∎

11institutetext:  Department of Applied Mathematics
College of Science and Technology
Andhra University, Visakhapatnam, 530003, India
⋆Corresponding author<EMAIL_ADDRESS>
# Positive Radial Solutions for an Iterative System of Nonlinear Elliptic
Equations in an Annulus

Mahammad Khuddush⋆ K. Rajendra Prasad

(Received: date / Accepted: date)

###### Abstract

This paper deals with the existence of positive radial solutions to the
iterative system of nonlinear elliptic equations of the form

$\displaystyle\triangle{\mathtt{u}_{{\dot{\iota}}}}-\frac{(\mathtt{N}-2)^{2}r_{0}^{2\mathtt{N}-2}}{|x|^{2\mathtt{N}-2}}\mathtt{u}_{\dot{\iota}}+\ell(|x|)\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1})=0,~{}\mathtt{R}_{1}<|x|<\mathtt{R}_{2},$

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.

###### Keywords:

Nonlinear elliptic equation annulus positive radial solutionfixed point
theoremBanach spaceRus’s theoremmetric space

###### MSC:

35J6635J6034B1847H10

##  1 Introduction

The semilinear elliptic equation of the form

$\triangle\mathtt{u}+\mathtt{g}(|x|)\mathtt{u}+\mathtt{h}(|x|)\mathtt{u}^{\mathtt{p}}=0$
(1)

arise in various fields of pure and applied mathematics such as Riemannian
geometry, nuclear physics, astrophysics and so on. For more details of the
background of (1), see ni ; yana . Study of nonlinear elliptic system of
equations,

$\left.\begin{aligned}
&\triangle{\mathtt{u}_{{\dot{\iota}}}}+\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1})=0~{}~{}\text{in}~{}~{}\Omega,\\\
&\hskip
28.45274pt\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}\partial\Omega,\\\
\end{aligned}\right\\}$ (2)

where ${\dot{\iota}}\in\\{1,2,3,\cdot\cdot\cdot,{\mathtt{n}}\\},$
$\mathtt{u}_{1}=\mathtt{u}_{\mathtt{n}+1},$ and $\Omega$ is a bounded domain
in $\mathbb{R}^{N}$, has an important applications in population dynamics,
combustion theory and chemical reactor theory. The recent literature for the
existence, multiplicity and uniqueness of positive solutions for (2), see dal
; hai1 ; ali1 ; hai2 ; hai3 ; ali2  and references therein.

In mei , Mei studied the structure of radial solutions to the quasilinear
elliptic problem of the form

$\displaystyle\triangle_{\mathtt{p}}\mathtt{u}-\frac{\uplambda}{\mathtt{u}^{\mathtt{q}}}=0~{}\text{in}~{}\Omega,$
$\displaystyle\hskip 11.38092pt0<\mathtt{u}<1~{}\text{in}~{}\Omega,$
$\displaystyle\hskip 17.07182pt\mathtt{u}=1~{}\text{on}~{}\partial\Omega,$

where $1<\mathtt{p}\leq\mathtt{N}\,(\mathtt{N}\geq
2),\mathtt{q}>0,\uplambda>0$ and
$\Omega=\\{x\in\mathbb{R}^{\mathtt{N}}:|x|<1\\}.$ In li1 , Li studied the
existence of positive radial solutions of the elliptic equation with nonlinear
gradient term

$\displaystyle-\triangle\mathtt{u}=\mathtt{g}\big{(}|x|,\mathtt{u},\frac{x}{|x|}\cdot\nabla\mathtt{u}\big{)}=0~{}~{}\text{in}~{}~{}\Omega_{a}^{b},$
$\displaystyle\hskip
45.52458pt\mathtt{u}=0~{}~{}\text{on}~{}~{}\partial\Omega_{a}^{b},$

by using fixed point index theory in cones. In chro , Chrouda and Hassine
established the uniqueness of positive radial solutions to the following
Dirichlet boundary value problem for the semilinear elliptic equation in an
annulus,

$\displaystyle\triangle\mathtt{u}=\mathtt{g}(\mathtt{u})$
$\displaystyle~{}~{}\text{on}~{}~{}\Omega=\\{x\in\mathbb{R}^{\mathtt{d}}:a<|x|<b\\},$
$\displaystyle\mathtt{u}=0~{}~{}\text{on}~{}~{}\mathtt{u}\in\partial\Omega,$

for any dimension $\mathtt{d}\geq 1.$ In kaji , R. Kajikiya and E. Ko
established the existence of positive radial solutions for a semipositone
elliptic equation of the form,

$\displaystyle\triangle\mathtt{u}+\uplambda\mathtt{g}(\mathtt{u})=0~{}~{}\text{in}~{}~{}\Omega,$
$\displaystyle\hskip
19.91684pt\mathtt{u}=0~{}~{}\text{on}~{}~{}\partial\Omega,$

where $\Omega$ is a ball or an annulus in $\mathbb{R}^{\mathtt{N}}.$ Recently,
Son and Wang son  established positive radial solutions to the nonlinear
elliptic systems,

$\displaystyle\triangle{\mathtt{u}_{{\dot{\iota}}}}+\uplambda\mathtt{K}_{\dot{\iota}}(|x|)\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1})=0~{}\text{in}~{}\Omega_{\mathtt{E}},$
$\displaystyle\hskip
35.56593pt\mathtt{u}_{{\dot{\iota}}}=0~{}\text{on}~{}|x|=r_{0},$
$\displaystyle\hskip 29.87547pt\mathtt{u}_{{\dot{\iota}}}\to
0~{}\text{as}~{}|x|\to+\infty,$

where ${\dot{\iota}}\in\\{1,2,3,\cdot\cdot\cdot,{\mathtt{n}}\\},$
$\mathtt{u}_{1}=\mathtt{u}_{\mathtt{n}+1},$ $\uplambda>0,$ $N>2,$ $r_{0}>0,$
and $\Omega_{\mathtt{E}}$ is an exterior of a ball. Motivated by the above
works, in this paper we study the existence of infinitely many positive radial
solutions for the following iterative system of nonlinear elliptic equations
in an annulus,

$\triangle{\mathtt{u}_{{\dot{\iota}}}}-\frac{(\mathtt{N}-2)^{2}r_{0}^{2\mathtt{N}-2}}{|x|^{2\mathtt{N}-2}}\mathtt{u}_{\dot{\iota}}+\ell(|x|)\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1})=0,~{}\mathtt{R}_{1}<|x|<\mathtt{R}_{2},$
(3)

with one of the following sets of boundary conditions:

$\left.\begin{aligned} &\hskip
25.6073pt\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=\mathtt{R}_{1}~{}\text{and}~{}|x|=\mathtt{R}_{2},\\\
&\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=\mathtt{R}_{1}~{}\text{and}~{}\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}\text{on}~{}|x|=\mathtt{R}_{2},\\\
&\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}~{}\text{on}~{}~{}|x|=\mathtt{R}_{1}~{}\text{and}~{}\mathtt{u}_{{\dot{\iota}}}=0~{}\text{on}~{}|x|=\mathtt{R}_{2},\end{aligned}\right\\}$
(4)

where ${\dot{\iota}}\in\\{1,2,3,\cdot\cdot\cdot,\mathtt{n}\\},$
$\mathtt{u}_{1}=\mathtt{u}_{\mathtt{n}+1},$
$\triangle\mathtt{u}=\mathtt{div}(\triangledown\mathtt{u}),$ $N>2,$
$\ell=\prod_{i=1}^{m}\ell_{i},$ each
$\ell_{i}:(\mathtt{R}_{1},\mathtt{R}_{2})\to(0,+\infty)$ is continuous,
$r^{\mathtt{N}-1}\ell$ is integrable, by an application of various fixed point
theorems in a Banach space. Further, we also study existence of unique
solution by using Rus’s theorem in a complete metric space.

The study of positive radial solutions to (3) reduces to the study of positive
solutions to the following iterative system of two-point boundary value
problems,

$\displaystyle\mathtt{u}^{\prime\prime}_{{\dot{\iota}}}(\mathtt{s})-r_{0}^{2}\mathtt{u}_{\dot{\iota}}(\mathtt{s})+\ell(\mathtt{s})\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1}(\mathtt{s}))=0,~{}0<\mathtt{s}<1,$
(5)

where ${\dot{\iota}}\in\\{1,2,3,\cdot\cdot\cdot,\mathtt{n}\\},$
$\mathtt{u}_{1}=\mathtt{u}_{\mathtt{n}+1},$ $r_{0}>0$ and
$\ell(\mathtt{s})=\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\mathtt{s}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}\prod_{i=1}^{m}\ell_{i}(\mathtt{s}),$
$\ell_{i}(\mathtt{s})=\ell_{i}(r_{0}\mathtt{s}^{\frac{1}{2-\mathtt{N}}})$ by a
Kelvin type transformation through the change of variables $r=|x|$ and
$\mathtt{s}=\left(\frac{r}{r_{0}}\right)^{2-\mathtt{N}}.$ The detailed
explanation of the transformation from the equation (7) to (5) see ali3 ; lan
; lee . By suitable choices of nonnegative real numbers
$\upalpha,\upbeta,\upgamma$ and $\updelta,$ the set of boundary conditions (5)
reduces to

$\left\\{\begin{aligned}
&\upalpha\mathtt{u}_{\dot{\iota}}(0)-\upbeta\mathtt{u}_{\dot{\iota}}^{\prime}(0)=0,\\\
&\,\upgamma\mathtt{u}_{\dot{\iota}}(1)+\updelta\mathtt{u}_{\dot{\iota}}^{\prime}(1)=0,\end{aligned}\right.$
(6)

we assume that the following conditions hold throughout the paper:

  * $(\mathcal{J}_{1})$

$\mathtt{g}_{\dot{\iota}}:[0,+\infty)\to[0,+\infty)$ is continuous.

  * $(\mathcal{J}_{2})$

$\ell_{i}\in L^{\mathtt{p}_{i}}[0,1],1\leq\mathtt{p}_{i}\leq+\infty$ for
$1\leq i\leq n.$

  * $(\mathcal{J}_{3})$

There exists $\ell_{i}^{\star}>0$ such that
$\ell_{i}^{\star}<\ell_{i}(\mathtt{s})<\infty$ a.e. on $[0,1].$

The rest of the paper is organized in the following fashion. In Section 2, we
convert the boundary value problem (5)–(6) into equivalent integral equation
which involves the kernel. Also, we estimate bounds for the kernel which are
useful in our main results. In Section 3, we develop a criteria for the
existence of atleast one positive radial solution by applying Krasnoselskii’s
cone fixed point theorem in a Banach space. In Section 4, we derive necessary
conditions for the existence of atleast two positive radial solution by an
application of Avery-Henderson cone fixed point theorem in a Banach space. In
Section 5, we establish the existence of atleast three positive radial
solution by utilizing Legget-William cone fixed point theorem in a Banach
space. Further, we also study uniqueness of solution in the final section.

##  2 Kernel and Its Bounds

In order to study BVP (5), we first consider the corresponding linear boundary
value problem,

$-\mathtt{u}_{1}^{\prime\prime}(\mathtt{s})+r_{0}^{2}\mathtt{u}_{1}(\mathtt{s})=u(\mathtt{s}),~{}0<\mathtt{s}<1,$
(7) $\left\\{\begin{aligned}
&\upalpha\mathtt{u}_{1}(0)-\upbeta\mathtt{u}_{1}^{\prime}(0)=0,\\\
&\,\upgamma\mathtt{u}_{1}(1)+\updelta\mathtt{u}_{1}^{\prime}(1)=0,\end{aligned}\right.$
(8)

where $u\in\mathcal{C}[0,1]$ is a given function.

