Ground State Solution of Kirchhoff Problems with Hartree Type Nonlinearity

Abstract

For the Kirchhoff type equation

$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^3}\left| \nabla u\right| ^2\,dx\right) \Delta u+V(x)u = (I_{\alpha }*|u|^p) |u|^{p-2}u\ \ \text {in}\ \mathbb {R}^3, \end{aligned}$$

where $a,\,b>0$, $0<\alpha <3$, $2<p<3+\alpha$ and $I_\alpha$ is the Riesz potential, we establish the existence of a positive ground state solution by using Nehari manifold technique and concentration compactness argument. The main novelty in our context is that the potential V exhibits a mixed behavior, i.e., V is periodic in some directions while tends to a positive constant in the remaining ones.