Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition

Abstract

Abstract. The existence of homoclinic orbits is obtained for a class of the second order Hamiltonian systemsü(t) − L(t)u(t) + ∇W (t,u(t)) = 0, ∀t ∈ R , by the mountain pass theorem, where W (t,x) needs not to satisfy the global (AR) condition.

Subharmonic and homoclinic solutions for second order Hamiltonian systems with new superquadratic conditions

“…In last decades, the existence and multiplicity of homoclinic orbits have been intensively studied by many mathematicians with variational methods [26][27][28][29][30][31][35][36][37] and the reference therein.…”

“…This condition is well known as the global Ambrosetti-Rabinowitz condition which can help prove the compact condition. In recent years, there are many papers [7,8,28,30,31,35] obtained the existence and multiplicity of homoclinic solutions of problem (1) with some other superquadratic conditions on W instead of ðA 1 Þ. Subsequently, we set f W ðt; xÞ ¼ ðrWðt; xÞ; xÞ À 2Wðt; xÞ:…”

“…It is easy to see that (4) satisfies the conditions of Theorem 1.2, but not the results in [8,13,21,[27][28][29][30][31] since (4) satisfies (3).…”

Periodic Solutions for Second Order Hamiltonian Systems with Impulses via the Local Linking Theorem

“…Recently, applying the local linking theorem (see [26]), the works in [27][28][29][30] obtained the existence of periodic solutions or homoclinic solutions with (3) superquadratic condition under different systems. As shown in [25], condition (B2) is a local superquadratic condition; this situation has been considered only by a few authors.…”

Existence of two almost homoclinic solutions for p(t)-Laplacian Hamiltonian systems with a small perturbation

“…Whereas, there are many potentials which are superquadratic as |u| → ∞ but do not satisfy the (AR) condition. So, many authors have been focusing their attention on deriving the existence of homoclinic solutions under the conditions weaker than the (AR) condition, see recent papers [10,25,26,47,48,55] and the references therein. In addition, to check the (PS) condition for the corresponding functional of (HS), the following coercive assumption on L(t) is frequently required.…”