Existence and concentration of positive solutions for a Schrödinger logarithmic equation

This article concerns with the existence and concentration of positive solutions for the following logarithmic elliptic equation -ϵ2Δu+V(x)u=ulogu2,inRN,u∈H1(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lc} -{\epsilon }^2\Delta u+ V(x)u=u \log u^2, &{} \text{ in } \quad \mathbb {R}^{N}, \\ u \in H^1(\mathbb {R}^{N}), &{} \; \\ \end{array}\right. \end{aligned}$$\end{document}where ϵ>0,N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0, N \ge 3$$\end{document} and V is a continuous function with a global minimum. Using variational method developed by Szulkin (Ann Inst H Poincaré Anal Non Linéaire 3:77–109, 1986) for functionals which are sum of 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} functional with a convex lower semicontinuous functional, we prove, for small enough ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0$$\end{document}, the existence of positive solutions and concentration around of a minimum point of V, when ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} approaches zero. We also study the cases when V is periodic or asymptotically periodic.