Nontrivial solutions of secondorder singular Dirichlet systems
 Jin Zhao^{1}Email author and
 Yanchao Wang^{1}
Received: 2 August 2017
Accepted: 27 November 2017
Published: 4 December 2017
Abstract
We study the existence of nontrivial solutions for secondorder singular Dirichlet systems. The proof is based on a wellknown fixed point theorem in cones and the LeraySchauder nonlinear alternative principle. We consider a very general singularity and generalize some recent results.
Keywords
MSC
1 Introduction
2 Preliminaries
Let us first recall the following inequality, which can be found in [11].
Lemma 2.1
To prove our main results, we shall apply the following two wellknown results.
Lemma 2.2
([27])
 (I)
T has at least one fixed point in Ω̅.
 (II)
There exists \(u\in\partial\Omega\) and \(0<\lambda<1\) such that \(u=\lambda Tu+(1\lambda)p\).
Let K be a cone in X, and let D be a subset of X. We set \(D_{K}=D\cap K\) and \(\partial_{K}D=(\partial D)\cap K\).
Lemma 2.3
([28])
 (i)
\(u\neq\lambda Su\) for \(\lambda\in[0,1)\) and \(u \in\partial_{K}\Omega^{1}\), and
 (ii)
there exists \(w\in K\setminus\{0\}\) such that \(u \neq Su+\lambda w\) for all \(u \in \partial_{K}\Omega^{2}\) and all \(\lambda> 0\).
 (D_{1}):

\(\langle v,f(t,u)\rangle:[0,1]\times {\mathbb {R}}^{N}\setminus\{0\}\rightarrow {\mathbb {R}}_{+}\) is continuous;
 (D_{2}):

\(q(t)\in {\mathbb {C}}(0,1)\), \(q(t)>0\) on (0,1), and \(\int_{0}^{1}t(1t)q(t)\,\mathrm{d}t<\infty \);
 (D_{3}):

\(\langle v,e(t)\rangle:[0,1]\rightarrow {\mathbb {R}}\) is continuous, and \(\int_{0}^{1}t(1t)\langle v,e(t)\rangle\,\mathrm {d}t<\infty \).
3 Main results
In this section, we always assume that (D_{1})(D_{3}) are satisfied and \(\Gamma_{*}=0\).
Theorem 3.1
 (H_{1}):

there exists a continuous nonnegative function \(\phi_{r+\Lambda^{*}}(t)\) on \([0,1]\) such thatfor all \(t\in(0,1)\) and \(u\in {\mathbb {R}}_{+}^{N}\) with \(0<u_{v}\leq r+\Lambda^{*}\);$$\bigl\langle v,f(t,u) \bigr\rangle \geq\phi_{r+\Lambda^{*}}(t) $$
 (H_{2}):

there exist two continuous nonnegative functions \(g(\cdot)\) and \(h(\cdot)\) on \((0,\infty)\) such thatfor all \(t\in(0,1)\) and \(u\in {\mathbb {R}}_{+}^{N}\) with \(0<u_{v}\leq r+\Lambda^{*}\), where \(g(\cdot)>0\) is nonincreasing and \(h(\cdot)/g(\cdot)\) is nondecreasing;$$0 \leq \bigl\langle v,f(t,u) \bigr\rangle \leq g\bigl(u_{v}\bigr)+h\bigl(u_{v}\bigr) $$
 (H_{3}):

the following inequality is satisfied:where$$\biggl\{ 1+\frac{h(r+\Lambda^{*})}{g(r+\Lambda^{*})} \biggr\} b< \int_{0}^{r}\frac{1}{g(x)}\,\mathrm{d}x, $$$$b=\max \biggl\{ 2 \int_{0}^{1/2}t(1t)q(t)\,\mathrm{d}t, 2 \int^{1}_{1/2}t(1t)q(t)\,\mathrm{d}t \biggr\} . $$
Proof
Theorem 3.2
 (H_{4}):

there exist two continuous nonnegative functions \(g_{1}(\cdot)\), \(h_{1}(\cdot )\) on \((0,\infty)\) such thatfor all \(t\in(0,1)\) and \(u\in {\mathbb {R}}_{+}^{N}\), where \(g_{1}(\cdot)>0\) is nonincreasing, and \(h_{1}(\cdot)/g_{1}(\cdot)\) is nondecreasing;$$\bigl\langle v,f(t,u) \bigr\rangle \geq g_{1}\bigl(u_{v}\bigr)+h_{1}\bigl(u_{v}\bigr) $$
 (H_{5}):

there exists a positive constant \(R>r\) such thatwhere \(a\in(0,\frac{1}{2})\) is fixed, \(\sigma=a(1a)\), and \(0\leq\xi\leq1\) is such that$$ \frac{R}{g_{1}(R+\Lambda^{*})(1+\frac{h_{1}(\sigma R)}{g_{1}(\sigma R)})} \leq \int_{a}^{1a} G(\xi,s)q(s)\,\mathrm{d}s, $$$$ \int_{a}^{1a}G(\xi,s)q(s)\,\mathrm{d}s=\sup _{0\leq t\leq1} \int_{a}^{1a}G(t,s)q(s)\,\mathrm{d}s. $$
Proof
First, we return to the beginning of the proof of Theorem 3.1. Similarly, we only need to prove that (3.1) has a nontrivial solution u, which satisfies \(r<u_{v}\leq R\) and \(\langle v,u(t)+\gamma(t) \rangle>0\) for all \(t\in(0,1)\).
 (i)
\(u\neq\lambda Su\) for \(\lambda\in[0,1]\) and \(u \in\partial_{K}\Omega^{1}\), and
 (ii)
there exists a vector \(w\in K\setminus\{0\}\) such that \(u \neq Su+\lambda w\) for all \(\lambda> 0\) and all \(u \in\partial_{K}\Omega^{2}\).
We start with (i). Assume that there exiss \(\lambda\in[0,1]\) and \(u\in \partial_{K}\Omega^{1}\) such that \(u=\lambda Su\). Suppose that \(\lambda \neq0\). Now \(u=\lambda Su\) can lead to a contradiction following the same ideas in proving (3.5), and so (i) holds. We omit the details.
The following multiplicity result is a direct consequence of Theorems 3.1 and 3.2.
Theorem 3.3
Assume that (H_{2})(H_{5}) are satisfied. Then (1.1) has at least two nontrivial solutions u, ũ with \(\langle v,u(t) \rangle>0\), \(\langle v,\tilde{u}(t) \rangle>0\) for \(t\in(0,1)\) and \(u\gamma_{v}< r<\tilde{u}\gamma_{v}\leq R\).
Corollary 3.4
 (i)
For each \(\mu>0\), system (1.2) has at least one nontrivial solution if \(\beta<1\).
 (ii)
For each \(0< \mu< \mu_{1}\), system (1.2) has at least one nontrivial solution if \(\beta\ge1\), where \(\mu_{1}\) is a positive constant.
 (iii)
For each \(0< \mu< \mu_{1}\), system (1.2) has at least two nontrivial solutions if \(\beta>1\).
Proof
Since \(\beta>1\), we obtain that the righthand side of (3.17) tends to zero as \(R\to+\infty\). Therefore, for any \(0< \mu< \mu_{1}\), we can find R large enough such that inequality (3.17) is satisfied. Therefore, system (1.2) has another nontrivial solution. □
Similarly, we can prove the following result for system (1.3).
Corollary 3.5
Suppose that \(\alpha>0\), \(\beta>1\), \(\Gamma _{*}=0\), and \(e_{1},e_{2}\in {\mathbb {C}}([0,1],{\mathbb {R}})\). Then there exists a positive constant \(\mu_{2}\) such that system (1.3) has at least two nontrivial solutions for each \(0< \mu< \mu_{2}\).
4 Conclusions
In this paper, we established the multiplicity of nontrivial solutions for a secondorder Dirichlet system by a wellknown fixed point theorem in cones and the LeraySchauder alternative principle. Some recent results in the literature are generalized and improved. We do not need each component of the nonlinear term \(f(t,u)\) to be singular at the origin, and therefore we can deal with some new systems.
Declarations
Funding
This research is partially supported by National Natural Science Foundation of China (Grant No. 11671118).
Authors’ contributions
Both authors contributed to each part of this study equally and read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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