Báo cáo hóa học: "Research Article Existence of Four Solutions of Some Nonlinear Hamiltonian System" docx

Hindawi Publishing Corporation Boundary Value Problems Volume 2008, Article ID 293987, 17 pages doi:10.1155/2008/293987 Research Article Existence of Four Solutions of Some Nonlinear Hamiltonian System Tacksun Jung 1 and Q-Heung Choi 2 1 Department of Mathematics, Kunsan National University, Kunsan 573-701, South Korea 2 Department of Mathematics Education, Inha University, Incheon 402-751, South Korea Correspondence should be addressed to Q-Heung Choi, qheung@inha.ac.kr Received 25 August 2007; Accepted 3 December 2007 Recommended by Kanishka Perera We show the existence of four 2π-periodic solutions of the nonlinear Hamiltonian system with some conditions. We prove this problem by investigating the geometry of the sublevels of the functional and two pairs of sphere-torus variational linking inequalities of the functional and applying the critical point theory induced from the limit relative category. Copyright q 2008 T. Jung and Q H. Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and statements of main results Let Ht, z be a C 2 function defined on R 1 × R 2n which is 2π-periodic with respect to the first variable t.In this paper, we investigate the number of 2π-periodic nontrivial solutions of the following nonlinear Hamiltonian system ˙z  J  H z t, zt  , 1.1 where z : R → R 2n ,˙z  dz/dt, J   0 −I n I n 0  , 1.2 I n is the identity matrix on R n , H : R 1 × R 2n → R,andH z is the gradient of H.Letz p,q, p z 1 , ,z n ,qz n1 , ,z 2n  ∈ R n . Then 1.1 can be rewritten as ˙p  −H q t, p, q, ˙q  H p t, p, q. 1.3 We assume that H ∈ C 2 R 1 × R 2n ,R 1  satisfies the following conditions. 2 Boundary Value Problems H1 There exist constants α<βsuch that αI ≤ d 2 z Ht, z ≤ βI ∀t, z ∈ R 1 × R 2n . 1.4 H2 Let j 1 ,j 2  j 1  1andj 3  j 2  1 be integers and α, β be any numbers without loss of generality, we may assume α, β / ∈ Z such that j 1 − 1 <α<j 1 <j 2 <β<j 2  1  j 3 . Suppose that there exist γ>0andτ>0 such that j 2 <γ<βand Ht, z ≥ 1 2 γz 2 L 2 − τ ∀t, z ∈ R 1 × R 2n . 1.5 H3 Ht, 00, H z t, 00, and j ∈ j 1 ,j 2  ∩ Z such that jI < d 2 z Ht, 0 < j  1I ∀t ∈ R 1 . 1.6 H4 H is 2π-periodic with respect to t. We are looking for the weak solutions of 1.1.LetE  W 1/2,2 0, 2π,R 2n .The2π- periodic weak solution z p, q ∈ E of 1.3 satisfies  2π 0  ˙p  H q t, zt  · ψ −  ˙q − H p t, zt  · φ  dt  0 ∀ζ φ, ψ ∈ E 1.7 and coincides with the critical points of the induced functional Iz  2π 0 p ˙qdt−  2π 0 Ht, ztdt  Az −  2π 0 Ht, ztdt, 1.8 where Az1/2  2π 0 ˙z · Jzdt. Our main results are the following. Theorem 1.1. Assume that H satisfies conditions (H1)–(H4). Then there exists a number δ>0 such that for any α and β with j 1 − 1 <α<j 1 <j 2 <β<j 2  δ<j 2  1  j 3 , α>0,system1.1 has at least four nontrivial 2π-periodic solutions. Theorem 1.2. Assume that H satisfies conditions (H1)–(H4). Then there exists a number δ>0 such that for any α and β,andj 1 − 1 <α<j 1 <j 2 <β<j 2  δ<j 2  1  j 3 , β<0,system 1.1 has at least four nontrivial 2π-periodic solutions. Changprovedin1 that, under conditions H1–H4, system 1.1 has at least two non- trivial 2π-periodic solutions. He proved this result by using the finite dimensional variational reduction method. He first investigate the critical points of the functional on the finite dimen- sional subspace and the P.S. condition of the reduced functional and find one critical point of the mountain pass type. He also found another critical point by the shape of graph of the reduced functional. T. Jung and Q H. Choi 3 For the proofs of Theorems 1.1 and 1.2, we first separate the whole space E into the four mutually disjoint four subspaces X 0 , X 1 , X 2 , X 3 which are introduced in Section 3 and then we investigate two pairs of sphere-torus variational linking inequalities of the reduced functional  I and ˇ I of I on the submanifold with boundary  C and ˇ C, respectively, and translate these two pairs of sphere-torus variational links of  I and ˇ I into the two pairs of torus-sphere variational links of −  I and − ˇ I,where  I and ˇ I are the restricted functionals of I to the manifold with bound- ary  C and ˇ C, respectively. Since  I and ˇ I are strongly indefinite functinals, we use the notion of the P.S. ∗ c condition and the limit relative category instead of the notion of P.S. c condition and the relative category, which are the useful tools for the proofs of the main theorems. We also investigate the limit relative category of torus in torus, boundary of torus on  C and ˇ C, respectively. By the critical point theory induced from the limit relative category theory we obtain two nontrivial 2π-periodic solutions in each subspace X 1 and X 2 , so we obtain at least four nontrivial 2π-periodic solutions of 1.1. In Section 2, we introduce some notations and some notions of P.S. ∗ c condition and the limit relative category and recall the critical point theory on the manifold with boundary. We also prove some propositions. In Section 3,weproveTheorem 1.1 and in Section 4,weprove Theorem 1.2. 2. Recall of the critical point theory induced from the limit relative category Let E  W 1/2,2 0, 2π,R 2n . The scalar product in L 2 naturally extends as the duality pairing between E and E   W −1/2,2 0, 2π,R 2n .Itisknownthatifz ∈ C ∞ R, R 2n  is 2π-periodic, then it has a Fourier expansion zt  k∞ k−∞ a k e ikn with a k ∈ C 2n and a −k  a k : E is the closure of such functions with respect to the norm z    k∈Z 1  |k||a k | 2  1/2 . 