A NOTE ON REGULARITY FOR A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS

The following regularity of weak solutions of a class of elliptic equations of the form are investigated, div A(x,u,Du)+B(x,u,Du)=0 in Ω.(*) Here Ω R n is a bounded domain, A(x,z,p)=(A 1(x,z,p),A 2(x,z,p),...,A n(x,z,p)) and B(x,z,p) satisfy 1Λ(κ+|p|) m|ξ| 2≤ ο A i ο p j(x,z,p)ξ iξ j≤Λ(κ+|p|) m|ξ| 2, |A(x,z,p)-A(y,w,p)|≤(1+|p|) m+1 φ(|x-y|+|z-w| and |B(x,z,p)|≤Λ(1+|p|) m+2 for all (x,z,p),(y,m,p)∈Ω× R×R n and all ξ∈ R n,where m≥0,κ≥0,Λ>0 and φ(r) is a bounded increasing function in [0,∞). The results of the paper are:a) if lim r→0 + φ(r)=φ(0)=0 ,then any bounded solution of (*) belongs to C β loc ( Ω ) for any β∈(0,1) ;b) if m =0,and φ(r) is Dini continuous,that is,lim r→0 + φ(r)=φ(0)=0, ∫ 1 0φ(r)r d r<+∞, then any bounded solution of (*) ∈ C 1 loc ( Ω ).