Abstract
In this paper, we establish several new existence theorems for positive solutions of systems of \((2n,2m)\)-order of two p-Laplacian equations. The results are based on the Krasnosel’skii fixed point theorem and mainly complement those of Djebali, Moussaoui, and Precup.
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1 Introduction
Quasilinear elliptic systems have been used in a great variety of applications, and existence results and a priori estimates of positive solutions for quasilinear elliptic systems have been broadly investigated. For instance, D’Ambrosio and Mitidieri [1] studied Liouville theorems for a class of possibly singular quasilinear elliptic equations and inequalities in the framework of Carnot groups, and their results are new even in the canonical Euclidean setting. In [2] the authors proved a priori estimates for the solutions of elliptic systems involving quasilinear operators in divergence form in an open set \(\varOmega \subset R^{N}\) and, as a consequence, obtained theorems on nonexistence of positive solutions in the case \(\varOmega =R ^{N}\). More related results can be found in [3,4,5]. Equations of the p-Laplacian form occur in the study of non-Newtonian fluid theory and the turbulent flow of a gas in a porous medium. To our best knowledge, there are many papers devoted to the study of differential equations with p-Laplacian. We refer the readers to [6] (one-dimensional p-Laplacian), [7,8,9,10] (fourth-order p-Laplacian), and [11] (2nd-order p-Laplacian). Most of these results are based upon the quadrature method, topological degree, the fixed point theorem on a cone, or the lower and upper solution method. Especially, Djebali et al. [12] have shown some existence results for the fourth-order p-Laplacian nonlinear system
The first existence result is obtained via the classical Krasnosel’skii fixed point theorem of cone compression and expansion under the following notation and assumptions.
-
(H1)
For \(i = 1, 2\), there exist nonnegative constants \(h_{i}^{0}\) and \(h_{i}^{\infty }\) defined as
$$ h_{i}^{0}=\lim_{u+v\rightarrow 0}\frac{h_{i}(u,v)}{(u+v)^{p-1}} \quad \text{and}\quad h_{i}^{\infty }=\lim_{u+v\rightarrow \infty } \frac{h_{i}(u,v)}{(u+v)^{p-1}}. $$
The second existence result is obtained via the vector versions of the Krasnosel’skii fixed point theorem [13] under the following notation and assumptions.
-
(H2)
For \(\lambda = 0\) or \(\lambda = +\infty \), there exist nonnegative constants \(H_{1}^{\lambda }\) and \(H_{2}^{\lambda }\) defined as
$$\begin{aligned}& H_{1}^{\lambda }=\lim_{u\rightarrow \lambda } \frac{h_{1}(u,v)}{u^{p-1}} \quad \text{uniformly with respect to } v \text{ on compact subsets of } R^{+}, \\& H_{2}^{\lambda }=\lim_{v\rightarrow \lambda } \frac{h_{2}(u,v)}{v^{p-1}} \quad \text{uniformly with respect to } u \text{ on compact subsets of } R^{+}. \end{aligned}$$
A comparison of the obtained results to those from the literature is provided.
In addition, there are some papers concerned with the existence and multiplicity of positive solutions of systems of \((2n,2m)\)th-order equations under assumption (H1); see [14,15,16,17]. The proof is based on the classical Krasnosel’skii fixed point theorem of cone compression and expansion [14, 15, 17], fixed point index arguments and upper and lower solutions method [16]. However, in [12, 14,15,16,17] the uniqueness of positive solutions is not considered. Therefore, via the classical Krasnosel’skii fixed point theorem of cone compression and expansion, the results we are going to present reveal how the behavior of the functions \(f_{i}\) (\(i=1,2\)) at zero and infinity have a profound effect on the existence, uniqueness, and multiplicity of a positive solution of the following system:
where \(n_{1}+n_{2}=n\), \(m_{1}+m_{2}=m\), \(n, m\in {\mathbf{N}}\), \(n,m\geq 2\). Here \(\varphi _{p}(s)=s{|s|}^{p-2}\) (\(p>1\)) refers to the p-Laplacian, and \(f_{i}:[0,1]\times R^{+}\rightarrow R^{+}\) are continuous (\(i=1,2\)) with \(f_{2}(t,0)=0\).
