Semilinear fractional elliptic equations involving measures

Abstract We study the existence of weak solutions to (E) ( − Δ ) α u + g ( u ) = ν in a bounded regular domain Ω in R N ( N ≥ 2 ) which vanish in R N ∖ Ω , where ( − Δ ) α denotes the fractional Laplacian with 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 − 1 r 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.