Liouville type theorems for elliptic equations involving Grushin operator and advection
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In this article, we study the equation
-Gαu + ∇Gw ∙ ∇Gu = ∥x∥s|u|p-1u, x = (x, y) ∈ ℝN = ℝN1 x ℝN2,
where Gα (resp., ∇G) is Grushin operator (resp. Grushin gradient), p > 1 and s ≥ 0. The scalar function w satisfies a decay condition, and ∥x∥ is the norm corresponding to the Grushin distance. Based on the approach by Farina , 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  is still valid for the above equation.
CitationDuong, A. T., & Nguyen, N. T. (2017). Liouville type theorems for elliptic equations involving Grushin operator and advection. Electronic Journal of Differential Equations, 2017(108), pp. 1-11.
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