Semilinear fractional elliptic equations with gradient nonlinearity involving measures

We study the existence of solutions to the fractional elliptic equation (E1) (−∆)u + ǫg(|∇u|) = ν in a bounded regular domain Ω of R (N ≥ 2), subject to the condition (E2) u = 0 in Ω, where ǫ = 1 or −1, (−∆) denotes the fractional Laplacian with α ∈ (1/2, 1), ν is a Radon measure and g : R+ 7→ 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 ν is Dirac mass, g(s) = s, p ≥ 1 and ǫ = 1.