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

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 \(\begin{aligned} S_{k} (\lambda (D_{i}(|Du|^{p-2}D_{j}u) + \alpha I)) =a^{k}(x)f^{k}(u),~x\in \mathbb {R}^n, \end{aligned}\) $$\begin{array}{c}\hfill S_k(\lambda (D_i{(|Du|}^{p-2}{D}_{j}u)+\alpha I))=a^k(x)f^k(u),x\in {\mathbb{R}}^n,\end{array}$$ and p-k-Hessian systems 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} S_{k}(\lambda (D_{i}(|Du|^{p-2}D_{j}u) + \alpha I)) =a^{k}(x)f^{k}(v),~x\in \mathbb {R}^n,\\ S_{k}(\lambda (D_{i}(|Dv|^{p-2}D_{j}v) + \alpha I)) = b^{k}(x)g^{k}(u),~x\in \mathbb {R}^n. \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{S}_k(\lambda (D_i{(|Du|}^{p-2}{D}_{j}u)+\alpha I))=a^k(x)f^k(v),x\in {\mathbb{R}}^n,\hfill \\ S_k(\lambda (D_i{(|Dv|}^{p-2}{D}_{j}v)+\alpha I))=b^k(x)g^k(u),x\in {\mathbb{R}}^n.\hfill \end{array}\right)\end{array}$$