Fonctions critiques et \'equations aux d\'eriv\'ees partielles sur les vari\'et\'es Riemanniennes compactes

TL;DR

This paper investigates the existence of solutions to a critical nonlinear PDE on compact Riemannian manifolds, introducing a new approach using the concept of 'critical functions' to handle cases not solvable by traditional variational methods.

Contribution

The work extends the theory of critical PDEs on manifolds by applying the concept of 'critical functions' to establish existence results in challenging limit cases.

Findings

Proves existence of solutions in non-variational limit cases.

Develops estimates for concentration phenomena with non-constant functions.

Introduces the use of 'critical functions' in this context.

Abstract

We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle_{\textbf{g}}u+h.u = f.u^{\frac{n+2}{n-2}}$ on a compact riemannian manifold in the limit case normally not solved by variational methods. For this purpose, we use a concept of "critical function" that was originally introduced by E. Hebey and M. Vaugon for the study of second best constant in the Sobolev embeddings. Along the way, we prove an important estimate concerning concentration phenomena's when $f$ is a non-constant function. We give here intuitive details.