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Liouville theorems for the weighted Lane–Emden equation with finite Morse indices
Author(s) -
Chen Caisheng,
Wang Hui
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4333
Subject(s) - mathematics , identity (music) , morse code , mathematical analysis , space (punctuation) , boundary (topology) , neumann boundary condition , nonlinear system , differential equation , energy (signal processing) , pure mathematics , statistics , electrical engineering , engineering , linguistics , philosophy , physics , quantum mechanics , acoustics
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.