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November  2013, 12(6): 2361-2380. doi: 10.3934/cpaa.2013.12.2361

Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance

Shiwang Ma 1,

1. 

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071

Received  July 2011 Revised  May 2012 Published  May 2013

In this paper, we consider the existence of nontrivial $1$-periodic solutions of the following Hamiltonian systems \begin{eqnarray} -J\dot{z}=H'(t,z), z\in R^{2N}, \end{eqnarray} where $J$ is the standard symplectic matrix of $2N\times 2N$, $H\in C^2 ( [0,1] \times R^{2N}, R)$ is $1$-periodic in its first variable and $H'(t,z)$ denotes the gradient of $H$ with respect to the variable $z$. Furthermore, $H'(t,z)$ is asymptotically linear both at origin and at infinity. Based on the precise computations of the critical groups, Maslov-type index theory and Galerkin approximation procedure, we obtain some existence results for nontrivial $1$-periodic solutions under new classes of conditions. It turns out that our main results improve sharply some known results in the literature.
Keywords: periodic solution, Critical group, resonance., Galerkin approximation, Maslov-type index.
Mathematics Subject Classification: Primary: 34B15, 34C25, 37J4.
Citation: Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361
References:
[1]

H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Sup. Pisa Cl. Sci. Ser. IV, 7 (1980), 539-603.  Google Scholar

[2]

H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math., 32 (1980), 149-189. doi: 10.1007/BF01298187.  Google Scholar

[3]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some problems with strong resonance at infinity, Nonlinear Anal. TMA, 7 (1983), 241-273. doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[4]

T. Bartsch and S. J. Li, Critical point theory for asymptotically quadratic functionals with applications to problems at resonance, Nonlinear Anal. TMA, 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

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G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336 (in Italian).  Google Scholar

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K. C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure. Appl. Math., 34 (1981), 693-712. doi: 10.1002/cpa.3160340503.  Google Scholar

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K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solutions Problems," Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[8]

K. C. Chang, J. Q. Liu and M. J. Liu, Nontrivial periodic solutions for strong resonance Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 14 (1997), 103-117. doi: 10.1016/S0294-1449(97)80150-3.  Google Scholar

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C. C. Conley and E. Zehnder, Morse type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37 (1984), 207-253. doi: 10.1002/cpa.3160370204.  Google Scholar

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G. Fei and Q. Qiu, Periodic solutions of asymptotically linear Hamiltonian systems, Chinese Ann. Math. Ser. B, 18 (1997), 359-372. doi: 10.1006/jdeq.1995.1124.  Google Scholar

[11]

G. Fei, Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity, J. Differential Equations, 121 (1995), 121-133. doi: 10.1006/jdeq.1995.1124.  Google Scholar

[12]

D. Gromoll and W. Meyer, On differentiable functions with isolated critical point, Topology, 8 (1969), 361-369. doi: 10.1016/0040-9383(69)90022-6.  Google Scholar

[13]

Y. X. Guo, "Morse Theory for Strongly Indefinite Functional and Its Applications," Doctoral thesis, Institute of Mathematics, Peking University, Beijing, 1999. doi: 10.1142/9789812704283_0013.  Google Scholar

[14]

Y. X. Guo, Nontrivial periodic solutions for asymptotically linear Hamiltonian systems with resonance, J. Differential Equations, 175 (2001), 71-87. doi: 10.1006/jdeq.2000.3966.  Google Scholar

[15]

N. Hirano and T. Nishimura, Multiplicity results for semilinear elliptic problems at resonance and with jumping non-linearities, J. Math. Anal. Appl., 180 (1993), 566-586. doi: 10.1006/jmaa.1993.1417.  Google Scholar

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S. Li and J. Q. Liu, Morse theory and asymptotically linear Hamiltonian systems, J. Differential Equations, 78 (1989), 53-73. doi: 0022-0396(89)90075-2.  Google Scholar

[17]

S. Li and J. Q. Liu, Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance, Houston J. Math., 25 (1999), 563-582.  Google Scholar

[18]

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: 10.1006/jmaa.1999.6396.  Google Scholar

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Y. Long and E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, in "Stochastic Processes, Physics and Geometry" (S. Albeverio, et al. Eds.), Proceedings of Conference in Asconal/Locarno, Switzerland, World Scientific, Singapore, 1990, pp. 528-563.  Google Scholar

[20]

Y. Long, Maslov-type index, degenerate critical points and asymptotically linear Hamiltonian systems, Sci. China Ser. A, 33 (1990), 1409-1419.  Google Scholar