2.1 Let us set the functional Az 1 2  2π 0 ˙z · Jzdt   2π 0 p ˙qdt, zp, q ∈ E, p, q ∈ R n , 2.2 so that IzAz −  2π 0 Ht, ztdt. 2.3 Let e 1 , ,e 2n denote the usual bases in R 2n and set E 0  span  e 1 , ,e 2n  , E   span  sin jte k − cos jte kn , cos jte k sin jte kn | j ∈ N, 1 ≤ k ≤ n  , E −  span  sin jte k cos jte kn , cos jte k − sin jte kn | j ∈ N, 1 ≤ k ≤ n  . 2.4 4 Boundary Value Problems Then E  E 0 ⊕E  ⊕E − and E 0 ,E  ,E − are the subspaces of E on which A is null, positive definite and negative definite, and these spaces are orthogonal with respect to the bilinear form Bz, ζ ≡  2π 0 p · ˙ψ  φ · ˙qdt 2.5 associated with A. Here, z p, q and ζ φ, ψ. If z ∈ E  and ζ ∈ E − , then the bilinear form is zero and Az  ζAzAζ.WealsonotethatE 0 ,E  ,andE − are mutually orthogonal in L 2 0, 2π,R 2n .LetP  be the projection from E onto E  and P − the one from E onto E − .Then the norm in E is given by z 2    z 0   2  A  z   − A  z −     z 0   2    P  z   2    P − z   2 2.6 which is equivalent to the usual one. The space E with this norm is a Hilbert space. We need the following facts which are proved in 2. Proposition 2.1. For each s ∈ 1, ∞, E is compactly embedded in L s 0, 2π,R 2n . In particular, there is an α s > 0 such that z L s ≤ α s z 2.7 for all z ∈ E. Proposition 2.2. Assume that Ht, z ∈ C 2 R 1 × R 2n ,R.ThenIz is C 1 ,thatis,Iz is continuous and Fr ´ echet differentiable in E with Fr ´ echet derivative DIzω   2π 0  ˙z − JH z t, z  · Jω   2π 0  ˙p  H q t, z  · ψ −  ˙q − H p t, z  · φ  dt, 2.8 where z p, q and ω φ, ψ ∈ E. Moreover, the functional z →  2π 0 Ht, zdt is C 1 . Proof. For z, w ∈ E,   Iz  w − Iz − DIzw        1 2  2π 0  ˙z ˙w · Jzw −  2π 0 Ht, zw− 1 2  2π 0 ˙z · Jz  2π 0 Ht, z −  2π 0  ˙z−J  H z t, z  · Jw          1 2  2π 0  ˙z · Jw  ˙w · Jz  ˙w · Jw  −  2π 0  Ht, z  w − Ht, z  −  2π 0  ˙z − J  H z t, z  · Jw      . 2.9 We have      2π 0 Ht, z  w − Ht, z     ≤      2π 0 H z t, z · w  o|w|dt      O|w|. 2.10 T. Jung and Q H. Choi 5 Thus, we have   Iz  w − Iz − DIzw    O|w| 2 . 2.11 Next, we prove that Iz is continuous. For z, w ∈ E,   Iz  w − Iz        1 2  2π 0  ˙z  ˙w · Jz  w −  2π 0 Ht, z  w − 1 2  2π 0 ˙z · Jz   2π 0 Ht, z          1 2  2π 0  ˙z · Jw  ˙w · Jz  ˙w · Jw  −  2π 0  Ht, z  w − Ht, z       O|w|. 