An Existence Result for a Class of Magnetic Problems in Exterior Domains

Vincenzo Ambrosio,Claudianor O Alves,César E Torres Ledesma

doi:10.1007/s00032-021-00340-z

Abstract

In this paper we deal with the existence of solutions for the following class of magnetic semilinear Schrödinger equation (P)(-i∇+A(x))2u+u=|u|p-2u,inΩ,u=0on∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (P) \\qquad \\qquad \\left\\{ \\begin{aligned}&(-i\\nabla + A(x))^2u +u = |u|^{p-2}u,\\;\\;\\text{ in }\\;\\;\\Omega ,\\\\&u=0\\;\\;\\text{ on }\\;\\;\\partial \\Omega , \\end{aligned} \\right. \\end{aligned}$$\\end{document}where N ge 3, Omega subset {mathbb {R}}^N is an exterior domain, pin (2, 2^*) with 2^*=frac{2N}{N-2}, and A: {mathbb {R}}^Nrightarrow {mathbb {R}}^N is a continuous vector potential verifying A(x) rightarrow 0;;text{ as };;|x|rightarrow infty .

Highlights

In this paper we investigate the existence of solutions for the following magnetic semilinear Schrodinger equation
During the past years there has been a considerable interest in the existence of solutions for elliptic equations in exterior domains, more precisely, for problems of the type
In [2], Alves and de Freitas studied the existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth

Summary

Introduction

During the past years there has been a considerable interest in the existence of solutions for elliptic equations in exterior domains, more precisely, for problems of the type. In the above-mentioned papers, a key point to prove the results of existence is the uniqueness, up to a translation, of the positive solution for the “equation at infinity” associated with (1.3) given by. In [2], Alves and de Freitas studied the existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. After a careful bibliography review, we did not find any paper concerned with magnetic semilinear Schrodinger equations in exterior domains Motivated by this fact and the above-mentioned papers, the aim of this paper is to give a first existence result for (P ).

Preliminary Results and the Limit Problem

Limit Problem

A Compactness Result for Energy Functional

A Compactness Lemma

Technical Estimates