Positive Radial Solutions for an Iterative System of Nonlinear Elliptic Equations in an Annulus

Abstract

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.