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We provide the classification of the positive solutions to $-\Delta_p u=u^{p^*-1}$ in $\mathcal {D}^{1,p}(\R^N)$ in the case $2<p<N$. Since the case$1<p\leq2$ is already known this provides the complete classification for$1<p<N$.

Classification of positive $\mathcal {D}^{1,p}(\R^N)$-solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$