On the anisotropic critical $p$-Laplace equation: classification, decomposition, and stability results

TL;DR

This paper classifies and analyzes solutions to the anisotropic critical p-Laplace equation, establishing decomposition, stability, and interaction estimates for solutions in Euclidean space.

Contribution

It introduces an anisotropic version of Struwe's decomposition, provides a simplified classification proof, and proves quantitative stability of solutions.

Findings

Established an anisotropic Struwe's decomposition.

Proved that solutions close to bubbles are stable.

Provided interaction estimates for bubble families.

Abstract

We investigate both qualitative and quantitative issues related to the classification of non-negative energy solutions to the anisotropic critical $p$-Laplace equation in $\mathbb{R}^n$, for $1<p<n$. Specifically, we establish an anisotropic version of Struwe's decomposition, along with the interaction estimate for the family of bubbles in this decomposition. Moreover, we provide a short proof of the classification result as well as a quantitative stability result, proving that every energy solution to a perturbation of the anisotropic critical equation must be closed to a bubble, in the absence of bubbling.