A priori estimates for semistable solutions of semilinear elliptic equations

A priori estimates for semistable solutions of semilinear elliptic equations

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Joel Spruck, Johns Hopkins University
Fine Hall 314

In this talk we will discuss semistable solutions of the boundary value problem $Lu+f(u)=0$ in $\Omega$ and $u=0$ on $\partial\Omega$, where $Lu:=\partial_i(a^{ij}u_j)$ is uniformly elliptic. By semistability we mean that the lowest Dirichlet eigenvalue of the linearized operator at u is nonnegative. The basic problem (which has a long history) is to obtain a priori $L^{\infty}$ bounds on a solution under minimal assumptions on $f(t)$.  A basic and standard assumption is that $u>0$ in $\Omega$  and $f\in C^2$ is positive, nondecreasing, and superlinear at infinity, i.e. $f(0)>0$, $f' \geq 0$ and $f(t)/t$ tends to infinity as $t$ tends to infinity. For radially symmetric solutions,  an $L^{\infty}$ bound for $u$ is known for $n\leq 9$. On the other hand there exists unbounded semistable solutions when $n\geq 10$ for $f(u)=e^u$. This problem, like many other semilinear elliptic problems studied in recent years, seems to be related to minimal surface stability but this still remains mysterious.