Nonexistence results for a class of fractional elliptic boundary value problems

In this paper we study a class of fractional elliptic problems of the form $$ \Ds u= f(x,u) \quad \textrm{in} Øu=0\quad \textrm{in} \R^N \setminus Ø,$$ where $s\in(0,1)$. We prove nonexistence of positive solutions when $Ø$ 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.

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