Existence and Concentration of Solutions for a Class of Kirchhoff–Boussinesq Equation with Exponential Growth in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^4$$\end{document}

This paper is concerned with the existence and concentration of ground state solutions for the following class of elliptic Kirchhoff–Boussinesq type problems given by Δ2u±Δpu+(1+λV(x))u=f(u)inR4,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta ^{2} u \pm \Delta _{p} u +(1+\lambda V(x))u= f(u)\quad \text {in}\ {\mathbb {R}}^{4}, \end{aligned}$$\end{document}where 2<p<4,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2< p< 4,$$\end{document}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}$$f\in C( {\mathbb {R}}, {\mathbb {R}})$$\end{document} is a nonlinearity which has subcritical or critical exponential growth at infinity and V∈C(R4,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in C({\mathbb {R}}^4,{\mathbb {R}})$$\end{document} is a potential that vanishes on a bounded domain Ω⊂R4.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^4.$$\end{document} Using variational methods, we show the existence of ground state solutions, which concentrates on a ground state solution of a Kirchhoff–Boussinesq type equation in Ω.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega .$$\end{document}