Supercritical elliptic problems on a perturbation of the ball

Abstract

We examine the H\'enon equation $ -\Delta u =|x|^\alpha u^p$ in $ \Omega \subset \mathbb{R}^N$ with $u=0$ on $ \partial \Omega$ where $ 0 < \alpha$. We show there exists a sequence $ \{p_k\}_k \subset [ \frac{N+2}{N-2}, p_{\alpha}(N)]$ with $p_1 < p_2 <p_3 < ...$, $ p_k \nearrow p_{\alpha}(N)$ such that for any $ \frac{N+2}{N-2} \le p < p_{\alpha}(N)$, which avoids $ \{p_k\}_k $, there exists a positive classical solution of the H\'enon equation, provided $ \Omega$ is a sufficiently small perturbation of the unit ball. We also examine the Lane-Emden-Fowler equation in the case of an exterior domain; ie. $ -\Delta u = u^p$ in $ \Omega$, an exterior domain, with $ u=0 $ on $ \partial \Omega$. We show the existence of $ \frac{N+2}{N-2} \le p_1 < p_2 < p_3<...$ with $ p_k \rightarrow \infty$ such that if $ \frac{N+2}{N-2} < p$, which avoids $\{p_k\}_k$, then there exists a positive \emph{fast decay} classical solution, provided $ \Omega$ is a sufficiently small perturbation of the exterior of the unit ball.