Abstract
This paper investigates the existence, multiplicity, and nonexistence of symmetric positive solutions for the fourth-order n-dimensional m-Laplace system
The vector-valued function x is defined by \(\mathbf {x}=[x_{1},x_{2},\dots,x_{n}]^{\top}\), \(\Psi(t)=\operatorname{diag}[\psi_{1}(t), \ldots, \psi _{i}(t), \ldots, \psi_{n}(t)]\), where \(\psi_{i}\in L^{p}[0,1]\) for some \(p\geq1\). Our methods employ the fixed point theorem in a cone and the inequality technique. Finally, an example illustrates our main results.
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1 Introduction
Consider the fourth-order n-dimensional m-Laplace system
subject to the following boundary conditions:
where
Here, we understand that \(f_{i}(t,\mathbf{x})\) means that \(f_{i}(t,x_{1},x_{2},\ldots,x_{n})\), \(i=1,2,\ldots,n\).
Therefore, system (1.1) means that
Similarly, (1.2) means that
And then it follows respectively from (1.3) and (1.4) that
From above, we know that system (1.1)–(1.2) is equivalent to system (1.3)–(1.4), and system (1.3)–(1.4) is equivalent to system (1.5)–(1.6); thus, system (1.1)–(1.2) is equivalent to (1.5)–(1.6).
A vector-valued function x is called a solution of (1.1)–(1.2) if \(\mathbf{x}\in C^{2}([0,1],\mathcal{R}^{n})\) with \(\phi_{m}(\mathbf{x}{''})\in C^{2}((0,1),\mathcal{R}^{n})\), and satisfies (1.1) and (1.2). If, for each \(i=1,2,\ldots ,n\), \(x_{i}(t)\ge0\) for all \(t\in(0,1)\) and there is at least one nontrivial component of \(\bf{x}\), then we say that \(\mathbf {x}(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))^{T}\) is positive on J.
For the case of \(n=1\) and \(\Psi(t)\equiv1\) for \(t\in J\), system (1.1)–(1.2) reduces to the problem studied by Zhang and Liu in [1]. By using the upper and lower solution method and fixed-point theorems, the authors obtained some sufficient conditions for the existence of positive solutions for the above problem. For the case of \(n=1\), \(m=2\), \(g(t)\equiv0\), \(h(t)\equiv0\) for \(t\in J\) and \(\Psi\in C[0,1]\) not \(\Psi\in L^{p}[0,1]\), system (1.1)–(1.2) reduces to the problem studied by Graef et al. in [2]. By using Krasnosel’skii’s fixed-point theorem, the authors obtained some existence and nonexistence results. For other related results on system (1.1)–(1.2), we refer the reader to the references [3–27]. Moreover, for the latest development direction of the fourth order differential equations, see the references [28–31].
At the same time, we notice that a class of boundary value problems with integral boundary conditions has attracted many authors (see [20, 32–42]). It is an important and interesting problem, which contains two-point, three-point, and multi-point boundary value problems as special cases; for instance, see [43–58] and the references cited therein.
Here we point out that our problem is new in the sense of fourth-order n-dimensional m-Laplace systems with integral boundary conditions introduced here. To the best of our knowledge, the existence of single or multiple positive solutions for fourth-order n-dimensional m-Laplace systems (1.1)–(1.2) has not yet been studied, especially for the case
where \(\psi_{i}\in L^{p}[0,1]\) for some \(p\geq1\). In consequence, our main results of the present work will be a useful contribution to the existing literature on the topic of fourth-order n-dimensional m-Laplace systems with integral boundary conditions. The existence, multiplicity, and nonexistence of symmetric positive solutions for the given problem are new, though they are proved by applying the well-known method based on the fixed point theorem of cone expansion and compression of norm type.
Throughout this paper, we use \(i = 1, 2,\ldots,n\), unless otherwise stated.
