Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems

In this paper, we present some new results of homoclinic solutions for second-order Hamiltonian systems ü−λL(t)u+Wu(t,u)=0; here λ>0λ>0 is a parameter, L∈C(R,RN×N)L∈C(R,RN×N) and W∈C1(R×RN,R)W∈C1(R×RN,R). Unlike most other papers on this problem, we require that L(t)L(t) is a positive semi-definite symmetric matrix for all t∈Rt∈R, that is, L(t)≡0L(t)≡0 is allowed to occur in some finite interval TT of RR. Under some suitable assumptions on WW, we prove the existence of two different homoclinic solutions uλ(1), uλ(2), which both vanish on R∖TR∖T as λ→∞λ→∞, and converge to u0(1), u0(2) in H1(R)H1(R), respectively; here u0(1)≠u0(2)∈H01(T) are two nontrivial solutions of the Dirichlet BVP for Hamiltonian systems on the finite interval TT.