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

2021/10/14 by Mousomi Bhakta, Bhakta, Mousomi, Moshe Marcus +3
Mathematics · Computer Science · #Advanced Mathematical Physics Problems #Differential Equations and Boundary Problems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2110.07445

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, where L_V= Δ+V, f∈ C(ℝ) is 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 by Brezis, Marcus and Ponce 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 for the case V=0, 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.

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