In this article, we have two parts. In the first part, we are concerned with the locally Holder continuity of quasi-minima of the following integral functional
∫Ω f(x, u, Du) dx, (1)
where Ω is an open subset of Euclidean N-space (N ≥3), u: Ω→R, the Carath\'eodory function f satisfies the critical Sobolev exponent growth condition
|Du|p-|u|p*-a(x)l≤ f(x, u, Du)≤ L(|Du|p+|u|p*+a(x)), (2)
where L≤1, 1<p<N, p*=Np/N-p, and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally H\"{o}lder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland's variational principal.
