Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions of (E) $(-\Delta)^\alpha u+g(u)=\nu$ in a bounded regular domain $\Omega$ in $\R^N (N\ge2)$ which vanish on $\R^N\setminus\Omega$, where $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ is a Radon measure and $g$ is a nondecreasing function satisfying some extra hypothesis. When $g$ satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where $\nu$ is Dirac measure, we characterize the asymptotic behavior of the solution. When $g(r)=|r|^{k-1}r$ with $k$ supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved

Semilinear fractional elliptic equations with gradient nonlinearity involving measures

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 $\R^N (N\ge2)$, 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 $g:\R_+\mapsto\R_+$ 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, $g(s)=s^p$, $p\geq 1$ and $\epsilon=1$.Comment: \`a para\^itre, J. Funct. Ana