Existence and Concentration of Positive Solutions For Coupled Nonlinear Schrödinger Systems in ℝN

2007 ◽  
Vol 7 (1) ◽  
Author(s):  
Youyan Wan
Keyword(s):  
Positive Solutions ◽  
Small Positive Number ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems ◽  
Positive Functions ◽  
Schrödinger System

AbstractThe aim of this paper is to study the existence and concentration of positive solutions for the coupled nonlinear Schrödinger systemwhere ε is a small positive number, N ≥ 3, p, q > 1 satisfyand W(x), Q(x), K(x) are continuous and bounded positive functions defined in ℝ

Compactness of Minimizing Sequences in Nonlinear Schrödinger Systems Under Multiconstraint Conditions

2014 ◽  
Vol 14 (1) ◽  
Author(s):  
Norihisa Ikoma
Keyword(s):  
Nonlinear Optics ◽  
Orbital Stability ◽  
Minimizing Sequences ◽  
Minimizing Sequence ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems ◽  
Schrödinger System

AbstractIn this paper, the precompactness of minimizing sequences under multiconstraint conditions are discussed. This minimizing problem is related to a coupled nonlinear Schrödinger system which appears in the field of nonlinear optics. As a consequence of the compactness of each minimizing sequence, the orbital stability of the set of all minimizers is obtained.

scholarly journals Minimal Energy Solutions and Infinitely Many Bifurcating Branches for a Class of Saturated Nonlinear Schrödinger Systems

2016 ◽  
Vol 16 (1) ◽  
pp. 95-113 ◽  
Author(s):  
Rainer Mandel
Keyword(s):  
Symmetric Case ◽  
Solution Curve ◽  
Minimal Energy ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems ◽  
Alpha Beta ◽  
Schrödinger System

AbstractWe prove a conjecture recently formulated by Maia, Montefusco and Pellacci saying that minimal energy solutions of the saturated nonlinear Schrödinger system$\left\{\begin{aligned} \displaystyle-\Delta u+\lambda_{1}u&\displaystyle=\frac% {\alpha u(\alpha u^{2}+\beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&% \displaystyle\text{in }\mathbb{R}^{n},\\ \displaystyle-\Delta v+\lambda_{2}v&\displaystyle=\frac{\beta v(\alpha u^{2}+% \beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&\displaystyle\text{in }\mathbb{R% }^{n}\end{aligned}\right.$are necessarily semitrivial whenever ${\alpha,\hskip 0.5pt\beta,\hskip 0.5pt\lambda_{1},\hskip 0.5pt\lambda_{2}>0}$ and ${0<s<\max\{\alpha/\lambda_{1},\hskip 0.5pt\beta/\lambda_{2}\}}$ except for the symmetric case ${\lambda_{1}=\lambda_{2}}$, ${\alpha=\beta}$. Moreover, it is shown that for most parameter samples ${\alpha,\beta,\lambda_{1},\lambda_{2}}$, there are infinitely many branches containing seminodal solutions which bifurcate from a semitrivial solution curve parametrized by s.

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.

scholarly journals Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems

2019 ◽  
Vol 12 (7) ◽  
pp. 2143-2161
Author(s):  
Jiabao Su ◽  
◽  
Rushun Tian ◽  
Zhi-Qiang Wang ◽  
◽  
...  
Keyword(s):  
Positive Solutions ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems
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