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

This paper studies the existence of entire radial solutions to the p-k-Hessian equation with nonlinear gradient term \(\begin{aligned} \sigma _{k}\left. (\lambda \left( D_{i}\left( |D u|^{p-2} D_{j}(u)\right) \right) +\alpha |\nabla u|^{(p-1) k}\right. =a(|x|) f^{k}(u), ~~x \in \mathbb {R}^{n}, \end{aligned}\) $$\begin{array}{c}\hfill {\sigma }_k\left((,\lambda ,\left(D_i,\left({|Du|}^{p-2},D_j,(u)\right)\right),+,\alpha ,|,\mathrm{\nabla },u,{|}^{(p-1)k}\right)=a(|x|)f^k(u),x\in {\mathbb{R}}^n,\end{array}$$ and system with nonlinear gradient term \(\begin{aligned} \left\{ \begin{array}{l} \sigma _{k}\left. (\lambda \left( D_{i}\left( |D u|^{p-2} D_{j}(u)\right) \right) +\alpha |\nabla u|^{(p-1) k}\right. =a(|x|) f^{k}(v), ~~x \in \mathbb {R}^{n}, \\ \sigma _{k}\left. (\lambda \left( D_{i}\left( |D v|^{p-2} D_{j}(v)\right) \right) +\beta |\nabla v|^{(p-1) k}\right. =b(|x|) g^{k}(u), ~~x \in \mathbb {R}^{n}. \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\sigma }_k\left((,\lambda ,\left(D_i,\left({|Du|}^{p-2},D_j,(u)\right)\right),+,\alpha ,|,\mathrm{\nabla },u,{|}^{(p-1)k}\right)=a(|x|)f^k(v),x\in {\mathbb{R}}^n,\hfill \\ {\sigma }_k\left((,\lambda ,\left(D_i,\left({|Dv|}^{p-2},D_j,(v)\right)\right),+,\beta ,|,\mathrm{\nabla },v,{|}^{(p-1)k}\right)=b(|x|)g^k(u),x\in {\mathbb{R}}^n.\hfill \end{array}\right)\end{array}$$ By adopting monotone iteration method, we derive the existence and asymptotic behavior of the radial solutions.