Can someone give me a proof for a Caccioppoli type inequality looking like $$ \int_M |\nabla u|^2 \psi^2 \leq C \int_M u^2 |\nabla \psi| ^2 $$ where $M$ is a Riemannian manifold.
I have seen multiple times this inequality but I can't find a proof and I don't know what are the hypothesis. I also don't see how to prove it.
The inequality holds only if $u$ satisfies additional assumptions. The simplest is if $u$ satisfies $u\ge 0$ (my comment above was wrong) and $$ -\Delta u \le 0. $$ Also, assume that $\psi$ is compactly supported. In that case, \begin{align*} \int |\nabla u|^2\psi^2 &= \int \nabla^i(u\psi^2\nabla_i u)- u\Delta u\psi^2 - 2u\psi\nabla\psi\cdot\nabla u\\ &\le \int -2(u\nabla\psi)\cdot(\psi\nabla u)\\ &\le 2\left(\int u^2|\nabla\psi|^2\right)^{1/2}\left(\int \psi^2|\nabla u|^2\right)^{1/2}\\ &\le 2\left(\delta^{-2}\int u^2|\nabla\psi|^2\right)^{1/2}\left(\delta^2\int \psi^2|\nabla u|^2\right)^{1/2}\\ &\le \delta^{-2}\int u^2|\nabla\psi|^2 + \delta^2\int \psi^2|\nabla u|^2 \end{align*} If we set $\delta$ to any positive value less than $1$, then we get $$ \int |\nabla u|^2\psi^2 \le \frac{\delta^{-2}}{1-\delta^2}\int u^2|\nabla\psi|^2. $$