Abstract
This paper deals with the existence of positive radial solutions to the iterative system of nonlinear elliptic equations of the form
where \(\mathtt {j}\in \{1,2,3,\cdot \cdot \cdot ,\ell \},\) \( \mathtt {z}_1= \mathtt {z}_{\ell +1},\) \(\triangle {\mathtt {z}}=\mathtt {div}(\triangledown \mathtt {z}),\) \(\mathtt {N}>2,\) \(0<r_0<\uppi /2,\) \(\varphi =\prod _{i=1}^{n}\varphi _i,\) each \(\varphi _i:(r_0,+\infty )\rightarrow (0,+\infty )\) is continuous, \(r^{\mathtt {N}-1}\varphi \) is integrable, and \(\mathtt {g}_\mathtt {j}:[0,+\infty )\rightarrow {\mathbb {R}}\) is continuous, by an application of various fixed point theorems in a Banach space. Further, we also establish uniqueness of the solution for the addressed system by using Rus’s theorem in a complete metric space.
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1 Introduction
The semilinear elliptic equation of the form
arises 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 [9, 10, 19, 24]. Study of nonlinear elliptic system of equations,
where \(\mathtt {j}\in \{1,2,3,\cdot \cdot \cdot ,{\ell }\},\) \( \mathtt {z}_1= \mathtt {z}_{\ell +1},\) and \(\Omega \) is a bounded domain in \({\mathbb {R}}^N\), has an important applications in population dynamics, combustion theory and chemical reactor theory. For the recent literature for the existence, multiplicity and uniqueness of positive solutions for (2), see [1, 3, 7, 12,13,14] and references therein.
In [6], 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,
for any dimension \(\mathtt {d}\ge 1.\) In [8], Dong and Wei established the existence of radial solutions for the following nonlinear elliptic equations with gradient terms in annular domains,
by using Schauder’s fixed point theorem and contraction mapping theorem. In [15], R. Kajikiya and E. Ko established the existence of positive radial solutions for a semipositone elliptic equation of the form,
where \(\Omega \) is a ball or an annulus in \({\mathbb {R}}^\mathtt {N}.\) Recently, Son and Wang [22] established positive radial solutions to the nonlinear elliptic systems,
where \(\mathtt {j}\in \{1,2,3,\cdot \cdot \cdot ,{\ell }\},\) \( \mathtt {z}_1= \mathtt {z}_{\ell +1},\) \(\uplambda >0,\) \(N>2,\) \(r_0>0,\) 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,
with one of the following sets of boundary conditions:
where \(\mathtt {j}\in \{1,2,3,\cdot \cdot \cdot ,\ell \},\) \( \mathtt {z}_1= \mathtt {z}_{\ell +1},\) \(\triangle \mathtt {z}=\mathtt {div}(\triangledown \mathtt {z}),\) \(N>2,\) \(0<r_0<\uppi /2,\) \(\varphi =\prod _{i=1}^{n}\varphi _i,\) each \(\varphi _i:(\mathtt {R}_1,\mathtt {R}_2)\rightarrow (0,+\infty )\) is continuous, \(r^{\mathtt {N}-1}\varphi \) 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,
where \(\mathtt {j}\in \{1,2,3,\cdot \cdot \cdot ,\ell \},\) \( \mathtt {z}_1= \mathtt {z}_{\ell +1},\) \(0<r_0<\uppi /2,\) and \(\varphi (\uptau )=\frac{r_0^2}{(\mathtt {N}-2)^2}\uptau ^\frac{2(\mathtt {N}-1)}{2-\mathtt {N}}\prod _{i=1}^{n}\varphi _i(\uptau ),\) \(\varphi _i(\uptau )=\varphi _i(r_0\uptau ^\frac{1}{2-\mathtt {N}})\) by a Kelvin-type transformation through the change of variables \(r=\vert x\vert \) and \(\uptau =\left( \frac{r}{r_0}\right) ^{2-\mathtt {N}}.\) For the detailed explanation of the transformation from equations (7) to (5), see [2, 16, 17]. By suitable choices of nonnegative real numbers \(\upalpha , \upbeta , \upgamma \) and \(\updelta \) with \(r_0^2\le \frac{\upalpha \upgamma }{\upbeta \updelta },\) the set of boundary conditions (5) reduces to
we assume that the following conditions hold throughout the paper:
- \(({\mathcal {H}}_1)\):
-
\(\mathtt {g}_\mathtt {j}:[0,+\infty )\rightarrow [0,+\infty )\) is continuous.
- \(({\mathcal {H}}_2)\):
-
\(\varphi _i\in L^{\mathtt {p}_i}[0,1], 1\le \mathtt {p}_i\le +\infty \) for \(1\le i\le n.\)
- \(({\mathcal {H}}_3)\):
-
There exists \(\varphi _i^\star >0\) such that \(\varphi _i^\star<\varphi _i(\uptau )<\infty \) a.e. on [0, 1].
The rest of the paper is organized in the following fashion. In Sect. 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 Sect. 3, we develop criteria for the existence of at least one positive radial solution by applying Krasnoselskii’s cone fixed point theorem in a Banach space. In Sect. 4, we derive necessary conditions for the existence of at least two positive radial solution by an application of Avery–Henderson cone fixed point theorem in a Banach space. In Sect. 5, we establish the existence of at least three positive radial solution by utilizing Leggett-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,
where \(y\in {\mathcal {C}}[0, 1]\) is a given function.
