Supercritical elliptic problems on a perturbation of the ball

We examine the H'enon equation $-\mathrm{\Delta }u=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }{u}^p$ 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 $\mathrm{\Omega }$ 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. $-\mathrm{\Delta }u=u^p$ in $\mathrm{\Omega }$, an exterior domain, with $u=0$ on $ \partial \Omega$\mathrm{.}Weshowtheexistenceof$ \frac{N+2}{N-2} \le p_1 < p_2 < p_3<…$with$ p_k \rightarrow \infty$suchthatif$ \frac{N+2}{N-2} < p$, which avoids $\{p_k{\}}_k$, then there exists a positive \emph{fast decay} classical solution, provided $\mathrm{\Omega }$ is a sufficiently small perturbation of the exterior of the unit ball.