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

In this article, we have two parts. In the first part, we are concerned with the locally Hölder continuity of quasi-minima of the following integral functionalequation(1)∫Ωf(x,u,Du)dx,∫Ωf(x,u,Du)dx,where Ω is an open subset of Euclidean N-space (N ≥ 3), u : Ω → R, the Carathéodory function f satisfies the critical Sobolev exponent growth conditionequation(2)|Dup|-|u|p*-a(x)≤f(x,u,Du)≤L(|Dup|+|u|p*+a(x)),|Du|p-|u|p*-a(x)≤f(x,u,Du)≤L(|Du|p+|u|p*+a(x)),where L≥1,1