A Class of De Giorgi Type and Hölder Continuity for Some Problems in Musielak-Orlicz-Sobolev Spaces

In this paper, we introduce a new class of De Giorgi type functions, denoted by ${\mathcal{B}}_{G(x,t)}$, and establish the Hölder continuity of its elements under suitable additional assumptions on the generalized $\text{N}$-function $G(x,t)$. As an application, we prove the Hölder continuity of solutions to quasilinear equations whose principal part is in divergence form with $G(x,t)$-growth conditions, including both critical and standard growth cases. The novelty of our work lies in the generalization of the Hölder continuity results previously known for variable exponent \cite[X, Fan and D. Zhao]{Fan1999} and Orlicz \cite[G. M. Lieberman]{Li1991} problems. Moreover, our results encompass a wide variety of quasilinear equations.