Boundary value problems with signed measure data for semilinear Schrödinger equations

Abstract

Consider operators $L_V:=\Delta +V$ in a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^N$. Assume that $V\in C^{\alpha }(\Omega )$ satisfies $|V(x)|\le \bar{a}\mathrm{dist}(x,\partial \Omega {)}^{-2}$ in $\Omega$ and that $L_V$ has a (minimal) ground state ${\Phi }_V$ in $\Omega$. We derive a representation formula for signed supersolutions (or subsolutions) of $L_{V}u=0$ possessing an $L_V$ boundary trace. We apply this formula to the study of some questions of existence and uniqueness for an associated semilinear boundary value problem with signed measure data.

14 pages