In this paper, we analyze the existence of nontrivial p- \(k_i\) -convex radial solutions for a coupled system of p- \(k_i\) -Hessian equations \(\begin{aligned} {\left\{ \begin{array}{ll} S_{k_1}(D(|Du_1|^{p-2}Du_1))=\lambda _1g_1(|x|)f_1(-u_2) \ \ \text {in} \ B,\\ S_{k_2}(D(|Du_2|^{p-2}Du_2))=\lambda _2g_2(|x|)f_2(-u_3) \ \ \text {in} \ B,\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \\ S_{k_{n-1}}(D(|Du_{n-1}|^{p-2}Du_{n-1}))=\lambda _{n-1}g_{n-1}(|x|)f_{n-1}(-u_n) \ \ \text {in} \ B,\\ S_{k_n}(D(|Du_n|^{p-2}Du_n))=\lambda _ng_n(|x|)f_n(-u_1) \ \ \text {in} \ B,\\ u_i=0,\ \text {on }\partial B,\ i\in I_n=\{1,2,\cdots ,n\}, \end{array}\right. } \end{aligned}\) where \(\lambda _i>0\) are parameters and B is the unit ball in \(\mathbb {R}^N\ (N\ge 2)\) . In addition, the asymptotic behaviors of nontrivial p- \(k_i\) -convex radial solutions on the parameter \(\lambda _i\) \((i\in I_n)\) are also studied using the eigenvalue theory. This is probably the first time that a system of p- \(k_i\) -Hessian equations has been studied by employing this technique. New nonexistence conclusions are also derived in this article. As an application, we present several new sufficient conditions for the existence and asymptotic behavior of nontrivial p- \(k_i\) -convex radial solutions for the power-type coupled system of p- \(k_i\) -Hessian equations.
