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Let $\Omega$ be and bounded open set in $\mathbb{R}^n$ and $u\in H^1(\Omega)\cap L^{\infty}(\Omega)$ be the weak subsolution of the following nonlinear and heterogeneous elliptic equation: $$-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+c(x)|u|^{p-2}u=f$$ where $0\le c(x)\le M$ and $f\in L^\infty(\Omega)$. That is, for every $\phi\in H^1(\Omega)\cap L^{\infty}(\Omega)$, $$\int_\Omega|\nabla u|^{p-2}\nabla u\cdot\nabla\phi+c(x)|u|^{p-2}u\phi\,dx\le\int_\Omega f\phi\,dx$$

Show that there exists a constant $R_0=R_0(M)>0$, such that for every $0<R\le R_0$ and $\forall x_0\in\Omega$, $$\sup_{B_{R/2}}u\le C\left(\frac{1}{R^n}\int_{B_R}u^p\,dx\right)^{1/p}+C\|f\|_\infty$$ , where $B_R=B_R(x_0)\subset\Omega$ and $C=C(n,R_0,M)$.

I think this can be done by a generalized version of Moser iteration, extended from linear case to nonlinear one. Can anyone give some clues or references? Thank you~

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Probably you can find good ideas in this paper: mai.liu.se/~toada/harnack_final.pdf –  Siminore Jul 14 '12 at 8:01

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