# American Institute of Mathematical Sciences

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 & 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. [2] H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems,, Manuscripta Math., 32 (1980), 149. 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. 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. 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. [6] K. C. Chang, Solutions of asymptotically linear operator equations via Morse theory,, Comm. Pure. Appl. Math., 34 (1981), 693. doi: 10.1002/cpa.3160340503. [7] K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solutions Problems,", Birkh\, (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. Poincar\'e Anal. Nonlin\'eaire, 14 (1997), 103. 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. 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. 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. doi: 10.1006/jdeq.1995.1124. [12] D. Gromoll and W. Meyer, On differentiable functions with isolated critical point,, Topology, 8 (1969), 361. doi: 10.1016/0040-9383(69)90022-6. [13] Y. X. Guo, "Morse Theory for Strongly Indefinite Functional and Its Applications,", Doctoral thesis, (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. 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. 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. 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. [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. doi: 10.1006/jmaa.1999.6396. [19] Y. Long and E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems,, in, (1990), 528. [20] Y. Long, Maslov-type index, degenerate critical points and asymptotically linear Hamiltonian systems,, Sci. China Ser. A, 33 (1990), 1409. [21] S. Ma, Infinitely many periodic solutions for asymptotically linear Hamiltonian systems,, Rocky Mountain J. Math., (). [22] S. Ma, Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems,, J. Differential Equations, 248 (2010), 2435. 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. doi: 10.1016/j.na.2010.08.013. [24] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Appl. Math. Sci., (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., (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. 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. 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. doi: 10.1006/jdeq.1997.3360. [29] A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals,, Math. Z., 209 (1992), 375. 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. 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. [32] W. Zou, Solutions for resonant elliptic systems with nonodd or odd nonlinearities,, J. Math. Anal. Appl., 223 (1998), 397. 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. 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. 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. 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. [2] H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems,, Manuscripta Math., 32 (1980), 149. 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. 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. 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. [6] K. C. Chang, Solutions of asymptotically linear operator equations via Morse theory,, Comm. Pure. Appl. Math., 34 (1981), 693. doi: 10.1002/cpa.3160340503. [7] K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solutions Problems,", Birkh\, (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. Poincar\'e Anal. Nonlin\'eaire, 14 (1997), 103. 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. 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. 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. doi: 10.1006/jdeq.1995.1124. [12] D. Gromoll and W. Meyer, On differentiable functions with isolated critical point,, Topology, 8 (1969), 361. doi: 10.1016/0040-9383(69)90022-6. [13] Y. X. Guo, "Morse Theory for Strongly Indefinite Functional and Its Applications,", Doctoral thesis, (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. 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. 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. 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. [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. doi: 10.1006/jmaa.1999.6396. [19] Y. Long and E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems,, in, (1990), 528. [20] Y. Long, Maslov-type index, degenerate critical points and asymptotically linear Hamiltonian systems,, Sci. China Ser. A, 33 (1990), 1409. [21] S. Ma, Infinitely many periodic solutions for asymptotically linear Hamiltonian systems,, Rocky Mountain J. Math., (). [22] S. Ma, Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems,, J. Differential Equations, 248 (2010), 2435. 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. doi: 10.1016/j.na.2010.08.013. [24] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Appl. Math. Sci., (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., (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. 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. 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. doi: 10.1006/jdeq.1997.3360. [29] A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals,, Math. Z., 209 (1992), 375. 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. 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. [32] W. Zou, Solutions for resonant elliptic systems with nonodd or odd nonlinearities,, J. Math. Anal. Appl., 223 (1998), 397. 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. 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. 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. doi: 10.1016/S0362-546X(01)00115-8.
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