# Moser interation for nonlinear elliptic equation

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~

-

1
– Siminore Jul 14 '12 at 8:01
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 11, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9217755794525146, "perplexity_flag": "head"}