Continuity and Harnack inequalities for local minimizers of non uniformly elliptic functionals with generalized Orlicz growth under the non-logarithmic conditions

Abstract

We study the qualitative properties of functions belonging to the corresponding De Giorgi classes \begin{equation*} \int\limits_{B_{r(1-\sigma)}(x_{0})}\,\varPhi(x, |\nabla(u-k)_{\pm}|)\,dx \leqslant \gamma\,\int\limits_{B_{r}(x_{0})}\,\varPhi\bigg(x, \frac{(u-k)_{\pm}}{\sigma r}\bigg)\,dx, \end{equation*} where $\sigma$, $r \in (0,1)$, $k\in \mathbb{R}$ and the function $\varPhi$ satisfies the non-logarithmic condition \begin{equation*} \bigg(r^{-n}\int\limits_{B_{r}(x_{0})}[\varPhi\big(x,\frac{v}{r}\big)]^{s}\,dx\bigg)^{\frac{1}{s}}\bigg(r^{-n}\int\limits_{B_{r}(x_{0})}[\varPhi\big(x,\frac{v}{r}\big)]^{-t}\,dx\bigg)^{\frac{1}{t}}\leqslant c(K) \Lambda(x_{0},r),\quad r\leqslant v\leqslant K\,\lambda(r), \end{equation*} under some assumptions on the functions $\lambda(r)$ and $\Lambda(x_{0}, r)$ and the numbers $s$, $t >1$. These conditions generalize the known logarithmic, non-logarithmic and non uniformly elliptic conditions. In particular, our results cover new cases of non uniformly elliptic double-phase, degenerate double-phase functionals and functionals with variable exponents.