Boundary value problems for semilinear Schrödinger equations with singular potentials and measure data

Abstract

We study boundary value problems with measure data in smooth bounded domains Ω, for semilinear equations involving Hardy type potentials. Specifically we consider problems of the form −L V u + f (u) = τ in Ω and tr * u = ν on ∂Ω, whereis monotone increasing with f (0) = 0 and tr * u denotes the normalized boundary trace of u associated with L V . The potential V is typically a Hölder continuous function in Ω that explodes as dist (x, F ) −2 for some F ⊂ ∂Ω. In general the above boundary value problem may not have a solution. We are interested in questions related to the concept of 'reduced measures', introduced in [4] for V = 0. For positive measures, the reduced measures τ * , ν * are the largest measures dominated by τ and ν respectively such that the boundary value problem with data (τ * , ν * ) has a solution. Our results extend results of [4] and [6], including a relaxation of the conditions on f . In the case of signed measures, some of the present results are new even for V = 0.