A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

TL;DR

This paper establishes necessary and sufficient conditions for the existence of positive radial solutions to weighted Hessian equations and systems, advancing the theoretical understanding of such solutions with a focus on large solutions.

Contribution

It provides a complete characterization of conditions for positive radial solutions, improving upon previous work by Zhang and Zhou.

Findings

Derived necessary conditions for large positive radial solutions.

Established sufficient conditions ensuring existence of solutions.

Enhanced theoretical framework for Hessian equations with weights.

Abstract

In this article we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation. Our results complete and improve a recently work published by Zhang and Zhou (Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights, Applied Mathematics Letters, Volume 50, December 2015, Pages 48--55).