Journal of Fixed Point Theory and Applications ( IF 1.8 ) Pub Date : 2023-04-10 , DOI: 10.1007/s11784-023-01057-9 Anh Tuan Duong , Dao Trong Quyet , Nguyen Van Biet
In this paper, we are concerned with the following system:
$$\begin{aligned} {\left\{ \begin{array}{ll}-w\Delta _\lambda u-\nabla _\lambda w.\nabla _\lambda u=\rho v^p\\ -w\Delta _\lambda v-\nabla _\lambda w.\nabla _\lambda v=\rho u^q\end{array}\right. } \text{ in } {\mathbb {R}}^N, \end{aligned}$$where \(w,\rho \) are nonnegative continuous functions satisfying some growth conditions at infinity and \(p,q>1\). Here, \(\Delta _\lambda \) is the sub-elliptic operator introduced in [A.E. Kogoj and E. Lanconelli. Nonlinear Anal. 2012;75(12): 4637–4649] and is of the form
$$\begin{aligned} \Delta _\lambda =\sum _{i=1}^{N}\partial _{x_i}(\lambda _i^2 \partial _{x_i}). \end{aligned}$$Our purpose is to establish a Liouville-type theorem for the class of positive stable solutions of the system. On one hand, our result generalizes the result in Duong and Nguyen (Electron J Differ Equ Paper No. 108, 11 pp, 2017) from the equation to the system, and on the other hand, it extends that of Hu (NoDEA Nonlinear Differ Equ Appl 25(1):7, 2018) to the sub-elliptic case.
中文翻译：
涉及 $$\Delta _\lambda $$ -Laplacian 和平流项的非线性子椭圆系统的 Liouville 型定理
在本文中，我们关注以下系统：
$$\begin{aligned} {\left\{ \begin{array}{ll}-w\Delta _\lambda u-\nabla _\lambda w.\nabla _\lambda u=\rho v^p\\ -w\Delta _\lambda v-\nabla _\lambda w.\nabla _\lambda v=\rho u^q\end{array}\right. } \text{ 在 } {\mathbb {R}}^N, \end{aligned}$$其中\(w,\rho \)是在无穷大和\(p,q>1\)处满足某些增长条件的非负连续函数。这里，\(\Delta _\lambda \)是 [AE Kogoj 和 E. Lanconelli 中引入的子椭圆算子。非线性肛门。2012;75(12): 4637–4649 ] 形式
$$\begin{aligned} \Delta _\lambda =\sum _{i=1}^{N}\partial _{x_i}(\lambda _i^2 \partial _{x_i})。\end{对齐}$$我们的目的是为系统的正稳定解类建立刘维尔型定理。一方面，我们的结果将 Duong 和 Nguyen (Electron J Differ Equ Paper No. 108, 11 pp, 2017) 中的结果从方程推广到系统，另一方面，它扩展了 Hu (NoDEA Nonlinear Differ Equ Appl 25(1):7, 2018) 到子椭圆的情况。
京公网安备 11010802027423号