Existence and multiplicity of positive solutions of a one-dimensional mean curvature equation in Minkowski space
- Minghe Pei^{1},
- Libo Wang^{1}Email author and
- Xuezhe Lv^{1}
Received: 27 November 2017
Accepted: 21 March 2018
Published: 27 March 2018
Abstract
In this paper, we consider a one-dimensional mean curvature equation in Minkowski space and the corresponding one-parameter problem. By using a fixed point theorem of cone expansion and compression of norm type, the existence and multiplicity of positive solutions for the above problems are obtained. Meanwhile, as applications of our results, some examples are given.
Keywords
MSC
1 Introduction
The rest of the paper is organized as follows. By means of a fixed point theorem of cone expansion and compression of norm type (see [25]), in Sect. 2 we show the existence and multiplicity of positive solutions of (1.1) and (1.2). In Sect. 3, we give some examples to illustrate our results.
2 Main results
In order to introduce our main theorem, we need some lemmas.
Simple computations lead to the following lemma.
Lemma 2.1
Lemma 2.2
([25])
- (i)
\(\|Tx\|\leq\|x\|\) for \(x\in K\cap\partial\Omega_{1}\) and \(\|Tx\|\geq\|x\|\) for \(x\in K\cap\partial\Omega_{2}\), or
- (ii)
\(\|Tx\|\geq\|x\|\) for \(x\in K\cap\partial\Omega_{1}\) and \(\|Tx\|\leq\|x\|\) for \(x\in K\cap\partial\Omega_{2}\).
Now, we state and prove the existence and multiplicity of positive solutions of problem (1.1) and (1.2) by using a fixed point theorem of cone expansion and compression of norm type.
Theorem 2.1
Proof
Corollary 2.1
Theorem 2.2
- (i)
\(f^{0}<1\) and \(f^{1}<1\);
- (ii)there exist a compact subinterval \([r_{0},r_{1}]\subset [0,1)\) and \(\rho\in(0,1)\) such that \(\sigma\rho\in(0,1)\) andwhere \(\sigma=\frac{1}{(r_{1}-r_{0})^{2}}(\sqrt{1+(r_{1}-r_{0})^{2}}+1)\).$$f(r,s)>\phi(\sigma\rho), \quad \forall(r,s)\in[r_{0},r_{1}] \times\bigl[(1-r_{1})\rho ,\rho\bigr], $$
Proof
Remark 2.1
Corollary 2.2
- (i)there exist a compact subinterval \([r_{0},r_{1}]\subset [0,1)\) and \(\rho\in(0,1)\) such that \(\sigma\rho\in(0,1)\) andwhere \(\sigma=\frac{1}{(r_{1}-r_{0})^{2}}(\sqrt{1+(r_{1}-r_{0})^{2}}+1)\);$$f(r,s)>0,\quad \forall(r,s)\in[r_{0},r_{1}]\times \bigl[(1-r_{1})\rho,\rho\bigr]=:D, $$
- (ii)
\(\frac{\phi(\sigma\rho)}{\min_{(r,s)\in D}f(r,s)}=:\Lambda_{1}<\Lambda_{2}:= \min\{\frac{1}{f^{0}},\frac{1}{f^{1}}\}\).
Proof
Theorem 2.3
- (i)
\(f_{0}^{J}=\infty\) and \(f^{1}<1\);
- (ii)there exists \(\rho\in(0,1)\) with \(\sigma\rho\in(0,1)\) such thatwhere \(\sigma=\frac{1}{(r_{1}-r_{0})^{2}}(\sqrt{1+(r_{1}-r_{0})^{2}}+1)\);$$f(r,s)>\phi(\sigma\rho), \quad \forall(r,s)\in J\times\bigl[(1-r_{1}) \rho,\rho\bigr], $$
- (iii)there exists \(\rho_{0}\in(0,\rho)\) such that$$f(r,s)< \phi(\rho_{0}),\quad \forall(r,s)\in[0,1]\times[0, \rho_{0}]. $$
Proof
Corollary 2.3
- (i)
\(f_{0}^{J}=\infty\);
- (ii)there exists \(\rho\in(0,1)\) with \(\sigma\rho\in(0,1)\) such thatwhere \(\sigma=\frac{1}{(r_{1}-r_{0})^{2}}(\sqrt{1+(r_{1}-r_{0})^{2}}+1)\);$$f(r,s)>0,\quad \forall(r,s)\in J\times\bigl[(1-r_{1})\rho,\rho \bigr]=:D_{\rho}, $$
- (iii)there exists \(\rho^{*}\in(0,\rho)\) such thatwhere \(D_{\rho^{*}}:= [0,1]\times[0,\rho^{*}]\).$$\frac{\phi(\sigma\rho)}{\min_{(r,s)\in D_{\rho}}f(r,s)} =:\Lambda_{1}< \Lambda_{2}:=\min\biggl\{ \frac{1}{f^{1}},\frac{\phi(\rho^{*})}{\max_{(r,s)\in D_{\rho^{*}}}f(r,s)}\biggr\} , $$
3 Examples
In this section, we give some examples to demonstrate the applications of the our results.
Example 3.1
Let \(f(r,s)=(r-\frac{1}{2})^{2}s^{p}/(1-s^{2})^{q}\) on \([0,1]\times[0,1)\). Take \(J=[\frac{3}{4},1]\), it is easy to see that \(f_{0}^{J}=\infty\) and \(f^{1}=0\). By Theorem 2.1, the problem (3.1) has at least one positive solution. We note that Theorem 1 of [2] cannot guarantee this conclusion since \(f(\frac{1}{2},s)=0\), \(\forall s\in[0,1)\).
Example 3.2
Example 3.3
Declarations
Acknowledgements
The authors thank the referee for valuable suggestions, which led to improvement of the original manuscript.
Funding
This work was supported by the Education Department of JiLin Province ([2016]45).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final 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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