Existence and multiplicity of solutions for discrete NeumannSteklov problems with singular ϕLaplacian
 Yanqiong Lu^{1}Email author,
 Ruyun Ma^{1} and
 Bo Lu^{2}
https://doi.org/10.1186/s1366201713731
© The Author(s) 2017
Received: 4 July 2017
Accepted: 21 September 2017
Published: 5 October 2017
Abstract
This paper establishes the existence and multiplicity of solutions for the discrete NeumannSteklov problem with singular ϕLaplacian by using the method of lower and upper solutions, a priori estimates and Brouwer degree theory.
Keywords
MSC
1 Introduction
The rest of this paper is organized as follows. In Section 2, we state some notations and preliminary results. Section 3 contains the proof of the existence of one solution of (1.1) when the nonlinearity f and \(h_{0}\), \(h_{N}\) satisfy some suitable sign conditions. In Section 4, we extend the classical method of upper and lower solutions to the NeumannSteklov problem, and we obtain AmbrosettiProdi type results for the NeumannSteklov problem (1.1) in Section 5.
2 Preliminaries
For convenience, we list a few notations that will be used throughout this paper. Let \(a,b\in\mathbb{N}\) with \(a< b\), we denote \([a,b]_{\mathbb{Z}}:=\{a,a+1,\ldots, b\}\). In addition, we denote \(\sum_{s=a}^{b}u_{s}=0\) with \(b< a\) and \(\prod_{s=a}^{b}u_{s}=1\) with \(b< a\).
 (\(H_{\phi}\)):

\(\phi:(a,a)\to\mathbb{R}\) (\(0< a<\infty\)) is an increasing homeomorphism with \(\phi(0)=0\).
Lemma 2.1
Proof
Remark 2.2
Lemma 2.1 means that \((f,A,B)\) belongs to the range of the nonlinear mapping \(\mathbf{u}\rightarrow[\nabla(\phi(\Delta\mathbf{u})), \phi (\Delta u_{1}), \phi(\Delta u_{N1})]\) if and only if \(\tilde{Q}(f,A,B)=0\).
Lemma 2.3
Proof
3 The existence of solutions for nonlinear NeumannSteklov problems
Now, we give the following technical results to obtain the main result.
Lemma 3.1
Proof
Lemma 3.2
Proof
Lemma 3.2 may imply the existence of a positive solution of (3.1). A limiting argument allows to weaken the sign condition; however, this generalization can also be proved directly using another way based on the following lemma, see Section 4.
Lemma 3.3
Proof
Theorem 3.4
Proof
Corollary 3.5
Corollary 3.6
Example 3.7
4 Upper and lower solutions for the nonlinear NeumannSteklov problem
Definition 4.1
Theorem 4.2
Proof
Similarly, we also obtain that \(u_{k}\leq\beta_{k}\), \(k\in[1,N]_{\mathbb{Z}}\). Notice that if α, β are strict, then, by the same reasoning, we get that \(\alpha<\mathbf{u}<\beta\) with \(\alpha<\beta\). Moreover, from the definition of strict lower and upper solution, neither α nor β can be a solution of (4.4). So (4.4) has no solution on the boundary of \(\Omega_{\alpha, \beta}\).
By a similar argument in [8], Theorem 4, we can conclude that the existence result in Theorem 4.2 also is true when the lower and upper solutions are not ordered.
Theorem 4.3
Assume that (4.1) has a lower solution α and an upper solution β, then (4.1) has at least one solution.
Proof
If we choose the constant lower and upper solutions for (4.1) in Theorem 4.2 and Theorem 4.3, then we get the following simple existence condition.
Corollary 4.4
Notice that Theorem 4.3 can deal with the case \(a=+\infty\), the key point is the following a priori estimation result.
Lemma 4.5
Proof
Theorem 4.6
Let \(\varphi:\mathbb{R}\to\mathbb{R}\) be an increasing homeomorphism with \(\varphi(0)=0\). Suppose that all conditions of Lemma 4.5 hold and problem (4.7) has a lower solution α and an upper solution β. Then (4.7) has at least one solution.
Proof
Let c be given in Lemma 4.5, \(a_{1}=\max\{ \Vert \Delta \alpha \Vert _{\infty}, \Vert \beta \Vert _{\infty}, c\}+1\) and \(a=a_{1}+1\). Set \(\phi:(a,a)\to\mathbb{R}\) be an increasing homeomorphism such that \(\phi=\varphi\) on \([a_{1},a_{1}]\). It is easy to verify that α is a lower solution of (4.1) and β is an upper solution of (4.1). From Theorem 4.3, (4.1) has a solution u, which is also a solution of (4.7) by Lemma 4.5. □
Remark 4.7
This result is new even in the case φ is identity operator, i.e., \(\varphi=\mathit{id}_{\mathbb{R}}\).
5 AmbrosettiProdi type results for the nonlinear NeumannSteklov problem
Now we shall obtain the existence and multiplicity of the solutions of (5.1) in terms of the value of the parameter s.
Lemma 5.1
Suppose that f, \(h_{1}\), \(h_{N}\) satisfy conditions (5.2), (5.3), (5.4), respectively. Then, for each \(b\in \mathbb{R}\), there exists \(\rho=\rho(b)>0\) such that any possible solution u of (5.1) with \(s\geq b\) belongs to the open ball \(B_{\rho}\).
Proof
Theorem 5.2
Suppose that f, \(h_{1}\), \(h_{N}\) satisfy conditions (5.2), (5.3), (5.4), respectively. Then there exists \(s_{1}\in\mathbb{R}\) such that problem (5.1) has no solution with \(s>s_{1}\), at least one solution with \(s=s_{1}\), or at least two solutions with \(s< s_{1}\).
Proof
Let \(S_{i}=\{s\in\mathbb{R}\mid(5.1) \text{ has at least } i \text{ solutions}\}\) (\(i\geq1\)). We shall divide the proof into five steps to obtain the conclusion.
Step 1. We show that \(S_{1}\neq\emptyset\).
Set \(s^{\star}<\min_{k\in[2,N1]_{\mathbb{Z}}}f(k,0,0)\). From (5.2), there exists \(R^{\star}<0\) such that \(\max_{k\in[2,N1]_{\mathbb{Z}}}f(k, R^{\star},0)< s^{\star}\). Hence, \(\alpha\equiv R^{\star}\) is a strict lower solution and \(\beta\equiv0\) is a strict upper solution for (5.1) with \(s=s^{\star}\). \(s^{\star}\in S_{1}\) follows from Theorem 4.2.
Step 4. \(S_{2}\supset(\infty, s_{1})\).
Step 5. We claim that \(s_{1}\in S_{1}\).
Last, we shall give a similar result for the following dual AmbrosettiProdi condition.
Theorem 5.3
Corollary 5.4
Example 5.5
Declarations
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work is supported by NSFC (No.11626188, No.11671322, No.11501451), Gansu provincial National Science Foundation of China (No.1606RJYA232) and NWNULKQN1516.
Authors’ contributions
LY and MR completed the main study, carried out the results of this article and drafted the manuscript, LB checked the proofs and verified the calculation. All the authors read and approved the manuscript.
Competing interests
The authors confirm that they have read SpringerOpen’s guidance on competing interests and have included these in the manuscript. The authors also declare that there is no conflict of interests regarding the publication of this paper.
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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