Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions to (E) (−Δ)αu+g(u)=ν(−Δ)αu+g(u)=ν in a bounded regular domain Ω   in RN(N≥2)RN(N≥2) which vanish in RN∖ΩRN∖Ω, where (−Δ)α(−Δ)α denotes the fractional Laplacian with α∈(0,1)α∈(0,1), ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν   is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|k−1rg(r)=|r|k−1r 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.