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In this paper we prove the boundedness and H\"older continuity of quasilinearelliptic problems involving variable exponents for a homogeneous Dirichlet anda nonhomogeneous Neumann boundary condition, respectively. The novelty of ourwork is the fact that we allow critical growth even on the boundary and so weclose the gap in the papers of Fan-Zhao [Nonlinear Anal. 36 (1999), no. 3,295--318.] and Winkert-Zacher [Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no.4, 865--878.] in which the critical cases are excluded. Our approach is basedon a modified version of De Giorgi's iteration technique along with thelocalization method. As a consequence of our results, the$C^{1,\alpha}$-regularity follows immediately.

The boundedness and H\"older continuity of weak solutions to elliptic equations involving variable exponents and critical growth