On a class of Nonlinear Grushin equations

TL;DR

This paper investigates nonlinear degenerate elliptic equations involving the Grushin operator, establishing symmetry, decay, nonexistence, and existence results through advanced analytical methods.

Contribution

It introduces new symmetry and nonexistence results for Grushin equations and develops a priori estimates and existence proofs for broader classes of these equations.

Findings

Solutions exhibit radial symmetry and specific decay rates.

Nonexistence of solutions in Euclidean half space under certain conditions.

Existence of positive solutions for generalized Grushin equations.

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.