Furthermore, in Sect. 4, we consider the existence of positive solutions of the system
under the following assumption:
-
(H3)
there exist two pairs of nonnegative functions \(F_{1}\), \(F_{2}\) and \(G_{1}\), \(G_{2}\) such that
$$\begin{aligned} \textstyle\begin{cases} F_{1}(v)\leq f(u,v)\leq F_{2}(v) \quad \text{for all } u\in [0,+\infty ], \\ G_{1}(u)\leq g(u,v)\leq G_{2}(u) \quad \text{for all }v\in [0,+\infty ], \end{cases}\displaystyle \end{aligned}$$
or
- (\(\widetilde{\text{H}}\)3):
-
there exist two pairs of nonnegative functions \(F_{1}\), \(F_{2}\) and \(G_{1}\), \(G_{2}\) such that
$$\begin{aligned} \textstyle\begin{cases} F_{1}(u)\leq f(u,v)\leq F_{2}(u), \quad \text{for all }v\in [0,+\infty ], \\ G_{1}(v)\leq g(u,v)\leq G_{2}(v), \quad \text{for all }u\in [0,+\infty ]. \end{cases}\displaystyle \end{aligned}$$
At the end of Sect. 4, we also give examples where \(f(u,v)\) and \(g(u,v)\) satisfying assumption (H3) do not satisfy assumptions (H1) and (H2).
2 Preliminaries
Let \(G_{n}(t,s)\) be the Green function of the linear boundary value problem
By induction the Green function \(G_{n}(t,s)\) can be expressed as (see[15])
where
Lemma 2.1
([15])
The function \(G_{n} (t,s)\) has the following properties:
-
(1)
\(G_{n}(t,s)>0\), \((t,s)\in (0,1)\times (0,1)\).
-
(2)
For any \((t,s)\in [0,1]\times [0,1]\), \(G_{n}(t,s)\leq \frac{1}{6^{n-1}}s(1-s)\).
-
(3)
If \(\delta \in (0,\frac{1}{2})\), then for any \((t,s)\in [\delta,1- \delta ]\times [0,1]\),
$$\begin{aligned} G_{n}(t,s) \geq &\delta ^{n} \biggl(\frac{4\delta ^{3}-6\delta ^{2}+1}{6} \biggr)^{n-1}s(1-s) \\ \geq & 6^{n-1}\delta ^{n} \biggl(\frac{4\delta ^{3}-6\delta ^{2}+1}{6} \biggr)^{n-1}\max_{0\leq t\leq 1}G_{n}(t,s). \end{aligned}$$
Lemma 2.2
For any \(m,n\in \mathbf{N^{+}}\) and \(h\in C[0,1]\) with \(h(t)\geq 0\), the solution of the boundary value problem
can be expressed by \(u(t)=\int _{0}^{1} G_{n}(t,s)\varphi _{q}(\int _{0} ^{1} G_{m}(s,\tau)h(\tau)\,d\tau)\,ds\), where \(\varphi _{q}\) stands for the inverse function \(\varphi _{q}=\varphi ^{-1}_{p}\) with conjugates p, q, that is, \(\frac{1}{p}+\frac{1}{q}=1\). Moreover, the solution satisfies the estimate
where \(\|u\|=\max_{t\in [0,1]}|u(t)|\), the norm of \(C[0,1]\).
Proof
For \(t\in [\delta,1-\delta ]\), we have
□
Definition 2.3
Let K be a cone in a real Banach space X. With some positive \(u_{0}\in K\backslash \{0\}\), \(A:K \rightarrow K\) is called \(u_{0}\)-sublinear if
-
(a)
for any \(x>0\), there exist \(\theta _{1}>0\) and \(\theta _{2}>0\) such that \(\theta _{1}u_{0}\leq Ax \leq \theta _{2}u_{0}\);
-
(b)
for any \(\theta _{1}u_{0}\leq x \leq \theta _{2}u_{0}\) and \(t\in (0,1)\), there exists \(\eta =\eta (x,t)>0\) such that \(A(tx) \geq (1+\eta)tAx\).
Lemma 2.4
([14], Theorem 2.2.2)
An increasing and \(u_{0}\)-sublinear operator A has at most one positive fixed point.
Lemma 2.5
([18])
Let E be a Banach space, and let \(K\subset E\) be a cone in E. Let \(\varOmega _{1}\) and \(\varOmega _{2}\) be open subsets of E with \(0\in \varOmega _{1}\) and \(\overline{\varOmega }_{1} \subset \varOmega _{2}\), and let \(A:K\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1})\rightarrow K\) be a completely continuous operator such that either
-
(i)
\(\|Au\|\leq \|u\|\), \(u\in K\cap \partial \varOmega _{1}\), and \(\|Au\|\geq \|u\|\), \(u\in K\cap \partial \varOmega _{2}\); or
-
(ii)
\(\|Au\|\geq \|u\|\), \(u\in K\cap \partial \varOmega _{1}\), and \(\|Au\|\leq \|u\|\), \(u\in K\cap \partial \varOmega _{2}\).
Then A has a fixed point in \(K\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1})\).
