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

Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance

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.
Citation: Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure and 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.

[2]

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

[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.

[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.

[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).

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

S. Ma, Infinitely many periodic solutions for asymptotically linear Hamiltonian systems, Rocky Mountain J. Math., to appear.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[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).

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

S. Ma, Infinitely many periodic solutions for asymptotically linear Hamiltonian systems, Rocky Mountain J. Math., to appear.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[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.

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