g$, 
and integrate over $\Omega \setminus
B_{\epsilon}(x)$ we obtain, 
\begin{equation}
-\int_{\Omega \setminus B_{\epsilon}(x)} g \, \Delta u  \, dy +
\int_{\Omega \setminus B_{\epsilon}(x)} g \, u^3 \, dy =
\lambda \int_{\Omega \setminus B_{\epsilon}(x)} g\, u \, dy.
\label{eq:e8}
\end{equation}
Using Green's formula and the boundary conditions (\ref{eq:e2}) 
and (\ref{eq:e5})
satisfied by $u$ and $g$ repectively, we have
\begin{equation}
\int_{\Omega \setminus B_{\epsilon}(x)} (g \Delta u - u \Delta g) \, dy
= - \int_{ \partial B_{\epsilon}(x)} (g \nabla u-u\nabla g) \cdot \hat n \, dS,
\label{eq:e9}
\end{equation}
where $\hat n$ is the exterior normal to the surface of the ball 
$B_{\epsilon}(x)$ and $dS$ its surface element. Since $u \in C^2(\Omega)$, 
it follows from (\ref{eq:e6}) that the limit of the right side of (\ref{eq:e9})
as $\epsilon$ goes to zero is given by $-4 \pi u(x)$. Thus,
\begin{equation}
\lim_{\epsilon \to 0} \int_{\Omega \setminus B_{\epsilon}(x)}
(g \Delta u - u \Delta g) \, dy = - 4 \pi u(x).
\label{eq:e10}
\end{equation} 
Hence, using (\ref{eq:e8}), (\ref{eq:e9}), and (\ref{eq:e10}) we have
\begin{equation}
4 \pi u(x) = \lim_{\epsilon \to 0}
\int_{\Omega \setminus B_{\epsilon}(x)}
\left((\Delta g + \lambda g) u - g u^3 \right) \, dy.
\label{eq:e11}
\end{equation}
Let us denote $h \equiv \Delta g + \lambda g$, which is positive by assumption
(\ref{eq:e7}). For fixed $y$, consider the integrand of (\ref{eq:e11}) as
a function of $u$. Maximizing the integrand with respect to $u$ we get
\begin{equation}
h \, u - u^3 \, g \le \frac{2}{3\sqrt{3}} \frac{h^{3/2}}{g^{1/2}}
\label{eq:e12}
\end{equation}
and (\ref{eq:e7}) follows from here.
%\end{proof}


\paragraph{Remark.}
For general facts about bifurcation problems defined through 
ordinary differential equations see \cite{Ra70}. For 
general bifurcation problems see \cite{Ra77}. 
\smallskip



As an application, consider $\Omega$ to be the unit ball in
$\mathbb{R}^3$. We will use Theorem \ref{thm:t2} to find an estimate 
for the principal branch $(\lambda,u_m)$ (with $u(x) \ge 0$) 
of the nonlinear eigenvalue problem (\ref{eq:e1}), (\ref{eq:e2}), 
in this case. Here $u_m$ denotes the sup--norm of the solution, 
which occurs at zero (the center of the ball). In fact, for any balanced, 
smooth domain $\Omega$, the sup--norm of the positive solution of 
(\ref{eq:e1}), (\ref{eq:e2}), is attained at the center of balance, 
in this case the origin of the ball.

For our purpose, take $g(x) = \cos (\pi r /2)/r$, with $r=\vert x\vert$. 
Clearly, this $g$ satisfies the hypothesis (\ref{eq:e4}), (\ref{eq:e5}), and (\ref{eq:e6}). 
An elementary computation shows that 
\begin{equation}
\Delta g = - \frac{\pi^2}{4} g, \quad \mbox{for $r \neq 0$.}
\label{eq:e12a}
\end{equation}
Thus, the corresponding $h$ will be nonnegative  as long as $\lambda \ge \pi^2/4$. 
It turns out that this is not a restriction, since the nonlinear
eigenvalue problem (\ref{eq:e1}), (\ref{eq:e2}) has a nontrivial 
positive solution if and only if $\lambda$ is larger than the first Dirichlet eigenvalue of 
$\Omega$, which for the case of the unit ball 
is $\pi^2$. Setting $x=0$, and using the function $g$ picked above, 
we conclude from (\ref{eq:e3})
\begin{equation}
u_m \equiv u(0) \le \frac{4 \sqrt{3}}{9 \pi}(1-\frac{2}{\pi})
\left(\lambda - \frac{\pi^2}{4}\right)^{3/2},
\label{eq:e13}
\end{equation}
which gives the following lower bound for the {\it nonlinear eigenvalue}
$\lambda$,
\begin{equation}
\lambda \ge \frac{\pi^2}{4} + \left[\frac{3 \sqrt{3}}{4} \pi
 \frac{u_m}{(1-2/\pi)} \right]^{2/3}
=\frac{\pi^2}{4}+ c u_m^{2/3},
\label{eq:e13b}
\end{equation}
with $c \approx 5.015$. A better multiplicative constant can be obtained
in (\ref{eq:e13b}) using a different trial function $g$. In fact 
take
\begin{equation}
g(x)=\frac{1}{r}(1-r), 
\label{eq:13c}
\end{equation}
where  $r=\vert x \vert$, as before. This function $g$ satisfies 
(\ref{eq:e4}), (\ref{eq:e5}), and (\ref{eq:e6}). Moreover, 
$\Delta g =0$ for $r>0$, and $g>0$ 
for $0<r<1$. Hence, from (\ref{eq:e3}) we get, $u_m \equiv u(0) \le 
\lambda^{3/2}/(9 \sqrt{3})$,
i.e.,
\begin{equation}
\lambda \ge 3^{5/3} u_m^{2/3} \approx 6.240 u_m^{2/3},
\label{eq:e13d}
\end{equation}
which is better than (\ref{eq:e13b}) for large values of $u_m$. 


