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 corresponding De Giorgi classes ?Br(1??)(x0)?(x,|?(u?k)±|)dx???Br(x0)?(x,(u?k)±?r)dx,where ?, r?(0,1), k?R and function ? satisfies non-logarithmic condition (r?n?Br(x0)[?(x,vr)]sdx)1s(r?n?Br(x0)[?(x,vr)]?tdx)1t?c(K)?(x0,r),r?v?K?(r), under some assumptions on ?(r) ?(x0,r) numbers s, t>1. These conditions generalize known logarithmic, non uniformly elliptic conditions. In particular, our results cover new cases double-phase, degenerate double-phase functionals with variable exponents.