Existence of Solutions for Planar Kirchhoff–Choquard Problems

Abstract

In this article, we are interested in the study of the following Kirchhoff–Choquard equations: −a+b∫R2|∇u|2dxΔu+V(x)u=λ(ln|x|∗u2)u+f(u),x∈R2, where λ>0,a>0,b>0, V and f are continuous functions with some appropriate assumptions. We prove that when the parameter λ is sufficiently small, the above problem has a mountain pass solution, a least energy solution and a ground state solution by applying the variational methods and building some subtle inequalities.