Existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity

IF 2.1 2区 数学 Q1 MATHEMATICS Advanced Nonlinear Studies Pub Date : 2022-01-01 DOI:10.1515/ans-2022-0010
Jun Wang, Xuan Wang, Song Wei
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引用次数: 4

Abstract

Abstract In the present paper, we study the existence of the normalized solutions for the following coupled elliptic system with quadratic nonlinearity − Δ u − λ 1 u = μ 1 ∣ u ∣ u + β u v in R N , − Δ v − λ 2 v = μ 2 ∣ v ∣ v + β 2 u 2 in R N , \left\{\begin{array}{ll}-\Delta u-{\lambda }_{1}u={\mu }_{1}| u| u+\beta uv\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ -\Delta v-{\lambda }_{2}v={\mu }_{2}| v| v+\frac{\beta }{2}{u}^{2}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where u , v u,v satisfying the additional condition ∫ R N u 2 d x = a 1 , ∫ R N v 2 d x = a 2 . \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={a}_{1},\hspace{1em}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{v}^{2}{\rm{d}}x={a}_{2}. On the one hand, we prove the existence of minimizer for the system with L 2 {L}^{2} -subcritical growth ( N ≤ 3 N\le 3 ). On the other hand, we prove the existence results for different ranges of the coupling parameter β > 0 \beta \gt 0 with L 2 {L}^{2} -supercritical growth ( N = 5 N=5 ). Our argument is based on the rearrangement techniques and the minimax construction.
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二次非线性耦合椭圆系统归一化解的存在性
摘要在本文中,我们研究了以下具有二次非线性的耦合椭圆系统的归一化解的存在性——R N中的−Δu−λ1 u=μ1ÜuÜu+βu v,R N中−Δv−λ2 v=μ2ÜvÜv+β2 u 2,\ left \{\ begin{array}{ll}-\增量u{\lambda}_{1}u={\mu}_{1}|u|u+\beta-uv\hspace{1.0em}&\hspace{0.1em}\text{in}\space{0.1em}\hspace}0.33em}{\mathbb{R}}^{N},\\-\Delta v-{\lambda}_{2}v={\mu}_{2}|v|v+\frac{\beta}{2}{u}^{2}\space{1.0em}&\space{{0.1em}\text{in}\space{0.1em}\ hspace{0.33em}{\mathbb{R}}^{N},\end{array}\right。式中u,vu,v满足附加条件:。\mathop{\int}\limits_{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}x={a}_{1} ,\space{1em}\mathop{\int}\limits_{{\mathbb{R}}}^{N}}{v}^{2}{\rm{d}x={a}_{2} 。一方面,我们证明了具有L2{L}^{2}-次临界增长(N≤3N\le3)系统的极小值的存在性。另一方面,我们证明了在L2{L}^{2}-超临界生长(N=5 N=5)的不同范围内,耦合参数β>0β>0的存在性结果。我们的论点是基于重排技术和极小极大构造。
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来源期刊
CiteScore
3.00
自引率
5.60%
发文量
22
审稿时长
12 months
期刊介绍: Advanced Nonlinear Studies is aimed at publishing papers on nonlinear problems, particulalry those involving Differential Equations, Dynamical Systems, and related areas. It will also publish novel and interesting applications of these areas to problems in engineering and the sciences. Papers submitted to this journal must contain original, timely, and significant results. Articles will generally, but not always, be published in the order when the final copies were received.
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