Classification of positive solutions of critical anisotropic Sobolev equation without the finite volume constraint

• DOI: 10.48550/arxiv.2404.11628
• Publication Date: 2024-01-01
• Authors: Chen, Lu; Yang, Yabo
• Keywords: Mathematics - Analysis of PDEs; FOS: Mathematics; Analysis of PDEs (math.AP)

In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation*} -Δ^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation*} without the finite volume constraint for $n \geq 2$ and $\frac{(n+1)}{3} \leq p < n$, where $p^{*} = \frac{np}{n-p}$ denotes the critical Sobolev exponent and $-Δ^{H}_{p}=-div(H^{p-1}(\cdot)\nabla H(\cdot))$ denotes the anisotropic $p$-Laplace operator. This result removes the finite volume assumption on the classification of critical anisotropic $p$-Laplace equation which was obtained by Ciraolo-Figalli-Roncoroni in the literature \cite{CFR}. The method is based on constructing suitable vector fields integral inequality and using Newton's type inequality.