Let \(J=[0,1]\), \(\mathcal{R}_{+}=[0,+\infty)\), \(\mathcal {R}_{+}^{n}=\underbrace{\mathcal{R}_{+}\times\mathcal{R}_{+}\times\cdots \times\mathcal{R}_{+}}_{n}\), and
In addition, let the components of Ψ, f, h, and g satisfy the following conditions:
- \((H_{1})\) :
-
\(\psi_{i}(t)\in L^{p}(J)\) for some \(1\le p\le+\infty \), and \(\psi_{i}(t)\) is nonnegative, symmetric on J, and there exists \(N>0\) such that \(\psi_{i}(t)\ge N\) a.e. on J;
- \((H_{2})\) :
-
\(f_{i}:J\times\mathcal{R}_{+}^{n}\rightarrow\mathcal {R}_{+}\) is continuous, and for all \(\mathbf{x}\in\mathcal{R}_{+}^{n}\), \(f_{i}(t,\mathbf{x})\) is symmetric on J;
- \((H_{3})\) :
-
\(g_{i},h_{i}\in L^{1}(J)\) are nonnegative, symmetric on J with
$$ \mu_{i}:= \int_{0}^{1}g_{i}(s)\,ds\in[0,1),\qquad \nu _{i}:= \int_{0}^{1}h_{i}(s)\,ds \in[0,1). $$(1.7)
The organization of this article is as follows. In Sect. 2, we present some new properties of Green’s function associated with system (1.1)–(1.2), and we list some definitions and lemmas that will be used to prove our main results. Section 3 is devoted to prove the existence, multiplicity, and nonexistence of symmetric positive solutions for system (1.1)–(1.2). Finally, in Sect. 4, an example illustrating our main results is also presented.
2 Preliminaries
In this part, we give some properties of Green’s function associated with system (1.1)–(1.2), and we present some definitions and lemmas which are needed throughout this paper.
Definition 2.1
(see [59])
Let E be a real Banach space over \(\mathcal{R}\). A nonempty closed set \(P\subset E\) is said to be a cone provided that
-
(i)
\(au+bv\in P\) for all \(u, v\in P \) and all \(a\geq0\), \(b\geq0\) and
-
(ii)
\(u,-u\in P\) implies \(u=0\).
Every cone \(P\subset E\) induces a semi-ordering in E given by \(u\leq v\) if and only if \(v-u\in P\).
Definition 2.2
If \(x(t)=x(1-t)\), \(t\in J\), then x is said to be symmetric in J.
In our discussion, \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})^{T}\) is symmetric on J if and only if \(x_{i}\) is symmetric on J.
Next, we reduce system (1.1)–(1.2) to an integral system. It follows from system (1.5)–(1.6) that system (1.1)–(1.2) can be written as follows:
where \(\phi_{m}(s)=|s|^{m-2}s\), \(m>1\), \(\phi_{m^{*}}=\phi_{m}^{-1}\), \(\frac {1}{m}+\frac{1}{m^{*}}=1\).
Firstly, by means of the transformation
we can convert system (2.1) into
and
Lemma 2.1
Assume that \((H_{3})\) holds. Then system (2.4) has a unique solution \(x_{i}(t)\) and \(x_{i}(t)\) can be expressed in the form
where
Proof
The proof is similar to that of Lemma 2.1 in [60]. □
By (2.6) and (2.7), we can show that \(H^{i}(t,s)\) and \(G(t,s)\) have the following properties.
Proposition 2.1
Assume that \((H_{3})\) holds. Then we have
Proposition 2.2
For all \(t,s\in J\), we have
Proposition 2.3
Assume that \((H_{3})\) holds. Then, for all \(t,s\in J\), we have
where
Proof
In addition, noticing that \(G(t,s)\le s(1-s)\), we have
□
Proposition 2.4
Assume that (\(H_{3}\)) holds. Then, for all \(t,s\in J\), we have
Proof
The proof is similar to that of Proposition 2.1 of [60]. □
Lemma 2.2
Assume that \((H_{1})\)–\((H_{3})\) hold. Then system (2.3) has a unique solution
where
Proof
The proof is similar to Lemma 2.1 of [60]. □
Remark 2.1
Assume that (\(H_{3}\)) holds. Then, for all \(t,s\in J\), it follows from (2.17) that
where
Assume that \(x_{i}\) is a solution of system (2.1). Then it follows from Lemma 2.1 that
and then, it follows from Lemma 2.2 that
Let \(E=C[0,1]\), \(X=\underbrace{E\times E\times\cdots\times E}_{n}\), and for all \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})^{T}\in X\), the norm in X is defined as
Then \((X,\|\cdot\|)\) is a real Banach space.
Define a cone K in X by
where
We also define two sets \(K_{r}\), \(K_{r,R}\) by
where \(0< r< R\).
