数学与统计学院知识库

本文研究带磁场的 Schrödinger 方程解的存在性. 在 A 和 V 满足适当条件时, 利用 Nehari 流形技巧, 我们给出了如下方程
−Δ_Au + V (x)u = |u|^p−2u + λ|u|^r−2u, u(x) ∈ H1A(R^N)基态解的存在性, 其中 2 < r < p < 2*, λ ∈ R. 进一步, 我们给出了带磁场的Sobolev 空间上集中紧性原理的一种抽象形式, 并利用这一框架给出了一类带磁场 Schrödinger 方程解的存在性.

In this paper we consider the existence of solution for nonlinear Schrödinger equation with an external magnetic field. Firstly, under appropriate assumptions on A and V , applying Nehari manifold technique, we prove the existence of ground state solution for the following equation
−Δ_Au + V (x)u = |u|^p−2u + λ|u|^r−2u, u(x) ∈ H1A(R^N)

where 2 < r < p < 2*, λ ∈ R. Furthermore, we give an abstract form for the concentration-compacteness principle on the space H1A(R^N), and give an example as an application.