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1, c +2 ε u ∈ Ic −2ε \Nr/8 ,
∈ C([0, 1] × X, X) such that
(1) (0, u) = u, ∀ u ∈ X . (2) I ( (·, u)) is nonincreasing, ∀ u ∈ X . (3) I ( (·, u)) < c, ∀ t ∈ (0, 1], u ∈ I c \Nr . c+ε c−ε , R > r > 0. (4) (1, (Ic −ε \Nr ) ∩ BR ) ∈ I (t, u) − u r/2, ∀ t ∈ [0, 1], u ∈ X . (5) Proof. Set f (u) := g(u) := u − Nr/8 , u − Nr/8 + u − I c \Nr/4 u − X \A , u − X \A + u − B
(H3 ) There are constants a1 , a2 > 0, p ∈ (1, 2) such that |∇ H (t, u)|p 0 < H (t, u) a1 + a2 u · ∇ H (t, u). 0 such that, for |u| R u · ∇ H (t, u). (H4 ) There are constants > 2 and R
Ju ˙ − A(t)u + ∇ H (t, u) = 0,
uHale Waihona Puke Baidu∈ R2 N , t ∈ R.
We prove an abstract result on the existence of a critical point for a real-valued functional on a Hilbert space via a new deformation theorem. Different from the works in the literature, the new deformation theorem is constructed under the Cerami-type condition instead of Palais–Smale-type condition. In addition, the main assumption here is weaker than the usual Ambrosetti–Rabinowitz-type condition: 0 < H (t, u) u · ∇ H (t, u), > 2, |u| R > 0.
1414
S. Luan, A. Mao / Nonlinear Analysis 61 (2005) 1413 – 1426
1. Introduction This paper deals with the existence of periodic solutions of Hamiltonian system: ( H)
S. Luan, A. Mao / Nonlinear Analysis 61 (2005) 1413 – 1426
1415
We can easily give an example which satisfies (H3 ) but not (H4 ), see [5]. It should be pointed out that (H3 )–(H4 ) implies (H3 ). (H3 )—(1) is apparent. To check ˜ (t, u) c2 |u|q , where q = p/(p − 1) ∈ (2, ∞), (H3 )—(2), we note that (H3 ) implies H ˜ (t, u) −1 /|u|2 c3 |u|(q −2) −q c3 . By (H4 ), one can take 1 < < q/(q − 2) such that H ˜ (t, u), hence |∇ H (t, u)||u| 2 /( − 2)H |∇ H (t, u)| |u| 2 −2 ˜ (t, u) H |u|2
Ju ˙ − A(t)u + ∇ H (t, u) = 0,
u ∈ R2N , t ∈ R,
where A(t) is a symmetric 2N × 2N matrix continuous and 2 -period in t, H ∈ C1 (R2N +1 , R) is 2 -period in t, ∇ H := ∇u H ∈ C(R2N +1 , R2N ) and J is standard symplectic matrix. In their paper [9,10], the authors deal with the situation where H satisfies the following assumptions: (H1 ) H (t, u) = o(|u|2 ), |u| → 0 uniformly on R. (H2 ) For some > 0, either H (t, u) or H (t, u) 0 for |u| , t ∈ R. 0 for |u| , t ∈R
This result extends theorems given by Li and Willem (J. Math. Anal. Appl. 189 (1995) 6–32) and Li and Szulkin (J. Differential Equations 112 (1994) 226–238). ᭧ 2005 Elsevier Ltd. All rights reserved.
MSC: 58E05; 58E50 Keywords: Hamiltonian system; Periodic solutions; Cerami condition; Local linking
ଁ
Supported by NSFC (10471075) and NSFSP(Y2003A01) and NSFQN(xj0503).
