Existence of positive solutions of symmetric variational eigenvalue problems

A symmetric variational eigenvalue problem in the Hilbert space with a cone is investigated. New sufficient conditions on the bilinear forms, the Hilbert space, and the cone of the variational problem guaranteeing the existence of a unique normalized positive eigenelement corresponding to a positive simple minimal eigenvalue are proposed and justified. The obtained abstract results are illustrated by the example of the generalized eigenvalue problem for the second order self-adjoint elliptic differential operator.