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

TL;DR

The paper addresses the classification of positive local weak solutions to the critical $p$-Laplace equation $\Delta_p u + u^{p^*-1}=0$ in $\mathbb{R}^n$, with $1<p<n$ and $n\ge3$, by establishing optimal or near-optimal integral estimates that constrain the behavior of $u$ and its transform. It introduces the auxiliary function $v = u^{-\frac{p}{n-p}}$ and the $P$-function, derives two regimes of integral bounds for $u^\gamma$ and $v^{-q}$, and proves that $u \in L^{p^*-1}(\mathbb{R}^n)$, leading to higher integrability results. Under global boundedness (i.e., $u\in L^\infty(\mathbb{R}^n)$) or certain $L^q$ conditions, the solutions are shown to be $p$-bubbles of the explicit form $U_p[z,\lambda]$, providing a complete classification in these cases. The work also presents an alternative $P$-function based proof that does not rely on $L^\infty$ bounds and discusses corollaries related to growth and uniform continuity of $u^{p^*-1}$, broadening the known landscape beyond energy solutions. These results sharpen our understanding of the rigidity of positive solutions to the critical $p$-Laplace equation and have implications for related geometric and variational problems.

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 $n\geq 3$, must be a bubble.

Theorems & Definitions (22)

• Theorem 1
• Theorem 1.1
• Theorem 1.2
• Corollary 1.2.1
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• proof
• Proposition 2.2
• proof