∗ Corresponding author. Institute of Mathematics, Academy of Math and System Sciences, Chinese Academy
of Sciences, Beijing 100080, PR China. E-mail address: luanshx@mail.amss.ac.cn (S. Luan). 0362-546X/$ - see front matter ᭧ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.01.108
They establish the existence of nontrivial solution of (H) via a new linking theorem and variational argument. We emphasize that the results in the papers mentioned above were obtained under the Ambrosetti–Rabinowitz-type condition (H4 ), which implies that H (t, u) grows at a superquadratic rate as |u| → ∞. This kind of technical condition often appears as necessary to use variational methods when solving super-linear differential equations such as elliptic problems, Dirac equations, Hamiltonian systems, wave equations and Schrödinger equations. See also [1–4,6–8,11,12]. In the present paper, a new deformation theorem is given under the (C)∗ condition instead of (P S)∗ condition. Following the deformation theorem, a linking result is established. So we only need the following conditions instead of (H3 )(H4 ): ˜= (H3 ) H (t, u)/|u|2 → ∞ as |u| → ∞, and H
a Department of Mathematics, Qufu Normal University, Shandong 273165, PR China b Institute of Mathematics, Academy of Math and System Sciences, Chinese Academy of Sciences, Beijing
100080, PR China Received 1 December 2004; accepted 28 January 2005
Abstract We consider the existence of nontrivial periodic solutions for a superlinear Hamiltonian system: (H)
Kc = {u ∈ X, I (u) = c, I (u) = 0}. Theorem 2.1. Let I be a functional of class C1 defined on a real Banach space X. Nr ={u ∈ X : u − Kc < r }, r > 0. Let ε > 0, 1 > 0, c ∈ R be such that (1 + u ) I (u) then exists
−1
˜ (t, u) H
˜ (t, u). c4 H
The rest of the paper is organized as follows. In Section 2 we give the proof of deformation theorem and the existence theorem of critical points. In Section 3, the abstract results are applied to first-order non-autonomous Hamiltonian system. 2. Abstract results Let us recall some standard notions: BR = {u ∈ X : u < R }, I c = {u ∈ X : I (u) Ic = {u ∈ X : I (u) c}, c},
1 2
∇ H (t, u) · u − H (t, u) satisfying
˜ a3 |u|2 if |u| R . (1) H ˜ (t, u) if |u| R , (2) |∇ H (t, u)| /|u| a4 H where a3 , a4 > 0 and 1 < < q/(q − 2), q ∈ (2, ∞). We have the main existence result. Theorem 1.1. Suppose that H satisfies (H1 ), (H2 ) and (H3 ). If 0 is an eigenvalue of L (with period boundary conditions). Then (H) has at least one nontrivial 2 -period solutions.
Nonlinear Analysis 61 (2005) 1413 – 1426 www.elsevier.com/locate/na
Periodic solutions for a class of non-autonomous Hamiltonian systemsଁ
Shixia Luana, b , ∗ , Anmin Maoa, b
∈ C([0, 1] × X, X) such that
(1) (0, u) = u, ∀ u ∈ X . (2) I ( (·, u)) is nonincreasing, ∀ u ∈ X . (3) I ( (·, u)) < c, ∀ t ∈ (0, 1], u ∈ I c \Nr . c+ε c−ε , R > r > 0. (4) (1, (Ic −ε \Nr ) ∩ BR ) ∈ I (t, u) − u r/2, ∀ t ∈ [0, 1], u ∈ X . (5) Proof. Set f (u) := g(u) := u − Nr/8 , u − Nr/8 + u − I c \Nr/4 u − X \A , u − X \A + u − B
(H3 ) There are constants a1 , a2 > 0, p ∈ (1, 2) such that |∇ H (t, u)|p 0 < H (t, u) a1 + a2 u · ∇ H (t, u). 0 such that, for |u| R u · ∇ H (t, u). (H4 ) There are constants > 2 and R
Ju ˙ − A(t)u + ∇ H (t, u) = 0,
uHale Waihona Puke Baidu∈ R2 N , t ∈ R.
We prove an abstract result on the existence of a critical point for a real-valued functional on a Hilbert space via a new deformation theorem. Different from the works in the literature, the new deformation theorem is constructed under the Cerami-type condition instead of Palais–Smale-type condition. In addition, the main assumption here is weaker than the usual Ambrosetti–Rabinowitz-type condition: 0 < H (t, u) u · ∇ H (t, u), > 2, |u| R > 0.
