Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

TL;DR

This paper investigates the existence and multiplicity of solutions to supercritical elliptic problems, revealing that domain topology influences solution existence and showing examples where solutions are absent or numerous depending on the domain's homology and the exponent.

Contribution

It constructs domains with complex homology where solutions to supercritical elliptic problems either do not exist or are multiple, extending previous results to richer topologies and specific dimensions.

Findings

Domains with richer homology can lack solutions for certain supercritical exponents.

Existence of multiple solutions is demonstrated in domains related to Hopf fibrations for specific dimensions.

The topology of the domain critically affects solution existence in supercritical elliptic problems.

Abstract

We consider the supercritical problem -\Delta u = |u|^{p-2}u in \Omega, u=0 on \partial\Omega, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ Bahri and Coron showed that if $\Omega$ has nontrivial homology this problem has a positive solution for $p=2^{*}.$ However, this is not enough to guarantee existence in the supercritical case. For $p\geq 2(N-1)/(N-3)$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as $p$ increases. More precisely, we show that for $p> 2(N-k)/(N-k-2)$ with $1\leq k\leq N-3$ there are bounded smooth domains in $\mathbb{R}^{N}$ whose cup-length is $k+1$ in which this problem does not have a nontrivial solution. For $N=4,8,16$ we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.