American Institute of Mathematical Sciences

Advanced
February  2020, 25(2): 545-554. doi: 10.3934/dcdsb.2019253

Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach

José Ángel Cid 1,, and  Luís Sanchez 2,

1. 

Departamento de Matemáticas, Universidade de Vigo, 32004, Campus de Ourense, Spain
2. 

CMAFcIO - Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

* Corresponding author: José Ángel Cid

Dedicated to Professor Juan José Nieto on the occasion of his 60th birthday.

Received  January 2019 Revised  May 2019 Published  November 2019

Fund Project: J. Á. Cid was partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2017-85054-C2-1-P and L. Sanchez was supported by Fundação para a Ciência e a Tecnologia, UID/MAT/04561/2019.

We deal with the existence of nonnegative and nontrivial $ T $–periodic solutions for the equation $ x'' = r(t)x^{\alpha}-s(t)x^{\beta} $ where $ r $ and $ s $ are continuous $ T $–periodic functions and $ 0<\alpha<\beta<1 $. This equation has been studied in connection with the valveless pumping phenomenon and we will take advantage of its variational structure in order to guarantee its solvability by means of the mountain pass theorem of Ambrosetti and Rabinowitz.

Keywords: Periodic solution, regular equation, critical point theory, mountain pass theorem, Liebau phenomenon.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34B18.
Citation: José Ángel Cid, Luís Sanchez. Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 545-554. doi: 10.3934/dcdsb.2019253
References:
[1]

C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations, 69 (1987), 310-321.  doi: 10.1016/0022-0396(87)90122-7.  Google Scholar

[2]

A. R. Chouikha, Monotonicity of the period function for some planar differential systems. Ⅰ. Conservative and quadratic systems, Appl. Math., 32 (2005), 305-325.  doi: 10.4064/am32-3-5.  Google Scholar

[3]

S.-N. Chow and D. Wang, On the monotonicity of the period function of some second order equations, Časopis Pěst. Mat., 111 (1986), 14–25, 89.  Google Scholar

[4]

J. A. Cid, G. Infante, M. Tvrdý and M. Zima, A topological approach to periodic oscillations related to the Liebau phenomenon, J. Math. Anal. Appl., 423 (2015), 1546-1556.  doi: 10.1016/j.jmaa.2014.10.054.  Google Scholar

[5]

J. A. Cid, G. Infante, M. Tvrdý and M. Zima, New results for the Liebau phenomenon via fixed point index, Nonlinear Anal. Real World Appl., 35 (2017), 457-469.  doi: 10.1016/j.nonrwa.2016.11.009.  Google Scholar

[6]

J. A. Cid, G. Propst and M. Tvrdý, On the pumping effect in a pipe/tank flow configuration with friction, Phys. D, 273/274 (2014), 28-33.  doi: 10.1016/j.physd.2014.01.010.  Google Scholar

[7]

I. Coelho and L. Sanchez, Travelling wave profiles in some models with nonlinear diffusion, Appl. Math. Comput., 235 (2014), 469-481.  doi: 10.1016/j.amc.2014.02.104.  Google Scholar

[8]

P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[9]

A. Gavioli and L. Sanchez, Positive homoclinic solutions to some Schrödinger type equations, Differential Integral Equations, 29 (2016), 665-682.   Google Scholar

[10]

G. Propst, Pumping effects in models of periodically forced flow configurations, Phys. D, 217 (2006), 193-201.  doi: 10.1016/j.physd.2006.04.007.  Google Scholar

[11]

R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098346.  Google Scholar

[12]

A. Sfecci, From isochronous potentials to isochronous systems, J. Differential Equations, 258 (2015), 1791-1800.  doi: 10.1016/j.jde.2014.11.013.  Google Scholar

[13] P. J. Torres, Mathematical Models with Singularities. A Zoo of Singular Creatures, Atlantis Briefs in Differential Equations, 1. Atlantis Press, Paris, 2015.  doi: 10.2991/978-94-6239-106-2.  Google Scholar
[14]

F. Wang, J. A. Cid and M. Zima, Lyapunov stability for regular equations and applications to the Liebau phenomenon, Discrete Contin. Dyn. Syst., 38 (2018), 4657-4674.  doi: 10.3934/dcds.2018204.  Google Scholar

