Existence and concentration of solution for a fractional Hamiltonian systems with positive semi-definite matrix
aa r X i v : . [ m a t h . A P ] A ug EXISTENCE AND CONCENTRATION OF SOLUTION FOR AFRACTIONAL HAMILTONIAN SYSTEMS WITH POSITIVESEMI-DEFINITE MATRIX
C´ESAR TORRES, ZIHENG ZHANG, AND AMADO MENDEZ
Abstract.
We study the existence of solutions for the following fractional Hamiltoniansystems ( − t D α ∞ ( −∞ D αt u ( t )) − λL ( t ) u ( t ) + ∇ W ( t, u ( t )) = 0 ,u ∈ H α ( R , R n ) , (FHS) λ where α ∈ (1 / , t ∈ R , u ∈ R n , λ > L ∈ C ( R , R n ) is a symmetric matrixfor all t ∈ R , W ∈ C ( R × R n , R ). Assuming that L ( t ) is a positive semi-definite symmetricmatrix for all t ∈ R , that is, L ( t ) ≡ T of R , W ( t, u )satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, weshow that (FHS) λ has a solution which vanishes on R \ T as λ → ∞ , and converges to some˜ u ∈ H α ( R , R n ). Here, ˜ u ∈ E α is a solution of the Dirichlet BVP for fractional systems on thefinite interval T . Our results are new and improve recent results in the literature even in thecase α = 1. Introduction
Fractional Hamiltonian systems are a significant area of nonlinear analysis, since they appearin many phenomena studied in several fields of applied science, such as engineering, physics,chemistry, astronomy and control theory. On the other hand, the theory of fractional calculusis a part that intensively developing during the last decades; see [1, 10, 11, 16, 19, 23] andthe references therein. The existence of homoclinic solutions for Hamiltonian systems andtheir importance in the study of behavior of dynamical systems can be recognized fromPoincar´e [22]. Since then, the investigation of the existence and multiplicity of homoclinicsolutions became one of the main important problems of research in dynamical systems. Criticalpoint theorem was first used by Rabinowitz [24] to obtain the existence of periodic solutionsfor first order Hamiltonian systems, while the first multiplicity result is due to Ambrosettiand Zelati [2]. Therefore, a large number of mathematicians used critical point theory andvariational methods to prove the existence of homoclinic solutions for Hamiltonian systems;see for instance [5–7, 12, 13, 21, 25, 26] and the reference therein.The critical point theory has become an effective tool in investigating the existenceand multiplicity of solutions for fractional differential equations by constructing fractionalvariational structures. Especially, in [15] the authors firstly dealt with a class of fractionalboundary value problem via critical point theory. From then on, Variational methods andcritical point theory are shown to be effective in determining the solutions for fractionaldifferential equations with variational structure. We also mention the work by Torres [30],
Mathematics Subject Classification.
Primary 34C37; Secondary 35A15, 35B38.
Key words and phrases.
Fractional Hamiltonian systems, Fractional Sobolev space, Critical point theory,Concentration phenomena. where the author considered the following fractional Hamiltonian systems (cid:26) t D α ∞ ( −∞ D αt u ( t )) + L ( t ) u ( t ) = ∇ W ( t, u ( t )) ,u ∈ H α ( R , R n ) , (FHS)where α ∈ (1 / , t ∈ R , u ∈ R n , L ∈ C ( R , R n ) is a symmetric and positive definite matrixfor all t ∈ R , W ∈ C ( R × R n , R ) and ∇ W ( t, u ) is the gradient of W ( t, u ) at u . Assuming that L ( t ) satisfied the following coercivity condition(L) there exists an l ∈ C ( R , (0 , ∞ )) with l ( t ) → ∞ as | t | → ∞ such that(1.1) ( L ( t ) u, u ) ≥ l ( t ) | u | for all t ∈ R and u ∈ R n . and that W ( t, u ) satisfies the Ambrosetti-Rabinowitz condition(FHS ) W ∈ C ( R × R n , R ) and there is a constant θ > < θW ( t, u ) ≤ ( ∇ W ( t, u ) , u ) for all t ∈ R and u ∈ R n \{ } , and other suitable conditions, the author showed that (FHS) possesses at least one nontrivialsolution via Mountain Pass Theorem. Note that (FHS) , implies that W ( t, u ) is ofsuperquadratic growth as | u | → ∞ . Since then, many researchers dealt with (FHS) for thecases that W ( t, u ) is superquadratic or subquadratic at infinity; see for instance [18, 36, 38]. Inaddition, some perturbed fractional Hamiltonian systems are discussed in [31, 36].In [39] the authors focused on weakening the coercivity condition ( L ), more precisely theyassumed that L ( t ) is bounded in the following sense:(L) ′ L ∈ C ( R , R n ) is a symmetric and positive definite matrix for all t ∈ R and there areconstants 0 < τ < τ < ∞ such that τ | u | ≤ ( L ( t ) u, u ) ≤ τ | u | for all ( t, u ) ∈ R × R n , By supposed that W ( t, u ) is subquadratic as | u | → + ∞ , the authors also showed that (FHS)possessed infinitely many solutions, which has been generalized in [20, 41].In the present paper we deal with the following fractional Hamiltonian systems (cid:26) − t D α ∞ ( −∞ D αt u ( t )) − λL ( t ) u ( t ) + ∇ W ( t, u ( t )) = 0 ,u ∈ H α ( R , R n ) , (FHS) λ where α ∈ (1 / , t ∈ R , u ∈ R n , λ > W ∈ C ( R × R n , R ) and L satisfiesthe following conditions( L ) L ∈ C ( R , R n × n ) is a symmetric matrix for all t ∈ R ; there exists a nonnegativecontinuous function l : R → R and a constant c > L ( t ) u, u ) ≥ l ( t ) | u | , and the set { l < c } := { t ∈ R | l ( t ) < c } is nonempty with meas { l < c } < C ∞ ,where meas {·} is the Lebesgue measure and C ∞ is the best Sobolev constant for theembedding of X α into L ∞ ( R );( L ) J = int ( l − (0)) is a nonempty finite interval and J = l − (0);( L ) there exists an open interval T ⊂ J such that L ( t ) ≡ t ∈ T .In particular, if α = 1 in (FHS) λ , then it reduces to the following well-known second orderHamiltonian systems ¨ u − λL ( t ) u + ∇ W ( t, u ) = 0 . (HS)Recently a second order Hamiltonian systems like (HS) with positive semi-definite matrix wasconsidered in [29]. Assuming that W ∈ C ( R × R n , R ) is an indefinite potential satisfyingasymptotically quadratic condition at infinity on u , Sun and Wu, with a little mistake in their RACTIONAL HAMILTONIAN SYSTEMS WITH POSITIVE SEMI-DEFINITE MATRIX 3 embedding results, have proved the existence of two homoclinic solutions of (FHS λ ). For morerelated works, we refer the reader to [5, 7, 12, 13, 21, 26] and the references mentioned there.Here we must point out, to obtain the existence or multiplicity of solutions for Hamiltoniansystems, all the papers mentioned above need the assumption that the symmetric matrix L ( t )is positive definite, see (L) and (L) ′ . Therefore, recently the authors in [4, 35, 40] consideredthe case that L ( t ) is positive semi-definite satisfying ( L ) . In [4], the author dealt with (FHS)for the case that ( L ) is satisfied and W ( t, u ) involves a combination of superquadratic andsubquadratic terms and is allowed to be sign-changing. In [35, 40], we have considered theexistence of solutions of (FHS) λ and the concentration of its solutions when ( L ) -( L ) aresatisfied and W ( t, u ) meets with some classes of superquadratic hypothesis.Motivated by these previous results, the main purpose of this paper is to investigate (FHS) λ without Ambrosetti-Rabinowitz condition (FHS ). More precisely, we suppose that W ( t, u )satisfy the following assumptions( W ) |∇ W ( t, u ) | = o ( | u | ) as | u | → t ∈ R .( W ) W ( t, u ) ≥ t, u ) ∈ R × R N and H ( t, u ) ≥ t, u ) ∈ R × R N , where H ( t, u ) := 12 h∇ W ( t, u ) , u i − W ( t, u ) . ( W ) W ( t,u ) | u | → + ∞ as | u | → + ∞ uniformly in t ∈ R .( W ) There exist C , R >
0, and σ > |∇ W ( t, u ) | σ | u | σ ≤ C H ( t, u ) if | u | ≥ R. Note that, according to [6] the nonlinearity W ( t, u ) = g ( t )( | u | p + ( p − | u | p − ǫ sin ( | u | ǫ ǫ )) , where g ( t ) > T -periodic in t , 0 < ǫ < p − p >
2, satisfies ( W ) − ( W ), but (FHS )is not satisfied.Now we are in the position to state our main result. Theorem 1.1.
Suppose that ( L ) - ( L ) , ( W ) − ( W ) are satisfied, then there exists Λ ∗ > such that for every λ > Λ ∗ , (FHS) λ has at least one nontrivial solution. On the concentration of solutions obtained above, for technical reason, we consider thatthere exists 0 < ̺ < + ∞ , such that T = [ − ̺, ̺ ], where T is given by ( L ) . We have thefollowing result. Theorem 1.2.
Let u λ be a solution of problem (FHS) λ obtained in Theorem 1.1, then u λ → ˜ u strongly in H α ( R ) as λ → ∞ , where ˜ u is a nontrivial solution of the following boundary valueproblem ( t D α̺ ( − ̺ D αt ) u = ∇ W ( t, u ) , t ∈ ( − ̺, ̺ ) u ( − ̺ ) = u ( ̺ ) = 0 , (1.2) where − ̺ D αt and t D α̺ are left and right Riemann-Liouville fractional derivatives of order α on [ − ̺, ̺ ] respectively. Remark 1.