###### Lemma 1

Let
$\varrho=r_{0}^{2}(\upalpha\updelta+\upbeta\upgamma)\cosh(r_{0})+r_{0}(\upalpha\upgamma+\upbeta\updelta
r_{0}^{2})\sinh(r_{0}).$ For every $u\in\mathcal{C}[0,1],$ the linear boundary
value problem (7)–(8) has a unique solution

$\mathtt{u}_{1}(\mathtt{s})=\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t})u(\mathtt{t})\mathtt{d}\mathtt{t},$
(9)

where

$\displaystyle\Xi_{r_{0}}(\mathtt{s},\mathtt{t})=\frac{1}{\varrho}\left\\{\begin{array}[]{ll}\big{(}\upalpha\sinh(r_{0}\mathtt{s})+\upbeta
r_{0}\cosh(r_{0}\mathtt{s})\big{)}\big{(}\upgamma\sinh(r_{0}(1-\mathtt{t}))+\updelta
r_{0}\cosh(r_{0}(1-\mathtt{t}))\big{)},\hskip
11.38092pt0\leq\mathtt{s}\leq\mathtt{t}\leq 1,\vspace{1.2mm}\\\
\big{(}\upalpha\sinh(r_{0}\mathtt{t})+\upbeta
r_{0}\cosh(r_{0}\mathtt{t})\big{)}\big{(}\upgamma\sinh(r_{0}(1-\mathtt{s}))+\updelta
r_{0}\cosh(r_{0}(1-\mathtt{s}))\big{)},\hskip
11.38092pt0\leq\mathtt{t}\leq\mathtt{s}\leq 1.\end{array}\right.$

###### Lemma 2

Let $\displaystyle\wp=\max\left\\{\frac{\upbeta
r_{0}}{\upalpha\sinh(r_{0})+\upbeta r_{0}\cosh(r_{0})},\frac{\updelta
r_{0}}{\upgamma\sinh(r_{0})+\updelta r_{0}\cosh(r_{0})}\right\\}.$ The kernel
$\Xi_{r_{0}}(\mathtt{s},\mathtt{t})$ has the following properties:

  * (i)

$\Xi_{r_{0}}(\mathtt{s},\mathtt{t})$ is nonnegative and continuous on
$[0,1]\times[0,1],$

  * (ii)

$\Xi_{r_{0}}(\mathtt{s},\mathtt{t})\leq\Xi_{r_{0}}(\mathtt{t},\mathtt{t})$ for
$\mathtt{s},\mathtt{t}\in[0,1],$

  * (iii)

$\wp\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\leq\Xi_{r_{0}}(\mathtt{s},\mathtt{t})$
for $\mathtt{s},\mathtt{t}\in[0,1].$

###### Proof

From the definition of kernel $\Xi_{r_{0}}(\mathtt{s},\mathtt{t}),$ it is
clear that $(i)$ holds. To prove $(ii),$ consider

$\displaystyle\frac{\Xi_{r_{0}}(\mathtt{s},\mathtt{t})}{\Xi_{r_{0}}(\mathtt{t},\mathtt{t})}=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\frac{\upalpha\sinh(r_{0}\mathtt{s})+\upbeta
r_{0}\cosh(r_{0}\mathtt{s})}{\upalpha\sinh(r_{0}\mathtt{t})+\upbeta
r_{0}\cosh(r_{0}\mathtt{t})},\hskip
58.18582pt0\leq\mathtt{s}\leq\mathtt{t}\leq 1,\vspace{1.2mm}\\\
\displaystyle\frac{\upgamma\sinh(r_{0}(1-\mathtt{s}))+\updelta
r_{0}\cosh(r_{0}(1-\mathtt{s}))}{\upgamma\sinh(r_{0}(1-\mathtt{t}))+\updelta
r_{0}\cosh(r_{0}(1-\mathtt{t}))},\hskip
11.38092pt0\leq\mathtt{t}\leq\mathtt{s}\leq 1,\end{array}\right.$
$\displaystyle\leq$ $\displaystyle\left\\{\begin{array}[]{ll}\displaystyle
1,\hskip 172.13925pt0\leq\mathtt{s}\leq\mathtt{t}\leq 1,\vspace{1.2mm}\\\
1,\hskip 172.13925pt0\leq\mathtt{t}\leq\mathtt{s}\leq 1,\end{array}\right.$

which proves $(ii).$ Finally for $(iii),$ consider

$\displaystyle\frac{\Xi_{r_{0}}(\mathtt{s},\mathtt{t})}{\Xi_{r_{0}}(\mathtt{t},\mathtt{t})}=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\frac{\upalpha\sinh(r_{0}\mathtt{s})+\upbeta
r_{0}\cosh(r_{0}\mathtt{s})}{\upalpha\sinh(r_{0}\mathtt{t})+\upbeta
r_{0}\cosh(r_{0}\mathtt{t})},\hskip
69.70915pt0\leq\mathtt{s}\leq\mathtt{t}\leq 1,\vspace{1.2mm}\\\
\displaystyle\frac{\upgamma\sinh(r_{0}(1-\mathtt{s}))+\updelta
r_{0}\cosh(r_{0}(1-\mathtt{s}))}{\upgamma\sinh(r_{0}(1-\mathtt{t}))+\updelta
r_{0}\cosh(r_{0}(1-\mathtt{t}))},\hskip
22.76228pt0\leq\mathtt{t}\leq\mathtt{s}\leq 1,\end{array}\right.$
$\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\frac{\upbeta
r_{0}}{\upalpha\sinh(r_{0})+\upbeta r_{0}\cosh(r_{0})},\hskip
80.23697pt0\leq\mathtt{s}\leq\mathtt{t}\leq 1,\vspace{1.2mm}\\\
\displaystyle\frac{\updelta r_{0}}{\upgamma\sinh(r_{0})+\updelta
r_{0}\cosh(r_{0})},\hskip 81.65926pt0\leq\mathtt{t}\leq\mathtt{s}\leq
1.\end{array}\right.$

This completes the proof. ∎

From Lemma 1, we note that an $\mathtt{n}$-tuple
$(\mathtt{u}_{1},\mathtt{u}_{2},\cdot\cdot\cdot,\mathtt{u}_{\mathtt{n}})$ is a
solution of the boundary value problem (5)–(6) if and only, if

$\displaystyle\mathtt{u}_{1}(\mathtt{s})=$
$\displaystyle\,\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}.$

In general,

$\displaystyle\mathtt{u}_{\dot{\iota}}(\mathtt{s})=$
$\displaystyle\,\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},s)\ell(s)\mathtt{g}_{\dot{\iota}}\big{(}\mathtt{u}_{{\dot{\iota}}+1}(s)\big{)}ds,~{}{\dot{\iota}}=1,2,3,\cdot\cdot\cdot,{\mathtt{n}},$
$\displaystyle\mathtt{u}_{1}(\mathtt{s})=$
$\displaystyle\,\mathtt{u}_{\mathtt{n}+1}(\mathtt{s}).$

We denote the Banach space $\mathtt{C}((0,1),\mathbb{R})$ by $\mathtt{B}$ with
the norm
$\|\mathtt{u}\|=\displaystyle\max_{\mathtt{s}\in[0,1]}|\mathtt{u}(\mathtt{s})|.$
The cone $\mathtt{E}\subset\mathtt{B}$ is defined by

$\mathtt{E}=\Big{\\{}\mathtt{u}\in\mathtt{B}:\mathtt{u}(\mathtt{s})\geq
0~{}\text{on}~{}[0,1]~{}\text{and}~{}\min_{\mathtt{s}\in{[0,\,1]}}\mathtt{u}(\mathtt{s})\geq\wp\|\mathtt{u}\|\Big{\\}}.$

For any $\mathtt{u}_{1}\in\mathtt{E},$ define an operator
$\aleph:\mathtt{E}\rightarrow\mathtt{B}$ by

$\displaystyle(\aleph\mathtt{u}_{1})(\mathtt{s})=$
$\displaystyle\,\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}.$
(10)

###### Lemma 3

$\aleph(\mathtt{E})\subset\mathtt{E}$ and
$\aleph:\mathtt{E}\rightarrow\mathtt{E}$ is completely continuous.

###### Proof

Since $\mathtt{g}_{\dot{\iota}}(\mathtt{u}_{{\dot{\iota}}+1}(\mathtt{s}))$ is
nonnegative for $\mathtt{s}\in[0,1],$ $\mathtt{u}_{1}\in\mathtt{E}.$ Since
$\Xi_{r_{0}}(\mathtt{s},s),$ is nonnegative for all
$\mathtt{s},\mathtt{t}\in[0,1],$ it follows that
$\aleph(\mathtt{u}_{1}(\mathtt{s}))\geq 0$ for all
$\mathtt{s}\in[0,1],\,\mathtt{u}_{1}\in\mathtt{E}.$ Now, by Lemma 1 and 2, we
have

$\displaystyle\min_{\mathtt{s}\in[0,1]}(\aleph\mathtt{u}_{1})(\mathtt{s})$
$\displaystyle=\,\min_{\mathtt{s}\in[0,1]}\Bigg{\\{}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
39.83368pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}\Bigg{\\}}$
$\displaystyle\geq\wp\Bigg{\\{}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
39.83368pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}\Bigg{\\}}$
$\displaystyle\geq\wp\Bigg{\\{}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
39.83368pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}\Bigg{\\}}$
$\displaystyle\geq\wp\,\max_{\mathtt{s}\in[0,1]}\,|\aleph\mathtt{u}_{1}(\mathtt{s})|.$

Thus $\aleph(\mathtt{E})\subset\mathtt{E}.$ Therefore, by the means of Arzela-
Ascoli theorem, the operator $\aleph$ is completely continuous. ∎

##  3 Existence of at Least One Positive Radial Solution

In this section, we establish the existence of at least one positive radial
solution for the system (5)–(6) by an application of following theorems.

###### Theorem 3.1

guo  Let $\mathtt{E}$ be a cone in a Banach space $\mathtt{B}$ and let
$\mathtt{G},\,\mathtt{F}$ be open sets with
$0\in\mathtt{G},\overline{\mathtt{G}}\subset\mathtt{F}.$ Let
$\aleph:\mathtt{E}\cap(\overline{\mathtt{F}}\backslash\mathtt{G})\rightarrow\mathtt{E}$
be a completely continuous operator such that

  * (i)

$\|\aleph\mathtt{u}\|\geq\|\mathtt{u}\|,\,\mathtt{u}\in\mathtt{E}\cap\partial\mathtt{G},$
and
$\|\aleph\mathtt{u}\|\leq\|\mathtt{u}\|,\,\mathtt{u}\in\mathtt{E}\cap\partial\mathtt{F},$
or

  * (ii)

$\|\aleph\mathtt{u}\|\leq\|\mathtt{u}\|,\,\mathtt{u}\in\mathtt{E}\cap\partial\mathtt{G},$
and
$\|\aleph\mathtt{u}\|\geq\|\mathtt{u}\|,\,\mathtt{u}\in\mathtt{E}\cap\partial\mathtt{F}.$

Then $\aleph$ has a fixed point in
$\mathtt{E}\cap(\overline{\mathtt{F}}\backslash\mathtt{E}).$

###### Theorem 3.2

(Hölder’s) Let $\mathtt{f}\in L^{\mathtt{p}_{i}}[0,1]$ with
$\mathtt{p}_{i}>1,$ for $i=1,2,\cdots,n$ and
$\displaystyle\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}=1.$ Then
$\prod_{i=1}^{m}\mathtt{f}_{i}\in L^{1}[0,1]$ and
$\left\|\prod_{i=1}^{m}\mathtt{f}_{i}\right\|_{1}\leq\prod_{i=1}^{m}\|\mathtt{f}_{i}\|_{\mathtt{p}_{i}}.$
Further, if $\mathtt{f}\in L^{1}[0,1]$ and $\mathtt{g}\in L^{\infty}[0,1].$
Then $\mathtt{fg}\in L^{1}[0,1]$ and
$\|\mathtt{fg}\|_{1}\leq\|\mathtt{f}\|_{1}\|\mathtt{g}\|_{\infty}.$

Consider the following three possible cases for $\ell_{i}\in
L^{\mathtt{p}_{i}}[0,1]:$

$\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}<1,~{}\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}=1,~{}\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}>1.$

Firstly, we seek positive radial solutions for the case
$\displaystyle\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}<1.$

###### Theorem 3.3

Suppose $(\mathcal{J}_{1})$–$(\mathcal{J}_{3})$ hold. Further, assume that
there exist two positive constants $a_{2}>a_{1}>0$ such that

  * $(\mathcal{J}_{4})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u}(\mathtt{s}))\leq\mathfrak{Q}_{2}a_{2}$
for all $0\leq\mathtt{s}\leq 1,\,0\leq\mathtt{u}\leq a_{2},$ where
$\displaystyle\mathfrak{Q}_{2}=\left[\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}\right]^{-1}$
and
$\widehat{\Xi}_{r_{0}}(\mathtt{t})=\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}.$

