Symmetric positive solutions for fourth-order n-dimensional m-Laplace systems

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.

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.

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

1. (i)

   \(au+bv\in P\) for all \(u, v\in P \) and all \(a\geq0\), \(b\geq0\) and

2. (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

By (2.6) and (2.9), we have

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

1. (i)

   \(\|Ax\|\le\|x\|\), \(\forall x\in P\cap\partial\Omega_{1}\); \(\|Ax\| \ge\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\) or

2. (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 \(0< r< R\), \(\|x\|=\max_{t\in[a,b]}|x(t)|\).

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 \(f_{i\infty}=\infty\)),

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. □

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.

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.

Acknowledgements

The authors are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.

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Funding

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).

Affiliations

1. School of Applied Science, Beijing Information Science & Technology University, Beijing, People’s Republic of China

   • Meiqiang Feng
   •  & Ping Li

2. College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, People’s Republic of China

   • Sujing Sun

Contributions

The authors contributed equally in this article. They have all read and approved the final manuscript.

Corresponding author

Correspondence to Meiqiang Feng.

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The authors declare that there is no conflict of interest regarding the publication of this manuscript. The authors declare that they have no competing interests.

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