arXiv:1906.10937v2 [math.AP] 23 Apr 2020 MULTIPLICITY AND CONCENTRATION RESULTS FOR A MAGNETIC SCHR ¨ ODINGER EQUATION WITH EXPONENTIAL CRITICAL GROWTH IN R 2 PIETRO D’AVENIA AND CHAO JI Abstract. In this paper we study the following nonlinear Schr¨ odinger equation with magnetic field  ε i ∇− A(x)  2 u + V (x)u = f (|u| 2 )u, x ∈ R 2 , where ε> 0 is a parameter, V : R 2 → R and A : R 2 → R 2 are continuous potentials and f : R → R has exponential critical growth. Under a local assumption on the potential V , by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration of solutions for ε small. Contents 1. Introduction and main results 1 Notation 4 2. The variational framework and the limit problem 4 3. The modified problem 9 4. Multiple solutions for the modified problem 15 5. Proof of Theorem 1.1 20 Acknowledgements 24 References 24 1. Introduction and main results In this paper, we are concerned with multiplicity and concentration results for the following nonlinear magnetic Schr¨ odinger equation (1.1)  ε i ∇− A(x)  2 u + V (x)u = f (|u| 2 )u in R 2 , where u ∈ H 1 (R 2 , C), ε> 0 is a parameter, V : R 2 → R is a continuous function, f : R → R, and the magnetic potential A : R 2 → R 2 is H¨ older continuous with exponent α ∈ (0, 1]. Equation (1.1) arises when one looks for standing wave solutions ψ(x,t) := e −iEt/ u(x), with E ∈ R, of i ∂ψ ∂t =   i ∇− A(x)  2 ψ + U (x)ψ − f (|ψ| 2 )ψ in R 2 × R. From a physical point of view, the existence of such solutions and the study of their shape in the semiclassical limit, namely, as  → 0 + , or, equivalently, as ε → 0 + in (1.1), is of the greatest importance, since the transition from Quantum Mechanics to Classical Mechanics can be formally performed by sending the Planck constant  to zero. For equation (1.1), there is a vast literature concerning the existence and multiplicity of bound state solutions, in particular for the case with A ≡ 0. The first result in this direction was given by Floer and Weinstein in [28], where the case N = 1 and f = i R is considered. Later, many authors generalized this 2010 Mathematics Subject Classification. 35J20, 35J60, 35B33. Key words and phrases. Nonlinear Schr¨ odinger equation, Magnetic field, Exponential critical growth, Trudinger-Moser inequality, Penalization technique. P. d’Avenia is supported by PRIN project 2017JPCAPN Qualitative and quantitative aspects of nonlinear PDEs. C. Ji is partially supported by Shanghai Natural Science Foundation (18ZR1409100). 1