Liouville theorems for the weighted Lane-Emden equation with finite Morse indices

Abstract

In this paper, we study the nonexistence result for the weighted Lane–Emden equation: −Δu=f(|x|)|u|p−1u,x∈RN(0.1) and the weighted Lane–Emden equation with nonlinear Neumann boundary condition: −Δu=f(|x|)|u|p−1u,x∈R+N,∂u∂ν=g(|x|)|u|−1u,x∈∂R+N,(0.2) where f(|x|) and g(|x|) are the radial and continuously differential functions, R+N={x=(x′,xN)∈RN−1×R+} is an upper half space in RN, and ∂R+N={x=(x′,0),x′∈RN−1}. Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems (0.1) and (0.2) under appropriate assumptions on f(|x|) and g(|x|). Copyright © 2017 John Wiley & Sons, Ltd.