Given $(M, g)$ a compact Riemannian manifold of dimension $n \geq 3$, we consider the following equation for the function $\phi$:
$$ -\Delta \phi + R \phi + \tau^2 \phi^{N-1} = \frac{A^2}{\phi^{N+1}} $$
where $R,\ \tau$ and $A$ are smooth given functions and $N = \frac{2n}{n-2}$.
Assume that $\tau > 0$, $R$ has constant sign and $\|A\|_{L^2} \geq \epsilon$ for some given $\epsilon > 0$.
I would like to show that $\frac{A}{\phi^N}$ is bounded in $L^2$ by a constant which does not depend on $A$.
The idea is to multiply the equation by $\phi^{N+1}$ and integrate over $M$:
$$ \int_M \left(\left|d\phi^{1-\frac{N}{2}}\right|^2 - R \phi^{2-N} + \frac{A^2}{\phi^{2N}}\right) = \int_M \tau^2. $$
(I removed the constant in front of $\left|d\phi^{1-\frac{N}{2}}\right|^2$ to simplify)
If $R$ is negative, everything go smoothly. But if $R$ is nonnegative, things become harder. The only information I have is that
A possible strategy would be to try to find a region $S$ where $A$ is not small (because its $L^2$-norm is bounded from below by $\epsilon$) and use an inequality similar to (7.45) in Gilbarg and Trudinger's book:
$$ \|u - u_S\|_{L^p} \leq C |S|^{1/n - 1} \|\nabla u\|_{L^p} $$
where $|S|$ the Lebesgue measure of $S$ and $u_S = \frac{1}{|S|} \int_S u$. But I was unable to implement it.
Any help would be invaluable!
