Multi-peak positive solutions for a logarithmic Schrödinger equation via variational methods

In this paper, using the variational methods, we show the existence and multiplicity of multi-peak positive solutions for the following logarithmic Schrödinger 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}$$\left\{ {\matrix{{ - \varepsilon{^2}\Delta u + V(x)u = u\log {u^2},} \hfill \;\;\;\;\;\;\;\;\;\; {\rm{in}\,\,{\mathbb{R}^N},} \hfill \cr {u \in {H^1}({\mathbb{R}^N}),} \hfill \;\;\;\;\; {} \hfill \cr} } \right.$$\end{document} where ϵ > 0, N ≥ 2 and V: ℝN → ℝ is a multi-well potential.