Lemma 1
Let \(\wp =r_0^2(\upalpha \updelta +\upbeta \upgamma )\cos (r_0)+r_0(\upalpha \upgamma -\upbeta \updelta r_0^2)\sin (r_0).\) For every \(y\in {\mathcal {C}}(0, 1),\) the linear boundary value problem (7)–(8) has a unique solution
where
Lemma 2
Let \(\displaystyle \upsigma =\max \left\{ \frac{\upalpha +\upbeta r_0}{\upbeta r_0\cos (r_0)},\frac{\upgamma +\updelta r_0}{\updelta r_0\cos (r_0)}\right\} .\) The kernel \(\aleph _{r_0}(\uptau ,\mathtt {s})\) has the following properties:
-
(i)
\(\aleph _{r_0}(\uptau ,\mathtt {s})\) is nonnegative and continuous on \([0, 1] \times [0, 1],\)
-
(ii)
\(\aleph _{r_0}(\uptau ,\mathtt {s})\le \upsigma \aleph _{r_0}(\mathtt {s},\mathtt {s})\) for \(\uptau ,\mathtt {s}\in [0, 1],\)
-
(iii)
\(\frac{1}{\upsigma } \aleph _{r_0}(\mathtt {s},\mathtt {s})\le \aleph _{r_0}(\uptau ,\mathtt {s})\) for \(\uptau , \mathtt {s}\in [0, 1].\)
Proof
Since \(r_0^2\le \frac{\upalpha \upgamma }{\upbeta \updelta },\) it follows that \(\wp >0.\) So, from the definition of kernel, \(\aleph _{r_0}(\mathtt {s},\mathtt {s})>0\) and continuous on \([0,1]\times [0,1].\) This proves (i). To prove (ii), consider
which proves (ii). Finally for (iii), consider
This completes the proof. \(\square \)
From Lemma 1, we note that an \(\ell \)-tuple \(( \mathtt {z}_1, \mathtt {z}_2,\cdot \cdot \cdot , \mathtt {z}_\ell )\) is a solution of the boundary value problem (5)–(6) if and only, if
In general,
We denote the Banach space \(\mathtt {C}((0, 1),{\mathbb {R}})\) by \(\mathtt {B}\) with the norm \(\Vert \mathtt {z}\Vert =\displaystyle \max _{\uptau \in [0,1]}\vert \mathtt {z}(\uptau )\vert .\) The cone \(\mathtt {E} \subset \mathtt {B}\) is defined by
For any \( \mathtt {z}_1\in \mathtt {E},\) define an operator \({\mathcal {P}}:\mathtt {E}\rightarrow \mathtt {B}\) by
Lemma 3
\({\mathcal {P}}(\mathtt {E})\subset \mathtt {E}\) and \({\mathcal {P}}:\mathtt {E}\rightarrow \mathtt {E}\) is completely continuous.
Proof
Since \(\mathtt {g}_\mathtt {j}( \mathtt {z}_{\mathtt {j}+1}(\uptau ))\) is nonnegative for \(\uptau \in [0, 1],\) \( \mathtt {z}_1 \in \mathtt {E}.\) Since \(\aleph _{r_0}(\uptau , s),\) is nonnegative for all \(\uptau , \mathtt {s} \in [0, 1],\) it follows that \({\mathcal {P}}( \mathtt {z}_1(\uptau ))\ge 0\) for all \(\uptau \in [0, 1],\, \mathtt {z}_1 \in \mathtt {E}\) Now, by Lemmas 1 and 2, we have
Thus, \({\mathcal {P}}(\mathtt {E})\subset \mathtt {E}.\) Therefore, by the means of Arzela–Ascoli theorem, the operator \({\mathcal {P}}\) is completely continuous. \(\square \)
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 1
[11] 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 \({\mathcal {P}}:\mathtt {E}\cap (\overline{\mathtt {F}}\backslash \mathtt {G})\rightarrow \mathtt {E}\) be a completely continuous operator such that
-
(i)
\(\Vert {\mathcal {P}} \mathtt {z}\Vert \le \Vert \mathtt {z}\Vert ,\, \mathtt {z}\in \mathtt {E}\cap \partial \mathtt {G},\) and \(\Vert {\mathcal {P}} \mathtt {z}\Vert \ge \Vert \mathtt {z}\Vert ,\, \mathtt {z}\in \mathtt {E}\cap \partial \mathtt {F},\) or
-
(ii)
\(\Vert {\mathcal {P}} \mathtt {z}\Vert \ge \Vert \mathtt {z}\Vert ,\, \mathtt {z}\in \mathtt {E}\cap \partial \mathtt {G},\) and \(\Vert {\mathcal {P}} \mathtt {z}\Vert \le \Vert \mathtt {z}\Vert ,\, \mathtt {z}\in \mathtt {E}\cap \partial \mathtt {F}.\)
Then, \({\mathcal {P}}\) has a fixed point in \(\mathtt {E}\cap (\overline{\mathtt {F}}\backslash \mathtt {G}).\)
Theorem 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 \(\displaystyle \sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}=1.\) Then, \(\prod _{i=1}^{n}\mathtt {f}_i\in L^{1}[0, 1]\) and \(\left\| \prod _{i=1}^{n}\mathtt {f}_i\right\| _1\le \prod _{i=1}^{n}\Vert \mathtt {f}_i\Vert _{\mathtt {p}_i}.\) 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 \(\Vert \mathtt {fg}\Vert _1\le \Vert \mathtt {f}\Vert _1\Vert \mathtt {g}\Vert _\infty .\)
Consider the following three possible cases for \(\varphi _i\in L^{\mathtt {p}_i}[0,1]:\)
Firstly, we seek positive radial solutions for the case \(\displaystyle \sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}<1.\)
Theorem 3
Suppose \(({\mathcal {H}}_1)\)–\(({\mathcal {H}}_3)\) hold. Further, assume that there exist two positive constants \(a_2>a_1>0\) such that
- \(({\mathcal {H}}_4)\):
-
\(\mathtt {g}_\mathtt {j}( \mathtt {z}(\uptau ))\le \mathtt {Q}_2a_2\) for all \(0\le \uptau \le 1,\, 0\le \mathtt {z} \le a_2,\) where \(\displaystyle \mathtt {Q}_2=\left[ \frac{\upsigma r_0^2}{(\mathtt {N}-2)^2}\Vert {\widehat{\aleph }}_{r_0}\Vert _\mathtt {q} \prod _{i=1}^{n}\Vert \varphi _i\Vert _{\mathtt {p}_i}\right] ^{-1}\) and \({\widehat{\aleph }}_{r_0}(\mathtt {s})=\aleph _{r_0}(\mathtt {s}, \mathtt {s})\mathtt {s}^\frac{2(\mathtt {N}-1)}{2-\mathtt {N}}.\)
- \(({\mathcal {H}}_5)\):
-
\(\mathtt {g}_\mathtt {j}( \mathtt {z}(\uptau ))\ge \mathtt {Q}_1a_1\) for all \(0\le \uptau \le 1,\, 0\le \mathtt {z} \le a_1,\) where \(\displaystyle \mathtt {Q}_1=\left[ \frac{r_0^2}{\upsigma (\mathtt {N}-2)^2} \prod _{i=1}^{n}\varphi _i^\star \int _{0}^{1}\aleph _{r_0}(\mathtt {s}, \mathtt {s})\mathtt {s}^\frac{2(\mathtt {N}-1)}{2-\mathtt {N}} d\mathtt {s}\right] ^{-1}.\)
Then, iterative system (5)–(6) has at least one positive radial solution \((\mathtt {z}_1,\mathtt {z}_2,\cdot \cdot \cdot ,\mathtt {z}_\ell )\) such that \(a_1\le \Vert \mathtt {z}_\mathtt {j}\Vert \le a_2,\,\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell .\)
