Periodic solutions of a class of nonautonomous second order differential systems with. Daniel Paşca

Transcription

1 Periodic solutions of a class of nonautonomous second order differential systems with (q, p) Laplacian Daniel Paşca Abstract Some existence theorems are obtained by the least action principle for periodic solutions of nonautonomous second-order differential systems with (q, p) Laplacian. 1 Introduction In the last years many authors starting with Mawhin and Willem (see [1]) proved the existence of solutions for problem ü(t) = F(t, u(t)) a.e. t [, ], u() u() = u() u() =, (1) under suitable conditions on the potential F (see [7]-[19]). Also in a series of papers (see [2]-[4]) we have generalized some of these results for the case when the potential F is just locally Lipschitz in the second variable x not continuously differentiable. Very recent (see [5] and [6]) we have considered the second order Hamiltonian inclusions systems with p Laplacian. he aim of this paper is to show how the results obtained in [14] can be generalized. More exactly our results represent the extensions to second-order differential systems with (q, p) Laplacian. Received by the editors August 29. Communicated by J. Mawhin. 2 Mathematics Subject Classification : 34C25, 49J24. Key words and phrases : (q, p) Laplacian, second order systems, (λ, µ) subconvex, periodic solutions. Bull. Belg. Math. Soc. Simon Stevin 17 (21),

2 842 D. Paşca Consider the second order system d dt( u1 (t) q 2 u 1 (t) ) = u1 F(t, u 1 (t), u 2 (t)), d dt( u2 (t) p 2 u 2 (t) ) = u2 F(t, u 1 (t), u 2 (t)) a.e. t [, ], u 1 () u 1 () = u 1 () u 1 () =, u 2 () u 2 () = u 2 () u 2 () =, (2) where 1 < p, q <, >, and F : [, ] R N R N R satisfy the following assumption (A): F is measurable in t for each(x 1, x 2 ) R N R N ; F is continuously differentiable in (x 1, x 2 ) for a.e. t [, ]; there exist a 1, a 2 C(R, R ) and b L 1 (, ; R ) such that F(t, x 1, x 2 ), x1 F(t, x 1, x 2 ), x2 F(t, x 1, x 2 ) [ a 1 ( x 1 )a 2 ( x 2 ) ] b(t) for all (x 1, x 2 ) R N R N and a.e. t [, ]. Definition 1. (see [14]) A function G : R N R is called to be (λ, µ) subconvex if for some λ, µ > and all x, y R N. G(λ(x y)) µ(g(x)g(y)) Remark 1. (see [14]) When λ = µ = 1 2, a function ( 2 1, 2 1 ) subconvex is called convex. When λ = µ = 1, a function (1, 1) subconvex is called subadditive. When λ = 1, µ >, a function (1, µ) subconvex is called µ subadditive. 2 Main results heorem 1. Assume that F = F 1 F 2, where F 1, F 2 satisfy assumption (A) and the following conditions: (i) F 1 (t,, ) is (λ, µ) subconvex with λ > 1/2 and < µ < 2 r 1 λ r for a.e. t [, ] where r = min(q, p); (ii) there exist f i, g i L 1 (, ; R ), i = 1, 2 and α 1 [, q 1), α 2 [, p 1) such that x1 F 2 (t, x 1, x 2 ) f 1 (t) x 1 α 1 g 1 (t) x2 F 2 (t, x 1, x 2 ) f 2 (t) x 2 α 2 g 2 (t) for all(x 1, x 2 ) R N R N and a.e. t [, ]; (iii) 1 [ 1 ] F x 1 q α 1 x2 p α 1 (t, λx 1, λx 2 )dt F 2 (t, x 1, x 2 )dt 2 µ as x = x 1 2 x 2 2, where 1 q 1 q = 1 and 1 p 1 p = 1.

