## Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign.(English)Zbl 0894.34036

The author studies the system $$\ddot x= W'(z,t)= 0$$, with $$x(0)= x(T)$$, $$\dot x(0)= \dot x(T)$$, where $$W(x,t)= \langle A(t)x,x\rangle+ b(t)V(x)$$. Here, $$b$$ is a continuous $$T$$-periodic function, $$V\in C^2(\mathbb{R}^N\times \mathbb{R})$$ has superquadratic behavior, $$A$$ is a continuous $$T$$-periodic symmetric matrix function, with $$\langle A(t)\xi,\xi\rangle> 0$$, for all $$\xi\in \mathbb{R}^N$$, and $$|\xi|= 1$$.
The author shows that if $$\lim_{x\to 0} V(x)/| x^2|= 0$$, and $$\int^T_0 b(t)dt> 0$$ then the system has a $$T$$-periodic solution. With some additional assumptions including $$\int^T_0 b(t)dt< 0$$ (reversing the sign) it also follows that a $$T$$-periodic solution exists. Using a variational approach, she proves that the corresponding Lagrangian action integral ${1\over 2}\int^T_0| \dot x^2| dt-{1\over 2}\int^T_0 [\langle A(t)x,x\rangle+ b(t)V(x)]dt$ has a critical point in an appropriately chosen finite subspace of the original Hilbert space $$H^1$$.
Reviewer: V.Komkov (Roswell)

### MSC:

 34C25 Periodic solutions to ordinary differential equations
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### References:

 [1] Lassoued, L., Solutions periodiques d’un systeme differentiel non lineaire du second order avec changement de sign, Ann. Math. Pura Appl., 156, 76-111 (1990) · Zbl 0724.34051 [2] Lassoued, L., Periodic solutions of a second order superquadratic system with a change of sign in the potential, J. Diff. Eq., 93, 1-18 (1991) · Zbl 0736.34041 [3] Ben Naoum, A. K.; Trostler, C.; Willem, M., Existence and multiplicity results for homogeneous second order differential equations, J. Diff. Equations, 112, 239-249 (1994) · Zbl 0808.58013 [4] Alama, S.; Tarantello, G., On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Diff. Eq., 1, 439-475 (1993) · Zbl 0809.35022 [5] Girardi, M.; Matzeu, M., Existence and multiplicity results for periodic solutions for superquadratic Hamiltonian systems where the potential changes sign, No. D.E.A., 2, 35-61 (1995) · Zbl 0821.34041 [6] Ambrosetti, A.; Mancini, G., Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann., 255, 405-421 (1981) · Zbl 0466.70022 [7] Ambrosetti, A.; Coti Zelati, V., Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova, 89, 177-194 (1993) · Zbl 0806.58018 [8] Coti Zelati, V.; Rabinowitz, P. H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potential, J. Ann. Math. Soc., 4, 693-727 (1991) · Zbl 0744.34045 [9] Ding, Y. H.; Girardi, M., Periodic and homoclinic solutions to a class of Hamiltonian systems with the potential changing sign, Dynamical Syst. and Appl., 2, 131-145 (1993) · Zbl 0771.34031 [10] Caldiroli, P.; Montecchiari, P., Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. on Appl. Nonlinear Anal., 1, 97-129 (1994) · Zbl 0867.70012 [11] Antonacci, F., Periodic and homoclinic solutions to a class of Hamiltonian systems with indefinite potential in sign, Boll. Unione Mat. Ital., 10-B, 303-324 (1996) · Zbl 1013.34038 [12] Rabinowitz, P. H., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31, 157-184 (1978) · Zbl 0358.70014 [13] Benci, V.; Rabinowitz, P. H., Critical point theorems for indefinite functionals, Invent. Math., 52, 241-273 (1979) · Zbl 0465.49006
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