Orbital stability property for coupled nonlinear Schrödinger equations
aa r X i v : . [ m a t h . A P ] O c t Orbital stability property forcoupled nonlinear Schr ¨odinger equations
Liliane Maia ∗ , Eugenio Montefusco † , Benedetta Pellacci † Abstract
Orbital stability property for weakly coupled nonlinear Schr¨odinger equations is investigated.Di ff erent families of orbitally stable standing waves solutions will be found, generated by di ff er-ent classes of solutions of the associated elliptic problem. In particular, orbitally stable standingwaves can be generated by least action solutions, but also by solutions with one trivial componentwhether or not they are ground states. Moreover, standing waves with components propagatingwith the same frequencies are orbitally stable if generated by vector solutions of a suitable singleSchr¨odinger weakly coupled system, even if they are not ground states. sec.introd We consider the following Cauchy problem for two coupled nonlinear Schr¨odinger equations schr (1.1) i ∂ t φ + ∆ φ + (cid:16) | φ | p − + β | φ | p | φ | p − (cid:17) φ = , i ∂ t φ + ∆ φ + (cid:16) | φ | p − + β | φ | p | φ | p − (cid:17) φ = ,φ (0 , x ) = φ ( x ) , φ (0 , x ) = φ ( x ) , where Φ = ( φ , φ ) and φ i : ’ × ’ n → ƒ , φ i : ’ n → ƒ , p > β is a real positive constant.Coupled nonlinear Schr¨odinger equations appear in the study of many physical processes. For in-stance, such equations with cubic nonlinearity model the nonlinear interaction of two wave packets,optical pulse propagation in birefringent fibers or wavelength-division-multiplexed optical systems(see man,men [16, 17], aa,ka [1, 10] and the references therein).A soliton or standing wave solution is a solution of the form Φ ( x , t ) = ( u ( x ) e i ω t , u ( x ) e i ω t ) where U ( x ) = ( u ( x ) , u ( x )) : ’ n → ƒ is a solution of the elliptic system ellittico (1.2) − ∆ u + ω u = (cid:16) | u | p − + β | u | p − | u | p (cid:17) u , − ∆ u + ω u = (cid:16) | u | p − + β | u | p − | u | p (cid:17) u . Among all the standing waves we can distinguish between ground and bound states. A ground statecorresponds to a least action solution U of ( ellitticoellittico schrschr schrschr bs [4]). On the other hand, vector multi-hump solitons are of much interest in the applications, for ∗ Research partially supported by Projeto Universal / CNP q and FAPDF. † Research supported by MIUR project
Metodi Variazionali ed Equazioni Di ff erenziali non lineari . msc [18].When investigating stability properties of a given set of solution, it is natural to take into accountthe rotation invariance of the problem, and this is done by the orbital stability. Roughly speaking,this means that if an initial datum Φ is close to a ground state U then all the orbit generated by Φ remains close to the soliton generated by U up to translations or phase rotations.For the single Schr¨odinger equation it has been proved that the orbital stability property is enjoyedby standing waves raised by least action solutions. This result can be deduced from the two followingfacts (see cl [6] and Section 8 in caz [5]):(a) every least action solution can be associated, by a bijective correspondence, to a minimumpoint of the energy constrained to the L sphere with a suitable choice of the radius.(b) Conservation laws and compactness properties of this minimization problem imply that theset of minimum points of the energy on this sphere manifold generates stable standing waves.Moreover, in gss,ss [9, 24] it is proved that every critical point of the action with Morse index larger thanone give rise to instability. Taking into consideration the result of k [11] the stability of the standingwave e i ω t z ω of the single Schr¨odinger equation holds if and only if z ω is the minimum point of theassociated energy functional constrained to the L sphere of radius k z ω k L .This and (a) are the reasons why ground states are the most desirable solution for the single Schr¨odingerequation.With this situation in mind, large e ff ort has been done in the last few years to find ground statesof ( ellitticoellittico ka,shm [10, 26] numerical