Interior continuity, continuity up to the boundary and Harnacks inequality for double-phase elliptic equations with non-logarithmic conditions

We prove continuity and Harnacks inequality for bounded solutions to elliptic equations of the type $$ begin{aligned} {rm div}big(| abla u|^{p-2}, abla u+a(x)| abla u|^{q-2}, abla ubig)=0,& quad a(x)geqslant0, |a(x)-a(y)|leqslant A|x-y|^{alpha}mu(|x-y|),& quad x eq y, {rm div}Big(| abla u|^{p-2}, abla u big[1+ln(1+b(x), | abla u|) big] Big)=0,& quad b(x)geqslant0, |b(x)-b(y)|leqslant B|x-y|,mu(|x-y|),& quad x eq y, end{aligned} $$ $$ begin{aligned} {rm div}Big(| abla u|^{p-2}, abla u+ c(x)| abla u|^{q-2}, abla u big[1+ln(1+| abla u|) big]^{beta} Big)=0,& quad c(x)geqslant0, , betageqslant0,phantom{=0=0} |c(x)-c(y)|leqslant C|x-y|^{q-p},mu(|x-y|),& quad x eq y, end{aligned} $$ under the precise choice of $mu$.