H\"older continuity of quasiminimizers with nonstandard growth

We show the H"older continuity of quasiminimizers of the energy functionals $\int f(x,u,\mathrm{\nabla }u)dx$ with nonstandard growth under the general structure conditions $$ |z|^{p(x)} - b(x)|y|^{r(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)}

• b(x)|y|^{r(x)} + g(x). The result is illustrated by showing that weak solutions to a class of $(A,B)$-harmonic equations -{\rm div} A(x,u,\nabla u) = B(x,u,\nabla u), $$ are quasiminimizers of the variational integral of the above type and, thus, are H"older continuous. Our results extend works by Chiad`o Piat-Coscia, Fan-Zhao and Giusti-Giaquinta.