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

Dipartimento di Matematica, Politecnico di Milano, Italy

^* Corresponding author: stefano.biagi@polimi.it

Received: 29 January 2025
Accepted: 27 September 2025

Abstract

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.