Supercritical elliptic problems on a perturbation of the ball

TL;DR

This paper proves the existence of positive solutions for supercritical elliptic equations on domains close to a ball, identifying specific ranges of the exponent where solutions exist, including for exterior domains.

Contribution

It establishes existence results for supercritical elliptic problems on perturbed ball domains, introducing sequences of exponents where solutions are guaranteed, extending previous results.

Findings

Existence of solutions for certain supercritical exponents on perturbed ball domains.

Identification of sequences of exponents approaching critical or infinity where solutions exist.

Solutions exhibit fast decay in exterior domain cases.

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.