Moser iteration applied to elliptic equations with critical growth on the boundary

Abstract

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.