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} -Δ^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation} without the finite volume constraint for $n\ge 3$ and $p_n(\Lambda )<p<n$, where $p^{\ast }=\frac{np}{n-p}$ denotes the critical Sobolev exponent, $-{\Delta }_p^H=-div(H^{p-1}(\cdot )\nabla H(\cdot ))$ denotes the anisotropic $p$-Laplace operator and $\Lambda =\lambda \underset{\begin{array}{c}\xi \in {\mathbb{R}}^n\\ 1\le i,j\le n\end{array}}{\max }\left\{\frac{|\xi {|}^2({\nabla }_{ij}^2{H}^p(\xi ))}{p(p-1)H^p(\xi )}\right\}$. 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 $p_n(\Lambda )\le p<n$, 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 $\Lambda$.

The earlier version contained errors in the proof of Lemma 3.6, which we have corrected in this version. Given the asymmetry of the anisotropic operator, diagonalization cannot be performed simultaneously.. And we don't have trace inequality. We have refined the treatment of this section to overcome this difficulty