Soliton dynamics for the generalized Choquard equation
Claudio Bonanno, Pietro d'Avenia, Marco Ghimenti, Marco Squassina
aa r X i v : . [ m a t h . A P ] O c t SOLITON DYNAMICS FOR THE GENERALIZED CHOQUARD EQUATION
CLAUDIO BONANNO, PIETRO D’AVENIA, MARCO GHIMENTI, AND MARCO SQUASSINA
Abstract.
We investigate the soliton dynamics for a class of nonlinear Schrödinger equationswith a non-local nonlinear term. In particular, we consider what we call generalized Choquardequation where the nonlinear term is ( | x | θ − N ∗ | u | p ) | u | p − u . This problem is particularlyinteresting because the ground state solutions are not known to be unique or non-degenerate. Introduction
The soliton dynamics of the nonlinear Schrödinger equation iε ∂ψ∂t = − ε m ∆ ψ + V ( x ) ψ − f ( | ψ | ) ψ in (0 , ∞ ) × R N in the last decade has been the object of many mathematical studies. In the case of pure powernonlinearities we just mention the fundamental papers [5, 12, 17]. Even if the results are accom-plished by completely different methods, in all these papers the non-degeneracy of the groundstates of the stationary equation plays a fundamental role in getting the modulational equationoriginally devised by Weinstein in [31, 32]. Recently, a new approach was developed in [3, 4] notrequiring the non-degeneracy of the ground states.Another important class of nonlinearities are the non-local Hartree type nonlinearities, i.e. f ( | ψ | ) ψ = (cid:16) | x | ∗ | ψ | (cid:17) ψ. Hartree nonlinearities arise in several examples of mathematical physics, as the mean field limitof weakly interacting molecules (see [22] and the references therein), in the Pekar theory ofpolarons (see [28,29] and [20] for further references), in Schrödinger-Newton systems [14] or, witha semi-relativistic differential operator, in boson stars modeling [13]. In the Hartree case, thenon-degeneracy of ground states has been investigated only recently by Lenzmann in [19] and thesolitonic dynamics in [10].The goal of this paper is to obtain a soliton dynamics behavior for a general class of Hartree typenonlinearities for which, currently, neither uniqueness nor non-degeneracy of ground states areknown, by exploiting the techniques of [3,4]. This further corroborates the usefulness and impactof the ideas developed in these papers on a problem which has recently attracted the attentionof many researchers, especially in the stationary case.We consider the following generalized Choquard equation ( GC ) iε ∂ψ∂t = − ε m ∆ ψ + V ( x ) ψ − ( I θ ∗ | ψ | p ) | ψ | p − ψ with ( t, x ) ∈ (0 , ∞ ) × R N , where N ≥ , ψ : [0 , ∞ ) × R N → C , ε is the Planck constant, m > and I θ ( x ) := Γ( N − θ )Γ( θ ) π N/ θ | x | N − θ , Mathematics Subject Classification.
Key words and phrases.
Soliton dynamics, Choquard equation, Hartree equation, modulational stability,ground states.The authors were supported by 2009 MIUR project: “Variational and Topological Methods in the Study ofNonlinear Phenomena”. This work has been partially carried out during a stay of M. Squassina in Pisa. He wouldlike to express his deep gratitude to the Dipartimento di Matematica for the warm hospitality. with θ ∈ (0 , N ) a real parameter and(1.1) p ∈ (cid:16) θN , θN (cid:17) . Moreover let V : R N → R be a C -function satisfying(V0) V ≥ ;(V1) |∇ V ( x ) | ≤ ( V ( x )) b for | x | > R > and b ∈ (0 , ;(V2) V ( x ) ≥ | x | a for | x | > R > and a > .By the rescaling ˆ ψ ( t, x ) = m − θ p − ε α − γ (2 p − p − ψ ( t, x/ √ m ) , equation ( GC ) can be written as(1.2) iε ∂ ˆ ψ∂t = − ε ψ + ˆ V ( x ) ˆ ψ − ε γ (2 p − − α ( I θ ∗ | ˆ ψ | p ) | ˆ ψ | p − ˆ ψ, where α and γ are real parameters and ˆ V ( x ) = V ( x/ √ m ) . We remark that in [10] it has beentreated the physical case ( N = 3 , θ = 2 , p = 2 ), passing from ( GC ) to (1.2) by using the samerescaling with γ = 0 and α = 2 . Hence in this paper we study the problem( P ε ) iε ∂ψ∂t = − ε ψ + V ( x ) ψ − ε γ (2 p − − α ( I θ ∗ | ψ | p ) | ψ | p − ψψ (0 , x ) = U ε ( x ) e iε x · v , where v ∈ R N and(1.3) U ε ( x ) = ε − γ U ( ε − β x ) ,U being a real solution of(1.4)
12 ∆ U + ( I θ ∗ | U | p ) | U | p − U = ωU, with ω > , and β ∈ R . Concerning local and global well-posedness of solutions in H ( R N ) to( P ε ), as well as conservation laws, in the case θ = 2 , we refer the reader to [16]. In the generalcase θ = 2 , we shall assume that local well-posedness holds (being global existence easy to showin the range (1.1) of values of p ). The solutions to problem (1.4) have recently been object ofvarious deep investigations from the point of view of regularity, qualitative properties such assymmetry and asymptotic behaviour and concentration properties of semiclassical states. Werefer the reader to [8, 18, 25–27]. Concerning uniqueness of positive radial solutions to (1.4), toour knowledge, after the original contribution due to Lieb [21], a result can be found in [16] inthe particular case θ = 2 . Finally, about the nondegeneracy of the ground states of (1.4), theonly case where it is known, is to our knowledge, when N = 3 , θ = 2 and p = 2 , see [19, 30].A problem similar to ( P ε ) arises in the study of equation ( GC ) with so-called semi-classical wavepackets (or coherent states) as initial data, see for example [6], and also [7] where the sameproblem has been studied for the nonlinear Schrödinger equation with local nonlinear term. Themain difference with our approach is that in the papers [6, 7] the idea is to fix initial conditionswith β = and γ = N in (1.3), and then to study the behaviour of the solution varying thepower of ε in front of the nonlinear term. Instead, we choose the initial conditions according tothe values of γ and α , see conditions (2.2), (2.8) and (2.9) below.The following is the main result of the paper. Theorem 1.1.
Assume that conditions (V0)–(V2) hold, that the solution ψ to problem ( P ε ) isin C ([0 , ∞ ) , H ( R N )) ∩ C ((0 , ∞ ) , L ( R N )) , that β = α + 2 − γθ + 2 > , and p is as in (1.1) . Then the barycenter (1.5) q ε ( t ) := 1 k ψ ( t ) k Z R N x | ψ ( t, x ) | dx, OLITON DYNAMICS FOR THE GENERALIZED CHOQUARD EQUATION 3 of the solution ψ to problem ( P ε ) satisfies the Cauchy problem (1.6) ¨ q ε ( t ) + ∇ V ( q ε ( t )) = H ε ( t ) ,q ε (0) = 0 , ˙ q ε (0) = v, where k H ε k L ∞ (0 , ∞ ) → as ε → + . As explained above, we obtain as a corollary the same result for equation ( GC ), namely Corollary 1.2.
Under the same assumptions of Theorem 1.1, the solution ψ ( t, x ) to equation ( GC ) with initial condition ψ (0 , x ) = ε γ − α p − U ( ε − β x ) e iε x · v has a barycenter q ε ( t ) which satisfies equation (1.6) with k H ε k L ∞ (0 , ∞ ) → as ε → + . We remark that, contrarily to the results obtained for example in [5], we have no informationabout the shape of the solution. This is due to the fact that we use no information about theuniqueness or non-degeneracy of the ground states, so we cannot conclude that the solutionstays close to some specific function. However we show that the solution is concentrated (seeProposition 3.1) and we have information about the motion of its barycenter. The first partof the resut is achieved by using the only information that the minimizing sequences for theconstrained variational problem associated to equation (1.4) are relatively compact.The paper is organized as follows. In Section 2, we give some preliminary results on the relationsbetween the parameters, on the first integrals of our equation and on the existence and propertiesof the ground states. In particular, the ground states of equation (1.4) are constrained minimizersfor a functional J on the set of functions with fixed L -norm (see Lemmas 2.3, 2.4 and also [26])and, as explained above, we show the pre-compactness, up to translations, of the minimizingsequences for J (see Lemma 2.5). Finally, in Section 3 we show the concentration behaviour inthe semi-classical limit and we conclude by Section 4 proving Theorem 1.1.In the paper we denote by C a generic positive constant which can change from line to line.2. Preliminary tools
Relations between the parameters.
Let ω ε ∈ R and U ε as in (1.3). We require that ψ ( t, x ) = U ε ( x ) e i ωεε t solves ( P ε ) with V ≡ , so that V can be interpreted as a perturbation term. Hence we ask that ψ solves(2.1) iε ∂ψ∂t = − ε ψ − ε γ (2 p − − α ( I θ ∗ | ψ | p ) | ψ | p − ψ, i.e. that U ε solves ε U ε + ε γ (2 p − − α ( I θ ∗ | U ε | p ) | U ε | p − U ε = ω ε U ε . So we establish a relation between β and the other parameters. Since [( I θ ∗ | U ε | p ) | U ε | p − U ε ]( x ) = e i ωεε t ε βθ − γ (2 p − (cid:2) ( I θ ∗ | U | p ) | U | p − U (cid:3) ( ε − β x ) , then ε − γ − β U + ε βθ − α ( I θ ∗ | U | p ) | U | p − U = ω ε ε − γ U. Thus, ψ ( t, x ) = U ε ( x ) e i ωεε t is a solution of (2.1) if(2.2) β = α + 2 − γθ + 2 and ω ε = ωε − β . In the following we always assume (2.2).
C. BONANNO, P. D’AVENIA, M. GHIMENTI, AND M. SQUASSINA
The first integrals of NSE.
