H\"older continuity of quasiminimizers with nonstandard growth

We show the H\"older continuity of quasiminimizers of the energy functionals $\int f(x,u,\nabla u)\,dx$ 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.