Regularity for double phase problems at nearly linear growth

Minima of functionals of the type $$w\mapsto {\int }_{\mathrm{\Omega }}[\text{snr}Dw\log (1+\text{snr}Dw)+a(x)\text{snr}{Dw}^q]\text{dx},0\le a(\cdot )\in C^{0,\alpha },$$ with $\mathrm{\Omega }\subset {\text{er}}^n$, have locally Hölder continuous gradient provided $1<q<1+\alpha \mathrm{/}n$.