To make our research significant, let \(g_{i}(t)\not\equiv0\), \(h_{i}(t)\not\equiv0\) for any \(t\in J\), \(i=1,2,\ldots,n\).
Remark 2.2
By the definition of \(\rho^{i}\), \(\rho _{1}^{i}\), \(\gamma^{i}\), \(\gamma_{1}^{i}\), we have \(0<\delta_{i}<1\), and then \(0<\delta<1\).
Let \(\mathbf{T}:K\rightarrow X\) be a map with components \((T_{1},\ldots ,T_{i}, \ldots,T_{n})\). Here, we understand \(\mathbf{Tx}= (T_{1}\mathbf{x},\ldots,T_{i}\mathbf{x},\ldots,T_{n}\mathbf{x} )^{T}\), where
From the proof of Lemma 2.1 and Lemma 2.2, we have the following remark.
Remark 2.3
From (2.21), we know that \(\mathbf {x}\in X\) is a solution of system (1.1)–(1.2) if and only if x is a fixed point of the map \(\bf{T}\).
Lemma 2.3
Assume that \((H_{1})\)–\((H_{3})\) hold. Then we have \(\mathbf{T}(K)\subset K\), and \(\mathbf{T}:K\rightarrow K\) is completely continuous.
Proof
For all \(\mathbf{x}\in K\), from (2.21), we know that
which implies that \(T_{i}\mathbf{x}\) is concave on J.
In addition, it follows from (2.21) that
Thus, for all \(t\in J\), we have \((T_{i}\mathbf{x})(t)\ge0\). Noticing that \(\psi_{i}(t)\) is symmetric on \((0,1)\), \(x_{i}(t)\) is symmetric on J, and \(f_{i}(\cdot,\bf{x})\) is symmetric on J, we have
which shows that \((T_{i}\mathbf{x})(1-t)=(T_{i}\mathbf{x})(t)\), \(t\in J\). And hence \((T_{i}\mathbf{x})(t)\) is symmetric on J.
In addition, according to (2.14), we know that
Then
Similarly, according to (2.13), we know that
Thus, we have \(T_{i}\mathbf{x}\in K\), then there is \(\mathbf {T}(K)\subset K\).
Next, we show T is completely continuous, and we need to show \(T_{i}\) is completely continuous.
Let \(l>0\) and define
We show that \(T_{i}\) is compact.
For each \(l>0\), let \(B_{l}=\{\mathbf{x}\in K:\|\mathbf{x}\|\leq l\}\). Then \(B_{l}\) is a bounded closed convex set in K. \(\forall(\mathbf{x}_{m})_{m\in\mathcal{N}}\in K\), it follows from (2.21) that
Therefore, \((T_{i}(B_{l}))\) is uniformly bounded.
Next we show the equicontinuity of \((T_{i}\mathbf{x}_{m})_{m\in \mathcal{N}}\). Due to \(H^{i}(t,s)\) is continuous on \(J\times J\), then \(H^{i}(t,s)\) is uniformly continuous. Thus, for any \(\varepsilon>0\), there exist \(l_{1}>0\), \(t_{1},t_{2}\in J\), if \(|t_{1}-t_{2}|< l_{1}\), we have
which shows that \((T_{i}\mathbf{x}_{m})_{m\in\mathcal{N}}\) is equicontinuous on J. Therefore, it follows from the Arzelà–Ascoli theorem that there exist a function \(T^{1}_{i}\in C[0,1]\) and a subsequence of \((T_{i}\mathbf{x}_{m})_{m\in\mathcal{N}}\) converging uniformly to \(T^{1}_{i}\) on J.
We prove the continuity of \(T_{i}\). Let \((\mathbf{x}_{m})_{m\in \mathcal{N}}\) be any sequence converging on K to \(\mathbf{x}\in K\), and let \(L>0\) be such that \(\|\mathbf{x}_{m}\|\le L\) for all \(m\in \mathcal{N}\). Note that \(f_{i}(t,\mathbf{x})\) is continuous on \(J\times K_{L}\). It is not difficult to see that the dominated convergence theorem guarantees that
for each \(t\in J\). Moreover, the compactness of \(T_{i}\) implies that \((T_{i}\mathbf{x}_{m})(t)\) converges uniformly to \((T_{i}\mathbf {x})(t)\) on J. If not, then there exist \(\varepsilon_{0}>0\) and a subsequence \((\mathbf{x}_{m_{j}})_{j\in\mathcal{N}}\) of \((\mathbf {x}_{m})_{m\in\mathcal{N}}\) such that
Now, it follows from the compactness of \(T_{i}\) that there exists a subsequence of \((\mathbf{x}_{m_{j}})_{j\in\mathcal{N}}\) (without loss of generality, assume that the subsequence is \((\mathbf {x}_{m_{j}})_{j\in\mathcal{N}}\)) such that \((T_{i}\mathbf {x}_{m_{j}})_{j\in\mathcal{N}}\) converges uniformly to \(y_{0}\in C[0,1]\). Thus, from (2.24), we easily see that
On the other hand, from the pointwise convergence (2.23) we obtain
This is a contradiction to (2.25). Therefore \(T_{i}\) is continuous.
