Abstract
In this paper, we will establish some existence results of positive stationary solutions for a reaction-diffusion system
The main method used here is the well-known fixed point theorem of cone expansion and compression.
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1 Introduction
The purpose of this work is to study the existence of positive solutions for the following system:
where \(\lambda>-\pi^{2}\), \(\alpha>0\) are constants. By a positive solution, we mean a pair of functions \((u,\varphi)\) with \(u,\varphi\in C^{2}(0,1)\cap C[0,1]\) is a positive solution of (1), if \((u,\varphi)\) satisfies (1), and \(u, \varphi\geq0\), \(t\in[0,1]\), \(u,\varphi\not\equiv0\).
During the last few decades, concerning local equations, the question of the existence of solutions is one of the important topics, and it therefore has attracted much attention [1–7]. Usually, the proof is based on either topological methods or a variational approach. For example, in [1], Gu and Wang applied an abstract fixed point theorem and the estimate of the weighted \(L^{p}\) norm to study a reaction-diffusion system as follows:
where \(\Omega\in R^{N}\) is a smooth bounded domain, \(a, b>0\) are constants and \(u_{10}\), \(u_{20}\) are continuous nonnegative functions on Ω̅. In [2], by using the fixed point theorem of cone expansion and compression, Wang and An investigated the existence and multiplicity of positive solutions for the following system:
More recently, Chen and Ma obtained some existence results for (1) using the bifurcation techniques in [3, 4]. In addition, in [5–7], the authors used the variational approach to study the existence, nonexistence, multiplicity, and qualitative behavior of the solutions in the semiclassical limit for the Schrödinger-Poisson system as follows:
where \(1< p<5\).
We are mainly motivated by the recent nice works [2, 8, 9] and focus on the general local equations (1): (a) the first equation of (1) has an external force such as \(f(u,\varphi)\); (b) the second equation of (1) has the nonlinearity of arbitrary growth \(u^{\alpha}\). By employing the fixed point theorem of cone expansion and compression, we will establish some existence results of positive solutions for (1), if the exponent α satisfies two cases: (i) \(\alpha\geq1\); (ii) \({0<\alpha<1}\).
The remaining part of this paper is organized as follows. In Section 2, some preliminary results are given. In Section 3, we show the main results. In Section 4, to illustrate the new results, some applications are also given.
2 Preliminaries
Let us describe some known results established by Li in [11].
Lemma 2.1
If \(\lambda>-\pi^{2}\), then the linear boundary value problem
has a nonnegative Green function \(G(t,s)\).
Lemma 2.2
The function \(G(t,s)\) has the following properties:
-
(i)
\(G(t,s)>0\), \(\forall t, s\in(0,1)\);
-
(ii)
\(G(t,s)\leq C G(s,s)\), \(\forall t, s\in[0,1]\);
-
(iii)
\(G(t,s)\geq\delta G(t,t)G(s,s)\), \(\forall t, s\in[0,1]\).
For convenience, we assume the following conditions hold throughout this paper:
-
(H1)
\(q(t) , p(t):[0,1]\rightarrow[0,+\infty)\) are continuous, and satisfy
$$C \int_{0}^{1} \int_{0}^{1}G(s,s)K(s,\tau)p(s)q(\tau)\,d\tau\,ds< \frac{1}{2}, $$where \(K(t,s)\) denotes the Green function \(G(t,s)\) when \(\lambda=0\);
-
(H2)
\(h(t):[0,1]\rightarrow[0,+\infty)\) is continuous, and
$$0< \int_{0}^{1}G(s,s)h(s)\,ds< \infty. $$
Using the idea in [2], Lemma 2.1, and Lemma 2.2, it is easy to convert system (1) into a fixed point equation as follows:
Now define a mapping \(T:C[0,1]\rightarrow C[0,1]\) by
and define a positive cone \(P\subset C[0,1]\) as
where \(\theta\in(0,\frac{1}{2})\) is a fixed constant, \(\sigma\in (0,1)\).
Through a standard proof process, we have the following.
Lemma 2.3
Assume that (H1)-(H2) hold. Then \(T(P)\subset P\) and \(T:P\rightarrow P\) is completely continuous.
The proofs of our main results are based on the following fixed point, which can be found in [10].
Lemma 2.4
Let E be a Banach space, and \(K\subset E\) be a cone in E. Assume \(\Omega_{1}\), \(\Omega_{2}\) are open subsets of E with \(0\in\Omega_{1}\), \(\overline{\Omega}_{1}\subset\Omega_{2}\), and let \(T:K\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\rightarrow K\) be a completely continuous operator such that either:
-
(i)
\(\|Tu\|\leq\|u\|\), \(u\in K\cap\partial\Omega_{1}\) and \(\|Tu\|\geq\|u\|\), \(u\in K\cap\partial\Omega_{2}\); or
-
(ii)
\(\|Tu\|\geq\|u\|\), \(u\in K\cap\partial\Omega_{1}\) and \(\|Tu\|\leq\|u\|\), \(u\in K\cap\partial\Omega_{2}\).
