Liouville type theorems for elliptic equations involving Grushin operator and advection

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.

Keywords

• Liouville type theorem
• stable weak solution
• Grushin operator
• degenerate elliptic equation