3 Periodic solutions of differential systems with (q, p) Laplacian 843 hen the problem (2) has at least one solution which minimizes the function ϕ : W = R given by W 1,q W1,p ϕ(u 1, u 2 ) = 1 q u 1 (t) q dt 1 p u 2 (t) p dt F(t, u 1 (t), u 2 (t))dt. heorem 2. Assume that F = F 1 F 2, where F 1, F 2 satisfy assumption (A) and the following conditions: (iv) F 1 (t,, ) is (λ, µ) subconvex with λ > 1/2 and < µ < 2 r 1 λ r for a.e. t [, ] where r = min(q, p), and there exists γ L 1 (, ; R), h 1, h 2 L 1 (, ; R N ) with h i(t)dt =, i = 1, 2 such that F 1 (t, x 1, x 2 ) ((h 1 (t), h 2 (t)),(x 1, x 2 ))γ(t) for all (x 1, x 2 ) R N R N and a.e. t [, ]; (v) there exist g 1, g 2 L 1 (, ; R ), c R such that (vi) x1 F 2 (t, x 1, x 2 ) g 1 (t) x2 F 2 (t, x 1, x 2 ) g 2 (t) for all (x 1, x 2 ) R N R N and a.e. t [, ], and for all x R N ; F 2 (t, x 1, x 2 )dt c 1 F 1 (t, λx 1, λx 2 )dt F 2 (t, x 1, x 2 )dt µ as x = x 1 2 x 2 2. hen the problem (2) has at least one solution which minimizes ϕ on W. heorem 3. Assume that F = F 1 F 2, where F 1, F 2 satisfy assumption (A) and the following conditions: (vii) F 1 (t,, ) is (λ, µ) subconvex with λ > 1/2 and < µ < 2 r 1 λ r for a.e. t [, ] where r = min(q, p), and there exists γ L 1 (, ; R), h 1, h 2 L 1 (, ; R N ) with h i(t)dt =, i = 1, 2 such that F 1 (t, x 1, x 2 ) ((h 1 (t), h 2 (t)),(x 1, x 2 ))γ(t) for all (x 1, x 2 ) R N R N and a.e. t [, ];

4 844 D. Paşca (viii) there exist f i, g i L 1 (, ; R ), i = 1, 2 and α 1 [, q 1), α 2 [, p 1) such that x1 F 2 (t, x 1, x 2 ) f 1 (t) x 1 α 1 g 1 (t) x2 F 2 (t, x 1, x 2 ) f 2 (t) x 2 α 2 g 2 (t) for all(x 1, x 2 ) R N R N and a.e. t [, ]; (ix) 1 x 1 q α 1 x2 p α 2 as x = x 1 2 x 2 2. F 2 (t, x 1, x 2 )dt hen the problem (2) has at least one solution which minimizes ϕ on W. Remark 2. heorems 1, 2 and 3 generalizes the corresponding heorems 1, 2 and 3 of [14]. In fact, it follows from these theorems letting p = q = 2 and F(t, x 1, x 2 ) = F 1 (t, x 1 ). 3 he proofs of the theorems We introduce some functional spaces. Let > be a positive number and 1 < q, p <. We use to denote the Euclidean norm in R N. We denote by W 1,p the Sobolev space of functions u L p (, ; R N ) having a weak derivative u L p (, ; R N ). he norm in W 1,p is defined by u W 1,p ( = ( u(t) p u(t) p) ) 1 p dt. Moreover, we use the space W defined by W = W 1,q with the norm (u 1, u 2 ) W = u 1 1,q W Banach space. We recall that ( u p = W1,p u 2 1,p W. It is clear that W is a reflexive u(t) p dt) 1 p and u = max t [,] u(t). For our aims it is necessary to recall some very well know results (for proof and details see [1]). Proposition 4. Each u W 1,p can be written as u(t) = ūũ(t) with ū = 1 u(t)dt, ũ(t)dt =.