  * $(\mathcal{J}_{5})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u}(\mathtt{s}))\geq\mathfrak{Q}_{1}a_{1}$
for all $0\leq\mathtt{s}\leq 1,\,0\leq\mathtt{u}\leq a_{1},$ where
$\displaystyle\mathfrak{Q}_{1}=\left[\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}\mathtt{d}\mathtt{t}\right]^{-1}.$

Then iterative system (5)–(6) has atleast one positive radial solution
$(\mathtt{u}_{1},\mathtt{u}_{2},\cdot\cdot\cdot,\mathtt{u}_{\mathtt{n}})$ such
that $a_{1}\leq\|\mathtt{u}_{\dot{\iota}}\|\leq
a_{2},\,{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$

###### Proof

Let $\mathtt{G}=\\{\mathtt{u}\in\mathtt{B}:\|\mathtt{u}\|<a_{2}\\}.$ For
$\mathtt{u}_{1}\in\partial\mathtt{G},$ we have $0\leq\mathtt{u}\leq a_{2}$ for
all $\mathtt{s}\in[0,1].$ It follows from $(\mathcal{J}_{4})$ that for
$\mathtt{t}_{\mathtt{n}-1}\in[0,1],$

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\mathfrak{Q}_{2}a_{2}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\mathfrak{Q}_{2}a_{2}\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\prod_{i=1}^{m}\ell_{i}(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}.$

There exists a $\mathtt{q}>1$ such that
$\displaystyle\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}+\frac{1}{\mathtt{q}}=1.$
By the first part of Theorem 3.2, we have

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\mathfrak{Q}_{2}a_{2}\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}$
$\displaystyle\leq a_{2}.$

It follows in similar manner for $0<\mathtt{t}_{\mathtt{n}-2}<1,$

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-2},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{g}_{\mathtt{n}-1}$
$\displaystyle\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\leq\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{g}_{\mathtt{n}-1}(a_{2})\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\leq\mathfrak{Q}_{2}a_{2}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\leq\mathfrak{Q}_{2}a_{2}\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}$
$\displaystyle\leq a_{2}.$

Continuing with this bootstrapping argument, we reach

$\displaystyle(\aleph\mathtt{u}_{1})(t)=$
$\displaystyle\,\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
28.45274pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}$
$\displaystyle\leq$ $\displaystyle\,a_{2}.$

Since $\mathtt{G}=\|\mathtt{u}_{1}\|$ for
$\mathtt{u}_{1}\in\mathtt{E}\cap\partial{\mathtt{G}},$ we get

$\|\aleph\mathtt{u}_{1}\|\leq\|\mathtt{u}_{1}\|.$ (11)

Next, let $\mathtt{F}=\\{\mathtt{u}\in\mathtt{B}:\|\mathtt{u}\|<a_{1}\\}.$ For
$\mathtt{u}_{1}\in\partial\mathtt{F},$ we have $0\leq\mathtt{u}\leq a_{1}$ for
all $\mathtt{s}\in[0,1].$ It follows from $(\mathcal{J}_{5})$ that for
$\mathtt{t}_{\mathtt{n}-1}\in[0,1],$

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\wp\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\wp\mathfrak{Q}_{1}a_{1}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\mathfrak{Q}_{1}a_{1}\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\prod_{i=1}^{m}\ell_{i}(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\mathfrak{Q}_{1}a_{1}\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq a_{1}.$

It follows in similar manner for $0<\mathtt{t}_{\mathtt{n}-2}<1,$

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-2},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{g}_{\mathtt{n}-1}$
$\displaystyle\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq\wp\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{g}_{\mathtt{n}-1}(a_{1})\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq\wp\mathfrak{Q}_{1}a_{1}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq\mathfrak{Q}_{1}a_{1}\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}-1}\prod_{i=1}^{m}\ell_{i}(\mathtt{t}_{\mathtt{n}-1})\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq\mathfrak{Q}_{1}a_{1}\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}-1}\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq a_{1}.$

Continuing with bootstrapping argument, we get

$\displaystyle(\aleph\mathtt{u}_{1})(\mathtt{s})=$
$\displaystyle\,\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
17.07182pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}$
$\displaystyle\geq$ $\displaystyle\,a_{1}.$

Thus, for $\mathtt{u}_{1}\in\mathtt{E}\cap\partial\mathtt{F},$ we have

$\|\aleph\mathtt{u}_{1}\|\geq\|\mathtt{u}_{1}\|.$ (12)

It is clear that
$0\in\mathtt{F}\subset\overline{\mathtt{F}}\subset\mathtt{G}.$ It follows from
(11), (12) and Theorem 3.1 that the operator $\aleph$ has a fixed point
$\mathtt{u}_{1}\in\mathtt{E}\cap\big{(}\overline{\mathtt{G}}\backslash\mathtt{F}\big{)}$
such that $\mathtt{u}_{1}(\mathtt{s})\geq 0$ on $(0,1).$ Next setting
$\mathtt{u}_{\mathtt{n}+1}=\mathtt{u}_{1},$ we obtain infinitely many positive
solutions
$(\mathtt{u}_{1},\mathtt{u}_{2},\cdot\cdot\cdot,\mathtt{u}_{\mathtt{n}})$ of
(5)–(6) given iteratively by

$\displaystyle\mathtt{u}_{\dot{\iota}}(\mathtt{s})$
$\displaystyle=\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},s)\ell(s)\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1}(s))ds,\,{\dot{\iota}}=1,2,\cdot\cdot\cdot,{\mathtt{n}}-1,\mathtt{n},$
$\displaystyle\mathtt{u}_{\mathtt{n}+1}(\mathtt{s})$
$\displaystyle=\mathtt{u}_{1}(\mathtt{s}),\,\mathtt{s}\in(0,1).$

This completes the proof.∎

For $\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}=1$ and
$\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}>1,$ we have following results.

###### Theorem 3.4

Suppose $(\mathcal{J}_{1})$–$(\mathcal{J}_{3})$ hold. Further, assume that
there exist two positive constants $b_{2}>b_{1}>0$ such that
$\mathtt{g}_{\dot{\iota}}\,({\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n})$
satisfies $(\mathcal{J}_{5})$ and

  * $(\mathcal{J}_{6})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u}(\mathtt{s}))\leq\mathfrak{N}_{2}b_{2}$
for all $0\leq\mathtt{s}\leq 1,\,0\leq\mathtt{u}\leq b_{2},$ where
$\displaystyle\mathfrak{N}_{2}=\left[\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\infty}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}\right]^{-1}$
and
$\widehat{\Xi}_{r_{0}}(\mathtt{t})=\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}.$

Then iterative system (5)–(6) has atleast one positive radial solution
$(\mathtt{u}_{1},\mathtt{u}_{2},\cdot\cdot\cdot,\mathtt{u}_{\mathtt{n}})$ such
that $b_{1}\leq\|\mathtt{u}_{\dot{\iota}}\|\leq
b_{2},\,{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$

###### Theorem 3.5

Suppose $(\mathcal{J}_{1})$–$(\mathcal{J}_{3})$ hold. Further, assume that
there exist two positive constants $c_{2}>c_{1}>0$ such that
$\mathtt{g}_{\dot{\iota}}\,({\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n})$
satisfies $(\mathcal{J}_{5})$ and

  * $(\mathcal{J}_{7})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u}(\mathtt{s}))\leq\mathfrak{M}_{2}c_{2}$
for all $0\leq\mathtt{s}\leq 1,\,0\leq\mathtt{u}\leq c_{2},$ where
$\displaystyle\mathfrak{M}_{2}=\left[\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\infty}\prod_{i=1}^{m}\|\ell_{i}\|_{1}\right]^{-1}$
and
$\widehat{\Xi}_{r_{0}}(\mathtt{t})=\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}.$

Then iterative system (5)–(6) has atleast one positive radial solution
$(\mathtt{u}_{1},\mathtt{u}_{2},\cdot\cdot\cdot,\mathtt{u}_{\mathtt{n}})$ such
that $c_{1}\leq\|\mathtt{u}_{\dot{\iota}}\|\leq
c_{2},\,{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$

###### Example 1

Consider the following nonlinear elliptic system of equations,

$\triangle{\mathtt{u}_{{\dot{\iota}}}}+\frac{(\mathtt{N}-2)^{2}r_{0}^{2\mathtt{N}-2}}{|x|^{2\mathtt{N}-2}}\mathtt{u}_{\dot{\iota}}+\ell(|x|)\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1})=0,~{}1<|x|<3,$
(13) $\left.\begin{aligned} &\hskip
25.6073pt\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}|x|=3,\\\
&\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}\text{on}~{}|x|=3,\\\ &\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}\mathtt{u}_{{\dot{\iota}}}=0~{}\text{on}~{}|x|=3,\end{aligned}\right\\}$
(14)

where $r_{0}=1,$ $\mathtt{N}=3,$
${\dot{\iota}}\in\\{1,2\\},\,\mathtt{u}_{3}=\mathtt{u}_{1},$
$\ell(\mathtt{s})=\frac{1}{\mathtt{s}^{4}}\prod_{i=1}^{2}\ell_{i}(\mathtt{s}),$
$\ell_{i}(\mathtt{s})=\ell_{i}\left(\frac{1}{\mathtt{s}}\right),$ in which

$\ell_{1}(t)=\frac{1}{t^{2}+1}~{}~{}~{}~{}\text{~{}~{}and~{}~{}}~{}~{}~{}~{}\ell_{2}(t)=\frac{1}{\sqrt{t+2}},$

then it is clear that

$\ell_{1},\ell_{2}\in
L^{\mathtt{p}}[0,1]~{}~{}\text{~{}~{}and~{}~{}}\prod_{i=1}^{2}\ell_{i}^{*}=\sqrt{2}.$

Let
$\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})=1+\frac{1}{5}\cos(1+\mathtt{u})+\frac{1}{1+\mathtt{u}}.$
Let $\upalpha=\upbeta=\gamma=\updelta=1,$ $\varrho=2\cosh(1)+2\sinh(1)\approx
5.436563658,$

$\displaystyle\Xi_{r_{0}}(\mathtt{s},\mathtt{t})=\frac{1}{2\cosh(1)+2\sinh(1)}\left\\{\begin{array}[]{ll}\big{(}\sinh(\mathtt{s})+\cosh(\mathtt{s})\big{)}\big{(}\sinh(1-\mathtt{t})+\cosh(1-\mathtt{t})\big{)},\hskip
17.07182pt0\leq\mathtt{s}\leq\mathtt{t}\leq 1,\vspace{1.2mm}\\\
\big{(}\sinh(\mathtt{t})+\cosh(\mathtt{t})\big{)}\big{(}\sinh(1-\mathtt{s})+\cosh(1-\mathtt{s})\big{)},\hskip
17.07182pt0\leq\mathtt{t}\leq\mathtt{s}\leq 1,\end{array}\right.$

and $\wp=\frac{1}{\sinh(1)+\cosh(1)}.$ Also,

$\displaystyle\mathfrak{Q}_{1}=\left[\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}\mathtt{d}\mathtt{t}\right]^{-1}\approx
0.1153270463\times 10^{-4}.$

Let $\mathtt{p}_{1}=2,\mathtt{p}_{2}=3$ and $\mathtt{q}=6,$ then
$\frac{1}{\mathtt{p}_{1}}+\frac{1}{\mathtt{p}_{2}}+\frac{1}{\mathtt{q}}=1$ and

$\displaystyle\mathfrak{Q}_{2}=\left[\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}\right]^{-1}\approx
0.4577977612\times 10^{-7}.$

Choose $a_{1}=10^{3}$ and $a_{2}=10^{8}.$ Then,

$\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})$
$\displaystyle\,=1+\frac{1}{5}\cos(1+\mathtt{u})+\frac{1}{1+\mathtt{u}}\leq
4.577977612=\mathfrak{Q}_{2}a_{2},\,\mathtt{u}\in[0,10^{8}],$
$\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})$
$\displaystyle\,=1+\frac{1}{5}\cos(1+\mathtt{u})+\frac{1}{1+\mathtt{u}}\geq
0.011532704=\mathfrak{Q}_{1}a_{1},\,\mathtt{u}\in[0,10^{3}].$

Therefore, by Theorem 3.3, the boundary value problem (13)–(14) has at least
one positive solution $(\mathtt{u}_{1},\mathtt{u}_{2})$ such that
$10^{3}\leq\|\mathtt{u}_{\dot{\iota}}\|\leq 10^{8}$ for ${\dot{\iota}}=1,2.$

##  4 Existence of at Least Two Positive Radial Solutions

In this section, we establish the existence of at least two positive radial
solutions for the system (5)–(6) by an application of following Avery-
Henderson fixed point theorem.

