论文标题
具有临界频率的耦合非线性schrödinger方程的半经典状态
Semiclassical states for coupled nonlinear Schrödinger equations with a critical frequency
论文作者
论文摘要
在本文中，我们关注的是耦合的非线性schrödinger系统\ begin {align*} \ begin {cases} - \ varepsilon^{2}ΔU+a（x）x） \mathbb{R}^{N},\\ -\varepsilon^{2}Δv+b(x)v=μ_{2}v^{3}+βu^{2}v \ \ \ \ \ \mbox{in}\ \mathbb{R}^{N}, \end{cases} \ end {align*}其中$ 1 \ leq n \ leq3 $，$μ__{1}，μ_{2}，β> 0 $，$ a（x）$和$ b（x）$是非阴性连续电位，$ \ varepsilon> 0 $是小参数。我们展示了上述系统的积极基础解决方案的存在，还将集中行为确定为$ \ varepsilon \ rightArrow0 $，当$ a（x）$和$ b（x）$实现0时，以同质行为或消失在某些具有平稳边界的非公开开放设置中。
In this paper, we are concerned with the coupled nonlinear Schrödinger system \begin{align*} \begin{cases} -\varepsilon^{2}Δu+a(x)u=μ_{1}u^{3}+βv^{2}u \ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\ -\varepsilon^{2}Δv+b(x)v=μ_{2}v^{3}+βu^{2}v \ \ \ \ \ \mbox{in}\ \mathbb{R}^{N}, \end{cases} \end{align*} where $1\leq N\leq3$, $μ_{1},μ_{2},β>0$, $a(x)$ and $b(x)$ are nonnegative continuous potentials, and $\varepsilon>0$ is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as $\varepsilon\rightarrow0$, when $a(x)$ and $b(x)$ achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.
