Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2024-04-10 , DOI: 10.1007/s00526-024-02695-8 Yuxia Guo , Yichen Hu , Shaolong Peng
In this paper, we consider the following elliptic system
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |v|^{p-1}v +\epsilon (\alpha u + \beta _1 v), &{}\quad {\hbox {in}}\; \Omega , \\ -\Delta v = |u|^{q-1}u+\epsilon (\beta _2 u +\alpha v), &{}\quad {\hbox {in}}\;\Omega , \\ u=v=0,&{}\quad {\hbox {on}}\; \partial \Omega , \end{array}\right. } \end{aligned}$$where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^{N}\), \(N\ge 3\), \(\epsilon \) is a small parameter, \(\alpha \), \( \beta _1\) and \( \beta _2\) are real numbers, (p, q) is a pair of positive numbers lying on the critical hyperbola
$$\begin{aligned} \begin{aligned} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}. \end{aligned} \end{aligned}$$We first revisited the blowing-up solutions constructed in Kim and Pistoia (J Funct Anal 281(2):58, 2021) and then we proved its non-degeneracy. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems involving critical Halmitonian system and the construction of new solutions.
中文翻译：
具有线性扰动的临界 Lane-Emden 系统解的非简并性
在本文中，我们考虑以下椭圆系统
$$\begin{对齐} {\left\{ \begin{array}{ll} -\Delta u = |v|^{p-1}v +\epsilon (\alpha u + \beta _1 v), & {}\quad {\hbox {in}}\; \Omega , \\ -\Delta v = |u|^{q-1}u+\epsilon (\beta _2 u +\alpha v), &{}\quad {\hbox {in}}\;\Omega , \\ u=v=0,&{}\quad {\hbox {on}}\; \partial \Omega ， \end{array}\right。 } \end{对齐}$$其中\(\Omega \)是\(\mathbb {R}^{N}\)中的平滑有界域，\(N\ge 3\)，\(\epsilon \)是一个小参数，\(\ alpha \)、\( \beta _1\)和\( \beta _2\)是实数， ( p , q ) 是位于临界双曲线上的一对正数
$$\begin{对齐} \begin{对齐} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}。 \end{对齐} \end{对齐}$$我们首先重新审视了 Kim 和 Pistoia 构建的爆破解决方案 (J Funct Anal 281(2):58, 2021)，然后证明了其非简并性。我们相信，本文中使用的各种新思想和技术计算对于处理涉及关键哈米顿系统和构建新解决方案的其他相关问题非常有用。
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