Homoclinic Orbits For Second-order Hamiltonian Systems

This dissertation investigates existence and multiplicity of homoclinic or-bits for sccond-order Hamiltonian systems via variational methods and analysis techniques.In Chapter2,we consider the following second-order superquadratic Hamil-tonian systems u(t)-L(t)u(t)+▽W(t,u(t))=0(2-1) where L∈C(R,RN2)is a symmetric matrix valued function.W∈C1(R×RN,R) and▽W(t,x)=((?)W/(?)x)(t,x). We say that a nonzero solution u of problem (2-1)is homoclinic(to0) if u(t)→0and u(t)→0as|t|→∞.An existence and a multiplicity result are obtained as following.Theorem2.1Assume that L satisfies(L) L∈C(R,RN2)is a symmetric and positively definite matrix for all t∈R and there exists a continuous function l:R→R such that l(t)>0for all t∈R and (L(t)x,x≥l(t)|x|2,l(t)→∞as|t|→∞.(L') For some a>0and r>0.one of the following is true:(ⅰ) L∈C1(R.RN2)and|L'(t)|≥a|L(t)|,for all|t|≥r.or (ⅱ) L∈C2(R.RN2)and|L"(t)≤aL(t),for all|t|≥r. where L'(t)=(d/dt)L(t) and L"(t)=(d2/dt2)L(t). and W satisfies (W1) For any T>0, uniformly in t∈[-T.T];(W2) W(t,0)=0for all t∈R and▽W(t,x)=ο(|x|) as|x|→0uniformly in t∈R;(W3) there exists θ≥1, such that θF(t,x)≥F(t,sx) for all (t,x)∈R×RN and s∈[0,1], where F(t,x)=(▽W(t,x),x)-2W(t,x), Then problem (2-1) possesses a nontrivial homoclinic orbit.Theorem2.2Assume that L satisfies (L),(L') and W satisfies (W1),(W2).(W3) and (W4) W(t,-x)=W(t,x), for all (t,x)∈R×RN. Then problem((2-1)) has infinitely many homoclinic orbits {uk} satisfying as k→∞.In Theorem2.1, we consider the existence of homoclinic orbits for problem (1) under a class of local superquadratic condition and without any periodici-ty assumptions on both L and W. There are functions L and W which satisfy Theorem1. but do not satisfy the corresponding results in [2-10].[14-16].[18-23]. Theorem2.2generalizes Theorem1.2in Yang and Han [38]. We consider the multiplicity of homoclinic orbits for problem (2-1) by using the Fountain Theorem in Willem [8], while in Yang and Han [38] a variant Fountain Theorem Zou [41] is used.In the third chapter. we consider the following second-order nonautonomous Hamiltonian systems with subquadratic potentials where L∈C(R,RN2)is a symmetric matrix valued function.W∈C1(R×RN,R) and▽W(t,x)=((?)W/(?)x)(t,x). We say that a nonzero solution u of problem (2-1)is homoclinic(to0) if u(t)→0as|t|→∞.We obtain existence and a multiplicity result as following.Theorem3.1Assume that L satisfies(L") L∈C(R,RN×N)is definite symmetric matrix for all t∈R and there exists a constant β>0such that (L(t)x,x)≥β|x|2. for all(t,x)∈R×RN; and W satisfies th3following conditions (V1) There exist three constants δ>0,r1∈(1.2).s1∈(2,2/(2-r1)]and a function a1∈Ls1(R,[0,+∞.)) such that|▽W(t,x)|≤a1(t)|x|r1-1for all t∈R and x∈RN with|x|≤δ;(V2) There exist three constants M>0. r2∈(1.2).s2∈(2,2/(2-r2)] and a function a2∈Ls2(R,[0,+∞))such that|W(t,x)|≤a2(t)|x|r2for all t∈R and x∈RN with|x|≥M;(V3) For every m>δ,there exist s3>2and bm∈Ls3(R,[0,+∞))such that|▽W(t,x)|≤bm(t) for all t∈R and x∈RN with|x|≤m;(V4) There exist constants r4∈(1,2),η>O and ζ>0such that W(t,x)≥η|x|r4for all t∈Ω and x∈RN with|x|≤ζ.where meas{Ω}>0. Then problem(3-1)possesses a nontrivial homoclinic orbit. Theorem3.2Assume that L satisfies(L")and W satisfies(V1),(V2),(V3),(V4) and (V5) W(t,-x)=W(t,x) for all t∈R and x∈RN. Then problem(3-1)has infinitely many homoclinic orbits.Theorem3.1and Theorem3.2generalize and improve the results in Zhang and Yuan[31].Sun,Chen and Nieto[32]Tang and Lin[42]. There are some functions W which satisfy Theorem1and Theorem2,but do not satisfy the corresponding results in Zhang and Yuan[31],Sun.Chen and Nieto[32],Ding [33],Yang and Han[38] and Tang and Lin[42].Lastly.we consider the existence and multiplicity of homoclinic orbits for following second-order nonautonomous Hamiltonian systems with asymptotically quadratic potentials u(t)-λu(t)+▽W(t,u(t))=0(4-1) where λ>0,W∈C1(R×RN,R)and▽W(t,u)=((?)W/(?)u)(t,u).As usual we say that a nonzero solution u of(4-1)is homoclinc(to0) if u(t)→0and u(t)→0as|t|→∞.The existence and multiplicity results are the following theorems.Theorem4.1Assume that W satisfies(U1) W∈C(R×RN)N,R+).lim(▽W(t,x)/|x|)=0uniformy in t∈R;(U2)((▽W(t,τx))/τ,x) is nondecreasing with respect,to τ∈(0,+∞)for all t∈R and x∈(R)N fixed,and there exists a function9∈C(R,R+)such that uniformly in t∈R and x∈RN;(U3) There exists a function W∞∈C((R+)N,R+) such that uniformly in x∈RN; (U4) There exists l∞∈(0,∞) such that(U5)(▽W(t,x),x)≥(▽W∞(x),x) for all t∈R,x∈(R+)N. and (▽W(t,x),x)>(▽W∞(x),x) for t∈ω and x∈(R+)N, where ω∈R is a set of positive measure. If0<λ<l∞. then (4-1) has a nontrivial homoclinic orbit.Theorem4.2Assume conditions (W1),(W2),(W3),(W4).0<λ<l∞. and(U6) W(t,-x)=W(t,x)for all t∈R and x∈RN.(U7) There exists k functions which have disjoint support Φ1,Φ2,...Φk∈Ⅱ1(R) such that for all1≤i≤k, where I is the energy functional and m∞is defined in (4-2). Then problem (4-1) has at least k pairs homoclinic orbits.Except Ding and Lee [19]. Wu and Liu [43]. Zhang and Yuan [44], all pa-pers above treat either the superquadratic case or the subquadratic case. Ding and Lee [19] treats the the asymptotically quadratic case by assuming W(t,q) be T-periodic in t. Wu and Liu [43] treats the asymptotically quadratic case with asymptotically quadratic potentials only at infinity under a cocrcivity con-dition。Theorem4.1and Theorem4.2treat the asymptotically quadratic case of problem (4-1) with asymptotically quadratic potentials both at infinity and at origin without any periodic and coercivity conditions. There are some functions W which satisfy Theorem1.1, but do not satisfy the corresponding results in Wu and Liu [43]. Ding and Lee [19] and Zhang and Yuan [44].