Boundary value problems with signed measure data for semilinear Schrödinger equations

Consider operators $L_V:=\mathrm{\Delta }+V$ in a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$. Assume that $V\in C^{\alpha }(\mathrm{\Omega })$ satisfies $\mathrm{\mid }V(x)\mathrm{\mid }\le \bar{a}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{-2}$ in $\mathrm{\Omega }$ and that $L_V$ has a (minimal) ground state ${\mathrm{\Phi }}_V$ in $\mathrm{\Omega }$. 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.