A note on Kirchhoff-type equations with Hartree-type nonlinearities

Abstract

We consider the following Kirchhoff-type equation in R3−(a+b∫R3|∇u|2dx)Δu+(1+μg(x))u=(1|x|α∗|u|p)|u|p−2u, where a>0, b≥0 are constants, α∈(0,3), p∈(2,6−α), μ>0 is a parameter and g(x) is a nonnegative continuous potential satisfying some conditions. By using the Nehari manifold and the concentration compactness principle, we establish the existence of ground state solutions for the equation if the parameter μ is large enough. Moreover, some concentration behaviors of these solutions as μ→+∞ are discussed.