Proof
Let \(\mathtt {G}=\{\mathtt {z}\in \mathtt {B}:\Vert \mathtt {z}\Vert <a_2\}.\) For \(\mathtt {z}_1\in \partial \mathtt {G},\) we have \(0\le \mathtt {z}\le a_2\) for all \(\uptau \in [0,1].\) It follows from \(({\mathcal {H}}_4)\) that for \(\mathtt {s}_{\ell -1}\in [0,1],\)
There exists a \(\mathtt {q}>1\) such that \(\displaystyle \sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}+\frac{1}{\mathtt {q}}=1.\) By the first part of Theorem 2, we have
It follows in similar manner for \(0<\mathtt {s}_{\ell -2}<1,\)
Continuing with this bootstrapping argument, we reach
Since \(\mathtt {G}=\Vert \mathtt {z}_1\Vert \) for \( \mathtt {z}_1\in \mathtt {E}\cap \partial {\mathtt {G}},\) we get
Next, let \(\mathtt {F}=\{\mathtt {z}\in \mathtt {B}:\Vert \mathtt {z}\Vert <a_1\}.\) For \(\mathtt {z}_1\in \partial \mathtt {F},\) we have \(0\le \mathtt {z}\le a_1\) for all \(\uptau \in [0,1].\) It follows from \(({\mathcal {H}}_5)\) that for \(\mathtt {s}_{\ell -1}\in [0,1],\)
It follows in similar manner for \(0<\mathtt {s}_{\ell -2}<1,\)
Continuing with bootstrapping argument, we get
Thus, for \( \mathtt {z}_1\in \mathtt {E}\cap \partial \mathtt {F},\) we have
It is clear that \(0\in \mathtt {F}\subset \overline{\mathtt {F}}\subset \mathtt {G}\) and by Lemma 3, \({\mathcal {P}}:\mathtt {E}\cap (\overline{\mathtt {F}}\backslash \mathtt {G})\rightarrow \mathtt {E}\) is completely continuous operator. Also from (11) and (12) that \({\mathcal {P}}\) satisfies (i) of Theorem 1. Hence, from Theorem 1, \({\mathcal {P}}\) has a fixed point \( \mathtt {z}_1\in \mathtt {E}\cap \big (\overline{\mathtt {F}}\backslash \mathtt {G}\big )\) such that \( \mathtt {z}_1(\uptau )\ge 0\) on (0, 1). Next setting \( \mathtt {z}_{\ell +1}= \mathtt {z}_1,\) we obtain infinitely many positive solutions \(( \mathtt {z}_1, \mathtt {z}_2,\cdot \cdot \cdot ,\mathtt {z}_\ell )\) of (5)–(6) given iteratively by
This completes the proof.\(\square \)
For \(\sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}=1\) and \(\sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}>1,\) we have following results.
Theorem 4
Suppose \(({\mathcal {H}}_1)\)–\(({\mathcal {H}}_3)\) hold. Further, assume that there exist two positive constants \(b_2>b_1>0\) such that \(\mathtt {g}_\mathtt {j}\,(\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell )\) satisfies \(({\mathcal {H}}_5)\) and
- \(({\mathcal {H}}_6)\):
-
\({\mathtt {g}_\mathtt {j}( \mathtt {z}(\uptau ))\le {\mathfrak {N}}_2b_2}\) for all \(0\le \uptau \le 1,\, 0\le \mathtt {z} \le b_2,\) where \({\displaystyle {\mathfrak {N}}_2=\left[ \frac{\upsigma r_0^2}{(\mathtt {N}-2)^2}\Vert {\widehat{\aleph }}_{r_0}\Vert _\infty \prod _ {i=1}^{n}\Vert \varphi _i\Vert _{\mathtt {p}_i}\right] ^{-1}}\) and \({\widehat{\aleph }}_{r_0}(\mathtt {s})=\aleph _{r_0}(\mathtt {s}, \mathtt {s})\mathtt {s}^\frac{2(\mathtt {N}-1)}{2-\mathtt {N}}.\)
Then, iterative system (5)–(6) has at least one positive radial solution \((\mathtt {z}_1,\mathtt {z}_2,\cdot \cdot \cdot ,\mathtt {z}_\ell )\) such that \(b_1\le \Vert \mathtt {z}_\mathtt {j}\Vert \le b_2,\,\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell .\)
Theorem 5
Suppose \(({\mathcal {H}}_1)\)–\(({\mathcal {H}}_3)\) hold. Further, assume that there exist two positive constants \(c_2>c_1>0\) such that \(\mathtt {g}_\mathtt {j}\,(\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell )\) satisfies \(({\mathcal {H}}_5)\) and
- \(({\mathcal {H}}_7)\):
-
\({\mathtt {g}_\mathtt {j}( \mathtt {z}(\uptau ))\le {\mathfrak {M}}_2c_2}\) for all \(0\le \uptau \le 1,\, 0\le \mathtt {z} \le c_2,\) where \({\displaystyle {\mathfrak {M}}_2=\left[ \frac{\upsigma r_0^2}{(\mathtt {N}-2)^2}\Vert {\widehat{\aleph }}_{r_0}\Vert _\infty \prod _{i=1}^ {n}\Vert \varphi _i\Vert _1\right] ^{-1}}\) and \({\widehat{\aleph }}_{r_0}(\mathtt {s})=\aleph _{r_0}(\mathtt {s}, \mathtt {s})\mathtt {s}^\frac{2(\mathtt {N}-1)}{2-\mathtt {N}}.\)
Then, iterative system (5)–(6) has at least one positive radial solution \((\mathtt {z}_1,\mathtt {z}_2,\cdot \cdot \cdot ,\mathtt {z}_\ell )\) such that \(c_1\le \Vert \mathtt {z}_\mathtt {j}\Vert \le c_2,\,\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell .\)
Example 1
Consider the following nonlinear elliptic system of equations,
where \( r_0=1,\) \(\mathtt {N}=3,\) \(\mathtt {j}\in \{1,2\},\, \mathtt {z}_3= \mathtt {z}_1,\) \(\varphi (\uptau )=\frac{1}{\uptau ^4}\prod _{i=1}^{2}\varphi _i(\uptau ),\) \(\varphi _i(\uptau )=\varphi _i\left( \frac{1}{\uptau }\right) ,\) in which
then it is clear that
Let \(\mathtt {g}_1(\mathtt {z})=1+\frac{1}{3}\sin (1+\mathtt {z})+\frac{1}{1+\mathtt {z}},\, \mathtt {g}_2(\mathtt {z})=1+\frac{2}{5}\cos (\sqrt{1+\mathtt {z}})+\frac{1}{1+\mathtt {z}^2}.\) Let \(\upalpha =\upbeta =\gamma =1,\updelta =\frac{1}{2},\) then \(1=r_0^2<2=\frac{\upalpha \upgamma }{\upbeta \updelta },\) \(\wp =\frac{3}{2}\cos (1)+\frac{1}{2}\sin (1)\approx 1.231188951,\)
and \(\upsigma =\frac{3}{\cos (1)}.\) Also,
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
Choose \(a_1=0.5\) and \(a_2=10^6.\) Then,
and
Therefore, by Theorem 3, the boundary value problem (13)–(14) has at least one positive solution \((\mathtt {z}_1,\mathtt {z}_2)\) such that \(0.5\le \Vert \mathtt {z}_\mathtt {j}\Vert \le 10^6\) for \(\mathtt {j}=1,2.\)
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 \(a'\) and \(c',\) we define the sets
and
Theorem 6
