Normalized Solutions for Schrödinger System with Subcritical Sobolev Exponent and Combined Nonlinearities

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Maoding Zhen
Keyword(s):  
Normalized Solutions ◽  
Sobolev Exponent ◽  
Schrödinger System

scholarly journals Existence of Solutions for a Modified Nonlinear Schrödinger System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yujuan Jiao ◽  
Yanli Wang
Keyword(s):  
Perturbation Method ◽  
Existence Of Solutions ◽  
Critical Sobolev Exponent ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Schrödinger System

We are concerned with the following modified nonlinear Schrödinger system:-Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu,  x∈Ω,  -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v,  x∈Ω,  u=0,  v=0,  x∈∂Ω, whereα>2,  β>2,  α+β<2·2*,  2*=2N/(N-2)is the critical Sobolev exponent, andΩ⊂ℝN  (N≥3)is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

scholarly journals Ground State Solution for an Autonomous Nonlinear Schrödinger System

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Min Liu ◽  
Jiu Liu
Keyword(s):  
Ground State ◽  
Critical Sobolev Exponent ◽  
Growth Conditions ◽  
Ground State Solution ◽  
State Solution ◽  
Subcritical Growth ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Sobolev Exponent ◽  
Schrödinger System

In this paper, we study the following autonomous nonlinear Schrödinger system (discussed in the paper), where λ , μ , and ν are positive parameters; 2 ∗ = 2 N / N − 2 is the critical Sobolev exponent; and f satisfies general subcritical growth conditions. With the help of the Pohožaev manifold, a ground state solution is obtained.

scholarly journals Normalized solutions for a coupled Schrödinger system with saturable nonlinearities

2018 ◽  
Vol 459 (1) ◽  
pp. 247-265 ◽  
Author(s):  
Xiaofei Cao ◽  
Junxiang Xu ◽  
Jun Wang ◽  
Fubao Zhang
Keyword(s):  
Coupled Schrödinger System ◽  
Normalized Solutions ◽  
Schrödinger System

scholarly journals Normalized solutions to the Chern-Simons-Schrödinger system

2021 ◽  
Vol 280 (5) ◽  
pp. 108894
Author(s):  
Tianxiang Gou ◽  
Zhitao Zhang
Keyword(s):  
Normalized Solutions ◽  
Chern Simons ◽  
Schrödinger System

scholarly journals Normalized solutions for a coupled fractional Schrödinger system in low dimensions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Meng Li ◽  
Jinchun He ◽  
Haoyuan Xu ◽  
Meihua Yang
Keyword(s):  
Coupling Parameter ◽  
Low Dimensions ◽  
Normalized Solutions ◽  
Normalized Solution ◽  
Schrödinger System

Abstract We consider the following coupled fractional Schrödinger system: $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \vert ^{2p-2}u+ \beta \vert v \vert ^{p} \vert u \vert ^{p-2}u, \\ (-\Delta )^{s}v+\lambda _{2}v=\mu _{2} \vert v \vert ^{2p-2}v+\beta \vert u \vert ^{p} \vert v \vert ^{p-2}v \end{cases}\displaystyle \quad \text{in } {\mathbb{R}^{N}}, $$ { ( − Δ ) s u + λ 1 u = μ 1 | u | 2 p − 2 u + β | v | p | u | p − 2 u , ( − Δ ) s v + λ 2 v = μ 2 | v | 2 p − 2 v + β | u | p | v | p − 2 v in  R N , with $0< s<1$ 0 < s < 1 , $2s< N\le 4s$ 2 s < N ≤ 4 s and $1+\frac{2s}{N}< p<\frac{N}{N-2s}$ 1 + 2 s N < p < N N − 2 s , under the following constraint: $$ \int _{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx=a_{1}^{2} \quad \text{and}\quad \int _{ \mathbb{R}^{N}} \vert v \vert ^{2}\,dx=a_{2}^{2}. $$ ∫ R N | u | 2 d x = a 1 2 and ∫ R N | v | 2 d x = a 2 2 . Assuming that the parameters $\mu _{1}$ μ 1 , $\mu _{2}$ μ 2 , $a_{1}$ a 1 , $a_{2}$ a 2 are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter $\beta >0$ β > 0 .

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