Abstract
In this paper we prove the existence of nontrivial convex solutions to the Monge-Ampère equation det(D 2u) = ƒ(x, u, Du) with Dirichlet or Neumann boundary condition, where ƒ(x, 0, 0) = 0. We also deal with the existence and multiplicity of convex solutions to the equation det(D 2u) = λƒ(x, u, Du) . Our results show that the existence of convex solutions of Mongé-Ampère equations is analogous to that of positive solutions of semilinear elliptic equations.
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