Noether’s theorem states that any invariance for a one-parameter group of the Lagrangian implies the existence of an integral of motion (see e.g. [15]).Now we describe the first integrals for ( P ε ) which will be relevant for this paper, namely the hylenic charge and the energy .Following [1], the hylenic charge (or simply charge ) is defined as the quantity which is preservedby the invariance of the Lagrangian with respect to the action ψ e iθ ψ. For the equation in ( P ε ) the charge is nothing else but the L -norm, namely C ( ψ ) = Z | ψ | = Z u . The energy , by definition, is the quantity which is preserved by the time invariance of the La-grangian. It has the form E ε ( ψ ) = ε Z |∇ ψ | + Z V ( x ) | ψ | − ε γ (2 p − − α p Z ( I θ ∗ | ψ | p ) | ψ | p . Writing ψ in the polar form ue iε S we get(2.3) E ε ( ψ ) = ε Z |∇ u | − ε γ (2 p − − α p Z ( I θ ∗ | u | p ) | u | p + Z (cid:18) |∇ S | + V ( x ) (cid:19) u . Thus the energy has two components: the internal energy (which, sometimes, is also called bindingenergy ) J ε ( u ) = ε Z |∇ u | − ε γ (2 p − − α p Z ( I θ ∗ | u | p ) | u | p and the dynamical energy G ( u, S ) = Z (cid:18) |∇ S | + V ( x ) (cid:19) u which is composed by the kinetic energy Z |∇ S | u and the potential energy Z V ( x ) u . Finally we define the momentum (2.4) p ε ( t, x ) := 1 ε N − Im ( ¯ ψ ( t, x ) ∇ ψ ( t, x )) , x ∈ R N , t ∈ [0 , ∞ ) . Arguing as in [10, Lemma 3.3], if ψ ∈ C ([0 , ∞ ) , H ( R N )) ∩ C ((0 , ∞ ) , L ( R N )) , the map t Z p ε ( t, x ) dx is C and, on the solutions, the following identities hold: ε − N ∂ t | ψ ( t, x ) | = − div( p ε ( t, x )) , t ∈ [0 , ∞ ) , x ∈ R N , (2.5) ∂ t Z p ε ( t, x ) dx = − ε − N Z ∇ V ( x ) | ψ ( t, x ) | dx, t ∈ [0 , ∞ ) . (2.6) OLITON DYNAMICS FOR THE GENERALIZED CHOQUARD EQUATION 5
Rescaling of internal energy and charge.
If we consider again U ε as in (1.3), since Z ( I θ ∗ | U ε | p ) | U ε | p = ε β ( N + θ ) − pγ Z ( I θ ∗ | U | p ) | U | p by (2.2), we have(2.7) J ε ( U ε ) = ε − γ + β ( N − J ( U ) where J ( u ) = 12 Z |∇ u | − p Z ( I θ ∗ | u | p ) | u | p . As pointed out in [26], J is of class C on H ( R N ) and for every u, v ∈ H ( R N ) h J ′ ( u ) , v i = Z ∇ u · ∇ v − Z ( I θ ∗ | u | p ) | u | p − uv. Moreover, computing the charge of a rescaled function, we have C ( U ε ) = ε Nβ − γ C ( U ) We can choose, without loss of generality, that(2.8)
N β − γ = 0 , in order to have the same charge for any rescaling and to simplify the notations.Thus, combining (2.7) and (2.8) we get J ε ( U ε ) = ε − β ) J ( U ) , so, when(2.9) β > we have that J ε ( U ε ) → + ∞ for ε → + which will be the key tool for the main result of thispaper. Remark 2.1.
In the physical case ( N = 3 and θ = 2 ), in order to satisfy conditions (2.2) , (2.8) and (2.9) , we have can any couple ( α, γ ) on the line α + 6 − γ = 0 with γ > / . This choiceimplies that for p = 2 the power γ (2 p − − α is, in the notation of [6], super-critical, indeed γ (2 p − − α − < . Ground states.
Let ω > . A ground state for (1.4) is a solution that realizes the minimumof the energy E ω ( u ) = 12 Z |∇ u | + ω Z | u | − p Z ( I θ ∗ | u | p ) | u | p on the set N ω = (cid:26) u ∈ H ( R N ) \ { } | Z |∇ u | + ω Z | u | = Z ( I θ ∗ | u | p ) | u | p (cid:27) . A ground state can be found in several ways. In the recent paper [26], for instance, the authorsminimize S θ,p ( u ) = k∇ u k + ω k u k (cid:0)R ( I θ ∗ | u | p ) | u | p (cid:1) /p in H ( R N ) \ { } . This way allows to obtain a sharp result on the existence with respect to the parameter p . In thefollowing lemma we summarize some results obtained in [26]. Lemma 2.2.
Let N ≥ , θ ∈ (0 , N ) and p ∈ (1 + θ/N, ( N + θ ) / ( N − . We have that (1.4) admits a ground state solution U in H ( R N ) . Moreover each ground state U of (1.4) isin L ( R N ) ∩ C ∞ ( R N ) , it has fixed sign and there exist x ∈ R N and a monotone real function v ∈ C ∞ (0 , ∞ ) such that U ( x ) = v ( | x − x | ) a.e. in R N . C. BONANNO, P. D’AVENIA, M. GHIMENTI, AND M. SQUASSINA
In the following, we consider only positive ground state.Another way to look for ground states is to minimize J on Σ ν = { u ∈ H ( R N ) | k u k = ν } for ν > (cf. Lemma 2.4). In fact, under our assumptions on p , for every u ∈ H ( R N ) ,(2.10) < N p − θ − N < and(2.11) N pN + θ ∈ (2 , ∗ ) , with ∗ = 2 NN − , so that | u | p ∈ L NN + θ ( R N ) . Thus, by Hardy-Littlewood-Sobolev and Gagliardo-Nirenberg inequal-ities, we have(2.12) Z ( I θ ∗ | u | p ) | u | p ≤ C k u k p Np/ ( N + θ ) ≤ C k∇ u k Np − θ − N k u k p − Np + N + θ . Hence, for all u ∈ Σ ν ,(2.13) J ( u ) ≥ k∇ u k − Cν p − Np + N + θ k∇ u k Np − θ − N and so, by (2.10), we get that J is bounded from below on Σ ν . Moreover we notice that if p ∈ [1 + (2 + θ ) /N, ( N + θ ) / ( N − , J is unbounded from below on Σ ν and if p satisfies (1.1), inf u ∈ H ( R N ) \{ } S θ,p ( u ) can be written in terms of inf u ∈ Σ ν J ( u ) and any minimizer of S θ,p in H ( R N ) \ { } is, up to suitable dilation and rescaling, a minimizer of J on Σ ν . This last methodseems to be the best for our arguments. So, for the sake of completeness we give some details.First of all we give the following preliminary result. Lemma 2.3.
