The regularity of quasi-minima and ω-minima of integral functionals

In this article, weof Euclidean N-space (N ≥ 3), u: Ω → R, the Carathéodory have two parts. In the first part, we are concerned with the locally Hölder continuity of quasi-minima of the following integral functional. ∫_Ωf(x,u,Du)dx, where Ω is an open subset function f satisfies the critical Sobolev exponent growth condition. |Du|^p-|u|^p*-a(x)≤f(x,u,Du)≤L(|Du|^p+|u|^p*+a(x)), where L≥1,1<p<N,p*=N_p/N_-p, and a(x) is a nonnegative function that lies in a suitable L^p space. In the second part, we study the locally Hölder 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.