Nonexistence results for a class of fractional elliptic boundary value problems
Abstract
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∈(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.