Existence Of Solutions Of Multiple Homoclinic Orbitals For A Class Of Second-order Quadratic Hamiltonian System Resonances

In this paper,the existence of homoclinic solutions to second-order Hamiltonian System-u(t)+L(t)u(t)=μu(t)+Wu(t,u(t))(t∈R)(HS)is studied.Here L(t)∈ C(R,RN2)is a symmetric matrix-valued function satisfying coercive condition(?)μ∈R is a parameter,the potential W E C1(R × RN)satisfies the following condition:(?)0≤a≤1,| Wu(t,u)|≤b1(t)|u|a+b2(t),0<b1∈L2/1-a(R)∩L∞(R),0≤b2∈L2(R).(HS)is said to be subquadratic under this condition.Under condition(L),the corresponding eigenvalue problem of(HS)is-u(t)+L(t)u(t)=λu(t).It has eigenvalues with-∞<λ1<λ2≤…λn≤0≤λn+1≤…→+∞and the eigenspaces corresponding to these eigenvalues are finite dimensional.The corresponding energy functional for(HS)is(?)(?)The critical points of this functional are weak solutions to(HS).They are also homoclinic solutions to(HS)under suitable conditions.By Z2 index theory,the multiplicity of homoclinic solutions to(HS)is proved.This paper consists of four chapters:The first chapter introduces the research background of homoclinic solutions to Hamiltonian System,literature review and main results of this paper.The second chapter introduces some relevant theories used in this paper.In the third chapter,we study the eigenvalue problem of(HS)and prove that the(P.S)condition holds for the functional Φ in the following cases:(i)μ<λ1;(ii)λk<μ<λk+1;(iii)μ=λk(Resonance).In the fourth chapter,the multiplicity of homoclinic solutions to(HS)is proved.In case(i),this is proved by using Clark theorem.In case(ii)and(iii),this is proved by using Z2 index theorem.The third and the fourth chapter are key chapters in this paper.