Bloch estimates in non-doubling generalized Orlicz spaces

We study minimizers of non-autonomous functionals \begin{align*} \inf_u \int_Ω\varphi(x,|\nabla u|) , dx \end{align*} when $\phi$ has generalized Orlicz growth. We consider the case where the upper growth rate of $\phi$ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on “truncating” the function $\phi$ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.