Then T has a fixed point in \(K\cap(\overline{\Omega}_{2}\setminus \Omega_{1})\).
3 Main result
Theorem 3.1
Assume (H1)-(H2) hold, and \(\alpha\geq1\). In addition, the following conditions hold:
-
(H3)
\(q(t)\) satisfies
$$\min_{t\in[\theta,1-\theta]} \int_{\theta}^{1-\theta}K(t,s)q(s)\,ds>0; $$ -
(H4)
\(h(t)\) satisfies
$$0< \int_{\theta}^{1-\theta} G\biggl(\frac{1}{2},s\biggr)h(s) \,ds; $$ -
(H5)
\(f\in C[R^{+}\times R^{+}, R^{+}]\), and \(f(u,\varphi)\) are nondecreasing in u and φ for \(u, \varphi>0\), moreover,
$$ \textstyle\begin{cases} 0< u\leq\overline{u},\\ 0< \varphi\leq\overline{\varphi}, \end{cases}\displaystyle \quad \Rightarrow\quad f(u,\varphi)\leq f(\overline{u}, \overline{\varphi}), $$in addition, \(f(u,\varphi)\) satisfies the following conditions:
$$\lim_{s\rightarrow\infty}\frac{f(s,m_{1}s)}{s}=\infty, \quad\quad\lim_{s\rightarrow 0} \frac{f(s,M_{1} s)}{s}=0, $$where
$$m_{1}=\sigma^{\alpha-1}\min_{s\in[\theta,1-\theta]} \int_{\theta}^{1-\theta }K(s,\tau)q(\tau)\,d\tau,\quad\quad M_{1}= \int_{0}^{1}K(\tau,\tau)q(\tau)\,d\tau. $$
Then system (1) has at least one positive solution.
Proof
On one hand, since \(\lim_{s\rightarrow 0}\frac{f(s,M_{1} s)}{s}=0\), there exists a \(\eta>0\) with \(\eta<1\) such that
where ϵ satisfies
Taking \(r\in(0,\eta)\) and setting \(\Omega_{r}=\{u(t)\in C[0,1]:\|u\|< r\}\), then, for any \(u(t)\in\partial\Omega_{r}\cap P\), we have
Consequently,
On the other hand, since \(\lim_{s\rightarrow\infty}\frac{f(s,m_{1}s)}{s}=\infty\), there exists a \(\overline{R}>0\) such that
where ρ satisfies
Choosing \(R>\{\frac{\overline{R}}{\sigma}, 2\}\), and setting \(\Omega_{R}=\{u\in C[0,1]:\|u\|< R\}\), it is easy to verify that
Then, for any \(u(t)\in\partial\Omega_{R}\cap P\), we have
Consequently,
Then by Lemma 2.4, system (1) has a positive solution \((u(t), \int_{0}^{1}K(t,s)q(s)u^{\alpha}(s)\,ds)\). □
In the condition (H5) of Theorem 3.1, the function \(f(u,\varphi)\) is required to be nondecreasing in u and φ for \(u, \varphi>0\). If the function \(f(u,\varphi)\) is mixed monotone in the condition (H7), we impose a stronger requirement on the function \(h(t)\) in the condition (H6). Then, similar to the proof of Theorem 3.1, we can obtain the following theorem.
Theorem 3.2
Assume (H1)-(H4) hold, and \(\alpha\geq1\). In addition, the following conditions hold:
-
(H6)
\(h(t)\) satisfies \(h(t)\equiv0\) for \(t\in[0,\theta]\cup [1-\theta,1]\), and
$$0< \int_{\theta}^{1-\theta} G\biggl(\frac{1}{2},s\biggr)h(s) \,ds; $$ -
(H7)
\(f\in C[R^{+}\times R^{+}, R^{+}]\), and \(f(u,\varphi)\) is nonincreasing in u and nondecreasing in φ for \(u, \varphi>0\). In addition, \(f(u,\varphi)\) satisfies the following conditions:
$$\lim_{s\rightarrow\infty}\frac{f(s,m_{2}s)}{s}=\infty, \quad\quad\lim_{ s\rightarrow0} \frac{f(\sigma s,M_{2} s)}{s}=0, $$where
$$m_{2}=\sigma^{\alpha}\min_{s\in[\theta,1-\theta]} \int_{\theta}^{1-\theta }K(s,\tau)q(\tau)\,d\tau,\quad\quad M_{2}= \int_{0}^{1}K(\tau,\tau)q(\tau)\,d\tau. $$
Then system (1) has at least one positive solution.
