Let \(N\geq 3\) be an integer, let \(p\in (2,N)\) and let \(p^*:=\frac{Np}{N-p}\) be the critical exponent for the Sobolev embedding. In this interesting paper, the author considers the following critical problem
\[ \begin{aligned} -\Delta_p u = u^{p^*-1} \text{ in }\mathbb{R}^N,\quad u>0\text{ in }\mathbb{R}^N,\\ u\in \mathcal{D}^{1,p}(\mathbb{R}^N):=\{u\in L^{p^*}(\mathbb{R}^N):|\nabla u|\in L^p(\mathbb{R}^N)\}.\end{aligned}\tag{P} \]
and proves that every solution \(u\) to problem (P) is of the form \[ u(x)=U_{\lambda,x_0}(x):=\left[\frac{\lambda^{\frac{1}{p-1}}N^{\frac{1}{p}}\left(\frac{N-p}{p-1}\right)^{\frac{p-1}{p}}} {\lambda^{\frac{p}{p-1}}+|x-x_0|^{\frac{p}{p-1}}}\right]^{\frac{N-p}{p}},\quad x\in \mathbb{R}^N, \] where \(\lambda\in (0,\infty)\) and \(x_0\in \mathbb{R}^N\). In particular, all the solutions to (P) are radial, radially decreasing, and minimizers to \[ \min_{\varphi\in\mathcal{D}^{1,p}(\mathbb{R}^N)\setminus \{0\}}\frac{\int_{\mathbb{R}^N}|\nabla \varphi|^pdx}{(\int_{\mathbb{R}^N}|\varphi|^{p^*})^{\frac{p}{p^*}}}. \] For \(p=2\) and for \(p\in (1,2)\) , this result has been proved, respectively, in [L. A. Caffarelli et al., Commun. Pure Appl. Math. 42, No. 3, 271–297 (1989; Zbl 0702.35085); J. Vétois, J. Differ. Equations 260, No. 1, 149–161 (2016; Zbl 1327.35117)]. Thanks to the result of the present paper, a complete classification of the solutions to (P) is now available for \(p\) in the whole range \((1,N)\). The key ingredients of the proof are a positive lower bound for \(|\nabla u(x)||x|^{-\frac{N-1}{p-1}}\), with \(|x|\geq R_0>0\), and a moving plane technique.