Fractional Kirchhoff equation with a general critical nonlinearity

Abstract In this paper, for any dimension N > 2 s ( 0 s 1 ) , we study the fractional Kirchhoff equation a + b ∫ R N | ( − Δ ) s 2 u | 2 d x ( − Δ ) s u + u = f ( u ) in R N , with a critical nonlinearity, where ( − Δ ) 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 > 2 s . Our result improves the study made in the low dimension 2 s N 4 s .

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