Existence and concentration of positive solutions for a logarithmic Schrödinger equation via penalization method

By Claudianor Oliveira Alves, Chao Ji. Calculus of Variations and Partial Differential Equations, 2019

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In this article we are concerned with 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}$$\begin{aligned} \left\{ \begin{array}{lc} -{\epsilon }^2\Delta u+ V(x)u=u \log u^2, &{}\quad \text{ in } \,\, {\mathbb {R}}^{N}, \\ u \in H^1({\mathbb {R}}^{N}), &{} \; \\ \end{array} \right. \end{aligned}$$\end{document}where ϵ>0,N≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0, N \ge 1$$\end{document} and 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 continuous potential. Under a local assumption on the potential V, we use the variational methods to prove the existence and concentration of positive solutions for the above problem.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/efd6443eb233faa2eb8b37672e04d74da56d09b9