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

In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation}\label{0.1} -Δ^{H}_{p}u = u^{p^{*}-1}, \ \ x\in $\mathbb{R}$^n \end{equation} without the finite volume constraint for $n\ge 3$ and $p_n(\mathrm{\Lambda })<p<n$, where $p^{\ast }=\frac{np}{n-p}$ denotes the critical Sobolev exponent, $-{\mathrm{\Delta }}_p^H=-div(H^{p-1}(\cdot )\mathrm{\nabla }H(\cdot ))$ denotes the anisotropic $p$-Laplace operator and $\mathrm{\Lambda }=\lambda \underset{\begin{array}{c}\xi \in {\mathbb{R}}^n\\ 1\le i,j\le n\end{array}}{\max }\left\{\frac{\mathrm{\mid }\xi {\mathrm{\mid }}^2({\mathrm{\nabla }}_{ij}^2{H}^p(\xi ))}{p(p-1)H^p(\xi )}\right\}$. 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 $p_n(\mathrm{\Lambda })\le p<n$, where $p_n(\mathrm{\Lambda })$ depends explicitly on $\mathrm{\Lambda }$. 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 $\mathrm{\Lambda }$.