arguments or analytical expansions have been employed to producedi ff erent families of solitons. The investigation has been improved by means of variational methods.In ac,bw,dfl,lw,mmp,si [2, 3, 7, 12, 15, 25]) assumptions on the constant β are stated in order to distinguish betweenground states with both nontrivial components ( vector ground states ) and ground states with onetrivial component ( scalar ground states ). It has been discovered that there exists vector ground statesfor the constant β su ffi ciently large in dependence on the frequencies ratio, while if β is small leastaction solutions have necessarily one trivial component. Moreover, in ac [2] it is clarified the di ff erencebetween scalar and vector positive solutions in dependence to di ff erent geometrical properties of theaction functional. For β small the scalar ground states are critical points of the action functional withMorse index equal to one, while for β large these kind of solutions have larger Morse index. Sincestable standing waves should be generated only by ground states, these results suggested the idea thatstable standing waves should be given by scalar solutions for β small and by vector ground states for β large. This opinion is confirmed also in l [14] where this topic has been studied for di ff erent evolutionsystems, and the orbital stability property is shown to be enjoyed by standing waves associated tosolution of the corresponding elliptic system with Morse index equal to one.For the cubic NLS systems in the one dimensional case, this subject has been recently studied insome interesting papers using numerical and analytical methods. In y [27] it is conjectured, basedon numerical evidence, that single-hump soliton are stable while multi-hump vector solitons are alllinearly unstable and this is proved by numerical and analytical arguments in py [23] for p = schrschr p for special families of multi-hump vector solitons. In pk [22] a stability criterion is foundto study the stability property of some families of single-hump vector solitons. When tackling thismatter for weakly coupled Schr¨odinger equations by means of variational methods, one has to takeinto account that the L norms of the components are conserved separately (see fm,o [8, 21]). So that wecan consider di ff erent constrains on which minimize the energy. When we choose the sphere withrespect of the L × L norm, we obtain ground states, however we do not know whether or not theyare scalar or vector solutions. Otherwise, we could try to minimize the energy constraining the L norms separately, this approach will permit us to know in advance if we will find scalar or vector2olutions even if they may be not least action solutions. The first approach consists in solving theminimization problem intromini (1.3) E ( u ) = inf M γ E where M γ = n U = ( u , u ) ∈ H × H : ω k u k L + ω k u k L = γ o and E ( U ) = E ( u , u ) = k∇ u k + k∇ u k − p (cid:16) k u k p p + k u k p p + β k u u k pp (cid:17) , for 1 < p < + / n in order to have global existence of ( schrschr fm [8]).For a suitable choice of γ , we will find that this problem has a solution corresponding, in a bijectivecorrespondence, to ground states of ( ellitticoellittico mainstabmainstab intromini2 (1.4) E ( u ) = inf M ( δ ,δ E where M ( δ ,δ ) = n U = ( u , u ) ∈ H × H : k u k L = δ , k u k L = δ o . When δ (or δ ) is equal to zero we obtain as minimum point the couple ( z ω ,
0) (or (0 , z ω )) where z ω ( z ω ) is the unique positive solution of the first (second) equation in ( ellitticoellittico ellitticoellittico β small. However, we will show that they still produce orbitally stablestanding waves for any β > mainstab2mainstab2 y [27]. But they are in contrast with the expectation that only ground states should give rise to orbitallystable waves, since they have Morse index greater than one for β large. The case δ = δ has beentackled in oh [20] for p = , n = β =
