Positive vortex solutions and phase separation for coupled Schrodinger system with singular potential
Positive vortex solutions and phase separation for coupled Schrodinger system with singular potential
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DOI：
10.58997/ejde.2020.108
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发表时间：
2020-10
影响因子：
0.7
通讯作者：
Jin Deng; Aliang Xia- ;Jianfu Yang
中科院分区：
文献类型：
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作者：
Jin Deng; Aliang Xia- ;Jianfu Yang
通过求解奇异方程组$$\displaystyle { -\Delta u+\frac{u}+\frac{u}}，我们考虑了耦合薛定谔方程的旋转孤波（涡）的存在性。|X| ^2}=\mu_1 u^3+\beta u v^2，\quad x\in\mathbb{R}^2，\cr -\Delta v+\enda_2 v+\frac{v}{|X| ^2}=\mu_2 v^3+\beta u^2 v，\quad x\in\mathbb{R}^2，\cr u，v \geq 0，\quad x\in\mathbb{R}^2，}$$其中\（\mathda_1，\mathda_2，\mu_1，\mu_2\）为正参数，\（\beta\neq 0\）。我们证明了当\（\beta\）为负或\（\beta\）为正且为小或大时，该系统存在正的最小能量解.此外，如果\（\mada_1 =\mada_2\），则解是唯一的。我们还研究了最小能量解的极限行为在排斥的情况下\（\beta\to-\infty\），和相分离。欲了解更多信息，请访问https://ejde.math.txstate.edu/Volumes/2020/108/abstr.html
We consider the existence of rotating solitary waves (vortices) for a coupled Schrodinger equations by finding solutions to the singular system $$\displaylines{ -\Delta u+\lambda_1 u+\frac{u}{|x|^2}=\mu_1 u^3+\beta u v^2, \quad x\in\mathbb{R}^2, \cr -\Delta v+\lambda_2 v+\frac{v}{|x|^2}=\mu_2 v^3+\beta u^2 v, \quad x\in\mathbb{R}^2, \cr u,v \geq 0,\quad x\in\mathbb{R}^2, }$$ where \(\lambda_1,\lambda_2,\mu_1, \mu_2\) are positive parameters, \(\beta\neq 0\). We show that this system has a positive least energy solution for the cases When either \(\beta\) is negative or \(\beta\) is positive and small or large. Moreover, if \(\lambda_1=\lambda_2\), then the solution is unique. We also study the limiting behavior of the least energy solutions in the repulsive case for \(\beta\to-\infty\), and phase separation. For more information see https://ejde.math.txstate.edu/Volumes/2020/108/abstr.html