A Symmetry problem for some quasi-linear equations in Euclidean space

TL;DR

This paper establishes precise asymptotic estimates for the gradient of positive solutions to specific nonlinear p-Laplace equations in Euclidean space by demonstrating symmetry and uniqueness in related limiting problems.

Contribution

It introduces new symmetry and uniqueness results for solutions to certain quasi-linear equations, leading to sharp gradient estimates.

Findings

Proved symmetry of solutions to the equations.

Established uniqueness of positive solutions.

Derived sharp asymptotic gradient estimates.

Abstract

We prove sharp asymptotic estimates for the gradient of positive solutions to certain nonlinear $p$-Laplace equations in Euclidean space by showing symmetry and uniqueness of positive solutions to associated limiting problems.