In this paper, we study the ellipticity of k-Hessian operators with respect to (k − 1)-convex admissible solutions. Moreover, we research the strictly convex solutions to degenerate k-Hessian equations. We first analyze the strictly convex solutions to degenerate k-Hessian equations. By the virtue of this analysis of the ellipticity under this condition, we give a classification of k-Hessian operators. According to this classification, we construct strictly convex local solutions to degenerate k-Hessian equations.
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