Existence and concentration of solutions for a quasilinear elliptic field equation

In this paper, we study the existence and concentration of solution for the following class of quasilinear problem -ħ2Δv+V(x)v-ħpΔpv+W′(v)=0x∈RN,(Pħ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\hbar ^{2} \Delta v+ V(x)v-\hbar ^{p}\Delta _{p} v + W^{\prime }(v)=0\, \,x\in {\mathbb {R}}^{N}, \qquad \qquad \qquad \qquad (P_{\hbar }) \end{aligned}$$\end{document}when ħ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar >0$$\end{document} is small enough, where v:RN→RN+1,3≤N<p,ħ>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N+1}, 3\le N<p, \hbar >0,$$\end{document}W is a C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} singular functional that satisfies some technical conditions and the potential V is continuous functions with lim inf|x|→+∞V(x)≡V∞>infx∈RNV(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \liminf _{\vert x\vert \rightarrow +\infty }V(x)\equiv V_{\infty }>\inf _{x \in {\mathbb {R}}^N}V(x)>0$$\end{document} . Moreover, we also consider the existence of solution for all ħ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar >0$$\end{document} when V is a radial function.