5 Periodic solutions of differential systems with (q, p) Laplacian 845 We have the Sobolev s inequality and Wirtinger s inequality ũ C 1 u p for each u W 1,p, ũ p C 2 ũ p for each u W 1,p. In [11] the authors have proved the following result (see Lemma 3.1) which generalize a very well known result proved by Jean Mawhin and Michel Willem (see heorem 1.4 in [1]): Lemma 5. Let L : [, ] R N R N R N R N R, (t, x 1, x 2, y 1, y 2 ) L(t, x 1, x 2, y 1, y 2 ) be measurable in t for each (x 1, x 2, y 1, y 2 ) for a.e. t [, ]. If there exist a i C(R, R ), b L 1 (, ; R ), and c 1 L q (, ; R ), c 2 L p (, ; R ), 1 < q, p <, 1 q 1 q = 1, 1 p 1 p = 1, such that for a.e. t [, ] and every (x 1, x 2, y 1, y 2 ) R N R N R N R N, one has L(t, x 1, x 2, y 1, y 2 ) [ a 1 ( x 1 )a 2 ( x 2 ) ][ b(t) y 1 q y 2 p], D x1 L(t, x 1, x 2, y 1, y 2 ) [ a 1 ( x 1 )a 2 ( x 2 ) ][ b(t) y 2 p], D x2 L(t, x 1, x 2, y 1, y 2 ) [ a 1 ( x 1 )a 2 ( x 2 ) ][ b(t) y 1 q], D y1 L(t, x 1, x 2, y 1, y 2 ) [ a 1 ( x 1 )a 2 ( x 2 ) ][ c 1 (t) y 1 q 1], D y2 L(t, x 1, x 2, y 1, y 2 ) [ a 1 ( x 1 )a 2 ( x 2 ) ][ c 2 (t) y 2 p 1], then the function ϕ : W 1,q W1,p R defined by ϕ(u 1, u 2 ) = is continuously differentiable on W 1,q ϕ (u 1, u 2 ),(v 1, v 2 ) = L(t, u 1 (t), u 2 (t), u 1 (t), u 2 (t))dt W1,p and [ (Dx1 L(t, u 1 (t), u 2 (t), u 1 (t), u 2 (t)), v 1 (t)) (D y1 L(t, u 1 (t), u 2 (t), u 1 (t), u 2 (t)), v 1 (t)) (D x2 L(t, u 1 (t), u 2 (t), u 1 (t), u 2 (t)), v 2 (t)) (D y2 L(t, u 1 (t), u 2 (t), u 1 (t), u 2 (t)), v 2 (t)) ] dt. Corollary 6. Let L : [, ] R N R N R N R N R be defined by L(t, x 1, x 2, y 1, y 2 ) = 1 q y 1 q 1 p y 2 p F(t, x 1, x 2 ) where F : [, ] R N R N R satisfy condition (A). If (u 1, u 2 ) W 1,q W1,p is a solution of the corresponding Euler equation ϕ (u 1, u 2 ) =, then (u 1, u 2 ) is a solution of (2). Remark 3. he function ϕ is weakly lower semi-continuous (w.l.s.c.) on W as the sum of two convex continuous functions and of a weakly continuous one.

