Generalized Monge-Ampère functionals and related variational problems

In this paper, we introduce a family of real Monge-Ampère functionals and study their variational properties. We prove a Sobolev type inequality for these functionals and use this to study the existence and uniqueness of some associated Dirichlet problems. In particular, we prove the existence of solutions for a nonlinear eigenvalue problem associated to this family of functionals.