 Advances in Pure Mathematics, 2011, 1, 235-237 doi:10.4236/apm.2011.14041 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Extremum Principle for Very Weak Solutions of A-Harmonic Equation with Weight* Hong-Ya Gao, Chao Liu, Yu Zhang College of Mathematics and Computer Science, Hebei University, Baoding, China E-mail: 578232915@qq.com Received March 2, 2011; revised April 11, 2011; accepted April 20, 2011 Abstract Extremum principle for very weak solutions of A-harmonic equation ,div Axu0 is obtained, where the operator :nnARR satisfies some coercivity and controllable growth conditions with Muckenhoupt weight. Keywords: A-Harmonic Equation, Muckenhoupt Weight, Extremum Principle, Hodge Decomposition 1. Introduction Throughout this paper will stands for a bounded regular domain in , . By a regular domain we understand any domain of finite measure for which the estimates (1.6) and (1.7) for the Hodge decomposition are justified, see . A Lipschitz domain, for example, is regular. nnR2Given a nonnegative locally integrable function , we say that belongs to the wwpA class of Mucken- houpt, , if 1< p<11111dd=suppppQQQwxwxA wQQ <    (1) where the supremum is taken over all cubes of . When , replace the inequality (1.1) with QnR=1p Mwx cwx for some fixed constant and a.e. cnxR, where M is the Hardy-Littlewood maximal operator. It is well-known that 1pAA whenever , see . We will denote by >1pLw,p, 1<ux vx. We say xux v on  in sobolev sense, or symbolically, uv if the function  defined above lies in 1,0rW. Consider the following second order divergence type elliptic equation (also called A-harmonic equation or Leray-Lions equation) div,=0Ax u (2) where :nnARR is a Carathéodory function and satisfies 1)  ,,pAx wx , 2)  1,pAx wx , where 1<