Liouville type theorems for elliptic equations involving Grushin operator and advection

Abstract

<p>In this article, we study the equation</p> <pre>-G_αu + ∇_Gw ∙ ∇_Gu = ∥<b>x</b>∥^s|u|^p-1u, <b>x</b> = (x, y) ∈ ℝ^N = ℝ^N₁ x ℝ^N₂,</pre> <p>where G_α (resp., ∇_G) is Grushin operator (resp. Grushin gradient), p > 1 and s ≥ 0. The scalar function w satisfies a decay condition, and ∥<b>x</b>∥ 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.</p>