At the end of this section, for any \(\alpha _{i}\), \(\tilde{\alpha } _{i}\), \(\beta _{i}\), \(\tilde{\beta }_{i}\in R^{+}\) (\(i=1,2\)), we give some notations:
For any \(\alpha,\beta,\sigma \in R^{+}\) and \(m,n,l\in \mathbf{N^{+}}\), let
3 Main results of Problem (1.1)
Define the mapping \(A: C[0,1]\rightarrow C[0,1]\) by
where
Let \(P=\{u\in C[0,1], u\geq 0\}\). The pair \((u,v)\in C[0,1]\times C[0,1]\) is a positive solution of (1.1) if and only if \((u,v)\) belongs to \(P\setminus \{0\}\times P\setminus \{0\}\) and satisfies \(u=A_{1}v\), \(v=A_{2}u\). If \(u\in P\setminus \{0\}\) is a fixed point of A, then define \(v=A_{2}u\). Then \(v\in P\setminus \{0\}\), so that \((u,v)\in C[0,1]\times C[0,1]\) solves (1.1). So our main goal is to look for nonzero fixed points of A in the subcone
Since \(u(t)\in K\) with \(u(t)\geq 0\), this means that the corresponding solutions of (1.1) are nonnegative.
For any given \(r>0\), let
Lemma 3.1
For any \(0< r< R\), the operator \(A:K\cap (\overline{ \varOmega }_{R}\setminus \varOmega _{r})\rightarrow K\) is completely continuous.
Proof
For any \(u(t)\in K\cap (\overline{\varOmega }_{R} \setminus \varOmega _{r})\), by Lemma 2.2 we get
which implies that \(A:K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\rightarrow K\).
Now, we verify that A is completely continuous.
First, for the continuity of A, we only need to prove that \(\|A(u_{n})-A(u)\|\rightarrow 0\) if \(u_{n}\rightarrow u\) as \(n\rightarrow \infty \). Let us consider
From the continuity of \(f_{1}\) and \(f_{2}\) it follows that \(\|A(u_{n})-A(u) \|\rightarrow 0\) as \(n\rightarrow \infty \).
Second, we show that the operator A is uniformly bounded. For any \(u(t)\in K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\), by Lemma 2.1(2) we have
Since \(f_{1}\) and \(f_{2}\) are continuous, it is clear that \(A(u)(t)\) is uniformly bounded on \(K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\).
Finally, we show the equicontinuity of the operator A. From the expression of \(G(t,s)\) we easily obtain that
which implies that \(|\frac{\partial G(t,\tau)}{\partial t}|\) is bounded. There exists a constant \(M>0\) such that \(|\frac{\partial G(t, \tau)}{\partial t}|< M\). For any \(u(t)\in K\cap (\overline{\varOmega } _{R}\setminus \varOmega _{r})\), \(t_{1},t_{2}\in [0,1]\), we have
which implies that A is equicontinuous. By the Arzelà–Ascoli theorem we get that \(A:K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\rightarrow K\) is compact. Consequently, it follows that \(A:K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\rightarrow K\) is completely continuous. □
Theorem 3.2
Assume that \(\alpha _{i}, \beta _{i}>0\) (\(i=1,2\)) with
In addition, let the functions \(f_{i}\) (\(i=1,2\)) satisfy the following assumptions:
Then (1.1) has at least one positive solution.
Proof
On one hand, from the assumption \(\overline{\varphi }_{1}^{0}<+\infty \) we have that there exist \(\varepsilon >0\) and \(\widehat{r}\in (0,1)\) such that
Furthermore, from the assumption \(\overline{\varphi }_{2}^{0}=0\) we have that there exist \(\varepsilon _{1}>0\) and \(\overline{r}\in (0,\widehat{r})\) such that
where \(\varepsilon _{1}\) satisfies
Set \(r=\bar{r}\). Then for any \(u\in K\cap \partial \varOmega _{r}\), we have
Furthermore, we have
that is, \(\|A(u)(t)\|\leq \|u\|\) for \(u\in K\cap \partial \varOmega _{r}\).
On the other hand, from the assumptions \(\underline{\psi }_{1}^{ \infty }>0\) and \(\underline{\psi }_{2}^{\infty }=+\infty \) it follows that there exist \(C_{1}>0\) and \(\overline{R}>1\) such that
where \(C_{1}\) satisfies
Set \(R=\max \{\frac{\overline{R}}{\sigma },\overline{R}^{\frac{1}{ \beta _{1}(q-1)}}\}\). Then for any \(u\in K\cap \partial \varOmega _{R}\), we have \(u(t)\geq \sigma R\geq \sigma \frac{\overline{R}}{\sigma }= \overline{R}\) for \(t\in [\delta,1-\delta ]\), and
Furthermore, for \(t\in [\delta,1-\delta ]\), we also get
Then from the above inequalities, for \(t\in [\delta,1-\delta ]\), we have
which yields that \(\|A(u)(t)\|\geq \|u\|\) for \(u\in K\cap \partial \varOmega _{R}\).
