Electron. J. Differential Equations, Vol. 2017 (2017), No. 108, pp. 1-11.

Liouville type theorems for elliptic equations involving Grushin operator and advection

Anh Tuan Duong, Nhu Thang Nguyen

Abstract:
In this article, we study the equation
$$ -G_{\alpha}u+\nabla_G w\cdot\nabla_Gu=\|\mathbf{x}\|^{s}|u|^{p-1}u , \quad \mathbf{x}=(x,y)\in \mathbb{R}^N= \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, $$
where $ G_\alpha$ (resp., $\nabla_G$) is Grushin operator (resp. Grushin gradient), p>1 and $s\geq 0$. The scalar function w satisfies a decay condition, and $\|\mathbf{x}\|$ is the norm corresponding to the Grushin distance. Based on the approach by Farina [8], we establish a Liouville type theorem for the class of stable sign-changing weak solutions. In particular, we show that the nonexistence result for stable positive classical solutions in [4] is still valid for the above equation.

Submitted January 19, 2017. Published April 25, 2017.
Math Subject Classifications: 35J61, 35B53.
Key Words: Liouville type theorem; stable weak solution; Grushin operator; degenerate elliptic equation.

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Anh Tuan Duong
Department of Mathematics
Hanoi National University of Education
136 Xuan Thuy, Cau Giay, Ha noi, Viet Nam
email: tuanda@hnue.edu.vn
Nhu Thang Nguyen
Department of Mathematics
Hanoi National University of Education
136 Xuan Thuy, Cau Giay, Ha noi, Viet Nam
email: thangnn@hnue.edu.vn

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