Infinitely many homoclinic solutions for a class of subquadratic ∗ second-order Hamiltonian systems 6 1 Xiang Lv† 0 2 Department of Mathematics, Shanghai Normal University, Shanghai 200234, PR China t c O 1 Abstract : ] In this paper, we mainly consider the existence of infinitely many homoclinic solutions for a class of S D subquadratic second-order Hamiltonian systems u¨ −L(t)u+Wu(t,u) = 0, where L(t) is not necessarily . positive definite and the growth rate of potential function W can be in (1,3/2). Using the variant fountain h t theorem, we obtain the existence of infinitely many homoclinic solutions for the second-order Hamiltonian a m systems. [ MSC: primary 34C37; 70H05; 58E05 3 Keywords: Homoclinic solutions; Hamiltonian systems; Variational methods v 7 5 5 6 0 1 Introduction and main results . 1 0 The aim of this paper is to study the following second-order Hamiltonian systems 5 1 : u¨−L(t)u+W (t,u)=0, ∀ t∈R (HS) v u i X where u =(u ,u ,...,u )∈RN, W ∈ C1(R×RN,R) and L∈ C(R,RN×N) is a symmetric matrix-valued 1 2 N r a function. We usually say that a solution u of (HS) is homoclinic (to 0) if u ∈ C2(R,RN), u(t) → 0 and u˙(t)→0 as t→±∞. Furthermore, if u6≡0, then u is called nontrivial. In the applied sciences, Hamiltonian systems can be used in many practical problems regarding gas dy- namics, fluid mechanics and celestial mechanics. It is clear that the existence of homoclinic solutions is one ∗Supported by the National Natural Science Foundation of China (NSFC) under Grants No.11371252 and No.11501369; Research and Innovation Project of Shanghai Education Committee under Grant No.14zz120; Yangfan Program of Shanghai (14YF1409100); Chen Guang Project(14CG43) of Shanghai Municipal Education Commission, Shanghai Education Develop- ment Foundation and the Research Program of Shanghai Normal University (SK201403) and Shanghai Gaofeng Project for UniversityAcademicProgramDevelopment. †E-mail:lvxiang@shnu.edu.cn. 1 of the most important problems in the theory of Hamiltonian systems. Recently, more and more mathe- maticians have paid their attention to the existence and multiplicity of homoclinic orbits for Hamiltonian systems, see [1-21]. Forthe case ofthat L(t)and W(t,x) areeither independent oft orperiodic in t, there havebeen several excellentresults, see [1–3,7,8,12–16]. More precisely,in the paper [16], Rabinowitz has provedthe existence of homoclinic orbits as a limit of 2kT-periodic solutions of (HS). Later, using the same method, several results for general Hamiltonian systems were obtained by Izydorek and Janczewska [8], Lv et al. [12]. If L(t) and W(t,x) are not periodic with respect to t, it will become more difficult to consider the existence of homoclinic orbits for (HS). This