Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger equation

Abstract

AbstractIn this paper, we study the following nonlinear magnetic Schrödinger equation $$\begin{aligned} \left\{ \begin{aligned}&\Big (\frac{\varepsilon }{i}\nabla -A(x)\Big )^{2}u+V(x)u=f(|u|^{2})u \quad \hbox {in }{\mathbb {R}}^N\ (N\ge 2),\\&u\in H^{1}({\mathbb {R}}^{N}, {\mathbb {C}}), \end{aligned} \right. \end{aligned}$$ ( ε i ∇ - A ( x ) ) 2 u + V ( x ) u = f ( | u | 2 ) u in R N ( N ≥ 2 ) , u ∈ H 1 ( R N , C ) , where $$\epsilon $$ ϵ is a positive parameter, and $$V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$$ V : R N → R , $$A: {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}$$ A : R N → R N are continuous potentials. Under a local assumption on the potential V, by combining variational methods, penalization techniques, and the Ljusternik–Schnirelmann theory, we prove multiplicity and concentration properties of solutions for $$\varepsilon >0$$ ε > 0 small. In our problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.