AbstractIn this paper we generalise the results proved in N. Katzourakis (SIAM J. Math. Anal.51, 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 orderL2“viscosity term” for the$L^{\infty }$L∞minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.