Nonexistence results for a class of fractional elliptic boundary value problems

TL;DR

This paper establishes nonexistence of positive solutions for certain fractional elliptic boundary value problems in star-shaped and unbounded domains, using the method of moving spheres and the Caffarelli-Silvestre extension.

Contribution

It extends nonexistence results to fractional elliptic problems with supercritical and subcritical nonlinearities, employing a novel approach suitable for fractional cases.

Findings

Nonexistence of positive solutions in star-shaped domains for supercritical nonlinearities.

Nonexistence results in some unbounded domains for subcritical nonlinearities.

Application of the method of moving spheres to fractional problems using Caffarelli-Silvestre extension.

Abstract

In this paper we study a class of fractional elliptic problems of the form $$ \Ds u= f(x,u) \quad \textrm{in} \O u=0\quad \textrm{in} \R^N \setminus \O,$$ where $s\in(0,1)$. We prove nonexistence of positive solutions when $\O$ is star-shaped and $f$ is supercritical. We also derive a nonexistence result for subcritical $f$ in some unbounded domains. The argument relies on the method of moving spheres applied to a reformulated problem using the Caffarelli-Silvestre extension \cite{CSilv} of a solution of the above problem. The standard approach in the case $s=1$ using Pohozaev type identities does not carry over to the case $0<s<1$ due to the lack of boundary regularity of solutions.