\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 $$\frac{d^2u}{dx^2}+\lambda \, u = N(u) \label{eq:i1}$$ 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, $$\lambda[u_m]=\max_{g \in D} \left( \int_0^{u_m} N(u) g(u) \, du + \frac{1}{2} \left(\int_0^{u_m} g'(u)^{1/3} \, du\right)^3 \right)/\int_0^{u_m} u g(u) \, du, \label{eq:i2}$$ 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 $$-\Delta u + u^3 = \lambda u \quad \mbox{in \Omega \subset \mathbb{R}^3,} \label{eq:e1}$$ with $$u=0 \quad \mbox{in \partial \Omega}. \label{eq:e2}$$ 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 $$4 \pi u(x) \le \frac{2}{3\sqrt{3}} \lim_{\epsilon \to 0} \int_{\Omega \setminus B_{\epsilon}(x)} \frac{(\Delta g + \lambda g)^{3/2}}{g^{1/2}} \, dy, \label{eq:e3}$$ where $B_{\epsilon}(x) \subset \Omega$ is a ball of radius $\epsilon$ centered at $x$. Here the function $g$ satisfies, $$g \in C^2(\Omega \setminus B_{\epsilon}(x)) \cap C^0(\overline{\Omega \setminus B_{\epsilon}(x)}), \label{eq:e4}$$ $$g=0 \quad \mbox{in \partial \Omega,} \label{eq:e5}$$ $$g(y) \approx \frac{1}{\vert x -y \vert} \quad \mbox{in the neighborhood of x,} \label{eq:e6}$$ (i.e., $g(y)$ behaves like the fundamental solution around $x$), $$g(y) > 0 \quad \mbox{and} \quad \Delta g + \lambda g >0 \quad \mbox{in \Omega \setminus B_{\epsilon}(x),} \label{eq:e7}$$ 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, $$-\int_{\Omega \setminus B_{\epsilon}(x)} g \, \Delta u \, dy + \int_{\Omega \setminus B_{\epsilon}(x)} g \, u^3 \, dy = \lambda \int_{\Omega \setminus B_{\epsilon}(x)} g\, u \, dy. \label{eq:e8}$$ Using Green's formula and the boundary conditions (\ref{eq:e2}) and (\ref{eq:e5}) satisfied by $u$ and $g$ repectively, we have $$\int_{\Omega \setminus B_{\epsilon}(x)} (g \Delta u - u \Delta g) \, dy = - \int_{ \partial B_{\epsilon}(x)} (g \nabla u-u\nabla g) \cdot \hat n \, dS, \label{eq:e9}$$ 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, $$\lim_{\epsilon \to 0} \int_{\Omega \setminus B_{\epsilon}(x)} (g \Delta u - u \Delta g) \, dy = - 4 \pi u(x). \label{eq:e10}$$ Hence, using (\ref{eq:e8}), (\ref{eq:e9}), and (\ref{eq:e10}) we have $$4 \pi u(x) = \lim_{\epsilon \to 0} \int_{\Omega \setminus B_{\epsilon}(x)} \left((\Delta g + \lambda g) u - g u^3 \right) \, dy. \label{eq:e11}$$ 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 $$h \, u - u^3 \, g \le \frac{2}{3\sqrt{3}} \frac{h^{3/2}}{g^{1/2}} \label{eq:e12}$$ 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 $$\Delta g = - \frac{\pi^2}{4} g, \quad \mbox{for r \neq 0.} \label{eq:e12a}$$ 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}) $$u_m \equiv u(0) \le \frac{4 \sqrt{3}}{9 \pi}(1-\frac{2}{\pi}) \left(\lambda - \frac{\pi^2}{4}\right)^{3/2}, \label{eq:e13}$$ which gives the following lower bound for the {\it nonlinear eigenvalue} $\lambda$, $$\lambda \ge \frac{\pi^2}{4} + \left[\frac{3 \sqrt{3}}{4} \pi \frac{u_m}{(1-2/\pi)} \right]^{2/3} =\frac{\pi^2}{4}+ c u_m^{2/3}, \label{eq:e13b}$$ 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 $$g(x)=\frac{1}{r}(1-r), \label{eq:13c}$$ 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