Semilinear fractional elliptic equations with gradient nonlinearity involving measures

Abstract

We study the existence of solutions to the fractional elliptic equation (E1) $(-\Delta {)}^{\alpha }u+\epsilon g(|\nabla u|)=\nu$ in a bounded regular domain $\Omega$ of ${\mathbb{R}}^N(N\ge 2)$, subject to the condition (E2) $u=0$ in ${\Omega }^c$, where $\epsilon =1$ or $-1$, $(-\Delta {)}^{\alpha }$ denotes the fractional Laplacian with $\alpha \in (1/2,1)$, $\nu$ is a Radon measure and $g:{\mathbb{R}}_{+}\mapsto {\mathbb{R}}_{+}$ is a continuous function. We prove the existence of weak solutions for problem (E1)-(E2) when $g$ is subcritical. Furthermore, the asymptotic behavior and uniqueness of solutions are described when $\nu$ is Dirac mass, $g(s)=s^p$, $p\ge 1$ and $\epsilon =1$.

à paraître, J. Funct. Anal