Research Article Antiperiodic Solutions for p-laplacian Systems via Variational Approach

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1 Applied Mahemaics Volume 214, Aricle ID , 7 pages hp://dx.doi.org/1.1155/214/ Research Aricle Aniperiodic Soluions for p-laplacian Sysems via Variaional Approach Lifang Niu and Kaimin eng Deparmen of Mahemaics, aiyuan Universiy of echnology, aiyuan, Shanxi 324, China Correspondence should be addressed o Lifang Niu; niulifangfly@163.com Received 2 June 214; Acceped 11 Augus 214; Published 24 Augus 214 Academic Edior: Xian Wu Copyrigh 214 L. Niu and K. eng. his is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. We esablish he exisence of soluions for p-laplacian sysems wih aniperiodic boundary condiions hrough using variaional mehods. 1. Inroducion In his paper we invesigae he exisence of soluions of he following second order p-laplacian sysems wih aniperiodic boundary condiion: ( u p 2 u ) = F(, u ) a.e. [, ], u () = u(), u () = u (), where >, F:[,] R N R (N 1) saisfies he following fundamenal assumpion: (H) F(, x) is measurable in for each x R N,coninuous differeniable in x for almos every [,],and here exiss a C(R +, R + ) and b L 1 (, ; R + ) such ha F (, x) a( x ) b, F (, x) a( x ) b (2) for all x R N and a.e. [,]. In he las few decades, he following second order sysems involving periodic boundary condiion: u= F(, u ) a.e. [, ], u () =u(), u () = u (), have aced as one of he mainsream research problems in he field of differenial equaion. Under various assumpions of (1) (3) he poenial F(, x), here have been los of exisence and mulipliciy of resuls in he lieraures by using he ool of nonlinear analysis, such as degree heory, minimax mehods, andmorseheory.herewedonoevenryoreviewhe huge bibliography, bu we only lis some references for our purpose; for example, we refer he readers o see [1 6] and he references herein. Comparing problem (1)wihproblem(3), we observe ha he only difference is he boundary condiions. In order o use he variaional mehods, one of he main difficulies is he variaional principle. For his maer, we ry o modify some work space such ha he variaional principle can be esablished. hanks o he work of ian and Henderson [7], we borrow heir ideas o give he variaional principle for problem (1). Noe ha he sudy of aniperiodic soluions for nonlinear differenial sysems of he form ( u p 2 u ) = F(, u ) a.e. R (4) is closely relaed o he sudy of is periodic soluions. Indeed, if we assume ha F( +, x) = F(, x) and F(, x) = F(, x),henle u, [, ], V ={ (5) u ( ), [, 2], and we ge ha V is a soluion of sysems (4) wih condiions V() = V(2) and V () = V (2).SinceF(, x) is 2-periodic

2 2 Applied Mahemaics in, V can be exended o be 2-periodic over R, and hence V is a 2-periodic soluion of sysems (4). In his paper, we will esablish he exisence of soluions for problem (1) by variaional mehod. As far as we know, here are few papers sudying he second order sysems wih aniperiodic boundary condiions by variaional mehods. his paper is organized as follows. In Secion 2, we inroduce a variaional principle for problem (1). In Secion 3, we prove our main resuls. 