(Theorem 3.1, Corollary 3.2) Let $\varphi$ be a smooth solution to (1.1), (1.2), then for any $1<p<\infty$, there exists a constant $\alpha(p)>0$, depending only on $p$, and another constant $C$, depending only on $||\varphi||_{0}$, the background metric $g$, and $p$, such that

(1.8)

$$\int_{M}e^{-\alpha(p)F}(n+\Delta\varphi)^{p}\leq C.$$

In particular, $||n+\Delta\varphi||_{L^{p}(dvol_{g})}\leq C^{\prime}$, where $C^{\prime}$ has the same dependence as $C$ in this theorem, but additionally on $||F||_{0}$.