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The weak Harnack inequality for strong solutions goes as follows (Taking $Lu = a^{ij}(x)D_{ij}u + b^i(x)D_iu+c(x)u$ to be elliptic)

Let $u\in W^{2,n}(\Omega)$ satisfy $Lu\leq f$ in $\Omega$ for some function $f\in L^n(\Omega)$ and suppose $u\geq 0$ in some $B = B_{2R}(y)\subseteq\Omega$. Then $$\left(\frac{1}{|B_R|}\int_{B_R} u^p\right)^{1/p}\leq C\left(\inf_{B_R} u + \frac{R}{\lambda}\|f\|_{L^n(B)}\right)$$ where $p$ and $C$ are positive constants.

I am to show that $p\leq\frac{n}{(n-1)\frac{\Lambda}{\lambda}-1}$ where $\lambda,\Lambda$ denotes the least and greates eigenvalues of $L$, but somehow the proof escapes me. Any hints?

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