  This paper deals with boundedness results for weak solutions of an elliptic
equation where the functions are Carath\'eodory functions satisfying certain
$p$-structure conditions that have critical growth even on the boundary. Based
on a modified version of the Moser iteration we are able to prove that every
weak solution of our problem is bounded up to the boundary. Under some
additional assumptions this leads directly to $C^{1,\alpha}$-regularity for
weak solutions of the problem.
