Concentration of solutions for non-autonomous double-phase problems with lack of compactness

The present paper is devoted to the study of the following double-phase equation -div(|del u|p-2 del u+mu epsilon(x)|del u|q-2 del u)+V epsilon(x)(|u|p-2u+mu epsilon(x)|u|q-2u)=f(u)inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\text {div}(|\nabla u|<^>{p-2}\nabla u+\mu _{\varepsilon }(x)|\nabla u|<^>{q-2}\nabla u)+V_{\varepsilon }(x)(|u|<^>{p-2}u+\mu _{\varepsilon }(x)|u|<^>{q-2}u)=f(u)\quad \text{ in }\quad \mathbb {R}<^>{N}, \end{aligned}$$\end{document}where N >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document}, 1<p<q<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<q<N$$\end{document}, q<p & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<p<^>{*}$$\end{document} with p & lowast;=NpN-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p<^>{*}=\frac{Np}{N-p}$$\end{document}, mu:RN -> R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu :\mathbb {R}<^>{N}\rightarrow \mathbb {R}$$\end{document} is a continuous non-negative function, mu epsilon(x)=mu(epsilon x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon }(x)=\mu (\varepsilon x)$$\end{document}, V:RN -> R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V:\mathbb {R}<^>{N}\rightarrow \mathbb {R}$$\end{document} is a positive potential satisfying a local minimum condition, V epsilon(x)=V(epsilon x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{{{\,\mathrm{\varepsilon }\,}}}(x)=V({{\,\mathrm{\varepsilon }\,}}x)$$\end{document}, and the nonlinearity f:R -> R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\mathbb {R}\rightarrow \mathbb {R}$$\end{document} is a continuous function with subcritical growth. Under natural assumptions on mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}, V and f, by using penalization methods and Lusternik-Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions.