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Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity
Nonlinear Analysis 60 (2005) 325 – 335 www.elsevier.com/locate/na
Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity夡 Fukun Zhao∗ , Xian Wu Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, People’s Republic of China Received 28 May 2004; accepted 24 August 2004
Abstract The purpose of this paper is to study the existence and multiplicity of periodic solution for the following non-autonomous second-order systems: u(t) ¨ = ∇F (t, u(t)) a.e. t ∈ [0, T ], u(0) − u(T ) = u(0) ˙ − u(T ˙ ) = 0. Some new existence and multiplicity theorems are obtained by using the least action principle and the minimax method. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: 34C25; 58E20; 47H04 Keywords: Periodic solution; Non-autonomous second-order systems; (PS) condition; Sobolev’s inequality; Wirtinger’s inequality
夡 This work is supported by the Foundation of Education Commission of Yunnan Province and the Foundation for Youth of Yunnan Normal University, China.
∗ Corresponding author. Tel./fax: +86-8715516070.
E-mail address: [email protected] (F. Zhao). 0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.08.031
326
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
1. Introduction and preliminaries Consider the non-autonomous second-order systems u(t) ¨ = ∇F (t, u(t)) a.e. t ∈ [0, T ], u(0) − u(T ) = u(0) ˙ − u(T ˙ ) = 0,
(1)
where T > 0, F : [0, T ] × R N → R satisfies the following assumption: (A) F (t, x) is measurable in t for every x ∈ R N and continuously differentiable in x for a.e. t ∈ [0, T ], and there exist a ∈ C(R + , R + ), b ∈ L1 (0, T ; R + ), such that |F (t, x)| a(|x|)b(t),
|∇F (t, x)| a(|x|)b(t)
for all x ∈ R N and a.e. t ∈ [0, T ]. The corresponding functional on HT1 given by
(u) =
1 2
T
T
|u(t)| ˙ 2 dt +
0
F (t, u(t)) dt
0
is continuously differentiable and weakly lower semicontinuous on HT1 (see [6]), where HT1 = {u : [0, T ] →R N |u is absolute continuous, u(0) = u(T ) and u˙ ∈ L2 (0, T )} is a Hilbert space with the norm
T
u = 0
T
|u(t)| dt + 2
1/2 |u(t)| ˙ dt 2
0
for each u ∈ HT1 . It is well known that the solutions of problem (1) correspond to the critical points of (see [6]). A function G : R N → R is called to be (, )-quasiconcave if G((x + y)) (G(x) + G(y)) for some , > 0 and all x, y ∈ R N . It has been shown by the least action principle that problem (1) has at least one solution which minimizes on HT1 (see [1,4–6,8–11]). When F (t, ·) is convex for a.e. t ∈ [0, T ], Mawhin–Willem [6] have studied the existence of solution which minimizes on HT1 for problem (1). For non-convex potential cases, using the least action principle and the minimax method, the existence of solution which minimizes on HT1 has been also researched by many people; for example, see [1,4,5,8–11] and their references. In non-compact cases, recently, Ekeland–Ghoussoub [3] deal with second-order systems with a super-quadratic potential, the action function satisfies the Palais–Smale condition, so that one can appeal to the Mountain Pass Lemma (see Proposition 3.2 on p. 226). The multiplicity of periodic solution was studied by Brezis–Nirenberg [2] and Tang [8]. In [12], by using the saddle point reduction method, we prove that problem (1) has at least one periodic solution with saddle point character in HT1 .
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
327
Inspired and motivated by the results due to Ma–Tang [4], Mawhin–Willem [6], Tang [8] and Wu–Tang [10], in this paper, we consider problem (1) with linear nonlinearity and sublinear nonlinearity. Some new solvability conditions are obtained by using the least action principle and minimax methods. Some results in this paper develop and generalize corresponding results, the others are new results.
2. Main results and proofs
Theorem 1. Suppose F satisfies assumption (A) and the following conditions: T (i) there exist f, g ∈ L1 (0, T ; R + ) with 0 f (t) dt < 12/T such that |∇F (t, x)| f (t)|x| + g(t) for all x ∈ R N and a.e. t ∈ [0, T ]; (ii) 1 |x|2
T
F (t, x) dt → −∞
as |x| → ∞.
0
Then problem (1) has at least one solution in HT1 . In addition, if the following condition holds: (iii) there exist r > 0, > 0 such that −|x|2 F (t, x) for all |x| r and a.e. t ∈ [0, T ], and F (t, x) − ( + )|x|2 for all |x| r and a.e. t ∈ [0, T ], where 1/2k 2 2 , 1/2(k + 1)2 2 for some integer k > 0 and = 2/T . Then problem (1) has at least two distinct solutions in HT1 . Proof. First we prove that satisfies the (PS) condition. Suppose that {un } is a (PS) T sequence for , that is, (un )→0 as n→∞ and (un ) is bounded. Since 0 f (t)dt<12/T , T T let A = 12/T − 0 f (t) dt, then A > 0 and 24/T − 2 0 f (t) dt − A > 0. By (ii) and Sobolev’s inequality one has (∇F (t, un (t)), u˜ n (t)) dt 0 T |(∇F (t, un (t)), u˜ n (t))| dt
T
0
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F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
T
T
f (t)|u¯ n + u˜ n (t)| · |u˜ n (t)| dt + g(t)|u˜ n (t)| dt 0 0 T T f (t) dt + u˜ n ∞ g(t) dt (|u¯ n | + u˜ n ∞ )u˜ n ∞ 0 0 T T ( 0 f (t) dt)2 24 1 2 2 f (t) dt − A u˜ n ∞ + −2 |u¯ n | T 24 2 T 0 T − 2 0 f (t) dt − A T T f (t) dt + u˜ n ∞ g(t) dt + u˜ n 2∞ 0
0 T 12 A g(t) dt − u˜ n 2∞ + C1 |u¯ n |2 + u˜ n ∞ = T 2 0
AT u˙ n 22 + C1 |u¯ n |2 + C2 u˙ n 2 1− 24
(2)
for all n. Hence, one has u˜ n (un ), u˜ n T = |u˙ n (t)|2 dt + 0
T 0
(∇F (t, un (t)), u˜ n (t)) dt
AT u˙ n 22 − C1 |u¯ n |2 − C2 u˙ n 2 24 for large n. It follows from Wirtinger’s inequality that
1/2 T2 u˙ n 2 u˜ n 1 + 2 u˙ n 2 4
(3)
(4)
for all n. We only need to consider the case: there exists some C > 0 such that C|u¯ n | u˙ n 2
(5)
for large n. In fact, if (5) is false, then there exists some subsequence of {un }, still denoted by {un }, such that |u¯ n | →0 u˙ n 2
(n → ∞).
(6)
It follows from (3) and (4) that AT u˙ n 22 − C3 u˙ n 2 24 for some C3 > 0 and for all large n. Hence we have
AT |u¯ n | 2 C3 C1 + 24 u˙ n 2 u˙ n 2 C1 |u¯ n |2
for large n. The above two inequalities and (6) imply that {u˙ n 2 } and {|u¯ n |} are both bounded. Hence {un } is bounded. Arguing then as Proposition 4.1 in [6], we conclude that the (PS) condition is satisfied.
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
If (5) is true for large n, in a similar way to (2) one can prove that T [F (t, un (t)) − F (t, u¯ n )] dt 0 T 1 = (∇F (t, u¯ n + s u˜ n (t)), u˜ n (t)) ds dt 0 0 T 1 T 1 g(t)|u˜ n (t)| ds dt f (t)|u¯ n + s u˜ n (t)| · |u˜ n (t)| ds dt + 0 0 0 0 T T 1 f (t) dt + u˜ n ∞ g(t) dt (|u¯ n | + u˜ n ∞ )u˜ n ∞ 2 0 0 T T ( 0 f (t) dt)2 1 A 12 2 2 f (t) dt − − u˜ n ∞ + |u¯ n | T 12 2 T 2 0 T − 0 f (t) dt − A/2 T T 1 f (t) dt + u˜ n ∞ g(t) dt + u˜ n 2∞ 2 0
0 T 6 A = − u˜ n 2∞ + C4 |u¯ n |2 + u˜ n ∞ g(t) dt T 4 0
1 AT − u˙ n 22 + C4 |u¯ n |2 + C5 u˙ n 2 2 48
329
(7)
for all n. It follows from the boundedness of {(un )}, (5) and (7) that C6 (un ) T T 1 T 2 |u˙ n (t)| dt + [F (t, un (t)) − F (t, u¯ n )] + F (t, u¯ n ) dt = 2 0 0 0 T AT F (t, u¯ n ) dt u˙ n 22 + C4 |u¯ n |2 + C5 u˙ n 2 + 48 0 T AT 2 F (t, u¯ n ) dt C |u¯ n |2 + C4 |u¯ n |2 + C5 C|u¯ n | + 48 0
T 1 C5 C = |u¯ n |2 F (t, u ¯ ) dt + C + n 7 |u¯ n |2 0 |u¯ n | for all large n and some real constant C6 and C7 . The above inequality and condition (ii) imply that {|u¯ n |} is bounded. Hence {un } is bounded by (5). Arguing then as Proposition 4.1 in [6], we conclude that the (PS) condition is satisfied. Since HT1 = H˜ T1 R N , where 1 1 ˜ HT = u ∈ HT
T
u(t) dt = 0 .
0
Next we shall prove that
(u) → +∞
(8)
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F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
as u → ∞ in H˜ T1 . In fact, in a similar way to (7) one has T
1 AT − u ˙ 22 + C5 u [F2 (t, u(t)) − F2 (t, 0)] dt ˙ 2 2 48 0 for all u ∈ H˜ T1 . Hence we have T T 1 T 2 dt + [F (t, u(t)) − F (t, 0)] dt − F (t, 0) dt (u) = |u(t)| ˙ 2 0 0 0 T AT ˙ 2− F (t, 0) dt u ˙ 22 − C5 u 48 0 for all u ∈ H˜ T1 . By Wirtinger’s inequality, one has u → ∞ ⇐⇒ u ˙ 2→∞ on H˜ T1 . Hence (8) follows from the above inequality. On the other hand, by condition (ii) one has
(u) → −∞
(9)
as |u| → ∞ in R N . Now we can obtain our theorem by (8), (9) and Saddle Point Theorem (see Theorem 4.6 in [7]). In addition, if condition (iii) holds. Let k Hk = (aj cos j t + bj sin t) aj , bj ∈ R N , j = 1, 2, . . . , k j =1
and = −. Then ∈ C 1 (E, R) satisfies the (PS) condition. By Theorem 5.29 and Example 5.26 in [7], we only need to prove that (1 ) lim inf u−2 (u) > 0 as u → 0 in Hk , (2 ) (u) 0 for all u ∈ Hk⊥ , and ⊥ . (3 ) (u) → −∞ as u → ∞ in Hk−1 Notice that 1 F (t, x) − F (t, 0) = (∇F (t, sx), x) ds 0
for all x ∈ R N and a.e. t ∈ [0, t]. By condition (i) we have F (t, x) − F (t, 0)
f (t) 2 |x| + g(t)|x| h(t)|x|3 2
for all |x| and a.e. t ∈ [0, T ] and some h ∈ L1 ([0, T ]; R + ) given by h(t) =
−1 f (t) + −2 g(t). 2
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
331
Now it follows from condition (iii) that F (t, x) − F (t, 0) − ( + )|x|2 + h(t)|x|3 − (1/2k 2 2 1 + )|x|2 + h(t)|x|3 for all x ∈ R N and a.e. t ∈ [0, T ]. Hence we obtain 1 (u) − u ˙ 2 + (1/2k 2 2 + ) 2 T − F (t, 0) dt 0 T 2 3 |u(t)| dt − u(t) 0
u2 − C8 u3 k 2 2 + 1
T
0
T
T
|u(t)|2 dt −
h(t)|u(t)|3 dt
0
T
h(t) dt −
0
F (t, 0) dt
0
for all u ∈ Hk and some positive constant C8 . Now (1 ) follows from the above inequality. By condition (iii), one has 1 (u) − 2
0
T
1 |u(t)| ˙ dt + (k + 1)2 2 2
T
2
|u(t)|2 dt 0
0
for all u ∈ Hk⊥ , (2 ) is obtained. At last (3 ) follows from (8). Hence the proof is completed.
Remark 1. In [6], Mawhin and Willem have proved the corresponding result when nonlinearity is bounded, i.e., there exist some g ∈ L1 (0, T ; R + ) such that |∇F (t, x)| g(t) (see Theorem 4.8 in [6]). Tang [8] generalizes this result to the case that nonlinearity ∇F (t, x) is sublinear, i.e., there exist some ∈ [0, 1) and f, g ∈ L1 (0, T ; R + ) such that |∇F (t, x)| f (t)|x| +g(t) (see Theorems 2 and 3 in [8]). Our Theorem 1 is a complement and development of these results corresponding to = 1. Replacing (ii) by 1 |x|2
T
F (t, x) dt → +∞
0
as |x| → ∞, the existence of solution which minimizes on HT1 has been studied in [11]. Theorem 2. Suppose F satisfies assumption (A) and the following conditions: T (i) there exists some k ∈ L1 (0, T ; R + ) with 0 < 0 k(t) dt < 12/T such that (∇F (t, x) − ∇F (t, y), x − y) − k(t)|x − y|2 for all x, y ∈ R N and a.e. t ∈ [0, T ]; (ii) F (t, ·) is (, )-quasiconcave, and ∇F (t, 0) = 0; there exist f, g ∈ L1 (0, T ; k + ) such that F (t, x) − f (t)|x|2 − g(t) for all x ∈ |R N and a.e. t ∈ [0, T ];
332
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
(iii)
T
F (t, x) dt → −∞
as |x| → ∞.
0
Then problem (1) has at least one solution in HT1 . Proof. First we prove that satisfies the (PS) condition. Suppose that {un } is a (PS) T sequence for . Let A = 1 − (T /12) 0 k(t) dt, then A > 0. Using condition (i), (ii) and Sobolev’s inequality one has u˜ n (un ), u˜ n T T = |u˙ n (t)|2 dt + (∇F (t, un (t)), u˜ n (t)) dt 0 0 T T = (∇F (t, un (t)) − ∇F (t, u¯ n ), u˜ n (t)) dt |u˙ n (t)|2 dt + 0 0 T T |u˙ n (t)|2 dt − k(t)|u˜ n (t))|2 dt 0 0 T T |u˙ n (t)|2 dt − u˜ n 2∞ k(t) dt 0 0 T |u˙ n (t)|2 dt A
(10)
0
for large n. It follows from Wirtinger’s inequality that
T T T2 2 2 |u˙ n (t)|2 dt. |u˙ n (t)| dt u˜ n 1 + 2 4 0 0
(11)
Eqs. (10) and (11) imply that u˜ n C1
(12)
for all n and some positive constant C1 . By the boundedness of {(un )}, condition (ii), and Sobolev’s inequality one has C2 (un ) T 1 T 2 = |u˙ n (t)| dt + F (t, un (t)) dt 2 0 0 T T T 1 1 |u˙ n (t)|2 dt + F (t, u¯ n ) dt − F (t, −u˜ n (t)) dt 2 0 0 0 T 1 T 1 T |u˙ n (t)|2 dt + F (t, u¯ n ) dt + [f (t)|u˜ n (t)|2 + g(t)] dt 2 0 0 0 1 1 T 2 2 u˙ n + F (t, u¯ n ) dt + C3 u˙ n + C4 2 0
(13)
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
333
for all n and some C2 , C3 and C4 . So, using (11)–(13) and condition (iii) we obtain |u¯ n | C5 for all n and some C5 . Hence {un } is a bounded sequence, and (PS) condition is satisfied. Next we shall prove that
(u) → +∞
(14)
as u → ∞ in H˜ T1 . It follows from Sobolev’s inequality and ∇F (t, 0) = 0 that T 1 2 (u) = u F (t, u(t)) dt ˙ 2+ 2 0 T 1 T 1 2 = u (∇F (t, su(t)), su(t)) ds dt + F (t, 0) dt ˙ 2+ 2 0 0 0 T 1 1 1 = u ˙ 22 + (∇F (t, su(t)) − ∇F (t, 0), su(t)) ds dt 2 0 0 s T + F (t, 0) dt 0 T T 1 1 ˙ 22 − u2∞ u k(t) dt + F (t, 0) dt 2 2 0 0 Au ˙ 22 − C6 for all u ∈ H˜ T1 and some constant C6 , where A = 1 − (T /12) inequality, one has
T 0
k(t) dt > 0. By Wirtinger’s
u → ∞ ⇐⇒ u ˙ 2→∞ on H˜ T1 . Since A > 0, (14) follows from the above inequality. On the other hand, by condition (iii) one has
(u) → −∞
(15)
as |u| → ∞ in R N . Now we can obtain our theorem by (14), (15) and Saddle Point Theorem in [7]. Remark 2. Theorem 2 is a new result. Replacing (i) by the following condition: T (i ) there exist some k ∈ L1 (0, T ; R + ) with 0 < 0 k(t) dt and some ∈ [1, 2) such that (∇F (t, x) − ∇F (t, y), x − y) − k(t)|x − y| for all x, y ∈ R N and a.e. t ∈ [0, T ], we can obtain the same result. Theorem 3. Suppose F satisfies assumption (A), (i) in Theorem 2, and T F (t, x) dt → +∞ as |x| → ∞. 0
334
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
Then problem (1) has at least one solution which minimizes on HT1 . In addition, if the following condition holds: (i) there exist r > 0 such that −|x|2 F (t, x) − |x|2 for all |x| r and for a.e. t ∈ [0, T ], where 1/2k 2 2 , 1/2(k + 1)2 2 for some integer k > 0 and = 2/T . Then problem (1) has at least three distinct solutions in HT1 . Proof. It follows from condition (i) and Sobolev’s inequality that T T 1 (∇F (t, u¯ + s u(t)), ˜ u(t)) ˜ ds dt [F (t, u(t)) − F (t, u)] ¯ dt = 0 0 0 T 1 = (∇F (t, u¯ + s u(t)) ˜ − ∇F (t, u), ¯ u(t)) ˜ ds dt 0 0 T 1 1 = (∇F (t, u¯ + s u(t)) ˜ 0 0 s − ∇F (t, u), ¯ s u(t)) ˜ ds dt T 1 1 (−k(t)|s u(t)| ˜ 2 ) ds dt 0 0 s T 1 ˜ 2∞ − u k(t) dt 2 0 T T − u k(t) dt ˙ 22 24 0 T for all u ∈ HT1 . Let A = 1/2 − (T /24) 0 k(t) dt, then A > 0. Hence we have T T 1 2 ˙ 2+ (u) = u [F (t, u(t)) − F (t, u)] ¯ dt + F (t, u) ¯ dt 2 0 0 T T T 1 k(t) dt + F (t, u) ¯ dt ˙ 22 − u ˙ 22 u 2 24 0 0 T = Au ˙ 22 + F (t, u) ¯ dt 0
for all u ∈
HT1 ,
which implies that
(u) → +∞ as u → ∞ by (ii) because A > 0 and u → ∞ ⇐⇒ (|u| ¯ 2 + u ˙ 22 )1/2 → ∞. By Theorem 1.1 and Corollary 1.1 in [6] we complete our proof.
In addition, if condition (i) holds. In a similar way to Tang [8], we can obtain the multiplicity results.
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
335
Remark 3. Theorem 3 is a new result. If F has control function from below, we can weaken the coercivity conditions from global coercivity to local coercivity. Replacing (i) in Theorem 2 by (i ) in Remark 2, we can obtain the following theorem. Theorem 4. Suppose F satisfies assumption (A), (i ) in Remark 2, and there are some E ⊂ [0, T ] with meas E > 0 and ∈ L1 ([0, T ], R) such that F (t, x) (t) for all x ∈ R N and a.e. t ∈ [0, T ], and F (t, x) dt → +∞ E
as |x| → ∞. Then problem (1) has at least one solution which minimizes on HT1 . In addition, if the condition (i) in Theorem 3 holds, then problem (1) has at least three distinct solutions in HT1 . Acknowledgements The authors express their thanks to the referee for his helpful suggestions. References [1] M.S. Berger, M. Schechter, On the solvability of semi-linear gradient operator equations, Adv. Math. 25 (1977) 97–132. [2] H. Brezis, L. Nirenberg, Remarks on finding critical points, Commun. Pur. Appl. Math. 44 (1991) 939–963. [3] I. Ekeland, N. Ghoussoub, Selected new aspects of the calculus variational in the large, Bull. Amer. Math. Soc. 39 (2002) 207–265. [4] J. Ma, C.L. Tang, Periodic solutions for some nonautonomous second-order systems, J. Math. Anal. Appl. 275 (2002) 482–494. [5] J. Mawhin, Semi-coercive monotone variational problems, Acad. Roy. Belg. Bull. Cl. Sci. (5) 73 (1987) 118 –130. [6] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, Berlin/New York, 1989. [7] P.H. Rabinowitz, Minimax methods in critical point theory with application to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Amer. Math. Soc., Providence, RI, 1986. [8] C.L. Tang, Periodic solutions for non-autonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc. 126 (1998) 3263–3270. [9] M. Willem, Oscillations forcées de systèmes hamiltoniens, in: Public. Sémin. Analyse Nonlinéaire, Univ. Besancon, 1981. [10] X.P. Wu, C.L. Tang, Periodic solutions of a class of non-autonomous second order systems, J. Math. Anal. Appl. 236 (1999) 227–235. [11] F. Zhao, X. Wu, Periodic solutions for a class of non-autonomous second order systems, J. Math. Anal. Appl. 296 (2004) 422–434. [12] F. Zhao, X. Wu, Saddle point reduction method for some non-autonomous second order systems, J. Math. Anal. Appl. 291 (2004) 653–665.
Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity夡 Fukun Zhao∗ , Xian Wu Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, People’s Republic of China Received 28 May 2004; accepted 24 August 2004
Abstract The purpose of this paper is to study the existence and multiplicity of periodic solution for the following non-autonomous second-order systems: u(t) ¨ = ∇F (t, u(t)) a.e. t ∈ [0, T ], u(0) − u(T ) = u(0) ˙ − u(T ˙ ) = 0. Some new existence and multiplicity theorems are obtained by using the least action principle and the minimax method. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: 34C25; 58E20; 47H04 Keywords: Periodic solution; Non-autonomous second-order systems; (PS) condition; Sobolev’s inequality; Wirtinger’s inequality
夡 This work is supported by the Foundation of Education Commission of Yunnan Province and the Foundation for Youth of Yunnan Normal University, China.
∗ Corresponding author. Tel./fax: +86-8715516070.
E-mail address: [email protected] (F. Zhao). 0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.08.031
326
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
1. Introduction and preliminaries Consider the non-autonomous second-order systems u(t) ¨ = ∇F (t, u(t)) a.e. t ∈ [0, T ], u(0) − u(T ) = u(0) ˙ − u(T ˙ ) = 0,
(1)
where T > 0, F : [0, T ] × R N → R satisfies the following assumption: (A) F (t, x) is measurable in t for every x ∈ R N and continuously differentiable in x for a.e. t ∈ [0, T ], and there exist a ∈ C(R + , R + ), b ∈ L1 (0, T ; R + ), such that |F (t, x)| a(|x|)b(t),
|∇F (t, x)| a(|x|)b(t)
for all x ∈ R N and a.e. t ∈ [0, T ]. The corresponding functional on HT1 given by
(u) =
1 2
T
T
|u(t)| ˙ 2 dt +
0
F (t, u(t)) dt
0
is continuously differentiable and weakly lower semicontinuous on HT1 (see [6]), where HT1 = {u : [0, T ] →R N |u is absolute continuous, u(0) = u(T ) and u˙ ∈ L2 (0, T )} is a Hilbert space with the norm
T
u = 0
T
|u(t)| dt + 2
1/2 |u(t)| ˙ dt 2
0
for each u ∈ HT1 . It is well known that the solutions of problem (1) correspond to the critical points of (see [6]). A function G : R N → R is called to be (, )-quasiconcave if G((x + y)) (G(x) + G(y)) for some , > 0 and all x, y ∈ R N . It has been shown by the least action principle that problem (1) has at least one solution which minimizes on HT1 (see [1,4–6,8–11]). When F (t, ·) is convex for a.e. t ∈ [0, T ], Mawhin–Willem [6] have studied the existence of solution which minimizes on HT1 for problem (1). For non-convex potential cases, using the least action principle and the minimax method, the existence of solution which minimizes on HT1 has been also researched by many people; for example, see [1,4,5,8–11] and their references. In non-compact cases, recently, Ekeland–Ghoussoub [3] deal with second-order systems with a super-quadratic potential, the action function satisfies the Palais–Smale condition, so that one can appeal to the Mountain Pass Lemma (see Proposition 3.2 on p. 226). The multiplicity of periodic solution was studied by Brezis–Nirenberg [2] and Tang [8]. In [12], by using the saddle point reduction method, we prove that problem (1) has at least one periodic solution with saddle point character in HT1 .
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
327
Inspired and motivated by the results due to Ma–Tang [4], Mawhin–Willem [6], Tang [8] and Wu–Tang [10], in this paper, we consider problem (1) with linear nonlinearity and sublinear nonlinearity. Some new solvability conditions are obtained by using the least action principle and minimax methods. Some results in this paper develop and generalize corresponding results, the others are new results.
2. Main results and proofs
Theorem 1. Suppose F satisfies assumption (A) and the following conditions: T (i) there exist f, g ∈ L1 (0, T ; R + ) with 0 f (t) dt < 12/T such that |∇F (t, x)| f (t)|x| + g(t) for all x ∈ R N and a.e. t ∈ [0, T ]; (ii) 1 |x|2
T
F (t, x) dt → −∞
as |x| → ∞.
0
Then problem (1) has at least one solution in HT1 . In addition, if the following condition holds: (iii) there exist r > 0, > 0 such that −|x|2 F (t, x) for all |x| r and a.e. t ∈ [0, T ], and F (t, x) − ( + )|x|2 for all |x| r and a.e. t ∈ [0, T ], where 1/2k 2 2 , 1/2(k + 1)2 2 for some integer k > 0 and = 2/T . Then problem (1) has at least two distinct solutions in HT1 . Proof. First we prove that satisfies the (PS) condition. Suppose that {un } is a (PS) T sequence for , that is, (un )→0 as n→∞ and (un ) is bounded. Since 0 f (t)dt<12/T , T T let A = 12/T − 0 f (t) dt, then A > 0 and 24/T − 2 0 f (t) dt − A > 0. By (ii) and Sobolev’s inequality one has (∇F (t, un (t)), u˜ n (t)) dt 0 T |(∇F (t, un (t)), u˜ n (t))| dt
T
0
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F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
T
T
f (t)|u¯ n + u˜ n (t)| · |u˜ n (t)| dt + g(t)|u˜ n (t)| dt 0 0 T T f (t) dt + u˜ n ∞ g(t) dt (|u¯ n | + u˜ n ∞ )u˜ n ∞ 0 0 T T ( 0 f (t) dt)2 24 1 2 2 f (t) dt − A u˜ n ∞ + −2 |u¯ n | T 24 2 T 0 T − 2 0 f (t) dt − A T T f (t) dt + u˜ n ∞ g(t) dt + u˜ n 2∞ 0
0 T 12 A g(t) dt − u˜ n 2∞ + C1 |u¯ n |2 + u˜ n ∞ = T 2 0
AT u˙ n 22 + C1 |u¯ n |2 + C2 u˙ n 2 1− 24
(2)
for all n. Hence, one has u˜ n (un ), u˜ n T = |u˙ n (t)|2 dt + 0
T 0
(∇F (t, un (t)), u˜ n (t)) dt
AT u˙ n 22 − C1 |u¯ n |2 − C2 u˙ n 2 24 for large n. It follows from Wirtinger’s inequality that
1/2 T2 u˙ n 2 u˜ n 1 + 2 u˙ n 2 4
(3)
(4)
for all n. We only need to consider the case: there exists some C > 0 such that C|u¯ n | u˙ n 2
(5)
for large n. In fact, if (5) is false, then there exists some subsequence of {un }, still denoted by {un }, such that |u¯ n | →0 u˙ n 2
(n → ∞).
(6)
It follows from (3) and (4) that AT u˙ n 22 − C3 u˙ n 2 24 for some C3 > 0 and for all large n. Hence we have
AT |u¯ n | 2 C3 C1 + 24 u˙ n 2 u˙ n 2 C1 |u¯ n |2
for large n. The above two inequalities and (6) imply that {u˙ n 2 } and {|u¯ n |} are both bounded. Hence {un } is bounded. Arguing then as Proposition 4.1 in [6], we conclude that the (PS) condition is satisfied.
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
If (5) is true for large n, in a similar way to (2) one can prove that T [F (t, un (t)) − F (t, u¯ n )] dt 0 T 1 = (∇F (t, u¯ n + s u˜ n (t)), u˜ n (t)) ds dt 0 0 T 1 T 1 g(t)|u˜ n (t)| ds dt f (t)|u¯ n + s u˜ n (t)| · |u˜ n (t)| ds dt + 0 0 0 0 T T 1 f (t) dt + u˜ n ∞ g(t) dt (|u¯ n | + u˜ n ∞ )u˜ n ∞ 2 0 0 T T ( 0 f (t) dt)2 1 A 12 2 2 f (t) dt − − u˜ n ∞ + |u¯ n | T 12 2 T 2 0 T − 0 f (t) dt − A/2 T T 1 f (t) dt + u˜ n ∞ g(t) dt + u˜ n 2∞ 2 0
0 T 6 A = − u˜ n 2∞ + C4 |u¯ n |2 + u˜ n ∞ g(t) dt T 4 0
1 AT − u˙ n 22 + C4 |u¯ n |2 + C5 u˙ n 2 2 48
329
(7)
for all n. It follows from the boundedness of {(un )}, (5) and (7) that C6 (un ) T T 1 T 2 |u˙ n (t)| dt + [F (t, un (t)) − F (t, u¯ n )] + F (t, u¯ n ) dt = 2 0 0 0 T AT F (t, u¯ n ) dt u˙ n 22 + C4 |u¯ n |2 + C5 u˙ n 2 + 48 0 T AT 2 F (t, u¯ n ) dt C |u¯ n |2 + C4 |u¯ n |2 + C5 C|u¯ n | + 48 0
T 1 C5 C = |u¯ n |2 F (t, u ¯ ) dt + C + n 7 |u¯ n |2 0 |u¯ n | for all large n and some real constant C6 and C7 . The above inequality and condition (ii) imply that {|u¯ n |} is bounded. Hence {un } is bounded by (5). Arguing then as Proposition 4.1 in [6], we conclude that the (PS) condition is satisfied. Since HT1 = H˜ T1 R N , where 1 1 ˜ HT = u ∈ HT
T
u(t) dt = 0 .
0
Next we shall prove that
(u) → +∞
(8)
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F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
as u → ∞ in H˜ T1 . In fact, in a similar way to (7) one has T
1 AT − u ˙ 22 + C5 u [F2 (t, u(t)) − F2 (t, 0)] dt ˙ 2 2 48 0 for all u ∈ H˜ T1 . Hence we have T T 1 T 2 dt + [F (t, u(t)) − F (t, 0)] dt − F (t, 0) dt (u) = |u(t)| ˙ 2 0 0 0 T AT ˙ 2− F (t, 0) dt u ˙ 22 − C5 u 48 0 for all u ∈ H˜ T1 . By Wirtinger’s inequality, one has u → ∞ ⇐⇒ u ˙ 2→∞ on H˜ T1 . Hence (8) follows from the above inequality. On the other hand, by condition (ii) one has
(u) → −∞
(9)
as |u| → ∞ in R N . Now we can obtain our theorem by (8), (9) and Saddle Point Theorem (see Theorem 4.6 in [7]). In addition, if condition (iii) holds. Let k Hk = (aj cos j t + bj sin t) aj , bj ∈ R N , j = 1, 2, . . . , k j =1
and = −. Then ∈ C 1 (E, R) satisfies the (PS) condition. By Theorem 5.29 and Example 5.26 in [7], we only need to prove that (1 ) lim inf u−2 (u) > 0 as u → 0 in Hk , (2 ) (u) 0 for all u ∈ Hk⊥ , and ⊥ . (3 ) (u) → −∞ as u → ∞ in Hk−1 Notice that 1 F (t, x) − F (t, 0) = (∇F (t, sx), x) ds 0
for all x ∈ R N and a.e. t ∈ [0, t]. By condition (i) we have F (t, x) − F (t, 0)
f (t) 2 |x| + g(t)|x| h(t)|x|3 2
for all |x| and a.e. t ∈ [0, T ] and some h ∈ L1 ([0, T ]; R + ) given by h(t) =
−1 f (t) + −2 g(t). 2
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Now it follows from condition (iii) that F (t, x) − F (t, 0) − ( + )|x|2 + h(t)|x|3 − (1/2k 2 2 1 + )|x|2 + h(t)|x|3 for all x ∈ R N and a.e. t ∈ [0, T ]. Hence we obtain 1 (u) − u ˙ 2 + (1/2k 2 2 + ) 2 T − F (t, 0) dt 0 T 2 3 |u(t)| dt − u(t) 0
u2 − C8 u3 k 2 2 + 1
T
0
T
T
|u(t)|2 dt −
h(t)|u(t)|3 dt
0
T
h(t) dt −
0
F (t, 0) dt
0
for all u ∈ Hk and some positive constant C8 . Now (1 ) follows from the above inequality. By condition (iii), one has 1 (u) − 2
0
T
1 |u(t)| ˙ dt + (k + 1)2 2 2
T
2
|u(t)|2 dt 0
0
for all u ∈ Hk⊥ , (2 ) is obtained. At last (3 ) follows from (8). Hence the proof is completed.
Remark 1. In [6], Mawhin and Willem have proved the corresponding result when nonlinearity is bounded, i.e., there exist some g ∈ L1 (0, T ; R + ) such that |∇F (t, x)| g(t) (see Theorem 4.8 in [6]). Tang [8] generalizes this result to the case that nonlinearity ∇F (t, x) is sublinear, i.e., there exist some ∈ [0, 1) and f, g ∈ L1 (0, T ; R + ) such that |∇F (t, x)| f (t)|x| +g(t) (see Theorems 2 and 3 in [8]). Our Theorem 1 is a complement and development of these results corresponding to = 1. Replacing (ii) by 1 |x|2
T
F (t, x) dt → +∞
0
as |x| → ∞, the existence of solution which minimizes on HT1 has been studied in [11]. Theorem 2. Suppose F satisfies assumption (A) and the following conditions: T (i) there exists some k ∈ L1 (0, T ; R + ) with 0 < 0 k(t) dt < 12/T such that (∇F (t, x) − ∇F (t, y), x − y) − k(t)|x − y|2 for all x, y ∈ R N and a.e. t ∈ [0, T ]; (ii) F (t, ·) is (, )-quasiconcave, and ∇F (t, 0) = 0; there exist f, g ∈ L1 (0, T ; k + ) such that F (t, x) − f (t)|x|2 − g(t) for all x ∈ |R N and a.e. t ∈ [0, T ];
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F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
(iii)
T
F (t, x) dt → −∞
as |x| → ∞.
0
Then problem (1) has at least one solution in HT1 . Proof. First we prove that satisfies the (PS) condition. Suppose that {un } is a (PS) T sequence for . Let A = 1 − (T /12) 0 k(t) dt, then A > 0. Using condition (i), (ii) and Sobolev’s inequality one has u˜ n (un ), u˜ n T T = |u˙ n (t)|2 dt + (∇F (t, un (t)), u˜ n (t)) dt 0 0 T T = (∇F (t, un (t)) − ∇F (t, u¯ n ), u˜ n (t)) dt |u˙ n (t)|2 dt + 0 0 T T |u˙ n (t)|2 dt − k(t)|u˜ n (t))|2 dt 0 0 T T |u˙ n (t)|2 dt − u˜ n 2∞ k(t) dt 0 0 T |u˙ n (t)|2 dt A
(10)
0
for large n. It follows from Wirtinger’s inequality that
T T T2 2 2 |u˙ n (t)|2 dt. |u˙ n (t)| dt u˜ n 1 + 2 4 0 0
(11)
Eqs. (10) and (11) imply that u˜ n C1
(12)
for all n and some positive constant C1 . By the boundedness of {(un )}, condition (ii), and Sobolev’s inequality one has C2 (un ) T 1 T 2 = |u˙ n (t)| dt + F (t, un (t)) dt 2 0 0 T T T 1 1 |u˙ n (t)|2 dt + F (t, u¯ n ) dt − F (t, −u˜ n (t)) dt 2 0 0 0 T 1 T 1 T |u˙ n (t)|2 dt + F (t, u¯ n ) dt + [f (t)|u˜ n (t)|2 + g(t)] dt 2 0 0 0 1 1 T 2 2 u˙ n + F (t, u¯ n ) dt + C3 u˙ n + C4 2 0
(13)
F. Zhao, X. Wu / Nonlinear Analysis 60 (2005) 325 – 335
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for all n and some C2 , C3 and C4 . So, using (11)–(13) and condition (iii) we obtain |u¯ n | C5 for all n and some C5 . Hence {un } is a bounded sequence, and (PS) condition is satisfied. Next we shall prove that
(u) → +∞
(14)
as u → ∞ in H˜ T1 . It follows from Sobolev’s inequality and ∇F (t, 0) = 0 that T 1 2 (u) = u F (t, u(t)) dt ˙ 2+ 2 0 T 1 T 1 2 = u (∇F (t, su(t)), su(t)) ds dt + F (t, 0) dt ˙ 2+ 2 0 0 0 T 1 1 1 = u ˙ 22 + (∇F (t, su(t)) − ∇F (t, 0), su(t)) ds dt 2 0 0 s T + F (t, 0) dt 0 T T 1 1 ˙ 22 − u2∞ u k(t) dt + F (t, 0) dt 2 2 0 0 Au ˙ 22 − C6 for all u ∈ H˜ T1 and some constant C6 , where A = 1 − (T /12) inequality, one has
T 0
k(t) dt > 0. By Wirtinger’s
u → ∞ ⇐⇒ u ˙ 2→∞ on H˜ T1 . Since A > 0, (14) follows from the above inequality. On the other hand, by condition (iii) one has
(u) → −∞
(15)
as |u| → ∞ in R N . Now we can obtain our theorem by (14), (15) and Saddle Point Theorem in [7]. Remark 2. Theorem 2 is a new result. Replacing (i) by the following condition: T (i ) there exist some k ∈ L1 (0, T ; R + ) with 0 < 0 k(t) dt and some ∈ [1, 2) such that (∇F (t, x) − ∇F (t, y), x − y) − k(t)|x − y| for all x, y ∈ R N and a.e. t ∈ [0, T ], we can obtain the same result. Theorem 3. Suppose F satisfies assumption (A), (i) in Theorem 2, and T F (t, x) dt → +∞ as |x| → ∞. 0