Let $\uppsi$ be a nonnegative continuous functional on a cone $\mathtt{E}$ of
the real Banach space $\mathcal{B}.$ Then for a positive real numbers
$a^{\prime}$ and $c^{\prime},$ we define the sets

$\mathtt{E}(\uppsi,c^{\prime})=\\{\mathtt{u}\in\mathtt{E}:\uppsi(\mathtt{u})<c^{\prime}\\},$

and

$\mathtt{E}_{a^{\prime}}=\\{\mathtt{u}\in\mathtt{E}:\|\mathtt{u}\|<a^{\prime}\\}.$

###### Theorem 4.1

(Avery-Hendersonavery ) Let $\mathtt{E}$ be a cone in a real Banach space
$\mathtt{B}.$ Suppose $\ss_{1}$ and $\ss_{2}$ are increasing, nonnegative
continuous functionals on $\mathtt{E}$ and $\ss_{3}$ is nonnegative continuous
functional on $\mathtt{E}$ with $\ss_{3}(0)=0$ such that, for some positive
numbers $c^{\prime}$ and $k,$
$\ss_{2}(\mathtt{u})\leq\ss_{3}(\mathtt{u})\leq\ss_{1}(\mathtt{u})$ and
$\|\mathtt{u}\|\leq k\ss_{2}(\mathtt{u}),$ for all
$\mathtt{u}\in\overline{\mathtt{E}(\ss_{2},c^{\prime})}.$ Suppose that there
exist positive numbers $a^{\prime}$ and $b^{\prime}$ with
$a^{\prime}<b^{\prime}<c^{\prime}$ such that
$\ss_{3}(\uplambda\mathtt{u})\leq\uplambda\ss_{3}(\mathtt{u}),$ for all
$0\leq\uplambda\leq 1$ and
$\mathtt{u}\in\partial\mathtt{E}(\ss_{3},b^{\prime}).$ Further, let
$\aleph:\overline{\mathtt{E}(\ss_{2},c^{\prime})}\to\mathtt{E}$ be a
completely continuous operator such that

  * $(a)$

$\ss_{2}(\aleph\mathtt{u})>c^{\prime},$ for all
$\mathtt{u}\in\partial\mathtt{E}(\ss_{2},c^{\prime}),$

  * $(b)$

$\ss_{3}(\aleph\mathtt{u})<b^{\prime},$ for all
$\mathtt{u}\in\partial\mathtt{E}(\ss_{3},b^{\prime}),$

  * $(c)$

$\mathtt{E}(\ss_{1},a^{\prime})\neq\emptyset$ and
$\ss_{1}(\aleph\mathtt{u})>a^{\prime},$ for all
$\partial\mathtt{E}(\ss_{1},a^{\prime}).$

Then, $\aleph$ has at least two fixed points
${}^{1}\mathtt{u},{}^{2}\mathtt{u}\in\mathtt{P}(\ss_{2},c^{\prime})$ such that
$a^{\prime}<\ss_{1}({}^{1}\mathtt{u})$ with
$\ss_{3}({}^{1}\mathtt{u})<b^{\prime}$ and
$b^{\prime}<\ss_{3}({}^{2}\mathtt{u})$ with
$\ss_{2}({}^{2}\mathtt{u})<c^{\prime}.$

Define the nonnegative, increasing, continuous functional $\ss_{2},\ss_{3},$
and $\ss_{1}$ by

$\displaystyle\ss_{2}(\mathtt{u})=\min_{\mathtt{s}\in[0,1]}\mathtt{u}(\mathtt{s}),\,\ss_{3}(\mathtt{u})=\max_{\mathtt{s}\in[0,1]}\mathtt{u}(\mathtt{s}),\,\ss_{1}(\mathtt{u})=\max_{\mathtt{s}\in[0,1]}\mathtt{u}(\mathtt{s}).$

It is obvious that for each $\mathtt{u}\in\mathtt{E},$

$\ss_{2}(\mathtt{u})\leq\ss_{3}(\mathtt{u})=\ss_{1}(\mathtt{u}).$

In addition, by Lemma 1, for each $\mathtt{u}\in\mathtt{P},$

$\ss_{2}(\mathtt{u})\geq\wp\|\mathtt{u}\|.$

Thus

$\|\mathtt{u}\|\leq\frac{1}{\wp}\ss_{2}(\mathtt{u})~{}~{}\textnormal{for~{}all}~{}~{}\mathtt{u}\in\mathtt{E}.$

Finally, we also note that

$\ss_{3}(\uplambda\mathtt{u})=\uplambda\ss_{3}(\mathtt{u}),~{}~{}0\leq\uplambda\leq
1~{}~{}\textnormal{and}~{}~{}\mathtt{u}\in\mathtt{E}.$

###### Theorem 4.2

Assume that $(\mathcal{J}_{1})$–$(\mathcal{J}_{3})$ hold and Suppose there
exist real numbers $a^{\prime},b^{\prime}$ and $c^{\prime}$ with
$0<a^{\prime}<b^{\prime}<c^{\prime}$ such that
$\mathtt{g}_{\dot{\iota}}({\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n})$
satisfies

  * $(\mathcal{J}_{8})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})>\frac{c^{\prime}}{\Bbbk_{1}}$, for all
$c^{\prime}\leq\mathtt{u}\leq\frac{c^{\prime}}{\wp},$ where
$\Bbbk_{1}=\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}\mathtt{d}\mathtt{t},$

  * $(\mathcal{J}_{9})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})<\frac{b^{\prime}}{\Bbbk_{2}}$, for all
$0\leq\mathtt{u}\leq\frac{b^{\prime}}{\wp},$ where
$\Bbbk_{2}=\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}},$

  * $(\mathcal{J}_{10})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})>\frac{a^{\prime}}{\Bbbk_{1}}$, for all
$a^{\prime}\leq\mathtt{u}\leq\frac{a^{\prime}}{\wp}.$

Then the boundary value problem (5)–(6) has at least two positive radial
solutions
$\\{({}^{1}\mathtt{u}_{1},{}^{1}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{1}\mathtt{u}_{\mathtt{n}})\\}$
and
$\\{({}^{2}\mathtt{u}_{1},{}^{2}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{2}\mathtt{u}_{\mathtt{n}})\\}$
satisfying

$a^{\prime}<\ss_{1}\big{(}{}^{1}\mathtt{u}_{{\dot{\iota}}}\big{)}~{}~{}\textit{with}~{}~{}\ss_{3}\big{(}{}^{1}\mathtt{u}_{{\dot{\iota}}}\big{)}<b^{\prime},~{}{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n},$

and

$b^{\prime}<\ss_{3}\big{(}{}^{2}\mathtt{u}_{{\dot{\iota}}}\big{)}~{}~{}\textit{with}~{}~{}\ss_{2}\big{(}{}^{2}\mathtt{u}_{{\dot{\iota}}}\big{)}<c^{\prime},~{}{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$

###### Proof

We begin by defining the completely continuous operator $\aleph$ by (10). So
it is easy to check that
$\aleph:\overline{\mathtt{E}(\ss_{2},c^{\prime})}\to\mathtt{E}.$ Firstly, we
shall verify that condition $(a)$ of Theorem 4.1 is satisfied. So, let us
choose $\mathtt{u}_{1}\in\partial\mathtt{E}(\ss_{2},c^{\prime}).$ Then
$\ss_{2}(\mathtt{u}_{1})=\min_{\mathtt{s}\in[0,1]}\mathtt{u}_{1}(\mathtt{s})=c^{\prime}$
this implies that $c^{\prime}\leq\mathtt{u}_{1}(\mathtt{s})$ for
$\mathtt{s}\in[0,1].$ Since
$\|\mathtt{u}_{1}\|\leq\frac{1}{\wp}\ss_{2}(\mathtt{u}_{1})=\frac{1}{\wp}c^{\prime}.$
So we have

$c^{\prime}\leq\mathtt{u}_{1}(\mathtt{s})\leq\frac{c^{\prime}}{\wp},~{}\mathtt{s}\in[0,1].$

Let $\mathtt{t}_{\mathtt{n}-1}\in[0,1].$ Then by $(\mathcal{J}_{8}),$ we have

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\wp\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\frac{\wp
c^{\prime}}{\Bbbk_{1}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\frac{\wp
c^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\Bbbk_{1}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\prod_{i=1}^{m}\ell_{i}(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\frac{\wp
c^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\Bbbk_{1}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq c^{\prime}.$

It follows in similar manner for $0<\mathtt{t}_{\mathtt{n}-2}<1,$

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-2},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{g}_{\mathtt{n}-1}$
$\displaystyle\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq\wp\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{g}_{\mathtt{n}-1}(c^{\prime})\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq\frac{\wp
c^{\prime}}{\Bbbk_{1}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq\frac{\wp
c^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\Bbbk_{1}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}-1}\prod_{i=1}^{m}\ell_{i}(\mathtt{t}_{\mathtt{n}-1})\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq\frac{\wp
c^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\Bbbk_{1}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}-1})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}-1}\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\geq c^{\prime}.$

Continuing with bootstrapping argument, we get

$\displaystyle\ss_{2}\left(\aleph\mathtt{u}_{1}\right)=$
$\displaystyle\min_{\mathtt{s}\in[0,1]}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
45.52458pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}$
$\displaystyle\geq$ $\displaystyle\,c^{\prime}.$

This proves $(i)$ of Theorem 4.1. We next address $(ii)$ of Theorem 4.1. So,
we choose $\mathtt{u}_{1}\in\partial\mathtt{E}(\ss_{3},b^{\prime}).$ Then
$\ss_{3}(\mathtt{u}_{1})=\max_{\mathtt{s}\in[0,1]}\mathtt{u}_{1}(\mathtt{s})=b^{\prime}$
this implies that $0\leq\mathtt{u}_{1}(\mathtt{s})\leq b^{\prime}$ for
$\mathtt{s}\in[0,1].$ Since
$\|\mathtt{u}_{1}\|\leq\frac{1}{\wp}\ss_{2}(\mathtt{u}_{1})\leq\frac{1}{\wp}\ss_{3}(\mathtt{u}_{1})=\frac{b^{\prime}}{\wp}.$
So we have

$0\leq\mathtt{u}_{1}(\mathtt{s})\leq\wp^{2}b^{\prime},~{}\mathtt{s}\in[0,1].$

Let $0<\mathtt{t}_{\mathtt{n}-1}<1.$ Then by $(\mathcal{J}_{9}),$ we have

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\wp\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\frac{\wp
b^{\prime}}{\Bbbk_{2}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\frac{\wp
b^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\Bbbk_{2}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\prod_{i=1}^{m}\ell_{i}(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}.$

There exists a $\mathtt{q}>1$ such that
$\displaystyle\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}+\frac{1}{\mathtt{q}}=1.$
By the first part of Theorem 3.2, we have

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\frac{\wp
b^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\Bbbk_{2}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}$
$\displaystyle\leq b^{\prime}.$

Continuing with this bootstrapping argument, we get

$\displaystyle\ss_{3}\left(\aleph\mathtt{u}_{1}\right)=$
$\displaystyle\max_{\mathtt{s}\in[0,1]}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
28.45274pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}$
$\displaystyle\leq$ $\displaystyle\,b^{\prime}.$

