Semiclassical states for a magnetic nonlinear Schrödinger equation with exponential critical growth in ℝ2

Abstract

This paper is devoted to the magnetic nonlinear Schrödinger equation

$${\left( {{\varepsilon \over i}\nabla - A(x)} \right)^2}u + V(x)u = f(|u{|^2})u\,\,{\rm{in}}\,\,{\mathbb{R}^2},$$

where ε > 0 is a parameter, V: ℝ² → ℝ and A: ℝ² → ℝ² are continuous functions and f: ℝ → ℝ is a C¹ 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.