Semilinear elliptic Schrödinger equations involving singular potentials and source terms

Abstract

Let Ω⊂RN (N>2) be a C2 bounded domain and Σ⊂Ω be a compact, C2 submanifold without boundary, of dimension k with 0≤k<N−2. Put Lμ=Δ+μdΣ−2 in Ω∖Σ, where dΣ(x)=dist(x,Σ) and μ is a parameter. We study the boundary value problem (P) −Lμu=g(u)+τ in Ω∖Σ with condition u=ν on ∂Ω∪Σ, where g:R→R is a nondecreasing, continuous function and τ and ν are positive measures. The interplay between the inverse-square potential dΣ−2, the nature of the source term g(u) and the measure data τ,ν yields substantial difficulties in the research of the problem. We perform a deep analysis based on delicate estimate on the Green kernel and Martin kernel and fine topologies induced by appropriate capacities to establish various necessary and sufficient conditions for the existence of a solution in different cases.