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 $\Omega$, for semilinear equations involving Hardy type potentials. Specifically we consider problems of the form $-L_{V}u+f(u)=\tau$ in $\Omega$ and ${\mathrm{tr}}^{\ast }u=\nu$ on $\partial \Omega$, where $L_V=\Delta +V$, $f\in C(\mathbb{R})$ is monotone increasing with $f(0)=0$ and ${\mathrm{tr}}^{\ast }u$ denotes the normalized boundary trace of $u$ associated with $L_V$. The potential $V$ is typically a Hölder 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 ${\tau }^{\ast },{\nu }^{\ast }$ are the largest measures dominated by $\tau$ and $\nu$ respectively such that the boundary value problem with data $({\tau }^{\ast },{\nu }^{\ast })$ 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 $V=0$.

In the second version, we used the notion of L_V boundary trace instead of normalized boundary trace, made several major modifications, and changed the structure of the paper