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Existence result for periodic solutions of a class of Hamiltonian systems with super quadratic potential
Nonlinear Analysis 63 (2005) 565 – 574 www.elsevier.com/locate/na
Existence result for periodic solutions of a class of Hamiltonian systems with super quadratic potential Leonard Karshima Shilgba∗ Hirano Laboratory, Division of Information Media and Environment Sciences, Graduate School of Environment and Information Sciences, Yokohama National University, 79-5, Tokiwadai, Hodogaya-Ku, Yokohama-shi, Kanagawa 240-8501, Japan Received 27 May 2003; accepted 18 May 2005
Abstract We have presented a variant existence result for periodic solutions to a class of Hamiltonian systems with a potential which is both super quadratic and sign-indefinite. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 34B15 Keywords: Periodic solutions; Hamiltonian systems; Potential; Linking
1. Introduction Our objective is to investigate the existence of periodic solutions to the class of Hamiltonian systems: x¨ + A(t)x + b(t)V (x) = 0
on [0, T ],
x(0) = x(T ), x(0) ˙ = x(T ˙ ),
∗ Tel.: +81 45 50 24 115; fax: +81 45 33 94 207.
E-mail address: [email protected]. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.05.018
(P)
566
L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
where 1.1 A ∈ C 0 (R, RN×N ) is a symmetric T -periodic matrix function that is not sign-definite on [0, T ]; 1.2 b ∈ C 0 (R, R) is T -periodic and hanges sign on [0, T ]; 1.3 V ∈ C 2 (RN , R) with V (x) V (0) = 0 for all x ∈ RN , and has super quadratic behaviour. The above problem has been studied by several authors: Case A: A ≡ 0. While Lassoued [14] obtained existence of T -periodic solutions when V is strictly convex and homogeneous, Ben Naoum et al. [7] provided existence results by relaxing condition on V to only homogeneity. Girardi and Matzeu [13] used the Alama–Tarantello condition in [1] and proved some existence and multiplicity results for T -periodic and subharmonic solutions. The condition is given by: There exist
c > 0, > 2 : |V (x)x − V (x)| c|x|2
for all x ∈ RN .
(1.4)
Case B: A = 0. In this case, when b(.) > 0, refer to [3,2,10] (with the references contained therein), severally for existence of periodic, homoclinic, and subharmonic solutions. Besides, when we assume b changes sign and A is a negative definite matrix, there are some existence and multiplicity results of periodic, subharmonic, and homoclinic solutions (refer to [12,8,4]). When b(.) changes sign and A is sign-indefinite, the author is aware of only the results of Antonacci [5], Xu and Guo [19], and the author in [18]. Antonacci proved the existence of non-trivial periodic solutions under the assumption that T (A(t), ) dt > 0 for all ∈ RN , || = 1. (1.5) 0
Antonacci observed the vital role played by assumption (1.5) as he posed two open problems: Given that A(.) is not negative-definite, to 1. study the existence of T -periodic solutions of (P) in the case (1.5) holds and A(.) is sign-indefinite in any interval [0, T ]; 2. find some existence results in the case that A does not satisfy (1.5). As mentioned above, Antonnaci attempted an elegant solution of problem 1 in [5]. Xu and Guo, in attempting the solution of open problem 2 in [18], relied, among other assumptions, on the glooseh or weaker version of (1.5), that: T There exist > 0, > 2, > 0, : 0 < < ( 0 b(t) dt) such that T (A(t), ) dt − for all ∈ RN , || = 1, (1.6) 0
where V (x) |x| − for all sufficiently large x ∈ RN .
L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
567
T It is clear from (1.6) that 0 (A(t), ) dt, ∈ RN , || = 1, is neither assumed to be definitely positive nor negative, which is a relaxation on condition (1.5.); however, it is only T valid when 0 b(t) dt > 0. In [18] we resolved the open problem 2 posed by Antonnaci by assuming some super-convex quadratic condition on the potential V : There exist real numbers, R > 0, , : > > 1 such that b(t) dt + b(t) dt > 0, I¯
I
and for all z1 , z2 ∈ RN , z1 = z2 , and |z1 + z2 | > R, we have V (z1 + z2 ) (V (z1 ) + V (z2 )) − |z1 − z2 |2 .
(V∗ )
In this paper we drop the super-convex quadratic condition (V∗ ).
2. Existence results We shall investigate the periodic solutions of (P) in the Sobolev space HT1 = H 1 ([0, T ], RN ) = {u : [0, T ] → RN is absolutely continuous, u(0) = u(T ), u˙ ∈ L2 (0, T ; RN )} with the norm 1/2 T T 2 2 |u| ˙ + |u| for all u ∈ HT1 , u = 0
0
assuming conditions 1.1–1.3 are verified in addition to the following assumptions: a.1 0 < l = max |A(t)| < (1+442 )T 2 : (A(t)x, x) l|x|2 , for all x ∈ RN , t ∈ [0, T ] t∈[0,T ] T b.1 0 b(t) dt = 0. m.1 There exist > 0, : 0 < < l, t0 ∈ [0, T ], R1 > 0: (i) b(t) > 0 for all t ∈ I , I = [t0 − , t0 + ] ⊂ [0, T ]. (ii) I (A(t)x, x) dt I |x|2 dt for all x ∈ HT1 . (iii) I¯ (A(t), ) dt T for all ∈ RN , || = 1, (I¯ = [0, T ]\I ). V .1 There exist > 2, a1 > 0, R2 > 0: (i) V (x) V (x)x for all x ∈ RN , |x| R2 , (ii) V (x) a1 |x| for all x ∈ RN , |x| R2 . V .2 There exist , R3 > 0 : V (x) |V (x)||x| for all x ∈ RN , |x| R3 (xy denotes the inner product of the pair of vectors x, y ∈ RN ). V .3 There exists R4 > 0 : V (x) − (V (x), x) c|x|2 for all x ∈ RN , |x| R4 where 4 2 −2 − l , m = max b(t). c< t∈[0,T ] 2m (1 + 42 )T 2 2
V (x) 2 |x|→0 |x|
V .4 lim
= 0.
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L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
Remark. Assumption m.1 indicates that T (A(t), ) dt (2 − T ) for all
∈ RN , || = 1; 2 − T < 0.
0
This is an improvement on condition (0.2) in [5]. Hamiltonian action: We reduce problem (P) to finding critical points of the functional: T T T 1 I (x) = |x| ˙ 2− A(t)x.x − b(t)V (x) x ∈ HT1 (∗ ) 2 0 0 0 associated with (P). We note that the critical points of this functional correspond to periodic solutions of (P). Definition 2.1 (Cerami–Palais–Smale condition). Given a real Banach space X, we say that I ∈ C 1 (X, R) satisfies the Cerami–Palais–Smale condition at level d ∈ R (i.e condition (CP S)d for short) if for any sequence (xn ) ∈ X such that I (xn ) → d and (1 + xn )I (xn ) → 0 it implies that (xn ) contains a strongly convergent subsequence in X. Remark. The Cerami–Palais–Smale condition was introduced by Cerami [9] as a weaker variant of the classical Palais–Smale condition. We shall rely on the following linking theorem of Benci–Rabinowitz (see [17,6]). Theorem 2.2 (Linking theorem). Let E be a Banach space and I ∈ C 1 (E, R) such that (a) E = E− ⊕ E+ , dim E− < ∞ and E+ is closed in E. (b) There exist r, R : 0 < r < R, and ∈ E+ with =1 : sup{I (x) : x ∈ jQ} inf{I (x) : x ∈ D} where Q = {u + : 0, u ∈ E− , u + R} ⊂ E− ⊕ R ; D = {x ∈ E+ : x = r}. (c) I satisfies the Cerami–Palais–Smale condition (CP S)d , d = inf sup I (g(y)), g∈ y∈Q
= {g ∈ C(Q, E) : g(y) = y
for all y ∈ jQ}.
Then d inf D I and d is a critical value of I. Moreover, if d = inf D I , there is a critical point xd ∈ D : I (xd ) = d. Lemma 2.3. Let conditions 1.1–1.3, a.1, b.1, V .1–V .3 be verified. Then I ∈ C 1 (HT1 , R) given as in (∗) satisfies the Cerami–Palais–Smale condition for all real d. Proof. We only need to show that given any sequence (xn ) ∈ HT1 such that I (xn ) is bounded and (1 + xn )I (xn ) → 0 as n → ∞ i.e sup{(1 + xn )I (xn ) : ∈ HT1 , = 1} → 0
L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
569
as n → ∞), then (xn ) contains a strongly convergent subsequence in HT1 . Boundedness of I (xn ) implies that there exists k > 0 such that T 1 T 1 T (A(t)xn , xn ) + b(t)V (xn ) + k. (2.1) |x˙n |2 2 0 2 0 0 Since (1 + xn )I (xn ) is linear, it follows from Riesz’s representation theorem that there exists a sequence (zn ) ∈ HT1 such that (1 + xn )I (xn )H −1 = zn = n → 0
as n → +∞
T
(1 + xn )I (xn ) = zn , H 1
and
for all ∈ HT1 .
T
Thus, we have from Cauchy–Schwartz inequality that T T T b+ (t)(V (xn ), xn ) |x˙n |2 (A(t)xn , xn ) + 0 0 0 T − b (t)(V (xn ), xn ) − n , − 0
where b+ (t) = max{0, b(t)} and b− (t) = − min{0, b(t)}, t ∈ [0, T ]. Clearly, b(t) = b+ (t) − b− (t) for all t ∈ [0, T ]. Hence, T T T T 2 (A(t)xn , xn ) + b(t)V (xn ) − mc |xn |2 − n . |x˙n | 0
0
0
0
Therefore, −
1
T
|x˙n |2 −
0
1
T 0
(A(t)xn , xn ) −
T 0
b(t)V (xn ) +
mc
T 0
|xn |2 +
Combining (2.1) and (2.2) results −2 T ( − 2)l + 2mc T
n |x˙n |2 |xn |2 + + k. 2 0 2 0 That is,
T
( − 2) 0
|x˙n |2 [( − 2)l + 2mc]
T 0
¯ |xn |2 + k,
(0 < k¯ < ∞).
Or, according to Wirtinger’s inequality, we have that T T2 ¯ |x˙n |2 T [( − 2)l + 2mc]|x¯n |2 + k, ( − 2) − [( − 2)l + 2mc] 2 4 0 where x¯n =
1 T
0
T
xn (t) dt,
n ∈ N.
n . (2.2)
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L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
Clearly from condition V .3, T d¯ = ( − 2) − [( − 2)l + 2mc] 2 > 0. 4 2
Thus, setting d1 = we obtain T 0
T [( − 2)l + 2mc] (> 0) d¯
and
d2 =
k¯ (> 0), d¯
|x˙n |2 d1 |x¯n |2 + d2 .
(2.3)
Moreover, applying Sobolev inequality and from (2.3), it is not difficult to verify that min |xn (t)| d3 |x¯n | − d4 ,
t∈[0,T ]
(d3 = 1 − (T d 1 )1/2 ,
d4 = (T d 2 )1/2 ).
We claim that 1 − T d 1 > 0. Suppose the contrary, that 1 − T d 1 0, i.e if and only if 42 2mc ( − 2) −l . (1 + 42 )T 2 This is a contradiction. Hence, we have, min |xn (t)| d3 |x¯n | − d4 ,
t∈[0,T ]
(d3 , d4 > 0).
(2.4)
Clearly, if the sequence (|x¯n |) of real numbers is bounded, then so is (xn ) in HT1 . Suppose |x¯n | → +∞ as n → +∞. Then V (xn ) a1 |xn | → +∞. So, we can define a sequence in HT1 : xn (t) , (1 + xn )V (xn (t))
n (t) =
t ∈ [0, T ]
which is well defined at infinity. Furthermore, V (xn )x˙n − xn [V (xn )x˙n ] ˙ n = . (1 + xn )(V (xn ))2 So, |˙n |2
( + 1)2 |x˙n |2 [(1 + xn )V (xn )]2
for sufficiently large n, and n 2
[{T 2 /42 + ( + 1)2 }d1 + T ]|x¯n |2 + const. a12 (1 + xn )2 (d3 |x¯n | − d4 )2
→0
as n → +∞.
L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
571
Hence, (1 + xn )I (xn )n → 0
as n → +∞.
It is clear that
T ˙ → 0 as n → +∞. (i) (1 + xn ) 0 x˙n n T (ii) (1 + xn ) 0 A(t)xn n → 0 as n → +∞. Now, (1 + xn )
T 0
b(t)V (xn )n
T
T
b(t) dt −
0
b+ (t)
0
|xn |2 dt. V (xn )
Thus, (1 + xn ) since
0
T
T 0
b(t)V (xn )n →
T
b(t) dt = 0
as n → +∞
0
|xn |2 m(T 2 /42 )d1 + T )|x¯n |2 + const. dt < b (t) →0 V (xn ) a1 (d3 |x¯n | − d4 ) +
as n → +∞.
This contradicts our supposition. Therefore, (xn ) is bounded, and so by a similar proof in [4], admits a strongly convergent subsequence. Theorem 2.4 (Main result). Given that assumptions 1.1–1.3 and a.1–V .4 hold, then (P ) has at least one periodic solution.
⊕ RN , H
= {x ∈ H 1 : T x(t) dt = 0}. Proof. HT1 = H T 0
. Then, Let x ∈ H T 1 T 2 1 T I (x) = |x| ˙ − (A(t)x, x) − b(t)V (x) 2 0 2 0 0 T T 1 4 2 2 − l |x| − m V (x). 2 T2 0 0 According to V .4, we can choose 1 (42 /T 2 − l) such that there exists r ( ) > 0 (sufficiently small): 2m T 1 4 2
, x∞ = r . I (x) − l − m
|x|2 > 0 for all x ∈ H (1) 2 T2 0
, 0 < <
Clearly, there exists c1 , c2 > 0 such that T I (x) c1 |x| ˙ 2 and I (x) c2 0
T 0
|x|2
, x∞ = r . for all x ∈ H
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L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
So, I (x) cm x2 ,
cm = min{c1 , c2 }
, x∞ = r . for all x ∈ H
, x = r. Thus, we can find r(r ) > 0 : I (x) > 0 for all x ∈ H Furthermore, choose a constant vector 0 ∈ RN : |0 |2 = /(2 + 2 ). Define, 0 cos (t + − t0 ) t ∈ I , (t) = 0 t ∈ I¯ = [0, T ]\I .
.) Besides, Supp{ } ⊂ I . (t) ∈ H So, we have (t) ∈ HT1 , = 1.(Particularly,
N Let H¯ = + z : 0, z ∈ R . Thus, for every + z ∈ H¯ , we have I ( + z)
2 2
T
(A(t), ) − 2 I¯ 0 − b(t)V ( + z) − V (z) b(t) |˙ |2 −
|z|2 2
I
| + z|2
I¯
I
|z|2 |˙ |2 + T − | + z|2 2 2 I I | + z| − V (z) b(t), − a1 ˜
2 2
I¯
I
where ˜ = minI b(t) > 0 and ∈ RN , = z/|z|(|| = 1). Now, defining some two norms on H¯ : 1/2 1/ 2 + z1 = | + z| , + z2 = | + z| , 2, I
I
it is clear that they are equivalent since dim H¯ < ∞. Therefore, for our fixed > 2, we have that there exists k1 () > 0 such that + z2 k1 + z1 for all 0, z ∈ RN . Besides, we can find some constant k2 > 0 (independent of z ∈ RN , 0): + z21 − k1 a1 ˜ + z − V (z) b(t). (2.5) I ( + z) k2 − 1 2 I¯ We have two cases, namely, i.e Case 1 : b(t) > 0 I¯
Case 2 :
I¯
b(t) < 0
b(t) > 0 .
0
T
if and only if −
I¯
b(t) > 0 .
Case 1: We have from (2.5) that
I ( + z) k2 + z21 − + z21 − k1 a1 ˜ + z . 1 2
L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
573
Clearly, we can find R > r (sufficiently large) such that I ( + z) 0
for all + z ∈ H¯ : + z = R.
Therefore,
, x = r}. z ∈ RN , + z = R} inf{I (x) : x ∈ H
sup{I ( + z) : 0
Hence the conditions of the linking theorem are satisfied. Consequently, I has a non-constant critical point in HT1 which is a solution of (P). Case 2: Without loss of generality, we assume that: > r, a2 > a1 : V (z) a1 |z| + a2 |z|2 for all z ∈ RN , |z| R. 1. There exists R 2 2 ¯ > 0 : − ¯ ¯ 2 /(2) ˜ . 2. There exists I¯ b(t) < 0 and k1 > Then, we have T 2 |z|
− | + z|2 + I ( + z) 2 2 2 2 I + − a1 ˜ | + z| + V (z) − b(t)
2
2
I¯
I
2 2 + 2
2(
(
2
22 − 2 2
2(2 + 2 )
− k1 a1 ˜
2
2 −
+ 2 )
+ 2 )/2
¯ a1 |z| + ¯ a2 |z|2
+ (2)/2 |z| +
1 ¯ a2 )|z|2
2 + (T − 2 + 2 2
− a1
2 + (T − 2)|z|2 2
˜ k1
(2 + 2 )/2
¯ ]|z|
− a1 [k1 ˜ (2)/2 −
D1 2 + D2 |z|2 − D3 − D4 |z| , where D1 , D2 , D3 , D4 ∈ R+ . + z2 = 2 + T |z|2
for all 0, z ∈ RN .
Thus, as + z → +∞, I ( + z) → −∞. Therefore, we can find R > r (sufficiently large) such that I ( + z) < 0
for all + z ∈ H¯ : + z = R.
This verifies the Linking theorem and hence completes proof of Theorem 2.4.
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L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
References [1] S. Alama, G. Tarantello, On semilinear elliptica equations with indefinite non linearities, Calc. Var. Partial Diff. Eq. 93 (1991) 1–18. [2] A. Ambrosetti, V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, Rend. Semin. Mat. Univ. Padova. 89 (1993) 177–194. [3] A. Ambrosetti, G. Mancini, Solutions of minimal period to a class of convex Hamiltonian systems, Math. Ann. 255 (1981) 405–421. [4] F. Antonacci, Periodic and Homoclinic solutions to a class of Hamiltonian systems with potential indefinite in sign, Boll. Unione. Mat. Ital. 10-B (1996) 303–324. [5] F. Antonacci, Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign, Nonlinear Anal. TMA 29 (1997) 1353–1364. [6] V. Benci, P.H. Rabinowitz, Critical point theorem for indefinite functionals, Invent. Math. 52 (1979) 241–273. [7] A.K. Ben Naoum, C. Trostler, M. Willem, Existence and multiplicity results for homogeneous second order differential equations, J. Diff. Equations 112 (1994) 239–249. [8] P. Caldiroli, P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. Appl. Anal. 1 (1994) 97–129. [9] G. Cerami, Un criterio di esistenza per in punti critici su varieta illimit ate, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978) 332–336. [10] V. Coti Zelati, P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing super quadratic potential, J. Ann. Math. Soc. 4 (1991) 693–727. [12] Y.H. Ding, M. Girardi, Periodic and homoclinic solutions to a class of Hamiltonian systems with the potential changing sign, Dynamical systems Appl. 2 (1993) 131–145. [13] M. Girardi, M. Matzeu, Existence and multiplicity results for periodic solutions for super quadratic Hamiltonian systems where the potential changes sign, Nonlinear Diff. Equations Appl. 2 (1995) 35–61. [14] L. Lassoued, Periodic solutions to a second order super quadratic system with a change of sign in the potential, J. Diff. Equations 93 (1991) 1–18. [17] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, American Mathematical Society, vol. 65, Providence, RI, 1986. [18] L.K. Shilgba, A variant existence result for periodic solutions of a class of Hamiltonian systems with indefinte potential, Journal of Nonlinear and convex analysis JNCA, to appear. [19] Y.-T. Xu, Z.-M. Guo, Existence of periodic solutions to second order Hamiltonian systems with potential indefinite in sign, Nonlinear Anal. TMA 51 (7) (2002) 1273–1283.
Further reading [11] M. Degiovani, Basic tools of critical point theory, 2002. [15] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. [16] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math. 31 (1978) 157–184.
Existence result for periodic solutions of a class of Hamiltonian systems with super quadratic potential Leonard Karshima Shilgba∗ Hirano Laboratory, Division of Information Media and Environment Sciences, Graduate School of Environment and Information Sciences, Yokohama National University, 79-5, Tokiwadai, Hodogaya-Ku, Yokohama-shi, Kanagawa 240-8501, Japan Received 27 May 2003; accepted 18 May 2005
Abstract We have presented a variant existence result for periodic solutions to a class of Hamiltonian systems with a potential which is both super quadratic and sign-indefinite. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 34B15 Keywords: Periodic solutions; Hamiltonian systems; Potential; Linking
1. Introduction Our objective is to investigate the existence of periodic solutions to the class of Hamiltonian systems: x¨ + A(t)x + b(t)V (x) = 0
on [0, T ],
x(0) = x(T ), x(0) ˙ = x(T ˙ ),
∗ Tel.: +81 45 50 24 115; fax: +81 45 33 94 207.
E-mail address: [email protected]. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.05.018
(P)
566
L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
where 1.1 A ∈ C 0 (R, RN×N ) is a symmetric T -periodic matrix function that is not sign-definite on [0, T ]; 1.2 b ∈ C 0 (R, R) is T -periodic and hanges sign on [0, T ]; 1.3 V ∈ C 2 (RN , R) with V (x) V (0) = 0 for all x ∈ RN , and has super quadratic behaviour. The above problem has been studied by several authors: Case A: A ≡ 0. While Lassoued [14] obtained existence of T -periodic solutions when V is strictly convex and homogeneous, Ben Naoum et al. [7] provided existence results by relaxing condition on V to only homogeneity. Girardi and Matzeu [13] used the Alama–Tarantello condition in [1] and proved some existence and multiplicity results for T -periodic and subharmonic solutions. The condition is given by: There exist
c > 0, > 2 : |V (x)x − V (x)| c|x|2
for all x ∈ RN .
(1.4)
Case B: A = 0. In this case, when b(.) > 0, refer to [3,2,10] (with the references contained therein), severally for existence of periodic, homoclinic, and subharmonic solutions. Besides, when we assume b changes sign and A is a negative definite matrix, there are some existence and multiplicity results of periodic, subharmonic, and homoclinic solutions (refer to [12,8,4]). When b(.) changes sign and A is sign-indefinite, the author is aware of only the results of Antonacci [5], Xu and Guo [19], and the author in [18]. Antonacci proved the existence of non-trivial periodic solutions under the assumption that T (A(t), ) dt > 0 for all ∈ RN , || = 1. (1.5) 0
Antonacci observed the vital role played by assumption (1.5) as he posed two open problems: Given that A(.) is not negative-definite, to 1. study the existence of T -periodic solutions of (P) in the case (1.5) holds and A(.) is sign-indefinite in any interval [0, T ]; 2. find some existence results in the case that A does not satisfy (1.5). As mentioned above, Antonnaci attempted an elegant solution of problem 1 in [5]. Xu and Guo, in attempting the solution of open problem 2 in [18], relied, among other assumptions, on the glooseh or weaker version of (1.5), that: T There exist > 0, > 2, > 0, : 0 < < ( 0 b(t) dt) such that T (A(t), ) dt − for all ∈ RN , || = 1, (1.6) 0
where V (x) |x| − for all sufficiently large x ∈ RN .
L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
567
T It is clear from (1.6) that 0 (A(t), ) dt, ∈ RN , || = 1, is neither assumed to be definitely positive nor negative, which is a relaxation on condition (1.5.); however, it is only T valid when 0 b(t) dt > 0. In [18] we resolved the open problem 2 posed by Antonnaci by assuming some super-convex quadratic condition on the potential V : There exist real numbers, R > 0, , : > > 1 such that b(t) dt + b(t) dt > 0, I¯
I
and for all z1 , z2 ∈ RN , z1 = z2 , and |z1 + z2 | > R, we have V (z1 + z2 ) (V (z1 ) + V (z2 )) − |z1 − z2 |2 .
(V∗ )
In this paper we drop the super-convex quadratic condition (V∗ ).
2. Existence results We shall investigate the periodic solutions of (P) in the Sobolev space HT1 = H 1 ([0, T ], RN ) = {u : [0, T ] → RN is absolutely continuous, u(0) = u(T ), u˙ ∈ L2 (0, T ; RN )} with the norm 1/2 T T 2 2 |u| ˙ + |u| for all u ∈ HT1 , u = 0
0
assuming conditions 1.1–1.3 are verified in addition to the following assumptions: a.1 0 < l = max |A(t)| < (1+442 )T 2 : (A(t)x, x) l|x|2 , for all x ∈ RN , t ∈ [0, T ] t∈[0,T ] T b.1 0 b(t) dt = 0. m.1 There exist > 0, : 0 < < l, t0 ∈ [0, T ], R1 > 0: (i) b(t) > 0 for all t ∈ I , I = [t0 − , t0 + ] ⊂ [0, T ]. (ii) I (A(t)x, x) dt I |x|2 dt for all x ∈ HT1 . (iii) I¯ (A(t), ) dt T for all ∈ RN , || = 1, (I¯ = [0, T ]\I ). V .1 There exist > 2, a1 > 0, R2 > 0: (i) V (x) V (x)x for all x ∈ RN , |x| R2 , (ii) V (x) a1 |x| for all x ∈ RN , |x| R2 . V .2 There exist , R3 > 0 : V (x) |V (x)||x| for all x ∈ RN , |x| R3 (xy denotes the inner product of the pair of vectors x, y ∈ RN ). V .3 There exists R4 > 0 : V (x) − (V (x), x) c|x|2 for all x ∈ RN , |x| R4 where 4 2 −2 − l , m = max b(t). c< t∈[0,T ] 2m (1 + 42 )T 2 2
V (x) 2 |x|→0 |x|
V .4 lim
= 0.
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Remark. Assumption m.1 indicates that T (A(t), ) dt (2 − T ) for all
∈ RN , || = 1; 2 − T < 0.
0
This is an improvement on condition (0.2) in [5]. Hamiltonian action: We reduce problem (P) to finding critical points of the functional: T T T 1 I (x) = |x| ˙ 2− A(t)x.x − b(t)V (x) x ∈ HT1 (∗ ) 2 0 0 0 associated with (P). We note that the critical points of this functional correspond to periodic solutions of (P). Definition 2.1 (Cerami–Palais–Smale condition). Given a real Banach space X, we say that I ∈ C 1 (X, R) satisfies the Cerami–Palais–Smale condition at level d ∈ R (i.e condition (CP S)d for short) if for any sequence (xn ) ∈ X such that I (xn ) → d and (1 + xn )I (xn ) → 0 it implies that (xn ) contains a strongly convergent subsequence in X. Remark. The Cerami–Palais–Smale condition was introduced by Cerami [9] as a weaker variant of the classical Palais–Smale condition. We shall rely on the following linking theorem of Benci–Rabinowitz (see [17,6]). Theorem 2.2 (Linking theorem). Let E be a Banach space and I ∈ C 1 (E, R) such that (a) E = E− ⊕ E+ , dim E− < ∞ and E+ is closed in E. (b) There exist r, R : 0 < r < R, and ∈ E+ with =1 : sup{I (x) : x ∈ jQ} inf{I (x) : x ∈ D} where Q = {u + : 0, u ∈ E− , u + R} ⊂ E− ⊕ R ; D = {x ∈ E+ : x = r}. (c) I satisfies the Cerami–Palais–Smale condition (CP S)d , d = inf sup I (g(y)), g∈ y∈Q
= {g ∈ C(Q, E) : g(y) = y
for all y ∈ jQ}.
Then d inf D I and d is a critical value of I. Moreover, if d = inf D I , there is a critical point xd ∈ D : I (xd ) = d. Lemma 2.3. Let conditions 1.1–1.3, a.1, b.1, V .1–V .3 be verified. Then I ∈ C 1 (HT1 , R) given as in (∗) satisfies the Cerami–Palais–Smale condition for all real d. Proof. We only need to show that given any sequence (xn ) ∈ HT1 such that I (xn ) is bounded and (1 + xn )I (xn ) → 0 as n → ∞ i.e sup{(1 + xn )I (xn ) : ∈ HT1 , = 1} → 0
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as n → ∞), then (xn ) contains a strongly convergent subsequence in HT1 . Boundedness of I (xn ) implies that there exists k > 0 such that T 1 T 1 T (A(t)xn , xn ) + b(t)V (xn ) + k. (2.1) |x˙n |2 2 0 2 0 0 Since (1 + xn )I (xn ) is linear, it follows from Riesz’s representation theorem that there exists a sequence (zn ) ∈ HT1 such that (1 + xn )I (xn )H −1 = zn = n → 0
as n → +∞
T
(1 + xn )I (xn ) = zn , H 1
and
for all ∈ HT1 .
T
Thus, we have from Cauchy–Schwartz inequality that T T T b+ (t)(V (xn ), xn ) |x˙n |2 (A(t)xn , xn ) + 0 0 0 T − b (t)(V (xn ), xn ) − n , − 0
where b+ (t) = max{0, b(t)} and b− (t) = − min{0, b(t)}, t ∈ [0, T ]. Clearly, b(t) = b+ (t) − b− (t) for all t ∈ [0, T ]. Hence, T T T T 2 (A(t)xn , xn ) + b(t)V (xn ) − mc |xn |2 − n . |x˙n | 0
0
0
0
Therefore, −
1
T
|x˙n |2 −
0
1
T 0
(A(t)xn , xn ) −
T 0
b(t)V (xn ) +
mc
T 0
|xn |2 +
Combining (2.1) and (2.2) results −2 T ( − 2)l + 2mc T
n |x˙n |2 |xn |2 + + k. 2 0 2 0 That is,
T
( − 2) 0
|x˙n |2 [( − 2)l + 2mc]
T 0
¯ |xn |2 + k,
(0 < k¯ < ∞).
Or, according to Wirtinger’s inequality, we have that T T2 ¯ |x˙n |2 T [( − 2)l + 2mc]|x¯n |2 + k, ( − 2) − [( − 2)l + 2mc] 2 4 0 where x¯n =
1 T
0
T
xn (t) dt,
n ∈ N.
n . (2.2)
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Clearly from condition V .3, T d¯ = ( − 2) − [( − 2)l + 2mc] 2 > 0. 4 2
Thus, setting d1 = we obtain T 0
T [( − 2)l + 2mc] (> 0) d¯
and
d2 =
k¯ (> 0), d¯
|x˙n |2 d1 |x¯n |2 + d2 .
(2.3)
Moreover, applying Sobolev inequality and from (2.3), it is not difficult to verify that min |xn (t)| d3 |x¯n | − d4 ,
t∈[0,T ]
(d3 = 1 − (T d 1 )1/2 ,
d4 = (T d 2 )1/2 ).
We claim that 1 − T d 1 > 0. Suppose the contrary, that 1 − T d 1 0, i.e if and only if 42 2mc ( − 2) −l . (1 + 42 )T 2 This is a contradiction. Hence, we have, min |xn (t)| d3 |x¯n | − d4 ,
t∈[0,T ]
(d3 , d4 > 0).
(2.4)
Clearly, if the sequence (|x¯n |) of real numbers is bounded, then so is (xn ) in HT1 . Suppose |x¯n | → +∞ as n → +∞. Then V (xn ) a1 |xn | → +∞. So, we can define a sequence in HT1 : xn (t) , (1 + xn )V (xn (t))
n (t) =
t ∈ [0, T ]
which is well defined at infinity. Furthermore, V (xn )x˙n − xn [V (xn )x˙n ] ˙ n = . (1 + xn )(V (xn ))2 So, |˙n |2
( + 1)2 |x˙n |2 [(1 + xn )V (xn )]2
for sufficiently large n, and n 2
[{T 2 /42 + ( + 1)2 }d1 + T ]|x¯n |2 + const. a12 (1 + xn )2 (d3 |x¯n | − d4 )2
→0
as n → +∞.
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571
Hence, (1 + xn )I (xn )n → 0
as n → +∞.
It is clear that
T ˙ → 0 as n → +∞. (i) (1 + xn ) 0 x˙n n T (ii) (1 + xn ) 0 A(t)xn n → 0 as n → +∞. Now, (1 + xn )
T 0
b(t)V (xn )n
T
T
b(t) dt −
0
b+ (t)
0
|xn |2 dt. V (xn )
Thus, (1 + xn ) since
0
T
T 0
b(t)V (xn )n →
T
b(t) dt = 0
as n → +∞
0
|xn |2 m(T 2 /42 )d1 + T )|x¯n |2 + const. dt < b (t) →0 V (xn ) a1 (d3 |x¯n | − d4 ) +
as n → +∞.
This contradicts our supposition. Therefore, (xn ) is bounded, and so by a similar proof in [4], admits a strongly convergent subsequence. Theorem 2.4 (Main result). Given that assumptions 1.1–1.3 and a.1–V .4 hold, then (P ) has at least one periodic solution.
⊕ RN , H
= {x ∈ H 1 : T x(t) dt = 0}. Proof. HT1 = H T 0
. Then, Let x ∈ H T 1 T 2 1 T I (x) = |x| ˙ − (A(t)x, x) − b(t)V (x) 2 0 2 0 0 T T 1 4 2 2 − l |x| − m V (x). 2 T2 0 0 According to V .4, we can choose 1 (42 /T 2 − l) such that there exists r ( ) > 0 (sufficiently small): 2m T 1 4 2
, x∞ = r . I (x) − l − m
|x|2 > 0 for all x ∈ H (1) 2 T2 0
, 0 < <
Clearly, there exists c1 , c2 > 0 such that T I (x) c1 |x| ˙ 2 and I (x) c2 0
T 0
|x|2
, x∞ = r . for all x ∈ H
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L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
So, I (x) cm x2 ,
cm = min{c1 , c2 }
, x∞ = r . for all x ∈ H
, x = r. Thus, we can find r(r ) > 0 : I (x) > 0 for all x ∈ H Furthermore, choose a constant vector 0 ∈ RN : |0 |2 = /(2 + 2 ). Define, 0 cos (t + − t0 ) t ∈ I , (t) = 0 t ∈ I¯ = [0, T ]\I .
.) Besides, Supp{ } ⊂ I . (t) ∈ H So, we have (t) ∈ HT1 , = 1.(Particularly,
N Let H¯ = + z : 0, z ∈ R . Thus, for every + z ∈ H¯ , we have I ( + z)
2 2
T
(A(t), ) − 2 I¯ 0 − b(t)V ( + z) − V (z) b(t) |˙ |2 −
|z|2 2
I
| + z|2
I¯
I
|z|2 |˙ |2 + T − | + z|2 2 2 I I | + z| − V (z) b(t), − a1 ˜
2 2
I¯
I
where ˜ = minI b(t) > 0 and ∈ RN , = z/|z|(|| = 1). Now, defining some two norms on H¯ : 1/2 1/ 2 + z1 = | + z| , + z2 = | + z| , 2, I
I
it is clear that they are equivalent since dim H¯ < ∞. Therefore, for our fixed > 2, we have that there exists k1 () > 0 such that + z2 k1 + z1 for all 0, z ∈ RN . Besides, we can find some constant k2 > 0 (independent of z ∈ RN , 0): + z21 − k1 a1 ˜ + z − V (z) b(t). (2.5) I ( + z) k2 − 1 2 I¯ We have two cases, namely, i.e Case 1 : b(t) > 0 I¯
Case 2 :
I¯
b(t) < 0
b(t) > 0 .
0
T
if and only if −
I¯
b(t) > 0 .
Case 1: We have from (2.5) that
I ( + z) k2 + z21 − + z21 − k1 a1 ˜ + z . 1 2
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Clearly, we can find R > r (sufficiently large) such that I ( + z) 0
for all + z ∈ H¯ : + z = R.
Therefore,
, x = r}. z ∈ RN , + z = R} inf{I (x) : x ∈ H
sup{I ( + z) : 0
Hence the conditions of the linking theorem are satisfied. Consequently, I has a non-constant critical point in HT1 which is a solution of (P). Case 2: Without loss of generality, we assume that: > r, a2 > a1 : V (z) a1 |z| + a2 |z|2 for all z ∈ RN , |z| R. 1. There exists R 2 2 ¯ > 0 : − ¯ ¯ 2 /(2) ˜ . 2. There exists I¯ b(t) < 0 and k1 > Then, we have T 2 |z|
− | + z|2 + I ( + z) 2 2 2 2 I + − a1 ˜ | + z| + V (z) − b(t)
2
2
I¯
I
2 2 + 2
2(
(
2
22 − 2 2
2(2 + 2 )
− k1 a1 ˜
2
2 −
+ 2 )
+ 2 )/2
¯ a1 |z| + ¯ a2 |z|2
+ (2)/2 |z| +
1 ¯ a2 )|z|2
2 + (T − 2 + 2 2
− a1
2 + (T − 2)|z|2 2
˜ k1
(2 + 2 )/2
¯ ]|z|
− a1 [k1 ˜ (2)/2 −
D1 2 + D2 |z|2 − D3 − D4 |z| , where D1 , D2 , D3 , D4 ∈ R+ . + z2 = 2 + T |z|2
for all 0, z ∈ RN .
Thus, as + z → +∞, I ( + z) → −∞. Therefore, we can find R > r (sufficiently large) such that I ( + z) < 0
for all + z ∈ H¯ : + z = R.
This verifies the Linking theorem and hence completes proof of Theorem 2.4.
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L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574
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