(Avery–Henderson [5]) 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 \(c'\) and k, \(\ss _2(\mathtt {z})\le \ss _3(\mathtt {z})\le \ss _1(\mathtt {z})\) and \(\Vert \mathtt {z}\Vert \le k\ss _2(\mathtt {z}),\) for all \(\mathtt {z}\in \overline{\mathtt {E}(\ss _2,c')}.\) Suppose that there exist positive numbers \(a'\) and \(b'\) with \(a'<b'<c'\) such that \(\ss _3(\uplambda \mathtt {z})\le \uplambda \ss _3(\mathtt {z}),\) for all \(0\le \uplambda \le 1\) and \(\mathtt {z}\in \partial \mathtt {E}(\ss _3,b').\) Further, let \({\mathcal {P}}:\overline{\mathtt {E}(\ss _2,c')}\rightarrow \mathtt {E}\) be a completely continuous operator such that
- (a):
-
\(\ss _2({\mathcal {P}}\mathtt {z})>c',\) for all \(\mathtt {z}\in \partial \mathtt {E}(\ss _2,c'),\)
- (b):
-
\(\ss _3({\mathcal {P}}\mathtt {z})<b',\) for all \(\mathtt {z}\in \partial \mathtt {E}(\ss _3,b'),\)
- (c):
-
\(\mathtt {E}(\ss _1,a')\ne \emptyset \) and \(\ss _1({\mathcal {P}}\mathtt {z})>a',\) for all \(\partial \mathtt {E}(\ss _1,a').\)
Then, \({\mathcal {P}}\) has at least two fixed points \({}^1\mathtt {z},{}^2\mathtt {z}\in \mathtt {P}(\ss _2,c')\) such that \(a'<\ss _1({}^1\mathtt {z})\) with \(\ss _3({}^1\mathtt {z})<b'\) and \(b'<\ss _3({}^2\mathtt {z})\) with \(\ss _2({}^2\mathtt {z})<c'.\)
Define the nonnegative, increasing, continuous functional \(\ss _2, \ss _3,\) and \(\ss _1\) by
It is obvious that for each \(\mathtt {z}\in \mathtt {E},\)
In addition, by Lemma 1, for each \(\mathtt {z}\in \mathtt {P},\)
Thus,
Finally, we also note that
Theorem 7
Assume that \(({\mathcal {H}}_1)\)–\(({\mathcal {H}}_3)\) hold and Suppose there exist real numbers \(a', b'\) and \(c'\) with \(0<a'<b'<c'\) such that \(\mathtt {g}_\mathtt {j}(\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell )\) satisfies
- \(({\mathcal {H}}_{8})\):
-
\(\mathtt {g}_\mathtt {j}(\mathtt {z})>\frac{ c'}{\complement _1}\), for all \(c'\le \mathtt {z}\le \upsigma ^2c',\) where \(\complement _1=\frac{r_0^2}{\upsigma (\mathtt {N}-2)^2}\prod _{i=1}^{n}\varphi _i^\star \int _{0}^{1}\aleph _{r_0}(\mathtt {s}, \mathtt {s})\mathtt {s}^\frac{2(\mathtt {N}-1)}{2-\mathtt {N}} d\mathtt {s},\)
- \(({\mathcal {H}}_{9})\):
-
\(\mathtt {g}_\mathtt {j}(\mathtt {z})<\frac{b'}{\complement _2}\), for all \(0\le \mathtt {z}\le \upsigma ^2b',\) where \(\complement _2=\frac{\upsigma r_0^2}{(\mathtt {N}-2)^2}\Vert {\widehat{\aleph }}_{r_0}\Vert _\mathtt {q}\prod _{i=1}^{n}\Vert \varphi _i\Vert _{\mathtt {p}_i},\)
- \(({\mathcal {H}}_{10})\):
-
\(\mathtt {g}_\mathtt {j}(\mathtt {z})>\frac{a'}{\complement _1}\), for all \(a'\le \mathtt {z}\le \upsigma ^2a'.\)
Then, the boundary value problem (5)–(6) has at least two positive radial solutions \(\{({}^1\mathtt {z}_1,{}^1\mathtt {z}_2,\cdot \cdot \cdot ,{}^1\mathtt {z}_\ell )\}\) and \(\{({}^2\mathtt {z}_1,{}^2\mathtt {z}_2,\cdot \cdot \cdot ,{}^2\mathtt {z}_\ell )\}\) satisfying
and
Proof
We begin by defining the completely continuous operator \({\mathcal {P}}\) by (10). So it is easy to check that \({\mathcal {P}}:\overline{\mathtt {E}(\ss _2,c')}\rightarrow \mathtt {E}.\) Firstly, we shall verify that condition (a) of Theorem 6 is satisfied. So, let us choose \(\mathtt {z}_1\in \partial \mathtt {E}(\ss _2,c').\) Then, \(\ss _2(\mathtt {z}_1)=\min _{\uptau \in [0,1]}\mathtt {z}_1(\uptau )=c'\) this implies that \(c'\le \mathtt {z}_1(\uptau )\) for \(\uptau \in [0,1].\) Since \(\Vert \mathtt {z}_1\Vert \le \upsigma ^2\ss _2(\mathtt {z}_1)=\upsigma ^2c'.\) So we have
Let \(\mathtt {s}_{\ell -1}\in [0,1].\) Then, by \(({\mathcal {H}}_{8}),\) we have
It follows in similar manner for \(0<\mathtt {s}_{\ell -2}<1,\)
Continuing with bootstrapping argument, we get
This proves (i) of Theorem 6. We next address (ii) of Theorem 6. So, we choose \(\mathtt {z}_1\in \partial \mathtt {E}(\ss _3,b').\) Then, \(\ss _3(\mathtt {z}_1)=\max _{\uptau \in [0,1]}\mathtt {z}_1(\uptau )=b'\) this implies that \(0\le \mathtt {z}_1(\uptau )\le b'\) for \(\uptau \in [0,1].\) Since \(\Vert \mathtt {z}_1\Vert \le \upsigma ^2\ss _2(\mathtt {z}_1)\le \upsigma ^2\ss _3(\mathtt {z}_1)=\upsigma ^2 b'.\) So we have
Let \(0<\mathtt {s}_{\ell -1}<1.\) Then, by \(({\mathcal {H}}_{9}),\) we have
There exists a \(\mathtt {q}>1\) such that \(\displaystyle \sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}+\frac{1}{\mathtt {q}}=1.\) By the first part of Theorem 2, we have
Continuing with this bootstrapping argument, we get
Hence, condition (b) is satisfied. Finally, we verify that (c) of Theorem 6 is also satisfied. We note that \(\mathtt {z}_1(\uptau )=a'/4,~\uptau \in [0,1]\) is a member of \(\mathtt {E}(\ss _1,a')\) and \(a'/4<a'.\) So \(\mathtt {E}(\ss _1,a')\ne \emptyset .\) Now let \(\mathtt {z}_1\in \mathtt {E}(\ss _1,a').\) Then, \(a'=\ss _1(\mathtt {z}_1)=\max _{\uptau \in [0,1]}\mathtt {z}_1(\uptau )=\Vert \mathtt {z}_1\Vert =\upsigma ^2\ss _2(\mathtt {z}_1)\le \upsigma ^2\ss _3(\mathtt {z}_1)=\upsigma ^2\ss _1(\mathtt {z}_1)=\upsigma ^2a',\) i.e., \(a'\le \mathtt {z}_1(\uptau )\le \upsigma ^2 a'\) for \(\uptau \in [0,1].\) Let \(0<\mathtt {s}_{\ell -1}<1.\) Then, by \(({\mathcal {H}}_{10}),\) we have
Continuing with this bootstrapping argument, we get
Thus, condition (c) of Theorem 6 is satisfied. Since all hypotheses of Theorem 6 are satisfied, the assertion follows.\(\square \)
For \(\sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}=1\) and \(\sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}>1,\) we have following results.
