Periodic solutions for a class of second-order Hamiltonian systems of prescribed energy

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1 Electroic Joural of Qualitative Theory of Differetial Equatios 215, No. 77, 1 1; doi: /ejqtde Periodic solutios for a class of secod-order Hamiltoia systems of prescribed eergy Dog-Lu Wu 1, Chu Li 2 ad Pegfei Yua 2 1 College of Sciece, Southwest Petroleum Uiversity, Chegdu, 615, P.R. Chia 2 School of Mathematics ad Statistics, Southwest Uiversity, Chogqig 4715, P.R. Chia Received 2 April 215, appeared 14 November 215 Commuicated by Gabriele Boao Abstract. I this paper, the existece of o-costat periodic solutios for a class of coservative Hamiltoia systems with prescribed eergy is obtaied by the saddle poit theorem. Keywords: periodic solutios, prescribed eergy, Hamiltoia systems, saddle poit theorem. 21 Mathematics Subject Classificatio: 34C25, 58E5. 1 Itroductio ad mai results Cosider the secod order Hamiltoia system such that ü(t) + V(u) =, (1.1) 1 2 u(t) 2 + V(u) = h, (1.2) where V : R N R is a C 1 -map ad V(x) deotes the gradiet with respect to the x variable, (, ) : R N R N R deotes the stadard ier product i R N ad is the iduced orm. Furthermore, h stads for the total eergy of system (1.1). Hamiltoia systems have may applicatios i applied sciece. There are may papers [1 8, 1 12, 14, 15] which obtaied the existece of periodic ad coected orbits for (1.1). As we kow, alog with a classical solutio of (1.1), the total eergy is a costat. I 1978, uder some costrait o the eergy sphere, Rabiowitz [1] used variatioal methods to prove the existece of periodic solutios for a class of first order Hamiltoia systems with prescribed eergy. After the, the prescribed eergy problems have bee studied by may mathematicias [1 4, 6, 7, 11] usig geometric, topological or variatioal methods. I 1984, Beci [4] obtaied the followig theorem. Correspodig author. wudl28@163.com

2 2 D-L. Wu, C. Li ad P. Yua Theorem A ([4]). Suppose that V C 2 (R N, R) satisfies: (A 1 ) Ω := {x R N : V(x) < h} is o-empty ad bouded. The system (1.1) (1.2) has at least oe periodic solutio. As show i [4], coditio (A 1 ) is ecessary for the existece of periodic solutios of system (1.1) (1.2). However, the periodic solutio may be costat i Theorem A. The author eeded the followig coditio to obtai the existece of o-costat periodic solutios, which is (A 2 ) V(x) = for every x Ω. Furthermore, it is assumed that V is of C 2 class i Theorem A. Recetly, Zhag [15] has proved the existece of o-costat periodic solutios for system (1.1) (1.2) with V beig oly required to be of C 1 class. He got the followig theorem. Theorem B ([15]). Suppose that V C 1 (R N, R) satisfies: (B 1 ) there are costats µ 1 > ad µ 2 > such that (x, V(x)) µ 1 V(x) µ 2, (B 2 ) V(x) h ad V(x), as x +, (B 3 ) V(x) a x µ 1 + b, a >, b R, (B 4 ) lim sup x V(x) < h. x R N The for ay h > µ 2 /µ 1, system (1.1) (1.2) has at least a o-costat C 2 -periodic solutio. This result ca be obtaied by the saddle poit theorem of Beci Rabiowitz. I 212, Che ad Xue [6] proved the existece of periodic solutios for system (1.1) (1.2) uder some weaker assumptios. They cosidered the eergy h to be a parameter ad used mootoicity method to obtai the existece of periodic solutios. The they obtaied the followig theorem. Subsequetly, let V = lim if x + V(x). Theorem C ([6]). Suppose that V C 1 (R N, R) satisfies (B 1 ) ad the followig coditios (C 1 ) V achieves a global miimum V at x ; (C 2 ) V > V. The for all h ( µ 2 µ 1, V ), there exists a o-costat periodic solutio of eergy h. But coditio (B 1 ) is still eeded for provig the compactess coditio. Motivated by these papers, we will obtai the existece of periodic solutios for system (1.1) (1.2) uder some differet coditios. The followig theorem is our mai result. Theorem 1.1. Suppose that V = + ad V C 1 (R N, R) satisfies (V 1 ) (x, V(x)) > for ay x R N \ {}; (V 2 ) lim if x + (x, V(x)) >. The for ay h > V(), system (1.1) (1.2) possesses at least oe o-costat periodic solutio. Remark 1.2. I Theorem 1.1, the total eergy could be egative if V() is smaller tha zero which is differet from Theorem B ad Theorem C. Furthermore, there are fuctios satisfyig (V 1 ), (V 2 ) but ot the coditios (B 1 ) ad (B 3 ). For example, let V(x) = l( x 2 + 1) 1.

3 Periodic solutios for Hamiltoia systems of prescribed eergy 3 2 Variatioal settigs Let us set H 1 = W 1,2 (R/Z, R N ). Ad we defie the equivalet orm i H 1 as follows. ( 1/2 u = u(t) dt) 2 + u(). The maximum orm is defied by u := max t [,1] u(t). I order to deal with the prescribed eergy situatio, let f : H 1 R be the fuctioal defied by f (u) = 1 u(t) 2 dt (h V(u(t)))dt. (2.1) 2 This fuctioal has bee used by va Groese [14] to study the existece of brake orbits for smooth Hamiltoia systems with prescribed eergy ad by A. Ambrosetti ad V. Coti Zelati [1, 2] to study the existece of periodic solutios of sigular Hamiltoia systems. It ca easily be checked that f C 1 (H 1, R) ad f (u), u = u(t) 2 dt (h V(u(t)) 12 ) ( V(u(t)), u(t)) dt. (2.2) I this paper, we still make use of the saddle poit theorem itroduced by Beci ad Rabiowitz i [5] to look for the critical poits of f. First, we recall that a fuctioal I is said to satisfy the (PS) + coditio, if ay sequece {u } H 1 satisfyig f (u ) C ad f (u ) as, with ay C >, implies a coverget subsequece. Lemma 2.1 ([5]). Let X be a Baach space ad let f C(X, R) satisfy (PS) + coditio. Let X = X 1 X2, dim X 1 <, B a = {x X x a}, S = B ρ X2, ρ >, ) ( )) Q = (B L X1 ( B L X 1 R + e, L > ρ, where e X 2, e = 1, ) B L (X 1 R + e = {x 1 + se (x 1, s) X 1 R +, x s 2 = L 2 }, If ad Q = {x 1 + se (x 1, s) X 1 R, s, x s 2 L 2 }. the f possesses a critical value c α give by where f S α > f Q, c = if g Γ max x Q f (g(x)), Γ = {g C(Q, X), g Q = id}.

4 4 D-L. Wu, C. Li ad P. Yua The followig lemma shows that the critical poits of f are o-costat periodic solutios after beig scaled. Lemma 2.2. Let f be defied as i (2.1) ad q H 1 such that f ( q) =, f ( q) >. Set T 2 = 1 q(t) 2 dt 2. (h V( q(t))dt The ũ(t) = q(t/t) is a o-costat T-periodic solutio for (1.1) (1.2). Proof. The proof of this lemma is similar to Lemma 3.1 of [2]. Here we sketch the proof for the readers coveiece. Sice f ( q) =, we ca deduce that f ( q), ν = for all ν H 1 which ca be writte as ( q(t), ν(t))dt (h V( q(t)))dt = 1 q(t) 2 dt ( V( q(t)), ν(t))dt. (2.3) 2 The we divide equatio (2.3) by (h V( q(t)))dt which is positive sice f ( q) > ad obtai that ( q(t), ν(t))dt = T 2 ( V( q(t)), ν(t))dt for all ν H 1, which implies that 1 T 2 q(t) + V( q(t)) =. (2.4) This shows ũ(t) = q(t/t) satisfies (1.1). The coservatio of eergy for (2.4) shows that there exists a costat K such that 1 2T 2 q(t) 2 + V( q(t)) = K. (2.5) By the defiitio of T, we itegrate (2.5) o [,1] ad get that We fiish the proof of this lemma. K = 1 2T 2 q(t) 2 dt + V( q(t))dt = h. 3 Proof of Theorem 1.1 It is kow that the deformatio lemma ca be proved whe the usual (PS) + coditio is replaced by (CPS) C coditio (see Lemma 3.1 for the defiitio of (CPS) C ) which meas that Lemma 2.1 holds uder (CPS) C coditio with positive level. Subsequetly, we apply Lemma 2.1 to obtai the critical poits of f uder (CPS) C coditio for ay C >. Lemma 3.1. Suppose that the coditios of Theorem 1.1 hold, the f satisfies (CPS) C coditio which meas that for all C >, ad {u j } j N H 1 such that f (u j ) C, f (u j ) (1 + u j ) as j, (3.1) the sequece {u j } j N has a strogly coverget subsequece.

5 Periodic solutios for Hamiltoia systems of prescribed eergy 5 Proof. By (3.1), we ca deduce that C 2 f (u j) C + 1, f (u j ) (1 + u j ) C (3.2) for j large eough. The it follows from (2.1), (2.2) ad (3.2) that 3C f (u j ) + f (u j ) (1 + u j ) 2 f (u j ) f (u j ), u j = 1 2 u j 2 L ( V(u 2 j (t)), u j (t))dt. (3.3) If u j L 2 is ubouded, the we ca choose a subsequece, still deoted by { u j }, such that u j L 2 as j. The it follows from (3.3) that ( V(u j (t)), u j (t))dt as j. By (V 1 ), we ca see that ( V(u j (t)), u j (t)) as j for a.e. t [, 1]. There exists a set Λ [, 1] such that ( V(u j (t)), u j (t)) as j for all t Λ with meas Λ = 1, where meas deotes the Lebesgue measure. Combiig (V 1 ) ad (V 2 ), we deduce that ( V(x), x) = if ad oly if x = which implies that u j (t) as j, for all t Λ. (3.4) Otherwise, there exists β 1 > such that N >, there exists j N > N ad t N Λ such that It follows from (V 1 ) ad (V 2 ) that there exists θ > such that u jn (t N ) β 1. (3.5) ( V(x), x) θ for all x β 1. Sice ( V(u j (t)), u j (t)) as j for all t Λ, there exists η > such that for ay j > η ad t Λ we have Let N > η i (3.5), we ca obtai ( V(u j (t)), u j (t)) 1 θ. (3.6) 2 ( V(u jn (t N )), u jn (t N )) θ, which cotradicts (3.6). The we obtai (3.4). By Egorov s theorem, we ca see that there exists Λ 1 Λ such that u j (t) as j uiformly i Λ 1 (3.7) with meas Λ 1 ( 1 4, 1 2 ). By V C1 (R N, R), h > V() ad (3.7), we ca deduce that there exists l > such that V(u j (t)) V() + ε for j > l ad t Λ 1, where ε = h V() 2 >, which implies that h V(u j (t))dt = h V(u j (t))dt h V() ε dt 1 Λ 1 Λ 1 4 ε

