Critical concave-convex problems in Carnot groups

TL;DR

This paper proves the existence of two positive solutions for a critical nonlinear Dirichlet problem in Carnot groups using variational methods and Sobolev inequalities.

Contribution

It extends variational techniques to Carnot groups with critical nonlinearities, addressing boundary regularity challenges.

Findings

Existence of two positive solutions established

Application of a variational Perron method in Carnot groups

Proper estimates of Sobolev minimizers used in proofs

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.