MULTIPLICITY OF HOMOCLINIC SOLUTIONS FOR FRACTIONAL DISCRETE LAPLACIAN EQUATIONS

We consider the existence and multiplicity of homoclinic solutions for the Schrödinger-type equation (−Δ1)su(n)+V(n)u(n)=f(n,u(n)) forn∈ℤ, where (−Δ1)s denotes the fractional discrete Laplacian, the sequence V(n) is the potential, and f(n,u) is a sequence of functions. Under some conditions on V and f, the existence of a ground state sign-changing homoclinic solution u1 and a ground state homoclinic solution u0 is obtained. We prove that the energy of u1 is more than twice of the energy of u0 and so u0≠u1. We also study the multiplicity of solutions in the case of concave-convex nonlinearity.