Theorem 3.3
Assume (H1)-(H3), (H6) hold and \(0<\alpha<1\). In addition,
-
(H8)
\(f\in C[R^{+}\times R^{+}, R^{+}]\), and \(f(u,\varphi)\) is nondecreasing in u and nonincreasing in φ for \(u, \varphi>0\). In addition, \(f(u,\varphi)\) satisfies the following conditions:
$$\lim_{s\rightarrow\infty}\frac{f(\sigma s,M_{3}s)}{s}=\infty,\quad\quad \lim_{s\rightarrow0} \frac{f(s,m_{3}s)}{s}=0, $$where
$$m_{3}=\sigma^{\alpha}\min_{s\in[\theta,1-\theta]} \int_{\theta}^{1-\theta }K(s,\tau)q(\tau)\,d\tau,\quad\quad M_{3}= \int_{0}^{1}K(\tau,\tau)q(\tau)\,d\tau. $$
Then system (1) has at least one positive solution.
Proof
Since \(\lim_{s\rightarrow0}\frac{f(s,m_{3} s)}{s}=0\), there exists a \(\eta>0\) with \(\eta<1\) such that
where ϵ satisfies
Taking \(r\in(0,\eta)\) and setting \(\Omega_{r}=\{u\in C[0,1]:\|u\|< r\}\), then, for any \(u(t)\in\partial\Omega_{r}\cap P\), we have
Consequently,
On the other hand, since \(\lim_{s\rightarrow\infty}\frac{f(\sigma s,M_{3}s)}{s}=\infty\), there exists a \(\overline{R}>0\) such that
where ρ satisfies
Choosing \(R>\{\frac{\overline{R}}{\sigma}, 2\}\) and setting \(\Omega_{R}=\{u\in C[0,1]:\|u\|< R\}\), it is easy to see that
Then, for all \(u(t)\in\partial\Omega_{R}\cap P\), we have
Consequently,
Therefore, by Lemma 2.4, system (1) has at least one positive solution. □
4 Examples
In this section, we only give some examples to illustrate the main theorems when \(\lambda=0\) and \(\theta=\frac{1}{4}\).
Example 4.1
Let us consider the following boundary value problem:
where \(\alpha=2\), \(p(t)=t^{2}\), \(q(t)=1\), \(h(t)=t\), \(f(u,\varphi)=u^{2}+\varphi^{3}\). Next via some simple computation, it is easy to see that \(h(t):[0,1]\rightarrow[0,+\infty)\) is continuous, and
Then (H3) and (H5) hold. Since \(p(t)=t^{2}\) and \(q(t)=1\), we obtain
So (H2) and (H4) hold. Finally, \(f(u,\varphi)\) are nondecreasing in u and φ for \(u, \varphi>0\), and
where \(m=\sigma^{\alpha-1}\min_{s\in[\frac{1}{4},\frac{3}{4}]}\int_{\frac {1}{4}}^{\frac{3}{4}}K(s,\tau)q(\tau)\,d\tau=\frac{3}{256}\), \(M=\int_{0}^{1}K(\tau,\tau)q(\tau)\,d\tau=\frac{1}{6}\). Thus (H6) holds. Therefore, from Theorem 3.1, the above problem has at least one positive solution.
Example 4.2
Let us consider the following boundary value problem:
where \(\alpha=2\), \(p(t)=t^{2}\), \(q(t)=1\), \(f(u,\varphi)=\frac{\varphi^{3}}{1+u}\), and
Therefore, from Theorem 3.2, it is easy to see that the above problem has at least one positive solution.
Example 4.3
Let us consider the following boundary value problem:
where \(\alpha=\frac{1}{2}\), \(p(t)=t^{2}\), \(q(t)=1\), \(h(t)=t\), \(f(u,\varphi)=\frac{u^{2}}{1+\varphi}\). Therefore, from Theorem 3.3, it is easy to see that the above problem has at least one positive solution.
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Acknowledgements
The first author is supported by NSF of China (No. 11501165), NSF of Jiangsu Province (No. BK20130825), the Fundamental Research Funds for the Central Universities. The second author is supported by Joint fund project of science and Technology Department of Guizhou Province, No. [2014]7366.
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Wang, F., Wang, Y. Existence of positive stationary solutions for a reaction-diffusion system. Bound Value Probl 2016, 11 (2016). https://doi.org/10.1186/s13661-015-0511-5
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DOI: https://doi.org/10.1186/s13661-015-0511-5