Fractional Kirchhoff equation with a general critical nonlinearity

In this paper, for any dimension N > 2s (0 < s < 1), we study the fractional Kirchhoff equation (a +b integral(RN) vertical bar(-Delta)(8/2) u vertical bar(2) dx) (-Delta)(s)u + u= f (u) in R-N, with a critical nonlinearity, where (Delta-)(s) is the fractional Laplacian. By using a perturbation approach, we prove the existence of solutions to the above problem without the Ambrosetti Rabinowitz condition when the parameter b is small. Moreover, we obtain the asymptotic behavior of solutions as b -> 0. The method we use and the result we get are applicable in any dimension N > 2s. Our result improves the study made in the low dimension 2s < N < 4s. (C) 2017 Elsevier Ltd. All rights reserved.