May  2012, 11(3): 945-958. doi: 10.3934/cpaa.2012.11.945

Multiple solutions of second-order ordinary differential equation via Morse theory

1. 

School of Mathematics Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

2. 

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083

Received  September 2010 Revised  September 2011 Published  December 2011

In this paper, we consider the the second-order ordinary differential equation with periodic boundary problem $ - \ddot{x}(t)=f(t,x(t))$, subject to $x(0)-x(2\pi)=\dot{x}(0)-\dot{x}(2\pi)=0$, where $f:C([0, 2\pi]\times R, R)$. The operator $K=(-\frac{d^2}{dt^2}+I)^{-1}$ plays an important role. By using Morse index, Leray-Schauder degree and Morse index theorem of the type Lazer-Solimini, we obtain that the equation has at least two or three nontrivial solutions without assuming nondegeneracy of critical points and has at least four nontrivial solutions assuming nondegeneracy of critical points.
Citation: Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945
References:
[1]

Academic Press, New York, 1975. Google Scholar

[2]

Birkhäuser, Boston, 1993.  Google Scholar

[3]

Topol. Methods Nonlinear Anal., 3 (1994), 179-187.  Google Scholar

[4]

Topol. Methods Nonlinear Anal., 71 (2009), 66-71. doi: 10.1016/j.na.2008.10.031.  Google Scholar

[5]

Topol. Methods Nonlinear Anal., 157 (1990), 99-116.  Google Scholar

[6]

Advanced Nonlinear Studies, 10 (2010), 819-836.  Google Scholar

[7]

Trans. of the AMS, 311 (1989), 711-726. doi: 10.1090/s0002-9947-1989-0951886-3.  Google Scholar

[8]

(A First Course) in Text and Reading in Mathematics, vol. 28, Hindustan Book Agency, India, 2004. Google Scholar

[9]

J. Math. Mech., 19 (1970), 609-623. Google Scholar

[10]

Topol. Methods Nonlinear Anal., 12 (1988), 761-775.  Google Scholar

[11]

Sichuan University Publishers, 1995. Google Scholar

[12]

Manuscripta Math., 124 (2007), 507-531. doi: 10.1007/s00229-007-0127-x.  Google Scholar

[13]

J. Math. Anal. Appl., 235 (1999), 237-259. doi: 10.1016/jmaa.1999.6396.  Google Scholar

[14]

J. Math. Anal. Appl., 354 (2009), 147-158. doi: 10.1016/j.jmaa.2008.12.053.  Google Scholar

[15]

J. Math. Anal. Appl., 336 (2007), 498-505. doi: 10.1016/j.jmaa.2007.01.051.  Google Scholar

[16]

Springer-Verlag, New York, 1989.  Google Scholar

[17]

CBMS Issues Math. Ed., vol. 65, 1986.  Google Scholar

[18]

Nonlinear Analysis, Theory, Methods and Applications, 21 (1993), 407-424.  Google Scholar

[19]

Electron. J. Differential Equations, 1 (1995), 1-14.  Google Scholar

[20]

Nonlinear Anal., 48 (2002), 881-895. doi: 10.1016/s0362-54x100100221-2.  Google Scholar

[21]

Nonlinear Anal., 70 (2009), 1520-1527. doi: 10.1016/j.na.2008.04.012.  Google Scholar

[22]

Proc. Amer. Math. Soc., 126 (1998), 3263-3270. doi: 10.1090/S0002-9939-98-04706-6.  Google Scholar

[23]

Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar

[24]

J. Math. Anal. Appl., 279 (2003) 290-307. doi: 10.1016/S0022-247X(03)00012-X.  Google Scholar

[25]

Journal of Differential Equations, 170 (2001), 68-95. doi: 10.1006/jdeq.2000.3812.  Google Scholar

show all references

References:
[1]

Academic Press, New York, 1975. Google Scholar

[2]

Birkhäuser, Boston, 1993.  Google Scholar

[3]

Topol. Methods Nonlinear Anal., 3 (1994), 179-187.  Google Scholar

[4]

Topol. Methods Nonlinear Anal., 71 (2009), 66-71. doi: 10.1016/j.na.2008.10.031.  Google Scholar

