WEIGHTED COMPLEX m-HESSIAN EQUATION

Abstract

Solving m-Hessian equations is an important problem in the theory of m-subharmonic functions. Recently, many authors have been interested in solving weighted m-Hessian equations  in the case when  is a non-decreasing function in the first variable and is a measure that puts no mass on all m-polar sets. In this article, we study the above-mentioned problem without the monotonicity assumption on the function  in the first variable. To achieve the above result, we apply the Schauder fixed point Theorem way by creating a suitable convex compact set and constructing a continuous map from the aforementioned convex compact set into itself. The techniques for solving weighted m-Hessian equations without the monotonicity assumption on the function  in the first variable are quite different from those used in the case with the monotonicity assumption on the function  in the first variable. We also solve the above-mentioned weighted m-Hessian equation in the case where the measure  is bounded by a suitable function of the m-capacity and provide an example of a measure  that satisfies this assumption.