Semilinear fractional elliptic equations involving measures

Original · EN

We study the existence of weak solutions of (E) (-Δ)αu+g(u)=ν in a bounded regular domain Ω in ⁿ (N≥2) which vanish on ⁿΩ, 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 hypothesis. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where ν is Dirac measure, we characterize the asymptotic behavior of the solution. When g(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.