Therefore, by Lemma 2.5 the operator A has at least one fixed point in \(K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\). □
Theorem 3.3
Assume that \(\alpha _{i}, \beta _{i}>0\) (\(i=1,2\)) with
In addition, let the functions \(f_{i}\) (\(i=1,2\)) satisfy the following assumptions:
Then (1.1) has at least one positive solution.
Proof
On one hand, from the assumptions \(\underline{ \varphi }_{1}^{0}>0\) and \(\underline{\varphi}_{2}^{0}=+\infty \) we have that there exist \(C_{3}>0\) and \(0<\overline{r}<1\) such that
where \(C_{3}\) satisfies
Since \(f_{2}\) is continuous and \(f_{2}(t,0)=0\), there exists \(\widehat{r}\in (0,\overline{r})\) such that
Set \(r=\widehat{r}\). For any \(u(t)\in K\cap \partial \varOmega _{r}\), we have
Then, for \(t\in [\delta,1-\delta ]\), we obtain
and
From these inequalities we have
On the other hand, by the assumptions \(\overline{\psi }_{1}^{\infty }<+ \infty \) and \(\overline{\psi }_{2}^{\infty }=0\) there exist \(\varepsilon _{2}>0\) and \(\overline{R}>0\) such that
where \(\varepsilon _{2}\) satisfies \(\varepsilon _{2}^{\beta _{1}(q-1)^{2}}(\overline{ \psi }_{1}^{\infty }+\varepsilon)^{q-1}L(n_{1},n_{2})L^{\beta _{1}(q-1)}(m _{1},m_{2})<1\). Since \(f_{i}\) is continuous, let
Then we have the estimates
Via some computations we obtain the inequalities
and
It is clear that the term with the highest index is
Thus there exists a sufficiently large \(R>0\) such that
Therefore by Lemma 2.5 the operator A has at least one fixed point in \(K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\). □
Theorem 3.4
Assume that the functions \(f_{i}\) (\(i=1,2\)) satisfy the following assumptions:
-
(i)
\(f_{i}(t,c)\) is nondecreasing on c uniformly for \(t\in [0,1]\);
-
(ii)
there exist four positive constants \(k_{1}< k_{2}\), \(l_{1}< l_{2}\) such that
$$\begin{aligned}& k_{1}v^{\alpha _{1}}\leq f_{1}(t,v)\leq k_{2} v^{\alpha _{1}} \quad \textit{uniformly in } t\in [0,1], v\in [0,+\infty); \\& l_{1}u^{\alpha _{2}}\leq f_{2}(t,u)\leq l_{2} u^{\alpha _{2}}\quad \textit{uniformly in } t\in [0,1], u\in [0,+\infty). \end{aligned}$$ -
(iii)
there exist two positive constants \(\alpha _{1}\), \(\alpha _{2}\) with \(\alpha _{1}\alpha _{2}<(p-1)^{2}\) such that
$$ f_{i}(t,\xi c)\geq \xi ^{\alpha _{i}} f_{i}(t,c) \quad \textit{for all } \xi \in (0,1). $$
Then (1.1) has a unique positive solution.
Proof
First, we give the existence result. On one hand, for \(u\in K\) and \(t_{0}\in [\delta,1-\delta ]\), we have
In the similar way, we also have
Combining these inequalities, for \(u\in K\), we get
Since \(\alpha _{1}\alpha _{2}<(p-1)^{2}\), there exists a sufficiently small \(r>0\) such that \(\|A(u)(t)\|>\|u\|\) for \(u\in K\cap \partial \varOmega _{r}\).
On the other hand, for \(u\in K\),
In a similar way, we have
Combining these inequalities, for \(u\in K\), we get
Since \(\alpha _{1}\alpha _{2}<(p-1)^{2}\), there exists a sufficiently large \(R>0\) such that \(\|A(u)(t)\|<\|u\|\) for \(u\in K\cap \partial \varOmega _{R}\).
Therefore by Lemma 2.5 the operator A has at least one fixed point in \(K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\). Finally, we prove that A has at most one fixed point in \(P\setminus \{ 0 \} \). It is easy to see that \(A_{1}\) and \(A_{2}\) are increasing operators with respect to the partial order induced by K. So is \(A=A_{1}A_{2}\). By Lemma 2.4 we only need to verify that A is \(u_{0}\)-sublinear for some positive \(u_{0}\in C[0,1]\). Take \(u_{0}=t(1-t)\). Set \(M=\max_{t\in [0,1]}F(t)\), where
Then we have
Furthermore, we have
So we can choose \(\theta _{1}=\int _{0}^{1} C G(s,s)F(s)\,ds\) and \(\theta _{2}=\frac{M}{2}\).
