  We consider a one-phase free boundary problem governed by doubly degenerate
fully non-linear elliptic PDEs with non-zero right hand side, which should be
understood as an analog (non-variational) of certain double phase functionals
in the theory of non-autonomous integrals. By way of brief elucidating example,
such non-linear problems in force appear in the mathematical theory of
combustion, as well as in the study of some flame propagation problems. In such
an environment we prove that solutions are Lipschitz continuous and they fulfil
a non-degeneracy property. Furthermore, we address the Caffarelli's
classification scheme: Flat and Lipschitz free boundaries are locally $C^{1,
\beta}$ for some $0 < \beta (universal) < 1$.
