The Yamabe problem with singularities

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\ge 3$. Under some assumptions, we prove that there exists a positive function $\phi$ solution of the following Yamabe type equation \Delta \varphi+ h\varphi= \tilde h \varphi^{\frac{n+2}{n-2}} where $h\in L^p(M)$, $p>n\mathrm{/}2$ and $\tilde h\in \mathbb R$\mathrm{.}Wegivetheregularityof$\varphi$withrespecttothevalueof$p$\mathrm{.}Finally,weconsidertheresultsingeometrywhen$g$ is a singular Riemannian metric and $h=\frac{n-2}{4(n-1)}{R}_g$, where $R_g$ is the scalar curvature of $g$.