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

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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

Abstract
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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 & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945

References:

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[2]	K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems,", Birkh\, (1993). Google Scholar
[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. Google Scholar
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[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. doi: 10.1090/s0002-9947-1989-0951886-3. Google Scholar
[8]	S. Kesavan, "Nonlinear Functional Analysis,", (A First Course) in Text and Reading in Mathematics, (2004). Google Scholar
[9]	E. Landesman and A. C. Lazer, Nonlinear perturbations of linear eigenvalues problem at resonance,, J. Math. Mech., 19 (1970), 609. Google Scholar
[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. Google Scholar
[11]	Wenduan Lu, "Variational Methods in Differential Equations,", Sichuan University Publishers, (1995). Google Scholar
[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. doi: 10.1007/s00229-007-0127-x. Google Scholar
[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. doi: 10.1016/jmaa.1999.6396. Google Scholar
[14]	Zhanping Liang and Jiabao Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance,, J. Math. Anal. Appl., 354 (2009), 147. doi: 10.1016/j.jmaa.2008.12.053. Google Scholar
[15]	Shibo Liu, Remarks on multiple solutions for elliptic resonant problems,, J. Math. Anal. Appl., 336 (2007), 498. doi: 10.1016/j.jmaa.2007.01.051. Google Scholar
[16]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989). Google Scholar
[17]	P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", CBMS Issues Math. Ed., (1986). Google Scholar
[18]	S. Robinson, Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains,, Nonlinear Analysis, 21 (1993), 407. Google Scholar
[19]	S. Robinson, Multiple solutions for semilinear elliptic boundary value problem at resonance,, Electron. J. Differential Equations, 1 (1995), 1. Google Scholar
[20]	J. B. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues,, Nonlinear Anal., 48 (2002), 881. doi: 10.1016/s0362-54x100100221-2. Google Scholar
[21]	J. B. Su and Leiga Zhao, Multiple periodic solutions of ordinary differential for equations with double resonance,, Nonlinear Anal., 70 (2009), 1520. doi: 10.1016/j.na.2008.04.012. Google Scholar
[22]	C. L. Tang, Periodic solutions of nonautonomous second order systems with sublinear nonlinearity,, Proc. Amer. Math. Soc., 126 (1998), 3263. doi: 10.1090/S0002-9939-98-04706-6. Google Scholar
[23]	J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191. doi: 10.1007/BF01449041. Google Scholar
[24]	Chiara Zanini, Rotation numbers, eigenvalues, and the Poincar-Birkhoff theorem,, J. Math. Anal. Appl., 279 (2003), 290. doi: 10.1016/S0022-247X(03)00012-X. Google Scholar
[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. doi: 10.1006/jdeq.2000.3812. Google Scholar

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References:

[1]	R. A. Adams, "Sobolev Spaces,", Academic Press, (1975). Google Scholar
[2]	K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems,", Birkh\, (1993). Google Scholar
[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. Google Scholar
[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. doi: 10.1016/j.na.2008.10.031. Google Scholar
[5]	C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance,, Topol. Methods Nonlinear Anal., 157 (1990), 99. Google Scholar
[6]	Leszek Gasiński and Nikolaos S. Papageorgiou, A multiplicity theorem for double resonant periodic problems,, Advanced Nonlinear Studies, 10 (2010), 819. Google Scholar
[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. doi: 10.1090/s0002-9947-1989-0951886-3. Google Scholar
[8]	S. Kesavan, "Nonlinear Functional Analysis,", (A First Course) in Text and Reading in Mathematics, (2004). Google Scholar
[9]	E. Landesman and A. C. Lazer, Nonlinear perturbations of linear eigenvalues problem at resonance,, J. Math. Mech., 19 (1970), 609. Google Scholar
[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. Google Scholar
[11]	Wenduan Lu, "Variational Methods in Differential Equations,", Sichuan University Publishers, (1995). Google Scholar
[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. doi: 10.1007/s00229-007-0127-x. Google Scholar
[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. doi: 10.1016/jmaa.1999.6396. Google Scholar
[14]	Zhanping Liang and Jiabao Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance,, J. Math. Anal. Appl., 354 (2009), 147. doi: 10.1016/j.jmaa.2008.12.053. Google Scholar
[15]	Shibo Liu, Remarks on multiple solutions for elliptic resonant problems,, J. Math. Anal. Appl., 336 (2007), 498. doi: 10.1016/j.jmaa.2007.01.051. Google Scholar
[16]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989). Google Scholar
[17]	P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", CBMS Issues Math. Ed., (1986). Google Scholar
[18]	S. Robinson, Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains,, Nonlinear Analysis, 21 (1993), 407. Google Scholar
[19]	S. Robinson, Multiple solutions for semilinear elliptic boundary value problem at resonance,, Electron. J. Differential Equations, 1 (1995), 1. Google Scholar
[20]	J. B. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues,, Nonlinear Anal., 48 (2002), 881. doi: 10.1016/s0362-54x100100221-2. Google Scholar
[21]	J. B. Su and Leiga Zhao, Multiple periodic solutions of ordinary differential for equations with double resonance,, Nonlinear Anal., 70 (2009), 1520. doi: 10.1016/j.na.2008.04.012. Google Scholar
[22]	C. L. Tang, Periodic solutions of nonautonomous second order systems with sublinear nonlinearity,, Proc. Amer. Math. Soc., 126 (1998), 3263. doi: 10.1090/S0002-9939-98-04706-6. Google Scholar
[23]	J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191. doi: 10.1007/BF01449041. Google Scholar
[24]	Chiara Zanini, Rotation numbers, eigenvalues, and the Poincar-Birkhoff theorem,, J. Math. Anal. Appl., 279 (2003), 290. doi: 10.1016/S0022-247X(03)00012-X. Google Scholar
[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. doi: 10.1006/jdeq.2000.3812. Google Scholar

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