[21]

S. Ma, Infinitely many periodic solutions for asymptotically linear Hamiltonian systems,, Rocky Mountain J. Math., ().   Google Scholar

[22]

S. Ma, Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems, J. Differential Equations, 248 (2010), 2435-2457. doi: 10.1016/j.jde.2009.11.013.  Google Scholar

[23]

S. Ma, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, Nonlinear Anal. TMA, 73 (2010), 3856-3872. doi: 10.1016/j.na.2010.08.013.  Google Scholar

[24]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Appl. Math. Sci., 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[25]

P. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," in CBMS Reg. Conf. Ser. in Math., Vol.65, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[26]

C.-L. Tang and X.-P. Wu, Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl., 259 (2001), 386-397. doi: 10.1006/jmaa.2000.7401.  Google Scholar

[27]

C.-L. Tang and X.-P. Wu, Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems, J. Math. Anal. Appl., 275 (2002), 870-882. doi: 10.1016/S0022-247X(02)00442-0.  Google Scholar

[28]

J. Su, Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity, J. Differential Equations, 145 (1998), 252-273. doi: 10.1006/jdeq.1997.3360.  Google Scholar

[29]

A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z., 209 (1992), 375-418. doi: 10.1007/BF02570842.  Google Scholar

[30]

A. Szulkin and W. Zou, Infinite dimensional cohomology groups and periodic solutions of asymptotically linear Hamiltonian systems, J. Differential Equations, 174 (2001), 369-391. doi: 10.1006/jdeq.2000.3942.  Google Scholar

[31]

J. R. Ward, Applications of critical point theory to weakly nonlinear boundary value problems at resonance, Houston J. Math., 10 (1984), 291-305.  Google Scholar

[32]

W. Zou, Solutions for resonant elliptic systems with nonodd or odd nonlinearities, J. Math. Anal. Appl., 223 (1998), 397-417. doi: 10.1006/jmaa.1998.5938.  Google Scholar

[33]

W. Zou, S. Li and J. Q. Liu, Nontrivial solutions for resonant cooperative elliptic systems via computations of critical groups, Nonlinear Anal. TMA, 38 (1999), 229-247. doi: 10.1016/S0362-546X(98)00191-6.  Google Scholar

[34]

W. Zou, Multiple solutions for second-order Hamiltonian systems via computation of the critical groups, Nonlinear Anal. TMA, 44 (2001), 975-989. doi: 10.1016/S0362-546X(99)00324-7.  Google Scholar

[35]

W. Zou, Computations of the critical groups and the nontrivial solutions for resonant type asymptotically linear Hamiltonian systems, Nonlinear Anal. TMA, 49 (2002), 481-499. doi: 10.1016/S0362-546X(01)00115-8.  Google Scholar

show all references

References:
[1]

H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Sup. Pisa Cl. Sci. Ser. IV, 7 (1980), 539-603.  Google Scholar

[2]

H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math., 32 (1980), 149-189. doi: 10.1007/BF01298187.  Google Scholar

[3]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some problems with strong resonance at infinity, Nonlinear Anal. TMA, 7 (1983), 241-273. doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[4]

T. Bartsch and S. J. Li, Critical point theory for asymptotically quadratic functionals with applications to problems at resonance, Nonlinear Anal. TMA, 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

[5]

G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336 (in Italian).  Google Scholar

[6]

K. C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure. Appl. Math., 34 (1981), 693-712. doi: 10.1002/cpa.3160340503.  Google Scholar

[7]

K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solutions Problems," Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[8]

K. C. Chang, J. Q. Liu and M. J. Liu, Nontrivial periodic solutions for strong resonance Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 14 (1997), 103-117. doi: 10.1016/S0294-1449(97)80150-3.  Google Scholar

[9]

C. C. Conley and E. Zehnder, Morse type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37 (1984), 207-253. doi: 10.1002/cpa.3160370204.  Google Scholar

[10]

G. Fei and Q. Qiu, Periodic solutions of asymptotically linear Hamiltonian systems, Chinese Ann. Math. Ser. B, 18 (1997), 359-372. doi: 10.1006/jdeq.1995.1124.  Google Scholar

[11]

G. Fei, Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity, J. Differential Equations, 121 (1995), 121-133. doi: 10.1006/jdeq.1995.1124.  Google Scholar

[12]

D. Gromoll and W. Meyer, On differentiable functions with isolated critical point, Topology, 8 (1969), 361-369. doi: 10.1016/0040-9383(69)90022-6.  Google Scholar

