arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1406.4577math.AP
收藏

具有非对称电磁场的非线性薛定谔方程的无穷多解

Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields

Weiming Liu, Chunhua Wang

备注：
39fages, 0 figures. arXiv admin note: text overlap with arXiv:1210.8209, arXiv:1209.2824 by other authors
发表信息：
Discrete Contin. Dyn. Syst. 36 (2016)
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文研究了具有非对称电磁场$$\Big(\frac{\nabla}{i}-A_ε x)\Big)^2 u+V_ε(x)u=f(u),\ u\in H^1 (\mathbb{R}^N,\mathbb{C}), $$的非线性薛定谔方程，其中$A_ε(x)=(A_{ε,1}(x),A_{ε,2}(x),\cdots,A_{ε,N}(x))$是磁场，满足$A_{ε,j}(x)(j=1,\ldots,N)$是$\mathbb{R}^{N}$上的实$C^{1}$有界函数，$V_ε(x)$是电势.它们都满足一定的衰减条件，而$f(u)$是满足一定非简并条件的非线性项。应用局部化能量方法，证明了存在某些$ε_{0 }> 0$，使得对于$0 < ε< ε_{0 }$，上述问题有无穷多个复值解.

英文摘要：

In this paper, we study the nonlinear Schrödinger equation with non-symmetric electromagnetic fields $$\Big(\frac{\nabla}{i}-A_ε x)\Big)^2 u+V_ε(x)u=f(u),\ u\in H^1 (\mathbb{R}^N,\mathbb{C}), $$ where $A_ε(x)=(A_{ε,1}(x),A_{ε,2}(x),\cdots,A_{ε,N}(x))$ is a magnetic field satisfying that $A_{ε,j}(x)(j=1,\ldots,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V_ε(x)$ is an electric potential. Both of them satisfy some decay conditions and $f(u)$ is a nonlinearity satisfying some nondegeneracy condition. Applying localized energy method, we prove that there exists some $ε_{0 }> 0$ such that for $0 < ε< ε_{0 }$, the above problem has infinitely many complex-valued solutions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