6 6 D-L. Wu, C. Li ad P. Yua for j > l. By u j L 2 as j ad the defiitio of f, we ca deduce that f (u j ) + as j, which cotradicts (3.1). The we get that u j L 2 is bouded. Next, we claim that u j () is still bouded. Otherwise, there is a subsequece, still deoted by {u j }, such that u j () + as j +. Sice u j L 2 is bouded, by Hölder s iequality, we ca deduce that mi u j(t) u j () u j 2 t 1 L + as j. 2 The it follows from lim if x V(x) = + that there exist ζ > h ad r > such that for all x r. By the defiitio of f, it follows from (3.8) that V(x) ζ (3.8) f (u j ) = 1 u j (t) 2 dt (h V(u j (t)))dt 2 h ζ u j 2 2 L 2 for j large eough. which cotradicts (3.2). Hece u j () is bouded, which implies that u j is bouded. The there is a weakly coverget subsequece, still deoted by {u j }, such that u j u i H 1. The followig proof is similar to that i [15]. The we have u j u strogly i H 1. Hece f satisfies (CPS) C coditio. Subsequetly, we use Lemma 2.1 to prove that the fuctioal f possesses at least oe critical poit. Lemma 3.2. Suppose that the coditios of Theorem 1.1 hold, the fuctioal f possesses at least oe critical poit i H 1. Proof. We set that { X 1 = R N, X 2 = u W 1,2 (R/Z, R N ), { ( } 1 1/2 S = u X 2 u(t) dt) 2 = ρ, } u(t)dt =, P = { u(t) = u 1 + se(t), u 1 X 1, e X 2, e = 1, s R +, u = ( u s 2 ) 1/2 = L > ρ }, Q = {u 1 R N u 1 = L} P. For all u X 2, by Poicaré Wirtiger s iequality, we obtai that there exists a costat C 1 > such that u(t) 2 dt C 1 u 2. (3.9) Moreover, if u X 2, the Sobolev s iequality shows that 12 u u. (3.1)

7 Periodic solutios for Hamiltoia systems of prescribed eergy 7 Sice h > V(), there exists δ > such that h V(u(t)) h V() 2 for u δ. For ay u S X 2, we ca choose ( u(t) 2 dt ) 1/2 = ρ 12C1 δ, the we ca deduce from (3.9) ad (3.1) that u δ. Thus we have ad f (u) = 1 2 h V() 4 u(t) 2 dt (h V(u(t)))dt f S h V() ρ 2 >. 4 u(t) 2 dt Whe u Q, there are two cases eeded to be discussed. Case 1. If u {u 1 R N u 1 = L}, it follows from V = + that h V(u(t))dt, as L large eough, (3.11) which implies that f Q for L large eough. Case 2. If u P. For σ >, set The there exists ε 1 > such that Γ σ (u) = {t [, 1] : u(t) σ u }. meas (Γ ε1 (u)) ε 1 (3.12) for all u P. Otherwise, there exists a sequece {u } N P such that ( ) meas Γ 1 (u ) 1. (3.13) Set v = u u, the v = 1 for all N. The there exists a v X 1 spa{e} such that v = 1 ad v v i L 2 ([, 1], R N ). The we have v (t) v (t) 2 dt as. (3.14) Furthermore, we claim that there exist costats τ 1, τ 2 > such that meas(γ τ1 (v )) τ 2. (3.15) If ot, we have meas ( Γ 1 (v ) ) = for all N. The by Sobolev s embeddig theorem, we have v (t) 3 dt v L v 2 L 2 C 2 2 v L C2 2

