A-priori bounds; De Giorgi iteration; Hölder continuity; Localization method; p(.),-Laplacian; Variable exponent Lebesgue and Sobolev spaces
In this paper we prove the boundedness and Hölder 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 (1999)  and Winkert-Zacher (2012)  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 -regularity follows immediately.