  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.
