Fractional elliptic equation with gradient nonlinearity and measure data

Marie-Françoise Bidaut-Veron (1) , Huyuan Chen (2) , Laurent Veron (1)

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

In this paper, we study fractional elliptic equation (E1) $ (-\Delta)^\alpha u+g(|\nabla u|)=\nu $ in a bounded regular domain $\Omega$ of $\R^N (N\ge2)$, satisfying (E2) $u=0$ in $\Omega^c$, where $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ is a Radon measure and $g:\R\mapsto\R$ is a continuous function satisfying. We prove the existence of weak solutions for problem (E1)-(E2) when $\nu$ is Radon measure and $g$ is subcritical. Furthermore, the asymptotic behaviors and uniqueness of solutions are analyzed when $\nu$ is Dirac measure.

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

Domaines: Equations aux dérivées partielles [math.AP]

Marie-Françoise Bidaut-Veron, Huyuan Chen, Laurent Veron. Fractional elliptic equation with gradient nonlinearity and measure data. 2013. ⟨hal-00856008v1⟩