Boundary value problems for semilinear Schr\"odinger equations with singular potentials and 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 $V=0$.