1, and it is proved that the set of solutions of ( intromini2intromini2 schrschr oh [20] for higher dimension andfor every β > mainstab3mainstab3 oh [20], we will show that, for a suitable choice of δ (and δ = δ = δ ) the set of solutions of ( intromini2intromini2 B = n ( e i θ z ωβ ( · − y ) , e i θ z ωβ ( · − y )) , θ , θ ∈ ’ , y ∈ ’ n o , where z ωβ is the unique positive solution of the problem − ∆ u + ω u = (1 + β ) | u | p − u in ’ n , u ( x ) → | x | → ∞ , Moreover, arguing as in oh [20] we will demonstrate that B also characterizes the set of least actionsolution of ( ellitticoellittico ω = ω and when one prescribes both of the components to be di ff erentfrom zero (see Theorem caraboundcarabound si [25] and in Theorem 2 in lw [12] for λ j = ω for every j . Let us stressagain that the set B is made of ground states only for β ≥
1. Then, for β large we have at least twofamilies of orbitally stable solution of ( schrschr β but for ω = ω : orbital stability is enjoyed by the standing waves gener-ated by scalar solutions and by vector solution solutions of ( intromini2intromini2 β small, the latter are ground states for β large. Unfortunately we cannot handle the case δ , δ , andto our knowledge the question of whether or not the set of solution of ( intromini2intromini2 δ , δ is open. 3inally, adapting the arguments in caz [5], we will also show an instability result in the supercritical case p = + / n , for ground state solutions, scalar solutions and for the set B (for ω = ω = ω ), as aconsequence of blowing up in finite time. While, for the critical case the instability is produced byevery solution of ( ellitticoellittico settingsetting preliminarypreliminary
3. In section proofsproofs setting
Our analysis will be carried out in the functional spaces Œ = L ( ’ n , ƒ ) × L ( ’ n , ƒ ) and ˆ = H ( ’ n , ƒ ) × H ( ’ n , ƒ ). We recall that the inner product between u , v ∈ ƒ is given by u · v = ℜ ( uv ) = / uv + vu ). Then for ω = ( ω , ω ), ω i ∈ ’ , ω i >
0, we can define an equivalent inner product in Œ given by ( Φ | Ψ ) ω = ℜ Z h ω φ ψ + ω φ ψ i , ∀ Φ = ( φ , φ ) , Ψ = ( ψ , ψ )and an equivalent norm k Φ k ,ω = k φ k ,ω + k φ k ,ω , where k φ i k ,ω i = ω i k φ i k = ω i Z φ i φ i for i = ,
2. It is known (see Remark 4.2.13 in caz [5]) that ( schrschr p < n / ( n −
2) when n > p for n = ,
2, in the space ˆ endowed with the norm k Φ k ˆ = k∇ Φ k + k Φ k ,ω for every Φ = ( φ , φ ) ∈ ˆ . Moreover we set the Œ p norm as k Φ k pp = k φ k pp + k φ k pp , for p ∈ [1 , + ∞ ). It is well known that the masses of the components of a solution andits total energy are preserved in time, that is the following conservation laws hold (see fm,o [8, 21]): mass (2.1) k φ k = k φ k , k φ k = k φ k , energy (2.2) E ( Φ ( t )) = k∇ Φ ( t ) k − F ( Φ ( t )) = (cid:13)(cid:13)(cid:13) ∇ Φ (cid:13)(cid:13)(cid:13) − F (cid:16) Φ (cid:17) = E (0) , where defF (2.3) F ( Φ ) = p (cid:16) k Φ k p p + β k φ φ k pp (cid:17) . In fm [8] it is proved that the solution of this Cauchy problem exists globally in time, under the assump-tion pzero (2.4) p < + n . In order to study orbital stability properties, we will use the functional energy (see ( energyenergy I ( U ) = k U k ˆ − F ( U ) = E ( U ) + k U k ,ω . round Definition 2.1
We will say that a ground state solution U of ( ellitticoellittico cne (2.5) I ( U ) = m N : = inf N I ( W ) where N : = { W ∈ ˆ \ { } : hI ′ ( W ) , W i = } . N is called in the literature Nehari manifold (see mmp,si [15, 25]). Moreover, we will denote with G the setof the ground state solutions. bound Definition 2.2
We will say that a positive bound state solution U of ( ellitticoellittico cnedue (2.6) I ( U ) = m : = inf N I ( W ) where N : = { W = ( w , w ) ∈ ˆ : w , w , , h ∂ I ( W ) , w i = h ∂ I ( W ) , w i = } , where ∂ I ( W ) ( ∂ I ( W )) is the partial derivative with respect to the first (second) component. N is called in the literature Nehari set (see lw,si [12, 25]). Moreover, we will denote with B the set of suchbound state solutions.It is well known (see Section 8 in caz [5], k [11]) that all the solutions of the elliptic problem schrequa (2.7) − ∆ u + ω u = (1 + β ) | u | p − u in ’ n , u ( x ) → | x | → ∞ , for ω > β ≥
