Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry

Abstract

As key problems we consider semilinear elliptic equations $\triangle u + \lambda f(u)=0\, \quad u|_{\partial {\Omega}} =0$ whose nonlinearities have supercritical growth near $u=\infty$ with $f(0)\ge 0$. For examples of these problems on ball domains it is known that there is a $\lambda^*>0$ such that there exists a unique small nonnegative solution for $0 <\lambda <\lambda^*$. This is in contrast to the situation where $f$ has superlinear but subcritical growth , in which case always also a large unstable solution can be found. We show that if $\Omega\subset{{\Bbb R}}^n$ is starshaped, $f(s)>0$ as $s\to\infty$ and $\lim\sup_{s\to\infty} \frac{F(s)}{sf(s)} <\frac{n-2}{2n}$ that a $\lambda^*$ as above always exists. We do not have to assume anything on the behavior of $f$ between zero and infinity. More generally we can assign to any domain $\Omega$ a number $M(\Omega)$ which in the case that $M(\Omega) < \frac12$ allows the same conclusion if in the above growth condition $\frac{n-2}{2n}$ is replaced by $\frac12-M(\Omega)$. There exist nonstarshaped domains in ${{\Bbb R}}^n$, $n \ge 3$ and topologically nontrivial ones in ${{\Bbb R}}^n$, $n\ge 4$ for which $M(\Omega) < \frac 12$. This indicates that for such domains a generalized critical exponent $\le\frac{1+2M(\Omega)}{ 1-2M(\Omega)}$ exists. Moreover, for starshaped domains $M(\Omega)=\frac 1 n$, so in this case the expression gives the usual critical exponent $\frac {n+2}{n-2}$.