problem is quite different from the case mentioned above, due to the lack of compactness of the Sobolev embedding. In [17], Rabinowitz and Tanaka investigated system (HS) without periodicity, both for L and W. Specifically, they assumed that the smallest eigenvalue of L(t) tends to +∞ as |t| → ∞, and showed that system (HS) admits a homoclinic orbit by using a variant of the Mountain Pass theorem without the Palais-Smale condition. Inspired by the work of Rabinowitz and Tanaka [17], many results [4,6,10,11,14,15,18,20,21] were obtained for the case of aperiodicity. Most of them were presented under the following condition that L(t) is positive definite for all t∈R, (L(t)u,u)>0, ∀ t∈R and u∈RN\{0}. Motivatedby [6,20], inthis articlewe willstudy the existence ofinfinitely manyhomoclinic solutions for (HS), where L(t) is not necessarily positive definite for all t ∈ R and the growth rate of potential function W can be in (1,3/2). The main tool is the variant fountain theorem established in [22]. Our main results are the following theorems. Theorem 1.1. Assume that L and W satisfy the following conditions: (L1) There exists an α<1 such that l(t)|t|α−2 →∞ as |t|→∞ where l(t):= inf (L(t)u,u) is the smallest eigenvalue of L(t); |u|=1,u∈RN (L2) There exist constants a¯>0 and r¯>0 such that (i) L∈C1(R,RN×N) and |L′(t)u|≤a¯|L(t)u|, ∀ |t|>r¯and u∈RN , or (ii) L∈C2(R,RN×N) and (L′′(t)−a¯L(t))u,u ≤0, ∀ |t|>r¯and u∈RN , where L′(t)=(d/dt)L(t) and L(cid:0)′′(t)=(d2/dt2)L(t);(cid:1) (W) W(t,u) = a(t)|u|ν where a : R → R+ is a continuous function such that a ∈ Lµ(R,R), 1 < ν < 2 is a constant, 2≤µ≤ν¯ and 2 3 , 1<ν < ν¯= 3−2ν 2  3  ∞, 2 ≤ν <2 Then (HS) possesses infinitely many homoclinic solutions. 2 Remark 1.2. When we choose ν ∈ (1,3), it is easy to see that W satisfies the condition (W) of Theorem 2 1.1 but does not satisfy the corresponding conditions in [6,20]. Furthermore, the constant µ can be change in [2,ν¯]. 2 Preliminaries In this section, for the purpose of readability and making this paper self-contained, we will show the variational setting for (HS), which can be found in [6,20]. In what follows, we will always assume that L(t) satisfies (L1). Let A be the selfadjoint extension of the operator −(d2/dt2)+L(t) with domain D(A) ⊂ L2 ≡L2(R,RN). Let us write {E(λ):−∞<λ<+∞} and |A| for the spectral resolution and the absolute value of A respectively, and denote by |A|1/2 the square root of |A|. Define U = I −E(0)−E(−0). Then U commutes with A, |A| and |A|1/2, and A = U|A| is the polar decomposition of A (see [9]). We write E =D(|A|1/2) and introduce the following inner product on E (u,v) =(|A|1/2u,|A|1/2v) +(u,v) 0 2 2 and norm kuk =(u,u)1/2. 