From the above discussion we know that \(A=A_{1}A_{2}\) satisfies (a) of Definition 2.3. The proof is complete if A satisfies (b) of Definition 2.3. To this end, let \(\theta _{1}u_{0}\leq u\leq \theta _{2}u_{0}\), \(\xi \in (0,1)\). Then a direct calculation gives \(A_{2}(\xi u)= \xi ^{\frac{\alpha _{2}}{p-1}}A_{2}(u)\), \(A_{1}(\xi v)= \xi ^{\frac{\alpha _{1}}{p-1}}A_{1}(v)\). Since \(\xi \in (0,1)\) and \(\alpha _{1}\alpha _{2}<(p-1)^{2}\), we get \(A(\xi u)=A_{1}( \xi ^{\frac{\alpha _{2}}{p-1}}A_{2}(u))= \xi ^{\frac{\alpha _{1}\alpha _{2}}{(p-1)^{2}}}A(u)\geq (1+\eta)\xi A(x)\) for some \(\eta >0\). □
Theorem 3.5
Assume that \(\alpha _{i}, \beta _{i}>0\) (\(i=1,2\)) with
In addition, let the functions \(f_{i}\) (\(i=1,2\)) satisfy the following assumptions::
-
(i)
\(\underline{\varphi}_{1}^{0}>0\), \(\underline{\varphi}_{2}^{0}=+ \infty \), \(\underline{\psi }_{1}^{\infty }>0\), \(\underline{\psi }_{2} ^{\infty }=+\infty\);
-
(ii)
there exists R̃ such that \(\frac{1}{6^{n_{1}}} \varphi _{q}(\frac{1}{6^{n_{2}}})\varphi _{q}(N^{1}_{\widetilde{R}}) \leq \widetilde{R}\), where
$$\begin{aligned}& N^{1}_{\widetilde{R}}=\max \bigl\{ f_{1}(t,v):0\leq t\leq 1,0\leq v\leq L(m _{1},m_{2})\varphi _{q} \bigl(N^{2}_{\widetilde{R}}\bigr)\bigr\} , \\& N^{2}_{\widetilde{R}}=\max \bigl\{ f_{2}(t,u):0\leq t\leq 1,\sigma (n_{1}) \widetilde{R}\leq u\leq \widetilde{R}\bigr\} . \end{aligned}$$
Then (1.1) has at least two positive solutions.
Proof
For any \(u\in K\cap \partial \varOmega _{\widetilde{R}}\), we have
Furthermore, we get
that is, \(\|A(u)(t)\|\leq \|u\|\) for \(u\in K\cap \partial \varOmega _{\widetilde{R}}\).
Since \(\alpha _{1}\alpha _{2}\leq (p-1)^{2}\), \(\beta _{1}\beta _{2}\geq (p-1)^{2}\), \(\underline{\varphi}_{1}^{0}>0\), \(\underline{\varphi}_{2}^{0}=+\infty \), \(\underline{\psi }_{1}^{\infty }>0\), and \(\underline{\psi }_{2}^{\infty }=+\infty \), from the proofs of Theorems 3.2 and 3.3 it follows that there exist \(r>0\) (sufficiently small) and \(R>0\) (sufficiently large) such that \(\|A(u)(t)\|\geq \|u \|\), \(u\in K\cap \partial \varOmega _{r} \), and \(\|A(u)(t)\|\geq \|u\|\), \(u\in K\cap \partial \varOmega _{R}\). Therefore by Lemma 2.5 the operator A has at least two fixed points in \(K\cap (\overline{ \varOmega }_{\widetilde{R}}\setminus \varOmega _{r})\) and \(K\cap (\overline{ \varOmega }_{R}\setminus \varOmega _{\widetilde{R}})\). □
Theorem 3.6
Assume that \(\alpha _{i}, \beta _{i}>0\) (\(i=1,2\)) with
In addition, let the functions \(f_{i}\) (\(i=1,2\)) satisfy the following assumptions:
-
(i)
\(\overline{\varphi }_{1}^{0}<+\infty \), \(\overline{\varphi } _{2}^{0}=0\), \(\overline{\psi }_{1}^{\infty }<\infty \), \(\overline{ \psi }_{2}^{\infty }=0\);
-
(ii)
there exists R̂ such that \(S(n_{1},n_{2},m_{1},0) \varphi _{q}(K^{1}_{\widehat{R}})\geq \widehat{R}\), where
$$\begin{aligned}& K^{1}_{\widehat{R}}=\min \bigl\{ f_{1}(t,v):\delta \leq t \leq 1-\delta, \\& \hphantom{K^{1}_{\widehat{R}}=}{}S(m _{1},m_{2},n_{1},0)\varphi _{q}\bigl(K^{2}_{\widehat{R}}\bigr)\leq v\leq L(m_{1},m _{2})\varphi _{q}\bigl(N^{2}_{\widetilde{R}} \bigr)\bigr\} , \\& N^{2}_{\widehat{R}}=\max \bigl\{ f_{2}(t,u):0\leq t\leq 1,\sigma (n_{1}) \widetilde{R}\leq u\leq \widetilde{R}\bigr\} , \\& K^{2}_{\widehat{R}}=\min \bigl\{ f_{2}(t,u):\delta \leq t \leq 1-\delta, \sigma (n_{1}) \widetilde{R}\leq u\leq \widetilde{R} \bigr\} . \end{aligned}$$
Then (1.1) has at least two positive solutions.
