  We study an equation governed by a discontinuous fully nonlinear operator.
Such discontinuities are solution-dependent, which introduces a free boundary.
Working under natural assumptions, we prove the existence of $L^p$-viscosity
and strong solutions to the problem. The operator does not satisfy the usual
structure conditions and to obtain the existence of solutions we resort to
solving approximate problems, combined with an iterative scheme. We believe our
strategy can be applied to other classes of non-variational free boundary
problems.
