Critical singular problems in Carnot groups

TL;DR

This paper investigates the existence of solutions to a critical semilinear PDE with singular perturbations in Carnot groups, introducing novel methods adapted from Euclidean techniques to this non-Euclidean setting.

Contribution

It is the first to analyze singular perturbations of power-type in semilinear PDEs within Carnot groups, employing variational and estimate-based methods.

Findings

Proved existence of two positive weak solutions.

Developed adaptation of Aubin-Talenti functions for Carnot groups.

Extended PDE analysis to non-Euclidean, singular perturbation context.

Abstract

We consider a power-type mild singular perturbation of a Dirichlet semilinear critical problem settled in an open and bounded set in a Carnot group. Here, the term critical has to be understood in the sense of the Sobolev embedding. We aim to prove the existence of two positive weak solutions: the first one is obtained by means of the variational Perron's method, while for the second one we adapt a classical argument relying on proper estimates of a family of functions which mimic the role of the classical Aubin-Talenti functions in the Euclidean setting. Our results fall in the framework of semilinear PDEs in Carnot group but, as far as we know, are the first ones dealing with singular perturbations of power-type.