Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials Original Research Article

In this paper, we deal with the existence and multiplicity of homoclinic solutions of the second-order Hamiltonian system View the MathML sourceü(t)−L(t)u(t)+∇W(t,u(t))=0, Turn MathJax on where L(t)L(t) and W(t,x)W(t,x) are neither autonomous nor periodic in tt. Under the assumption that W(t,x)W(t,x) is indefinite sign and subquadratic as |x|→+∞|x|→+∞ and L(t)L(t) is a N×NN×N real symmetric positive definite matrices for all t∈Rt∈R, we establish some existence criteria to guarantee that the above system has at least one or infinitely many homoclinic solutions by using the genus properties in critical theory.