Monotonicity and nonexistence results for some fractional elliptic problems in the half space

Abstract

We study a class of fractional elliptic problems of the form $\Ds u= f(u)$ in the half space $\R^N_+:=\{x \in \R^N\::\: x_1>0\}$ with the complementary Dirichlet condition $u \equiv 0$ in $\R^N \setminus \R^N_+$. Under mild assumptions on the nonlinearity $f$, we show that bounded positive solutions are increasing in $x_1$. For the special case $f(u)=u^q$, we deduce nonexistence of positive bounded solutions in the case where $q \ge 1$ and $q<\frac{N-1+2s}{N-1-2s}$ if $N \ge 1+2s$. We do not require integrability assumptions on the solutions we study.