Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger equation (Q776030)

In this paper, the authors study the following nonlinear magnetic Schrödinger equation \begin{align*} \begin{cases} \left(\displaystyle \frac{\varepsilon}{i}\nabla -A(x)\right)^2u+V(x)u= f\left(|u|^2\right)u &\text{in }\mathbb{R}^N \ \ (N\geq 2) ,\\ u\in H^1\left(\mathbb{R}^N,\mathbb{C} \right), \end{cases} \end{align*} where \(\varepsilon\) is a positive parameter, \(f\) is continuous and \(V \colon \mathbb{R}^N \to \mathbb{R}\) as well as \(A\colon \mathbb{R}^N\to\mathbb{R}^N\) are continuous potentials. Based on variational methods, penalization techniques and the Ljusternik-Schnirelmann theory, the authors prove multiplicity and concentration properties of solutions provided the parameter \(\varepsilon>0\) is small and under a local condition on the potential \(V\).