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

In this paper, we study the nonexistence result for the weighted Lane-Emden equation: -Delta u = f(vertical bar x vertical bar)vertical bar u vertical bar(p-1)u, x is an element of R-N (0.1) and the weighted Lane-Emden equation with nonlinear Neumann boundary condition: {-Delta u = f(vertical bar x vertical bar)vertical bar u vertical bar(p-1)u, x is an element of R-+(N) , partial derivative u/partial derivative v =g(vertical bar x vertical bar)vertical bar u vertical bar(-1) u, x is an element of R-+(N), (0.2) where f(vertical bar x vertical bar) and g(vertical bar x vertical bar) are the radial and continuously differential functions, R-+(N) = {x= (x',x(N)) is an element of RN-1 x R+} is an upper half space in R-N, and partial derivative R-+(N) = {x = (x', 0), x' is an element of 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(vertical bar x vertical bar) and g(vertical bar x vertical bar) Copyright (C) 2017 JohnWiley & Sons, Ltd.