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.
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