Abstract
This paper concerns solutions for the Hamiltonian system:z˙=𝒥Hz(t,z). HereH(t,z)=(1/2)z⋅Lz+W(t,z),Lis a2N×2Nsymmetric matrix, andW∈C1(ℝ×ℝ2N,ℝ). We consider the case that0∈σc(−(𝒥(d/dt)+L))andWsatisfies some superquadratic condition different from the type of Ambrosetti-Rabinowitz. We study this problem by virtue of some weak linking theorem recently developed and prove the existence of homoclinic orbits.
Highlights
Introduction and the Main ResultsIn this paper, we consider the existence of homoclinic orbits for the following Hamiltonian system:z JHz t, z, 1.1 where H t, z 1/2 z · Lz W t, z, L is a 2N × 2N symmetric matrix-valued function, and W ∈ C1 R × R2N, R is superquadratic both around 0 and at infinity in z ∈ R2N.A solution of 1.1 is called to be homoclinic to 0 if z t /≡ 0 and z t → 0 as |t| → ∞
We observed that just recently some abstract linking theorems were developed by Bartsch and Ding in 18
|u|2μ, 2.6 and let Eμ− denote the completion of D A ∩ L2− with respect to the norm · μ
Summary
We consider the existence of homoclinic orbits for the following Hamiltonian system:. Later, Ding and Girardi obtained infinitely many homoclinic orbits in 16 under the conditions of 14 with an additional evenness assumption on W Note that in both papers W satisfies a condition of the type of Ambrosetti-Rabinowitz see 17 , that is,. We observed that just recently some abstract linking theorems were developed by Bartsch and Ding in 18 These theorems are impactful to study the existence and multiplicity of solutions for the strongly indefinite problem. Besides C c condition, it seems hard to check the following condition necessary for the linking theorems in 19–21 : Φ1 for any c > 0, there exists ζ > 0 such that z < ζ PY z for all z ∈ Φc.
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