Proof
For any \(u\in K\cap \partial \varOmega _{\widehat{R}}\), we have
For \(t\in [\delta,1-\delta ]\),
Thus, for \(t\in [\delta,1-\delta ]\), we have the estimates
and
that is, \(\|A(u)(t)\|\geq \|u\|\) for \(u\in K\cap \partial \varOmega _{\widetilde{R}}\).
Since \(\alpha _{1}\alpha _{2}\geq (p-1)^{2}\), \(\beta _{1}\beta _{2}\leq (p-1)^{2}\), \(\overline{\varphi }_{1}^{0}<+ \infty \), \(\overline{\varphi }_{2}^{0}=0\), \(\overline{\psi }_{1}^{ \infty }<\infty \), and \(\overline{\psi }_{2}^{\infty }=0\), from the proof of Theorem 3.2 and Theorem 3.3 it follows that there exist \(r>0\) (sufficiently small) and \(R>0\) (sufficiently large) such that \(\|A(u)(t) \|\leq \|u\|\), \(u\in K\cap \partial \varOmega _{r} \), and \(\|A(u)(t)\| \leq \|u\|\), \(u\in K\cap \partial \varOmega _{R}\). Therefore by Lemma 2.5 the operator A has at least two fixed points in \(K\cap (\overline{ \varOmega }_{\widehat{R}}\setminus \varOmega _{r})\) and \(K\cap (\overline{ \varOmega }_{R}\setminus \varOmega _{\widehat{R}})\). □
Example 1
Assume that \(\alpha,\beta >0\). Then for the problem
we have the following existence, uniqueness, and nonexistence results:
-
(I)
If \(\alpha \beta \neq (p-1)^{2}\), then (3.1) has at least a positive solution.
-
(II)
If \(\alpha \beta <(p-1)^{2}\), then (3.1) has a unique positive solution.
-
(III)
If \(\alpha \beta =(p-1)^{2}\), then (3.1) has no positive solutions.
Proof
First, we give the proof of existence results.
(I), (II) On one hand, for \(u\in K\) and \(t_{0}\in [\delta,1- \delta ]\), we have
In a similar way, we also have
Combining these inequalities, for \(u\in K\), we get
On the other hand, for \(u\in K\),
In a similar way, we have
Combining these inequalities, for \(u\in K\), we get
We take into account the following two cases.
Case 1: If \(\alpha \beta <(p-1)^{2}\), then from Theorem 3.4 it follows that (3.1) has only a positive solution.
Case 2: If \(\alpha \beta >(p-1)^{2}\), then from (3.2) it follows that there exists \(R>1\) such that, for any \(u\in K\cap \partial \varOmega _{R}\),
By (3.3) there exists \(0< r<1\) such that, for any \(u\in K\cap \partial \varOmega _{r}\)
Therefore by Lemma 2.5 the operator A has at least one fixed point in \(K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\).
(III) We only need to show that A has no positive fixed point in K. On the contrary, if A has a positive fixed point \(u^{*}\in K\), then we have
which yields a contradiction. □
Example 2
If \(p=3\) and \(n_{1}=n_{2}=m_{1}=m_{2}=1\), then (1.1) is related to the fourth-order system
where \(f_{1}(t,v)=tv^{2}\) and \(f_{2}(t,u)=tu+tu^{3}\). Choosing \(\alpha _{1}=2\), \(\alpha _{2}=\frac{3}{2}\), \(\beta _{1}=2\), \(\beta _{2}= \frac{5}{2}\), and \(\delta =\frac{1}{4}\), it is easy to verify that
which implies that (i) of Theorem 3.5 holds.
Choosing \(\widetilde{R}=46{,}000\), via some computations we can get
which yields that (ii) of Theorem 3.5 holds. Therefore (3.4) has at least two positive solutions.
Example 3
If \(p=2\), \(n_{1}=m_{1}=1\), and \(n_{2}=m_{2}=1\), then (1.1) is related to the second-order system
where \(f_{1}(t,v)=(t+\frac{2^{58}\cdot 3^{6}}{11^{6}})v^{2}\) and
Choosing \(\alpha _{1}=2\), \(\alpha _{2}=\frac{3}{2}\), \(\beta _{1}=2\), \(\beta _{2}=\frac{1}{4}\), and \(\delta =\frac{1}{4}\), it is easy to verify that
which implies that (i) of Theorem 3.6 holds.
