Sobolev Minimizers for Critical Vectorial $p$-Laplacian Problems

Abstract

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

arXiv comment26 pages