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Liouville-type theorem for a nonlinear sub-elliptic system involving \(\Delta _\lambda \)-Laplacian and advection terms

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Abstract

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.

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Authors and Affiliations

  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Ha Noi, Viet Nam

    Anh Tuan Duong

  2. Academy of Finance, Dong Ngac, Bac Tu Liem, Ha Noi, Viet Nam

    Dao Trong Quyet

  3. Department of Education and Training of Phu Tho Province, Viet Tri, Phu Tho, Viet Nam

    Nguyen Van Biet

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Duong, A.T., Quyet, D.T. & Van Biet, N. Liouville-type theorem for a nonlinear sub-elliptic system involving \(\Delta _\lambda \)-Laplacian and advection terms. J. Fixed Point Theory Appl. 25, 52 (2023). https://doi.org/10.1007/s11784-023-01057-9

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