Sobolev Minimizers for Critical Vectorial $p$-Laplacian Problems

We study minimizers of the optimal Sobolev quotients for the critical embeddings $D^{1,p}(\mathbb R^N;\mathbb R^m)\hookrightarrow L^{p^*}(\mathbb R^N;\mathbb R^m)$ and $W^{1,p}(Ω;\mathbb R^m)\hookrightarrow L^{p^*}(Ω;\mathbb R^m)$. In the whole space, we classify all minimizers as $\boldsymbol u=\bo