Weighted Eigenvalue Problem for a Class of Singular/Degenerate $k$-Hessian Equations

Abstract

In this paper, we study the existence and uniqueness of solutions to the weighted eigenvalue problem for $k$-Hessian equation. To achieve this, we establish the uniform a priori estimates for gradient and second derivatives of solutions to Hessian equation with weight $|x{|}^{2sk}$ on the right-hand-side. We also prove that the eigenfunction is a minimizer of the corresponding functional among all $k$-admissible functions vanishing on the boundary.

31 pages, to appear in Indiana Univ. Math. J