For every ν > we have that m ν := inf u ∈ Σ ν J ( u ) ∈ ( −∞ , . Proof.
From the arguments above we know that J is bounded from below on Σ ν . So it remainsto prove that m ν < . To this end let u ∈ Σ ν and define u τ ( x ) := τ N/ u ( τ x ) for τ > and x ∈ R N . Then u τ ∈ Σ ν and m ν ≤ J ( u τ ) = τ Z |∇ u | − τ Np − θ − N p Z ( I θ ∗ | u | p ) | u | p . By (2.10), taking τ > sufficiently small we get m ν < . (cid:3) Moreover, following step by step [9, Proof of Lemma 2.6], we get
Lemma 2.4.
For every ν, ω > , the minimization problems min u ∈ Σ ν J ( u ) and min u ∈N ω E ω ( u ) are equivalent. Moreover the L -norm of any ground state U of (1.4) is √ σ where (2.14) σ := N + θ − ( N − p ω ( p −
1) min u ∈N ω E ω ( u ) and min u ∈ Σ σ E ω ( u ) = min u ∈N ω E ω ( u ) . Proof.
Let ν, ω > , K Σ ν = { m ∈ R − | ∃ u ∈ Σ ν s.t. J ′ | Σ ν ( u ) = 0 and J ( u ) = m } and K N ω = { c ∈ R | ∃ u ∈ N ω s.t. E ′ ω ( u ) = 0 and E ω ( u ) = c } . Let now u ∈ Σ ν such that J ′ | Σ ν ( u ) = 0 and J ( u ) = m with m < . Then there exists γ ∈ R suchthat(2.15)
12 ∆ u + ( I θ ∗ | u | p ) | u | p − u = γu OLITON DYNAMICS FOR THE GENERALIZED CHOQUARD EQUATION 7 and so(2.16) k∇ u k − Z ( I θ ∗ | u | p ) | u | p = − γν. Thus, since J ( u ) = m < , by (2.16) we get p − p k∇ u k − m = γνp and so γ > . Now let w ( x ) := τ θ +22( p − u ( τ x ) with τ = r ωγ . We have that w solves −
12 ∆ w + ωw − ( I θ ∗ | w | p ) | w | p − w = 0 and so w ∈ N ω , E ′ ω ( w ) = 0 and c = E ω ( w ) ∈ K N ω .Viceversa, if w ∈ N ω such that E ′ ω ( w ) = 0 and c = E ω ( w ) , we consider u ( x ) := τ θ +22( p − w ( τ x ) with τ = (cid:18) ν k w k (cid:19) p − θ +2 − N ( p − . We have that u ∈ Σ ν , (2.15) holds for γ = ωτ = ω (cid:18) ν k w k (cid:19) p − θ +2 − N ( p − and(2.17) m = τ θ +2 p − N ( p − p − ( c − ω k w k ) = (cid:18) ν k w k (cid:19) θ +2 p − N ( p − θ +2 − N ( p − ( c − ω k w k ) . By [26, Proposition 3.1] (Pohožaev identity) and since we have w ∈ N ω and E ω ( w ) = c we getthe system N − k∇ w k + ωN k w k − N + θp Z ( I θ ∗ | w | p ) | w | p = 012 k∇ w k + ω k w k − Z ( I θ ∗ | w | p ) | w | p = 012 k∇ w k + ω k w k − p Z ( I θ ∗ | w | p ) | w | p = c from which k w k = N + θ − ( N − p ω ( p − c. Thus (2.17) becomes m = N p − − N − θ p − (cid:18) ων ( p − N + θ − ( N − p (cid:19) θ +2 p − N ( p − θ +2 − N ( p − c − p ) θ +2 − N ( p − and the first conclusion easily follows. The second part is a trivial consequence of the calculationsof the first part. (cid:3) By combining Lemma 2.2 and Lemma 2.4, we get that for every ν > the minimum of J in Σ ν is attained. Furthermore, in order to obtain some uniform decay properties on the ground states,proceeding as in [2, Theorem 3.1], we prove the following result. Lemma 2.5.
For every ν > , every minimizing sequence for J in Σ ν is relatively compact in H ( R N ) up to a translation. C. BONANNO, P. D’AVENIA, M. GHIMENTI, AND M. SQUASSINA
Proof.
