In this paper we are concerned with elliptic equations in divergence form with a potential, posed in a bounded domain. 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, and A (x) loses ellipticity at. The boundary is assumed to be the (disjoint) union of a finite number p of submanifolds of dimension ᵢ \0, , n-1\\, (i=1, , p). Under suitable assumptions on the behavior of Q (x) and A (x), which also depend on 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.
Biagi et al. (Mon,) studied this question.