Existence of solutions for critical fractional Kirchhoff problems

Abstract

Consider the following fractional Kirchhoff equations involving critical exponent: urn:x-wiley:mma:media:mma4085:mma4085-math-0001 where (−Δ)α is the fractional Laplacian operator with α∈(0,1), , , λ2>0 and 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.