The bound embodied in (\ref{eq:e3}) is a local upper bound on 
the principal solution of the boundary-value problem (\ref{eq:e1}), 
(\ref{eq:e2}). To end with this example, we
will consider a function $g(y)$ depending  parametrically on $x$ to 
produce an $x$ dependent bound in (\ref{eq:e3}). What we will use as a 
trial $g$ will be the fundamental solution
of the Laplacian in a ball of radius 1. 
For the ball of radius $R$, the fundamental solution can be 
constructed using the method of images. 
It is given explicitly by
\begin{equation}
G_x(y)=\frac{1}{\vert y - x \vert} - \frac{R}{\vert x \vert}
\frac{1}{\vert y - R^2 x / \vert x \vert^2 \vert}
\label{eq:e14}
\end{equation}
(see, e.g., \cite{GT98}, pp. 19--20). Clearly this particular 
function satisfies all the hypothesis. Moreover, $\Delta G_x(y)=0$ 
for $y \neq x$. Thus, using $g(y)=G_x(y)$, with $R=1$ in (\ref{eq:e3}), 
together with Newton's theorem (i.e., 
$\int d \Omega_y (1/\vert y - x \vert)= 
4 \pi /\max(\vert x \vert, \vert y \vert)$, where the integral is over
the sphere of radius $\vert y \vert$ and $d\Omega_y$ is the invariant
measure on the sphere), we get at once
\begin{equation}
u(x) \le \frac{\lambda^{3/2}}{3\sqrt{3}}(1-\vert x \vert^2).
\label{eq:e15}
\end{equation}
The bound (\ref{eq:e15}), with a better constant, can be obtained using 
comparison theorems. In fact, using the comparison function 
$1-\vert x \vert^2$, and the maximum principle (see e.g., \cite{Sa73,PW67}), 
one can show that
\begin{equation}
u(x) \le \frac{\lambda^{3/2}}{9\sqrt{3}}(1-\vert x \vert^2).
\label{eq:e16}
\end{equation}
Notice that at $x=0$ the bound (\ref{eq:e16}) is precisely the bound
(\ref{eq:e13d}) obtained above. 


If instead of (\ref{eq:e1}), (\ref{eq:e2}), one considers a more general
boundary-value problem
\begin{equation}
-\Delta u + f(u) = \lambda u  \quad \mbox{in $\Omega \subset \mathbb{R}^3$,}
\label{eq:f1}
\end{equation}
with 
\begin{equation}
u=0 \quad \mbox{in $\partial \Omega$},
\label{eq:f2}
\end{equation}
where $\Omega$, as before, is a bounded, smooth, domain in $\mathbb{R}^3$,
and $f$ is a positive, continuous, increasing function, with 
$f(u)/u \to \infty$ as $u \to \infty$, then theorem \ref{thm:t2} holds with
(\ref{eq:e3}) replaced by
\begin{equation}
4 \pi u(x) \le
\lim_{\epsilon \to 0}
\int_{\Omega \setminus B_{\epsilon}(x)}
f_L(\frac{h}{g}) \, g  \, dy.
\label{eq:f3}
\end{equation}
Here, $f_L$ denotes the Legendre transform of the function $f$, and
$h \equiv \Delta g + \lambda g$ as before. 




\paragraph{Acknowledgements}

This work has been supported by Fondecyt (Chile) project 199--0429.
We thank the organizers of USA--Chile Conferece for their kind invitation
and for the opportunity to present these results. 



\begin{thebibliography}{0}

\bibitem{BD98}
R.D. Benguria and M.C. Depassier.
\newblock A variational method for nonlinear eigenvalue problems.
\newblock {\em Contemporary Mathematics}, 217:1--17, 1998. 

\bibitem{BY98}
R.D. Benguria and J.M. Y\'a\~nez.
\newblock Variational principle for the chemical potential in the 
Thomas--Fermi model.
\newblock {\em J.Phys. A: Math. Gen.}, 31:585--593, 1998.


\bibitem{GT98} 
D. Gilbarg and N.S. Trudinger.
\newblock  Partial Differential Equations. Second Edition. Grundlehren
der mathematischen Wissenschaften 224. 
\newblock Springer--Verlag, Berlin, 1998.  


\bibitem{PW67}
M.H. Protter and H.F. Weinberger.
\newblock Maximum principles in differential equations.
\newblock Prentice--Hall, Englewood Cliffs, NJ., 1967.

\bibitem{Ra70}
P. H. Rabinowitz.
\newblock Nonlinear Sturm--Liouville Problems for Second Order
Ordinary Differential Equations.
\newblock {\em Comm. Pure Applied Math.}, XXIII:939--961, 1970.

\bibitem{Ra77} 
P. H. Rabinowitz (Ed.)
\newblock Applications of Bifurcation Theory.
\newblock Academic Press, New York, 1977.


\bibitem{Sa73}
D.H. Sattinger.
\newblock Topics in stability and bifurcation theory.
\newblock Lecture notes in mathematics, vol. 309.
\newblock Springer--Verlag, Berlin, 1973.




\end{thebibliography}

\noindent{\sc Rafael D. Benguria}
\\ 
P. Universidad Cat\'olica de Chile \\
Casilla 306, Santiago 22, Chile \\ 
e-mail: rbenguri@fis.puc.cl
\smallskip
 
\noindent{\sc M. Cristina Depassier} \\
Departamento de F\'\i sica\\ 
P. Universidad Cat\'olica de Chile \\
Casilla 306, Santiago 22, Chile \\ 
e-mail: mcdepass@fis.puc.cl 

\end{document}