GLOBAL BIFURCATION PROBLEMS ASSOCIATED WITH k-HESSIAN OPERATORS

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. 0. Introduction Let Ω be a domain in R. If k ∈ {1, . . . , n} and u ∈ C(Ω), then the k-Hessian operator is defined by Sk(Du) = Sk(λ[Du]) = ∑ 1≤i1<...<ik≤n λi1 . . . λik , where λ[r] = (λ1, . . . , λn) denotes the eigenvalues of the symmetric matrix r and Sk is the k elementary symmetric polynomial in n variables. Notice that S1(Du) = ∆u and Sn(Du) = detDu. Thus, the k-Hessian operators form a discrete collection of partial differential operators, which includes the Laplace and Monge–Ampère operators. In this framework, it is natural to think of the 1991 Mathematics Subject Classification. Primary 35J65; Secondary 35P30.