Therefore \(T_{i}:K\rightarrow K\) is completely continuous. This completes the proof of Lemma 2.3. □
In the following lemma, we employ Hölder’s inequality to obtain some of the norm inequalities in our main results.
Lemma 2.4
(Hölder)
Let \(e\in L^{p}[a,b]\) with \(p>1\), \(h\in L^{q}[a,b]\) with \(q>1\), and \(\frac{1}{p}+\frac{1}{q}=1\). Then \(eh\in L^{1}[a,b]\), and
Let \(e\in L^{1}[a,b]\) and \(h\in L^{\infty}[a,b]\). Then \(eh\in L^{1}[a,b]\), and
Finally, we state the well-known fixed point theorem of cone expansion and compression of norm type.
Lemma 2.5
(see [59])
Let P be a cone in a real Banach space E. Assume \(\Omega_{1}\), \(\Omega_{2}\) are bounded open sets in E with \(0\in\overline{\Omega}_{1}\subset\Omega_{2}\), \(A:P\cap (\overline{\Omega}_{2}\setminus\Omega_{1})\rightarrow P\) is completely continuous such that either
-
(i)
\(\|Ax\|\le\|x\|\), \(\forall x\in P\cap\partial\Omega_{1}\); \(\|Ax\| \ge\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\) or
-
(ii)
\(\|Ax\|\ge\|x\|\), \(\forall x\in P\cap\partial\Omega_{1}\); \(\|Ax\| \le\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\).
Then A has at least one fixed point in \(P\cap(\overline{\Omega }_{2}\setminus\Omega_{1})\).
Remark 2.4
To make it clear for the reader what \(\Omega _{1}\), \(\Omega_{2}\), \(\partial\Omega_{1}\), \(\partial\Omega_{2}\), and \(\bar{\Omega}_{2}\setminus\Omega_{1}\) mean, we give typical examples of \(\Omega_{1}\) and \(\Omega_{2}\).
where \(0< r< R\), \(\|x\|=\max_{t\in[a,b]}|x(t)|\).
3 Main results
In this part, by using Lemmas 2.1–2.5, we show the existence, multiplicity, and nonexistence of symmetric positive solutions for system (1.1)–(1.2) under the following three cases for \(\psi_{i}\in L^{p}[0,1]:1< p<\infty\), \(p=1\), and \(p=\infty\).
For convenience’s sake, we introduce the notations:
where β is 0 or ∞.
The first existence theorem deals with the case \(1< p<\infty\).
Theorem 3.1
Assume that \((H_{1})\)–\((H_{3})\) hold. Furthermore, assume that one of the following conditions is satisfied.
- \((C_{1})\) :
-
There exist two constants r, R with \(0< r\le\delta R\) such that \(f_{i}(t,\mathbf{x})\le\phi_{m}(\frac{r}{D_{i}})\) for \(t\in J\), \(0\le\|\mathbf{x}\|\le r\), and \(f_{i}(t,\mathbf{x})\ge\phi_{m}(\frac {R}{\delta_{i}D_{1}^{i}})\) for \(t\in J\), \(\delta R\le\|\mathbf{x}\|\le R\);
- \((C_{2})\) :
-
\(f_{i0}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\) and \(f_{i}^{\infty}<\phi_{m}(\frac{1}{D_{i}})\) (particularly, \(f_{i0}=\infty \) and \(f_{i}^{\infty}=0\)).
Then, system (1.1)–(1.2) has at least one symmetric positive solution.