6 846 D. Paşca Proof of heorem 1. Like in [14] we obtain ( F 1 (t, x 1, x 2 ) 2 β 2 1 µ ( x 1 β x 2 β) ) 1 (a 1 a 2 )b(t) for all (x 1, x 2 ) R N R N and a.e. t [, ], where β < r, and a i = max s 1 a i (s), i = 1, 2. It follows from (i), (ii) and Sobolev s inequality that = [ F2 (t, u 1 (t), u 2 (t)) F 2 (t, ū 1, ū 2 ) ] dt [ F2 (t, u 1 (t), u 2 (t)) F 2 (t, u 1 (t), ū 2 ) ] dt [ F2 (t, u 1 (t), ū 2 ) F 2 (t, ū 1, ū 2 ) ] dt = ( x2 F 2 (t, u 1 (t), ū 2 sũ 2 (t)), ũ 2 (t))dsdt ( x1 F 2 (t, ū 1 sũ 1 (t), ū 2 ), ũ 1 (t))dsdt f 2 (t) ū 2 sũ 2 (t) α 2 ũ 2 (t) dsdt f 1 (t) ū 1 sũ 1 (t) α 1 ũ 1 (t) dsdt 2 ( ū 2 α 2 ũ 2 α ) 2 ũ2 1 1 g 2 (t) ũ 2 (t) dsdt g 1 (t) ũ 1 (t) dsdt f 2 (t)dt ũ 2 g 2 (t)dt 2 ( ū 1 α 1 ũ 1 α ) 1 ũ1 f 1 (t)dt ũ 1 g 1 (t)dt = = c 11 ũ 1 α 11 2c 12 ū 1 α 1 ũ 1 c 13 ũ 1 c 21 ũ 2 α 21 2c 22 ū 2 α 2 ũ 2 c 23 ũ 2 c 11 u 1 α 11 q 2 c 12 ū 1 α 1 u 1 q c 13 u 1 q c 21 u 2 α 21 p 2 c 22 ū 2 α 2 u 2 p c 23 u 2 p c 11 u 1 α 11 q 1 2q u 1 q q c 13 u 1 q c 12 ū 1 q α 1 c 21 u 2 α 21 p 1 2p u 2 p p c 23 u 2 p c 22 ū 2 p α 2 for all (u 1, u 2 ) W and some positive constants c 11,..., c 22. Hence we have ϕ(u 1, u 2 ) = 1 q u 1 (t) q dt 1 p u 2 (t) p dt F 1 (t, u 1 (t), u 2 (t))dt [ F2 (t, u 1 (t), u 2 (t)) F 2 (t, ū 1, ū 2 ) ] dt F 2 (t, ū 1, ū 2 )dt

7 Periodic solutions of differential systems with (q, p) Laplacian µ 1 2q u 1 q q c 11 u 1 α 11 q c 13 u 1 q c 12 ū 1 q α 1 1 2p u 2 p p c 21 u 2 α 21 p c 23 u 2 p c 22 ū 2 p α 2 1 µ F 1 (t, λū 1, λū 2 )dt F 2 (t, ū 1, ū 2 )dt F 1 (t, ũ 1 (t), ũ 2 (t))dt 1 2q u 1 q q c 11 u 1 α 11 q c 13 u 1 q c 12 ū 1 q α 1 1 2p u 2 p p c 21 u 2 α 21 p c 23 u 2 p c 22 ū 2 p α 2 F 1 (t, λū 1, λū 2 )dt [ 2 β 2 1 µ ( ũ 1 β ũ 2 β ) 1 ] (a1 a 2 ) 1 2q u 1 q q c 11 u 1 α 11 q c 13 u 1 q c 31 u 1 β q 1 2p u 2 p p c 21 u 2 α 21 p c 23 u 2 p c 31 u 2 β p c 32 1 µ F 2 (t, ū 1, ū 2 )dt b(t)dt F 2 (t, ū 1, ū 2 )dt F 1 (t, λū 1, λū 2 )dt max( c 12, c 22 ) ( ū 1 q α 1 c 22 ū 2 p α 2 ) = = 1 2q u 1 q q c 11 u 1 α 11 q c 13 u 1 q c 31 u 1 β q 1 2p u 2 p p c 21 u 2 α 21 p c 23 u 2 p c 31 u 2 β p c 32 ( ) { ū 1 q α 1 c 22 ū 2 p α 2 1 [ 1 F ū 1 q α 1 c22 ū 2 p α 1 (t, λū 1, λū 2 )dt 2 µ ] } F 2 (t, ū 1, ū 2 )dt max( c 12, c 22 ) for all (u 1, u 2 ) W, which imply that ϕ(u 1, u 2 ) as (u 1, u 2 ) W due to (iii). By heorem 1.1 in [1] and Corollary 6 we complete our proof. Proof of heorem 2. Let (u 1k, u 2k ) be a minimizing sequence of ϕ. It follows from (iv), (v) and Sobolev s inequality that ϕ(u 1k, u 2k ) = 1 q u 1k (t) q dt 1 p u 2k (t) p dt F 2 (t, u 1k (t), u 2k (t))dt 1 q u 1k q q 1 p u 2k p p ( (h1 (t), h 2 (t)),(u 1k (t), u 2k (t)) ) dt γ(t)dt F 1 (t, u 1k (t), u 2k (t))dt 1 ( x2 F 2 (t, u 1k (t), ū 2k sũ 2k (t)), ũ 2k (t))dsdt F 2 (t, ū 1k, ū 2k )dt

