The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

Abstract

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 [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,αregularity follows immediately.

A new class of double phase variable exponent problems: Existence and uniqueness

“…In the same way we have the embedding into the boundary Lebesgue space, see Fan [27, Proposition 2.1] and Ho-Kim-Winkert-Zhang [40,Proposition 2.5] for the continuous and Fan [25,Corollary 2.4] for the compact embedding. Proposition 2.2.…”

A multiplicity theorem for anisotropic Robin equations

“…Using (3.6), (3.7) and the anisotropic regularity theory, see Winkert-Zacher [22] (see also Ho-Kim-Winkert-Zhang [12]) and Fan [8], we have…”