Regularity for Very Weak Solutions of A-Harmonic Equation with Weight

. This paper deals with very weak solutions of the A-harmonic equationdivA(x,5u) = 0 (∗)with the operator A : Ω × R n → R n satises some coercivity and controllable growthconditions with Muckenhoupt weight. By using the Hodge decomposition with weight, aregularity property is proved: There exists an integrable exponent r 1 = r 1 (λ,n,p) < p,such that every very weak solution u ∈ W 1,rloc (Ω,w) with r 1 < r < p belongs to W 1,ploc (Ω,w).That is, u is a weak solution to (∗) in the usual sense. 1. Introduction and statement of resultLet w be a locally integrable, nonnegative function in R n . Then a Radonmeasure µ is canonically associated with the weight w,(1.1) µ(E) =Z E w(x)dx.Thus dµ(x) = w(x)dx, where dx is the n-dimensional Lebesgue measure. In whatfollows, the weight w and the measure µ are identied via (1.1). Let Ω be an opensubset of R n , n ≥ 2. Consider the following second order divergence type elliptic ∗ Corresponding author.Received 18 January 2007; accepted 26 December 2008.2000 Mathematics Subject Classication: 35J60.Key words and phrases: A-harmonic equation, Muckenhoupt weight, regularity, Hodgedecomposition.The rst author is supported by NSF of Hebei Province (07M003). The third author issupported by NSFC (60850005), NSF of Zhejiang Province (Y607128).