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Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance
1. | School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 |
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., ().
|
[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., ().
|
[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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