0 0 Here, (·,·) denotes the usual L2-inner product. Therefore, E is a Hilbert space. Since C∞(R,RN) is dense 2 0 inE,itisobviousthatE iscontinuousembeddedinH1(R,RN)(see[6]). Furthermore,wehavethefollowing lemmas by [6]. Lemma 2.1. If L satisfies (L1), then E is compactly embedded in Lp ≡Lp(R,RN) for all 1≤p∈(2/(3− α),∞]. Lemma 2.2. Let L satisfies (L1) and (L2), then D(A) is continuously embedded in W2,2(R,RN), and consequently, we have |u(t)|→0 and |u˙(t)|→0 as |t|→∞, ∀ u∈D(A). From [6], combining (L1) and Lemma 2.1, we can prove that A possesses a compact resolvent. Conse- quently, the spectrum σ(A) consists of eigenvalues, which can be arranged as λ ≤ λ ≤···→∞ (counted 1 2 with multiplicity), and the corresponding system of eigenfunctions {e : n ∈ N}, Ae = λ e , which forms n n n n an orthogonal basis in L2. Next, we define n− =#{i|λ <0}, n0 =#{i|λ =0}, n¯ =n−+n0 i i and E− =span{e1,··· ,en−}, E0 =span{en−+1,··· ,en¯}=KerA, E+ =span{en¯+1,···}. 3 Here, the closure is taken in E with respect to the norm k·k . Then 0 E =E−⊕E0⊕E+. Furthermore, we define on E the following inner product (u,v)=(|A|1/2u,|A|1/2v) +(u0,v0) , 2 2 and norm kuk2 =(u,u)=k|A|1/2uk2+ku0k2, 2 2 where u=u−+u0+u+ and v =v−+v0+v+ ∈E−⊕E0⊕E+. It is clear that the norms k·k and k·k 0 are equivalent by [6]. From now on, we will take (E,k·k)instead of (E,k·k ) as the workingspace without 0 loss of generality. Remark 2.3. We note that the decomposition E =E−⊕E0⊕E+ is also orthogonalwith respect to inner products (·,·) and (·,·) . Moreover, we will denote by E = E− ⊕E0 ⊕E+ the orthogonal decomposition 2 with respect to the inner products (·,·) unless otherwise stated. Remark 2.4. Since the norms k·k and k·k are equivalent, by Lemma 2.1,for any 1≤p∈(2/(3−α),∞], 0 there exists a constant β >0 such that p kuk ≤β kuk, ∀ u∈E, (2.1) p p where kuk denotes the usual norm of Lp and β is independent of u. p p Let O(u,v)=(|A|1/2Uu,|A|1/2v), ∀ u,v∈E be the quadratic form associated with A, where U is the polar decomposition of A. Given any u ∈ D(A) and v ∈E, we can get O(u,v)= ((u˙,v˙)+(L(t)u,v))dt. (2.2) R Z Note that D(A) is dense in E, we have (2.2) holds for all u,v ∈ E. Furthermore, by definition, it follows that O(u,v)= (P+−P−)u,v =ku+k2−ku−k2 (2.3) for all u=u−+u0+u+ ∈E, where P± :(cid:0)E →E± are th(cid:1)e respective orthogonal projections. Combining (2.2) and (2.3), we define the functional Φ on E by 1 Φ(u) = ku˙k2+(L(t)u,u) dt− W(t,u)dt 2 R R = 1kZu+(cid:0)k2− 1ku−k2− (cid:1)W(t,Zu)dt (2.4) 2 2 R 1 1 Z = ku+k2− ku−k2−Ψ(u), 2 2 where Ψ(u)= W(t,u)dt= a(t)|u|νdt for all u=u−+u0+u+ ∈E =E−⊕E0⊕E+. R R R R 4 Remark 2.5. From(W)withLemma2.1,wecaneasilyseethatΦandΨarewelldefined. Wewillconsider two cases as follows. Case (i) If 2≤µ<∞, then |Ψ(u)| = W(t,u)dt = a(t)|u|νdt ≤(cid:12)(cid:12)(cid:12)ZR |a(t)|µdt(cid:12)(cid:12)(cid:12) µ1 (cid:12)(cid:12)(cid:12)ZR |u|νµ∗dt (cid:12)(cid:12)(cid:12)µ1∗ (cid:12) (cid:12) (cid:12) (cid:12) R R (cid:18)Z (cid:19) (cid:18)Z (cid:19) =kakµkukννµ∗ <∞ where 1 + 1 =1, νµ∗ ≥1. µ µ∗ Case (ii) If µ=∞, then |Ψ(u)|≤kak kukν <∞. ∞ ν Lemma 2.6. Let (L1), (L2) and (W) hold. Then Ψ ∈ C1(E,R) and Ψ′ : E → E∗ is compact, and consequently Φ∈C1(E,R). Moreover, Ψ′(u)v = (W (t,u),u)dt= νa(t)|u|ν−2u,v dt (2.5) u R R Z Z (cid:0) (cid:1) Φ′(u)v =(u+,v+)−(u−,v−)−Ψ′(u)v (2.6) =(u+,v+)−(u−,v−)− (W (t,u),v)dt u R Z for all u = u−+u0+u+ and v = v−+v0+v+ ∈ E−⊕E0⊕E+. Moreover, all critical points of Φ on E are homoclinic solutions of (HS) satisfying u(t)→0 and u˙(t)→0 as |t|→∞. Proof. We first show that (2.5) holds by definition. If 2≤µ<∞, then 1<µ∗ ≤2, where 1 + 1 =1. For µ µ∗ any given u,v∈E, by the Mean Value Theorem and the Ho¨lder inequality, we have W(t,u+v)−W(t,u)− W (t,u),v dt u (cid:12)(cid:12)ZR(cid:2) 1 (cid:0) (cid:1)(cid:3) (cid:12)(cid:12) =(cid:12) W (t,u+θv)−W (t,u),v dθ dt (cid:12) (cid:12) u u (cid:12) (cid:12)ZR(cid:20)Z0 (cid:21) (cid:12) ≤(cid:12)(cid:12)2ν |a((cid:0)t)|(|u|+|v|)ν−1|v|dt (cid:1) (cid:12)(cid:12) (cid:12) R (cid:12) Z ≤2ν |a(t)|(|u|ν−1+|v|ν−1)|v|dt R ≤2νZ |a(t)|µdt µ1 |u|µ∗(ν−1)|v|µ∗dt µ1∗ (2.7) R R (cid:18)Z (cid:19)1 (cid:18)Z 1 (cid:19) +2ν |a(t)|µdt µ |v|µ∗νdt µ∗ R R (cid:18)Z (cid:19) (cid:18)νZ−1 (cid:19) 2+µ∗−µ∗ν ≤2νkakµ |u|2dt 2 |v|2+µ2∗µ−∗µ∗νdt 2µ∗ +2νkakµkvkνµ∗ν R R (cid:18)Z (cid:19) (cid:18)Z (cid:19) =2νkakµkuk2ν−1kvk2+µ2∗µ−∗µ∗ν +2νkakµkvkνµ∗ν ≤2νβ2+µ2∗µ−∗µ∗νkakµkuk2ν−1kvk+2νβµν∗νkakµkvkν →0, as v →0 in E 5 where 2µ∗ ≥1andthesecondinequalityholdsbythefactthatif0 3), this partismainly motivatedby the proofofLemma 2 2 in [14]. By the Ho¨lder inequality, we have 1 2 kΨ′(un)−Ψ′(u0)kE∗ ≤ sup |Wu(t,un)−Wu(t,u0)|2dt kvk2 kvk=1"(cid:18)ZR (cid:19) # (2.10) 1 2 ≤β |W (t,u )−W (t,u )|2dt , ∀ n∈N 2 u n u 0 R (cid:18)Z (cid:19) 6 We note that by Lemma 2.1, u →u in L2(ν−1) for ν > 3, passing to a subsequence if necessary, it can be n 0 2 assumed that ∞ ku −u k <+∞, n 0 2(ν−1) n=1 X which implies that ∞ |u (t)−u (t)|=g(t)∈L2(ν−1)(R,R). n 0 n=1 X Since ν > 3, then 2 |W (t,u )−W (t,u )|2dt u n u 0 R Z ≤ 2ν2|a(t)|2(|u |2(ν−1)+|u |2(ν−1))dt n 0 R Z ≤ 2ν2|a(t)|2(22ν−3|u −u |2(ν−1)+(22ν−3+1)|u |2(ν−1))dt n 0 0 R Z ≤22ν−1ν2kak2 (|g(t)|2(ν−1)+|u |2(ν−1))dt ∞ 0 R Z ≤22ν−1ν2kak2 (kgk2(ν−1)+β2(ν−1)ku k2(ν−1)) ∞ 2(ν−1) 2(ν−1) 0 Applying the Lebesgue’s Dominated Convergence Theorem, we have 1 2 |W (t,u )−W (t,u )|2dt →0, as n→∞. u n u 0 R (cid:18)Z (cid:19) Consequently, Ψ′ is weakly continuous, and so Ψ′ is continuous. Therefore Ψ ∈ C1(E,R) and hence Φ ∈ C1(E,R). Moreover,Ψ′ is compactdue to the weak continuity of Ψ′ and the fact that E is a Hilbert