当前位置: X-MOL 学术J. Fixed Point Theory Appl. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities
Journal of Fixed Point Theory and Applications ( IF 1.8 ) Pub Date : 2021-07-18 , DOI: 10.1007/s11784-021-00878-w
Houwang Li 1 , Wenming Zou 1
Affiliation  

In the present paper, we study the normalized solutions with least energy to the following system:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ -\Delta v+\lambda _2v=\mu _2 |v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ \int _{{{\mathbb {R}}^N}}u^2=a_1^2\quad \hbox {and}\quad \int _{{{\mathbb {R}}^N}}v^2=a_2^2, \end{array}\right. } \end{aligned}$$

where \(p,r_1+r_2<2^*\) and \(q\le 2^*\). To this purpose, we study the geometry of the Pohozaev manifold and the associated minimization problem. Under some assumptions on \(a_1,a_2\) and \(\beta \), we obtain the existence of the positive normalized ground state solution to the above system.



中文翻译:

具有临界和亚临界非线性的半线性椭圆系统的归一化基态

在本文中,我们研究了以下系统的能量最少的归一化解:

$$\begin{对齐} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 |u|^{p-2}u+\beta r_1|u|^{r_1 -2}|v|^{r_2}u\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ -\Delta v+\lambda _2v=\mu _2 |v| ^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad &{}\hbox {in}~{{\mathbb {R}}^N}, \\ \int _{{{\mathbb {R}}^N}}u^2=a_1^2\quad \hbox {and}\quad \int _{{{\mathbb {R}}^N}} v^2=a_2^2,\end{array}\right。} \end{对齐}$$

其中\(p,r_1+r_2<2^*\)\(q\le 2^*\)。为此,我们研究了 Pohozaev 流形的几何形状和相关的最小化问题。在\(a_1,a_2\)\(\beta \) 的一些假设下,我们得到上述系统的正归一化基态解的存在性。

更新日期:2021-07-19
down
wechat
bug