[15]

W. S. Zhou, X. L. Qin, G. K. Xu and X. D. Wei, On the one-dimensional p-Laplacian with a singular nonlinearity, Nonlinear Anal., 75 (2012), 3994-4005.  doi: 10.1016/j.na.2012.02.015.  Google Scholar

show all references

References:
[1]

C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations, 69 (1987), 310-321.  doi: 10.1016/0022-0396(87)90122-7.  Google Scholar

[2]

A. R. Chouikha, Monotonicity of the period function for some planar differential systems. Ⅰ. Conservative and quadratic systems, Appl. Math., 32 (2005), 305-325.  doi: 10.4064/am32-3-5.  Google Scholar

[3]

S.-N. Chow and D. Wang, On the monotonicity of the period function of some second order equations, Časopis Pěst. Mat., 111 (1986), 14–25, 89.  Google Scholar

[4]

J. A. Cid, G. Infante, M. Tvrdý and M. Zima, A topological approach to periodic oscillations related to the Liebau phenomenon, J. Math. Anal. Appl., 423 (2015), 1546-1556.  doi: 10.1016/j.jmaa.2014.10.054.  Google Scholar

[5]

J. A. Cid, G. Infante, M. Tvrdý and M. Zima, New results for the Liebau phenomenon via fixed point index, Nonlinear Anal. Real World Appl., 35 (2017), 457-469.  doi: 10.1016/j.nonrwa.2016.11.009.  Google Scholar

[6]

J. A. Cid, G. Propst and M. Tvrdý, On the pumping effect in a pipe/tank flow configuration with friction, Phys. D, 273/274 (2014), 28-33.  doi: 10.1016/j.physd.2014.01.010.  Google Scholar

[7]

I. Coelho and L. Sanchez, Travelling wave profiles in some models with nonlinear diffusion, Appl. Math. Comput., 235 (2014), 469-481.  doi: 10.1016/j.amc.2014.02.104.  Google Scholar

[8]

P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[9]

A. Gavioli and L. Sanchez, Positive homoclinic solutions to some Schrödinger type equations, Differential Integral Equations, 29 (2016), 665-682.   Google Scholar

[10]

G. Propst, Pumping effects in models of periodically forced flow configurations, Phys. D, 217 (2006), 193-201.  doi: 10.1016/j.physd.2006.04.007.  Google Scholar

[11]

R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098346.  Google Scholar

[12]

A. Sfecci, From isochronous potentials to isochronous systems, J. Differential Equations, 258 (2015), 1791-1800.  doi: 10.1016/j.jde.2014.11.013.  Google Scholar

[13] P. J. Torres, Mathematical Models with Singularities. A Zoo of Singular Creatures, Atlantis Briefs in Differential Equations, 1. Atlantis Press, Paris, 2015.  doi: 10.2991/978-94-6239-106-2.  Google Scholar
[14]

F. Wang, J. A. Cid and M. Zima, Lyapunov stability for regular equations and applications to the Liebau phenomenon, Discrete Contin. Dyn. Syst., 38 (2018), 4657-4674.  doi: 10.3934/dcds.2018204.  Google Scholar

[15]

W. S. Zhou, X. L. Qin, G. K. Xu and X. D. Wei, On the one-dimensional p-Laplacian with a singular nonlinearity, Nonlinear Anal., 75 (2012), 3994-4005.  doi: 10.1016/j.na.2012.02.015.  Google Scholar

Figure 1.  Graph of the potential $ V $ for the values $ r = 12, \, s = 8, \, \alpha = \frac 1 2 $ and $ \beta = \frac 3 4 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Phase plane for (3) with the values $ r = 12, \, s = 8, \, \alpha = \frac 1 2 $ and $ \beta = \frac 3 4 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[2]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[3]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[4]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[5]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[6]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349
[7]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065
[8]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053
[9]

Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391
[10]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[11]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[12]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153
[13]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[14]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[15]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[16]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035
[17]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[18]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[19]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[20]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (140)
  • HTML views (113)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences