Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in exterior domains

TL;DR

This paper investigates positive solutions of nonlinear elliptic equations in exterior domains and the whole space, establishing existence results for ground state and bound state solutions under minimal assumptions on the potential.

Contribution

It provides new existence results for ground and bound state solutions in exterior domains for nonlinear elliptic problems with critical growth, without size restrictions on the domain or potential.

Findings

Existence of ground state solutions under certain conditions.

Existence of bound state solutions for small perturbations.

No restrictions on the size of the exterior domain or potential deviations.

Abstract

The paper concerns with positive solutions of problems of the type $-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $\Omega\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $\Omega$ can be an exterior domain, i.e. $\mathbb{R}^N\setminus\Omega$ bounded, or the whole of $\mathbb{R}^N$. The potential $a\in L^{N/2}_{\rm loc}(\mathbb{R}^N)$ is assumed to be strictly positive and such that there exists $\lim_{|x|\to\infty}a(x):=a_\infty$, with $a_\infty>0$; in particular $a\equiv {\rm const}$ is allowed. First, some existence results of ground state solutions are proved. Then the case $a(x)\ge a_\infty$ is considered, with $a(x)\not\equiv a_\infty$ or $\Omega\neq\mathbb{R}^N$. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small $\varepsilon$. No hypotheses are assumed on the size of $\mathbb{R}^N\setminus\Omega$ and on $\|a-a_\infty\|_{L^{N/2}}$.