Homoclinic solutions for Hamiltonian systems with variable-order fractional derivatives

Abstract

ABSTRACTIn this paper, we are concerned with the multiplicity and concentration of solutions for a Hamiltonian system driven by the fractional Laplace operator with variable order derivative. More precisely, we consider where , , is a nonlocal fractional integro-differential operator with variable order derivative, is a parameter, is a real symmetric matrix and belongs to . Under some suitable assumptions, we show that the system admits at least two distinct homoclinic solutions. Moreover, we investigate the concentration of solutions as . This paper is the first time to deal with Hamiltonian systems with variable order fractional derivatives. Moreover, our system is anisotropic. Thus this is different from other papers in the literature.