Existence of solutions for critical fractional Kirchhoff problems

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.