Theorem 8
Assume that \(({\mathcal {H}}_1)\)–\(({\mathcal {H}}_3)\) hold and Suppose there exist real numbers \(a', b'\) and \(c'\) with \(0<a'<b'<c'\) such that \(\mathtt {g}_\mathtt {j}(\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell )\) satisfies \(({\mathcal {H}}_{8}),\) \(({\mathcal {H}}_{10})\) and
- \(({\mathcal {H}}_{9}')\):
-
\(\mathtt {g}_\mathtt {j}(\mathtt {z})<\frac{b'}{\complement _3}\), for all \(0\le \mathtt {z}\le \upsigma ^2b',\) where \(\complement _3=\frac{\upsigma r_0^2}{(\mathtt {N}-2)^2}\Vert {\widehat{\aleph }}_{r_0}\Vert _\infty \prod _{i=1}^{n}\Vert \varphi _i\Vert _{\mathtt {p}_i}.\)
Then, the boundary value problem (5)–(6) has at least two positive radial solutions \(\{({}^1\mathtt {z}_1,{}^1\mathtt {z}_2,\cdot \cdot \cdot ,{}^1\mathtt {z}_\ell )\}\) and \(\{({}^2\mathtt {z}_1,{}^2\mathtt {z}_2,\cdot \cdot \cdot ,{}^2\mathtt {z}_\ell )\}\) satisfying
and
Theorem 9
Assume that \(({\mathcal {H}}_1)\)–\(({\mathcal {H}}_3)\) hold and Suppose there exist real numbers \(a', b'\) and \(c'\) with \(0<a'<b'<c'\) such that \(\mathtt {g}_\mathtt {j}(\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell )\) satisfies \(({\mathcal {H}}_{8}),\) \(({\mathcal {H}}_{10})\) and
- \(({\mathcal {H}}_{9}'')\):
-
\(\mathtt {g}_\mathtt {j}(\mathtt {z})<\frac{b'}{\complement _4}\), for all \(0\le \mathtt {z}\le \upsigma ^2b',\) where \(\complement _4=\frac{\upsigma r_0^2}{(\mathtt {N}-2)^2}\Vert {\widehat{\aleph }}_{r_0}\Vert _\infty \prod _{i=1}^{n}\Vert \varphi _i\Vert _{1}.\)
Then, the boundary value problem (5)–(6) has at least two positive radial solutions \(\{({}^1\mathtt {z}_1,{}^1\mathtt {z}_2,\cdot \cdot \cdot ,{}^1\mathtt {z}_\ell )\}\) and \(\{({}^2\mathtt {z}_1,{}^2\mathtt {z}_2,\cdot \cdot \cdot ,{}^2\mathtt {z}_\ell )\}\) satisfying
and
Example 2
Consider the following nonlinear elliptic system of equations,
where \( r_0=1,\) \(\mathtt {N}=3,\) \(\mathtt {j}\in \{1,2\},\, \mathtt {z}_3= \mathtt {z}_1,\) \(\varphi (\uptau )=\frac{1}{\uptau ^4}\prod _{i=1}^{2}\varphi _i(\uptau ),\) \(\varphi _i(\uptau )=\varphi _i\left( \frac{1}{\uptau }\right) ,\) in which
then it is clear that
Let \(\mathtt {g}_1(\mathtt {z})= \mathtt {g}_2(\mathtt {z})=1+\frac{1}{\sqrt{1+\mathtt {z}^2}}.\) Let \(\upalpha =\updelta =\gamma =1,\upbeta =\frac{1}{2},\) then \(1=r_0^2<2=\frac{\upalpha \upgamma }{\upbeta \updelta },\) \(\wp =\frac{3}{2}\cos (1)+\frac{1}{2}\sin (1)\approx 1.231188951,\)
and \(\upsigma =\frac{3}{\cos (1)}.\) Also,
Let \(\mathtt {p}_1=6,\mathtt {p}_2=2\) and \(\mathtt {q}=3,\) then \(\frac{1}{\mathtt {p}_1}+\frac{1}{\mathtt {p}_2}+\frac{1}{\mathtt {q}}=1\) and
Choose \(a'=10^3,\,b'=2\times 10^{7}\) and \(c'=10^{8}.\) Then,
Therefore, by Theorem 3, the boundary value problem (15)–(16) has at least two positive radial solutions \(({}^\mathtt {j}\mathtt {z}_1,{}^\mathtt {j}\mathtt {z}_2),\,\mathtt {j}=1,2\) such that
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 Leggett-William fixed point theorem. Let \(a',b'\) be two real numbers such that \(0<a'<b'\) and \(\Bbbk \) a nonnegative, continuous, concave functional on \(\mathtt {E}.\) We define the following convex sets,
Theorem 10
(Leggett-William [18]) Let \(\mathtt {E}\) be a cone in a Banach space \(\mathtt {B}.\) Let \(\Bbbk \) a nonnegative, continuous, concave functional on \(\mathtt {E}\) satisfying for some \(c'>0\) such that \(\Bbbk (\mathtt {z})\le \Vert \mathtt {z}\Vert \) for all \(\mathtt {z}\in \overline{\mathtt {E}}_{c'}.\) Suppose there exists a completely continuous operator \({\mathcal {P}}:\overline{\mathtt {E}}_{c'}\rightarrow \overline{\mathtt {E}}_{c'}\) and \(0<a'<b'<d'\le c'\) such that
- (a):
-
\(\{\mathtt {z}\in \mathtt {E}(\Bbbk ,b',d'):\Bbbk (\mathtt {z})>a'\}\ne \emptyset \) and \(\Bbbk ({\mathcal {P}}\mathtt {z})>b'\) for \(\mathtt {z}\in \mathtt {E}(\Bbbk ,b',d'),\)
- (b):
-
\(\Vert {\mathcal {P}}\mathtt {z}\Vert <a'\) for \(\Vert \mathtt {z}\Vert <a',\)
- (c):
-
\(\Bbbk ({\mathcal {P}}\mathtt {z})>b'\) for \(\mathtt {z}\in \mathtt {E}(\Bbbk ,a',c'),\) with \(\Vert {\mathcal {P}}\mathtt {z}\Vert >d'\)
Then, \({\mathcal {P}}\) has at least three fixed points \({}^1\mathtt {z},{}^2\mathtt {z},{}^3\mathtt {z}\in \mathtt {E}_{c'}\) satisfying \(\Vert {}^1\mathtt {z}\Vert <a',\) \(b'<\Bbbk ({}^2\mathtt {z})\) and \(\Vert {}^3\mathtt {z}\Vert >a'\) and \(\Bbbk ({}^3\mathtt {z})<b'.\)
Theorem 11
Assume that \(({\mathcal {H}}_1)\)–\(({\mathcal {H}}_3)\) hold. Let \(0<a'<b'<c'\) and suppose that \(\mathtt {g}_\mathtt {j},\,\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell \) satisfies the following conditions,
- \(({\mathcal {H}}_{11})\):
-
\({\mathtt {g}_\mathtt {j}(\mathtt {z})<\frac{a'}{{\mathfrak {O}}_1}}\) for \(0\le \mathtt {z}\le a',\) where \({{\mathfrak {O}}_1=\frac{\upsigma r_0^2}{(\mathtt {N}-2)^2}\Vert {\widehat{\aleph }}_{r_0}\Vert _\mathtt {q} \prod _{i=1}^{n}\Vert \varphi _i\Vert _{\mathtt {p}_i}.}\)
- \(({\mathcal {H}}_{12})\):
-
\({\mathtt {g}_\mathtt {j}(\mathtt {z})>\frac{b'}{{\mathfrak {O}}_2}}\) for \(b'\le \mathtt {z}\le c',\) where \({{\mathfrak {O}}_2=\frac{r_0^2}{\upsigma (\mathtt {N}-2)^2}\prod _{i=1}^ {n}\varphi _i^\star \int _{0}^{1}\aleph _{r_0}(\mathtt {s}, \mathtt {s})\mathtt {s}^\frac{2(\mathtt {N}-1)}{2-\mathtt {N}} d\mathtt {s}.}\)
- \(({\mathcal {H}}_{13})\):
-
\({\mathtt {g}_\mathtt {j}(\mathtt {z})<\frac{c'}{{\mathfrak {O}}_1}}\) for \(0\le \mathtt {z}\le c'.\)
Then, the iterative system (5)–(6) has at least three positive radial solutions \(({}^1\mathtt {z}_1,{}^1\mathtt {z}_2,\cdot \cdot \cdot ,{}^1\mathtt {z}_\ell ),\) \(({}^2\mathtt {z}_1,{}^2\mathtt {z}_2,\cdot \cdot \cdot ,{}^2\mathtt {z}_\ell )\) and \(({}^3\mathtt {z}_1,{}^3\mathtt {z}_2,\cdot \cdot \cdot ,{}^3\mathtt {z}_\ell )\) with \(\Vert {}^\mathtt {j}\mathtt {z}_1\Vert <a',\) \(b'<\Bbbk ({}^\mathtt {j}\mathtt {z}_2),\) \(\Vert {}^\mathtt {j}\mathtt {z}_3\Vert >a'\) and \(\Bbbk ({}^\mathtt {j}\mathtt {z}_3)<b'\) for \(\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell .\)
Proof
From Lemma 3, \({\mathcal {P}}:\mathtt {E}\rightarrow \mathtt {E}\) is a completely continuous operator. If \(\mathtt {z}_1\in \overline{\mathtt {E}}_{c'},\) then \(\Vert \mathtt {z}_1\Vert \le c'\) and for \(0<\mathtt {s}_{\ell -1}<1\) and by \(({\mathcal {H}}_{13}),\) we have
There exists a \(\mathtt {q}>1\) such that \(\displaystyle \sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}+\frac{1}{\mathtt {q}}=1.\) By the first part of Theorem 2, we have
Continuing with this bootstrapping argument, we get
Hence, \({\mathcal {P}}:\overline{\mathtt {E}}_{c'}\rightarrow \overline{\mathtt {E}}_{c'}.\) In the same way, if \(\mathtt {z}_1\in \overline{\mathtt {E}}_{a'},\) then \({\mathcal {P}}:\overline{\mathtt {E}}_{a'}\rightarrow \overline{\mathtt {E}}_{a'}.\) Therefore, condition (b) of Theorem 10 satisfied. To check condition (a) of Theorem 10, choose \(\mathtt {z}_1(\uptau )=(b'+c')/2,\,\uptau \in [0,1].\) It is easy to see that \(\mathtt {z}_1\in \mathtt {E}(\Bbbk ,b',c')\) and \(\Bbbk (\mathtt {z}_1)=\Bbbk ((b'+c')/2)>b'.\) So, \(\{\mathtt {z}_1\in \mathtt {E}(\Bbbk ,b',c'):\Bbbk (\mathtt {z}_1)>b'\}\ne \emptyset .\) Hence, if \(\mathtt {z}_1\in \mathtt {E}(\Bbbk ,b',c')\) then \(b'<\mathtt {z}_1(\uptau )<c',\,\uptau \in [0,1].\) Let \(0<\mathtt {s}_{\ell -1}<1.\) Then, by \(({\mathcal {H}}_{12}),\) we have
Continuing with this bootstrapping argument, we get
Therefore, we have
This implies that condition (a) of Theorem 10 is satisfied.
