  Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Under
some assumptions, we prove that there exists a positive function $\varphi$
solution of the following Yamabe type equation \Delta \varphi+ h\varphi= \tilde
h \varphi^{\frac{n+2}{n-2}} where $h\in L^p(M)$, $p>n/2$ and $\tilde h\in
\mathbb R$. We give the regularity of $\varphi$ with respect to the value of
$p$. Finally, we consider the results in geometry when $g$ is a singular
Riemannian metric and $h=\frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar
curvature of $g$.
