Semilinear fractional elliptic equations with gradient nonlinearity involving measures

We study the existence of solutions to the fractional elliptic equation (E1) (−Δ)αu+ϵg(|∇u|)=ν(−Δ)αu+ϵg(|∇u|)=ν in an open bounded regular domain Ω   of RN(N⩾2)RN(N⩾2), subject to the condition (E2) u=0u=0 in ΩcΩc, where ϵ=1ϵ=1 or −1, (−Δ)α(−Δ)α denotes the fractional Laplacian with α∈(1/2,1)α∈(1/2,1), ν   is a Radon measure and g:R+↦R+g:R+↦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 ϵ=1ϵ=1, ν   is Dirac mass and g(s)=spg(s)=sp with p∈(0,NN−2α+1).