2. Variaional Principle In he sequel, we denoe by p he L p -norm (1 p ). Le C (, ; R N ) be he space of infiniely differeniable funcions from [, ] ino R N.LeX={u C (, ; R N ), u() = u()};obviously,cos ((2k+1)π/) and sin(((2k+1)π/)+ π/2) (k=,1,2,...)belongox,andhusx =. he following fundamenal lemma proved in [7]isessen- ial o esablish he variaional principle. Lemma 1. Le u, V L 1 (, ; R N ).If,foreveryf X, (u,f ) d = (V,f) d, (6) where (, ) denoes he inner produc on R N,hen 2 [u +V ] d = V d (7) and here exiss c R N such ha u = V (s) ds + c a.e. on [, ]. (8) Remark 2. From Lemma 1,wehavehefollowingfacs. (i) A funcion V saisfying (6) is called a weak derivaive of u. By a Fourier series argumen, he weak derivaive, if i exiss, is unique. We denoe by u he weak derivaive of u. (ii) We will idenify he equivalence class u and is coninuous represenaion u = u (s) ds + c. (9) (iii) Equaions (7) and(8) imply ha u() = u() = c. For his maer, we only show ha u()+c =. Indeed, using inegraion by pars, we have u () = V (s) ds + c = 2 [u +V ] d + c = 2 u d + 2 = 2 u d + 2 =2u() +c. V d 2 V (s) ds d + c V d 2 u d + 3c (1) By (9), we have u =u(τ) + u (s) ds, for, τ [, ]. (11) τ (iv) If u is coninuous on [, ],henby(9) u is he classical derivaive of u= u. (v) I follows from (9) and Rademacher heorem ha u is he classical derivaive of u a.e. on [, ]. he Sobolev space is he space of funcions u L p (, ; R N ) having a weak derivaive u L p (, ; R N ). Obviously, if u, u = u (s)ds + c and u() = u() = c.henormover is defined by u I is easy o see ha =( u p 1/p d + d). (12) is a reflexive Banach space and X Ẇih he proof of Lemma 3.3 in [7] and heorem 8.8 in [8], we have he following embedding heorem. Lemma 3. Le 1/p + 1/q = 1 (1 < p < + ).hen (i) he embedding Lq (, ; R N ) is compac; (ii) hereexisaconsan c>such ha Moreover, he embedding u c u. (13) C(, ; RN ) is compac. Proof. From heorem 8.2 in [8], for u, u = u() + u (s)ds and u = u() u (s)ds. ByHölder s inequaliy, we have u = 1 2 [u () +u() + 1 1/p 2 1/q ( d). u (s) ds u (s) ds] (14) On he oher hand, leing B be a uni ball in,foru B, we have u u(s) = u (τ) dτ s u p s 1/q s 1/q,,s (, ). (15) I follows from he Ascoli-Arzelà heoremhab has a compac closure in C(, ; R N ). By Lemma 1,henorm norm defined as in is equivalen o he 1/p u =( d). (16)

3 Applied Mahemaics 3 Indeed, by (14), one has u p d u p 1+p/q herefore, we have 2 1,p u W +2 2 p d. (17) u u, u (18) and he equivalence of he norm is proved. Remark 4. here is some differences beween aniperiodic boundary value problem and periodic boundary value problem. One is he norm on he work space, and he oher is he decomposiion of space where W 1,p = {u W 1,p : known ha he norms u p and u 1,p W W 1,p. no like W 1,p = R N W 1,p, ud = }. Andiiswell are equivalen o he energy funcional φ: R corresponding o problem (1) is defined by φ (u) = 1 p d F (, u ) d. (19) Under he assumpion of (H),we have he following. Lemma 5. Le (H) hold. hen φ C 1 (, R) and φ (u), V = 2 (u, V )d ( F (, u ), V ) d (2) for every V.Moreover,ifφ (u) =,henu is a soluion of problem (1); hais,u saisfies he equaion and aniperiodic condiion in (1). Proof. Similarlyasheproofofheorem1.4in[1], we obain ha φ C 1 (, R) and (28)holds.If = φ (u), V = 2 (u, V )d ( F (, u ), V ) d =, (21) for all V and hence for all V X,byLemma 1, here exiss a consan c R N such ha 2 u = F (s, u (s)) ds + c, a.e. on [, ]. (22) From Remark 2, weknowha u p 2 u has a weak derivaive ( u p 2 u ) = F(, u ) a.e. on [, ]. (23) By Remark 2,wege 2 () u () = 2 () u (). (24) By (24), if u () = u () =,clearly,wehaveu () = u (). If no, we see ha u () u () <.Hence,bycalculaion,we ge u () = u (). (25) his implies ha u () = u () andhenceheconclusionis proved. Lemma 6. he funcional φ : semiconinuous. R is weakly lower Proof. Assuming u n uin, hen by (ii) of Lemma 3,we have u n uin C([, ]; R N ).Byhypohesis(H),wehave [F (, u n ) F(, u )]d = 1 ( F (, u +s(u n u)), u n u)dsd 1 b a( u+s(u n u) ) u n u ds d max a (s) b L s [,M] 1 u n u L, (26) where M>is a consan such ha u n +s(u n u) Mfor every n N, alls [,1],and [,]. I follows from he weak lower semiconinuiy of he norm funcion in Banach space ha lim inf n u n p d d. (27) Accordingly, he conclusion is compleed. Nex, we sudy he eigenvalue problem ( u p 2 u ) =λ u p 2 u, [, ], u () = u(), u () = u (). (28) Definiion 7. One says λ Ris an eigenvalue of problem (28) if here exiss u, u,suchha ( 2 u, V )d =λ ( u p 2 u, V )d, for every V. (29)

4 4 Applied Mahemaics Lemma 8. he eigenvalue problem (28) possesses he following properies: (a) all he eigenvalues are posiive real numbers; (b) for each u,onehasλ 1 u p d u p d,whereλ 1 = inf u, u L p =1 u p d. Proof. (a) Leing e λ be he eigenfuncion corresponding o he eigenvalue λ,wehave e λ p d = λ e λ p d. (3) Hence, λ and λ=if and only if e λ =, implying ha e λ = C,a.e.on[, ],wherecis a consan. Since e λ (u() = u()), we see ha e λ, forall [,];his conradics wih he definiion of eigenfuncion. (b) By sandard minimizaion argumens, we can prove he conclusion. We omi i. Remark 9. Noe ha he eigenvalue of one dimensional vecor p-laplacian operaor under aniperiodic boundary condiion possesses some similar properies as he p- Laplacian operaor Δ p under Dirichle boundary condiion on bounded domain. 3. Main Resuls and Proof In his secion, we will give some exisence and mulipliciy of resuls for problem (1). heorem 1. Assume ha F(, u) saisfies hypohesis (H) and he following condiions. (F ) here exis f L 1 ([, ]; R + ) and g L 1 ([, ]; R + ) such ha F (, x) f x α +g, x R N, a.e. [, ] (31) wih α<p.henproblem(1) has a leas one soluion in. Proof. Firs we show ha φ is coercive on.infac,from (F ) and Lemma 3,wehave φ (u) = 1 p d F (, u ) d 1 p d u α L f d g d 1 p α/p d C( d) C. (32) herefore, φ(u) + as u. Asaresul, we ge a bounded minimizing sequence in.combining Lemma 6, byheorem1.1andcorollary1.1in[1], problem (1) has a leas one soluion which minimizes φ on. Remark 11. By hypohesis (H), we see ha F(, x) is summable over [,]in he neighborhood of zero, and hus he condiion (F ) canbeweakenoholdfor x large. If α=p, we may assume he funcion f in (F ) saisfies ha f d < 2p, (33) pp 1 ashesameproofofheorem 1, andwecangehefollowing heorem. heorem 12. Under he above assumpions of F(, x), hen problem (1) has a leas one soluion in. heorem 13. Assume ha F(, u) saisfies hypohesis (H) and he following condiions: (F 1 ) lim x (F(, x)/ x p ) =, and lim x + (F(, x)/ x p ) = + uniformly for a.e. [,]; (F 2 ) here exis consans r>pand μ>r psuch ha lim sup x + (F(, x)/ x r )<+ uniformly for a.e. [,], and lim inf x + ((( F(, x), x) pf(, x))/ x μ )>uniformly for a.e. [,]. hen problem (1) possesses a leas one nonrivial soluion in. In order o prove heorem 13, we need he following resuls. Lemma 14. Suppose (H), (F 1 ),and(f 2 ) hold. hen funcional φ saisfies he (C)-condiion; ha is, for every sequence {u n }, {u n} has a convergen subsequence if φ(u n )is bounded and (1 + u n ) φ (u n ) as n. Proof. Suppose {u n }, φ(u n) is bounded, and (1 + u n ) φ (u n ) as n, and hen here exiss a consanm >such ha φ(u n) M, (1+ u n ) φ (u n ) M (34) for every n N. On one hand, by (F 2 ), here exis consans C>and δ>such ha F (, x) C x r, x δ, a.e. [, ]. (35) I follows from (H) and (35)ha F (, x) max s [,δ] a (s) b +C x r, x R N, a.e. [, ]. (36)

