文献类型：期刊文献

英文题名：Existence and Multiplicity of Solutions for the Logarithmic Schr?dinger Equation with a Potential on Lattice Graphs

作者：He, Zhentao[1];Ji, Chao[1]

机构：[1]East China Univ Sci & Technol, Sch Math, Shanghai 200237, Peoples R China

年份：2024

卷号：34

期号：12

外文期刊名：JOURNAL OF GEOMETRIC ANALYSIS

收录：;EI(收录号：20240137245);WOS:【SCI-EXPANDED(收录号:WOS:001337736400001)】；

基金：The authors would like to thank the anonymous referee for several valuable suggestions and comments which helped to improve the paper. C. Ji was partially supported by National Natural Science Foundation of China (No. 12171152).

语种：英文

外文关键词：Logarithmic Schr & ouml;dinger equation; Lattice graphs; Ground states; Multiplicity of solutions; Nonsmooth critical point theory

摘要：In this paper, we consider the existence and multiplicity of solutions for the logarithmic Schr & ouml;dinger equation on lattice graphs ZN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}<^>N$$\end{document}-Delta u+V(x)u=ulogu2,x is an element of ZN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta u+V(x) u=u \log u<^>2, \quad x \in {\mathbb {Z}}<^>N, \end{aligned}$$\end{document}When the potential V is coercive, we obtain infinitely many solutions by adapting some arguments of the Fountain theorem. In the cases of periodic potential, asymptotically periodic potential and bounded potential, we first investigate the existence of ground state solutions via the variational methods, and then we generalize these results from ZN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}<^>N$$\end{document} to quasi-transitive graphs. Finally, we extend the main results of the paper to the p-Laplacian equation with the logarithmic nonlinearity.