HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS*

doi:10.1007/s10473-023-0312-3

Abstract

Abstract: In this paper, we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems $\dot{z}=JH_z(t,z)$, where the Hamiltonian function $H$ possesses the form $H(t,z)=\frac{1}{2}L(t)z\cdot z +G(t,z)$, and $G(t,z)$ is only locally defined near the origin with respect to $z$. Under some mild conditions on $L$ and $G$, we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense, which is essentially forced by the subquadraticity of $G$ near the origin with respect to $z$.

Key words: Hamiltonian systems, homoclinic solutions, variational method

Qingye Zhang, Chungen Liu. HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*[J].Acta mathematica scientia,Series B, 2023, 43(3): 1195-1210.

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HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*

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