A Liouville theorem for elliptic equations in divergence form with a potential

In this paper we are concerned with elliptic equations in divergence form with a potential, posed in a bounded domain Ω. We allow the coefficients of the diffusion matrix $A(x)$ and the potential $Q(x)$ to diverge at the boundary; in addition, we permit that $Q(x)$ vanishes inside Ω, and $A(x)$ loses ellipticity at ∂Ω. The boundary ∂ Ω is assumed to be the (disjoint) union of a finite number p of submanifolds of dimension κᵢ∈ \0, …, n-1\\, (i=1, …, p). Under suitable assumptions on the behavior of $Q(x)$ and $A(x)$, which also depend on κᵢ, we prove the validity of a Liouville-type theorem. Finally, we show an example for which our hypotheses on Q and A are sharp.