Multiplicity and concentration results for magnetic relativistic Schrödinger equations

Abstract

Abstract In this paper, we consider the following magnetic pseudo-relativistic Schrödinger equation $$\begin{array}{} \displaystyle \sqrt{\left(\frac{\varepsilon}{i}\nabla-A(x)\right)^2+m^2}u+V(x)u= f(|u|)u \quad {\rm in}\,\,\mathbb{R}^N, \end{array}$$ where ε > 0 is a parameter, m > 0, N ≥ 1, V : ℝN → ℝ is a continuous scalar potential satisfies V(x) ≥ − V0 > − m for any x ∈ ℝN and f : ℝN → ℝ is a continuous function. Under a local condition imposed on the potential V, we discuss the number of nontrivial solutions with the topology of the set where the potential attains its minimum. We proof our results via variational methods, penalization techniques and Ljusternik-Schnirelmann theory.