[5]

Topol. Methods Nonlinear Anal., 157 (1990), 99-116.  Google Scholar

[6]

Advanced Nonlinear Studies, 10 (2010), 819-836.  Google Scholar

[7]

Trans. of the AMS, 311 (1989), 711-726. doi: 10.1090/s0002-9947-1989-0951886-3.  Google Scholar

[8]

(A First Course) in Text and Reading in Mathematics, vol. 28, Hindustan Book Agency, India, 2004. Google Scholar

[9]

J. Math. Mech., 19 (1970), 609-623. Google Scholar

[10]

Topol. Methods Nonlinear Anal., 12 (1988), 761-775.  Google Scholar

[11]

Sichuan University Publishers, 1995. Google Scholar

[12]

Manuscripta Math., 124 (2007), 507-531. doi: 10.1007/s00229-007-0127-x.  Google Scholar

[13]

J. Math. Anal. Appl., 235 (1999), 237-259. doi: 10.1016/jmaa.1999.6396.  Google Scholar

[14]

J. Math. Anal. Appl., 354 (2009), 147-158. doi: 10.1016/j.jmaa.2008.12.053.  Google Scholar

[15]

J. Math. Anal. Appl., 336 (2007), 498-505. doi: 10.1016/j.jmaa.2007.01.051.  Google Scholar

[16]

Springer-Verlag, New York, 1989.  Google Scholar

[17]

CBMS Issues Math. Ed., vol. 65, 1986.  Google Scholar

[18]

Nonlinear Analysis, Theory, Methods and Applications, 21 (1993), 407-424.  Google Scholar

[19]

Electron. J. Differential Equations, 1 (1995), 1-14.  Google Scholar

[20]

Nonlinear Anal., 48 (2002), 881-895. doi: 10.1016/s0362-54x100100221-2.  Google Scholar

[21]

Nonlinear Anal., 70 (2009), 1520-1527. doi: 10.1016/j.na.2008.04.012.  Google Scholar

[22]

Proc. Amer. Math. Soc., 126 (1998), 3263-3270. doi: 10.1090/S0002-9939-98-04706-6.  Google Scholar

[23]

Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar

[24]

J. Math. Anal. Appl., 279 (2003) 290-307. doi: 10.1016/S0022-247X(03)00012-X.  Google Scholar

[25]

Journal of Differential Equations, 170 (2001), 68-95. doi: 10.1006/jdeq.2000.3812.  Google Scholar

[1]

Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021015

[2]

Marian Gidea. Leray functor and orbital Conley index for non-invariant sets. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 617-630. doi: 10.3934/dcds.1999.5.617

[3]

Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020081

[4]

Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003

[5]

Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure & Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507

[6]

Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985

[7]

Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107

[8]

Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823

[9]

Abdelbaki Selmi, Abdellaziz Harrabi, Cherif Zaidi. Nonexistence results on the space or the half space of $ -\Delta u+\lambda u = |u|^{p-1}u $ via the Morse index. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2839-2852. doi: 10.3934/cpaa.2020124

[10]

Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487

[11]

Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385

[12]

Kaifa Wang, Aijun Fan. Uniform persistence and periodic solution of chemostat-type model with antibiotic. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 789-795. doi: 10.3934/dcdsb.2004.4.789

[13]

Zhibo Cheng, Xiaoxiao Cui. Positive periodic solution for generalized Basener-Ross model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4361-4382. doi: 10.3934/dcdsb.2020101

[14]

Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365

[15]

Gui-Dong Li, Yong-Yong Li, Xiao-Qi Liu, Chun-Lei Tang. A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1351-1365. doi: 10.3934/cpaa.2020066

[16]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121

[17]

María Barbero Liñán, Hernán Cendra, Eduardo García Toraño, David Martín de Diego. Morse families and Dirac systems. Journal of Geometric Mechanics, 2019, 11 (4) : 487-510. doi: 10.3934/jgm.2019024

[18]

Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006

[19]

Pablo Amster, Pablo De Nápoli. Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 757-767. doi: 10.3934/dcds.2011.29.757

[20]

Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]