Supercritical elliptic problems on a perturbation of the ball

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.

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