论文标题
非线性狄拉克方程的强烈局部半经典状态
Strongly localized semiclassical states for nonlinear Dirac equations
论文作者
论文摘要
我们研究非线性狄拉克方程的半经典状态\ [ -i \ hbar \partial_tψ= ic \ hbar \ sum_ {k = 1}^3α_k\partial_kar_kψ-mc^2βψ-m（x）ψ+ f（|ψ|）ψ，\ quad t \ in \ ythbb {r}连续的潜在函数和非线性项$ f（|ψ|）ψ$是超线性的，可能是临界增长。我们的主要结果涉及将集中在电势临界点附近的常规溶液。适用于非线性schrödinger方程的标准方法，例如lyapunov-schmidt减少或惩罚，甚至不起作用，甚至不适合均质的非线性$ f（s）= s^p $。我们为与该问题相关的强烈不确定功能开发了一种变异方法。
We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_tψ= ic\hbar\sum_{k=1}^3α_k\partial_kψ- mc^2βψ- M(x)ψ+ f(|ψ|)ψ,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] where $V$ is a bounded continuous potential function and the nonlinear term $f(|ψ|)ψ$ is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity $f(s)=s^p$. We develop a variational method for the strongly indefinite functional associated to the problem.
