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We study the qualitative properties of functions belonging to thecorresponding 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 thenon-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}}\leqslantc(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 generalizethe known logarithmic, non-logarithmic and non uniformly elliptic conditions.In particular, our results cover new cases of non uniformly ellipticdouble-phase, degenerate double-phase functionals and functionals with variableexponents.

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