Hence condition $(b)$ is satisfied. Finally, we verify that $(c)$ of Theorem
4.1 is also satisfied. We note that
$\mathtt{u}_{1}(\mathtt{s})=a^{\prime}/4,~{}\mathtt{s}\in[0,1]$ is a member of
$\mathtt{E}(\ss_{1},a^{\prime})$ and $a^{\prime}/4<a^{\prime}.$ So
$\mathtt{E}(\ss_{1},a^{\prime})\neq\emptyset.$ Now let
$\mathtt{u}_{1}\in\mathtt{E}(\ss_{1},a^{\prime}).$ Then
$a^{\prime}=\ss_{1}(\mathtt{u}_{1})=\max_{\mathtt{s}\in[0,1]}\mathtt{u}_{1}(\mathtt{s})=\|\mathtt{u}_{1}\|=\frac{1}{\wp}\ss_{2}(\mathtt{u}_{1})\leq\frac{1}{\wp}\ss_{3}(\mathtt{u}_{1})=\frac{1}{\wp}\ss_{1}(\mathtt{u}_{1})=\frac{a^{\prime}}{\wp},$
i.e., $a^{\prime}\leq\mathtt{u}_{1}(\mathtt{s})\leq\frac{a^{\prime}}{\wp}$ for
$\mathtt{s}\in[0,1].$ Let $0<\mathtt{t}_{\mathtt{n}-1}<1.$ Then by
$(\mathcal{J}_{10}),$ we have

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\wp\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\frac{\wp
a^{\prime}}{\Bbbk_{1}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\frac{\wp
a^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\Bbbk_{1}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\prod_{i=1}^{m}\ell_{i}(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\frac{\wp
a^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\Bbbk_{1}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq a^{\prime}.$

Continuing with this bootstrapping argument, we get

$\displaystyle\ss_{1}\left(\aleph\mathtt{u}_{1}\right)=$
$\displaystyle\max_{\mathtt{s}\in[0,1]}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
28.45274pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}$
$\displaystyle\geq$
$\displaystyle\min_{\mathtt{s}\in[0,1]}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
28.45274pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}$
$\displaystyle\geq$ $\displaystyle\,a^{\prime}.$

Thus condition $(c)$ of Theorem 4.1 is satisfied. Since all hypotheses of
Theorem 4.1 are satisfied, the assertion follows.∎

For $\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}=1$ and
$\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}>1,$ we have following results.

###### Theorem 4.3

Assume that $(\mathcal{J}_{1})$–$(\mathcal{J}_{3})$ hold and Suppose there
exist real numbers $a^{\prime},b^{\prime}$ and $c^{\prime}$ with
$0<a^{\prime}<b^{\prime}<c^{\prime}$ such that
$\mathtt{g}_{\dot{\iota}}({\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n})$
satisfies $(\mathcal{J}_{8}),$ $(\mathcal{J}_{10})$ and

  * $(\mathcal{J}_{9}^{\prime})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})<\frac{b^{\prime}}{\Bbbk_{3}}$, for all
$0\leq\mathtt{u}\leq\frac{b^{\prime}}{\wp},$ where
$\Bbbk_{3}=\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\infty}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}.$

Then the boundary value problem (5)–(6) has at least two positive radial
solutions
$\\{({}^{1}\mathtt{u}_{1},{}^{1}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{1}\mathtt{u}_{\mathtt{n}})\\}$
and
$\\{({}^{2}\mathtt{u}_{1},{}^{2}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{2}\mathtt{u}_{\mathtt{n}})\\}$
satisfying

$a^{\prime}<\ss_{1}\big{(}{}^{1}\mathtt{u}_{{\dot{\iota}}}\big{)}~{}~{}\textit{with}~{}~{}\ss_{3}\big{(}{}^{1}\mathtt{u}_{{\dot{\iota}}}\big{)}<b^{\prime},~{}{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n},$

and

$b^{\prime}<\ss_{3}\big{(}{}^{2}\mathtt{u}_{{\dot{\iota}}}\big{)}~{}~{}\textit{with}~{}~{}\ss_{2}\big{(}{}^{2}\mathtt{u}_{{\dot{\iota}}}\big{)}<c^{\prime},~{}{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$

###### Theorem 4.4

Assume that $(\mathcal{J}_{1})$–$(\mathcal{J}_{3})$ hold and Suppose there
exist real numbers $a^{\prime},b^{\prime}$ and $c^{\prime}$ with
$0<a^{\prime}<b^{\prime}<c^{\prime}$ such that
$\mathtt{g}_{\dot{\iota}}({\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n})$
satisfies $(\mathcal{J}_{8}),$ $(\mathcal{J}_{10})$ and

  * $(\mathcal{J}_{9}^{\prime\prime})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})<\frac{b^{\prime}}{\Bbbk_{4}}$, for all
$0\leq\mathtt{u}\leq\frac{b^{\prime}}{\wp},$ where
$\Bbbk_{4}=\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\infty}\prod_{i=1}^{m}\|\ell_{i}\|_{1}.$

Then the boundary value problem (5)–(6) has at least two positive radial
solutions
$\\{({}^{1}\mathtt{u}_{1},{}^{1}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{1}\mathtt{u}_{\mathtt{n}})\\}$
and
$\\{({}^{2}\mathtt{u}_{1},{}^{2}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{2}\mathtt{u}_{\mathtt{n}})\\}$
satisfying

$a^{\prime}<\ss_{1}\big{(}{}^{1}\mathtt{u}_{{\dot{\iota}}}\big{)}~{}~{}\textit{with}~{}~{}\ss_{3}\big{(}{}^{1}\mathtt{u}_{{\dot{\iota}}}\big{)}<b^{\prime},~{}{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n},$

and

$b^{\prime}<\ss_{3}\big{(}{}^{2}\mathtt{u}_{{\dot{\iota}}}\big{)}~{}~{}\textit{with}~{}~{}\ss_{2}\big{(}{}^{2}\mathtt{u}_{{\dot{\iota}}}\big{)}<c^{\prime},~{}{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$

###### Example 2

Consider the following nonlinear elliptic system of equations,

$\triangle{\mathtt{u}_{{\dot{\iota}}}}+\frac{(\mathtt{N}-2)^{2}r_{0}^{2\mathtt{N}-2}}{|x|^{2\mathtt{N}-2}}\mathtt{u}_{\dot{\iota}}+\ell(|x|)\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1})=0,~{}1<|x|<3,$
(15) $\left.\begin{aligned} &\hskip
25.6073pt\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}|x|=3,\\\
&\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}\text{on}~{}|x|=3,\\\ &\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}\mathtt{u}_{{\dot{\iota}}}=0~{}\text{on}~{}|x|=3,\end{aligned}\right\\}$
(16)

where $r_{0}=1,$ $\mathtt{N}=3,$
${\dot{\iota}}\in\\{1,2\\},\,\mathtt{u}_{3}=\mathtt{u}_{1},$
$\ell(\mathtt{s})=\frac{1}{\mathtt{s}^{4}}\prod_{i=1}^{2}\ell_{i}(\mathtt{s}),$
$\ell_{i}(\mathtt{s})=\ell_{i}\left(\frac{1}{\mathtt{s}}\right),$ in which

$\ell_{1}(t)=\frac{1}{t+1}~{}~{}~{}~{}\text{~{}~{}and~{}~{}}~{}~{}~{}~{}\ell_{2}(t)=\frac{1}{t^{2}+1},$

then it is clear that

$\ell_{1},\ell_{2}\in
L^{\mathtt{p}}[0,1]~{}~{}\text{~{}~{}and~{}~{}}\prod_{i=1}^{2}\ell_{i}^{*}=1.$

Let $\upalpha=\upbeta=\gamma=\updelta=1,$ then
$\varrho=2\cosh(1)+2\sinh(1)\approx 5.436563658,$

$\displaystyle\Xi_{r_{0}}(\mathtt{s},\mathtt{t})=\frac{1}{2\cosh(1)+2\sinh(1)}\left\\{\begin{array}[]{ll}\big{(}\sinh(\mathtt{s})+\cosh(\mathtt{s})\big{)}\big{(}\sinh(1-\mathtt{t})+\cosh(1-\mathtt{t})\big{)},\hskip
17.07182pt0\leq\mathtt{s}\leq\mathtt{t}\leq 1,\vspace{1.2mm}\\\
\big{(}\sinh(\mathtt{t})+\cosh(\mathtt{t})\big{)}\big{(}\sinh(1-\mathtt{s})+\cosh(1-\mathtt{s})\big{)},\hskip
17.07182pt0\leq\mathtt{t}\leq\mathtt{s}\leq 1,\end{array}\right.$

and $\wp=\frac{1}{\sinh(1)+\cosh(1)}.$ Also,

$\displaystyle\Bbbk_{1}=\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}\mathtt{d}\mathtt{t}\approx
0.1630970729\times 10^{-4}.$

Let $\mathtt{p}_{1}=3,\mathtt{p}_{2}=6$ and $\mathtt{q}=2,$ then
$\frac{1}{\mathtt{p}_{1}}+\frac{1}{\mathtt{p}_{2}}+\frac{1}{\mathtt{q}}=1$ and

$\displaystyle\Bbbk_{2}=\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}\approx
4.388193758\times 10^{-8}.$

Let

$\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})=\left\\{\begin{array}[]{ll}10^{16},\hskip
54.06006pt\mathtt{u}\geq 1,\vspace{1.2mm}\\\
10^{16}\mathtt{u}^{2}-\mathtt{u}+1,\hskip
11.38092pt\mathtt{u}<1.\end{array}\right.$

Choose $a^{\prime}=10^{4},\,b^{\prime}=10^{9}$ and $c^{\prime}=10^{10}.$ Then,

$\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})$
$\displaystyle\geq 6.131317885\times
10^{14}=\frac{c^{\prime}}{\Bbbk_{1}},~{}~{}\mathtt{u}\in[10^{10},2.72\times
10^{10}],$
$\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})$
$\displaystyle\leq 2.278841945\times
10^{16}=\frac{b^{\prime}}{\Bbbk_{2}},~{}~{}\mathtt{u}\in[0,2.72\times
10^{9}],$ $\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})$
$\displaystyle\geq 6.131317885\times
10^{8}=\frac{a^{\prime}}{\Bbbk_{1}},~{}~{}\mathtt{u}\in[10^{4},2.72\times
10^{4}].$

Therefore, by Theorem 3.3, the boundary value problem (15)–(16) has at least
two positive radial solutions
$({}^{\dot{\iota}}\mathtt{u}_{1},{}^{\dot{\iota}}\mathtt{u}_{2}),\,{\dot{\iota}}=1,2$
such that

$10^{4}<\max_{\mathtt{s}\in[0,1]}{}^{\dot{\iota}}\mathtt{u}_{1}(\mathtt{s})~{}~{}\text{with}~{}~{}\max_{\mathtt{s}\in[0,1]}{}^{\dot{\iota}}\mathtt{u}_{1}(\mathtt{s})<10^{9},~{}~{}\text{for}~{}~{}{\dot{\iota}}=1,2,$
$10^{9}<\max_{\mathtt{s}\in[0,1]}{}^{\dot{\iota}}\mathtt{u}_{2}(\mathtt{s})~{}~{}\text{with}~{}~{}\min_{\mathtt{s}\in[0,1]}{}^{\dot{\iota}}\mathtt{u}_{2}(\mathtt{s})<10^{10},~{}~{}\text{for}~{}~{}{\dot{\iota}}=1,2.$

##  5 Existence of at Least Three Positive Radial Solutions

In this section, we establish the existence of at least three positive radial
solutions for the system (5)–(6) by an application of following Legget-William
fixed point theorem. Let $a^{\prime},b^{\prime}$ be two real numbers such that
$0<a^{\prime}<b^{\prime}$ and $\ss$ a nonnegative, continuous, concave
functional on $\mathtt{E}.$ We define the following convex sets,

$\mathtt{E}_{a^{\prime}}=\\{\mathtt{u}\in\mathtt{E}:\|\mathtt{u}\|<a^{\prime}\\},$
$\mathtt{E}(\ss,a^{\prime},b^{\prime})=\\{\mathtt{u}\in\mathtt{E}:a^{\prime}\leq\ss(\mathtt{u}),\,\|\mathtt{u}\|<b^{\prime}\\}.$

###### Theorem 5.1

(Legget-Williamlegget ) Let $\mathtt{E}$ be a cone in a Banach space
$\mathtt{B}.$ Let $\ss$ a nonnegative, continuous, concave functional on
$\mathtt{E}$ satisfying for some $c^{\prime}>0$ such that
$\ss(\mathtt{u})\leq\|\mathtt{u}\|$ for all
$\mathtt{u}\in\overline{\mathtt{E}}_{c^{\prime}}.$ Suppose there exists a
completely continuous operator
$\aleph:\overline{\mathtt{E}}_{c^{\prime}}\to\overline{\mathtt{E}}_{c^{\prime}}$
and $0<a^{\prime}<b^{\prime}<d^{\prime}\leq c^{\prime}$ such that

