Boundary value problems for semilinear Schr\"odinger equations with singular potentials and measure data

TL;DR

This paper investigates boundary value problems for semilinear Schrödinger equations with singular Hardy-type potentials and measure data, extending the concept of reduced measures and analyzing existence conditions for solutions.

Contribution

It extends the theory of reduced measures to Schrödinger equations with singular potentials, relaxing conditions on the nonlinearity and addressing signed measure data.

Findings

Extended reduced measure concept to singular potentials

Established existence criteria for boundary value problems

Provided new results for signed measure data

Abstract

We study boundary value problems with measure data in smooth bounded domains $\Omega$, for semilinear equations involving Hardy type potentials. Specifically we consider problems of the form $-L_V u + f(u) = \tau$ in $\Omega$ and $\mathrm{tr}^*u=\nu$ on $\partial \Omega$, where $L_V= \Delta+V$, $f\in C(\mathbb{R})$ is monotone increasing with $f(0)=0$ and $\mathrm{tr}^*u$ denotes the normalized boundary trace of $u$ associated with $L_V$. The potential $V$ is typically a H\"older continuous function in $\Omega$ that explodes as $\mathrm{dist}(x,F)^{-2}$ for some $F \subset \partial \Omega$. 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 $\tau^*, \nu^*$ are the largest measures dominated by $\tau$ and $\nu$ respectively such that the boundary value problem with data $(\tau^*,\nu^*)$ 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 $V=0$.