Classification results for bounded positive solutions to the critical $p$-Laplace equation

TL;DR

This paper proves that positive, bounded, or moderately growing solutions to the critical p-Laplace equation in higher dimensions are necessarily bubbles, based on integral estimates and behavior analysis.

Contribution

It establishes that solutions with certain growth conditions are exactly the bubble solutions, advancing understanding of the critical p-Laplace equation's solution structure.

Findings

Positive bounded solutions are bubbles.

Integral estimates are optimal or nearly optimal.

Solutions with proper infimum behavior are classified as bubbles.

Abstract

By providing optimal or nearly optimal integral estimates, we show that every positive, bounded or moderately growing, local weak solution to the critical $p$-Laplace equation in $\mathbb{R}^n$, with $n\geq 3$, and whose infimum over a ball behaves properly must be a bubble.