Existence and nonexistence of entire k-convex radial solutions to Hessian type system

Abstract

In this paper, a Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entire k-convex radial solutions is established by the monotone iterative method. Moreover, a nonexistence result is also obtained.

Existence of entire radial solutions to Hessian type system

“…By the way, on general k-Hessian equation (1.2) and general coupling k-Hessian system (1.4) when μ = ν = 0, Zhang and Zhou [20] obtained several results on the existence of entire positive k-convex radial solutions. In [3], we studied the coupling k-Hessian system (1.4) and obtained the existence and nonexistence of entire k-convex radial solutions. In the process of obtaining the existence of entire k-convex radial solutions, we utilized the monotone iterative method, and so we require the monotonicity of f and g. In the present paper, we want to remove the requirement for monotonicity and utilize a method different from the monotone iterative method.…”