# EXISTENCE OF SIX SOLUTIONS OF THE NONLINEAR HAMILTONIAN SYSTEM

• Jung, Tack-Sun ;
• Choi, Q-Heung
• Accepted : 2008.09.01
• Published : 2008.09.25
• 68 14

#### Abstract

We give a theorem of existence of six nontrivial solutions of the nonlinear Hamiltonian system $\.{z}$ = $J(H_z(t,z))$. For the proof of the theorem we use the critical point theory induced from the limit relative category of the torus with three holes and the finite dimensional reduction method.

#### Keywords

Hamiltonian system;critical point theory;limit relative category;$(P.S.)^*_c$ condition;finite dimensional reduction method

#### References

1. H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 127-166 (1979).
2. T. Bartsch and M. Klapp, Critical point theory for indefinite functionals with symmetries, J. Funct. Anal., 107-136 (1996).
3. K. C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhauser, (1993).
4. Q. H. Choi and T. Jung, An application of a variational reduction method to a nonlinear wave equation, J. Differential Equations, 117, 390-410 (1995). https://doi.org/10.1006/jdeq.1995.1058
5. M. Degiovanni, Homotopical properties of a class of nonsmooth functions, Ann. Mat. Pura Appl. 156, 37-71 (1990). https://doi.org/10.1007/BF01766973
6. M. Degiovanni, A. Marino, and M. Tosques, Evolution equation with lack of convexity, Nonlinear Anal. 9, 1401-1433 (1985). https://doi.org/10.1016/0362-546X(85)90098-7
7. G. Fournier, D. Lupo, M. Ramos, and M. WiIlem, Limit relative category and critical point theory, Dynam. Report, 3, 1-23 (1993).
8. T. Jung and Q. H. Choi, Existence of four solutions of the nonlinear Hamiltonian system with nonlinearity crossing two eigenvalues, Boundary Value Problems, Volume 2008, 1-17.
9. D. Lupo and A. M. Micheletti, Nontrivial solution for an asymptotically linear beam equation, Dynam. Systems Appl. 4, 147-156 (1995).
10. D. Lupo and A. M. Micheletti, Two applications of a three critical points theorem, J. Differential Equations 132, 222-238 (1996). https://doi.org/10.1006/jdeq.1996.0178
11. P. A. Marino, C. Sacconl, Some variational theorems of mixed type and elliptic problems with jumping nonlinearities, Ann. Scuola Norm. Sup. Pisa, 631-665 (1997).
12. A. M. Micheletti and A. Pistoia, On the number of solutions for a class of fourth order elliptic problems, Communications on Applied Nonlinear Analysis, 6, No. 2, 49-69 (1999).
13. P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS. Regional Conf. Ser. Math., 65, Amer. Math. Soc., Providence, Rhode Island (1986).
14. P. H. Rabinowitz, A variational method for finding periodic solutions of differential equations, Nonlinear Evolution Equations (M.G.Crandall.ed.), Academic Press, New York, 225-251 (1978).