The Existence And Multiplicity Of Homoclinic Orbits For Second Order Hamiltonian Systems

In this paper,we firstly study the following second order Hamiltonian systems u(t)-L(t)u(t)+▽W(t,u(t))=0, At∈R. (0-1) We make the following assumptions: (A1) L(t) and W(t.x)are 1-periodic in t (A2) L(t) is uniformly positive for t∈R; (W2)▽W(t,x)|/|x|→0 as x→0 uniformly in t∈R; (W3) W(t,x)/|x|2→+∞as |x|→∞uniformly in t∈R; (W4) There existα0>1,d1>0,d2>0 such that |▽W(t,x)|≤d1|x|α0+d2 for all t∈R and x∈RN; (W5)There existβ≥α0.d3>0,R1>0 such that(▽W(t,x),x)-2W(t,x)≥d3|x|βfor all t∈Rand |x|≥R1; (W5′)There existα>α0-1.d4>0,r1>0 such that(▽W(t,x),x)-2W(t,x)≥d4|x|αfor all t∈R and |x|≥r1; (W6)(▽W(t,x),x)≥2W(t,x)≥0 for all t∈R and x∈RN\{0}; (W6′)(▽W(t,x),x)＞2W(t,x)≥0 for all t∈R and x∈RN\{0}; (W7)There exist a bounded set B(?)R with int(B)≠(?),andμ>2,θ>μ/(μ-2) such that (ⅰ)0<μW(t,x)≤(▽W(t,x),x)for all t∈B and x∈RN\{0}; (ⅱ)0≤2W(t,x)≤(▽W(t,x),x)≤1/θ(L(t)x,x)for all t (?) B and x∈RN (W8)For any 00; (W11) W(t,x)=W(t,-x)for all t∈R and x∈RN: (L**)There existsγ>1 such that meas(t∈R||t|-γL(t)(?)MOIN)<+∞for all M0>0; (L1) For l(t)≡inf|x|=1(L(t)x,x),there existsγ>1 such that l(t)|t|-γ→+∞as |t|→∞; (L2)For someξ>0 and r>0,one of the following is true: (ⅰ)L∈C1(R,RN2),and |L′(t)x|≤ξ|L(t)x| for all |t|>r and x∈RN with |x|=1; (ⅱ)L∈C2(R,RN2),((ξL(t)-L″(t))x,x)≥0 for all |t|>r and x∈RN with |x|=1,where L′(t)=(d/dt)L(t),L″(t)=(d2/dt2)L(t): (L3)There exists l1≥0 such that l(t):=inf|x|=1(L(t)x,x)≥-l1 for all t∈R.We obtain the following results: Theorem 2.3 Assume that L∈C(R,RN2) and W∈C1(R×RN,R) satisfy (A1) (A2),(W2)-(W4),(W5′) and (W6′),then problem(0-1)has at least one nontrivial homoclinic orbits. Theorem 2.5 Assume that L∈C(R,RN2) and W∈C1(R×RN:R) satisfy (A2), (W2),(W7),then problem(0-1)has at least one nontrivial homoclinic orbits. Theorem 2.7 Assume that L∈C(R,RN2)and W∈C1(R×RN,R)satisfy(L1). (L2).(W2)-(W4),(W5′)and(W8),then problem(0-1)has at least one nontrivial homoclinie orbits. Theorem 3.2 Assume that L∈C(R,RN2)and W∈C1(R×RN,R)satisfy(L**), (L3),(W2)-(W6),(W11),then then problem(0-1)has infinitely many nontrivial homoclinic orbits. Then,we consider the following Schrodinger equation:-△u+V(x)u=f(x,u),x∈RN. (0-2) We obtain the following results: Theorem 4.1 Assume that V∈C1(RN,R)and f∈C(RN×R,R)satisfy (D1)There exists a constant M≥0 such that V(x)≥-M for all x∈RN: (D2)for any r>0 and any sequence(xn)CRN which goes to infinity, whereμn={u∈H01(Bn)|||u||L2(Bu)=1}and Bn=B(xn.r)is the open ball with centerxn and radius r; (D3) f(x,s)/s→+∞as |s|→∞uniforml¨in x; (D4) There exists aθ≥1 such thatθF(x,s)≥F(x,ts) for all(x,s)∈RN×R and t∈[0,1],where F(x,s)=f(x,s)s-2F(x,s).F(x,s)=∫0sf(x,z)dz. Besides,suppose that there exist a positive constant C and a function K∈LIoc∞(RN,R),with K(x)≥C>0 for all x∈RN satisfying: (D5)There exist constants d6>0,α1>1,R2>0 such that K(x)≤d6(1+(max{0,V(x)}1/(α1)) for all|x|≥R2; (D6)There exist constants d7>0,10,1