Proof
Case (1). Considering the condition \((C_{1})\), for all \(\mathbf{x}\in\partial K_{r}\), we have \(\|\mathbf{x}\|= r\) and \(f_{i}(t,\mathbf{x}(t))\le\phi_{m}(\frac{r}{D_{i}})\), \(i=1,2,\ldots,n\). Thus, for all \(t\in J\), we have
In addition, for all \(\mathbf{x}\in\partial K_{R}\), we have \(\|\mathbf {x}\|= R\), and then it follows from (2.20) and \((C_{1})\) that
Case (2). Considering condition \((C_{2})\), it follows from the definition of \(f_{i0}\) and \(f_{i0}>\phi_{m}(\frac{1}{\delta _{i}D_{1}^{i}})\) that there exists \(r_{1}>0\) such that
where \(\varepsilon_{1}^{i}>0\) satisfies \(D_{1}^{i}\delta _{i}(f_{i0}-\varepsilon_{1}^{i})^{m^{*}-1}\ge1\), \(i=1,2,\ldots,n\). Then, for all \(t\in J\), \(\mathbf{x}\in\partial K_{r_{1}}\), we have
Next, turning to \(f_{i}^{\infty}<\phi_{m}(\frac{1}{D_{i}})\), \(i=1,2,\ldots,n\), and we know that there exists \(\overline{R}_{1}>0\) such that
where \(\varepsilon_{2}^{i}>0\) satisfies \(D_{i}(f_{i}^{\infty }+\varepsilon_{2}^{i})^{m^{*}-1}\le1\), \(i=1,2,\ldots,n\).
Let
Thus, for all \(t\in J\), \(\mathbf{x}\in K\), we have \(f_{i}(t,\mathbf {x})\le M^{i}+(f_{i}^{\infty}+\varepsilon_{2}^{i})\phi_{m}(\|\mathbf {x}\|)\), \(i=1,2,\ldots,n\).
Letting
then, for all \(t\in J\), \(\mathbf{x}\in\partial K_{R_{1}}\), we have
Applying Lemma 2.5 to (3.1) and (3.2), or (3.3) and (3.4) yields that T has at least one fixed point \(\mathbf{x}^{*}\in\overline{K}_{r,R}\), or \(\mathbf {x}^{*}\in\overline{K}_{r_{1},R_{1}}\). Thus it follows from Remark 2.3 that system (1.1)–(1.2) has at least one symmetric positive solution. This finishes the proof of Theorem 3.1. □
The following theorem deals with the case \(p=\infty\).
Theorem 3.2
Assume that \((H_{1})\)–\((H_{3})\), \((C_{1})\) or \((H_{1})\)–\((H_{3})\), \((C_{2})\) hold. Then system (1.1)–(1.2) has at least one symmetric positive solution.
Proof
Let \(\|e\|_{1}\|\psi_{i}\|_{\infty}\) replace \(\|e\|_{q}\|\psi _{i}\|_{p}\) and repeat the argument above. □
Finally, we consider the case of \(p=1\).
Let
Theorem 3.3
Assume that \((H_{1})\)–\((H_{3})\), \((C_{1})\) or \((H_{1})\)–\((H_{3})\), \((C_{2})\) hold. Then system (1.1)–(1.2) has at least one symmetric positive solution.
Proof
Similar to the proof of Theorem 3.1. For all \(t\in J\), \(\mathbf{x}\in\partial K_{r}\), we have
Next, similar to the proof of Theorem 3.1, we can finish the proof. □
In the following theorems we only consider the case of \(1< p<\infty\).
Theorem 3.4
Assume that \((H_{1})\)–\((H_{3})\) hold. Furthermore, assume that one of the following conditions is satisfied.
- \((C_{3})\) :
-
There exists two constants r, R, which satisfy \(0< r\le \delta R\), such that \(f_{i}(t,\mathbf{x})\ge\phi_{m}(\frac{r}{\delta _{i}D_{1}^{i}})\) for \(t\in J\), \(0\le\|\mathbf{x}\|\le r\), and \(f_{i}(t,\mathbf{x})\le\phi_{m}(\frac{R}{D_{i}})\) for \(t\in J\), \(\delta R\le\|\mathbf{x}\|\le R\);
- \((C_{4})\) :
-
\(f_{i}^{0}<\phi_{m}(\frac{1}{D_{i}})\) and \(f_{i\infty }>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\) (particularly, \(f_{i}^{0}=0\) and \(f_{i\infty}=\infty\)),
where \(i=1,2,\ldots,n\). Thus, system (1.1)–(1.2) has at least one symmetric positive solution \(\mathbf{x}^{*}\).
