  This paper deals with the existence of positive radial solutions to the
iterative system of nonlinear elliptic equations of the form $$ \begin{aligned}
  \triangle{\mathtt{u}_{{\dot{\iota}}
}}-\frac{(\mathtt{N}-2)^2r_0^{2\mathtt{N}-2}}{\vert
x\vert^{2\mathtt{N}-2}}\mathtt{u}_{\dot{\iota}} +\ell(\vert
x\vert)\mathtt{g}_{{\dot{\iota}} }(\mathtt{u}_{{\dot{\iota}}
+1})=0,~\mathtt{R}_1<\vert x\vert<\mathtt{R}_2, \end{aligned} $$ where
${\dot{\iota}} \in\{1,2,3,\cdot\cdot\cdot,\mathtt{n}\},$ $ \mathtt{u}_1=
\mathtt{u}_{\mathtt{n}+1},$ $\triangle{\mathtt{u}}=\mathtt{div}(\triangledown
\mathtt{u}),$ $\mathtt{N}>2,$ $\ell=\prod_{i=1}^{m}\ell_i,$ each
$\ell_i:(r_0,+\infty)\to(0,+\infty)$ is continuous, $r^{\mathtt{N}-1}\ell$ is
integrable, and $\mathtt{g}_{\dot{\iota}} :[0,+\infty)\to\mathbb{R}$ is
continuous, by an application of various fixed point theorems in a Banach
space. Further, we also establish uniqueness of solution to the addressed
system by using Rus's theorem in a complete metric space.
