Sobolev Minimizers for Critical Vectorial p-Laplacian Problems
Abstract
We study minimizers of the optimal Sobolev quotients for the critical embeddings D1,p(ℝN;ℝm)↪ Lp*(ℝN;ℝm) and W1,p(Ω;ℝm)↪ Lp*(Ω;ℝm). In the whole space, we classify all minimizers as u=c Uδ,x0, where c≠0 and Uδ,x0 is an Aubin--Talenti bubble. On bounded domains and for λ>0, we equip W1,p(Ω;ℝm) with $ \|\boldsymbol u\|_λ^p :=\int_Ω \bigl(|D\boldsymbol u|^p+λ|\boldsymbol u|^p\bigr) $. Using scalar attainment results, we establish existence and synchronization of vectorial minimizers. At resonance, we consider the pure Neumann problem $ -\boldsymbolΔ_p\boldsymbol u =|\boldsymbol u|^{p^*-2}\boldsymbol u\ \text{in }Ω. $ For the associated quotient, we prove existence and global C1(Ω̅) regularity of minimizers. Finally, we prove synchronization of radial solutions and deduce symmetry breaking on balls.
Related
- Elliptic problems involving a diffusion operator of vectorial p&q-Laplacian type
- Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group
- The homogeneity function of the fractional p-Laplacian and optimal asymptotic exponents for critical Hardy-Sobolev equations
- Non-radial minimizers for the first eigenvalue of the Laplacian in cones and a related overdetermined problem
- A complete theory of local minimizers for periodic variational problems
BibTeX
@misc{mauro2026sobolev,
title = {{Sobolev Minimizers for Critical Vectorial $p$-Laplacian Problems}},
author = {Simone Mauro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18002},
eprint = {2610.18002},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18002},
}
RIS
TY - UNPB
TI - Sobolev Minimizers for Critical Vectorial p-Laplacian Problems
AU - Simone Mauro
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18002
UR - https://arxiv.org/abs/2610.18002
AB - We study minimizers of the optimal Sobolev quotients for the critical embeddings D^1,p(ℝ^N;ℝ^m)↪ L^p^*(ℝ^N;ℝ^m) and W^1,p(Ω;ℝ^m)↪ L^p^*(Ω;ℝ^m). In the whole space, we classify all minimizers as u=c U_δ,x_0, where c≠0 and U_δ,x_0 is an Aubin--Talenti bubble. On bounded domains and for λ>0, we equip W^1,p(Ω;ℝ^m) with $ \|\boldsymbol u\|_λ^p :=\int_Ω \bigl(|D\boldsymbol u|^p+λ|\boldsymbol u|^p\bigr) $. Using scalar attainment results, we establish existence and synchronization of vectorial minimizers. At resonance, we consider the pure Neumann problem $ -\boldsymbolΔ_p\boldsymbol u =|\boldsymbol u|^{p^*-2}\boldsymbol u\ \text{in }Ω. $ For the associated quotient, we prove existence and global C^1(Ω̅) regularity of minimizers. Finally, we prove synchronization of radial solutions and deduce symmetry breaking on balls.
ER -