In Theorem 1.1, we give some new superquadratic conditions on W ( t, u ) toguarantee the existence of solutions and investigate the concentration of these solutions in1.2. However, we must point out that the methods in [4, 35, 40] are not be valid for our newassumptions. To overcome this difficulty we apply the Mountain Pass Theorem with Ceramicondition, however, the direct application of the mountain pass theorem is not enough since C´ESAR TORRES, ZIHENG ZHANG, AND AMADO MENDEZ the Cerami sequences might lose compactness in the whole space R . Then it is necessary tointroduce a new compactness result to recover the convergence of Cerami sequence, for moredetails see Lemma 3.3.The remaining part of this paper is organized as follows. Some preliminary results arepresented in Section 2. In Section 3, we are devoted to accomplishing the proof of Theorem1.1 and in Section 4 we present the proof of Theorem 1.2.2. Preliminary Results
In this section, for the reader’s convenience, firstly we introduce some basic definitions offractional calculus. The Liouville-Weyl fractional derivative of order 0 < α < −∞ D αx u ( x ) = ddx −∞ I − αx u ( x ) and x D α ∞ u ( x ) = − ddx x I − α ∞ u ( x ) . where −∞ I αx and x I α ∞ are the left and right Liouville-Weyl fractional integrals of order 0 < α < −∞ I αx u ( x ) = 1Γ( α ) Z x −∞ ( x − ξ ) α − u ( ξ ) dξ and x I α ∞ u ( x ) = 1Γ( α ) Z ∞ x ( ξ − x ) α − u ( ξ ) dξ. Furthermore, for u ∈ L p ( R ), p ≥
1, we have F ( −∞ I αx u ( x )) = ( iω ) − α b u ( ω ) and F ( x I α ∞ u ( x )) = ( − iω ) − α b u ( ω ) , and for u ∈ C ∞ ( R ), we have F ( −∞ D αx u ( x )) = ( iω ) α b u ( ω ) and F ( x D α ∞ u ( x )) = ( − iω ) α b u ( ω ) , In order to establish the variational structure which enables us to reduce the existence ofsolutions of (FHS) λ to find critical points of the corresponding functional, it is necessary toconsider some appropriate function spaces. Denote by L p ( R , R n ) (1 ≤ p < ∞ ) the Banachspaces of functions on R with values in R n under the norms k u k L p = (cid:16)Z R | u ( t ) | p dt (cid:17) /p , and L ∞ ( R , R n ) is the Banach space of essentially bounded functions from R into R n equippedwith the norm k u k ∞ = ess sup {| u ( t ) | : t ∈ R } . Let −∞ < a < b < + ∞ , 0 < α ≤ < p < ∞ . The fractional derivative space E α,p isdefined by the closure of C ∞ ([ a, b ] , R n ) with respect to the norm(2.2) k u k α,p = (cid:18)Z ba | u ( t ) | p dt + Z ba | a D αt u ( t ) | p dt (cid:19) /p , ∀ u ∈ E α,p . Furthermore ( E α,p , k . k α,p ) is a reflexive and separable Banach space and can be characterizedby E α,p = { u ∈ L p ([ a, b ] , R n ) | a D αt u ∈ L p ([ a, b ] , R n ) and u ( a ) = u ( b ) = 0 } . Proposition 2.1. [15] Let < α ≤ and < p < ∞ . For all u ∈ E α,p , we have (2.3) k u k L p ≤ ( b − a ) α Γ( α + 1) k a D αt u k L p . If α > /p and p + q = 1 , then (2.4) k u k ∞ ≤ ( b − a ) α − /p Γ( α )(( α − q + 1) /q k a D αt u k L p . RACTIONAL HAMILTONIAN SYSTEMS WITH POSITIVE SEMI-DEFINITE MATRIX 5
By (2.3), we can consider in E α,p the following norm(2.5) k u k α,p = k a D αt u k L p , which is equivalent to (2.2). Proposition 2.2. [15] Let < α ≤ and < p < ∞ . Assume that α > p and { u k } ⇀ u in E α,p . Then u k → u in C [ a, b ] , i.e. k u k − u k ∞ → , k → ∞ . For α >
0, consider the Liouville-Weyl fractional spaces I α −∞ = C ∞ ( R , R n ) k·k Iα −∞ , where(2.6) k u k I α −∞ = (cid:16)Z R u ( x ) dx + Z R | −∞ D αx u ( x ) | dx (cid:17) / . Furthermore, we introduce the fractional Sobolev space H α ( R , R n ) of order 0 < α < H α = C ∞ ( R , R n ) k·k α , where k u k α = (cid:16)Z R u ( x ) dx + Z R | w | α b u ( w ) dw (cid:17) / . Note that, a function u ∈ L ( R , R n ) belongs to I α −∞ if and only if | w | α b u ∈ L ( R , R n ) . Therefore, I α −∞ and H α are equivalent with equivalent norm, for more details see [10]. Lemma 2.3. [30, Theorem 2.1] If α > / , then H α ⊂ C ( R , R n ) and there is a constant C ∞ = C α, ∞ such that (2.8) k u k ∞ = sup x ∈ R | u ( x ) | ≤ C ∞ k u k α . Remark 2.
