Approximate radial symmetry for $p$-Laplace equations via the moving planes method

TL;DR

This paper studies approximate radial symmetry in solutions to perturbed $p$-Laplace equations, extending symmetry results using the moving planes method and providing quantitative inequalities and comparison principles.

Contribution

It introduces a quantitative approach to approximate symmetry for perturbed $p$-Laplace equations and establishes new comparison principles for small domains.

Findings

Quantitative symmetry results for perturbed $p$-Laplace equations.

New comparison principle for small domains.

Extension of symmetry techniques to $p$-Laplace equations.

Abstract

We investigate quasi-symmetry for small perturbations of the Gidas-Ni-Nirenberg problem involving the $p$-Laplacian and for small perturbations the critical $p$-Laplace equation for $p>2$. To achieve these results, we provide a quantitative review of the work by Damascelli & Sciunzi (Calc. Var. Partial Differential Equations 25 (2006), no. 2, 139-159) concerning the weak Harnack comparison inequality and the local boundedness comparison inequality. Moreover, we prove a comparison principle for small domains.