Critical functions and elliptic PDE on compact riemannian manifolds

TL;DR

This paper investigates the existence of solutions to a critical elliptic PDE on compact Riemannian manifolds using a novel concept of 'critical function', addressing cases not solvable by traditional variational methods.

Contribution

It introduces the use of 'critical functions' for analyzing critical elliptic PDEs on manifolds, extending the understanding of concentration phenomena and solution existence.

Findings

Established conditions for existence of solutions in the critical case

Proved estimates related to concentration phenomena

Extended the concept of critical functions to elliptic PDEs on manifolds

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.