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

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