Finally, if \(\mathtt {z}_1\in \mathtt {E}(\Bbbk ,b',c'),\) then what we have already proved, \(\Bbbk ({\mathcal {P}}\mathtt {z}_1)>b',\) which proves the condition (c) of Theorem 10. To sum up, all the conditions of Theorem 10 are satisfied. Therefore, \({\mathcal {P}}\) has at least three fixed points, that is, problem (5)–(6) has at least three positive solutions \(({}^1\mathtt {z}_1,{}^1\mathtt {z}_2,\cdot \cdot \cdot ,{}^1\mathtt {z}_\ell ),\) \(({}^2\mathtt {z}_1,{}^2\mathtt {z}_2,\cdot \cdot \cdot ,{}^2\mathtt {z}_\ell )\) and \(({}^3\mathtt {z}_1,{}^3\mathtt {z}_2,\cdot \cdot \cdot ,{}^3\mathtt {z}_\ell )\) with \(\Vert {}^\mathtt {j}\mathtt {z}_1\Vert <a',\) \(b'<\Bbbk ({}^\mathtt {j}\mathtt {z}_2),\) \(\Vert {}^\mathtt {j}\mathtt {z}_3\Vert >a'\) and \(\Bbbk ({}^\mathtt {j}\mathtt {z}_3)<b'\) for \(\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell .\) \(\square \)
For \(\sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}=1\) and \(\sum _{i=1}^{n}\frac{1}{\mathtt {p}_i}>1,\) we have following results.
Theorem 12
Assume that \(({\mathcal {H}}_1)\)–\(({\mathcal {H}}_3)\) hold. Let \(0<a'<b'< c'\) and suppose that \(\mathtt {g}_\mathtt {j},\,\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell \) satisfies \(({\mathcal {H}}_{12}),\,({\mathcal {H}}_{13})\) and
- \(({\mathcal {H}}_{14})\):
-
\({\mathtt {g}_\mathtt {j}(\mathtt {z})<\frac{a'}{{\mathfrak {O}}_3}}\) for \(0\le \mathtt {z}\le a',\) where \({\mathfrak {O}}_3=\frac{\upsigma r_0^2}{(\mathtt {N}-2)^2}\Vert {\widehat{\aleph }}_{r_0}\Vert _{\infty }\prod _{i=1}^{n} \Vert \varphi _i\Vert _{\mathtt {p}_i}.\)
Then, the iterative system (5)–(6) has at least three positive solutions \(({}^1\mathtt {z}_1,{}^1\mathtt {z}_2,\cdot \cdot \cdot ,{}^1\mathtt {z}_\ell ),\) \(({}^2\mathtt {z}_1,{}^2\mathtt {z}_2,\cdot \cdot \cdot ,{}^2\mathtt {z}_\ell )\) and \(({}^3\mathtt {z}_1,{}^3\mathtt {z}_2,\cdot \cdot \cdot ,{}^3\mathtt {z}_\ell )\) with \(\Vert {}^\mathtt {j}\mathtt {z}_1\Vert <a',\) \(b'<\Bbbk ({}^\mathtt {j}\mathtt {z}_2),\) \(\Vert {}^\mathtt {j}\mathtt {z}_3\Vert >a'\) and \(\Bbbk ({}^\mathtt {j}\mathtt {z}_3)<b'\) for \(\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell .\)
Theorem 13
Assume that \(({\mathcal {H}}_1)\)–\(({\mathcal {H}}_3)\) hold. Let \(0<a'<b'< c'\) and suppose that \(\mathtt {g}_\mathtt {j},\,\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell \) satisfies \(({\mathcal {H}}_{12}),\,({\mathcal {H}}_{13})\) and
- \(({\mathcal {H}}_{15})\):
-
\({\mathtt {g}_\mathtt {j}(\mathtt {z})<\frac{a'}{{\mathfrak {O}}_4}}\) for \(0\le \mathtt {z}\le a',\) where \({{\mathfrak {O}}_4=\frac{\upsigma r_0^2}{(\mathtt {N}-2)^2}\Vert {\widehat{\aleph }}_{r_0}\Vert _{\infty }\prod _{i=1}^{n} \Vert \varphi _i\Vert _1.}\)
Then, the iterative system (5)–(6) has at least three positive solutions \(({}^1\mathtt {z}_1,{}^1\mathtt {z}_2,\cdot \cdot \cdot ,{}^1\mathtt {z}_\ell ),\) \(({}^2\mathtt {z}_1,{}^2\mathtt {z}_2,\cdot \cdot \cdot ,{}^2\mathtt {z}_\ell )\) and \(({}^3\mathtt {z}_1,{}^3\mathtt {z}_2,\cdot \cdot \cdot ,{}^3\mathtt {z}_\ell )\) with \(\Vert {}^\mathtt {j}\mathtt {z}_1\Vert <a',\) \(b'<\Bbbk ({}^\mathtt {j}\mathtt {z}_2),\) \(\Vert {}^\mathtt {j}\mathtt {z}_3\Vert >a'\) and \(\Bbbk ({}^\mathtt {j}\mathtt {z}_3)<b'\) for \(\mathtt {j}=1,2,\cdot \cdot \cdot ,\ell .\)
Example 3
Consider the following nonlinear elliptic system of equations,
where \( r_0=1,\) \(\mathtt {N}=3,\) \(\mathtt {j}\in \{1,2\},\, \mathtt {z}_3= \mathtt {z}_1,\) \(\varphi (\uptau )=\frac{1}{\uptau ^4}\prod _{i=1}^{2}\varphi _i(\uptau ),\) \(\varphi _i(\uptau )=\varphi _i\left( \frac{1}{\uptau }\right) ,\) in which
then it is clear that
Let
Let \(\upalpha =2,\upbeta =\gamma =\updelta =1,\) then \(1=r_0^2<2=\frac{\upalpha \upgamma }{\upbeta \updelta },\) \(\wp =2\cos (1)+\sin (1)\approx 1.922075596,\)
and \(\upsigma =\frac{3}{\cos (1)}.\) Also,
Let \(\mathtt {p}_1=6,\mathtt {p}_2=3\) and \(\mathtt {q}=2,\) then \(\frac{1}{\mathtt {p}_1}+\frac{1}{\mathtt {p}_2}+\frac{1}{\mathtt {q}}=1\) and
Choose \(a'=10^{10},\,b'=10^{12}\) and \(c'=10^{13}.\) Then,
Therefore, by Theorem 3, the boundary value problem (15)–(16) has at least two positive radial solutions \(({}^\mathtt {j}\mathtt {z}_1,{}^\mathtt {j}\mathtt {z}_2),\,\mathtt {j}=1,2\) such that
6 Existence of Unique Positive Radial Solution
In the next, for the existence of unique solution to the boundary value problem (5)–(6) where we employ two metrics under Rus’s theorem (see [4, 20, 23] for more details). In this regard, consider the set of real valued functions that are defined and continuous on [0, 1] and denote this space by \(\mathtt {X}= C([ 0, 1]).\) For functions \(\mathtt {y}_1,\mathtt {y}_2\in \mathtt {X},\) consider the following two metrics on \(\mathtt {X}:\)
For \(\mathtt {d}\) in (19), the pair \((C([ 0, 1]), \mathtt {d})\) forms a complete metric space. For \(\varrho \) in (20), the pair \((C([ 0, 1]), \varrho )\) forms a metric space. The relationship between the two metrics on \(\mathtt {X}\) is given by
Theorem 14
(Rus [21]) Let \(\mathtt {X}\) be a nonempty set and let \(\mathtt {d}\) and \(\varrho \) be two metrics on \(\mathtt {X}\) such that \((\mathtt {X}, \mathtt {d})\) forms a complete metric space. If the mapping \(\mho :\mathtt {X}\rightarrow \mathtt {X}\) is continuous with respect to \(\mathtt {d}\) on \(\mathtt {X}\) and
for some \(\mathtt {c}_1>0\) and for all \(\mathtt {y}_1,\mathtt {y}_2\in \mathtt {X},\)
for some \(0<\mathtt {c}_2<1\) for all \(\mathtt {y}_1,\mathtt {y}_2\in \mathtt {X},\) then there is a unique \(\mathtt {y}^*\in \mathtt {X}\) such that \(\mho \mathtt {y}^*=\mathtt {y}^*.\)
Denote \(\Psi (\mathtt {s})=\aleph _{r_0}(\mathtt {s}, \mathtt {s})\mathtt {s}^\frac{2(\mathtt {N}-1)}{2-\mathtt {N}}\prod _{i=1}^{n}\varphi _i(\mathtt {s})d\mathtt {s}.\)
Theorem 15
Assume that \(({\mathcal {H}}_1),\,({\mathcal {H}}_3)\) and the following condition are satisfied.
