Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger equation

By Chao Ji, Vicentiu D. Rădulescu. Calculus of Variations and Partial Differential Equations, 2020

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In this paper, we study the following nonlinear magnetic Schrödinger equation (εi∇-A(x))2u+V(x)u=f(|u|2)uinRN(N≥2),u∈H1(RN,C),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&\Big (\frac{\varepsilon }{i}\nabla -A(x)\Big )^{2}u+V(x)u=f(|u|^{2})u \quad \hbox {in }{\mathbb {R}}^N\ (N\ge 2),\\&u\in H^{1}({\mathbb {R}}^{N}, {\mathbb {C}}), \end{aligned} \right. \end{aligned}$$\end{document}where 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} is a positive parameter, 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}, A:RN→RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A: {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}$$\end{document} are continuous potentials. Under a local assumption on the potential V, by combining variational methods, penalization techniques, and the Ljusternik–Schnirelmann theory, we prove multiplicity and concentration properties of solutions for ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} small. In our problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

Subfields: Applied Mathematics, Mathematical Physics

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