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

Abstract

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).

A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

“…In addition, we also put forward some suitable growth conditions for nonlinearity for obtaining the results of nonexistence and existence of blow-up radial solutions for the k-Hessian equation. The conditions are easy to be checked, and the corresponding results improve and generalize some recent results such as those in [1,2,18,25,49].…”

“…Covei [2] considered the existence of positive radial solutions for the above Hessian equation, provided…”

Existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations

“…To establish the existence, nonexistence, and multiplicity of radial solutions to Dirichlet problems of k-Hessian equations with a nonlinear operator in a ball, we adopt the Leray-Schauder alternative theorem and the Leggett-Williams fixed point theorem as well as some suitable growth conditions for nonlinearity. Our work improves and generalizes some recent work such as [3,4,6,17,21,28].…”

“…In [20] and [12], the regularity for a more general class of fully nonlinear elliptic equations was obtained under nondivergence form. Recently, Covei [6] considered the existence of positive radial solutions for a Hessian equation with weights…”

“…Moreover, if B(x) = const = 0, then the k-Hessian equation becomes Hessian equation (3). Ji and Bao [17], Covei [6] considered the necessary and sufficient conditions for the existence of positive radial solutions. When B…”