Existence of homoclinic solutions for the second-order discrete \(p\)-Laplacian systems
The authors study the following second-order discrete \(p\)-Laplacian system \[ \Delta (\varphi_p(\Delta u(n-1)))-a(n)|u(n)|^{p-2}u(n)+\nabla W(n,u(n))=0,\tag{\(*\)} \] where \(p>1\), \(\varphi_p(s)=|s|^{p-2}s\), \(n\in \mathbb{Z}\), \(u\in \mathbb{R}^N\), \(a:\mathbb{Z}\to \mathbb{R}\) and \(W: \mathbb{Z}\times \mathbb{R}^N\to\mathbb{R}\). System \((*)\) can be regarded as the discretization of a second order Hamiltonian system. The focus is on the existence of homoclinic orbits of \((*)\) emanating from \(0\) since homoclinic orbits play an important role in analyzing the chaos of dynamical systems. With the help of critical point theory, several sufficient conditions on the existence of at least one or infinitely many homoclinic orbits are established. The obtained results improve some existing ones.
second-order discrete \(p\)-Laplacian system
scientific article; zbMATH DE number 5990176