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

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.

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

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

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

Abstract

Keywords:

MSC:

1 Introduction

2 Kernel and Its Bounds

Lemma 1

Lemma 2

Proof

Lemma 3

Proof

3 Existence of at Least One Positive Radial Solution

Theorem 3.1

Theorem 3.2

Theorem 3.3

Proof

Theorem 3.4

Theorem 3.5

Example 1

4 Existence of at Least Two Positive Radial Solutions

Theorem 4.1

Theorem 4.2

Proof

Theorem 4.3

Theorem 4.4

Example 2

5 Existence of at Least Three Positive Radial Solutions

Theorem 5.1

Theorem 5.2

Proof

Theorem 5.3

Theorem 5.4

Example 3

6 Existence of Unique Positive Radial Solution

Theorem 6.1 (Rus rus )

Theorem 6.2

Proof

Example 4

Compliance with ethical standards

References