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