Homoclinic Orbits for Second Order Discrete Hamiltonian Systems with General Potentials
In this paper, we study the second order discrete Hamiltonian system $$\begin{aligned} \Delta ^2 u(n-1)-L(n)u(n)+\nabla W(n,u(n))=0, \end{aligned}$$where \(L:\mathbb {Z}\rightarrow \mathbb {R}^{N\times N}\) is positive definite for sufficiently large \(|n|\in \mathbb {Z}\). By using the variant fountain theorem, we obtain an existence criterion to guarantee that the above system has infinitely many homoclinic solutions under the assumption that \(W(n,x)\) is only locally defined near the origin with respect to \(x\). Our results generalize and improve some existing results in the literature.
discrete hamiltonian systems variant fountain theorem homoclinic solutions variational methods 37j45 39a12 58e05 70h05