8 8 D-L. Wu, C. Li ad P. Yua as, for some C 2 >, which cotradicts v = 1. The (3.15) holds. By (3.13) ad (3.15), we obtai ( meas Γ c 1 (v ) ) ( ( Γ τ1 (v ) = meas Γ τ1 (v ) \ meas (Γ τ1 (v )) meas τ 2 1, Γ 1 (v ) )) Γ τ1 (v ) ( Γ 1 (v ) Γ τ1 (v ) ) where Γ c 1 (v ) = [, 1] \ Γ 1 (v ). For large eough, we ca deduce that v (t) v (t) 2 v (t) v (t) 2 ( τ 1 1 ) τ2 1 for all t Γ c 1 (v ) Γ τ1 (v ). Cosequetly, for large eough, we have v (t) v (t) 2 dt 1 4 τ2 1 Γ c 1 (v ) Γ τ1 (v ) ( τ 2 1 ) v (t) v (t) 2 dt 1 8 τ2 1 τ 2 >, which cotradicts (3.14). The we obtai (3.12). By the defiitio of Γ ε1 (u), we coclude that for ay u P, we have if u(t) ε 1 u = ε 1 L + as L +. (3.16) t Γ ε1 (u) Sice V is of C 1 class ad V = +, there exists a global miimum V mi R such that V(x) V mi for ay x R N. It follows from (3.16) ad lim if x + V(x) = + that h V(u(t))dt = h V(u(t))dt [,1]\Γ ε1 (u) h (1 meas(γ ε1 (u)))v mi Γ ε1 (u) V(u(t))dt Γ ε1 (u) V(u(t))dt as L +, which implies (3.11). Together with Lemma 3.1, we ca deduce from Lemma 2.1 that f possesses a critical value c. Hece there exists a u H 1 such that The we fiish the proof of this lemma. c = f (u ) h V() ρ 2 >, f (u ) =. 4 Fially, it follows from Lemma 2.2 that system (1.1) (1.2) possesses at least oe ocostat periodic solutio. The we fiish the proof of Theorem 1.1.

9 Periodic solutios for Hamiltoia systems of prescribed eergy 9 Ackowledgemets This work was supported by the Natioal Natural Sciece Foudatio of Chia (No ), the Youg scholars developmet fud of Southwest Petroleum Uiversity (SWPU) (Grat No ) ad the Fudametal Research Fuds for the Cetral Uiversities (No. XDJK214B41). Refereces [1] A. Ambrosetti, V. Coti Zelati, Closed orbits of fixed eergy for sigular Hamiltoia systems, Arch. Ratioal Mech. Aal. 112(199), MR177264; url [2] A. Ambrosetti, V. Coti Zelati, Closed orbits of fixed eergy for a class of N-body problems, A. Ist. H. Poicaré Aal. No Liéaire 9(1992), MR [3] A. Ambrosetti, Critical poits ad oliear variatioal problems, Mém. Soc. Math. Frace (N.S.) 49(1992), MR [4] V. Beci, Closed geodesics for the Jacobi metric ad periodic solutios of prescribed eergy of atural Hamiltoia systems, A. Ist. H. Poicaré Aal. No Liéaire 1(1984), MR [5] V. Beci, P. H. Rabiowitz, Critical poit theorems for idefiite fuctioals, Ivet. Math. 52(1979), MR53761; url [6] C. Che, X. Xue, Periodic solutios for secod order Hamiltoia systems o a arbitrary eergy surface, A. Polo. Math. 15(212), MR ; url [7] H. Gluck, W. Ziller, Existece of periodic motios of coservative systems, i: Semiar o miimal submaifolds, A. of Math. Stud., Vol. 13, Priceto Uiv. Press, Priceto, NJ, 1983, pp MR [8] K. Hayashi, Periodic solutios of classical Hamiltoia systems, Tokyo J. Math. 6(1983), MR73299; url [9] J. Mawhi, M. Willem, Critical poit theory ad Hamiltoia systems, Applied Mathematical Scieces, Vol. 74, Spriger-Verlag, New York, MR982267; url [1] P. H. Rabiowitz, Periodic solutios of Hamiltoia systems, Comm. Pure Appl. Math. 31(1978), MR [11] P. H. Rabiowitz, Periodic ad heterocliic orbits for a periodic Hamiltoia system, A. Ist. H. Poicaré Aal. No Liéaire 6(1989), MR13854 [12] P. H. Rabiowitz, Periodic solutios of a Hamiltoia systems o a prescribed eergy surface, J. Differetial Equatios 33(1979), MR54373; url [13] P. H. Rabiowitz, Miimax methods i critical poit theory with applicatios to differetial equatios, CBMS, Regioal Cof. Ser. i Math., Vol. 65, Amer. Math. Soc., Providece, RI, MR845785

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