Critical concave-convex problems in Carnot groups
Mattia Galeotti, Eugenio Vecchi
TL;DR
The paper extends the Brezis–Nirenberg paradigm to Dirichlet problems on Carnot groups with a concave–convex structure and a critical Sobolev nonlinearity. It develops a two-solution framework: a first solution obtained via a variational Perron method, and a second solution obtained through Tarantello-type variational techniques that leverage localized Sobolev minimizers and Ekeland’s principle. The authors establish a finite threshold $\Lambda$ for solvability, prove existence up to $\Lambda$, and demonstrate multiplicity for $0<\lambda<\Lambda$, with a limiting weak solution at $\lambda=\Lambda$. This work advances the understanding of critical, nonlocal PDEs in sub-Riemannian settings and highlights the role of boundary regularity and Sobolev minimizers in multiplicity results on Carnot groups.
Abstract
We consider a model Dirichlet problem with concave-convex and critical nonlinearity settled in Carnot groups. Our aim is to prove the existence of two positve solutions in the spirit of a famous result by Ambrosetti, Brezis and Cerami. To this aim we use a variational Perron method combined with proper estimates of a family of functions which are minimizers of the relevant Sobolev inequality. Due to the lack of boundary regularity, we also have to be careful while proving that the first solution found is a local minimizer in the proper topology.