Existence of Homoclinic Orbits for Hamiltonian Systems with Superquadratic Potentials

and Applied Analysis 3 We make the following assumptions. A1 W t, z ∈ C1 R × R2N,R is 1-periodic in t. W t, 0 0 for all t ∈ R. There exist constants c1 > 0 and μ > 2 such that Wz t, z z ≥ c1|z| for t, z ∈ R × R2N. A2 there exist c2, r > 0 such that |Wz t, z | ≤ c2|z|μ−1 for t ∈ R and |z| ≤ r. A3 there exist c3, R ≥ r and p ≥ μ such that |Wz t, z | ≤ c3|z|p−1 for t ∈ R and |z| ≥ R. A4 there exists b0 > 2 such that lim infz→ 0 Wz t, z z/W t, z ≥ b0 uniformly for t ∈ R; A5 ̃ W t, z : 1/2 Wz t, z z − W t, z > 0 for all t ∈ R, z ∈ R2N \ {0}. There exist constants b∞ > 0 and β > p p − 2 / p − 1 such that lim inf|z|→∞ W t, z /|z|β ≥ b∞ uniformly for t ∈ R. Theorem 1.1. Let A0 , A1 – A5 be satisfied, then 1.1 has at least one homoclinic orbit. Remark 1.2. We can easily check that the A-R condition implies A4 and A5 . But the converse proposition is not true. See the following example: W t, z |z| (μ − 2)|z|μ− sin2 ( |z| ) , 1.2 where 2 < μ < ∞, 0 < < min{μ − 2, μ/ μ − 1 } see 25 or 26 for details . IfWz t, z a|z|μ−2z Rz t, z , a > 0, μ ∈ 2,∞ with R satisfying B1 R ∈ C1 R × R2N,R is 1-periodic in t and Rz t, z o ( |z|μ−1 ) as |z| −→ 0, Rz t, z o ( |z|μ−1 )