Semilinear fractional elliptic equations with gradient nonlinearity involving measures

Huyuan Chen (1) , Laurent Veron (2)

1 Departamento de Ingeniea Matematica
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

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 $\epsilon=1$.

Mots clés: Fractional Laplacian; Radon measure; Green kernel; Dirac measure

Domaines: Equations aux dérivées partielles [math.AP]

Huyuan Chen, Laurent Veron. Semilinear fractional elliptic equations with gradient nonlinearity involving measures. 2013. ⟨hal-00856008v6⟩