From Lemma 2.3, we know that if u ∈ H α with / < α < , then u ∈ L p ( R , R n ) for all p ∈ [2 , ∞ ) , since Z R | u ( x ) | p dx ≤ k u k p − ∞ k u k L . Now, we introduce the fractional space which we will use to construct the variationalframework for (FHS) λ . Let X α = n u ∈ H α : Z R [ | −∞ D αt u ( t ) | + ( L ( t ) u ( t ) , u ( t ))] dt < ∞ o , then X α is a reflexive and separable Hilbert space with the inner product h u, v i X α = Z R [( −∞ D αt u ( t ) , −∞ D αt v ( t )) + ( L ( t ) u ( t ) , v ( t ))] dt and the corresponding norm is k u k X α = h u, u i X α . For λ >
0, we also need the following inner product h u, v i X α,λ = Z R [( −∞ D αt u ( t ) , −∞ D αt v ( t )) + λ ( L ( t ) u ( t ) , v ( t ))] dt C´ESAR TORRES, ZIHENG ZHANG, AND AMADO MENDEZ and the corresponding norm is k u k X α,λ = h u, u i X α,λ . Lemma 2.4. [40] Suppose L ( t ) satisfies ( L ) and ( L ) , then X α is continuously embedded in H α . Remark 3.
Under the same conditions of Lemma 2.4, for all λ ≥ cC ∞ meas { l The aim of this section is to establish the proof of Theorem 1.1. Consider the functional I : X α,λ → R given by(3.1) I λ ( u ) = Z R h | −∞ D αt u ( t ) | + 12 ( λL ( t ) u ( t ) , u ( t )) − W ( t, u ( t )) i dt = 12 k u k X α,λ − Z R W ( t, u ( t )) dt. Under the conditions of Theorem 1.1, we note that I ∈ C ( X α,λ , R ), and(3.2) I ′ λ ( u ) v = Z R h ( −∞ D αt u ( t ) , −∞ D αt v ( t )) + ( λL ( t ) u ( t ) , v ( t )) − ( ∇ W ( t, u ( t )) , v ( t )) i dt for all u , v ∈ X α . In particular we have(3.3) I ′ λ ( u ) u = k u k X α,λ − Z R ( ∇ W ( t, u ( t )) , u ( t )) dt. Remark 4. It follows from ( W ) and ( W ) that |∇ W ( t, u ) | σ ≤ C |∇ W ( t, u ) || u | σ +1 for | u | ≥ R. Thus, by ( W ), for any ǫ > , there is C ǫ > such that (3.4) |∇ W ( t, u ) | ≤ ǫ | u | + C ǫ | u | p − , ∀ ( t, u ) ∈ R × R N and (3.5) | W ( t, u ) | ≤ ǫ | u | + C ǫ p | u | p ∀ ( t, u ) ∈ R × R N , where p = σσ − > . Lemma 3.1. Suppose that ( L ) - ( L ) , (W ) and (W ) are satisfied. Then RACTIONAL HAMILTONIAN SYSTEMS WITH POSITIVE SEMI-DEFINITE MATRIX 7 i There exists ρ > and η > such that inf k u k Xα,λ = ρ I λ ( u ) > η f or all λ ≥ cC ∞ meas { l < c } . ii Let ρ > defined in ( i ) , then there exists e ∈ X α,λ with k e k X α,λ > ρ such that I λ ( e ) < for all λ > .Proof. i By (3.5) and Remark 3, we obtain I λ ( u ) = 12 k u k X α,λ − Z R W ( t, u ( t )) dt ≥ k u k X α,λ − ǫ Z R | u ( t ) | dt − C ǫ p Z R | u ( t ) | p dt ≥ (cid:16) − ǫ Θ (cid:17) k u k X α,λ − C ǫ p Θ p ( meas { l < c } ) p − k u k pX α,λ . Let ǫ > − ǫ Θ > k u k X α,λ = ρ . Since p > 2, taking ρ small enough such that12 (cid:16) − ǫ Θ (cid:17) − C ǫ p Θ p ( meas { l < c } ) p − ρ p − > . Therefore I λ ( u ) ≥ ρ " (cid:16) − ǫ Θ (cid:17) − C ǫ p Θ p ( meas { l < c } ) p − ρ p − := η > . ii By ( L ) and without loss of generality let T = ( − ̺, ̺ ) ⊂ J such that L ( t ) ≡ 0. Let ψ ∈ C ∞ ( R , R n ) such that supp ( ψ ) ⊂ ( − τ, τ ), for some τ < ̺ . Hence(3.6) 0 ≤ Z R h L ( t ) ψ, ψ i dt = Z supp ( ψ ) h L ( t ) ψ, ψ i dt ≤ Z τ − τ h L ( t ) ψ, ψ i dt ≤ Z T h L ( t ) ψ, ψ i dt = 0 . On the other hand, by ( W ), for any ǫ > 0, there exists R > W ( t, u ) > | u | ǫ − R ǫ for all | u | ≥ R. Then, by taking ǫ → | σ |→∞ Z supp ( ψ ) W ( t, σψ ) | σ | dt = + ∞ . Hence, by (3.6) and (3.7) we obtain(3.8) I λ ( σψ ) | σ | = 12 Z R | −∞ D αt ψ ( t ) | dt − Z R W ( t, σψ ) | σ | dt → −∞ , as | σ | → ∞ . Therefore, if σ is large enough and e = σ ψ one gets I λ ( e ) < (cid:3) Since we have loss of compactness we need the following compactness results to recover theCerami condition for I λ . C´ESAR TORRES, ZIHENG ZHANG, AND AMADO MENDEZ Lemma 3.2. Suppose that ( L ) − ( L ) , ( W ) − ( W ) be satisfied. If u n ⇀ u in X α,λ , then (3.9) I λ ( u n − u ) = I λ ( u n ) − I λ ( u ) + o (1) as n → + ∞ and (3.10) I ′ λ ( u n − u ) = I ′ λ ( u n ) − I ′ λ ( u ) + o (1) as n → + ∞ . In particular, if I λ ( u n ) → c and I ′ λ ( u n ) → , then I ′ λ ( u ) = 0 after passing to a subsequence.Proof. Since u n ⇀ u in X α,λ , we have h u n − u, u i X α,λ → n → ∞ , which implies that k u n k X α,λ = k u n − u k X α,λ + k u k X α,λ + o (1) . Therefore, to obtain (3.9) and (3.10) it suffices to check that(3.11) Z R [ W ( t, u n ) − W ( t, u n − u ) − W ( t, u )] dt = o (1)and(3.12) sup ϕ ∈ X α,λ , k ϕ k α,λ =1 Z R h∇ W ( t, u n ) − ∇ W ( t, u n − u ) − ∇ W ( t, u ) , ϕ i dt = o (1) . Here, we only prove (3.12), the verification of (3.11) is similar. In fact, let(3.13) A := lim n →∞ sup ϕ ∈ X α,λ , k ϕ k α,λ =1 Z R h∇ W ( t, u n ) − ∇ W ( t, u n − u ) − ∇ W ( t, u ) , ϕ i dt. If A > 0, then, there exists ϕ ∈ X α,λ with k ϕ k X α,λ = 1 such that (cid:12)(cid:12)(cid:12)(cid:12)Z R h∇ W ( t, u n ) − ∇ W ( t, u n − u ) − ∇ W ( t, u ) , ϕ i dt (cid:12)(cid:12)(cid:12)(cid:12) ≥ A n large enough. Now, from (3.4) and Young’s inequality, there exist C , C and C > |h∇ W ( t, u n ) − ∇ W ( t, u n − u ) , ϕ i| ≤ C (cid:0) ǫ | u | + ǫ | u n − u | + ǫ | ϕ | + C | u | p + ǫ | u n − u | p + C | ϕ | p (cid:1) . Hence, there exists C , C , C > |h∇ W ( t, u n ) − ∇ W ( t, u n − u ) − ∇ W ( t, u ) , ϕ i|≤ C (cid:0) ǫ | u | + ǫ | u n − u | + ǫ | ϕ | + C | u | p + ǫ | u n − u | p + C | ϕ | p (cid:1) . Let h n ( t ) = max {|h∇ W ( t, u n ) − ∇ W ( t, u n − u ) − ∇ W ( t, u ) , ϕ i| − C ǫ ( | u n − u | + | u n − u | p ) , } . So 0 ≤ h n ( t ) ≤ C ( ǫ | u | + ǫ | ϕ | + C | u | p + C | ϕ | p ) . By the Lebesgue dominated convergence Theorem and the fact u n → u a.e. in R , , we can get Z R h n ( t ) dt → n → ∞ . From where Z R |h∇ W ( t, u n ( t )) − ∇ W ( t, u n ( t ) − u ( t )) − ∇ W ( t, u ( t )) , ϕ ( t ) i| dt → n → ∞ , which is a contradiction. Hence A = 0.Furthermore, if I λ ( u n ) → c and I ′ λ ( u n ) → n → ∞ , by (3.9) and (3.10), we get I λ ( u n − u ) → c − I λ ( u ) + o (1) RACTIONAL HAMILTONIAN SYSTEMS WITH POSITIVE SEMI-DEFINITE MATRIX 9 and I ′ λ ( u n − u ) = − I ′ λ ( u ) as n → + ∞ . Now, for every ϕ ∈ C ∞ ( R , R n ) we have I ′ λ ( u ) ϕ = lim n →∞ I ′ λ ( u n ) ϕ = 0 . Consequently, I ′ λ ( u ) = 0. (cid:3) Lemma 3.3. Suppose that ( L ) − ( L ) , ( W ) − ( W ) be satisfied and let c ∈ R . Then each ( Ce ) c -sequence of I λ is bounded in X α,λ .Proof. Suppose that { u n } ⊂ X α,λ is a ( Ce ) c sequence for c > 0, namely(3.14) I λ ( u n ) → c, (1 + k u n k X α,λ ) I ′ λ ( u n ) → n → ∞ . Therefore(3.15) c − o n (1) = I λ ( u n ) − I ′ λ ( u n ) u n = Z R H ( t, u n ( t )) dt. By contradiction, suppose that there is a subsequence, again denoted by { u n } , such that k u n k X α,λ → + ∞ as n → + ∞ . Taking v n = u n k u n k Xα,λ , we get that { v n } is bounded in X α,λ and k v n k X α,λ = 1. Moreover, we have o (1) = h I ′ λ ( u n ) , u n ik u n k X α,λ = 1 − Z R h∇ W ( t, u n ) , u n ik u n k X α,λ , as n → ∞ , which implies(3.16) Z R h∇ W ( t, u n ) , v n i| u n | | v n | dt = Z R h∇ W ( t, u n ) , u n ik u n k X α,λ → . For r ≥ 0, let h ( r ) := inf { H ( t, u ) : t ∈ R , | u | ≥ r } . From ( W ) we have h ( r ) > r > 0. Furthermore, by ( W ) and ( W ), for | u | ≥ r ,(3.17) C H ( t, u ) ≥ |∇ W ( t, u ) | σ | u | σ = (cid:18) |∇ W ( t, u ) || u || u | (cid:19) σ ≥ (cid:18) h∇ W ( t, u ) , u i| u | (cid:19) σ ≥ (cid:18) W ( t, u ) | u | (cid:19) σ , it follows from ( W ) and the definition of h ( r ) that h ( r ) → ∞ as r → ∞ . For 0 ≤ a < b , let Ω n ( a, b ) := { t ∈ R : a ≤ | u n ( t ) | < b } and C ba := inf (cid:26) H ( t, u ) | u | : t ∈ R and u ∈ R N with a ≤ | u | < b (cid:27) . By ( W ), for any ǫ > 0, there is δ > |∇ W ( t, u ) | ≤ ǫ K | u | for all | t | ≤ δ. Consequently(3.18) Z Ω n (0 ,δ ) |∇ W ( t, u n ) || u n | | v n | dt ≤ Z Ω n (0 ,δ ) ǫ K | v n | dt ≤ ǫ K k v n k L ≤ ǫ, ∀ n. We note that H ( t, u n ) ≥ C ba | u n | for all t ∈ Ω n ( a, b ) , consequently, by (3.15) we get(3.19) c − o n (1) = Z Ω n (0 ,a ) H ( t, u n ) dt + Z Ω n ( a,b ) H ( t, u n ) dt + Z Ω n ( b, + ∞ ) H ( t, u n ) dt ≥ Z Ω n (0 ,a ) H ( t, u n ) dt + C ba Z Ω n ( a,b ) | u n | dt + Z Ω n ( b, + ∞ ) H ( t, u n ) dt = Z Ω n (0 ,a ) H ( t, u n ) dt + C ba Z Ω n ( a,b ) | u n | dt + h ( b ) meas (Ω n ( b, + ∞ )) . Since h ( r ) → + ∞ as r → + ∞ , for p < q < ∞ it follows from (3.19) that(3.20) Z Ω n ( b, + ∞ ) | v n | p dt ≤ Z Ω n ( b, + ∞ ) | v n | q dt ! pq meas (Ω n ( b + ∞ )) q − pq ≤ k v n k pL q (cid:18) c − o n (1) h ( b ) (cid:19) q − pp ≤ K pq (cid:18) c − o n (1) h ( b ) (cid:19) q − pp → b → + ∞ , where p = σσ − > 2. Furthermore, by ( W ) and the H¨older inequality, we canchoose R > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω n ( R, + ∞ ) h∇ W ( t, u n ) , u n ik u n k X α,λ dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω n ( R, + ∞ ) |∇ W ( t, u n ) || u n | | v n | dt ≤ Z Ω n ( R, + ∞ ) |∇ W ( t, u n ) | σ | u n | σ ! /σ Z Ω n ( R, + ∞ ) | v n | p dt ! σ − σ ≤ Z Ω n ( R, + ∞ ) C H ( t, u n ) dt ! /σ Z Ω n ( R, + ∞ ) | v n | p dt ! σ − σ ≤ C /σ ( c − o n (1)) /σ Z Ω n ( R, + ∞ ) | v n | p dt ! σ − σ < ǫ. Now, by using (3.19) again, we get Z Ω n ( δ,R ) | v n | dt = 1 k u n k X α,λ Z Ω n ( δ,R ) | u n | dt ≤ c − o n (1) C Rδ k u n k X α,λ → n → ∞ . Then, for n large enough, by the continuity of ∇ W one has(3.22) Z Ω n ( δ,R ) |∇ W ( t, u n ) || u n | | v n | dt ≤ K Z Ω n ( δ,R ) | v n | dt < ǫ. Hence, by (3.18), (3.21) and (3.22) we have Z R h∇ W ( t, u n ) , v n i| u n | | v n | dt ≤ Z R |∇ W ( t, u n ) || u n | | v n | dt ≤ ǫ < , for n large enough, a contradiction with (3.16) and then { u n } is bounded in X α,λ . (cid:3) Lemma 