- \(({\mathcal {H}}_{14})\):
-
there exists a number \(\mathtt {K}>0\) such that
$$\begin{aligned} \vert \mathtt {g}_\mathtt {j}(\mathtt {z})-\mathtt {g}_\mathtt {j}(\mathtt {y})\vert \le \mathtt {K}\vert \mathtt {z}-\mathtt {y}\vert ~~\text {for}~~\mathtt {z},\mathtt {y}\in \mathtt {X}. \end{aligned}$$
Further, assume that there are constants \(\mathtt {p}>1\) and \(\mathtt {q}>1\) such that \(1/\mathtt {p}+1/\mathtt {q}=1\) with
then the boundary value problem (5)–(6) has a unique positive radial solution in \(\mathtt {X}.\)
Proof
Let \(\mathtt {z}_1, \mathtt {y}_1\in C([ 0, 1])\) and \(\mathtt {s}\in [ 0, 1].\) Then, by Hölder’s inequality, we have
where
It follows in similar manner for \(0<\mathtt {s}_{\ell -2}<1,\)
where
Continuing with bootstrapping argument, we get
we see that
for some \(\mathtt {c}_1=\widehat{\mathtt {c}}_1^{\ell }\mathtt {c}_1^\star >0\) for all \(\mathtt {z}_1,\mathtt {y}_1\in \mathtt {X},\) and so the inequality (22) of Theorem 14 holds. Now, for all \(\mathtt {z}_1,\mathtt {y}_1\in \mathtt {X},\) we may apply (21) to (25) to obtain
Thus, given any \(\varepsilon >0\) we can choose \(\upeta =\varepsilon /\mathtt {c}_1\) so that \(\mathtt {d}(\mho \mathtt {z}_1,\mho \mathtt {y}_1)<\varepsilon ,\) whenever \(\mathtt {d}(\mathtt {z}_1,\mathtt {y}_1)<\upeta .\) Hence, \(\mho \) is continuous on \(\mathtt {X}\) with respect to the metric \(\mathtt {d}.\) Finally, we show that \(\mho \) is contractive on \(\mathtt {X}\) with respect to the metric \(\varrho .\) From (25), for each \(\mathtt {z}_1,\mathtt {y}_1\in \mathtt {X}\) consider
That is
From the assumption (24), we have
for some \(\mathtt {c}_2<1\) and all \(\mathtt {y}_1,\mathtt {y}_2\in \mathtt {X}.\) Thus, Theorem 14, the operator \(\mho \) has a unique fixed point in \(\mathtt {X}.\) Also, we note that the operator \(\mho \) is positive from Lemma 3. Therefore, the boundary value problem (2) has a unique positive radial solution.\(\square \)
Example 4
Consider the following nonlinear elliptic system of equations,
where \( r_0=1,\) \(\mathtt {N}=3,\) \(\mathtt {j}\in \{1,2\},\, \mathtt {z}_3= \mathtt {z}_1,\) \(\varphi (\uptau )=\frac{1}{\uptau ^4}\prod _{i=1}^{2}\varphi _i(\uptau ),\) \(\varphi _i(\uptau )=\varphi _i\left( \frac{1}{\uptau }\right) ,\) in which \(\varphi _1(t)=\varphi _2(t)=\frac{1}{t+1},\) then \(\prod _{i=1}^{2}\varphi _i^*=1.\) Let \(\mathtt {g}_1(\mathtt {z})=\frac{1}{10^{10}}\sin (\mathtt {z}),\,\mathtt {g}_2(\mathtt {z})=\frac{\mathtt {z}}{10^{10}(1+\mathtt {z})}\) and \(\upalpha =\upbeta =\updelta =1,\,\gamma =2,\) then \(1=r_0^2<2=\frac{\upalpha \upgamma }{\upbeta \updelta },\) \(\wp =2\cos (1)+\sin (1)\approx 1.922075596,\)
and \(\upsigma =\frac{3}{\cos (1)}.\) Then,
and
So, \(\mathtt {K}=\frac{1}{10^{10}}.\) Let \(\ell =2\) and \(\mathtt {p}=\mathtt {q}=2.\) Then,
Therefore, from Theorem 15, the iterative system of boundary value problems (26)–(27) has a unique positive radial solution.
Data Availability Statement
Data sharing is not applicable to this paper as no data sets were generated or analyzed during the current study.
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Khuddush, M., Prasad, K.R. Positive Solutions for an Iterative System of Nonlinear Elliptic Equations. Bull. Malays. Math. Sci. Soc. 45, 245–272 (2022). https://doi.org/10.1007/s40840-021-01183-y
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DOI: https://doi.org/10.1007/s40840-021-01183-y
Keywords
- Nonlinear elliptic equation
- Annulus
- Positive radial solution
- Fixed point theorem
- Banach space
- Rus’s theorem
- Metric space
- Continuous functions