2.12 Similarly, it is easily checked that I is C 1 . Now, we consider the critical point theory on the manifold with boundary induced from the limit relative category. Let E be a Hilbert space and X be the closure of an open subset of E such that X can be endowed with the structure of C 2 manifold with boundary. Let f : W → R be a C 1,1 functional, where W is an open set containing X.TheP.S. ∗ c condition and the limit relative category see 3 are useful tools for the proof of the main theorem. Let E n  n be a sequence of a closed finite dimensional subspace of E with the following assumptions: E n  E − n ⊕ E  n where E  n ⊂ E  , E − n ⊂ E − for all n E  n and E − n are subspaces of E, dim E n < ∞, E n ⊂ E n1 ,  n∈N E n are dense in E.LetX n  X ∩ E n , for any n, be the closure of an open subset of E n and has the structure of a C 2 manifold with boundary in E n . We assume that for any n there exists a retraction r n : X → X n .ForagivenB ⊂ E, we will write B n  B ∩E n . Let Y be a closed subspace of X. Definition 2.3. Let B be a closed subset of X with Y ⊂ B.Letcat X,Y  B be the relative category of B in X, Y . We define the limit relative category of B in X, Y, with respect to X n  n ,by cat ∗ X,Y  Blim sup n→∞ cat X n ,Y n  B n . 2.13 We set B i   B ⊂ X | cat ∗ X,Y  B ≥ i  , c i  inf B∈B i sup x∈B fx. 2.14 We have the following multiplicity theorem for the proof, see 4. Theorem 2.4. Let i ∈ N and assume that 1 c i < ∞, 2 sup x∈Y fx <c i , 3 the P.S. ∗ c i condition with respect to X n  n holds. 6 Boundary Value Problems Then there exists a lower critical point x such that fxc i .If c i  c i1  ···  c ik−1  c, 2.15 then cat X  x ∈ X | fxc, grad − X fx0  ≥ k. 2.16 Now, we state the following multiplicity result for the proof, see 4, Theorem 4.6 which will be used in the proofs of our main theorems. Theorem 2.5. Let H be a Hilbert space and let H  X 1 ⊕ X 2 ⊕ X 3 ,whereX 1 , X 2 , X 3 are three closed subspaces of H with X 1 , X 2 of finite dimension. For a given subspace X of H,letP X be the orthogonal projection from H onto X.Set C   x ∈ H |   P X 2 x   ≥ 1  , 2.17 and let f : W → R be a C 1,1 function defined on a neighborhood W of C.Let1 <ρ<R, R 1 > 0.One defines Δ  x 1  x 2 | x 1 ∈ X 1 ,x 2 ∈ X 2 ,   x 1   ≤ R 1 , 1 ≤   x 2   ≤ R  , Σ  x 1  x 2 | x 1 ∈ X 1 ,x 2 ∈ X 2 ,   x 1   ≤ R 1 ,   x 2    1  ∪  x 1  x 2 | x 1 ∈ X 1 ,x 2 ∈ X 2 ,   x 1   ≤ R 1 ,   x 2    R  ∪  x 1  x 2 | x 1 ∈ X 1 ,x 2 ∈ X 2 ,   x 1    R 1 , 1 ≤   x 2   ≤ R  , S   x ∈ X 2 ⊕ X 3 |x  ρ  , B   x ∈ X 2 ⊕ X 3 |x≤ρ  . 2.18 Assume that sup fΣ < inf fS2.19 and that the P.S. c condition holds for f on C, with respect to the sequrnce C n  n , for all c ∈ a, b, where a  inf fS,b sup fΔ. 2.20 Moreover, one assumes b<∞ and f| X 1 ⊕X 3 has no critical points z in X 1 ⊕ X 3 with a ≤ fz ≤ b. Then there exist two lower critical points z 1 , z 2 for f on C such that a ≤ fz i  ≤ b, i  1.2. 