Bifurcation problems associated with Monge-Ampere operators

This dissertation is concerned with global bifurcation phenomena for Monge-Ampère and k-Hessian equations and related problems in the theory of fully nonlinear equations. In addition to being of independent interest, the bifurcation phenomena studied here provide new methods for establishing various existence results concerning fully nonlinear elliptic equations. The main equation of study is defined by SkD²u =fl,u, x∈W,u=0, x∈6W, where Ω is a strictly (k – 1)-convex domain in $Rn.$; An extension of the Krein-Rutman theorem is developed, which is then applied to the nonlinear k-Hessian operators to establish the existence of a unique positive eigenfunction (of prescribed norm) for said operators, whose corresponding (positive) eigenvalue is unique. Furthermore, it is shown that the eigenvalue is simple and satisfies a monotonicity of domain property.; Global bifurcation results are then developed in two model cases, defined by either a superlinear perturbation near the origin or a sublinear perturbation at infinity. Techniques from nonlinear analysis, global bifurcation theory, convexity, and nonlinear elliptic equations are combined to study the behavior of the solution continua.; Applications to the theory of critical exponents and the geometry of k-convex functions are considered. In addition, a problem of Liouville-Gelfand type is analyzed.