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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 $L_V$. Thepotential $V$ is typically a H\"older continuous function in $\Omega$ thatexplodes as $\mathrm{dist}(x,F)^{-2}$ for some $F \subset \partial \Omega$. Ingeneral the above boundary value problem may not have a solution. We areinterested in questions related to the concept of 'reduced measures',introduced by Brezis, Marcus and Ponce for $V=0$. For positive measures, thereduced 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 signedmeasures, some of the present results are new even for $V=0$.

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