Abstract
In this article, we study the existence of homoclinic orbits for the first-order Hamiltonian system\begin{equation*} J\dot{u}(t)+\nabla H(t,u(t))=0,\quad t\in\mathbb{R}.\end{equation*}Under the Ambrosetti-Rabinowitz's superquadraticy condition, or no Ambrosetti-Rabinowitz's superquadracity condition, we present two results on the existence of infinitely many large energy homoclinic orbits when $H$ is even in $u$. We apply the generalized (variant) fountain theorems established recently by the author and Colin. Under no Ambrosetti-Rabinowitz's superquadracity condition, we also obtain the existence of a ground state homoclinic orbit by using the method of the generalized Nehari manifold for strongly indefinite functionals developed by Szulkin and Weth.
Highlights
We are interrested in the existence and the multiplicity of homoclinic orbits of the first order Hamiltonian system
0 ́IN IN 0 is the standard symplectic structure on R2N, H : R R2N Ñ R is 1 ́periodic with respect to the tvariable and ∇H is the gradient of H with respect to u
W pt, uq, where the dot denotes the inner product of R2N, A : R Ñ R4N2 is a 2N 2N
Summary
We are interrested in the existence and the multiplicity of homoclinic orbits of the first order Hamiltonian system. The existence of infinitely many homoclinic orbits of (HS) under pA0q and pW1qpW4q was first proved by Ding and Girardi [9] see [2] by using a generalized linking theorem. They allowed 0 to be an end point of the spectrum σpLq. They allowed 0 to be an end point of the spectrum σpLq
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