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Existence of solutions for critical fractional Kirchhoff problems
Author(s) -
Zhang Xia,
Zhang Chao
Publication year - 2016
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.4085
Subject(s) - mathematics , sobolev space , fractional laplacian , minimax , truncation (statistics) , compact space , operator (biology) , critical exponent , laplace operator , exponent , sobolev inequality , space (punctuation) , mathematical analysis , nonlinear system , mountain pass theorem , pure mathematics , geometry , mathematical optimization , biochemistry , statistics , chemistry , linguistics , philosophy , physics , repressor , quantum mechanics , scaling , transcription factor , gene
Consider the following fractional Kirchhoff equations involving critical exponent:1 + λ 1∫R N( | ( − Δ )α 2u | 2 + V ( x ) u 2 ) dx [ ( − Δ ) α u + V ( x ) u ] = k ( x ) f ( u ) + λ 2 | u |2 α ∗ − 2 u inR N ,where (−Δ) α is the fractional Laplacian operator with α ∈(0,1), N ≥ 2 ,λ 1 ≥ 0 , λ 2 >0 and2 α ∗ = 2 N / ( N − 2 α ) is the critical Sobolev exponent, V ( x ) and k ( x ) are functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space, the minimax arguments, Pohozaev identity, and suitable truncation techniques, we obtain the existence of a nontrivial weak solution for the previously mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity f . Copyright © 2016 John Wiley & Sons, Ltd.