3. Proof of Theorem 1.1 We assume that 0 <α<β.Lete 1 , ,e 2n denote the usual bases in R 2n and set X 0 ≡ span  sin jte k − cos jte kn , cos jte k sin jte kn , sin jte k cos jte kn , cos jte k − sin jte kn ,e 1 ,e 2 , ,e 2n | j ≤ j 1 − 1,j∈ N, 1 ≤ k ≤ n  , X 1 ≡ span  sin jte k − cos jte kn , cos jte k sin jte kn | j  j 1 , 1 ≤ k ≤ n  , X 2 ≡ span  sin jte k − cos jte kn , cos jte k sin jte kn | j  j 2 , 1 ≤ k ≤ n  , X 3 ≡ span  sin jte k − cos jte kn , cos jte k sin jte kn | j ≥ j 2  1  j 3 ,j∈ N, 1 ≤ k ≤ n  . 3.1 T. Jung and Q H. Choi 7 Then E is the topological direct sum of subspaces X 0 , X 1 , X 2 , and X 3 ,whereX 1 and X 2 are finite dimensional subspaces. We also set S 1 ρ  z ∈ X 1 |z  ρ  , S r 1  X 0 ⊕ X 1    z ∈ X 0 ⊕ X 1 |z  r 1  , B r 1  X 0 ⊕ X 1    z ∈ X 0 ⊕ X 1 |z≤r 1  , Σ R 1  S 1 ρ,X 2 ⊕ X 3    z  z 1  z 2  z 3 ∈ X 1 ⊕ X 2 ⊕ X 3 | z 1 ∈ S 1 ρ,   z 1  z 2  z 3    R 1  , Δ R 1  S 1 ρ,X 2 ⊕ X 3    z  z 1  z 2  z 3 ∈ X 1 ⊕ X 2 ⊕ X 3 | z 1 ∈ S 1 ρ,   z 1  z 2  z 3   ≤ R 1  , S 2 ρ{z ∈ X 2 |z  ρ}, S r 2  X 0 ⊕ X 1 ⊕ X 2    z ∈ X 0 ⊕ X 1 ⊕ X 2 |z  r 2  , B r 2  X 0 ⊕ X 1 ⊕ X 2    z ∈ X 0 ⊕ X 1 ⊕ X 2 |z≤r 2  , Σ R 2  S 2 ρ,X 3    z  z 2  z 3 ∈ X 2 ⊕ X 3 | z 2 ∈ S 2 ρ,   z 2  z 3    R 2  , Δ R 2  S 2 ρ,X 3    z  z 2  z 3 ∈ X 2 ⊕ X 3 | z 2 ∈ S 2 ρ,   z 2  z 3   ≤ R 2  . 3.2 We have the following two pairs of the sphere-torus variational linking inequalities. Lemma 3.1 first sphere-torus variational linking. Assume that H satisfies the conditions (H1), (H3), (H4), and the condition H2  suppose that there exist γ>0 and τ>0 such that j 1 <γ<βand Ht, z ≥ 1 2 γz 2 − τ ∀t, z ∈ R 1 × R 2n . 3.3 Then there exist δ 1 > 0, ρ>0, r 1 > 0,andR 1 > 0 such that r 1 <R 1 , and for any α and β with j 1 − 1 <α<j 1 <β<j 2  δ 1 <j 2  1  j 3 and α>0, sup z∈S r 1 X 0 ⊕X 1  Iz < 0 < inf z∈Σ R 1 S 1 ρ,X 2 ⊕X 3  Iz, inf z∈Δ R 1 S 1 ρ,X 2 ⊕X 3  Iz > −∞, sup z∈B r 1 X 0 ⊕X 1  Iz < ∞. 3.4 Proof. Let z  z 0  z 1 ∈ X 0 ⊕ X 1 .ByH2  ,wehave Iz 1 2  2π 0 ˙z · Jz dt −  2π 0 Ht, ztdt ≤ 1 2 z 0  z 1  2 − γ 2 z 0  z 1  2 L 2  τ ≤ 1 2 j 1 − γz 0  z 1  2 L 2  τ 3.5 8 Boundary Value Problems for some τ>0. Since j 1 − γ<0, there exists r 1 > 0 such that if z 0  z 1 ∈ S r 1 X 0 ⊕ X 1 , then Iz < 0. Thus, sup z∈S r 1 X 0 ⊕X 1  Iz < 0. Moreover, if z ∈ B r 1 X 0 ⊕ X 1 ,thenIz ≤ 1/2j 1 − γz 0  z 1  2 L 2  τ<τ<∞, so we have sup z∈B r 1 X 0 ⊕X 1  Iz < ∞. Next, we will show that there exist δ 1 > 0, ρ>0andR 1 > 0 such that if j 1 − 1 <α<j 1 <β<j 2  δ 1 <j 2  1  j 3 , then inf z∈Σ R 1 S 1 ρ,X 2 ⊕X 3  Iz > 0. Let z  z 1  z 2  z 3 ∈ X 1 ⊕ X 2 ⊕ X 3 with z 1 ∈ S 1 ρ, z 2 ∈ X 2 , z 3 ∈ X 3 ,whereρ is a small number. Let j 1 − 1 <α<j 1 <β<j 2  δ<j 2  1  j 3 for some δ>0 and α>0. Then X 1 ⊕ X 2 ⊕ X 3 ⊂ E  and P − z 1  z 2  z 3 0. By H1, there exists d>0 