Proof
The proof is similar to that of Theorem 3.1, so we omit it here. □
Next, we discuss the multiplicity of system (1.1)–(1.2).
Theorem 3.5
Assume that \((H_{1})\)–\((H_{3})\) and the following conditions hold.
- \((C_{5})\) :
-
\(f_{i0}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\) and \(f_{i\infty}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\) (particularly, \(f_{i0}=f_{i\infty}=\infty\));
- \((C_{6})\) :
-
there exists a constant \(b>0\) such that \(\max_{t\in J,\|\mathbf{x}\|=b}f_{i}(t,\mathbf{x})<\phi_{m}(\frac{b}{D_{i}})\).
Then system (1.1)–(1.2) has at least two symmetric positive solutions \(\mathbf{x}^{*}\), \(\mathbf{x}^{**}\) with
Proof
Choose two constants r, R with \(0< r< b< R\). It follows from \((C_{5})\) that
if \(f_{i0}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\), then by means of the proof of (3.3), we obtain that
if \(f_{i\infty}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\), then by means of the proof of (3.3), we obtain that
On the other hand, it follows from \((C_{6})\) that
Applying Lemma 2.5 to (3.7), (3.8), and (3.9) yields that T has a fixed point \(\mathbf{x}^{**}\in \overline{K}_{r,b}\) and a fixed point \(\mathbf{x}^{*}\in\overline {K}_{b,R}\). Then it follows from Remark 2.3 that system (1.1)–(1.2) has at least two symmetric positive solutions \(\mathbf{x}^{*}\) and \(\mathbf{x}^{**}\). Noticing (3.9), we obtain \(\|\mathbf{x}^{*}\|\neq b\) and \(\|\mathbf{x}^{**}\|\neq b\). Therefore (3.6) holds, and the proof of Theorem 3.5 is complete. □
Similarly, the following Theorem 3.6 can be obtained.
Theorem 3.6
Assume that \((H_{1})\)–\((H_{3})\) and the following conditions hold.
- \((C_{7})\) :
-
\(f_{i}^{0}<\phi_{m}(\frac{1}{D_{i}})\) and \(f_{i}^{\infty}<\phi_{m}(\frac{1}{D_{i}})\);
- \((C_{8})\) :
-
There exists a constant \(B>0\) such that \(\min_{t\in J,\|\mathbf{x}\|=B}f_{i}(t,\mathbf{x})>\phi_{m}(\frac{B}{\delta_{i}D_{1}^{i}})\).
Then system (1.1)–(1.2) has at least two symmetric positive solutions \(\mathbf{x}^{*}\) and \(\mathbf{x}^{**}\) with
Theorem 3.7
Assume that \((H_{1})\)–\((H_{3})\) hold. If there exist 2n positive numbers \(b_{k}\), \(d_{k}\), \(k=1,2,\ldots,n\), with \(b_{1}<\delta d_{1}<d_{1}<b_{2}<\delta d_{2}<d_{2}<\cdots<b_{n}<\delta d_{n}<d_{n}\), such that
- \((C_{9})\) :
-
\(f_{i}(t,\mathbf{x})\le\phi_{m}(\frac{b_{k}}{D_{i}})\) for \(t\in J\), \(\delta b_{k}\le\|\mathbf{x}\|\le b_{k}\) and \(f_{i}(t,\mathbf{x})\ge\phi_{m}(\frac{d_{k}}{\delta_{i}D_{1}^{i}})\) for \(t\in J\), \(\delta d_{k}\le\|\mathbf{x}\|\le d_{k}\), \(k=1,2,\ldots,n\); or
- \((C_{10})\) :
-
\(f_{i}(t,\mathbf{x})\ge\phi_{m}(\frac{b_{k}}{\delta _{i}D_{1}^{i}})\) for \(t\in J\), \(\delta b_{k}\le\|\mathbf{x}\|\le b_{k}\) and \(f_{i}(t,\mathbf{x})\le\phi_{m}(\frac{d_{k}}{D_{i}})\) for \(t\in J\), \(\delta d_{k}\le\|\mathbf{x}\|\le d_{k}\), \(k=1,2,\ldots,n\).