  * $(a)$

$\\{\mathtt{u}\in\mathtt{E}(\ss,b^{\prime},d^{\prime}):\ss(\mathtt{u})>a^{\prime}\\}\neq\emptyset$
and $\ss(\aleph\mathtt{u})>b^{\prime}$ for
$\mathtt{u}\in\mathtt{E}(\ss,b^{\prime},d^{\prime}),$

  * $(b)$

$\|\aleph\mathtt{u}\|<a^{\prime}$ for $\|\mathtt{u}\|<a^{\prime},$

  * $(c)$

$\ss(\aleph\mathtt{u})>b^{\prime}$ for
$\mathtt{u}\in\mathtt{E}(\ss,a^{\prime},c^{\prime}),$ with
$\|\aleph\mathtt{u}\|>d^{\prime}$

Then, $\aleph$ has at least three fixed points
${}^{1}\mathtt{u},{}^{2}\mathtt{u},{}^{3}\mathtt{u}\in\mathtt{E}_{c^{\prime}}$
satisfying $\|{}^{1}\mathtt{u}\|<a^{\prime},$
$b^{\prime}<\ss({}^{2}\mathtt{u})$ and $\|{}^{3}\mathtt{u}\|>a^{\prime}$ and
$\ss({}^{3}\mathtt{u})<b^{\prime}.$

###### Theorem 5.2

Assume that $(\mathcal{J}_{1})$–$(\mathcal{J}_{3})$ hold. Let
$0<a^{\prime}<b^{\prime}<c^{\prime}$ and suppose that
$\mathtt{g}_{\dot{\iota}},\,{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}$
satisfies the following conditions,

  * $(\mathcal{J}_{11})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})<\frac{a^{\prime}}{\mathfrak{O}_{1}}$ for
$0\leq\mathtt{u}\leq a^{\prime},$ where
$\mathfrak{O}_{1}=\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}.$

  * $(\mathcal{J}_{12})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})>\frac{b^{\prime}}{\mathfrak{O}_{2}}$ for
$b^{\prime}\leq\mathtt{u}\leq c^{\prime},$ where $\mathfrak{O}_{2}=\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}\mathtt{d}\mathtt{t}.$

  * $(\mathcal{J}_{13})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})<\frac{c^{\prime}}{\mathfrak{O}_{1}}$ for
$0\leq\mathtt{u}\leq c^{\prime}.$

Then the iterative system (5)–(6) has at least three positive solutions
$({}^{1}\mathtt{u}_{1},{}^{1}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{1}\mathtt{u}_{\mathtt{n}}),$
$({}^{2}\mathtt{u}_{1},{}^{2}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{2}\mathtt{u}_{\mathtt{n}})$
and
$({}^{3}\mathtt{u}_{1},{}^{3}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{3}\mathtt{u}_{\mathtt{n}})$
with $\|{}^{\dot{\iota}}\mathtt{u}_{1}\|<a^{\prime},$
$b^{\prime}<\ss({}^{\dot{\iota}}\mathtt{u}_{2}),$
$\|{}^{\dot{\iota}}\mathtt{u}_{3}\|>a^{\prime}$ and
$\ss({}^{\dot{\iota}}\mathtt{u}_{3})<b^{\prime}$ for
${\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$

###### Proof

From Lemma 3, $\aleph:\mathtt{E}\to\mathtt{E}$ is a completely continuous
operator. If $\mathtt{u}_{1}\in\overline{\mathtt{E}}_{c^{\prime}},$ then
$\|\mathtt{u}_{1}\|\leq c^{\prime}$ and for $0<\mathtt{t}_{\mathtt{n}-1}<1$
and by $(\mathcal{J}_{13}),$ we have

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\frac{c^{\prime}}{\mathfrak{O}_{1}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\frac{c^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\mathfrak{O}_{1}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\prod_{i=1}^{m}\ell_{i}(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}.$

There exists a $\mathtt{q}>1$ such that
$\displaystyle\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}+\frac{1}{\mathtt{q}}=1.$
By the first part of Theorem 3.2, we have

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\frac{c^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\mathfrak{O}_{1}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}$
$\displaystyle\leq c^{\prime}.$

Continuing with this bootstrapping argument, we get

$\displaystyle\left\|\aleph\mathtt{u}_{1}\right\|=$
$\displaystyle\max_{\mathtt{s}\in[0,1]}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
28.45274pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}$
$\displaystyle\leq$ $\displaystyle\,c^{\prime}.$

Hence,
$\aleph:\overline{\mathtt{E}}_{c^{\prime}}\to\overline{\mathtt{E}}_{c^{\prime}}.$
In the same way, if $\mathtt{u}_{1}\in\overline{\mathtt{E}}_{a^{\prime}},$
then
$\aleph:\overline{\mathtt{E}}_{a^{\prime}}\to\overline{\mathtt{E}}_{a^{\prime}}.$
Therefore, condition $(b)$ of Theorem 5.1 satisfied. To check condition $(a)$
of Theorem 5.1, choose
$\mathtt{u}_{1}(\mathtt{s})=(b^{\prime}+c^{\prime})/2,\,\mathtt{s}\in[0,1].$
It is easy to see that
$\mathtt{u}_{1}\in\mathtt{E}(\ss,b^{\prime},c^{\prime})$ and
$\ss(\mathtt{u}_{1})=\ss((b^{\prime}+c^{\prime})/2)>b^{\prime}.$ So,
$\\{\mathtt{u}_{1}\in\mathtt{E}(\ss,b^{\prime},c^{\prime}):\ss(\mathtt{u}_{1})>b^{\prime}\\}\neq\emptyset.$
Hence, if $\mathtt{u}_{1}\in\mathtt{E}(\ss,b^{\prime},c^{\prime})$ then
$b^{\prime}<\mathtt{u}_{1}(\mathtt{s})<c^{\prime},\,\mathtt{s}\in[0,1].$ Let
$0<\mathtt{t}_{\mathtt{n}-1}<1.$ Then by $(\mathcal{J}_{12}),$ we have

$\displaystyle\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\wp\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\frac{\wp
b^{\prime}}{\mathfrak{O}_{2}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\frac{\wp
b^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\mathfrak{O}_{2}}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\prod_{i=1}^{m}\ell_{i}(\mathtt{t}_{\mathtt{n}})\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq\frac{\wp
b^{\prime}r_{0}^{2}}{(\mathtt{N}-2)^{2}\mathfrak{O}_{2}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}_{\mathtt{n}}\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\geq b^{\prime}.$

Continuing with this bootstrapping argument, we get

$\displaystyle\min_{\mathtt{s}\in[0,1]}\left(\aleph\mathtt{u}_{1}\right)=$
$\displaystyle\min_{\mathtt{s}\in[0,1]}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{s},\mathtt{t}_{1})\ell(\mathtt{t}_{1})\mathtt{g}_{1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{1},\mathtt{t}_{2})\ell(\mathtt{t}_{2})\mathtt{g}_{2}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{2},\mathtt{t}_{3})\ell(\mathtt{t}_{3})\mathtt{g}_{4}\cdots$
$\displaystyle\hskip
28.45274pt\mathtt{g}_{\mathtt{n}-1}\Bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\Bigg{]}\cdot\cdot\cdot\Bigg{]}\mathtt{d}\mathtt{t}_{3}\Bigg{]}\mathtt{d}\mathtt{t}_{2}\Bigg{]}\mathtt{d}\mathtt{t}_{1}$
$\displaystyle\geq$ $\displaystyle\,b^{\prime}.$

Therefore, we have

$\ss(\aleph\mathtt{u}_{1})>b^{\prime},~{}\textnormal{for}~{}\mathtt{u}_{1}\in\mathtt{E}(\ss,b^{\prime},c^{\prime}).$

This implies that condition $(a)$ of Theorem 5.1 is satisfied.

Finally, if $\mathtt{u}_{1}\in\mathtt{E}(\ss,b^{\prime},c^{\prime}),$ then
what we have already proved, $\ss(\aleph\mathtt{u}_{1})>b^{\prime}.$ Which
proves the condition $(c)$ of Theorem 5.1. To sum up, all the conditions of
Theorem 5.1 are satisfied. Therefore, $\aleph$ has at least three fixed
points, that is, problem (5)–(6) has at least three positive solutions
$({}^{1}\mathtt{u}_{1},{}^{1}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{1}\mathtt{u}_{\mathtt{n}}),$
$({}^{2}\mathtt{u}_{1},{}^{2}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{2}\mathtt{u}_{\mathtt{n}})$
and
$({}^{3}\mathtt{u}_{1},{}^{3}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{3}\mathtt{u}_{\mathtt{n}})$
with $\|{}^{\dot{\iota}}\mathtt{u}_{1}\|<a^{\prime},$
$b^{\prime}<\ss({}^{\dot{\iota}}\mathtt{u}_{2}),$
$\|{}^{\dot{\iota}}\mathtt{u}_{3}\|>a^{\prime}$ and
$\ss({}^{\dot{\iota}}\mathtt{u}_{3})<b^{\prime}$ for
${\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$ ∎

For $\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}=1$ and
$\sum_{i=1}^{m}\frac{1}{\mathtt{p}_{i}}>1,$ we have following results.

###### Theorem 5.3

Assume that $(\mathcal{J}_{1})$–$(\mathcal{J}_{3})$ hold. Let
$0<a^{\prime}<b^{\prime}<c^{\prime}$ and suppose that
$\mathtt{g}_{\dot{\iota}},\,{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}$
satisfies $(\mathcal{J}_{12}),\,(\mathcal{J}_{13})$ and

  * $(\mathcal{J}_{14})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})<\frac{a^{\prime}}{\mathfrak{O}_{3}}$ for
$0\leq\mathtt{u}\leq a^{\prime},$ where
$\mathfrak{O}_{3}=\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\infty}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}.$

Then the iterative system (5)–(6) has at least three positive solutions
$({}^{1}\mathtt{u}_{1},{}^{1}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{1}\mathtt{u}_{\mathtt{n}}),$
$({}^{2}\mathtt{u}_{1},{}^{2}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{2}\mathtt{u}_{\mathtt{n}})$
and
$({}^{3}\mathtt{u}_{1},{}^{3}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{3}\mathtt{u}_{\mathtt{n}})$
with $\|{}^{\dot{\iota}}\mathtt{u}_{1}\|<a^{\prime},$
$b^{\prime}<\ss({}^{\dot{\iota}}\mathtt{u}_{2}),$
$\|{}^{\dot{\iota}}\mathtt{u}_{3}\|>a^{\prime}$ and
$\ss({}^{\dot{\iota}}\mathtt{u}_{3})<b^{\prime}$ for
${\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$

###### Theorem 5.4

Assume that $(\mathcal{J}_{1})$–$(\mathcal{J}_{3})$ hold. Let
$0<a^{\prime}<b^{\prime}<c^{\prime}$ and suppose that
$\mathtt{g}_{\dot{\iota}},\,{\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}$
satisfies $(\mathcal{J}_{12}),\,(\mathcal{J}_{13})$ and

  * $(\mathcal{J}_{15})$

$\mathtt{g}_{\dot{\iota}}(\mathtt{u})<\frac{a^{\prime}}{\mathfrak{O}_{4}}$ for
$0\leq\mathtt{u}\leq a^{\prime},$ where
$\mathfrak{O}_{4}=\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\infty}\prod_{i=1}^{m}\|\ell_{i}\|_{1}.$

Then the iterative system (5)–(6) has at least three positive solutions
$({}^{1}\mathtt{u}_{1},{}^{1}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{1}\mathtt{u}_{\mathtt{n}}),$
$({}^{2}\mathtt{u}_{1},{}^{2}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{2}\mathtt{u}_{\mathtt{n}})$
and
$({}^{3}\mathtt{u}_{1},{}^{3}\mathtt{u}_{2},\cdot\cdot\cdot,{}^{3}\mathtt{u}_{\mathtt{n}})$
with $\|{}^{\dot{\iota}}\mathtt{u}_{1}\|<a^{\prime},$
$b^{\prime}<\ss({}^{\dot{\iota}}\mathtt{u}_{2}),$
$\|{}^{\dot{\iota}}\mathtt{u}_{3}\|>a^{\prime}$ and
$\ss({}^{\dot{\iota}}\mathtt{u}_{3})<b^{\prime}$ for
${\dot{\iota}}=1,2,\cdot\cdot\cdot,\mathtt{n}.$