such that Iz 1 2  2π 0 ˙z · Jz dt −  2π 0 Ht, ztdt ≥ 1 2   P  z 1  z 2  z 3    2 − β 2   P  z 1  z 2  z 3    2 L 2 − d ≥ 1 2 j 1 − β   P  z 1   2 L 2  1 2 j 2 − β   P  z 2   2 L 2  1 2 j 3 − β   P  z 3   2 L 2 − d  1 2 j 1 − βρ 2 − 1 2 δ   P  z 2   2 L 2  1 2 j 3 − β   P  z 3   2 L 2 − d. 3.6 Since j 1 − β<0, j 2 − β>−δ,andj 3 − β>0, there exist a small number δ 1 > 0andR 1 > 0 with δ 1 <δand R 1 >r 1 such that if j 1 − 1 <α<j 1 <β<j 2  δ 1 <j 2  1  j 3 and z ∈ Σ R 1 S 1 ρ,X 2 ⊕ X 3 ,thenIz > 0. Thus, we have inf z∈Σ R 1 S 1 ρ,X 2 ⊕X 3  Iz > 0. Moreover, if j 1 − 1 <α<j 1 <β<j 2  δ 1 <j 2  1  j 3 and z ∈ Δ R 1 S 1 ρ,X 2 ⊕ X 3 ,thenwehave Iz > 1/2j 1 − βρ 2 − 1/2δ1P  z 2  2 L 2 − d>−∞. Thus, inf Δ R 1 S 1 ρ,X 2 ⊕X 3  Iz > −∞. Thus, we prove the lemma. Lemma 3.2. Let δ 1 be the number introduced in Lemma 3.1. Then for any α and β with j 1 − 1 <α< j 1 <β≤ j 2 <j 2  1  j 3 and α>0,ifu is a critical point for I| X 0 ⊕X 2 ⊕X 3  ,thenIu0. Proof. We notice that from Lemma 3.1,forfixedu 0 ∈ X 0 , the functional u 23 → Iu 0  u 23  is weakly convex in X 2 ⊕ X 3 ,while,forfixedu 23 ∈ X 2 ⊕ X 3 , the functional u 0 → Iu 0  u 23  is strictly concave in X 0 . Moreover, 0 is the critical point in X 0 ⊕ X 2 ⊕ X 3 with I00. So if u  u 0  u 23 is another critical point for I| X 0 ⊕X 2 ⊕X 3  ,thenwehave 0  I0 ≤ Iu 23  ≤ Iu 0  u 23  ≤ Iu 0  ≤ I00. 3.7 So we have IuI00. Let P X 1 be the orthogonal projection from E onto X 1 and  C   z ∈ E |   P X 1 z   ≥ 1  . 3.8 Then  C is the smooth manifold with boundary. Let  C n   C ∩ E n . Let us define afunctional  Ψ : E \{X 0 ⊕ X 2 ⊕ X 3 }→E by  Ψzz − P X 1 z   P X 1 z    P X 0 ⊕X 2 ⊕X 3  z   1 − 1   P X 1 z    P X 1 z. 3.9 T. Jung and Q H. Choi 9 We have ∇  Ψzww − 1   P X 1 z    P X 1 w −  P X 1 z   P X 1 z   ,w  P X 1 z   P X 1 z    . 3.10 Let us define the functional  I :  C → R by  I  I ◦  Ψ. 3.11 Then  I ∈ C 1,1 loc .Wenotethatifz is the critical point of  I and lies in the interior of  C,thenz   Ψz is the critical point of I.Wealsonotethat   grad −  C  Iz   ≥   P X 0 ⊕X 2 ⊕X 3  ∇I  Ψz   ∀z ∈ ∂  C. 3.12 Let us set  S r 1   Ψ −1  S r 1  X 0 ⊕ X 1  ,  B r 1   Ψ −1  B r 1  X 0 ⊕ X 1  ,  Σ R 1   Ψ −1  Σ R 1  S 1 ρ,X 2 ⊕ X 3  ,  Δ R 1   Ψ −1  Δ R 1  S 1 ρ,X 2 ⊕ X 3  . 3.13 We note that  S r 1 ,  B r 1 ,  Σ R 1 , and  Δ R 1 have the same topological structure as S r 1 , B r 1 , Σ R 1 , and Δ R 1 , respectively. Lemma 3.3. −  I satisfies the P.S. ∗ c condition with respect to   C n  n for every real number c such that 0 < inf z∈  Ψ −1 S r 1 X 0 ⊕X 1  −  Iz ≤ c ≤ sup z∈  Ψ −1 Δ R 1 S 1 ρ,X 2 ⊕X 3  −  Iz. 3.14 Proof. Let k n  n be a sequence such that k n → ∞, z n  n be a sequence in C such that z n ∈ C k n , for all n, −  Iz n  → c and grad − C −  I| E k n z n  → 0. Set z n Ψz n and hence z n ∈ E k n  and −Iz n  → c. We first consider the case in which z n / ∈ X 0 ⊕ X 2 ⊕ X 3 , for all n. Since for n large P E n ◦ P X 1  P X 1 ◦ P E n  P X 1 ,wehave P E k n ∇−  Iz n P E k n Ψ  z n ∇−Iz n   Ψ  z n P E k n ∇−Iz n  −→ 0. 