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Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. (English) Zbl 0840.34044
The existence of solutions homoclinic to an equilibrium $0\in \bbfR^N$ for second-order time-dependent Hamiltonian systems of the type $$\ddot q- L(t) q+ {\partial\over \partial q} W(t, q)= 0,\quad q\in \bbfR^N,\tag1$$ where $L\in C(\bbfR, \bbfR^{N^2})$ is a symmetric matrix, $W\in C^1(\bbfR\times \bbfR^N, \bbfR)$, is studied. As usual, a solution $q(t)$ of system (1) is said to be homoclinic (to 0) if $q(t)\ne 0$, $q(t)\to 0$, and $\dot q(t)\to 0$ as $|t|\to \infty$. The existence and multiplicity of homoclinic solutions for Hamiltonian systems of this type have been studied in many recent papers via the critical point theory under the assumptions that $L(t)$ is positive definite for all $t\in \bbfR$, $W(t, q)$ is globally superquadratic in $q$ and the potential $V(t, q)= -{1\over 2} L(t)q\cdot q+ W(t, q)$ is periodic in $t$. The author studies the existence of homoclinic (to 0) solutions for a class of systems (1) when the global positive definiteness of $L(t)$ is not necessarily satisfied and $L$ and $W$ are not periodic in $t$. Both the case that $W(t, q)$ is superquadratic in $q$ and the one that $W(t, q)$ is of subquadratic growth as $|q|\to \infty$ are considered.

MSC:
34C37Homoclinic and heteroclinic solutions of ODE
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References:
[1] Ambrosetti, A.; Bertotti, M.L.: Homoclinics for a second order conservative systems. Proc. conf. In honour of L. Nirenberg (1990) · Zbl 0804.34046
[2] AMBROSETTI A. & COTI ZELATI V., Multiple homoclinic orbits for a class of conservative systems, preprint. · Zbl 0806.58018
[3] CHANG K. C. & LIU J.Q., A remark on the homoclinic orbits for Hamiltonian systems, preprint. · Zbl 0940.37023
[4] Coti Zelati, V.; Rabinowitz, P.H.: Homoclinic orbits for a second order Hamiltonian systems possessing superquadratic potentials. J. am. Math. soc. 4, 693-727 (1991) · Zbl 0744.34045
[5] Ding, Y.H.; Girardi, M.: Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign. Dynam. syst. Applic. 2, 131-145 (1993) · Zbl 0771.34031
[6] BESSI U., Multiple homoclinics for autonomous, singular potentials, preprint. · Zbl 0812.58088
[7] Rabinowitz, P.H.: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. edinb. 114, 33-38 (1990) · Zbl 0705.34054
[8] Rabinowitz, P.H.; Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 473-499 (1991) · Zbl 0707.58022
[9] Coti Zelati, V.; Ekeland, I.; Séré, E.: A variational approach to homoclinic orbits in Hamiltonian systems. Math. ann. 288, 133-160 (1990) · Zbl 0731.34050
[10] Hofer, H.; Kysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. ann. 288, 483-503 (1990)
[11] Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 27-42 (1992) · Zbl 0725.58017
[12] Tanaka, K.: Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits. J. diff. Eqns 4, 315-339 (1991) · Zbl 0787.34041
[13] Reed, M.; Simon, B.: Methods of modern mathematical physics IV. Analysis of operators. (1978) · Zbl 0401.47001
[14] Simon, B.: Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, II, III. Proc. am. Math. soc. 45, 454-456 (1974) · Zbl 0292.35061
[15] Kato, T.: Perturbation theory for linear operators. (1966) · Zbl 0148.12601
[16] Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS reg. Conf. ser. In math. 65 A.M.S. (1986)
[17] Omana, W.; Willem, M.: Homoclinic orbits for a class of Hamiltonian systems. Diff. integral eqns 5, 1115-1120 (1992) · Zbl 0759.58018