A supercritical elliptic problem in a cylindrical shell

TL;DR

This paper investigates a supercritical elliptic PDE in a cylindrical shell domain, establishing the critical exponent for solution existence and nonexistence depending on the power p.

Contribution

It identifies the precise critical exponent for the elliptic problem in a cylindrical shell and proves existence and nonexistence results based on this exponent.

Findings

Critical exponent for the problem is 2_{N,m}^*

Existence of solutions for 2<p<2_{N,m}^*

Nonexistence of solutions for p≥2_{N,m}^*

Abstract

We consider the problem \[ -\Delta u=|u|^{p-2}u in \Omega, u=0 on \partial\Omega, \] where $\Omega:=\{(y,z)\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}: 0<a<|y|<b<\infty\}$, $0\leq m\leq N-1$ and $N\geq2$. Let $2_{N,m}^{\ast}:=2(N-m)/(N-m-2)$ if $m<N-2$ and $2_{N,m}^{\ast}:=\infty$ if $m=N-2$ or $N-1$. We show that $2_{N,m}^{\ast}$ is the true critical exponent for this problem, and that there exist nontrivial solutions if $2<p<2_{N,m}^{\ast}$ but there are no such solutions if $p\geq2_{N,m}^{\ast}$.