1414
S. Luan, A. Mao / Nonlinear Analysis 61 (2005) 1413 – 1426
1. Introduction This paper deals with the existence of periodic solutions of Hamiltonian system: ( H)
S. Luan, A. Mao / Nonlinear Analysis 61 (2005) 1413 – 1426
1415
We can easily give an example which satisfies (H3 ) but not (H4 ), see [5]. It should be pointed out that (H3 )–(H4 ) implies (H3 ). (H3 )—(1) is apparent. To check ˜ (t, u) c2 |u|q , where q = p/(p − 1) ∈ (2, ∞), (H3 )—(2), we note that (H3 ) implies H ˜ (t, u) −1 /|u|2 c3 |u|(q −2) −q c3 . By (H4 ), one can take 1 < < q/(q − 2) such that H ˜ (t, u), hence |∇ H (t, u)||u| 2 /( − 2)H |∇ H (t, u)| |u| 2 −2 ˜ (t, u) H |u|2
Ju ˙ − A(t)u + ∇ H (t, u) = 0,
u ∈ R2N , t ∈ R,
where A(t) is a symmetric 2N × 2N matrix continuous and 2 -period in t, H ∈ C1 (R2N +1 , R) is 2 -period in t, ∇ H := ∇u H ∈ C(R2N +1 , R2N ) and J is standard symplectic matrix. In their paper [9,10], the authors deal with the situation where H satisfies the following assumptions: (H1 ) H (t, u) = o(|u|2 ), |u| → 0 uniformly on R. (H2 ) For some > 0, either H (t, u) or H (t, u) 0 for |u| , t ∈ R. 0 for |u| , t ∈R
This result extends theorems given by Li and Willem (J. Math. Anal. Appl. 189 (1995) 6–32) and Li and Szulkin (J. Differential Equations 112 (1994) 226–238). ᭧ 2005 Elsevier Ltd. All rights reserved.
MSC: 58E05; 58E50 Keywords: Hamiltonian system; Periodic solutions; Cerami condition; Local linking
ଁ
Supported by NSFC (10471075) and NSFSP(Y2003A01) and NSFQN(xj0503).
∗ Corresponding author. Institute of Mathematics, Academy of Math and System Sciences, Chinese Academy
of Sciences, Beijing 100080, PR China. E-mail address: luanshx@mail.amss.ac.cn (S. Luan). 0362-546X/$ - see front matter ᭧ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.01.108
They establish the existence of nontrivial solution of (H) via a new linking theorem and variational argument. We emphasize that the results in the papers mentioned above were obtained under the Ambrosetti–Rabinowitz-type condition (H4 ), which implies that H (t, u) grows at a superquadratic rate as |u| → ∞. This kind of technical condition often appears as necessary to use variational methods when solving super-linear differential equations such as elliptic problems, Dirac equations, Hamiltonian systems, wave equations and Schrödinger equations. See also [1–4,6–8,11,12]. In the present paper, a new deformation theorem is given under the (C)∗ condition instead of (P S)∗ condition. Following the deformation theorem, a linking result is established. So we only need the following conditions instead of (H3 )(H4 ): ˜= (H3 ) H (t, u)/|u|2 → ∞ as |u| → ∞, and H
a Department of Mathematics, Qufu Normal University, Shandong 273165, PR China b Institute of Mathematics, Academy of Math and System Sciences, Chinese Academy of Sciences, Beijing
100080, PR China Received 1 December 2004; accepted 28 January 2005
Abstract We consider the existence of nontrivial periodic solutions for a superlinear Hamiltonian system: (H)
Kc = {u ∈ X, I (u) = c, I (u) = 0}. Theorem 2.1. Let I be a functional of class C1 defined on a real Banach space X. Nr ={u ∈ X : u − Kc < r }, r > 0. Let ε > 0, 1 > 0, c ∈ R be such that (1 + u ) I (u) then exists
−1
˜ (t, u) H
˜ (t, u). c4 H
The rest of the paper is organized as follows. In Section 2 we give the proof of deformation theorem and the existence theorem of critical points. In Section 3, the abstract results are applied to first-order non-autonomous Hamiltonian system. 2. Abstract results Let us recall some standard notions: BR = {u ∈ X : u < R }, I c = {u ∈ X : I (u) Ic = {u ∈ X : I (u) c}, c},
1 2
∇ H (t, u) · u − H (t, u) satisfying
˜ a3 |u|2 if |u| R . (1) H ˜ (t, u) if |u| R , (2) |∇ H (t, u)| /|u| a4 H where a3 , a4 > 0 and 1 < < q/(q − 2), q ∈ (2, ∞). We have the main existence result. Theorem 1.1. Suppose that H satisfies (H1 ), (H2 ) and (H3 ). If 0 is an eigenvalue of L (with period boundary conditions). Then (H) has at least one nontrivial 2 -period solutions.
Nonlinear Analysis 61 (2005) 1413 – 1426 www.elsevier.com/locate/na
Periodic solutions for a class of non-autonomous Hamiltonian systemsଁ
Shixia Luana, b , ∗ , Anmin Maoa, b