Homoclinic Solutions Of Second-order Hamiltonian Systems

In this paper, firstly, we consider the following second order Hamiltonian systemsü+â–½V(t, u)=0, (HS₁)where V∈C¹(R×R^N, R),â–½V(t, x)=((?)V/(?)x)(t, x). In our paper V(t, x)=-K(t, x)+W(t, x) and K(t, x) is not necessarily homogeneous with respect to x. The proof is done by using the Mountain Pass Theorem without (PS) condition.The main result is the following theoremTheorem 1 Suppose that(V1) V(t, x)=-K(t, x)+W(t, x);(V2) K∈C¹(R×R^N, R) and there exists a positive constantλsuch thatwhere L(t) is a positive definite symmetric matrix valued function for all t∈R;(V3) (K(t, x))/(|x|²)→+∞as |t|→∞uniformly in x∈R^N\{0};(V4) There exists a constant c₀＞0 such that(W1) W∈C¹(R×R^N, R) and there exists a constantμ＞2 such that (W2)â–½W(t, x) = o(|x|) as |x|→0 uniformly in t∈R;(W3) There is a W^*∈C(R^N, R) such that|W(t,x)| + |â–½W(t,x)|≤|W^*(x)|, (?)t∈R, (?)x∈R^N, then (HS₁ has at least one nontrivial homoclinic solution.Next we consider the existence of multiple homoclinic orbits for the following Hamiltonian systems-ü+ L(t)u =â–½W(t, u) + g(t), (HS₂) where L(t) is a symmetric matrix valued function, W∈C¹ (R x R^N, R), W(t,.) is even and g(t) (?) 0. When g(t)≡0, the existence of multiple solutions can be got by using the Symmetric Mountain Pass Theorem. If g(t) (?) 0, in spite of the loss of the symmetry of the problem, we will prove the existence of multiple solutions of (HS₂) using the perturbation method.The main result is the following theoremTheorem 2 Suppose that(L) the smallest eigenvalue of L(t)→+∞as |t|→∞, i. e.(L′) for someā＞0 and (?)＞0 one of the following is true(â…°) n∈C¹(R,R^N) and |L′(t)|≤ā|L(t)|, (?)|t|≥(?), or(â…±) L∈C²(R, R^N) and L″(t)≤āL(t), (?)|t|≥(?), where L′(t) =(d/dt)L(t) and L″(t) =(d²/dt²)n(t); assume that W(t, .) is even, (W1), (W2) and the following conditions hold(W4) there exist constants C₁, C₂＞0 such that(W5) there exist constants C₃＞0 and p≥μsuch that for every t∈R and |x|＞1W(t, x)≤C₃|x|^p;moreover, we have (G) given g∈L²(R, R^N)(?)L^μ'(R,R^N) (μ^' =μ/μ-1), for n large we have whereλ_n is the n-th eigenvalue of -d²/dt²+L(t) in L²(R, R^N)(for detail, see the proof), then (HS₂) has an unbounded sequence of homoclinic solutions.At last, we consider the following second order functionü-α(t)u +β(t)u² +γ(t)u³ = 0, (HS₃) whereα,β,γ∈C¹(R). We get positive homoclinic solutions of (HS₃) under conditions thatβ(?) 0 and the oefficient functions are not necessarily even.The main results are the following theoremsTheorem 3 Suppose that(H1) There exist positive constants a′, b′, c′, B and C such that0＜a′≤α(t), 0≤b′≤β(t)≤B, 0＜c′≤γ(t)≤C;(H2)α,β,γ,∈C¹(R) satisfy tα′(t)≥0, tβ′(t)≤0, tγ′(t)≤0, (?)t∈R, whereα′(t) =(d/dt)α(t);(H3)(for detail, see the proof), then (HS₃) has a positive homoclinic solution.Theorem 4 Assume that (H1) holds and suppose that(H4)then (HS₃) has a positive homoclinic solution.Corollary 1 Assume that (H1), (H2) hold and suppose that(H5)then (HS₃) has a positive homoclinic solution.