Space. Finally,wewillprovethatallcriticalpoints ofΦ onE arehomoclinic solutionsof(HS). By the standard procedure, we can see that any critical points of Φ on E satisfy (HS) and u ∈ C2(R,RN). We note that if 1<ν < 3, then 2≤µ≤ 2 . For µ=2, by (HS), we have 2 3−2ν kAuk2 = |W (t,u)|2dt 2 u R Z ≤ν2kuk2(ν−1) |a(t)|2dt (2.11) ∞ R Z ≤ν2β2(ν−1)kuk2(ν−1) |a(t)|µdt<∞. ∞ R Z In the case of 2<µ≤ 2 , then 3−2ν kAuk2 = |W (t,u)|2dt 2 u R Z 2 1 µ µ¯ ≤ν2 |a(t)|µdt |u|2µ¯(ν−1)dt R R (cid:18)Z (cid:19) (cid:18)Z 2 (cid:19) (2.12) µ ≤ν2kuk2(ν−1) |a(t)|µdt 2µ¯(ν−1) R (cid:18)Z (cid:19) 2 ≤ν2β2(ν−1) kuk2(ν−1) |a(t)|µdt µ <∞, 2µ¯(ν−1) R (cid:18)Z (cid:19) 7 where 2 + 1 =1 and2µ¯(ν−1)≥1 because ofµ≤ 2 . If 3 ≤ν <2,combining the factthat 2(ν−1)≥1 µ µ¯ 3−2ν 2 and Ho¨lder inequality, similar to the proof of (2.11) and (2.12), we can get the same result. Consequently, u∈D(A) and hence u is a homoclinic solution of (HS) by Lemma 2.2. The proof is complete. Inthe nextargument,the followingvariantfountaintheoremwillbe usedto proveour mainresults. Let E be a Banach space with the norm k·k and E = X with dimX < ∞ for any j ∈ N. We write j∈N j j Yk = kj=1Xj and Zk = j=kXj. The C1-functionaLl Φλ :E →R is given by L L Φ (u):=A(u)−λB(u), λ∈[1,2]. λ Theorem 2.7. ([22, Theorem2.2.]) Assume that the functional Φ defined above satisfies λ (F1) Φ maps bounded sets to bounded sets uniformly for λ∈[1,2]. Furthermore, Φ (−u)=Φ (u) for all λ λ λ (λ,u)∈[1,2]×E; (F2) B(u)≥0; B(u)→∞ as kuk→∞ on any finite dimensional subspace of E; (F3) There exist ρ >r >0 such that k k a (λ):= inf Φ (u)≥0>b (λ):= max Φ (u), ∀ λ∈[1,2] k λ k λ u∈Zk,kuk=ρk u∈Yk,kuk=rk and d (λ):= inf Φ (u)→0 as k→∞ uniformly for λ∈[1,2]. k λ u∈Zk,kuk≤ρk Then there exist λ →1, u ∈Y such that n λn n Φ′ | (u )=0, Φ (u )→c ∈[d (2),b (1)] as n→∞. λn Yn λn λn λn k k k In particular, if {u } has a convergent subsequence for every k, then Φ has infinitely many nontrivial λn 1 critical points {u }∈E\{0} satisfying Φ (u )→0− as k →∞. k 1 k In order to make use of Theorem 2.7, we consider the functionals A, B and Φ on the working space λ defined E =D(|A|1/2) by 1 1 A(u)= ku+k2, B(u)= ku−k2+ W(t,u)dt, (2.13) 2 2 R Z and 1 1 Φ (u)=A(u)−λB(u)= ku+k2−λ ku−k2+ W(t,u)dt (2.14) λ 2 2 R (cid:18) Z (cid:19) for all u=u−+u0+u+ ∈E and λ∈[1,2]. By Lemma 2.6, it is clear that Φ ∈C1(E,R) for all λ∈[1,2]. λ LetX :=Re =span{e }, j ∈N, where{e ,j ∈N} is the systemofeigenfunctions andthe orthogonalbasis j j j j in L2 below Lemma 2.2. Furthermore, it is evident that Φ =Φ, where Φ is the functional defined in (2.4). 