  We prove the weak Harnack inequality for the functions $u$ which belong to
the corresponding De Giorgi classes $DG^{-}(\Omega)$ under the additional
assumption that $u\in L^{s}_{loc}(\Omega)$ with some $s> 0$. In particular, our
result covers new cases of functionals with a variable exponent or double-phase
functionals under the non-logarithmic condition.
