# American Institute of Mathematical Sciences

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 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.

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##### 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.
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