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

We study the qualitative properties of functions belonging to the corresponding De Giorgi classes \begin{equation*} \int\limits_{B_{r(1-σ)}(x_{0})},\varPhi(x, |\nabla(u-k){\pm}|),dx \leqslant γ,\int\limits{B_{r}(x_{0})},\varPhi\bigg(x, \frac{(u-k){\pm}}{σr}\bigg),dx, \end{equation*} where $\sigma$, $r\in (0,1)$, $k\in \mathbb{R}$ and the function $\Phi$ 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) Λ(x_{0},r),\quad r\leqslant v\leqslant K,λ®, \end{equation*} under some assumptions on the functions $\lambda (r)$ and $\mathrm{\Lambda }(x_0,r)$ 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.