Periodic Solution Of Second Order Hamiltonian Systems With Superlinear

In this paper, by using bifurcation theory, the minimax and homological linking arguments we give some multiplicity results on existence of periodic solutions for the second order Hamiltonian systems(0.1)where let T > 0 be fixed, A(t) is a N×N order symmetric matrix which is continuous and T-periodic in t. V satisfies the following assumptions:(V₁) V∈C²(R×R^N;R) is T- periodic in t.(V₂) V(t, 0) = 0, V'_x(t, 0) = 0, V"_x(t, 0) = 0.(V₃) There is an r > 0 andθ> 2 such that(V₄) V(t, x)≥0,(?)t∈R,x∈R^N; V"_x(t, x) > 0, for |x| > 0 small.(V₅) V"_x(t, x) < 0, for |x| > 0 small.Denote S_T = R/(TZ), V^±(t,x) = max{±V(t, x), 0}. Denote byλ₁ <λ₂ <…<λ_m<…the distinct eigenvalues of the linear eigenvalue problem(0.2)The main results in this paper are the following three theorems:Theorem A Suppose that V satisfies (V₁)- (V₄) and let k≥1 be fixed. Thenthere isδ> 0, such that forλ∈(λ_k+1-δ,λ_k+1), (0.1) has at least three nontrivial T-periodic solutions. Theorem B Suppose that V satisfies (V₁) - (V₃)(V₅) and let k≥1 be fixed. Thenthere isδ> 0, such that when sup(?) V^-(t, x) <δ, forλ∈(λ_k+1,λ_k+1 +δ), (0.1) hasat least three nontrivial T- periodic solutions.Theorem C Suppose that V satisfies (V₁) - (V₃)(V₅) and let k≥1 be fixed. Then there isδ> 0, such that when sup (?) V^-(t, x) <δ, forλ∈(λ_k+1 -δ,λ_k+1], (0.1) hasat least two nontrivial T- periodic solutions.