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

TL;DR

This paper proves the existence and uniqueness of positive radial solutions for a nonlinear elliptic system in an annulus, using fixed point theorems and Rus's theorem.

Contribution

It introduces new conditions for existence and uniqueness of solutions to a nonlinear elliptic system in an annulus, extending previous results.

Findings

Existence of positive radial solutions established.

Uniqueness of solutions proved under certain conditions.

Application of fixed point theorems and Rus's theorem.

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.