  In this paper we prove the boundedness and H\"older continuity of quasilinear
elliptic problems involving variable exponents for a homogeneous Dirichlet and
a nonhomogeneous Neumann boundary condition, respectively. The novelty of our
work is the fact that we allow critical growth even on the boundary and so we
close 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 based
on a modified version of De Giorgi's iteration technique along with the
localization method. As a consequence of our results, the
$C^{1,\alpha}$-regularity follows immediately.
