Semilinear elliptic Schrodinger equations involving singular potentials and source terms

Let Omega subset of R-N (N > 2) be a C-2 bounded domain and Sigma subset of Omega be a compact, C-2 submanifold without boundary, of dimension k with 0 <= k < N - 2. Put L-mu = Delta + mu d(Sigma)(-2) in Omega \ Sigma, where d(Sigma)(x) = dist(x, Sigma) and mu is a parameter. We study the boundary value problem (P) -L(mu)u = g(u) + tau in Omega \ Sigma with condition u = nu on partial derivative Omega boolean OR Sigma, where g : R -> R is a nondecreasing, continuous function and tau and nu are positive measures. The interplay between the inverse-square potential d(Sigma)(-2), S, the nature of the source term g(u) and the measure data tau, nu 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. (c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

This article studies the boundary value problem with an inverse-square potential and measure data. By analyzing the Green kernel and Martin kernel and using appropriate capacities, necessary and sufficient conditions for the existence of a solution are established in different cases.