The existence and concentration of ground-state solutions for a class of Kirchhoff type problems in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{R}^{3}}$\end{document} involving critical Sobolev exponents

We are concerned with ground-state solutions for the following Kirchhoff type equation with critical nonlinearity: {−(ε2a+εb∫R3|∇u|2)Δu+V(x)u=λW(x)|u|p−2u+|u|4uin R3,u>0,u∈H1(R3),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} - ({\varepsilon^{2}}a + \varepsilon b\int_{{\mathbb{R}^{3}}} {{{ \vert {\nabla u} \vert }^{2}}} )\Delta u + V(x)u = \lambda W(x){ \vert u \vert ^{p - 2}}u + { \vert u \vert ^{4}}u\quad {\text{in }}{\mathbb{R}^{3}} ,\\ u > 0, \quad\quad u \in{H^{1}}({\mathbb{R}^{3}}) , \end{cases} $$\end{document} where ε is a small positive parameter, 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}, λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda > 0$\end{document}, 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}$2 < p \le4$\end{document}, V and W are two potentials. Under proper assumptions, we prove that, for ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon > 0$\end{document} sufficiently small, the above problem has a positive ground-state solution uε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${u_{\varepsilon}}$\end{document} by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to Gui (Commun. Partial Differ. Equ. 21:787-820, 1996) to show that uε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${u_{\varepsilon}}$\end{document} is concentrated around a set which is related to the set where the potential V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x)$\end{document} attains its global minima or the set where the potential W(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W(x)$\end{document} attains its global maxima as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon \to0$\end{document}.