Uniqueness of positive solutions to a Schrödinger system with linear and nonlinear couplings

2021 ◽  
pp. 103029
Author(s):  
Xinqiu Zhang ◽  
Lishan Liu
Keyword(s):  
Positive Solutions ◽  
Linear And Nonlinear ◽  
Schrödinger System ◽  
Nonlinear Couplings

Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system

2021 ◽  
pp. 1-33
Author(s):  
Yuxia Guo ◽  
Shaolong Peng
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Direct Method ◽  
Strict Monotonicity ◽  
Liouville Type ◽  
Moving Plane ◽  
Liouville Type Results ◽  
Image Position ◽  
Schrödinger System

In this paper, we are concerned with the physically engaging pseudo-relativistic Schrödinger system: \[ \begin{cases} \left(-\Delta+m^{2}\right)^{s}u(x)=f(x,u,v,\nabla u) & \hbox{in } \Omega,\\ \left(-\Delta+m^{2}\right)^{t}v(x)=g(x,u,v,\nabla v) & \hbox{in } \Omega,\\ u>0,v>0 & \hbox{in } \Omega, \\ u=v\equiv 0 & \hbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $s,t\in (0,1)$ and the mass $m>0.$ By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$ , $\mathbb {R}^{N}_{+}$ and a coercive epigraph domain $\Omega$ in $\mathbb {R}^{N}$ , respectively.

Existence of positive solutions to a linearly coupled Schrödinger system with critical exponent

2018 ◽  
Vol 20 (07) ◽  
pp. 1750082 ◽  
Author(s):  
Huiling Wu ◽  
Jianqing Chen ◽  
Yongqing Li
Keyword(s):  
Critical Exponent ◽  
Positive Solution ◽  
Positive Solutions ◽  
Nonlinear Schrödinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Coupled Schrödinger System ◽  
Nonlinear Schrodinger Equations ◽  
Existence Of Positive Solutions ◽  
Schrödinger System

We are concerned with the system of nonlinear Schrödinger equations [Formula: see text] The existence of a positive solution to the system is proved.

Linear and nonlinear couplings between orbital forcing and the marine δ18O record during the Late Neocene

1991 ◽  
Vol 6 (6) ◽  
pp. 729-746 ◽  
Author(s):  
Teresa Hagelberg ◽  
Nick Pisias ◽  
Steve Elgar
Keyword(s):  
Orbital Forcing ◽  
Linear And Nonlinear ◽  
Nonlinear Couplings

Existence and uniqueness of positive solutions of nonlinear Schrödinger systems

2015 ◽  
Vol 145 (2) ◽  
pp. 365-390 ◽  
Author(s):  
Haidong Liu ◽  
Zhaoli Liu ◽  
Jinyong Chang
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Ground State ◽  
Existence And Uniqueness ◽  
Unique Positive Solution ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems ◽  
Least Energy ◽  
Schrödinger System

We prove that the Schrödinger systemwhere n = 1, 2, 3, N ≥ 2, λ1 = λ2 = … = λN = 1, βij = βji > 0 for i, j = 1, …, N, has a unique positive solution up to translation if the βij (i ≠ j) are comparatively large with respect to the βjj. The same conclusion holds if n = 1 and if the βij (i ≠ j) are comparatively small with respect to the βjj. Moreover, this solution is a ground state in the sense that it has the least energy among all non-zero solutions provided that the βij (i ≠ j) are comparatively large with respect to the βjj, and it has the least energy among all non-trivial solutions provided that n = 1 and the βij (i ≠ j) are comparatively small with respect to the βjj. In particular, these conclusions hold if βij = (i ≠ j) for some β and either β > max{β11, β22, …, βNN} or n = 1 and 0 < β < min{β11, β22, …, βNN}.

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