Weighted Eigenvalue Problem for a Class of Hessian Equations

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 $\mathrm{\mid }x{\mathrm{\mid }}^{2sk}$ 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.