0, are given by v ( x ) = e i θ z ωβ ( x − y ) where θ ∈ ’ y ∈ ’ n and z ωβ is the unique leastenergy solution , where z ωβ is the unique positive least energy solution in H ( ’ n , ’ ) of ( schrequaschrequa defzbo (2.8) z ωβ ( x ) = ω + β ! / p − z (cid:16) √ ω x (cid:17) . scalari Definition 2.3
Problem ( ellitticoellittico U = ( u ,
0) (or (0 , u )). We will denotewith S the set of such solutions. The uniqueness result in k [11] for the single Schr¨odinger equation,gives us the following characterization for the set S . S = n ( e i θ z ω ( · − y ) , , θ ∈ ’ , y ∈ ’ n o ∪ n (0 , e i θ z ω ( · − y )) , θ ∈ ’ , y ∈ ’ n o where z ω is defined in ( defzbodefzbo ω = ω or ω = ω ). Remark 2.4
The results contained in ac,bw,dfl,mmp,lw,si [2, 3, 7, 15, 12, 25] show that, depending on the parameters ω , ω , β , the set G may coincide with either B or S .For β su ffi ciently large in dependence on ω , ω , G = B and the point in S are scalar bound statessolutions. While, G = S for β small.In the particular case ω = ω = β ≥ ac,bw,mmp,si [2, 3, 15, 25]), so that for β ≥ G = B , while for β < G = S .Let us recall the orbital stability property for a set of solutions F , introduced for the single equationcase in cl [6]. 5 efinition 2.5 A set
F ⊆ ˆ of solutions of Problem ( ellitticoellittico ε > δ = δ ( ε ) > U ∈F k Ψ − U k ˆ < δ, then sup t ≥ inf V ∈F k Ψ ( t , · ) − V k ˆ < ε, where Ψ is the global solution of ( schrschr Ψ . Remark 2.6
We call the property in the previous definition orbital stability of F because everyelement ( u , u ) of F generates an orbit given by the standing wave ( e i ω t u , e i ω t u ).Roughly speaking, a set F is orbitally stable if any orbit generated from an initial datum Ψ close toan element of F remains close to F uniformly with respect to the time.Up to now the uniqueness of the ground state solution is an open problem for system ( schrschr Ψ togo from a ground state U to a di ff erent ground state solution V ; with this respect, it would be veryinteresting to know, at least, if ground states are isolated. In addition, we will show that there existalso other sets of orbitally stable solutions, then our definition has to take into account this aspect.Our main results are the following ones. mainstab Theorem 2.7
Assume ( pzeropzero . For any β, ω , ω > the set G is orbitally stable. Di ff erent from the single equations case, we have other families of orbitally stable solutions for thesystem, as the next results show. mainstab2 Theorem 2.8
Assume ( pzeropzero . For any β, ω , ω > the set S is orbitally stable. Remark 2.9
The preceding results imply that the problem ( schrschr β su ffi ciently large, ground state standing waves and scalar ones.While, for β small the stability property of scalar standing waves is a consequence both of Theorems mainstabmainstab mainstab2mainstab2 G = S in this case.If ω = ω = ω the set B is completely characterized in the next result. carabound Theorem 2.10
Assume ω = ω = ω > . For any β ≥ it holds B = n ( e i θ z ωβ ( · − y ) , e i θ z ωβ ( · − y )) , θ , θ ∈ ’ , y ∈ ’ n , z ωβ defined in ( defzbodefzbo o In other words the set B is described by the standing wave of the single equation, up to translationsand phase shifts of the components. This characterization of the set B leads us to show that the set B is orbitally stable even for β small. mainstab3 Theorem 2.11
Assume ( pzeropzero . For any β ≥ and ω = ω = ω > the set B is orbitally stable. Remark 2.12
1. These results imply that solutions that starts from initial data close to groundstates with both nontrivial components remain close to orbits generated by ground states withboth nontrivial components. While, solutions that start close to S will stay close to orbitsgenerated by S . 6. ¿From the preceding results we deduce that, for ω = ω , B and S are always orbitally stablesets independently of β . When ω , ω , we have that S is always orbitally stable, while wecan prove that B is orbitally stable only when it coincides with G , that is for β large. It is anopen problem to study the stability property for B for any ω , ω . Our results cover thefollowing cases:(a) ω = ω positive; for any β ≥ ω , ω , β large (in dependence of ω /ω ) such that B = G .We will also prove an instability result for the sets G , S and B (for ω = ω ) in dependence of theexponent p . More precisely, we will show the following results. instap Theorem 2.13