1 8 3 Proof of theorems Lemma 3.1. Let (L1), (L2) and (W) hold, then B(u) ≥ 0. Moreover, B(u) → ∞ as kuk → ∞ on any finite dimensional subspace of E. Proof. By definitions of the functional B and (W), B(u) ≥ 0 holds obviously. Next we will prove that B(u) → ∞ as kuk → ∞ on any finite dimensional subspace of E. First we claim that for any finite dimensional subspace F ⊂E, there exists ε>0 such that meas{t∈R:a(t)|u(t)|ν ≥εkukν}≥ε, ∀ u∈F\{0}. (3.1) The proof of (3.1) is very similar as that of [18]. We omit it here. Now, let Ω ={t∈R:a(t)|u(t)|ν ≥εkukν}, ∀ u∈F\{0}, (3.2) u where ε is given in (3.1). From (3.1), we can obtain that meas(Ω )≥ε, ∀ u∈F\{0}, (3.3) u Combining (W) and (3.3), for all u∈F\{0}, we can see that 1 B(u) = ku−k2+ W(t,u)dt 2 R Z ≥ a(t)|u(t)|νdt (3.4) ZΩu ≥εkukνmeas(Ω )≥ε2kukν. u This implies B(u)→∞ as kuk→∞ on any finite dimensional subspace of E. If µ=∞, similar to the case of 2≤µ<∞, by the standard procedure, we can prove that there exists ε >0 such that 1 meas{t∈R:a(t)|u(t)|ν ≥ε kukν}≥ε , ∀ u∈F\{0}. (3.5) 1 1 Therefore, by (3.4), we can conclude that B(u)→∞ as kuk→∞ on any finite dimensional subspace of E. The proof is complete. Lemma 3.2. Under the conditions in Theorem 1.1, then there exists a sequence ρ → 0+ as k → ∞ such k that a (λ):= inf Φ (u)≥0, ∀ λ∈[1,2], k≥n¯+1, k λ u∈Zk,kuk=ρk and d (λ):= inf Φ (u)→0 as k→∞ uniformly for λ∈[1,2]. k λ u∈Zk,kuk≤ρk where Z = X for all k ∈N. k j=k j L 9 Proof. Bythedefinitionofn¯ belowtheLemma2.2,wecanknowthatZ ⊂E+ forallk ≥n¯+1. Therefore, k for all k ≥n¯+1, from (W) and (2.14), it follows that 1 Φ (u) = kuk2−λ W(t,u)dt λ 2 R Z 1 ≥ kuk2−2 W(t,u)dt (3.6) 2 R Z 1 = kuk2−2 a(t)|u|νdt, ∀ (λ,u)∈[1,2]×Z . k 2 R Z If 2≤µ<∞, let ηk := sup kukνµ∗, where µ1 + µ1∗ =1. By Lemma 2.1, we can conclude that ηk →0 u∈Zk,kuk=1 as k →∞. Therefore, combining (3.6) with (W), we have 1 1 Φλ(u)≥ 2kuk2−2kakµkukννµ∗ ≥ 2kuk2−2ηkνkakµkukν, ∀ (λ,u)∈[1,2]×Zk. (3.7) Let ρ := (8ηνkak )1/(2−ν), the rest of proof is very similar as that of [18]. We omit it here. For the case k k µ of µ = ∞, similar to the above procedure, the same result can be obtained. We omit it here. The proof is complete. Lemma 3.3. Assume that (L1), (L2) and (W) hold, then for the sequence {ρk}k∈N obtained in Lemma 3.2, there exists a sequence {rk}k∈N such that ρk >rk >0 for ∀ k ∈N and b (λ):= max Φ (u)<0, ∀ λ∈[1,2]. (3.8) k λ u∈Yk,kuk=rk where Y = k X =span{e ,...,e } for ∀ k ∈N. k j=1 j 1 k Proof. ForL∀ k ∈ N, it is clear that Y is a finite dimensional subspace of E. Therefore, for ∀ λ ∈ [1,2], k from (W), (3.2), (3.3) and (3.5), let ε =min{ε,ε }, we have 0 1 1 1 Φ (u) = ku+k2−λ ku−k2+ W(t,u)dt λ 2 2 R 1 (cid:18) Z (cid:19) ≤ kuk2− W(t,u)dt 2 R 1 Z ≤ kuk2− a(t)|u|νdt (3.9) 2 1 ZΩu ≤ kuk2−ε kukνmeas(Ω ) 0 u 2 1 ≤ kuk2−ε2kukν, ∀ u∈Y ,k∈N. 2 0 k 2 For ∀ k ∈N, we choose 0

Infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems

Similar Infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems