  In this paper we generalise the results proved in [N. Katzourakis, An
$L^\infty$ regularisation strategy to the inverse source identification problem
for elliptic equations, SIAM J. Math. Anal. 51:2, 1349-1370 (2019)] by studying
the ill-posed problem of identifying the source of a fully nonlinear elliptic
equation. We assume Dirichlet data and some partial noisy information for the
solution on a compact set through a fully nonlinear observation operator. We
deal with the highly nonlinear nonconvex nature of the problem and the lack of
weak continuity by introducing a two-parameter Tykhonov regularisation with a
higher order $L^2$ "viscosity term" for the $L^\infty$ minimisation problem
which allows to approximate by weakly lower semicontinuous cost functionals.
