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In the first part of this thesis, we study the Yamabe problem withsingularities, that we can announce as follow: Given a compact Riemannianmanifold $(M,g)$, find a constant scalar curvature metric, conformal to $g$,when $g$ has not necessarily the usual regularity (it can be $C^1$). To solvethis problem, we start the study of the Yamabe type equations. We show that allthe known properties in the smooth case are still valid. Under someassumptions, we prove the existence and uniqueness of solutions. The secondpart is dedicated to the Hebey-Vaugon conjecture, stated in their paper aboutthe equivariant Yamabe problem. We prove that this conjecture is true in somenew cases, after we generalize T. Aubin's theorem.

Le probl\'eme de Yamabe avec singularit\'es et la conjecture de Hebey-Vaugon