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.

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}.

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 The asymptotically linear Hénon problem

2020 ◽  
pp. 2050042
Author(s):  
Anna Lisa Amadori
Keyword(s):  
Boundary Conditions ◽  
Morse Index ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Radial Solutions ◽  
Asymptotically Linear ◽  
Planar Case ◽  
Asymptotic Profile ◽  
Least Energy Solutions ◽  
Least Energy

In this paper, we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent [Formula: see text] is close to [Formula: see text]. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection and rotations of a given angle [Formula: see text]. By considerations based on the Morse index we see that, depending on the values of [Formula: see text] and [Formula: see text], such least energy solutions can be radial, or nonradial and different one from another.

Ground state solutions of Hamiltonian elliptic systems in dimension two

2019 ◽  
Vol 150 (4) ◽  
pp. 1737-1768 ◽  
Author(s):  
Djairo G. de Figueiredo ◽  
João Marcos do Ó ◽  
Jianjun Zhang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Elliptic Systems ◽  
Nehari Manifold ◽  
Existence Results ◽  
Ground State Solutions ◽  
Variational Framework ◽  
Hamiltonian Elliptic System ◽  
Schrödinger Systems ◽  
Image Position

AbstractThe aim of this paper is to study Hamiltonian elliptic system of the form 0.1$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$ where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).

scholarly journals Ground state solutions for fractional Schrödinger equation with variable potential and Berestycki–Lions type nonlinearity

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jing Chen ◽  
Zu Gao
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlocal Problem ◽  
Minimization Method ◽  
Fractional Schrödinger Equation ◽  
Ground State Solutions ◽  
The One ◽  
Least Energy Solutions ◽  
Fractional Schrodinger Equation

Abstract We consider the following nonlinear fractional Schrödinger equation: $$ (-\triangle )^{s} u+V(x)u=g(u) \quad \text{in } \mathbb{R} ^{N}, $$ ( − △ ) s u + V ( x ) u = g ( u ) in  R N , where $s\in (0, 1)$ s ∈ ( 0 , 1 ) , $N>2s$ N > 2 s , $V(x)$ V ( x ) is differentiable, and $g\in C ^{1}(\mathbb{R} , \mathbb{R} )$ g ∈ C 1 ( R , R ) . By exploiting the minimization method with a constraint over Pohoz̆aev manifold, we obtain the existence of ground state solutions. With the help of Pohoz̆aev identity we also process the existence of the least energy solutions for the above equation. Our results improve the existing study on this nonlocal problem with Berestycki–Lions type nonlinearity to the one that does not need the oddness assumption.

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