8 848 D. Paşca ũ 1k h 1 (t) dt ũ 2k h 2 (t) dt 1 γ(t)dt c ( x1 F 2 (t, ū 1k sũ 1k (t), ū 2k ), ũ 1k (t))dsdt 1 q u 1k q q 1 p u 2k p p ũ 1k g 1 (t)dt ũ 2k g 2 (t)dt 1 q u 1k q q 1 p u 2k p p c 1 u 1k q c 2 u 2k p c 3 for all k and some constants c 1, c 2, c 3, which implies that(ũ 1k, ũ 2k ) is bounded. On the other hand, in a way similar to the proof of heorem 1, one has [ F2 (t, u 1 (t), u 2 (t)) F 2 (t, ū 1, ū 2 ) ] dt c 13 u 1 q c 23 u 2 p for all (u 1, u 2 ) W and some positive constants c 13, c 23, which implies that ϕ(u 1k, u 2k ) 1 q u 1k q q 1 p u 2k p p 1 F 1 (t, λū µ 1k, λū 2k )dt F 1 (t, ũ 1k (t), ũ 2k (t))dt [ F2 (t, u 1k (t), u 2k (t)) F 2 (t, ū 1k, ū 2k ) ] dt F 2 (t, ū 1k, ū 2k )dt 1 q u 1k q q c 13 u 1k q 1 p u 2k p p c 23 u 2k p [ a 1 ( ũ 1k ) a 2 ( ũ 2k ) ] b(t)dt F 2 (t, ū 1k, ū 2k )dt 1 µ F 1 (t, λū 1k, λū 2k )dt for all k. It follows from (vi) and the boundedness of (ũ 1k, ũ 2k ) that (ū 1k, ū 2k ) is bounded. Hence ϕ has a bounded minimizing sequence(u 1k, u 2k ). Now heorem 2 follows from heorem 1.1 in [1] and Corollary 6. Proof of heorem 3. From (vii) and Sobolev s inequality it follows like in the proof of heorem 1 that ϕ(u 1, u 2 ) 1 q u 1 q q 1 p u 2 p ( p (h1 (t), h 2 (t)),(u 1 (t), u 2 (t)) ) dt γ(t)dt [ F 2 (t, ū 1, ū 2 )dt F2 (t, u 1 (t), u 2 (t)) F 2 (t, ū 1, ū 2 ) ] dt 1 2q u 1 q q c 11 u 1 α 11 q c 13 u 1 q c 12 ū 1 q α 1 1 2p u 2 p p c 21 u 2 α 21 p c 23 u 2 p c 22 ū 2 p α 2 F 2 (t, ū 1, ū 2 )dt γ(t)dt ũ 1 h 1 (t) dt ũ 2 h 2 (t) dt

