Singular boundary value problems for the Monge–Ampère equation

Given a strictly convex, smooth, and bounded domain ΩΩ in RnRn we consider solving the Monge–Ampére equation det(D2u)=f(x,−u)det(D2u)=f(x,−u) for solutions in C∞(Ω)∩C(Ω¯) with zero boundary value, where the nonlinearity f(x,t)f(x,t) could be singular at t=0t=0. We will show that under some fairly general assumptions on ff the above Dirichlet problem admits a negative convex solution in ΩΩ. Uniqueness of such solutions is then established for a wide class of nonlinearities f(x,t)f(x,t) as a consequence of a comparison principle.