Existence of homoclinic solution for the second order Hamiltonian systems

Abstract

An existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order Hamiltonian systemsü(t) − L(t)u(t) + ∇W (t, u(t)) = 0, ∀t ∈ R, by the minimax methods in the critical point theory, specially, the generalized mountain pass theorem, where L(t) is unnecessary uniformly positively definite for all t ∈ R, and W (t, x) satisfies the superquadratic condition W (t, x)/|x| 2 → +∞ as |x| → ∞ uniformly in t, and need not satisfy the global AmbrosettiRabinowitz condition.

Periodic solutions for a class of nonautonomous second order Hamiltonian systems

“…Particularly, Fei [4] got the existence of 1-periodic solutions of systems (1) under some new superquadratic conditions, Wu [11] studied multiplicity of periodic solutions, and Tao and Tang [10] researched the subharmonic solutions. When B(t) ≡ 0, Zou and Li [13] studied the existence of infinitely many T -periodic solutions under the assumption that H (t, x) is even in x, Ou and Tang [7] get the existence of homoclinic solutions and Faraci [3] studies the existence of multiple periodic solutions.…”

Even homoclinic orbits for super quadratic Hamiltonian systems

“…Equation (2) is a whole super quadratic condition which is crucial for checking the (PS) condition. Later some papers weakened this condition [6,7]. There are also some other papers considered the sub-quadratic case [8,9] and the asymptotically quadratic case [10,11].…”

Ground state homoclinic orbits of a class of superquadratic damped vibration problems

“…In this case, the existence of the homoclinic solutions can be obtained by going to the limit of the periodic solutions of the approximating problems. If L(t) and W (t, x) are neither autonomous or periodic in t, the existence of the homoclinic solutions of (H S ) is quite different from the ones just described because of the lack of compactness of the Sobolev embedding, see, e.g., [4,10,11,[13][14][15][16][17]19,22] and the references therein.…”