9 Periodic solutions of differential systems with (q, p) Laplacian q u 1 q q c 11 u 1 α 11 q c 14 u 1 q 1 2p u 2 p p c 21 u 2 α 21 p c 24 u 2 p c max( c 12, c 22 ) ( ) ū 1 q α 1 ū 2 p α 2 F 2 (t, ū 1, ū 2 )dt = = 1 2q u 1 q q c 11 u 1 α 11 q c 14 u 1 q 1 2p u 2 p p c 21 u 2 α 21 p c 24 u 2 p c ( ) [ ū 1 q α 1 ū 2 p α 2 1 ] F ū 1 q α 1 ū2 p α 2 (t, ū 1, ū 2 )dt max( c 12, c 22 ) 2 for all(u 1, u 2 ) W and some positive constants c 11, c 14, c 21, c 24. Now follows like in the proof of heorem 1 that ϕ is coercive by (ix), which completes the proof. References [1] Jean Mawhin and Michel Willem - Critical Point heory and Hamiltonian Systems, Springer-Verlag, Berlin/New York, [2] Daniel Paşca - Periodic Solutions for Second Order Differential Inclusions, Communications on Applied Nonlinear Analysis, vol. 6, nr. 4 (1999) [3] Daniel Paşca - Periodic Solutions for Second Order Differential Inclusions with Sublinear Nonlinearity, PanAmerican Mathematical Journal, vol. 1, nr. 4 (2) [4] Daniel Paşca - Periodic Solutions of a Class of Non-autonomous Second Order Differential Inclusions Systems, Abstract and Applied Analysis, vol. 6, nr. 3 (21) [5] Daniel Paşca - Periodic solutions of second-order differential inclusions systems with p Laplacian, J. Math. Anal. Appl., vol. 325, nr. 1 (27) 9-1. [6] Daniel Paşca and Chun-Lei ang - Subharmonic solutions for nonautonomous sublinear second order differential inclusions systems with p Laplacian, Nonlinear Analysis: heory, Methods & Applications, vol. 69, nr. 3 (28) [7] Chun-Lei ang - Periodic Solutions of Non-autonomous Second-Order Systems with γ Quasisubadditive Potential, J. Math. Anal. Appl., 189 (1995), [8] Chun-Lei ang - Periodic Solutions of Non-autonomous Second Order Systems, J. Math. Anal. Appl., 22 (1996), [9] Chun-Lei ang - Periodic Solutions for Nonautonomous Second Order Systems with Sublinear Nonlinearity, Proc. AMS, vol. 126, nr. 11 (1998),

10 85 D. Paşca [1] Chun-Lei ang - Existence and Multiplicity of Periodic Solutions of Nonautonomous Second Order Systems, Nonlinear Analysis, vol. 32, nr. 3 (1998), [11] Yu ian, Weigao Ge - Periodic solutions of non-autonomous second-order systems with a p Laplacian, Nonlinear Analysis 66 (1) (27), [12] Jian Ma and Chun-Lei ang - Periodic Solutions for Some Nonautonomous Second-Order Systems, J. Math. Anal. Appl. 275 (22), [13] Xing-Ping Wu - Periodic Solutions for Nonautonomous Second-Order Systems with Bounded Nonlinearity, J. Math. Anal. Appl. 23 (1999), [14] Xing-Ping Wu and Chun-Lei ang - Periodic Solutions of a Class of Nonautonomous Second-Order Systems, J. Math. Anal. Appl. 236 (1999), [15] Xing-Ping Wu, Chun-Lei ang - Periodic Solutions of Nonautonomous Second- Order Hamiltonian Systems with Even-yped Potentials, Nonlinear Analysis 55 (23), [16] Fukun Zhao and Xian Wu - Saddle Point Reduction Method for Some Nonautonomous Second Order Systems, J. Math. Anal. Appl. 291 (24), [17] Chun-Lei ang and Xing-Ping Wu - Periodic Solutions for Second Order Systems with Not Uniformly Coercive Potential, J. Math. Anal. Appl. 259 (21), [18] Chun-Lei ang and Xing-Ping Wu - Notes on Periodic Solutions of Subquadratic Second Order Systems, J. Math. Anal. Appl. 285 (23), [19] Chun-Lei ang, Xing-Ping Wu - Subharmonic Solutions for Nonautonomous Second Order Hamiltonian Systems, J. Math. Anal. Appl. 34 (25), Department of Mathematics and Informatics, University of Oradea, University Street 1, 4187 Oradea, Romania dpasca@uoradea.ro