334
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Then problem (1) has at least one solution which minimizes on HT1 . In addition, if the following condition holds: (i) there exist r > 0 such that −|x|2 F (t, x) − |x|2 for all |x| r and for a.e. t ∈ [0, T ], where 1/2k 2 2 , 1/2(k + 1)2 2 for some integer k > 0 and = 2/T . Then problem (1) has at least three distinct solutions in HT1 . Proof. It follows from condition (i) and Sobolev’s inequality that T T 1 (∇F (t, u¯ + s u(t)), ˜ u(t)) ˜ ds dt [F (t, u(t)) − F (t, u)] ¯ dt = 0 0 0 T 1 = (∇F (t, u¯ + s u(t)) ˜ − ∇F (t, u), ¯ u(t)) ˜ ds dt 0 0 T 1 1 = (∇F (t, u¯ + s u(t)) ˜ 0 0 s − ∇F (t, u), ¯ s u(t)) ˜ ds dt T 1 1 (−k(t)|s u(t)| ˜ 2 ) ds dt 0 0 s T 1 ˜ 2∞ − u k(t) dt 2 0 T T − u k(t) dt ˙ 22 24 0 T for all u ∈ HT1 . Let A = 1/2 − (T /24) 0 k(t) dt, then A > 0. Hence we have T T 1 2 ˙ 2+ (u) = u [F (t, u(t)) − F (t, u)] ¯ dt + F (t, u) ¯ dt 2 0 0 T T T 1 k(t) dt + F (t, u) ¯ dt ˙ 22 − u ˙ 22 u 2 24 0 0 T = Au ˙ 22 + F (t, u) ¯ dt 0
for all u ∈
HT1 ,
which implies that
(u) → +∞ as u → ∞ by (ii) because A > 0 and u → ∞ ⇐⇒ (|u| ¯ 2 + u ˙ 22 )1/2 → ∞. By Theorem 1.1 and Corollary 1.1 in [6] we complete our proof.
In addition, if condition (i) holds. In a similar way to Tang [8], we can obtain the multiplicity results.
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Remark 3. Theorem 3 is a new result. If F has control function from below, we can weaken the coercivity conditions from global coercivity to local coercivity. Replacing (i) in Theorem 2 by (i ) in Remark 2, we can obtain the following theorem. Theorem 4. Suppose F satisfies assumption (A), (i ) in Remark 2, and there are some E ⊂ [0, T ] with meas E > 0 and ∈ L1 ([0, T ], R) such that F (t, x) (t) for all x ∈ R N and a.e. t ∈ [0, T ], and F (t, x) dt → +∞ E
as |x| → ∞. Then problem (1) has at least one solution which minimizes on HT1 . In addition, if the condition (i) in Theorem 3 holds, then problem (1) has at least three distinct solutions in HT1 . Acknowledgements The authors express their thanks to the referee for his helpful suggestions. References [1] M.S. Berger, M. Schechter, On the solvability of semi-linear gradient operator equations, Adv. Math. 25 (1977) 97–132. [2] H. Brezis, L. Nirenberg, Remarks on finding critical points, Commun. Pur. Appl. Math. 44 (1991) 939–963. [3] I. Ekeland, N. Ghoussoub, Selected new aspects of the calculus variational in the large, Bull. Amer. Math. Soc. 39 (2002) 207–265. [4] J. Ma, C.L. Tang, Periodic solutions for some nonautonomous second-order systems, J. Math. Anal. Appl. 275 (2002) 482–494. [5] J. Mawhin, Semi-coercive monotone variational problems, Acad. Roy. Belg. Bull. Cl. Sci. (5) 73 (1987) 118 –130. [6] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, Berlin/New York, 1989. [7] P.H. Rabinowitz, Minimax methods in critical point theory with application to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Amer. Math. Soc., Providence, RI, 1986. [8] C.L. Tang, Periodic solutions for non-autonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc. 126 (1998) 3263–3270. [9] M. Willem, Oscillations forcées de systèmes hamiltoniens, in: Public. Sémin. Analyse Nonlinéaire, Univ. Besancon, 1981. [10] X.P. Wu, C.L. Tang, Periodic solutions of a class of non-autonomous second order systems, J. Math. Anal. Appl. 236 (1999) 227–235. [11] F. Zhao, X. Wu, Periodic solutions for a class of non-autonomous second order systems, J. Math. Anal. Appl. 296 (2004) 422–434. [12] F. Zhao, X. Wu, Saddle point reduction method for some non-autonomous second order systems, J. Math. Anal. Appl. 291 (2004) 653–665.