Positive Radial Solutions for an Iterative System of Nonlinear Elliptic Equations in an Annulus (2109.09066v1)

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.