3.4. Suppose that ( L ) − ( L ) , ( W ) − ( W ) be satisfied. Then, for any C > , thereexists Λ = Λ( C ) > such that I λ satisfies ( Ce ) c condition for all c ≤ C and λ > Λ . RACTIONAL HAMILTONIAN SYSTEMS WITH POSITIVE SEMI-DEFINITE MATRIX 11 Proof. For any C > 0, suppose that { u n } ⊂ X α,λ is a ( Ce ) c sequence for c ≤ C , namely I λ ( u n ) → c, (1 + k u n k X α,λ ) I ′ λ ( u n ) → n → ∞ . By Lemma 3.3, { u n } is bounded. Therefore, there exists u ∈ X α,λ such that u n ⇀ u in X α,λ and u n → u a.e. in R .Let w n := u n − u . By Lemma 3.2 we get I ′ λ ( u ) = 0 , I λ ( w n ) → c − I λ ( u ) and I ′ λ ( w n ) → n → ∞ . Next(3.23) I λ ( u ) = I λ ( u ) − I ′ λ ( u ) u = Z R H ( t, u ) dt ≥ , and(3.24) Z R H ( t, w n ) dt → c − I λ ( u ) . Therefore, for c ≤ C , we get(3.25) Z R H ( t, w n ) dt ≤ C + o n (1) . On the other hand, by ( L ) and since w n → L loc ( R , R N ), we have(3.26) k w n k L ≤ λc Z { l ≥ c } λ h L ( t ) w n , w n i dt + o n (1) ≤ λc k w n k X α,λ + o n (1) . Let p < q < ∞ , where p = σσ − . Using Remark 3 and H¨older inequality we obtain(3.27) Z R | w n | p dt = Z R | w n | q − p ) q − | w n | q ( p − q − dt ≤ k w n k q − p ) q − L k w n k q ( p − q − L q ≤ K q ( p − q − q (cid:18) λc (cid:19) q − pq − k w n k pX α,λ + o n (1) . Furthermore, for | u | ≤ R (where R is defined in (W )), from (3.4), we get |∇ W ( t, u ) | ≤ ( ǫ + C ǫ R p − ) | u | = ˜ C | u | . It follows from (3.26) that Z { t ∈ R : | w n ( t ) |≤ R } h∇ W ( t, w n ) , w n i dt ≤ Z { t ∈ R : | w n ( t ) |≤ R } |∇ W ( t, w n ) || w n | dt ≤ ˜ C Z { t ∈ R : | w n ( t ) |≤ R } | w n | dt ≤ ˜ Cλc k w n k X α,λ + o n (1) . On the other hand, from (3.27) and the H¨older inequality we obtain Z { t ∈ R : | w n ( t ) | >R } h∇ W ( t, w n ) , w n i dt ≤ Z { t ∈ R : | w n ( t ) | >R } |∇ W ( t, w n ) || w n | dt ≤ Z { t ∈ R : | w n ( t ) | >R } |∇ W ( t, w n ) || w n | | w n | dt ≤ Z { t ∈ R : | w n ( t ) | >R } |∇ W ( t, w n ) | σ | w n | σ dt ! /σ Z { t ∈ R : | w n ( t ) | >A } | w n | p ! p ≤ (cid:18) C Z R H ( t, w n ) dt (cid:19) /σ k w n k p ≤ ( C C ) /σ K q ( p − p ( q − q (cid:18) λc (cid:19) q − p ) p ( q − k w n k X α,λ + o n (1) . Therefore o n (1) = h I ′ λ ( w n ) , w n i = k w n k X α,λ − Z R h∇ W ( t, w n ) , w n i dt = k w n k X α,λ − Z { t ∈ R : | w n ( t ) |≤ R } h∇ W ( t, w n ) , w n i dt − Z { t ∈ R : | w n ( t ) | >R } h∇ W ( t, w n ) , w n i dt ≥ − ˜ Cλc − C ∗ (cid:18) λc (cid:19) q − p ) p ( q − k w n k X α,λ + o n (1) , where C ∗ = ( C C ) /σ K q ( q − p ) p ( q − q . Now, we choose Λ = Λ( C ) > − ˜ Cλc − C ∗ (cid:18) λc (cid:19) q − p ) p ( q − > λ > Λ . Then w n → X α,λ for all λ > Λ . (cid:3) Proof of Theorem 1.1 By Lemmas 3.1, I λ has the mountain pass geometry and by Lemma3.4, I λ satisfies the ( Ce ) c -condition. Therefore, by using mountain pass lemma with Ceramicondition [9], for any c λ > c λ = inf g ∈ Γ max s ∈ [0 , I λ ( g ( s )) , where Γ = { g ∈ C ([0 , , X α,λ ) | g (0) = 0 , g (1) = e } , ( e is defined in Lemma 3.1-ii), there exists u λ ∈ X α,λ such that(3.28) I λ ( u λ ) = c λ and I ′ λ ( u λ ) = 0 . That is, (FHS) λ has at least one nontrivial solution for λ > Λ( c λ ) (defined in Lemma 3.4).4. Concentration phenomena In this section, we study the concentration of solutions for problem (FHS) λ as λ → ∞ . Thatis, we focus our attention on the proof of Theorem 1.2. RACTIONAL HAMILTONIAN SYSTEMS WITH POSITIVE SEMI-DEFINITE MATRIX 13 Remark 5. The main difficulty to proof Theorem 1.2, is to show that c λ is bounded form aboveindependent of λ . Thank to the proof of Lemma 3.1-ii, we can get a finite upper bound to c λ ,that is, choose ψ as in the proof of Lemma 3.1-ii, then by definition of c λ , we have c λ ≤ max σ ≥ I λ ( σψ )= max σ ≥ (cid:18) σ Z R | −∞ D αt ψ ( t ) | − Z R W ( t, σψ ) dt (cid:19) = ˜ c, where ˜ c < + ∞ is independent of λ .As a consequence of the above estimates, we have that Λ( c λ ) is bounded form below. Thatis, there exists Λ ∗ > such that the conclusion of Theorem 1.1 is satisfied for λ > Λ ∗ . Consider T = [ − ̺, ̺ ] and the following fractional boundary value problem(4.1) (cid:26) t D α̺ − ̺ D αt u = ∇ W ( t, u ) , t ∈ ( − ̺, ̺ ) ,u ( − ̺ ) = u ( ̺ ) = 0 . Associated to (4.1) we have the functional I : E α → R given by I ( u ) := 12 Z ̺ − ̺ | − ̺ D αt u ( t ) | dt − Z ̺ − ̺ W ( t, u ( t )) dt and we have that I ∈ C ( E α , R ) with I ′ ( u ) v = Z ̺ − ̺ h − ̺ D αt u ( t ) , − ̺ D αt v ( t ) i dt − Z ̺ − ̺ h∇ W ( t, u ( t )) , v ( t ) i dt. Following the ideas of the proof of Theorem 1.1, we can get the following existence result Theorem 4.1. Suppose that W satisfies ( W ) − ( W ) with t ∈ [ − ̺, ̺ ] , then (4.1) has at leastone weak nontrivial solution. Proof of Theorem 1.2 We follow the argument in [40]. For any sequence λ k → ∞ , let u k = u λ k be the critical point of I λ k , namely c λ k = I λ k ( u k ) and I ′ λ k ( u k ) = 0 , and, by (3.5), we get c λ k = I λ k ( u k ) = 12 k u k k X α,λ − Z R W ( t, u k ( t )) dt ≥ k u k k X α,λ − ǫ Z R | u k | dt − C ǫ p Z R | u k | p dt, which implies that { u k } is bounded, due to Remarks 2 and 3. Therefore, we may assume that u k ⇀ ˜ u weakly in X α,λ k . Moreover, by Fatou’s lemma, we have Z R l ( t ) | ˜ u ( t ) | dt ≤ lim inf k →∞ Z R l ( t ) | u k ( t ) | dt ≤ lim inf k →∞ Z R ( L ( t ) u k ( t ) , u k ( t )) dt ≤ lim inf k →∞ k u k k X α,λk λ k = 0 . Thus, ˜ u = 0 a.e. in R \ J . Now, for any ϕ ∈ C ∞ ( T, R n ), since I ′ λ k ( u k ) ϕ = 0, it is easy to seethat Z ̺ − ̺ ( − ̺ D αt ˜ u ( t ) , − ̺ D αt ϕ ( t )) dt − Z ̺ − ̺ ( ∇ W ( t, ˜ u ( t )) , ϕ ( t )) dt = 0 , that is, ˜ u is a solution of (4.1) by the density of C ∞ ( T, R n ) in E α . Now we show that u k → ˜ u in X α . 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Zhang, Existence and multiplicity results of homoclinic solutions for fractional Hamiltoniansystems , Comput. Math. with Appl., (2017), 1325-1345. 2(C´esar Torres) Departamento de Matem´aticasUniversidad Nacional de Trujillo,Av. Juan Pablo II s/n. Trujillo-Per´u E-mail address : ctl [email protected] (Ziheng Zhang) Department of Mathematics,Tianjin Polytechnic University,Tianjin 300387, China. E-mail address : [email protected] (Amado Mendez) Departamento de Matem´aticasUniversidad Nacional de Trujillo,Av. Juan Pablo II s/n. Trujillo-Per´u E-mail address ::