Semiclassical Ground State Solutions for a Class of Kirchhoff-Type Problem with Convolution Nonlinearity

In this paper, we discuss the following Kirchhoff-type problem with convolution nonlinearity -ε2+bε∫R3|∇v|2dx▵v+V(x)v=ε-α(Iα∗F(v))f(v),x∈R3,v∈H1(R3),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&-\left( \varepsilon ^{2}+ b\varepsilon \int _{ {\mathbb {R}}^{3}}|\nabla v|^{2} dx \right) \triangle v+ V(x)v=\varepsilon ^{-\alpha }(I_{\alpha }*F(v))f( v),&x\in {\mathbb {R}}^{3},\\&v\in H^{1}({\mathbb {R}}^{3}), \end{aligned} \right. \end{aligned}$$\end{document}where b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b>0$$\end{document}, Iα:R3→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{\alpha }:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}$$\end{document}, with α∈(0,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,3)$$\end{document}, is the Riesz potential, V is differentiable, f∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in {\mathbb {C}}({\mathbb {R}},{\mathbb {R}})$$\end{document} and F(t)=∫0tf(s)ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(t)=\int ^{t}_{0}f(s)ds$$\end{document}. By using variational and some analytical techniques, we establish the existence and concentration of the semiclassical ground state solutions for the above equation. It is worth mentioning that our results generalize and improve some ones in Gu and Tang (Adv Nonlinear Stud 19:779–795, 2019), Lü (Monatsh Math 182:335–358, 2017) and some other related literatures.