Some existence and concentration results for nonlinear Schrödinger equations
Some existence and concentration results for nonlinear Schrödinger equations
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非线性薛定谔方程的一些存在性和集中性结果
DOI：
10.3934/cpaa.2002.1.457
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发表时间：
2002
影响因子：
1
通讯作者：
T. D’Aprile
中科院分区：
文献类型：
--
作者：
T. D’Aprile
在本文中，我们关注的是角动量不为零的解的存在性
一类非线性薛定谔方程的形式
$ i \h$$ \frac{\partial\psi}{\partial t}=-$ $\frac{ \h^2}{2m}\Delta \psi+V(x)\psi-\gamma|\psi|^{p-2}\psi,$ $\gamma>0,$ $ x\in\mathbb R^{N}$
其中 $\h$$ >0$, $p>2$, $\psi:\mathbb R^{N}\rightarrow\mathbb C,$
并且潜在的$V$满足一些对称性质。特别是 $N=2$ 且 $V$ 径向对称的情况
讨论了$N=3$且$V$具有圆柱对称性。
我们的主要目的是研究此类解在半经典极限下的渐近行为
（即 $\hbar \rightarrow 0^+$）
当$\mathbb R^N$ 点周围出现集中现象时。
In this paper we are concerned with the existence of solutions with non-vanishing angular momentum for
a class of nonlinear Schrodinger equations of the form
$ i \h$$ \frac{\partial\psi}{\partial t}=-$ $\frac{ \h^2}{2m}\Delta \psi+V(x)\psi-\gamma|\psi|^{p-2}\psi,$ $\gamma>0,$ $ x\in\mathbb R^{N}$
where $\h$$ >0$, $p>2$, $\psi:\mathbb R^{N}\rightarrow\mathbb C,$
and the potential $V$ satisfies some symmetric properties. In particular the cases $N=2$ with $V$ radially symmetric
and $N=3$ with $V$ having a cylindrical symmetry are discussed.
Our main purpose is to study the asymptotic behaviour of such solutions in the semiclassical limit
(i.e. as $\hbar \rightarrow 0^+$)
when a concentration phenomenon around a point of $\mathbb R^N$ appears.