Supercritical elliptic problems on a perturbation of the ball

Abstract

We examine the Hénon equation $ -Δu =|x|^αu^p$ in $ Ω\subset \mathbb{R}^N$ with $u=0$ on $ \partial Ω$ where $ 0 < α$. We show there exists a sequence $ \{p_k\}_k \subset [ \frac{N+2}{N-2}, p_α(N)]$ with $p_1 < p_2 <p_3 < ...$, $ p_k \nearrow p_α(N)$ such that for any $ \frac{N+2}{N-2} \le p < p_α(N)$, which avoids $ \{p_k\}_k $, there exists a positive classical solution of the Hénon equation, provided $ Ω$ 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. $ -Δu = u^p$ in $ Ω$, an exterior domain, with $ u=0 $ on $ \partial Ω$. 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 $ Ω$ is a sufficiently small perturbation of the exterior of the unit ball.