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

Previous Article
Large time behavior for the full compressible magnetohydrodynamic flows
CPAA Home
This Issue
Next Article
Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities

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

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

Qiong Meng ^1, and X. H. Tang ^2,

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

Abstract
Related Papers

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.

Keywords: Leray-Schauder degree, Lazer Solimini., Periodic solution, Morse index.

Mathematics Subject Classification: 34C25, 37B30, 37J4.

Citation: Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945

References:

[1]	R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.
[2]	K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser, Boston, 1993.
[3]	K. C. Chang, S. J. Li and J. Q. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problem, Topol. Methods Nonlinear Anal., 3 (1994), 179-187.
[4]	Jorge Cossio, Sigifredo Herrón and Carlos Vélez, Existence of solutions for an asymptotically linear Dirichlet problem via Lazer-Solimini results, Topol. Methods Nonlinear Anal., 71 (2009), 66-71. doi: 10.1016/j.na.2008.10.031.
[5]	C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Topol. Methods Nonlinear Anal., 157 (1990), 99-116.
[6]	Leszek Gasiński and Nikolaos S. Papageorgiou, A multiplicity theorem for double resonant periodic problems, Advanced Nonlinear Studies, 10 (2010), 819-836.
[7]	R. Iannacci, M. N. Nkashama and J. R. Ward, Jr., Nonlinear second order elliptic partial differential equations at resonance, Trans. of the AMS, 311 (1989), 711-726. doi: 10.1090/s0002-9947-1989-0951886-3.
[8]	S. Kesavan, "Nonlinear Functional Analysis," (A First Course) in Text and Reading in Mathematics, vol. 28, Hindustan Book Agency, India, 2004.
[9]	E. Landesman and A. C. Lazer, Nonlinear perturbations of linear eigenvalues problem at resonance, J. Math. Mech., 19 (1970), 609-623.
[10]	A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Topol. Methods Nonlinear Anal., 12 (1988), 761-775.
[11]	Wenduan Lu, "Variational Methods in Differential Equations," Sichuan University Publishers, 1995.
[12]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations, Manuscripta Math., 124 (2007), 507-531. doi: 10.1007/s00229-007-0127-x.
[13]	S. Li and W. Zou, The computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue, J. Math. Anal. Appl., 235 (1999), 237-259. doi: 10.1016/jmaa.1999.6396.
[14]	Zhanping Liang and Jiabao Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158. doi: 10.1016/j.jmaa.2008.12.053.
[15]	Shibo Liu, Remarks on multiple solutions for elliptic resonant problems, J. Math. Anal. Appl., 336 (2007), 498-505. doi: 10.1016/j.jmaa.2007.01.051.
[16]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989.
[17]	P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Issues Math. Ed., vol. 65, 1986.
[18]	S. Robinson, Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains, Nonlinear Analysis, Theory, Methods and Applications, 21 (1993), 407-424.
[19]	S. Robinson, Multiple solutions for semilinear elliptic boundary value problem at resonance, Electron. J. Differential Equations, 1 (1995), 1-14.
[20]	J. B. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895. doi: 10.1016/s0362-54x100100221-2.
[21]	J. B. Su and Leiga Zhao, Multiple periodic solutions of ordinary differential for equations with double resonance, Nonlinear Anal., 70 (2009), 1520-1527. doi: 10.1016/j.na.2008.04.012.
[22]	C. L. Tang, Periodic solutions of nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc., 126 (1998), 3263-3270. doi: 10.1090/S0002-9939-98-04706-6.
[23]	J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.
[24]	Chiara Zanini, Rotation numbers, eigenvalues, and the Poincar-Birkhoff theorem, J. Math. Anal. Appl., 279 (2003) 290-307. doi: 10.1016/S0022-247X(03)00012-X.
[25]	W. Zou and J. Q. Liu, Multiple solutions for resonant elliptic equations via local linking theory and morse theory, Journal of Differential Equations, 170 (2001), 68-95. doi: 10.1006/jdeq.2000.3812.

show all references

References:

[1]	R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.
[2]	K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser, Boston, 1993.
[3]	K. C. Chang, S. J. Li and J. Q. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problem, Topol. Methods Nonlinear Anal., 3 (1994), 179-187.
[4]	Jorge Cossio, Sigifredo Herrón and Carlos Vélez, Existence of solutions for an asymptotically linear Dirichlet problem via Lazer-Solimini results, Topol. Methods Nonlinear Anal., 71 (2009), 66-71. doi: 10.1016/j.na.2008.10.031.
[5]	C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Topol. Methods Nonlinear Anal., 157 (1990), 99-116.
[6]	Leszek Gasiński and Nikolaos S. Papageorgiou, A multiplicity theorem for double resonant periodic problems, Advanced Nonlinear Studies, 10 (2010), 819-836.
[7]	R. Iannacci, M. N. Nkashama and J. R. Ward, Jr., Nonlinear second order elliptic partial differential equations at resonance, Trans. of the AMS, 311 (1989), 711-726. doi: 10.1090/s0002-9947-1989-0951886-3.
[8]	S. Kesavan, "Nonlinear Functional Analysis," (A First Course) in Text and Reading in Mathematics, vol. 28, Hindustan Book Agency, India, 2004.
[9]	E. Landesman and A. C. Lazer, Nonlinear perturbations of linear eigenvalues problem at resonance, J. Math. Mech., 19 (1970), 609-623.
[10]	A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Topol. Methods Nonlinear Anal., 12 (1988), 761-775.
[11]	Wenduan Lu, "Variational Methods in Differential Equations," Sichuan University Publishers, 1995.
[12]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations, Manuscripta Math., 124 (2007), 507-531. doi: 10.1007/s00229-007-0127-x.
[13]	S. Li and W. Zou, The computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue, J. Math. Anal. Appl., 235 (1999), 237-259. doi: 10.1016/jmaa.1999.6396.
[14]	Zhanping Liang and Jiabao Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158. doi: 10.1016/j.jmaa.2008.12.053.
[15]	Shibo Liu, Remarks on multiple solutions for elliptic resonant problems, J. Math. Anal. Appl., 336 (2007), 498-505. doi: 10.1016/j.jmaa.2007.01.051.
[16]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989.
[17]	P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Issues Math. Ed., vol. 65, 1986.
[18]	S. Robinson, Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains, Nonlinear Analysis, Theory, Methods and Applications, 21 (1993), 407-424.
[19]	S. Robinson, Multiple solutions for semilinear elliptic boundary value problem at resonance, Electron. J. Differential Equations, 1 (1995), 1-14.
[20]	J. B. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895. doi: 10.1016/s0362-54x100100221-2.
[21]	J. B. Su and Leiga Zhao, Multiple periodic solutions of ordinary differential for equations with double resonance, Nonlinear Anal., 70 (2009), 1520-1527. doi: 10.1016/j.na.2008.04.012.
[22]	C. L. Tang, Periodic solutions of nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc., 126 (1998), 3263-3270. doi: 10.1090/S0002-9939-98-04706-6.
[23]	J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.
[24]	Chiara Zanini, Rotation numbers, eigenvalues, and the Poincar-Birkhoff theorem, J. Math. Anal. Appl., 279 (2003) 290-307. doi: 10.1016/S0022-247X(03)00012-X.
[25]	W. Zou and J. Q. Liu, Multiple solutions for resonant elliptic equations via local linking theory and morse theory, Journal of Differential Equations, 170 (2001), 68-95. doi: 10.1006/jdeq.2000.3812.

[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 and Continuous Dynamical Systems - S, 2022, 15 (1) : 213-227. doi: 10.3934/dcdss.2021015
[2]	Marian Gidea. Leray functor and orbital Conley index for non-invariant sets. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 617-630. doi: 10.3934/dcds.1999.5.617
[3]	Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
[4]	Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure and Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507
[5]	Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations and Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081
[6]	Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 3805-3816. doi: 10.3934/era.2021062
[7]	Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985
[8]	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 and Continuous Dynamical Systems, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107
[9]	Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure and Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823
[10]	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 and Applied Analysis, 2020, 19 (5) : 2839-2852. doi: 10.3934/cpaa.2020124
[11]	Ningning Ye, Zengyun Hu, Zhidong Teng. Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1361-1384. doi: 10.3934/cpaa.2022022
[12]	Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure and Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487
[13]	Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385
[14]	Kaifa Wang, Aijun Fan. Uniform persistence and periodic solution of chemostat-type model with antibiotic. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 789-795. doi: 10.3934/dcdsb.2004.4.789
[15]	Zhibo Cheng, Xiaoxiao Cui. Positive periodic solution for generalized Basener-Ross model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4361-4382. doi: 10.3934/dcdsb.2020101
[16]	Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[17]	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 and Applied Analysis, 2020, 19 (3) : 1351-1365. doi: 10.3934/cpaa.2020066
[18]	Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
[19]	Pablo Amster, Pablo De Nápoli. Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator. Discrete and 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 and Continuous Dynamical Systems, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Qiong Meng X. H. Tang
on Google Scholar
Qiong Meng X. H. Tang

[Back to Top]