###### Example 3

Consider the following nonlinear elliptic system of equations,

$\triangle{\mathtt{u}_{{\dot{\iota}}}}+\frac{(\mathtt{N}-2)^{2}r_{0}^{2\mathtt{N}-2}}{|x|^{2\mathtt{N}-2}}\mathtt{u}_{\dot{\iota}}+\ell(|x|)\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1})=0,~{}1<|x|<2,$
(17) $\left.\begin{aligned} &\hskip
25.6073pt\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}|x|=2,\\\
&\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}\text{on}~{}|x|=2,\\\ &\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}\mathtt{u}_{{\dot{\iota}}}=0~{}\text{on}~{}|x|=2,\end{aligned}\right\\}$
(18)

where $r_{0}=1,$ $\mathtt{N}=3,$
${\dot{\iota}}\in\\{1,2\\},\,\mathtt{u}_{3}=\mathtt{u}_{1},$
$\ell(\mathtt{s})=\frac{1}{\mathtt{s}^{4}}\prod_{i=1}^{2}\ell_{i}(\mathtt{s}),$
$\ell_{i}(\mathtt{s})=\ell_{i}\left(\frac{1}{\mathtt{s}}\right),$ in which

$\ell_{1}(t)=\frac{1}{\sqrt{t+1}}~{}~{}~{}~{}\text{~{}~{}and~{}~{}}~{}~{}~{}~{}\ell_{2}(t)=\frac{1}{\sqrt{t^{2}+25}},$

then it is clear that

$\ell_{1},\ell_{2}\in
L^{\mathtt{p}}[0,1]~{}~{}\text{~{}~{}and~{}~{}}\prod_{i=1}^{2}\ell_{i}^{*}=5.$

Let

$\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})=\left\\{\begin{array}[]{ll}\frac{3}{2},\hskip
36.98866pt\mathtt{u}\geq 1,\vspace{1.2mm}\\\
\frac{1}{2}\mathtt{u}^{2}+1,\hskip 11.38092pt\mathtt{u}<1.\end{array}\right.$

Let $\upalpha=\upbeta=\gamma=\updelta=1,$ $\varrho=2\cosh(1)+2\sinh(1)\approx
5.436563658,$

$\displaystyle\Xi_{r_{0}}(\mathtt{s},\mathtt{t})=\frac{1}{2\cosh(1)+2\sinh(1)}\left\\{\begin{array}[]{ll}\big{(}\sinh(\mathtt{s})+\cosh(\mathtt{s})\big{)}\big{(}\sinh(1-\mathtt{t})+\cosh(1-\mathtt{t})\big{)},\hskip
17.07182pt0\leq\mathtt{s}\leq\mathtt{t}\leq 1,\vspace{1.2mm}\\\
\big{(}\sinh(\mathtt{t})+\cosh(\mathtt{t})\big{)}\big{(}\sinh(1-\mathtt{s})+\cosh(1-\mathtt{s})\big{)},\hskip
17.07182pt0\leq\mathtt{t}\leq\mathtt{s}\leq 1,\end{array}\right.$

and $\wp=\frac{3}{\cosh(1)}.$ Also,

$\displaystyle\mathfrak{O}_{1}=\frac{\wp
r_{0}^{2}}{(\mathtt{N}-2)^{2}}\prod_{i=1}^{m}\ell_{i}^{\star}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}\mathtt{d}\mathtt{t}\approx
4.627034665\times 10^{6}.$

Let $\mathtt{p}_{1}=3,\mathtt{p}_{2}=2$ and $\mathtt{q}=6,$ then
$\frac{1}{\mathtt{p}_{1}}+\frac{1}{\mathtt{p}_{2}}+\frac{1}{\mathtt{q}}=1$ and

$\displaystyle\mathfrak{O}_{2}=\frac{r_{0}^{2}}{(\mathtt{N}-2)^{2}}\|\widehat{\Xi}_{r_{0}}\|_{\mathtt{q}}\prod_{i=1}^{m}\|\ell_{i}\|_{\mathtt{p}_{i}}\approx
9.696074194\times 10^{7}.$

Choose $a^{\prime}=10^{7},\,b^{\prime}=10^{8}$ and $c^{\prime}=10^{9}.$ Then,

$\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})$
$\displaystyle\,\leq
2.161211386=\frac{a^{\prime}}{\mathfrak{O}_{1}},~{}\mathtt{u}\in[0,10^{7}],$
$\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})$
$\displaystyle\,\geq
1.031345243=\frac{b^{\prime}}{\mathfrak{O}_{2}},~{}\mathtt{u}\in[10^{8},10^{9}],$
$\displaystyle\mathtt{g}_{1}(\mathtt{u})=\mathtt{g}_{2}(\mathtt{u})$
$\displaystyle\,\leq
216.1211386=\frac{c^{\prime}}{\mathfrak{O}_{1}},~{}\mathtt{u}\in[0,10^{9}].$

Therefore, by Theorem 3.3, the boundary value problem (15)–(16) has at least
two positive radial solutions
$({}^{\dot{\iota}}\mathtt{u}_{1},{}^{\dot{\iota}}\mathtt{u}_{2}),\,{\dot{\iota}}=1,2$
such that

$\max_{\mathtt{s}\in[0,1]}{}^{\dot{\iota}}\mathtt{u}_{1}(\mathtt{s})<10^{7},~{}10^{8}<\min_{\mathtt{s}\in[0,1]}{}^{\dot{\iota}}\mathtt{u}_{2}(\mathtt{s})<\max_{\mathtt{s}\in[0,1]}{}^{\dot{\iota}}\mathtt{u}_{2}(\mathtt{s})<10^{9},~{}~{}\text{for}~{}~{}{\dot{\iota}}=1,2,$
$10^{7}<\max_{\mathtt{s}\in[0,1]}{}^{\dot{\iota}}\mathtt{u}_{3}(\mathtt{s})<10^{9},~{}~{}\min_{\mathtt{s}\in[0,1]}{}^{\dot{\iota}}\mathtt{u}_{3}(\mathtt{s})<10^{8},~{}~{}\text{for}~{}~{}{\dot{\iota}}=1,2.$

##  6 Existence of Unique Positive Radial Solution

In the next, for the existence of unique positive radial solution to the
boundary value problem (5)–(6), we employ two metrics under Rus’s theorem
(see, alm ; stin  for more details). Consider the set of real valued functions
that are defined and continuous on $[0,1]$ and denote this space by
$\mathtt{v}=C([0,1]).$ For functions
$\mathtt{v}_{1},\mathtt{v}_{2}\in\mathtt{v},$ consider the following two
metrics on $\mathtt{v}:$

$\mathtt{d}(\mathtt{v}_{1},\mathtt{v}_{2})=\max_{\mathtt{t}\in[0,1]}|\mathtt{v}_{1}(\mathtt{t})-\mathtt{v}_{2}(\mathtt{t})|,$
(19)
$\uprho(\mathtt{v}_{1},\mathtt{v}_{2})=\left[\int_{0}^{1}|\mathtt{v}_{1}(\mathtt{t})-\mathtt{v}_{2}(\mathtt{t})|^{\mathtt{p}}\mathtt{d}\mathtt{t}\right]^{\frac{1}{\mathtt{p}}},~{}~{}\mathtt{p}>1.$
(20)

For $\mathtt{d}$ in (19), the pair $(C([0,1]),\mathtt{d})$ forms a complete
metric space. For $\uprho$ in (20), the pair $(C([0,1]),\uprho)$ forms a
metric space. The relationship between the two metrics on $\mathtt{v}$ is
given by

$\uprho(\mathtt{v}_{1},\mathtt{v}_{2})\leq\mathtt{d}(\mathtt{v}_{1},\mathtt{v}_{2})~{}~{}\text{for~{}all}~{}~{}\mathtt{v}_{1},\mathtt{v}_{2}\in\mathtt{v}.$
(21)

######  Theorem 6.1 (Rus rus )

Let $\mathtt{v}$ be a nonempty set and let $\mathtt{d}$ and $\uprho$ be two
metrics on $\mathtt{v}$ such that $(\mathtt{v},\mathtt{d})$ forms a complete
metric space. If the mapping $\Lambda:\mathtt{v}\to\mathtt{v}$ is continuous
with respect to $\mathtt{d}$ on $\mathtt{v}$ and

$\mathtt{d}(\Lambda\mathtt{v}_{1},\Lambda\mathtt{v}_{2})\leq\alpha_{1}\uprho(\mathtt{v}_{1},\mathtt{v}_{2}),$
(22)

for some $\alpha_{1}>0$ and for all
$\mathtt{v}_{1},\mathtt{v}_{2}\in\mathtt{v},$

$\uprho(\Lambda\mathtt{v}_{1},\Lambda\mathtt{v}_{2})\leq\alpha_{2}\uprho(\mathtt{v}_{1},\mathtt{v}_{2}),$
(23)

for some $0<\alpha_{2}<1$ for all
$\mathtt{v}_{1},\mathtt{v}_{2}\in\mathtt{v},$ then there is a unique
$\mathtt{v}^{*}\in\mathtt{v}$ such that
$\Lambda\mathtt{v}^{*}=\mathtt{v}^{*}.$

Denote
$\Upsilon(\mathtt{t})=\Xi_{r_{0}}(\mathtt{t},\mathtt{t})\mathtt{t}^{\frac{2(\mathtt{N}-1)}{2-\mathtt{N}}}\prod_{i=1}^{m}\ell_{i}(\mathtt{t})\mathtt{d}\mathtt{t}.$

###### Theorem 6.2

Assume that $(\mathcal{J}_{1}),\,(\mathcal{J}_{3})$ and the following codition
are satisfied.

  * $(\mathcal{J}_{14})$

there exists a number $\mathtt{K}>0$ such that

$|\mathtt{g}_{\dot{\iota}}(\mathtt{u})-\mathtt{g}_{\dot{\iota}}(\mathtt{v})|\leq\mathtt{K}|\mathtt{u}-\mathtt{v}|~{}~{}\text{for}~{}~{}\mathtt{u},\mathtt{v}\in\mathtt{v}.$

Further, assume that there are constants $\mathtt{p}>1$ and $\mathtt{q}>1$
such that $1/\mathtt{p}+1/\mathtt{q}=1$ with

$\left[\frac{\wp\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\right]^{\mathtt{n}+1}\left[\int_{0}^{1}|\Upsilon(\mathtt{t})|\mathtt{d}\mathtt{t}\right]^{\mathtt{n}}\left[\int_{0}^{1}|\Upsilon(\mathtt{t})|^{\mathtt{q}}\mathtt{d}\mathtt{t}\right]^{\frac{1}{q}}<1,$
(24)

then the boundary value problem (5)–(6) has a unique positive radial solution
in $\mathtt{v}.$

###### Proof

Let $\mathtt{u}_{1},\mathtt{v}_{1}\in C([0,1])$ and $\mathtt{t}\in[0,1].$ Then
by Hölder’s inequality, we have

