Global bifurcation problems associated with $k$-Hessian operators

Abstract

In this paper we study global bifurcation phenomena for a class of nonlinear elliptic equations governed by the $h$-Hessian operator. The bifurcation phenomena considered provide new methods for establishing existence results concerning fully nonlinear elliptic equations. Applications to the theory of critical exponents and the geometry of $k$-convex functions are considered. In addition, a related problem of Liouville-Gelfand type is analyzed.