On the existence of radially symmetric solutions to p-k-Hessian equations and systems

Abstract

The main objective of this paper is to study the p-k-Hessian problems. To our knowledge, the problems that has additional term in the p-k-Hessian operator were seldom studied in the literature. By means of monotone iteration method and Arzelà-Ascoli theorem, this paper investigates the existence of positive radially symmetric solutions of the following augmented p-k-Hessian equations

$$\begin{aligned} S_{k} (\lambda (D_{i}(|Du|^{p-2}D_{j}u) + \alpha I)) =a^{k}(x)f^{k}(u),~x\in \mathbb {R}^n, \end{aligned}$$

and p-k-Hessian systems

$$\begin{aligned} {\left\{ \begin{array}{ll} S_{k}(\lambda (D_{i}(|Du|^{p-2}D_{j}u) + \alpha I)) =a^{k}(x)f^{k}(v),~x\in \mathbb {R}^n,\\ S_{k}(\lambda (D_{i}(|Dv|^{p-2}D_{j}v) + \alpha I)) = b^{k}(x)g^{k}(u),~x\in \mathbb {R}^n. \end{array}\right. } \end{aligned}$$

(0.1)