Let { u n } be a minimizing sequence for J on Σ ν . Without loss of generality, by EkelandVariational Principle [11], we can assume that { u n } is a Palais-Smale sequence for J . By (2.13)we have that { u n } is bounded in H ( R N ) and then there exists u ∈ H ( R N ) such that u n ⇀ u in H ( R N ) . Fixed R > , we have that there exist c > and a subsequence { u n } , such that(2.18) sup n ∈ N sup y ∈ R N Z B R ( y ) u n ≥ c. Indeed, if lim n sup y ∈ R N Z B R ( y ) u n = 0 , then, by [24, Lemma I.1], it follows that u n → in L q ( R N ) for q ∈ (2 , ∗ ) . Thus, by (2.11) and(2.12), we have that Z ( I θ ∗ | u n | p ) | u n | p → and this is a contradiction since m ν < . Hence, by (2.18), for every n ∈ N there exists y n ∈ R N such that Z B R ( y n ) u n ≥ c. So, if we take v n = u n ( · + y n ) , by using the compact embedding of H ( R N ) into L ( R N ) weobtain a minimizing sequence whose weak limit is nontrivial. Moreover, the weak convergenceimplies immediately that k u k ≤ ν , k u n − u k + k u k = k u n k + o n (1) , (2.19) k∇ u n − ∇ u k + k∇ u k = k∇ u n k + o n (1) (2.20)and, by [26, Lemma 2.4],(2.21) Z ( I θ ∗ | u n − u | p ) | u n − u | p + Z ( I θ ∗ | u | p ) | u | p = Z ( I θ ∗ | u n | p ) | u n | p + o n (1) . Assume by contradiction that k u k = τ < ν . Since, by (2.19), a n = √ ν − τ k u n − u k → and, by (2.20) and (2.21), J ( u n − u ) + J ( u ) = m ν + o n (1) , we have that J ( a n ( u n − u )) + J ( u ) = J ( u n − u ) + J ( u ) + o n (1) = m ν + o n (1) . Then, since k a n ( u n − u ) k = ν − τ , we get(2.22) m ν − τ + m τ ≤ m ν + o n (1) . But, if we consider, for µ > , Σ µν = (cid:8) u ∈ Σ ν | R ( I θ ∗ | u | p ) | u | p ≥ µ (cid:9) , we can prove that thereexists µ > such that(2.23) m ν = inf u ∈ Σ µν J ( u ) . Indeed, since Σ µν ⊂ Σ ν , we have m ν ≤ inf u ∈ Σ µν J ( u ) . If we suppose by contradiction that, forevery µ > , m ν < inf u ∈ Σ µν J ( u ) , then we can construct a minimizing sequence { u n } such that J ( u n ) → m ν and Z ( I θ ∗ | u n | p ) | u n | p → . Thus ≤ k∇ u n k = J ( u n ) + 1 p Z ( I θ ∗ | u n | p ) | u n | p → m ν < . Then, by using (2.23), it is easy to check that for every τ > m τν < τ m ν . OLITON DYNAMICS FOR THE GENERALIZED CHOQUARD EQUATION 9
Thus, as proved in [23, Lemma II.1], we have that for all τ ∈ (0 , ν ) m ν < m τ + m ν − τ which is in contradiction with (2.22). Hence u ∈ Σ ν , k u n − u k = o n (1) and, by applying theGagliardo-Nirenberg inequality as in the second part of (2.12), we have that(2.24) k u n − u k Np/ ( N + θ ) = o n (1) . It remains to show that k∇ u n − ∇ u k = o n (1) . Since { u n } is a Palais-Smale sequence, thereexists { λ n } ⊂ R such that for every v ∈ H ( R N ) h J ′ ( u n ) − λ n u n , v i = o n (1) and, since { u n } is bounded h J ′ ( u n ) − λ n u n , u n i = o n (1) . Then we obtain that { λ n } is bounded and h J ′ ( u n ) − J ′ ( u m ) − λ n u n + λ m u m , u n − u m i → as m, n → + ∞ . Since, by Hardy-Littlewood-Sobolev inequality and (2.24) (cid:12)(cid:12)(cid:12)(cid:12)Z ( I θ ∗ | u n | p ) | u n | p − u n ( u n − u m ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k u n k p +2 N ( p − / ( N + θ )2 Np/ ( N + θ ) k u n − u m k Np/ ( N + θ ) → and λ n h u n , u n − u m i → as m, n → + ∞ , we have that { u n } is a Cauchy sequence in H ( R N ) and we conclude. (cid:3) We close this section by showing the following uniform estimate on the ground states.
Lemma 2.6.
For every λ > there exists R > such that for every ground state U there exists q ( U ) ∈ R N such that Z R N \ B R ( q ( U )) U < λ. Proof.
Assume by contradiction that there exists λ > such that, for any n ∈ N , there exists aground state U n such that for every q ∈ R N Z R N \ B n ( q ) U n ≥ λ and so(2.25) inf q ∈ R N Z R N \ B n ( q ) U n ≥ λ. Then { U n } is a minimizing sequence and by virtue of Lemma 2.5 is relatively compact up to atranslation { q n } ⊂ R N . Thus there exists a ground state U with U n ( · − q n ) → U in H ( R N ) and inf q ∈ R N Z R N \ B n ( q ) U n ≤ Z R N \ B n ( − q n ) U n = Z R N \ B n (0) U n ( · − q n ) = Z R N \ B n (0) U + o n (1) = o n (1) , which is in contradiction with (2.25). (cid:3) Remark 2.7.
Of course, without loss of generality we can take q ( U ) = 0 in Lemma 2.6 forradially symmetric ground states U. Throughout the rest of the paper, we will consider radially symmetric ground states U . Concentration results
In this section we prove a concentration property of the solution of ( P ε ) with suitable initial data;more exactly, we prove that, fixed t ∈ (0 , ∞ ) , this solution is a function on R N with one peaklocalized in a ball with center depending on t and radius not depending on t . In order to provethis result, it is sufficient to assume that problem ( P ε ) admits global solutions ψ which satisfythe conservation of the energy and of the L -norm. Given K, ε > , let(3.1) B Kε = ψ (0 , x ) = u ε (0 , x ) e iε S ε (0 ,x ) with: u ε (0 , x ) = ε − γ (cid:2) ( U + w )( ε − β ( x − q )) (cid:3) ,U radial ground state solution of (1.4) ,q ∈ R N ,w ∈ H ( R N ) s.t. k U + w k = k U k = σ and k w k < Kε β − , k∇ S ε (0 , x ) k ∞ ≤ K, Z R N V ( x ) u ε (0 , x ) dx ≤ K the set of admissible initial data, where k · k denotes the H ( R N ) -norm. Of course, here σ satisfies(2.14). In the following, if m ∈ R we denote with J m the sublevels of J . The main result of thissection is Proposition 3.1.
