Liouville-type results for semilinear elliptic equations in unbounded domains

Abstract

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: $$ -a_{ij}(x)\partial_{ij}u(x) -- q_i(x)\partial_iu(x)=f(x,u(x)),\quad x\in{\mathbb R}^N. $$ The aim is to prove uniqueness of positive- bounded solutions—Liouville-type theorems. Along the way, we establish also various existence results. We first derive a sufficient condition, directly expressed in terms of the coefficients of the linearized operator, which guarantees the existence result as well as the Liouville property. Then, following another approach, we establish other results relying on the sign of the principal eigenvalue of the linearized operator about u= 0, of some limit operator at infinity which we define here. This framework will be seen to be the most general one. We also derive the large time behavior for the associated evolution equation.