Sobolev Minimizers for Critical Vectorial $p$-Laplacian Problems

We study minimizers of the optimal Sobolev quotients for the critical embeddings $D^{1,p}({\mathbb{R}}^N;{\mathbb{R}}^m)\hookrightarrow L^{p^{\ast }}({\mathbb{R}}^N;{\mathbb{R}}^m)$ and $W^{1,p}(\mathrm{\Omega };{\mathbb{R}}^m)\hookrightarrow L^{p^{\ast }}(\mathrm{\Omega };{\mathbb{R}}^m)$. In the whole space, we classify all minimizers as $\mathit{u}=\mathit{c}{U}_{\delta ,x_0}$, where $\mathit{c}\mathrm{\ne }\mathbf{0}$ and $U_{\delta ,x_0}$ is an Aubin–Talenti bubble. On bounded domains and for $\lambda >0$, we equip $W^{1,p}(\mathrm{\Omega };{\mathbb{R}}^m)$ with $\mathrm{\parallel }\mathit{u}{\mathrm{\parallel }}_{\lambda }^p:={\int }_{\mathrm{\Omega }}(\mathrm{\mid }D\mathit{u}{\mathrm{\mid }}^p+\lambda \mathrm{\mid }\mathit{u}{\mathrm{\mid }}^p)$. Using scalar attainment results, we establish existence and synchronization of vectorial minimizers. At resonance, we consider the pure Neumann problem $-{\mathbf{\Delta }}_p\mathit{u}=\mathrm{\mid }\mathit{u}{\mathrm{\mid }}^{p^{\ast }-2}\mathit{u}\text{ in }\mathrm{\Omega }\mathrm{.}$ For the associated quotient, we prove existence and global $C^1(\bar{\mathrm{\Omega }})$ regularity of minimizers. Finally, we prove synchronization of radial solutions and deduce symmetry breaking on balls.