Weighted Eigenvalue Problem for a Class of Hessian Equations

TL;DR

This paper investigates the existence and uniqueness of solutions to a weighted eigenvalue problem for the k-Hessian equation, establishing key estimates and variational characterizations.

Contribution

It introduces uniform a priori estimates for solutions with weighted right-hand sides and characterizes eigenfunctions as minimizers of a related functional.

Findings

Established existence and uniqueness of solutions.

Derived uniform gradient and second derivative estimates.

Proved eigenfunctions minimize the associated functional.

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.