Multi-bump solutions for the nonlinear magnetic Schr\xf6dinger equation with exponential critical growth in {mathbb {R}}^{2}

Abstract

In this paper, using variational methods, we establish the existence and multiplicity of multi-bump solutions for the following nonlinear magnetic Schrödinger equation -(∇+iA(x))2u+(λV(x)+Z(x))u=f(|u|2)uinR2,\\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}$$\\begin{aligned} -(\\nabla +\\mathrm{i} A(x))^2 u+(\\lambda V(x)+Z(x))u=f(\\vert u\\vert ^{2})u\\quad \\text {in}\\, \\,{\\mathbb {R}}^{2}, \\end{aligned}$$\\end{document}where lambda >0, f(t) is a continuous function with exponential critical growth, the magnetic potential A:{mathbb {R}}^2rightarrow {mathbb {R}}^2 is in L^{2}_{loc}({mathbb {R}}^2) and the potentials V, Z:{mathbb {R}}^{2}rightarrow {mathbb {R}} are continuous functions verifying some natural conditions. We show that if the zero set of the potential V has several isolated connected components Omega _{1}, ldots , Omega _{k} such that the interior of Omega _{j} is non-empty and partial Omega _{j} is smooth, then for lambda >0 large enough, the equation has at least 2^{k}-1 multi-bump solutions.