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

Abstract

In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation}\label{0.1} -\Delta^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation} without the finite volume constraint for $n \geq 3$ and $p_n(\Lambda) < p < n$, where $p^{*} = \frac{np}{n-p}$ denotes the critical Sobolev exponent, $-\Delta^{H}_{p}=-div(H^{p-1}(\cdot)\nabla H(\cdot))$ denotes the anisotropic $p$-Laplace operator and $\Lambda = \lambda\max\limits_{\substack{\xi \in \mathbb{R}^n\\1 \leq i, j \leq n}}\left\{\frac{|\xi|^{2}(\nabla^{2}_{ij}H^{p}(\xi))} {p(p-1)H^{p}(\xi)}\right\}$. By employing a novel approach based on invariant tensors technique, and using a Kato-type inequality, we prove that the positive solutions of \eqref{0.1} can be classified for $p_n(\Lambda) \leq p < n$, where $p_n(\Lambda)$ depends explicitly on $\Lambda$. 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}. In particular, this results capture the precise dependence of critical exponents $p$ on both $n$ and $\Lambda$.