Let V ∈ L ∞ loc ( R N ) , V ≥ , β > and fix K > . For all λ > , there exist ˆ R > and ε > such that, for any ε < ε , ψ solution of ( P ε ) with initial data ψ (0 , x ) ∈ B Kε and t ∈ (0 , ∞ ) , there exists ˆ q ε ( t ) ∈ R N for which k ψ ( t ) k Z R N \ B ˆ Rεβ (ˆ q ε ( t )) | ψ ( t, x ) | dx < λ. Here ˆ q ε ( t ) depends on ψ . For the proof of this proposition we need some technical results.
Lemma 3.2.
For any λ > there exist ˆ R = ˆ R ( λ ) > and δ = δ ( λ ) > such that, for any u ∈ J m σ + δ ∩ Σ σ , there exists ˆ q ∈ R N such that (3.2) σ Z R N \ B ˆ R (ˆ q ) u < λ. Proof.
First of all we prove that for any λ > , there exists δ > such that, for all u ∈ J m σ + δ ∩ Σ σ ,there exist ˆ q ∈ R N and a ground state U of (1.4) such that u = U ( · − ˆ q ) + w and k w k ≤ λ. Indeed, let us assume by contradiction that there exist λ > and a minimizing sequence { u n } such that for every q n ∈ R N and U ground state(3.3) k u n − U ( · − q n ) k > λ. Since, by Lemma 2.5, { u n } is relatively compact up to translations, there exists a ground state U ∈ H ( R N ) such that w n = u n − U ( · − q n ) → in H ( R N ) and this contradicts (3.3).Now, let us fix λ > . We can suppose that λ < . Then, for √ σλ , there exists δ > such that,for all u ∈ J m σ + δ ∩ Σ σ , there exists ˆ q ∈ R N and a ground state U such that u = U ( · − ˆ q ) + w and k w k ≤ √ σλ . Moreover, by Lemma 2.6, there exists ˆ R > such that, for every ground state U , Z R N \ B ˆ R (0) U < σλ (1 − √ λ ) . OLITON DYNAMICS FOR THE GENERALIZED CHOQUARD EQUATION 11
Thus, if u ∈ J m σ + δ ∩ Σ σ , we have σ Z R N \ B ˆ R (ˆ q ) u ≤ σ Z R N \ B ˆ R (ˆ q ) U ( · − ˆ q ) + 1 σ k w k + 2 σ k w k Z R N \ B ˆ R (ˆ q ) U ( · − ˆ q ) ! / = 1 σ Z R N \ B ˆ R (0) U + 1 σ k w k + 2 σ k w k Z R N \ B ˆ R (0) U ! / < λ (1 − √ λ ) + λ + 2 λ √ λ (1 − √ λ ) = λ which concludes the proof. (cid:3) As a consequence of the previous lemma, we can describe the concentration properties of thesolutions of ( P ε ). Lemma 3.3.
For any λ > , there exist δ = δ ( λ ) > and a ˆ R = ˆ R ( λ ) > such that for any ψ solution of ( P ε ) with ε γ | ψ ( t, ε β x ) | ∈ J m σ + δ ∩ Σ σ for all t ∈ (0 , ∞ ) , there exists ˆ q ε ( t ) ∈ R N ,which depends on λ , ε , t and ψ , for which σ Z R N \ B εβ ˆ R (ˆ q ε ( t )) | ψ ( t, x ) | dx < λ. Proof.
Let λ > be fixed. By Lemma 3.2 we have that there exist δ = δ ( λ ) > and a ˆ R =ˆ R ( λ ) > such that for any u ∈ J m σ + δ ∩ Σ σ , there exists ˆ q ∈ R N such that (3.2) holds. So wefix ε , t and ψ solution of ( P ε ), such that v ( x ) = ε γ | ψ ( t, ε β x ) | ∈ J m σ + δ ∩ Σ σ . We have that thereexists ¯ q = ¯ q ( v ) ∈ R N such that, using (2.8), σ Z R N \ B ˆ R (¯ q ) | v | = 1 σ Z R N \ B εβ ˆ R ( ε β ¯ q ) | ψ ( t, x ) | dx < λ. Then we conclude taking ˆ q ε ( t ) = ε β ¯ q , which depends on λ , ε , t and ψ , while ˆ R depends onlyupon the value of λ . (cid:3) Now we are ready to prove Proposition 3.1.
Proof of Proposition 3.1. If ψ is a solution of ( P ε ) with admissible initial datum, then, by theconservation of the energy E ε and by (3.1), (2.3) and (2.2), we have(3.4) E ε ( ψ ) ≤ ε − β ) J ( U + w ) + K σ K. Moreover, since J is C in H ( R N ) , we have(3.5) J ( U + w ) ≤ m σ + C k w k ≤ m σ + Cε β − . So, combining (3.4) and (3.5), we obtain(3.6) E ε ( ψ ) ≤ ε − β ) m σ + C. Thus, in light of (3.6) and because V ( x ) ≥ , if u ε ( t, x ) = | ψ ( t, x ) | , we get(3.7) J ε ( u ε ) = E ε ( ψ ) − G ( u ε , S ε ) ≤ ε − β ) m σ + C. Then, by (2.2), (2.8) and (3.7) we get J ( ε γ u ε ( t, ε β x )) = ε β − J ε ( u ε ) ≤ m σ + ε β − C. So, since, by the conservation of the hylenic charge, k ε γ u ε ( t, ε β x ) k = k ε γ u ε (0 , ε β x ) k = k U + w k = σ, if β > and for ε small we can apply Lemma 3.3 and we conclude. (cid:3) Proof of the main result
Barycenter and concentration point.
