On a class of Nonlinear Grushin equations

TL;DR

The paper analyzes nonlinear degenerate elliptic equations driven by the Grushin operator $-$\Delta_\gamma$, establishing symmetry, nonexistence, a priori estimates, and existence results. It leverages the moving plane method adapted to Grushin geometry to obtain radial symmetry in the $y$-variables and exponential decay for the model equation in $\mathbb{R}^{N+l}$, and proves Liouville-type nonexistence in the half-space $\mathbb{R}^{N+l}_+$. A blow-up analysis tailored to anisotropic Grushin scaling provides uniform a priori bounds for general nonlinearities $f(z,u)$, which, together with Schauder regularity, enables a topological degree argument to guarantee the existence of positive solutions on bounded domains. The work integrates symmetry, Liouville theorems, blow-up techniques, and degree theory to advance understanding of nonlinear Grushin-type problems and their hypoelliptic structure.

Abstract

In this paper, we study two kinds of nonlinear degenerate elliptic equations containing the Grushin operator. First, we prove radial symmetry and a decay rate at infinity of solutions to such a Grushin equation by using the moving plane method in combination with suitable integral inequalities. Applying similar methods, we obtain nonexistence results for solutions to a second type of Grushin equation in Euclidean half space. Finally, we derive a priori estimates and the existence for positive solutions to more general types of Grushin equations by employing blow up analysis and topological degree methods, respectively.

Theorems & Definitions (22)

• Proposition 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2