Existence of solutions to Yamabe-type equations on Carnot groups

This thesis investigates the existence of weak solutions to Dirichlet problems involving Yamabe-type equations on Carnot groups and Grushin spaces. Such equations are also called critical because they involve a nonlinear part that depends on the critical Sobolev exponent p∗ associated with the underlying geometric structure, a feature that makes standard variational methods inapplicable. We begin by recalling the classical Riemannian formulation of the Yamabe equation, which provides the main geometric motivation for this work. We then introduce Carnot groups, the associated gradient, p-Laplacian, and the Sobolev spaces built on this structure, together with their main properties. In particular, we prove a Sobolev-type inequality and the corresponding Rellich–Kondrachov compactness theorem needed for the variational argument used to establish an existence result in this first study case. Finally, we show how the same strategy can be adapted to Grushin spaces, after recovering the Grushin counterparts of the analytic and geometric tools developed in the Carnot setting.