scholarly journals Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

2020 ◽  
Vol 59 (1) ◽  
Author(s):  
Hugo Tavares ◽  
Song You
Keyword(s):  
Positive Solutions ◽  
Critical Case ◽  
Competition And Cooperation ◽  
Schrödinger Systems ◽  
Least Energy

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}.

Standing waves for weakly coupled nonlinear Schrödinger systems

2019 ◽  
Vol 22 (05) ◽  
pp. 1950006
Author(s):  
Claudiney Goulart ◽  
Elves A. B. Silva
Keyword(s):  
Positive Solutions ◽  
Standing Waves ◽  
Nonlinear Schrödinger ◽  
Weakly Coupled ◽  
Euclidian Space ◽  
Least Energy Solution ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems ◽  
Multiplicity Of Positive Solutions ◽  
Least Energy

This paper is concerned with the application of variational methods in the study of positive solutions for a system of weakly coupled nonlinear Schrödinger equations in the Euclidian space. The results on multiplicity of positive solutions are established under the hypothesis that the coupling is either sublinear or superlinear with respect to one of the variables. Conditions for the existence or nonexistence of a positive least energy solution are also considered.

Least Energy Solutions for Nonlinear Schrödinger Systems with Mixed Attractive and Repulsive Couplings

2015 ◽  
Vol 15 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Yohei Sato ◽  
Zhi-Qiang Wang
Keyword(s):  
Ground State ◽  
Nonlinear Elliptic System ◽  
Ground State Solutions ◽  
Nonlinear Elliptic ◽  
Asymptotic Profile ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems ◽  
Bose Einstein ◽  
Least Energy Solutions ◽  
Least Energy

AbstractIn this paper we study the ground state solutions for a nonlinear elliptic system of three equations which comes from models in Bose-Einstein condensates. Comparing with existing works in the literature which have been on purely attractive or purely repulsive cases, our investigation focuses on the effect of mixed interaction of attractive and repulsive couplings. We establish the existence of least energy positive solutions and study asymptotic profile of the ground state solutions, giving indication of co-existence of synchronization and segregation. In particular we show symmetry breaking for the ground state solutions.

Dynamics of Forced Dissipative Schrödinger Systems in the Critical Case

1991 ◽  
Vol 15 (2) ◽  
pp. 173-178 ◽  
Author(s):  
H Pietsch ◽  
R Blaha ◽  
E. W Laedke ◽  
A Kumar
Keyword(s):  
Critical Case ◽  
Schrödinger Systems

scholarly journals Structure of positive solutions to a class of Schrödinger systems

2018 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Xiyou Cheng ◽  
◽  
Zhitao Zhang ◽  
Keyword(s):  
Positive Solutions ◽  
Schrödinger Systems
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