An optimal constant for the existence of least energy solutions of a coupled Schrödinger system

2012 ◽  
Vol 48 (3-4) ◽  
pp. 695-711 ◽  
Author(s):  
Zhijie Chen ◽  
Wenming Zou
Keyword(s):  
Optimal Constant ◽  
Coupled Schrödinger System ◽  
Least Energy Solutions ◽  
Least Energy ◽  
Schrödinger System

scholarly journals Ground state solution of semilinear Schrödinger system with local super-quadratic conditions

2021 ◽  
pp. 1-18
Author(s):  
Jing Chen ◽  
Yiqing Li
Keyword(s):  
Variational Method ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
The Other ◽  
Ground State Solutions ◽  
Approximation Argument ◽  
Least Energy Solutions ◽  
Least Energy ◽  
Schrödinger System

In this paper, we dedicate to studying the following semilinear Schrödinger system equation*-Δu+V1(x)u=Fu(x,u,v)amp;mboxin~RN,r-Δv+V2(x)v=Fv(x,u,v)amp;mboxin~RN,ru,v∈H1(RN),endequation* where the potential Vi are periodic in x,i=1,2, the nonlinearity F is allowed super-quadratic at some x ∈ R N and asymptotically quadratic at the other x ∈ R N . Under a local super-quadratic condition of F, an approximation argument and variational method are used to prove the existence of Nehari–Pankov type ground state solutions and the least energy solutions.

scholarly journals Bifurcations for a coupled Schrödinger system with multiple components

2015 ◽  
Vol 66 (5) ◽  
pp. 2109-2123 ◽  
Author(s):  
Thomas Bartsch ◽  
Rushun Tian ◽  
Zhi-Qiang Wang
Keyword(s):  
Coupled Schrödinger System ◽  
Multiple Components ◽  
Schrödinger System

Schrödinger systems with quadratic interactions

2019 ◽  
Vol 21 (08) ◽  
pp. 1850077
Author(s):  
Rushun Tian ◽  
Zhi-Qiang Wang ◽  
Leiga Zhao
Keyword(s):  
Variational Formulation ◽  
Variational Problems ◽  
Nontrivial Solutions ◽  
New Techniques ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Coupled Schrödinger System ◽  
Schrödinger Systems ◽  
Bose Einstein ◽  
Schrödinger System

In this paper, we consider the existence and multiplicity of nontrivial solutions to a quadratically coupled Schrödinger system [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are constants and [Formula: see text], [Formula: see text]. Such type of systems stem from applications in nonlinear optics, Bose–Einstein condensates and plasma physics. The existence (and nonexistence), multiplicity and asymptotic behavior of vector solutions of the system are established via variational methods. In particular, for multiplicity results we develop new techniques for treating variational problems with only partial symmetry for which the classical minimax machinery does not apply directly. For the above system, the variational formulation is only of even symmetry with respect to the first component [Formula: see text] but not with respect to [Formula: see text], and we prove that the number of vector solutions tends to infinity as [Formula: see text] tends to infinity.

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