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The existence of a nontrivial solution to p‐Laplacian equations in R N with supercritical growth
Author(s) -
Li Gongbao,
Wang Chunhua
Publication year - 2012
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.2570
Subject(s) - mathematics , laplace operator , supercritical fluid , p laplacian , norm (philosophy) , operator (biology) , perturbation (astronomy) , mathematical physics , combinatorics , pure mathematics , mathematical analysis , physics , law , thermodynamics , chemistry , quantum mechanics , biochemistry , repressor , political science , transcription factor , gene , boundary value problem
In this paper, we consider the p ‐Laplacian equations inR Nwith supercritical growth− △ p u + V ( x ) | u | p − 2 u = f ( u ) ,x ∈R N ,u ∈W 1 , p(R N) ,where △ p u = div( | ∇ u | p − 2 ∇ u ),1 < p < N is the p ‐Laplacian operator. Under certain assumptions on V ( x ) and f ( u ) that will be given in Section 1, we prove that the problem has at least a nontrivial solution by using variational methods combined with perturbation arguments. The solutions to subcritical p‐Laplacian equations are estimated applying the L ∞ norm. Copyright © 2012 John Wiley & Sons, Ltd.