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

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