Multiplicity and concentration results for magnetic relativistic Schrödinger equations

Advances in Nonlinear Analysis

article

Aliang Xia¹ 

[1] Department of Mathematics, Jiangxi Normal University

关键词: magnetic pseudo-relativistic operators;    Nehari manifold;    Ljusternik-Schnirelmann theory;   

DOI  :  10.1515/anona-2020-0044

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

In this paper, we consider the following magnetic pseudo-relativistic Schrödinger equation ε i∇ − A(x)2+m2u+V(x)u=f(|u|)uinRN, $$\begin{array}{} \displaystyle \sqrt{\left(\frac{\varepsilon}{i}\nabla-A(x)\right)^2+m^2}u+V(x)u= f(|u|)u \quad {\rm in}\,\,\mathbb{R}^N, \end{array}$$ where ε > 0 is a parameter, m > 0, N ≥ 1, V : ℝ N → ℝ is a continuous scalar potential satisfies V ( x ) ≥ − V 0 > − m for any x ∈ ℝ N and f : ℝ N → 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.

【 授权许可】

CC BY