论文标题
在尺寸的哈密顿椭圆体系的局部半经典状态
Localized semiclassical states for Hamiltonian elliptic systems in dimension two
论文作者
论文摘要
在本文中，我们考虑尺寸的哈密顿省椭圆系统{equination} \ label {1.5} \ aligned \ left \ left \ {\ okent {array} {lll} {lll}-ε^2ΔU+v（x） -ε^2ΔV+v（x）v = f（u）\＆\ text {in} \ quad \ mathbb {r}^2，\ end {array} \ right。 C^1（\ Mathbb {r}）$假定在无穷大处是超线性或渐近线性，并且在Trudinger-Moser不平等的意义上是次临界指数增长。在$ v $的地方条件下，我们获得了一个半经典状态的家族，这些国家集中在$ v $的当地最低点附近。此外，如果$ f $ and $ g $在无穷大处是超级线，则还给出了半经典状态的衰减和积极性。该证明基于还原方法，变异方法和惩罚技术。
In this paper, we consider the Hamiltonian elliptic system in dimension two\begin{equation}\label{1.5}\aligned \left\{ \begin{array}{lll} -ε^2Δu+V(x)u=g(v)\ & \text{in}\quad \mathbb{R}^2,\\ -ε^2Δv+V(x)v=f(u)\ & \text{in}\quad \mathbb{R}^2, \end{array}\right.\endaligned \end{equation} where $V\in C(\mathbb{R}^2)$ has local minimum points, and $f,g\in C^1(\mathbb{R})$ are assumed to be either superlinear or asymptotically linear at infinity and of subcritical exponential growth in the sense of Trudinger-Moser inequality. Under only a local condition on $V$, we obtain a family of semiclassical states concentrating around local minimum points of $V$. In addition, in the case that $f$ and $g$ are superlinear at infinity, the decay and positivity of semiclassical states are also given. The proof is based on a reduction method, variational methods and penalization techniques.
