Some Liouville theorems for Hénon type equations in half-space with nonlinear boundary value conditions and finite Morse indices

Abstract

In this paper, we are concerned with Liouville-type theorems for Henon type equations in half-space with nonlinear boundary value conditions. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the integral estimates, the Pohozaev-type identity and the monotonicity formula of solutions and blowing down sequence.