Multiple positive solutions to critical p-Laplacian equations with vanishing potential

This paper deals with the following p-Laplacian equation -εpΔpu+V(x)|u|p-2u=|u|p∗-2u,u∈D1,p(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\varepsilon ^{p}\Delta _{p}u+V(x)|u|^{p-2}u=|u|^{p^{*}-2}u,\quad u\in D^{1,p}({\mathbb {R}}^N), \end{aligned}$$\end{document}where p∈(1,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1,N)$$\end{document}, p-Laplacian operator Δp:=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{p}{:}{=}$$\end{document}div(|∇u|p-2∇u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(|\nabla u|^{p-2}\nabla u) $$\end{document}, p∗=Np/(N-p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{*}=Np/(N-p)$$\end{document}, ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is a positive parameter, V(x)∈LN/p(RN)∩Lloc∞(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(x)\in L^{{N}/{p}}({\mathbb {R}}^N)\cap L^{\infty }_{loc}({\mathbb {R}}^N)$$\end{document} and V(x) is assumed to be zero in some region of RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^N$$\end{document}, which means it is of the vanishing potential case. In virtue of Ljusternik–Schnirelman theory of critical points, we succeed in proving the multiplicity of positive solutions. This result generalizes the result for semilinear Schrödinger equation by Chabrowski and Yang (Port. Math. 57 (2000), 273–284) to p-Laplacian equation.