Multiplicity of solutions for a class of Kirchhoff type equations with Hardy-Littlewood-Sobolev critical nonlinearity

In this paper, we study the following Kirchhoff type equations with Hardy-Littlewood-Sobolev critical nonlinearity {-m(integral(Omega)vertical bar del u vertical bar(2)dx)Delta u = lambda f(x, u) + (integral(Omega)vertical bar u(y)vertical bar(2 mu)*/vertical bar x-y vertical bar(mu)dy)vertical bar u vertical bar(2 mu)*(-2)u, x is an element of Omega, u = 0, x is an element of partial derivative Omega, where N >= 3, Omega subset of R-N is a bounded smooth domain, 2(mu)* = 2N-mu/N-2, lambda, mu > 0 and m is a locally increasing positive function. The nonlinearity f is odd in the second variable and satisfies some quasi-critical growth conditions. For any given k is an element of N, k pairs of nontrivial solutions are got for lambda large enough via variational methods. (C) 2019 Elsevier Ltd. All rights reserved.