In this subsection, we provide the dynamics ofthe barycenter and we estimate the distance between the concentration point and the barycenterof a solution ψ for a potential satisfying (V0) and (V2). Proposition 4.1.
Let ψ be a global solution of ( P ε ) with initial data ψ (0 , x ) such that Z | x || ψ (0 , x ) | dx < + ∞ . Then the map q ε : R → R N , where q ε ( t ) is given by (1.5) , is well defined, is C and ˙ q ε ( t ) = ε N k ψ ( t ) k Z p ε ( t, x ) dx (4.1) ¨ q ε ( t ) = − k ψ ( t ) k Z ∇ V ( x ) | ψ ( t, x ) | dx (4.2) Proof.
We prove that q ε is well defined by a regularization argument. Let λ > and k λ ( t ) = Z e − λ | x | | x || ψ ( t, x ) | dx. By (2.5) we have k ′ λ ( t ) = − ε N Z e − λ | x | | x | div( p ε ( t, x )) dx = ε N Z e − λ | x | (1 − λ | x | ) x | x | · p ε ( t, x ) dx. Thus, on account of (2.4), | k ′ λ ( t ) | ≤ ε k ψ ( t ) k k∇ ψ ( t ) k and then k λ ( t ) = k λ (0) + Z t k ′ λ ( s ) ds ≤ Z | x || ψ (0 , x ) | dx + ε Z t k ψ ( s ) k k∇ ψ ( s ) k ds. Hence, using Fatou’s Lemma, we get that for all t ∈ (0 , ∞ ) Z | x || ψ ( t, x ) | dx < + ∞ and so q ε is well defined for all t . With the same regularization technique, we can also prove that q ε is C and that (4.1) holds by (2.5). Finally, equation (4.2) is a straightforward consequence of(4.1) and (2.6). (cid:3) Now, for
K > fixed, let ψ be a global solution of ( P ε ) such that ψ ∈ C ([0 , ∞ ) , H ( R N )) ∩ C ((0 , ∞ ) , L ( R N )) and the initial data ψ (0 , x ) ∈ B Kε . Moreover let u ε ( t, x ) = | ψ ( t, x ) | . Lemma 4.2.
There exists a constant
C > such that, for all t ∈ R , Z V ( x ) u ε ( t, x ) dx ≤ C. Proof.
Since ε γ u ε ( t, ε β x ) ∈ Σ σ , then, by (2.2) and (2.8),(4.3) J ε ( u ε ( t, x )) = ε − β ) J ( ε γ u ε ( t, ε β x )) ≥ ε − β ) m σ . Moreover, as in the proof of Proposition 3.1, inequality (3.6) holds and so, using (4.3), we get Z V ( x ) u ε ( t, x ) dx = E ε ( ψ ) − J ε ( u ε ) − Z |∇ S | u ε ( t, x ) dx ≤ C. (cid:3) The following lemma shows the boundedness for the barycenter q h ( t ) defined in (1.5). Lemma 4.3.
There exists K > such that for all t ∈ [0 , ∞ ) , | q ε ( t ) | ≤ K . OLITON DYNAMICS FOR THE GENERALIZED CHOQUARD EQUATION 13
Proof.
By Lemma 4.2 and assumption (V2) we get that for any R ≥ R and for any t ∈ [0 , ∞ ) ,(4.4) C ≥ Z R N \ B R (0) V ( x ) u ε ( t, x ) dx ≥ R a − Z R N \ B R (0) | x | u ε ( t, x ) dx. Hence (cid:12)(cid:12)(cid:12)(cid:12)Z xu ε ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R N \ B R (0) | x | u ε ( t, x ) dx + Z B R (0) | x | u ε ( t, x ) dx ≤ CR a − + R k u ε ( t ) k , so that | q ε ( t ) | ≤ R + C/ ( R a − σ ) . (cid:3) Remark 4.4.
By the inequality (4.4) we have also that, if R is large enough, for all t ∈ [0 , ∞ )1 k u ε ( t ) k Z R N \ B R (0) u ε ( t, x ) dx ≤ CσR a < . Now we show the boundedness of the concentration point ˆ q ε ( t ) defined in Lemma 3.3. Lemma 4.5. If < λ < / and R large enough we get that(1) for ε small enough sup t ∈ [0 , ∞ ) | ˆ q ε ( t ) | < R + ˆ R ( λ ) ε β < R + 1; (2) for all R ≥ R and ε small enough sup t ∈ [0 , ∞ ) | q ε ( t ) − ˆ q ε ( t ) | < CσR a − + 3 R λ + ˆ R ( λ ) ε β . Proof.