Now, we will show that there exists a R̂ such that (ii) of Theorem 3.5 holds. For convenience, choose \(\widehat{R}<1\). Via some computations we can get
Choosing \((\frac{\frac{2^{58}\cdot 3^{6}}{11^{6}}}{\frac{1}{4}+\frac{2^{58} \cdot 3^{6}}{11^{6}}})^{\frac{1}{5}}<\widehat{R}<1\), we have
which yields that (ii) of Theorem 3.6 holds. Therefore (3.5) has at least two positive solutions.
4 Main results of Problem (1.2)
Theorem 4.1
Assume that (H3) or (\(\widetilde{\textrm{H}}\)3) holds. In addition, assume that the functions \(F_{i}\), \(G_{i}\) (\(i=1,2\)) satisfy the following condition:
there exist \(\alpha _{i}, \beta _{i}>0\) (\(i=1,2\)) with \(\alpha _{i} \leq (p-1)\) and \(\beta _{i}\leq (p-1)\) such that
Then (1.2) has at least one positive solution.
Proof
Let E denote the Banach space \(C[0,1]\times C[0,1]\) with norm \(\|(u,v)\|= \max \{|u(t)|_{1}, |v(t)|_{1}\}\), where \(|u|_{1}=\max_{t\in [0,1]}|u(t)|\). Define the mapping \(A: E\rightarrow E\) by
where
Let \(P=\{u\in C[0,1], u\geq 0\}\), and let K be its subcone defined by
As in the proof of Lemma 3.1, it is clear that \(A : K\rightarrow K\) is completely continuous.
On one hand, from the assumption \(\overline{F}_{2}^{\infty }= \overline{G}_{2}^{\infty }=0\) it follows that there exist \(\varepsilon >0\) and \(\overline{R}>0\) such that
where ε satisfies
For given R̅, let
Then we have
Furthermore, we have the the estimates
and
Therefore, combining them with the assumption \(\beta _{i}\leq p-1\), we get that there exists a sufficiently large \(R>0\) such that, for any \((u,v)\in \partial \varOmega _{R}\cap K\),
On the other hand, from the assumption \(\underline{F}_{1}^{0}= \underline{G}_{1}^{0}=+\infty \) it follows that there exist \(M>0\) and \(r<1\) such that
where M satisfies
Then for any \((u,v)\in \partial \varOmega _{r}\cap K\) and \(t_{0}\in [ \delta,1-\delta ]\), we have
In a similar way, we also have
Furthermore, we obtain
Therefore, for any \((u,v)\in \partial \varOmega _{r}\cap K\), we have \(\|A(u,v)\|>\|(u,v)\|\).
Therefore by Lemma 2.5 the operator A has at least one fixed point in \(K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\). □
Theorem 4.2
Assume that (H3) or (\(\widetilde{\textrm{H}}\)3) holds. In addition, let the functions \(F_{i}\), \(G_{i}\) (\(i=1,2\)) satisfy the following assumption:
-
there exist \(\tilde{\alpha }_{i}, \tilde{\beta }_{i}>0\) with \(\tilde{\alpha }_{i}\geq (p-1)\) and \(\tilde{\beta }_{i}\geq (p-1)\) such that
$$ \underline{F}_{1}^{\infty }=\underline{G}_{1}^{\infty }=+ \infty,\qquad \overline{F}_{2}^{0}=\overline{G}_{2}^{0}=0. $$
Then (1.2) has at least one positive solution.
Proof
On one hand, from the assumption \(\overline{F}_{2} ^{0}=\overline{G}_{2}^{0}=0\) it follows that there exist \(\varepsilon >0\) and \(r<1\) such that
where ε satisfies
Then, for any \((u,v)\in \partial \varOmega _{r}\cap K\), we have
and
Therefore, combining these inequalities with the assumption \(\tilde{\beta }_{i}\geq p-1\), we have
On the other hand, from the assumption \(\underline{F}_{1}^{\infty }= \underline{G}_{1}^{\infty }=+\infty \) it follows that there exist \(M>0\) and \(\overline{R}>r\) such that
where M satisfies
Set \(R=\max \{\frac{\overline{R}}{\sigma (n_{1})}+1,\frac{ \overline{R}}{\sigma (m_{1})}+1\}\). Let
Then, for any \((u,v)\in \partial \varOmega _{R}\cap K\) and \(t_{0}\in [ \delta,1-\delta ]\), we consider two cases.
