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 κ i ∈ {0, …, n − 1} ( i = 1, …, p ). Under suitable assumptions on the behavior of Q ( x ) and A ( x ), which also depend on κ i , we prove the validity of a Liouville-type theorem. Finally, we show an example for which our hypotheses on Q and A are sharp.