By Proposition 3.1, with λ < / , and by Remark 4.4, it is obvious that the ball B ˆ R ( λ ) ε β (ˆ q ε ( t )) R N \ B R (0) and B ˆ R ( λ ) ε β (ˆ q ε ( t )) ⊂ B R +2 ˆ R ( λ ) ε β (0) . Because ˆ R ( λ ) does not depend on ε , we can assume ε so small that R ( λ ) ε β < . Then | ˆ q ε ( t ) | < R + 2 ˆ R ( λ ) ε β < R + 1 , (4.5) B ˆ R ( λ ) ε β (ˆ q ε ( t )) ⊂ B R +1 (0) , (4.6)and (4.5) implies (1).To prove (2), first we estimate the difference between the barycenter and the concentration point.We have | q ε ( t ) − ˆ q ε ( t ) | = 1 k u ε ( t ) k (cid:12)(cid:12)(cid:12)(cid:12)Z ( x − ˆ q ε ( t )) u ε ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ I + I + I where I = 1 k u ε ( t ) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N \ B R (0) ( x − ˆ q ε ( t )) u ε ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I = 1 k u ε ( t ) k (cid:12)(cid:12)(cid:12)(cid:12)Z A ( x − ˆ q ε ( t )) u ε ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ,I = 1 k u ε ( t ) k (cid:12)(cid:12)(cid:12)(cid:12)Z A ( x − ˆ q ε ( t )) u ε ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ,A = B R (0) \ B ˆ R ( λ ) ε β (ˆ q ε ( t )) , A = B R (0) ∩ B ˆ R ( λ ) ε β (ˆ q ε ( t )) and R ≥ R . Obviously I ≤ ˆ R ( λ ) ε β . Moreover, by (1) and Proposition 3.1 we have I ≤ [2 R + 1] λ < R λ. Finally, by (4.4), (1) and Remark 4.4 we have I ≤ k u ε ( t ) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N \ B R (0) | x | u ε ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | ˆ q ε ( t ) |k u ε ( t ) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N \ B R (0) u ε ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < CσR a − + ( R + 1) CσR a < CσR a − and we conclude using the independence of t ∈ [0 , ∞ ) . (cid:3) We notice that R , R and R defined in this section do not depend on λ .4.2. Equation of the traveling soliton.
We prove that the barycenter dynamics is approxi-matively that of a point particle moving under the effect of an external potential V satisfying ourassumptions. Theorem 4.6.
Assume that V satisfies (V0), (V1), (V2). Given K > , let ψ be a global solutionof equation ( P ε ), with initial data in B Kε . If ε is small enough, then we have ¨ q ε ( t ) + ∇ V ( q ε ( t )) = H ε ( t ) where k H ε ( t ) k L ∞ (0 , ∞ ) → as ε → + .Proof. By (4.2) it is sufficient to estimate H ε ( t ) = [ ∇ V ( q ε ( t )) − ∇ V (ˆ q ε ( t ))] + 1 k u ε ( t ) k Z [ ∇ V (ˆ q ε ( t )) − ∇ V ( x )] u ε ( t, x ) dx. We set M = max {| ∂ α V ( τ ) | | α = 1 , and | τ | ≤ K + R + 1 } where K is defined in Lemma 4.3 and R is defined in Remark 4.4. By Lemma 4.3 and Lemma4.5, for any R ≥ R , we get(4.7) |∇ V ( q ε ( t )) − ∇ V (ˆ q ε ( t )) | ≤ M (cid:18) CσR a − + 3 R λ + ˆ R ( λ ) ε β (cid:19) . Moreover, we consider k u ε ( t ) k (cid:12)(cid:12)(cid:12)(cid:12)Z R N [ ∇ V (ˆ q ε ( t )) − ∇ V ( x )] u ε ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ L + L + L with L = 1 k u ε ( t ) k Z B ˆ R ( λ ) εβ (ˆ q ε ( t )) |∇ V (ˆ q ε ( t )) − ∇ V ( x ) | u ε ( t, x ) dx,L = 1 k u ε ( t ) k Z R N \ B ˆ R ( λ ) εβ (ˆ q ε ( t )) |∇ V ( x ) | u ε ( t, x ) dx,L = 1 k u ε ( t ) k Z R N \ B ˆ R ( λ ) εβ (ˆ q ε ( t )) |∇ V (ˆ q ε ( t )) | u ε ( t, x ) dx. By Proposition 3.1 and Lemma 4.5 we have(4.8) L < M λ and(4.9) L ≤ M ˆ R ( λ ) ε β . Finally,(4.10) L ≤ M λ + (cid:18) Cσ (cid:19) b λ − b , OLITON DYNAMICS FOR THE GENERALIZED CHOQUARD EQUATION 15 since, by (V1), (4.6), Proposition 3.1 and (4.4), for R ≥ R , k u ε ( t ) k Z R N \ B R (0) |∇ V ( x ) | u ε ( t, x ) dx ≤ k u ε ( t ) k Z R N \ B R (0) |∇ V ( x ) | /b u ε ( t, x ) dx ! b Z R N \ B R (0) u ε ( t, x ) dx ! − b ≤ k u ε ( t ) k Z R N \ B R (0) V ( x ) u ε ( t, x ) dx ! b λ − b ≤ (cid:18) C ( R + 1) a − σ (cid:19) b λ − b ≤ (cid:18) Cσ (cid:19) b λ − b and, again by Proposition 3.1, we have k u ε ( t ) k Z B R (0) \ B ˆ R ( λ ) εβ (ˆ q ε ( t )) |∇ V ( x ) | u ε ( t, x ) dx ≤ M λ.
So, by (4.7), (4.8), (4.9) and (4.10), we have | H ε ( t ) | ≤ CMσR a − + (cid:18) Cσ (cid:19) b λ − b + M (2 + 3 R ) λ + 2 M ˆ R ( λ ) ε β . At this point we can have k H ε ( t ) k L ∞ (0 , ∞ ) arbitrarily small choosing firstly R sufficiently large,secondly λ sufficiently small and, finally, ε small enough. (cid:3) Proof of Theorem 1.1.
By Theorem 4.6 we immediately conclude the proof of Theorem 1.1. (cid:3)
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