3.15 By 3.9 and 3.10, P E k n ∇−Iz n −→ 0or P X 0 ⊕X 2 ⊕X 3  P E k n ∇−Iz n  −→ 0,P X 1 z n −→ 0. 3.16 In the first case, the claim follows from the limit Palais-Smale condition for −I. In the second case, P X 0 ⊕X 2 ⊕X 3  P E k n ∇−Iz n  → 0. We claim that z n  n is bounded. By contradiction, we sup- pose that z n →∞ and set w n  z n /z n . Up to a subsequence w n w 0 weakly for some 10 Boundary Value Problems w 0 ∈ X 0 ⊕ X 2 ⊕ X 3 . By the asymptotically linearity of ∇−Iz n  we have  ∇−Iz n    z n   ,w n    P X 0 ⊕X 2 ⊕X 3  P E k n ∇−Iz n    z n   ,w n    ∇−Iz n    z n   2 ,P X 1 z n  −→ 0. 3.17 We have  ∇−Iz n    z n   ,w n   2−Iz n    z n   2   2π 0  − 2Ht, z n    z n   2  H z t, z n  · w n   z n    dt, 3.18 where z n z n  1 , ,z n  2n . Passing to the limit, we get lim n→∞  2π 0  2Ht, z n    z n   2 − H z t, z n  · w n   z n    dt  0. 3.19 Since H and H z t, z n ·z n are bounded and z n →∞in Ω, w 0  0. On the other hand, we have  P X 0 ⊕X 2 ⊕X 3  P E k n ∇−Iz n    z n   ,w n    2π 0  − ˙w n · Jw n   P X 0 ⊕X 2 ⊕X 3  P E k n H z t, z n    z n    · w n  dt. 3.20 Moreover, we have  P X 0 ⊕X 2 ⊕X 3  P E k n ∇−Iz n    z n   ,P  w n − P − w n   −   P X 2 ⊕X 3 P  w n   2 −   P X 0 P − w n   2 −  2π 0 P X 0 ⊕X 2 ⊕X 3  P E k n H z t, z n    z n   ·  P  w n − P − w n  dt. 3.21 Since w n converges to 0 weakly and H z t, z n  · P  w n − P − w n  is bounded, P X 2 ⊕X 3 P  w n  2  P X 0 P − w n  2 → 0. Since P X 1 w n  2 → 0, w n converges to 0 strongly, which is a contradiction. Hence, z n  n is bounded. Up to a subsequence, we can suppose that z n converges to z 0 for some z 0 ∈ X 0 ⊕ X 2 ⊕ X 3 . We claim that z n converges to z 0 strongly. We have  P X 0 ⊕X 2 ⊕X 3  P E k n ∇−Iz n ,P  z n − P − z n   −   P X 2 ⊕X 3 P E k n P  z n   2 −   P X 0 P E k n P − z n   2  P X 0 ⊕X 2 ⊕X 3  P E k n  2π 0 H z t, z n  ·  P  z n − P − z n  . 3.22 By H1 and the boundedness of H z t, z n P  z n − P − z n ,   P X 2 ⊕X 3 P E k n P  z n   2    P X 0 P E k n P − z n   2 −→ P X 0 ⊕X 2 ⊕X 3  P E k n  2π 0 H z t, z ·  P  z − P − z  . 3.23 [...]... topological direct sum of the subspaces X0 , X1 , X2 , and X3 , where X1 and X2 are finite dimensional subspaces Proof of Theorem 1.2 By the same arguments as that of the proof of Theorem 1.1, there exist δ > 0, ρ > 0, r 1 > 0, R 1 , r 2 > 0, and R 2 > 0 such that for any α and β with j1 − 1 < α < j1 < j2 < β < j2 δ, 1.1 has at least four nontrivial solutions, two of which are nontrivial solutions zi , i... nontrivial solutions wi , i 1, 2, in X2 for the functional I such that inf z∈ΔR 2 S2 ρ ,X3 I z ≤ I wi ≤ sup z∈Sr 2 X0 ⊕X1 ⊕X2 I z 0, 1.1 has at least four nontrivial solutions, two of which are in X1 and two of which are in X2 T Jung and Q.