Ground State Solution of Kirchhoff Problems with Hartree Type Nonlinearity

For the Kirchhoff type equation -a+b∫R3∇u2dxΔu+V(x)u=(Iα∗|u|p)|u|p-2uinR3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^3}\left| \nabla u\right| ^2\,dx\right) \Delta u+V(x)u = (I_{\alpha }*|u|^p) |u|^{p-2}u\ \ \text {in}\ \mathbb {R}^3, \end{aligned}$$\end{document}where a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,\,b>0$$\end{document}, 0<α<3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha <3$$\end{document}, 2<p<3+α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<p<3+\alpha $$\end{document} and Iα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_\alpha $$\end{document} 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.