Radial Solutions for p-k-Hessian Equations and Systems with Gradient Term

Abstract

This paper studies the existence of entire radial solutions to the p-k-Hessian equation with nonlinear gradient term

$$\begin{aligned} \sigma _{k}\left. (\lambda \left( D_{i}\left( |D u|^{p-2} D_{j}(u)\right) \right) +\alpha |\nabla u|^{(p-1) k}\right. =a(|x|) f^{k}(u), ~~x \in \mathbb {R}^{n}, \end{aligned}$$

and system with nonlinear gradient term

$$\begin{aligned} \left\{ \begin{array}{l} \sigma _{k}\left. (\lambda \left( D_{i}\left( |D u|^{p-2} D_{j}(u)\right) \right) +\alpha |\nabla u|^{(p-1) k}\right. =a(|x|) f^{k}(v), ~~x \in \mathbb {R}^{n}, \\ \sigma _{k}\left. (\lambda \left( D_{i}\left( |D v|^{p-2} D_{j}(v)\right) \right) +\beta |\nabla v|^{(p-1) k}\right. =b(|x|) g^{k}(u), ~~x \in \mathbb {R}^{n}. \end{array}\right. \end{aligned}$$

By adopting monotone iteration method, we derive the existence and asymptotic behavior of the radial solutions.