Then system (1.1)–(1.2) has at least n symmetric positive solutions \(\mathbf{x}_{k}\), and \(\mathbf{x}_{k}\) satisfy
Finally, we discuss the existence result corresponding to the case when system (1.1)–(1.2) has no symmetric positive solutions.
Theorem 3.8
Assume that conditions \((H_{1})\)–\((H_{3})\) hold and \(f_{i}(t,\mathbf{x})<\phi_{m} (\frac {\|\mathbf{x}\|}{D_{i}} )\), \(\forall t\in J\), \(\|\mathbf{x}\|>0\). Then system (1.1)–(1.2) has no positive solution.
Proof
Assume to the contrary that x is a positive solution of system (1.1)–(1.2), then for any \(0< t<1\), we have \(\mathbf{x}\in K\), \(x_{i}(t)>0\), and
This is a contradiction, and this completes the proof. □
Similarly, we have the following results.
Theorem 3.9
Assume that \((H_{1})\)–\((H_{3})\) hold and \(f_{i}(t,\mathbf{x})>\phi_{m} (\frac{\|\mathbf{x}\|}{\delta _{i}D_{1}^{i}} )\), \(\forall t\in J\), \(\|\mathbf{x}\|>0\), \(i=1,2,\ldots,n\). Then system (1.1)–(1.2) has no positive solution.
Proof
Assume that x is a positive solution of system (1.1)–(1.2). Then, for any \(0< t<1\), we have \(\mathbf {x}\in K\), \(x_{i}(t)>0\), and
This leads to a contradiction, and this finishes the proof. □
4 An example
In the following example, we select \(n=2\), \(m=2\), \(p=2 \), and \(N=1\).
Example 4.1
Consider the following system:
where
Then, by calculations, we obtain that \(m^{*}=2\), \(\mu_{1}=\mu_{2}=\nu _{1}=\frac{1}{2}\), \(\nu_{2}=\frac{1}{3}\), \(\gamma^{1}=\gamma^{2}=\gamma _{1}^{1}=2\), \(\gamma_{1}^{2}=\frac{3}{2}\), \(\rho^{1}=\rho^{2}=\rho _{1}^{1}=\frac{1}{6}\), \(\rho_{1}^{2}=\frac{1}{12}\), \(\delta_{1}=\frac {1}{144}\), \(\delta_{2}=\frac{1}{216}\), \(\delta=\frac{1}{216}\), \(D_{1}=\frac {8}{\sqrt{30}}\), \(D_{2}=\frac{3}{\sqrt{30}}\), \(D_{1}^{1}=\frac{2}{9}\), \(D_{1}^{2}=\frac{1}{6}\), and
where
Clearly, conditions \((H_{1})\)–\((H_{3})\) hold. Next, we show that the condition \((C_{1})\) of Theorem 3.1 holds. Choosing \(r=\frac {1}{2}\), \(R=6^{5}\times\sqrt[6]{3}\), we obtain that
Therefore, \(f_{i}(t,\mathbf{x})\le\phi_{2}(\frac{r}{D_{i}})\), for all \(t\in J\), \(0\le\|\mathbf{x}\|\le r\), and \(f_{i}(t,\mathbf{x})\ge\phi _{2}(\frac{R}{\delta_{i}D_{1}^{i}})\), for all \(t\in J\), \(\delta R\le\| \mathbf{x}\|\le R\), \(i=1,2\).
Therefore, it follows from Theorem 3.1 that system (4.1) has at least one symmetric positive solution.
5 Conclusion
In this paper, we obtained several sufficient conditions for the existence, multiplicity, and nonexistence of symmetric positive solutions for the fourth-order n-dimensional m-Laplace system with integral boundary conditions. Our results will be a useful contribution to the existing literature on the topic of fourth-order n-dimensional m-Laplace systems.
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The authors are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.
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This work is sponsored by the National Natural Science Foundation of China (11401031), the Beijing Natural Science Foundation (1163007), the Scientific Research Project of Construction for Scientific and Technological Innovation Service Capacity (KM201611232017), and the teaching reform project of Beijing Information Science & Technology University (2018JGZD41).
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Feng, M., Li, P. & Sun, S. Symmetric positive solutions for fourth-order n-dimensional m-Laplace systems. Bound Value Probl 2018, 63 (2018). https://doi.org/10.1186/s13661-018-0981-3
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DOI: https://doi.org/10.1186/s13661-018-0981-3