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In 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 order  L  2  “viscosity term” for the    $L^{\infty }$  L ∞ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.

On the Inverse Source Identification Problem in $L^{\infty }$ for Fully Nonlinear Elliptic PDE