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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 \(0\in \mathbb{R}^N\) for second-order time-dependent Hamiltonian systems of the type \[ \ddot q- L(t) q+ {\partial\over \partial q} W(t, q)= 0,\quad q\in \mathbb{R}^N,\tag{1} \] where \(L\in C(\mathbb{R}, \mathbb{R}^{N^2})\) is a symmetric matrix, \(W\in C^1(\mathbb{R}\times \mathbb{R}^N, \mathbb{R})\), is studied. As usual, a solution \(q(t)\) of system (1) is said to be homoclinic (to 0) if \(q(t)\neq 0\), \(q(t)\to 0\), 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 \(t\in \mathbb{R}\), \(W(t, q)\) 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.

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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[1] Ambrosetti, A.; Bertotti, M.L., Homoclinics for a second order conservative systems, (), Trento · 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, (), 33-38, (A) · 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) · Zbl 0702.34039
[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), Academic Press · Zbl 0401.47001
[14] Simon, B., Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, II, III, (), 454-456 · Zbl 0292.35061
[15] Kato, T., Perturbation theory for linear operators, (1966), Springer New York · Zbl 0148.12601
[16] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, (), Providence · Zbl 0152.10003
[17] Omana, W.; Willem, M., Homoclinic orbits for a class of Hamiltonian systems, Diff. integral eqns, 5, 1115-1120, (1992) · Zbl 0759.58018
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