Semiclassical states for a magnetic nonlinear Schr\"{o}dinger equation with exponential critical growth in $\mathbb{R}^{2}$

Abstract. This paper is devoted to the magnetic nonlinear Schrödinger equation (ε i ∇− A(x) )2 u+ V (x)u = f(|u|)u in R, where ε > 0 is a parameter, V : R → R and A : R → R are continuous functions and f : R → R is a C function having exponential critical growth. Under a global assumption on the potential V , we use variational methods and Ljusternick-Schnirelmann theory to prove existence, multiplicity, concentration, and decay of nontrivial solutions for ε > 0 small.

[1]  G. Arioli,et al.  A Semilinear Schrödinger Equation in the Presence of a Magnetic Field , 2003 .

[2]  S. Secchi,et al.  Multi-peak solutions for magnetic NLS equations without non{degeneracy conditions , 2007, 0710.3227.

[3]  S. Secchi,et al.  Semiclassical states for NLS equations with magnetic potentials having polynomial growths , 2005 .

[4]  B. Helffer,et al.  Magnetic Bottles in Connection with Superconductivity , 2001 .

[5]  M. Squassina,et al.  Ground states for fractional magnetic operators , 2016, 1601.04230.

[6]  K Fan,et al.  Minimax Theorems. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[7]  C. O. Alves,et al.  Existence of a ground state solution for a nonlinear scalar field equation with critical growth , 2012 .

[8]  Multiplicity of positive solutions for a class of problems with exponential critical growth in R2 ✩ , 2008 .

[9]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[10]  G. Figueiredo,et al.  An existence result for Schrödinger equations with magnetic fields and exponential critical growth , 2017, Journal of Elliptic and Parabolic Equations.

[11]  Otared Kavian,et al.  Introduction à la théorie des points critiques : et applications aux problèmes elliptiques , 1993 .

[12]  M. Pino,et al.  Local mountain passes for semilinear elliptic problems in unbounded domains , 1996 .

[13]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[14]  P. d’Avenia,et al.  Nonlinear fractional magnetic Schr\"odinger equation: existence and multiplicity , 2017, 1709.08207.

[15]  K. Kurata.  Existence and semi-classical limit of the least energy solution to a nonlinear Schrouml;dinger equation with electromagnetic fields , 2000 .

[16]  Tosis Kato,et al.  Schrödinger operators with singular potentials , 1972 .

[17]  S. Cingolani,et al.  Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations , 1997 .

[18]  P. Lions,et al.  Stationary solutions of nonlinear Schr?odinger equations with an external magnetic field , 1989 .

[19]  Sihua Liang,et al.  On multi-bump solutions of nonlinear Schrödinger equation with electromagnetic fields and critical nonlinearity in $$\mathbb {R}^N$$RN , 2017 .

[20]  Standing waves for nonlinear Schrödinger equations involving critical growth of Trudinger–Moser type , 2015 .

[21]  D. Cao.  Nontrivial Solution of Semilinear Elliptic Equations with Critical Exponent in R , 1992 .

[22]  V. Ambrosio.  On a fractional magnetic Schrödinger equation in R with exponential critical growth , 2018, Nonlinear Analysis.

[23]  C. O. Alves,et al.  Multiple Solutions for a Nonlinear Schrödinger Equation with Magnetic Fields , 2011 .

[24]  C. O. Alves,et al.  On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN , 2009 .

[25]  S. Cingolani,et al.  Multiple Positive Solutions to Nonlinear Schrödinger Equations with Competing Potential Functions , 2000 .

[26]  S. Cingolani.  Semiclassical stationary states of Nonlinear Schrodinger equations with an external magnetic field , 2003 .

[27]  Paul H. Rabinowitz,et al.  On a class of nonlinear Schrödinger equations , 1992 .

[28]  P. Sternberg,et al.  Boundary Concentration for Eigenvalue Problems Related to the Onset of Superconductivity , 2000 .

[29]  V. Benci,et al.  Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology , 1994 .

[30]  F. Lin,et al.  Solutions of perturbed Schrödinger equations with critical nonlinearity , 2007 .

[31]  Marion Kee,et al.  Analysis , 2004, Machine Translation.