[13]

Y. X. Guo, "Morse Theory for Strongly Indefinite Functional and Its Applications," Doctoral thesis, Institute of Mathematics, Peking University, Beijing, 1999. doi: 10.1142/9789812704283_0013.  Google Scholar

[14]

Y. X. Guo, Nontrivial periodic solutions for asymptotically linear Hamiltonian systems with resonance, J. Differential Equations, 175 (2001), 71-87. doi: 10.1006/jdeq.2000.3966.  Google Scholar

[15]

N. Hirano and T. Nishimura, Multiplicity results for semilinear elliptic problems at resonance and with jumping non-linearities, J. Math. Anal. Appl., 180 (1993), 566-586. doi: 10.1006/jmaa.1993.1417.  Google Scholar

[16]

S. Li and J. Q. Liu, Morse theory and asymptotically linear Hamiltonian systems, J. Differential Equations, 78 (1989), 53-73. doi: 0022-0396(89)90075-2.  Google Scholar

[17]

S. Li and J. Q. Liu, Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance, Houston J. Math., 25 (1999), 563-582.  Google Scholar

[18]

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: 10.1006/jmaa.1999.6396.  Google Scholar

[19]

Y. Long and E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, in "Stochastic Processes, Physics and Geometry" (S. Albeverio, et al. Eds.), Proceedings of Conference in Asconal/Locarno, Switzerland, World Scientific, Singapore, 1990, pp. 528-563.  Google Scholar

[20]

Y. Long, Maslov-type index, degenerate critical points and asymptotically linear Hamiltonian systems, Sci. China Ser. A, 33 (1990), 1409-1419.  Google Scholar

[21]

S. Ma, Infinitely many periodic solutions for asymptotically linear Hamiltonian systems,, Rocky Mountain J. Math., ().   Google Scholar

[22]

S. Ma, Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems, J. Differential Equations, 248 (2010), 2435-2457. doi: 10.1016/j.jde.2009.11.013.  Google Scholar

[23]

S. Ma, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, Nonlinear Anal. TMA, 73 (2010), 3856-3872. doi: 10.1016/j.na.2010.08.013.  Google Scholar

[24]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Appl. Math. Sci., 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[25]

P. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," in CBMS Reg. Conf. Ser. in Math., Vol.65, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[26]

C.-L. Tang and X.-P. Wu, Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl., 259 (2001), 386-397. doi: 10.1006/jmaa.2000.7401.  Google Scholar

[27]

C.-L. Tang and X.-P. Wu, Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems, J. Math. Anal. Appl., 275 (2002), 870-882. doi: 10.1016/S0022-247X(02)00442-0.  Google Scholar

[28]

J. Su, Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity, J. Differential Equations, 145 (1998), 252-273. doi: 10.1006/jdeq.1997.3360.  Google Scholar

[29]

A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z., 209 (1992), 375-418. doi: 10.1007/BF02570842.  Google Scholar

[30]

A. Szulkin and W. Zou, Infinite dimensional cohomology groups and periodic solutions of asymptotically linear Hamiltonian systems, J. Differential Equations, 174 (2001), 369-391. doi: 10.1006/jdeq.2000.3942.  Google Scholar

[31]

J. R. Ward, Applications of critical point theory to weakly nonlinear boundary value problems at resonance, Houston J. Math., 10 (1984), 291-305.  Google Scholar

[32]

W. Zou, Solutions for resonant elliptic systems with nonodd or odd nonlinearities, J. Math. Anal. Appl., 223 (1998), 397-417. doi: 10.1006/jmaa.1998.5938.  Google Scholar

[33]

W. Zou, S. Li and J. Q. Liu, Nontrivial solutions for resonant cooperative elliptic systems via computations of critical groups, Nonlinear Anal. TMA, 38 (1999), 229-247. doi: 10.1016/S0362-546X(98)00191-6.  Google Scholar

[34]

W. Zou, Multiple solutions for second-order Hamiltonian systems via computation of the critical groups, Nonlinear Anal. TMA, 44 (2001), 975-989. doi: 10.1016/S0362-546X(99)00324-7.  Google Scholar

[35]

W. Zou, Computations of the critical groups and the nontrivial solutions for resonant type asymptotically linear Hamiltonian systems, Nonlinear Anal. TMA, 49 (2002), 481-499. doi: 10.1016/S0362-546X(01)00115-8.  Google Scholar

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