I’m trying to obtain an $L^{p}$ estimate under certain conditions. Suppose $f\in C^{2}(B_{2})$ is a function satisfying $0<f<1$ and $f\Delta f>\frac{1}{n} |\nabla f|^{2}$, where $B_{2}\subset\mathbb{R}^{n}$. I want to prove the following: For any $\delta>0$, there exists $\epsilon>0$, such that if $\int_{B_{2}}|f-1|<\epsilon$, $\int_{B_{2}}\frac{|\nabla f|^{2}}{f}<\epsilon$, and $\int_{B_{2}}|\nabla f|^{2p}<\epsilon$, where $p>1$ is an integer, then $\int_{B_{1}}\frac{|\nabla f|^{2p}}{f^{p}}<\delta$. Note that I only need the inequality in the smaller ball. So one may use cut-off function.
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