$\displaystyle\bigg{|}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})$
$\displaystyle\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}-\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{v}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\bigg{|}$
$\displaystyle\leq\int_{0}^{1}|\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\,\ell(\mathtt{t}_{\mathtt{n}})||\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}-\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{v}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}|\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\int_{0}^{1}|\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}},\mathtt{t}_{\mathtt{n}})\,\ell(\mathtt{t}_{\mathtt{n}})|\,\mathtt{K}|\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})-\mathtt{v}_{1}(\mathtt{t}_{\mathtt{n}})|\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\frac{\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\int_{0}^{1}|\Upsilon(\mathtt{t}_{\mathtt{n}})||\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})-\mathtt{v}_{1}(\mathtt{t}_{\mathtt{n}})|\mathtt{d}\mathtt{t}_{\mathtt{n}}$
$\displaystyle\leq\frac{\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\left[\int_{0}^{1}|\Upsilon(\mathtt{t}_{\mathtt{n}})|^{\mathtt{q}}\mathtt{d}\mathtt{t}_{\mathtt{n}}\right]^{\frac{1}{\mathtt{q}}}\left[\int_{0}^{1}|\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})-\mathtt{v}_{1}(\mathtt{t}_{\mathtt{n}})|^{\mathtt{p}}\mathtt{d}\mathtt{t}_{\mathtt{n}}\right]^{\frac{1}{\mathtt{p}}}$
$\displaystyle\leq\frac{\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\left[\int_{0}^{1}|\Upsilon(\mathtt{t}_{\mathtt{n}})|^{\mathtt{q}}\mathtt{d}\mathtt{t}_{\mathtt{n}}\right]^{\frac{1}{\mathtt{q}}}\uprho(\mathtt{u}_{1},\mathtt{v}_{1})$
$\displaystyle\leq\alpha_{1}^{\star}\uprho(\mathtt{u}_{1},\mathtt{v}_{1}),$

where

$\alpha_{1}^{\star}=\frac{\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\left[\int_{0}^{1}|\Upsilon(\mathtt{t}_{\mathtt{n}})|^{\mathtt{q}}\right]^{\frac{1}{\mathtt{q}}}.$

It follows in similar manner for $0<\mathtt{t}_{\mathtt{n}-2}<1,$

$\displaystyle\bigg{|}\int_{0}^{1}\Xi_{r_{0}}($
$\displaystyle\mathtt{t}_{\mathtt{n}-2},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{g}_{\mathtt{n}-1}\bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\bigg{]}\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\hskip
28.45274pt-\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-2},\mathtt{t}_{\mathtt{n}-1})\ell(\mathtt{t}_{\mathtt{n}-1})\mathtt{g}_{\mathtt{n}-1}\bigg{[}\int_{0}^{1}\Xi_{r_{0}}(\mathtt{t}_{\mathtt{n}-1},\mathtt{t}_{\mathtt{n}})\ell(\mathtt{t}_{\mathtt{n}})\mathtt{g}_{\mathtt{n}}\big{(}\mathtt{u}_{1}(\mathtt{t}_{\mathtt{n}})\big{)}\mathtt{d}\mathtt{t}_{\mathtt{n}}\bigg{]}\mathtt{d}\mathtt{t}_{\mathtt{n}-1}\bigg{|}$
$\displaystyle\leq\frac{\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\int_{0}^{1}|\Upsilon(\mathtt{t}_{\mathtt{n}-1})|\alpha_{1}\uprho(\mathtt{u}_{1},\mathtt{v}_{1})\mathtt{d}\mathtt{t}_{\mathtt{n}-1}$
$\displaystyle\leq\widehat{\alpha}_{1}{\alpha}_{1}^{\star}\uprho(\mathtt{u}_{1},\mathtt{v}_{1}),$

where

$\widehat{\alpha}_{1}=\frac{\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\int_{0}^{1}|\Upsilon(\mathtt{t})|\mathtt{d}\mathtt{t}.$

Continuing with bootstrapping argument, we get

$\left|\Lambda\mathtt{u}_{1}(\mathtt{t})-\Lambda\mathtt{v}_{1}(\mathtt{t})\right|\leq\widehat{\alpha}_{1}^{\mathtt{n}}\alpha_{1}^{\star}\uprho(\mathtt{u}_{1},\mathtt{v}_{1}).$

we see that

$\mathtt{d}(\Lambda\mathtt{u}_{1},\Lambda\mathtt{v}_{1})\leq\alpha_{1}\uprho(\mathtt{u}_{1},\mathtt{v}_{1}),$
(25)

for some
$\alpha_{1}=\widehat{\mathtt{c}}_{1}^{\mathtt{n}}\alpha_{1}^{\star}>0$ for all
$\mathtt{u}_{1},\mathtt{v}_{1}\in\mathtt{v},$ and so the inequality (22) of
Theorem 6.1 holds. Now, for all $\mathtt{u}_{1},\mathtt{v}_{1}\in\mathtt{v},$
we may apply (21) to (25) to obtain

$\mathtt{d}(\Lambda\mathtt{u}_{1},\Lambda\mathtt{v}_{1})\leq\alpha_{1}\uprho(\mathtt{u}_{1},\mathtt{v}_{1})\leq\alpha_{1}\mathtt{d}(\mathtt{u}_{1},\mathtt{v}_{1}).$

Thus, given any $\varepsilon>0$ we can choose $\upeta=\varepsilon/\alpha_{1}$
so that $\mathtt{d}(\Lambda\mathtt{u}_{1},\Lambda\mathtt{v}_{1})<\varepsilon,$
whenever $\mathtt{d}(\mathtt{u}_{1},\mathtt{v}_{1})<\upeta.$ Hence $\Lambda$
is continuous on $\mathtt{v}$ with respect to the metric $\mathtt{d}.$
Finally, we show that $\Lambda$ is contractive on $\mathtt{v}$ with respect to
the metric $\uprho.$ From (25), for each
$\mathtt{u}_{1},\mathtt{v}_{1}\in\mathtt{v}$ consider

$\displaystyle\bigg{[}\int_{0}^{1}|(\Lambda\mathtt{u}_{1})(\mathtt{t})-(\Lambda\mathtt{v}_{1})(\mathtt{t})|^{\mathtt{p}}\mathtt{d}\mathtt{t}\bigg{]}^{\frac{1}{\mathtt{p}}}$
$\displaystyle\leq\left[\int_{0}^{1}\left|\widehat{\mathtt{c}}_{1}^{\mathtt{n}}\alpha_{1}^{\star}\uprho(\mathtt{u}_{1},\mathtt{v}_{1})\right|^{\mathtt{p}}\mathtt{d}\mathtt{t}\right]^{\frac{1}{\mathtt{p}}}$
$\displaystyle\leq\,\left[\frac{\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\right]^{\mathtt{n}+1}\left[\int_{0}^{1}|\Upsilon(\mathtt{t})|\mathtt{d}\mathtt{t}\right]^{\mathtt{n}}\left[\int_{0}^{1}|\Upsilon(\mathtt{t})|^{\mathtt{q}}\mathtt{d}\mathtt{t}\right]^{\frac{1}{q}}\uprho(\mathtt{v}_{1},\mathtt{v}_{2}).$

That is

$\displaystyle\uprho(\Lambda\mathtt{u}_{1},\Lambda\mathtt{v}_{1})\leq\left[\frac{\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\right]^{\mathtt{n}+1}\left[\int_{0}^{1}|\Upsilon(\mathtt{t})|\mathtt{d}\mathtt{t}\right]^{\mathtt{n}}\left[\int_{0}^{1}|\Upsilon(\mathtt{t})|^{\mathtt{q}}\mathtt{d}\mathtt{t}\right]^{\frac{1}{q}}\uprho(\mathtt{v}_{1},\mathtt{v}_{2}).$

From the assumption (24), we have

$\uprho(\Lambda\mathtt{v}_{1},\Lambda\mathtt{v}_{2})\leq\alpha_{2}\uprho(\mathtt{v}_{1},\mathtt{v}_{2})$

for some $\alpha_{2}<1$ and all $\mathtt{v}_{1},\mathtt{v}_{2}\in\mathtt{v}.$
Thus, Theorem 6.1, the operator $\Lambda$ has a unique fixed point in
$\mathtt{v}.$ Also, we note that the operator $\Lambda$ is positive from Lemma
3. Therefore, the boundary value problem (2) has a unique positive radial
solution.∎

###### Example 4

Consider the following nonlinear elliptic system of equations,

$\triangle{\mathtt{u}_{{\dot{\iota}}}}+\frac{(\mathtt{N}-2)^{2}r_{0}^{2\mathtt{N}-2}}{|x|^{2\mathtt{N}-2}}\mathtt{u}_{\dot{\iota}}+\ell(|x|)\mathtt{g}_{{\dot{\iota}}}(\mathtt{u}_{{\dot{\iota}}+1})=0,~{}1<|x|<2,$
(26) $\left.\begin{aligned} &\hskip
25.6073pt\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}|x|=2,\\\
&\mathtt{u}_{{\dot{\iota}}}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}\text{on}~{}|x|=2,\\\ &\frac{\partial\mathtt{u}_{\dot{\iota}}}{\partial
r}=0~{}~{}\text{on}~{}~{}|x|=1~{}\text{and}~{}\mathtt{u}_{{\dot{\iota}}}=0~{}\text{on}~{}|x|=2,\end{aligned}\right\\}$
(27)

where $r_{0}=1,$ $\mathtt{N}=3,$
${\dot{\iota}}\in\\{1,2\\},\,\mathtt{u}_{3}=\mathtt{u}_{1},$
$\ell(\mathtt{s})=\frac{1}{\mathtt{s}^{4}}\prod_{i=1}^{2}\ell_{i}(\mathtt{s}),$
$\ell_{i}(\mathtt{s})=\ell_{i}\left(\frac{1}{\mathtt{s}}\right),$ in which
$\ell_{1}(t)=\ell_{2}(t)=\frac{1}{t+1},$ then $\prod_{i=1}^{2}\ell_{i}^{*}=1.$
Let
$\mathtt{g}_{1}(\mathtt{u})=\frac{1}{10^{4}}\cos(\mathtt{u}),\,\mathtt{g}_{2}(\mathtt{u})=\frac{\mathtt{u}}{10^{4}(\mathtt{u}+1)}$
and $\upalpha=\upbeta=\upgamma=\updelta=1,$ then
$\uprho=2\cosh(1)+2\sinh(1)\approx 5.436563658,$

$\displaystyle\Xi_{r_{0}}(\mathtt{s},\mathtt{t})=\frac{1}{2\cosh(1)+2\sinh(1)}\left\\{\begin{array}[]{ll}\big{(}\sinh(\mathtt{s})+\cosh(\mathtt{s})\big{)}\big{(}\sinh(1-\mathtt{t})+\cosh(1-\mathtt{t})\big{)},\hskip
17.07182pt0\leq\mathtt{s}\leq\mathtt{t}\leq 1,\vspace{1.2mm}\\\
\big{(}\sinh(\mathtt{t})+\cosh(\mathtt{t})\big{)}\big{(}\sinh(1-\mathtt{s})+\cosh(1-\mathtt{s})\big{)},\hskip
17.07182pt0\leq\mathtt{t}\leq\mathtt{s}\leq 1,\end{array}\right.$

and $\wp=\frac{3}{\cosh(1)}.$ Then,

$|\mathtt{g}_{1}(\mathtt{u})-\mathtt{g}_{1}(\mathtt{v})|=\frac{|\cos(\mathtt{u})-\cos(\mathtt{v})|}{10^{4}}\leq\frac{1}{10^{4}}|\mathtt{u}-\mathtt{v}|,$

and

$|\mathtt{g}_{2}(\mathtt{u})-\mathtt{g}_{2}(\mathtt{v})|=\frac{1}{10^{4}}\left|\frac{\mathtt{u}}{\mathtt{u}+1}-\frac{\mathtt{v}}{\mathtt{v}+1}\right|\leq\frac{1}{10^{4}}|\mathtt{u}-\mathtt{v}|.$

So, $\mathtt{K}=\frac{1}{10^{4}}.$ Let $\mathtt{n}=2$ and
$\mathtt{p}=\mathtt{q}=2.$ Then,

$\left[\frac{\mathtt{K}r_{0}^{2}}{(\mathtt{N}-2)^{2}}\right]^{\mathtt{n}+1}\left[\int_{0}^{1}|\Upsilon(\mathtt{t})|\mathtt{d}\mathtt{t}\right]^{\mathtt{n}}\left[\int_{0}^{1}|\Upsilon(\mathtt{t})|^{\mathtt{q}}\mathtt{d}\mathtt{t}\right]^{\frac{1}{q}}\approx
0.3149700790<1.$

Therefore, from Theorem 6.2, the iterative system of boundary value problems
(26)–(27) has a unique positive radial solution.

Acknowledgement

Author contributions The study was carried out in collaboration of all
authors. All authors read and approved the final manuscript.

Funding Not Applicable.

Data availibility statement Data sharing not applicable to this paper as no
data sets were generated or analyzed during the current study.

## Compliance with ethical standards

Conflict of interest It is declared that authors has no competing interests.

Ethical approval This article does not contain any studies with human
participants or animals performed by any of the authors.

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