Bound States of 2-D Nonlinear Schrödinger Equations with Potentials Tending to Zero at Infinity
Bound States of 2-D Nonlinear Schrödinger Equations with Potentials Tending to Zero at Infinity
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无穷远势趋于零的二维非线性薛定谔方程的束缚态
DOI：
10.1137/110846919
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发表时间：
2013
影响因子：
2
通讯作者：
Huicheng Yin
中科院分区：
文献类型：
--
作者：
Mingwen Fei;Huicheng Yin
在本文中，我们重点研究 $\Bbb R^2$ 中以下具有临界或次临界指数的非线性薛定谔方程的正解的存在性和集中性： $-\varepsilon^2\triangle u_{\varepsilon}+V(x)u_{\varepsilon}=K(x) |u_{\varepsilon}|^{p-2}u_{\varepsilon}e^{\alpha_{0}|u_{\varepsilon}| ^{\gamma}},\x\in\Bbb R^2; u_{\varepsilon}>0,\ u_{\varepsilon}\in H^1(\Bbb R^2),$ 其中 $p>2$, $\alpha_{0}>0$, $V(x), K(x)>0$, $0 0$ 是一个小常数。当势能$V(x)$像$0 0$的$(1+|x|)^{-\alpha}$一样在无穷大衰减时，在一些必要的限制下允许无界，如果假设非线性薛定谔方程$ -\triangle的相关地面能量函数$G(\xi)$，我们将证明至少存在一个正的$H^1(\Bbb R^2)$-解u+V(\xi)u=K(\xi) |u|^{p-2}ue^{\alpha_{0}|u|^{\gamma}}$ 具有局部极小值点。同时，$u_{\varepsilon}$的浓度性质也成立...
In this paper, we focus on the existence and concentration of positive solutions to the following nonlinear Schrodinger equations with critical or subcritical exponents in $\Bbb R^2$: $-\varepsilon^2\triangle u_{\varepsilon}+V(x)u_{\varepsilon}=K(x) |u_{\varepsilon}|^{p-2}u_{\varepsilon}e^{\alpha_{0}|u_{\varepsilon}| ^{\gamma}},\ x\in\Bbb R^2; u_{\varepsilon}>0,\ u_{\varepsilon}\in H^1(\Bbb R^2),$ where $p>2$, $\alpha_{0}>0$, $V(x), K(x)>0$, $0 0$ is a small constant. When the potential $V(x)$ decays at infinity like $(1+|x|)^{-\alpha}$ with $0 0$ is permitted to be unbounded under some necessary restrictions, we will show that at least one positive $H^1(\Bbb R^2)$-solution exists if it is assumed that the related ground energy function $G(\xi)$ of the nonlinear Schrodinger equation $ -\triangle u+V(\xi)u=K(\xi) |u|^{p-2}ue^{\alpha_{0}|u|^{\gamma}}$ has local minimum points. Meanwhile, the concentration property of $u_{\varepsilon}$ is also establishe...