Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well

Abstract

In this paper, we study the following fractional Kirchhoff equations{(a+b∫RN|(−△)α2u|2dx)(−△)αu+λV(x)u=(|x|−μ⁎G(u))g(u),u∈Hα(RN),N≥3, where a,b>0 are constants, and (−△)α is the fractional Laplacian operator with α∈(0,1),2<2α,μ⁎=2N−μN−2α≤2α⁎=2NN−2α, 0<μ<2α, λ>0, is real parameter. 2α⁎ is the critical Sobolev exponent. g satisfies the Berestycki–Lions-type condition (see [2]). By using Pohožaev identity and concentration-compact theory, we show that the above problem has at least one nontrivial solution. Furthermore, the phenomenon of concentration of solutions is also explored. Our result supplements the results of Lü (see [8]) concerning the Hartree-type nonlinearity g(u)=|u|p−1u with p∈(2,6−α).