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Solutions for perturbed biharmonic equations with critical nonlinearity
Author(s) -
Wang Chunhua
Publication year - 2014
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.2846
Subject(s) - biharmonic equation , mathematics , sobolev space , bounded function , exponent , nonlinear system , critical exponent , mathematical analysis , operator (biology) , pure mathematics , scaling , geometry , boundary value problem , physics , philosophy , linguistics , biochemistry , chemistry , repressor , quantum mechanics , transcription factor , gene
In this paper, we study the perturbed biharmonic equationsε 4Δ 2 u + V ( x ) u = P ( x ) | u | p − 2 u + Q ( x ) | u |2 * * − 2 u , x ∈ R N ,u ∈ H 2(R N) , u ( x ) → 0 , as | x | → ∞ ,where Δ 2 is the biharmonic operator, N ≥ 5 , 2 * * = 2 N N − 4is the Sobolev critical exponent, p ∈ (2,2 * * ), P ( x ), and Q ( x ) are bounded positive functions. Under some given conditions on V , we prove that the problem has at least one nontrivial solution provided that ε ≤ E and that for anyn * ∈ N , it has at least n * pairs solutions if ε ≤ En *, where E andEn *are sufficiently small positive numbers. Moreover, these solutions u ε → 0 inH 2(R N) as ε → 0. Copyright © 2013 The authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.