5 Applied Mahemaics 5 Hence, by (34), (36), and Hölder inequaliy, we have 1 p u n p =φ(u n )+ F(,u n )d C u n r d + C. On he oher hand, by (F 2 ), here are consans C>and δ 1 >such ha ( F (, x),x) pf(, x) C x μ, (38) x δ 1, a.e. [, ]. By (H),onehas (37) ( F (, x),x) pf(, x) Cb, (39) x δ 1, a.e. [, ], where C=(p+δ 1 ) max s [,δ1 ]a(s).hence,from(38)and(39), we ge ( F (, x),x) pf(, x) C( x μ δ μ 1 ) Cb, (4) for all x R N and a.e. [,].Hence,by(34) and(4), one has (p + 1) M pφ (u n ) φ (u n ),u n = [( F (, u n ),u n ) pf(,u n )] d C u n μ d Cδ μ 1 C b d. (41) So {u n } is bounded in L μ (, ; R N ).Ifμ r,by(37) and Hölder inequaliy, i is easy o obain ha {u n } is bounded in.ifμ<r,bylemma 3,wehave u n r d u n r μ L u n μ d (42) C u n r μ u n μ d. Hence, by (37) and] >r p,weobain{u n } is bounded in oo. hus, {u n} is bounded in 1,p.Since W is a reflexive Banach space, by Lemma 3, here exiss a subsequence, sill denoed by {u n },suchha u n uin, u n u in C([, ] ; R N ), as n. (43) Nex, we will show ha u n u in. Indeed, from (43) and hypohesis (H),iiseasyoobainha φ (u n ) φ (u),u n u, as n, ( F (, u n ) F(, u ),u n u)d, as n. (44) Hence, by (44), we ge ( u n p 2 u n 2 u,u n u )d, as n. (45) herefore, i follows from (45) and Lemmas 3.2 and 3.3 in [9] ha u n u in. he proof of Lemma 14 is compleed. Proof of heorem 13. By (F 1 ),foreveryε>, here exiss δ 2 > such ha F (, x) ε x p, x δ 2, a.e. [, ]. (46) Combined wih (35)and(46), we have F (, x) ε x p +C x r, x R N, a.e. [, ]. (47) hus, by Lemma 3,foru,onehas φ (u) = 1 p d F (, u ) d (48) ( 1 p εc) u p C u r. Since r>p, hen here exis ρ>and α>such ha φ (u) α, u wih u =ρ. (49) On he oher hand, by (F 1 ),foranym 1 >, here exiss δ 3 > such ha F (, x) M 1 x p, x δ 3, a.e. [, ]. (5) I follows from (H) ha F (, x) max s [,δ 3] x δ 3, a.e. [, ]. (51) herefore, we obain F (, x) M 1 ( x p δ p 3 ) max s [,δ 3] (52) for all x R N and a.e. [,]. Now, we choose e being he eigenfuncion corresponding o λ 1 which is defined in Lemma 8.Forζ>, from (52), we have φ(ζe )= ζp p e p d F(,ζe )d ζp p e p d M1 ζ p e p d + M 1 δ p 3 +C =ζ p ( 1 p M 1 λ 1 ) e p +M 1 δ p 3 +C. (53) We choose M 1 =2(λ 1 /p), and he above inequaliy implies ha φ(ζe ), as ζ +. (54)

6 6 Applied Mahemaics In view of Lemma 14, (49), and (54), noing ha φ() = and applying he mounain pass heorem under he (C)- condiion, here exiss a criical poin u of φ, such ha φ(u) α.henceu is a nonrivial soluion of problem (1), and his complees he proof. Wecanweakenhecondiion(F 1 ) in he following condiion (F 1 ): pf (, x) pf (, x) lim sup x x p <λ 1, lim inf x + x p >λ 1 (55) uniformly for a.e. [,]. heorem 15. Suppose ha F(, x) saisfies hypoheses (H), (F 1 ),and(f 2).henproblem(1) has a leas one nonrivial soluion in. Proof. Checking he proof of heorem 13, we only need o verify ha (49)and(54)hold.Infac,by(F 1 ), here exis wo consans γ<λ 1 and δ 4 >such ha F (, x) γ p x p, x δ 4, a.e. [, ]. (56) From (35)and(56), we have F (, x) γ p x p +C x r, x R N, a.e. [, ]. (57) hus, for u,onehas φ (u) = 1 p d F (, u ) d 1 p (1 γ λ 1 ) u p C u r. (58) Since r>pand 1 γ/λ 1 >,hen(49)holds. By (F 1 ), here exis wo consans γ 1 >λ 1 and δ 5 >such ha F (, x) γ 1 p x p, x δ 5, a.e. [, ]. (59) Hence, by (H) and (59), we ge F (, x) γ 1 p ( x p δ p 5 ) max a (s) b (6) s [,δ 5] for all x R N and a.e. [,]. herefore, (6)andγ 1 >λ 1 imply ha φ(ζe ), as ζ +. (61) Hence, we complee he proof. Nex, we consider he asympoically quadraic case. For his purpose, we suppose F saisfies he following condiions: (F 3 ) lim sup x + (F(, x)/ x p )=+ uniformly for a.e. [,]; (F 4 ) here exiss l L 1 (, ; R + ) such ha ( F(, x), x) pf(, x) l for all x R N and a.e. [,]and lim x + [( F (, x),x) pf(, x)] =+ (62) uniformly for a.e. [,]. heorem 16. Suppose (H), (F 1 ), (F 3 ),and(f 4 ) hold. hen problem (1) possesses a leas one nonrivial soluion in. Proof. Paralleling o he proof of heorem 13, we only need o verify ha φ saisfies he (C)-condiion. Suppose {u n }, φ(u n ) is bounded, and (1 + u n ) φ (u n ) as n. Hence, i is easy o ge ha [( F (, u n ),u n ) pf(,u n )] d (63) =pφ(u n ) φ (u n ),u n C. We nex show ha {u n } is bounded in.ifno,wihou loss of generaliy, we can assume ha u n + as n. LeingV n =u n / u n,hen V n =1,andsogoingoa sequence if necessary, we assume ha V n V in and V n V in C(, ; R N ). By (F 3 ), here exis wo consans a 1 >and R 1 >such ha F (, x) a 1 x p, x R 1, a.e. [, ]. (64) From (H) and (64), we ge F (, x) a 1 x p +b max s [,R 1] a (s) (65) for all x R N and a.e. [,].Hence,by(65), one has φ(u n ) u n p = 1 p 1 u n p F(,u n )d (66) 1 p a 1 V n p d C u n p, which implies ha V L p C> (67) and hus V. herefore, here exiss a subse E [,] wih meas(e) > such ha V =on E. By (F 4 ),fromlemma2in[5], hen for ε = (1/2) meas(e) >, here exiss a subse E ε [,]wih meas([, ]\E ε )<ε such ha lim [( F (, x),x) pf(, x)] =+ (68) x + uniformly for E ε.obviously,meas(e E ε )>.Ifno,we assume meas(e E ε )=.SinceE=(E E ε ) (E\E ε ),hus we have < meas (E) meas (E E ε )+meas ([, ] \E ε ) <ε= 1 2 meas (E), (69)

7 Applied Mahemaics 7 which leads o a conradicion. Hence, we have proved ha u n +, as n, for a.e. E E ε. (7) From (F 4 ),wehave [( F (, u n ),u n ) pf(,u n )] d = E E ε [( F (, u n ),u n ) pf(,u n )] d [7] Y. ian and J. Henderson, hree ani-periodic soluions for second-order impulsive differenial inclusions via nonsmooh criical poin heory, Nonlinear Analysis: heory, Mehods & Applicaions,vol.75,no.18,pp ,212. [8] H. Brezis, Funcional analysis, Sobolev Spaces and Parial Differenial Equaions, Springer, 211. [9]M.M.Boureanu,B.Noris,andS.erracini, Subandsupersoluions, invarian cones and mulipliciy resuls for p-laplace equaions, In press, hp://arxiv.org/abs/ v1. + [,]\E E ε [( F (, u n ),u n ) pf(,u n )] d E E ε [( F (, u n ),u n ) pf(,u n )] d + l d. [,]\E E ε (71) By (68)and(7), we ge [( F (, u n ),u n ) pf(,u n )] d + (72) as n.hisconradicswih(63). he proof is compleed. Conflic of Ineress he auhors declare ha here is no conflic of ineress regarding he publicaion of his paper. Acknowledgmens Kaimin eng is suppored by he NSFC under Gran and he Shanxi Province Science Foundaion for Youhs under Gran References [1] J. Mawhin and M. Willem, Criical Poin heory and Hamilonian Sysems, Springer, New York, NY, USA, [2] M. Schecher, Periodic non-auonomous second-order dynamical sysems, Differenial Equaions, vol.223, no. 2, pp , 26. [3] C. ang, Periodic soluions for nonauonomous second order sysems wih sublinear nonlineariy, Proceedings of he American Mahemaical Sociey,vol.126,no.11,pp ,1998. [4] C. ang and X. Wu, Some criical poin heorems and heir applicaions o periodic soluion for second order Hamilonian sysems, Differenial Equaions, vol. 248, no. 4, pp , 21. [5] C.-L. ang and X.-P. Wu, Periodic soluions for second order sysems wih no uniformly coercive poenial, Mahemaical Analysis and Applicaions,vol.259,no.2,pp , 21. [6] X. Wu, Saddle poin characerizaion and mulipliciy of periodic soluions of non-auonomous second-order sysems, Nonlinear Analysis: heory, Mehods and Applicaions, vol.58, no. 7-8, pp , 24.

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