Fully degenerate Monge–Ampére equations

In this paper, we consider the following nonlinear eigenvalue problem for the Monge–Ampére equation: find a non-negative weakly convex classical solution f satisfying{detD2f=fpin Ω,f=φon ∂Ω for a strictly convex smooth domain Ω⊂R2Ω⊂R2 and 00}¯ and whose free-boundary ∂{f=0}∂{f=0} is also smooth.