New Results on the Radial Solutions to a Class of Nonlinear k -Hessian System

Abstract

This paper investigates the positive radial solutions of a nonlinear $k$ -Hessian system. $\left\{\begin{array}{c}\mathrm{\Lambda }\left(S_k^{1/k}\left(\lambda \left(D^2{z}_1\right)\right)\right)S_k^{1/k}\left(\lambda \left(D^2{z}_1\right)\right)=b\left(\left|x\right|\right)\phi \left(z_1,z_2\right),x\in {ℝ}^N\hfill \\ \mathrm{\Lambda }\left(S_k^{1/k}\left(\lambda \left(D^2{z}_2\right)\right)\right)S_k^{1/k}\left(\lambda \left(D^2{z}_2\right)\right)=h\left(\left|x\right|\right)\psi \left(z_1,z_2\right),x\in {ℝ}^N\hfill \end{array},\right)$ where $\mathrm{\Lambda }$ is a nonlinear operator and $b$ , $h$ , $\phi$ , $\psi$ are continuous functions. With the help of Keller–Osserman type conditions and monotone iterative technique, the positive radial solutions of the above problem are given in cases of finite, infinite, and semifinite. Our results complement the work in by Wang, Yang, Zhang, and Baleanu (Radial solutions of a nonlinear $k$ -Hessian system involving a nonlinear operator, Commun. Nonlinear Sci. Numer. Simul. 91(2020), 105396).