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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
- Mean curvature equations
- Positive solution
- Existence
- Multiplicity
- Fixed point theorem of cone expansion and compression of norm type
MSC
- 35J93
- 34B16
- 34B18
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
References
- Bartnik, R., Simon, L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87, 131–152. 1982–83 Google Scholar
- Bereanu, C., Jebelean, P., Torres, P.J.: Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space. J. Funct. Anal. 264, 270–287 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Bereanu, C., Jebelean, P., Torres, P.J.: Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. J. Funct. Anal. 265, 644–659 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Pei, M., Wang, L.: Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space. Appl. Math. Lett. 60, 50–55 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Pei, M., Wang, L.: Positive radial solutions of a mean curvature equation in Minkowski space with strong singularity. Proc. Am. Math. Soc. 145, 4423–4430 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Dai, G.: Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain. Indiana Univ. Math. J. (in press). http://www.iumj.indiana.edu/IUMJ/Preprints/7546.pdf
- Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces. Proc. Am. Math. Soc. 137, 161–169 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Bereanu, C., Jebelean, P., Mawhin, J.: Multiple solutions for Neumann and periodic problems with singular ϕ-Laplacian. J. Funct. Anal. 261, 3226–3246 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Bereanu, C., Jebelean, P., Mawhin, J.: The Dirichlet problem with mean curvature operator in Minkowski space—a variational approach. Adv. Nonlinear Stud. 14, 315–326 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Coelho, I., Corsato, C., Rivetti, S.: Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball. Topol. Methods Nonlinear Anal. 44, 23–39 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Corsato, C., Obersnel, F., Omari, P., Rivetti, S.: Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space. J. Math. Anal. Appl. 405, 227–239 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Dai, G.: Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space. Calc. Var. Partial Differ. Equ. 55, 72 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Dai, G.: Global bifurcation for problem with mean curvature operator on general domain. Nonlinear Differ. Equ. Appl. 24, 30 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Dai, G., Wang, J.: Nodal solutions to problem with mean curvature operator in Minkowski space. Differ. Integral Equ. 30, 463–480 (2017) MathSciNetMATHGoogle Scholar
- Kusahara, T., Usami, H.: A barrier method for quasilinear ordinary differential equations of the curvature type. Czechoslov. Math. J. 50, 185–196 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Bereanu, C., Jebelean, P., Mawhin, J.: Non-homogeneous boundary value problems for ordinary and partial differential equations involving singular ϕ-Laplacians. Mat. Contemp. 36, 51–65 (2009) MathSciNetMATHGoogle Scholar
- Bereanu, C., Jebelean, P., Mawhin, J.: Periodic solutions of pendulum-like perturbations of singular and bounded ϕ-Laplacians. J. Dyn. Differ. Equ. 22, 463–471 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Bereanu, C., Mawhin, J.: Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian. J. Differ. Equ. 243, 536–557 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Bereanu, C., Mawhin, J.: Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and ϕ-Laplacian. Nonlinear Differ. Equ. Appl. 15, 159–168 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Bereanu, C., Mawhin, J.: Nonhomogeneous boundary value problems for some nonlinear equations with singular ϕ-Laplacian. J. Math. Anal. Appl. 352, 218–233 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Brezis, H., Mawhin, J.: Periodic solutions of the forced relativistic pendulum. Differ. Integral Equ. 23, 801–810 (2010) MathSciNetMATHGoogle Scholar
- Herlea, D.-R., Precup, R.: Existence, localization and multiplicity of positive solutions to ϕ-Laplace equations and systems. Taiwan. J. Math. 20, 77–89 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Mawhin, J.: Periodic solutions of the forced pendulum: classical vs relativistic. Matematiche 65, 97–107 (2010) MathSciNetMATHGoogle Scholar
- Torres, P.: Nondegeneracy of the periodically forced Liénard differential equation with ϕ-Laplacian. Commun. Contemp. Math. 13, 283–292 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988) MATHGoogle Scholar