Case i: If \(\|(u,v)\|=|v|_{1}=R\), then \(\min_{t\in [\delta,1-\delta ]}v(t)\geq \sigma (m_{1})|v|_{1}>\overline{R}\), and we have
and
Furthermore, we obtain
Case ii: If \(\|(u,v)\|=|u|_{1}=R\), then \(\min_{t\in [\delta,1-\delta ]}u(t)\geq \sigma (n_{1})|u|_{1}>\overline{R}\), and we have
and
Furthermore, we obtain
So, for any \((u,v)\in \partial \varOmega _{R}\cap K\), we have \(\|A(u,v)\|> \|(u,v)\|\).
Therefore by Lemma 2.5 the operator A has at least one fixed point in \(K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})\). □
Example 4
Let \(\alpha,\beta >0\) and \(p=4\). Then, for the problem
it is obvious that
We choose \(\alpha _{1}=\widetilde{\alpha }_{1}=3\), \(\alpha _{2}= \tilde{\alpha }_{2}=\frac{\beta +3}{2}\), \(\beta _{1}=\tilde{\beta } _{1}=\frac{3\alpha +3}{4}\), \(\beta _{2}=\tilde{\beta }_{2}=\frac{4 \beta +3}{5}\).
Case I. If \(\alpha, \beta <3\), then it is easy to verify that
So by Theorem 4.1 the problem has at least one positive solution.
Case II. If \(\alpha, \beta >3\), then it is easy to verify that
So by Theorem 4.2 the problem has at least one positive solution.
References
D’Ambrosio, L., Mitidieri, E.: Quasilinear elliptic equations with critical potentials. Adv. Nonlinear Anal. 6(2), 147–164 (2017)
D’Ambrosio, L., Mitidieri, E.: Quasilinear elliptic systems in divergence form associated to general nonlinearities. Adv. Nonlinear Anal. 7(4), 425–447 (2018)
Djellit, A., Tas, S.: Quasilinear elliptic systems with critical Sobolev exponents in \(R^{N}\). Nonlinear Anal. 66, 1485–1497 (2007)
Aghajnt, A., Shamshiri, J.: Multiplicity of positive solutions for quasilinear elliptic p-Laplacian systems. Electron. J. Differ. Equ. 2012, 111 (2012)
Papageorgiou, N., Radulescu, V., Repovs, D.: Modern Nonlinear Analysis – Theory. Springer Monographs in Mathematics. Springer, Berlin (2019)
Lakmeche, A., Hammoudi, A.: Multiple positive solutions of the one-dimensional p-Laplacian. J. Math. Anal. Appl. 317, 43–49 (2006)
Bai, Z., Huang, B., Ge, W.: The iterative solutions for some fourth-order p-Laplace equation boundary value problems. Appl. Math. Lett. 19, 8–14 (2006)
Xu, J., Yang, Z.: Positive solutions for a fourth order p-Laplacian boundary value problem. Nonlinear Anal. 74, 2612–2623 (2011)
Zhang, X., Liu, L.: Positive solutions of fourth-order four point boundary value problems with p-Laplacian operator. J. Math. Anal. Appl. 336, 1414–1423 (2007)
Zhang, X., Liu, L.: A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with p-Laplacian. Nonlinear Anal. 68, 3127–3137 (2008)
Ding, Y., Xu, J., Zhang, X.: Positive solutions for a 2nth-order p-Laplacian boundary value problems involving all derivatives. Electron. J. Differ. Equ. 2013, 36 (2013)
Djebali, S., Moussaoui, T., Precup, R.: Fourth-order p-Laplacian nonlinear systems via the vector version of Krasnosel’skii’s fixed point theorem. Mediterr. J. Math. 6, 447–460 (2009)
Precup, R.: A vector version of Krasnosel’skii’s fixed point theorem in cones and positive periodic solutions of nonlinear systems. J. Fixed Point Theory Appl. 2, 141–151 (2007)
Prasad, K., Kameswararao, A.: Positive solutions for the system of higher order singular nonlinear boundary value problem. Math. Commun. 18, 49–60 (2013)
Ru, Y., An, Y.: Positive solutions for 2p-order and 2q-order nonlinear ordinary differential systems. J. Math. Anal. Appl. 324, 1093–1104 (2006)
Yang, X.: Existence of positive solutions for 2m-order nonlinear differential systems. Nonlinear Anal. 61, 77–95 (2005)
Kang, P., Xu, J., Wei, Z.: Positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions. Nonlinear Anal. 72, 2767–2786 (2010)
Krasnosel’skii, M.A.: Positive Solution of Operator Equations. Noordhoff, Groningnen (1964)
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Wang, P., Ru, Y. Some existence results of positive solutions for p-Laplacian systems. Bound Value Probl 2019, 9 (2019). https://doi.org/10.1186/s13661-019-1124-1
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DOI: https://doi.org/10.1186/s13661-019-1124-1