-H Choi 17 4 Proof of Theorem... 1 and two of which are nontrivial solutions wi , i inf z∈ΔR 2 S2 ρ ,X3 I z ≤ I wi ≤ I z 0, r 1 > 0, and R 1 > 0 such that for any α and β with j1 − 1 < α < j1 < β < j2 δ 1 < j2 1 j3 , 1.1 has at least two nontrivial solutions zi , i 1, 2, in X1 for the functional I such that inf z∈ΔR 1 S1 ρ ,X2 ⊕X3... Applications to Differential Equations, vol 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986 3 G Fournier, D Lupo, M Ramos, and M Willem, “Limit relative category and critical point theory,” in Dynamics Reported, vol 3, pp 1–24, Springer, Berlin, Germany, 1994 4 A M Micheletti and C Saccon, “Multiple nontrivial solutions for a floating beam equation... I z ≤ τ, then z 0 Proof By contradiction, we can suppose that there exist Λ > 0, a sequence αn n , βn n such that αn → α, βn → β with α ∈ j1 − 1, j1 , β ∈ j2 · Λ , and a sequence zn n in X0 ⊕ X1 ⊕ X3 such that I zn → 0 and P X0 ⊕X1 ⊕X3 ∇I zn 0 We claim that zn n is bounded If we do not suppose that zn → ∞, let us set wn zn / zn We have up to a subsequence, that wn w0 weakly for some w0 ∈ X0 ⊕ X1 ⊕... 2 L 2π 2 0 H t, z dt < β z 2 2 c2 , for some c1 and c2 If z ∈ X0 ⊕ X1 with PX0 ⊕X1 z 2 ≥ |j| z L j < 0 and |j| > β, 2 L2 |j| PX0 ⊕X1 z If z ∈ X3 , PX3 z 2 ≥ j 3 PX 3 z 2 L2 , ≤ PX0 ⊕X1 z 2 3.46 0 2 L2 ≤ β PX0 ⊕X1 z c2 c1 < for 2 L2 3.47 and 2 L2 j 3 PX 3 z ≤ PX 3 z 2 2 L2 ≤ β PX 3 z c2 3.48 Thus, we have |j| − β 2 L2 PX0 ⊕X1 z j3 − β which is absurd because of |j| > β and j3 > β Thus z PX 3 z 2 L2... ⊕X2 I z −∞, inf 2 3.32 X0 ⊕X1 ⊕X2 z2 ∈ X0 ⊕ X1 ⊕ X2 By H2 , we have z1 2π z · Jz dt − ˙ 0 H t, z t dt ≤ 0 1 z 2 2 − γ z 2 2 L2 τ≤ 1 j2 − γ z 2 2 L2 τ 3.34 for some τ Since j2 − γ < 0, there exists r 2 > 0 such that if z ∈ Sr 2 X0 ⊕ X1 ⊕ X2 , then I z < 0 Thus we have . 2008, Article ID 293987, 17 pages doi:10.1155/2008/293987 Research Article Existence of Four Solutions of Some Nonlinear Hamiltonian System Tacksun Jung 1 and Q-Heung Choi 2 1 Department of Mathematics,. Perera We show the existence of four 2π-periodic solutions of the nonlinear Hamiltonian system with some conditions. We prove this problem by investigating the geometry of the sublevels of the functional and. by the shape of graph of the reduced functional. T. Jung and Q H. Choi 3 For the proofs of Theorems 1.1 and 1.2, we first separate the whole space E into the four mutually disjoint four subspaces