Assume p < n / ( n − . For any ω , ω , β > the following conclusions hold:a) Let p > + / n, then the sets G , S are unstable in the following sense:For any U ∈ G (or U ∈ S ) and ε > there exists U ε with k U ε − U k ˆ ≤ ε such that the solution Φ ε satisfying Φ ε (0) = U ε blows up in a finite time in ˆ .b) Let p = + / n, then every solution of ( ellitticoellittico is unstable in the sense of the previous conclusion. instapB Theorem 2.14
Assume + / n < p < n / ( n − and ω = ω = ω . For any β > , the set B isunstable in the following sense:For any U ∈ B and ε > there exists U ε with k U ε − U k ˆ ≤ ε such that the solution Φ ε satisfying Φ ε (0) = U ε blows up in a finite time in ˆ . preliminary In this section we will present some general results which will be useful in proving Theorems mainstabmainstab mainstab2mainstab2 mainstab3mainstab3 instapinstap l2 Definition 3.1
Given γ >
0, let us consider the minimization problemsmin M γ I ( U ) = m γ , elledueI (3.1) min M γ E ( U ) = c γ , elledueE (3.2)where M γ = n U ∈ ˆ : k U k ,ω = γ o . Moreover, we denote with A the set of the solution of problem( elledueEelledueE elledueIE Remark 3.2
Notice that, solving problem ( elledueIelledueI elledueEelledueE V ∈ M γ we have I ( V ) = E ( V ) + γ/ reali Remark 3.3
It results A = { ( e i θ u , e i θ u ) , θ j ∈ ’ , ( u , u ) ∈ H ( ’ n , ’ ) solves ( elledueErelledueEr } , where elledueEr (3.3) σ ’ = inf n E ( V ) : V ∈ H ( ’ n ; ’ ) , ω k v k + ω k v k = γ o . elledueEelledueE c γ = σ ’ and if U = ( u , u )solves ( elledueEelledueE θ j ∈ ’ such that u j = e i θ j | u j | for j = ,
2. For more details, see Remark 3.12of mopesq [19].
Remark 3.4
The conservation laws of the problem suggest that orbital stability has to be studied byusing problem ( elledueIelledueI p < + / n , as for p > + / n , E (and then I )is not bounded from below on M γ . We will prove our stability result using problem ( cnecne p < n / ( n − p < n / ( n − instapinstap cnecne elledueEelledueE elledueIelledueI p < + / n . Indeed, in thisrange of exponents we can construct a bijective correspondence between the negative critical valuesof E on M γ and the critical values of I on N . While, for p > + / n we cannot derive this mapbetween these critical values.This suggests that for p < + / n the Nehari manifold and M γ have the same tangent planes, whilewhen p > + / n the tangent planes are di ff erent, so that a minimum point on N would probably giverise to a di ff erent critical point on M γ . This point is crucial in proving orbital stability properties,since the conservation laws show that the dynamical analysis has to be performed on M γ .In proving many of the results of this section we will make use of the following lemma the proof ofwhich is straightforward. ulamu Lemma 3.5
For any u ∈ H ( ’ n , ƒ ) and for any positive real numbers λ, µ , we can define the scalingu µ,λ ( x ) = µ u ( λ x ) such that the following equalities hold. scaling (3.4) k u µ,λ k = µ λ − n k u k , k∇ u µ,λ k = µ λ − n k∇ u k , k u µ,λ k p p = µ p λ − n k u k p p . Proof.
The proof is an immediate consequence of a change of variables.First, we want to show the equivalence between problems ( cnecne elledueEelledueE elledueIelledueI K E = n c < ∃ U ∈ M γ : E ( U ) = c , ∇ M γ E ( U ) = o , ˜ K E = n U ∈ M γ : ∇ M γ E ( U ) = , E ( U ) < o , defkI (3.5) K I = (cid:8) m ∈ ’ : ∃ U ∈ N : I ( U ) = m , I ′ ( U ) = (cid:9) , ˜ K I = (cid:8) U ∈ N : I ′ ( U ) = (cid:9) , where we have denoted with ∇ M γ E the tangential derivative of E on M γ . The following result holds. critici Theorem 3.6
Assume ( pzeropzero . For any β, ω , ω > the following conclusions hold:a ) there exists a bijective correspondence between the sets ˜ K E and ˜ K I ,b ) there exists a bijective map T : K I → K E given by mec (3.6) T ( m ) = − " p − − n γ p ′ − n p ′− n / ( p − − n " m / ( p − − n , where p ′ = p / ( p − stands for the conjugate exponent of p. Proof.
In order to prove assertion a) , take V = ( v , v ) ∈ M γ such that V satisfies evincolato (3.7) hE ′ ( V ) , V i = − νγ, E ( V ) = c < . h F ′ ( V ) , V i = pF ( V ), it follows − νγ − c = hE ′ ( V ) , V i − E ( V ) = F ( V )(1 − p ) < , showing that ν is a positive real number. Therefore it is well defined the map T µ,λ : M γ → N T µ,λ ( V ) = V µ,λ , where µ, λ , are given by parameters (3.8) µ = ν − / p − , λ = ν − / . Using this and ( evincolatoevincolato V µ,λ solves ( ellitticoellittico T µ,λ ( V ) belongs to ˜ K I . Vice-versa if U ∈ ˜ K I let us take ν > ν / ( p − − n / = γ k U k ,ω and λ, µ > λ = ν / , µ = ν / p − , so that U µ,λ belongs to M γ , ∇ M γ E ( U ) =
0. This shows that ( T µ,λ ) − = T /µ, /λ .In order to prove assertion b) , note first that any m ∈ K I is positive. Indeed, since there exists U ∈ N such that I ( U ) = m and I ′ ( U ) =
0, it follows m = I ( U ) − p hI ′ ( U ) , U i = − p ! k U k ˆ > , so that T is a well defined and injective map. Let us first show that if c ∈ K E ∩ ’ − , then c = T ( m ).Indeed take V ∈ M γ corresponding to such c , and take T µ,λ ( V ) = V µ,λ . Recalling Pohozaev identity(see (5 .
9) in mmp [15]) and since V µ,λ ∈ N we get − p ! (cid:16) k∇ V µ,λ k + k V µ,λ k ,ω (cid:17) = m , ( n − k∇ V µ,λ k + n k V µ,λ k ,ω = np (cid:16) k∇ V µ,λ k + k V µ,λ k ,ω (cid:17) , where m = I ( V µ,λ ). We derive poho (3.9) k∇ V µ,λ k = nm , F ( V µ,λ ) = mp − , k V µ,λ k ,ω = pp − − n ! m . Using ( scalingscaling scaling2 (3.10) µ λ n − k∇ V k = nm , µ p λ n F ( V ) = mp − , µ λ n k V k ,ω = pp − − n ! m . Since V is in M γ , ( parametersparameters γ = k V k ,ω = ν / ( p − − n / pp − − n ! m , scaling2scaling2 k∇ V k = n γ p ′ − n ! + p − − n ( p − m ! p − − n ( p − , F ( V ) = p − γ p ′ − n ! + p − − n ( p − m ! p − − n ( p − . All the above calculations imply that, if c is a negative constrained critical value of E on M γ and m is the corresponding critical value of I , than c is given by ( mecmec T − is surjective let us take m in K I and the corresponding U that satisfies theconditions in ( defkIdefkI ν > λ = ν / , µ = ν / p − and consider U µ,λ . Using ( pohopoho scalingscaling U µ,λ ∈ M γ imply that ν is related to γ bythe expression ν / ( p − − n / = γ m " p ′ − n . Moreover, since U is a free critical point of I we obtain that U µ,λ is a constrained critical pointof E with Lagrange multipliers equal to ν . In order to conclude the proof we have to impose that E ( U µ,λ ) = c . ¿From conditions ( pohopoho scalingscaling K I it follows that c , m and ν satisfy c = m " n − p − ν p ′ − n / and substituting the value of ν in dependence of γ implies that m = T − ( c ). equiground Corollary 3.7
There exists a bijective correspondence between the sets G and A . Proof.
Let V ∈ A and take T µ,λ ( V ); Theorem criticicritici T µ,λ ( V ) is a critical point of I , sothat we only have to show that I ( T µ,λ ( V )) = m = m N . Indeed, suppose by contradiction that m > m N . In mmp [15] it is proved that m N is achieved by a vector U , then U /µ, /λ , with µ, λ as in ( parametersparameters M γ and gives a negative critical value c given by( mecmec m N < m we get c < c γ which is a contradiction, so that the claim is true.Using the preceding result and Theorem 2 . mmp [15] we can prove the following statement. groundstate Theorem 3.8
Assume ( pzeropzero . For any β, ω , ω > , there exists a solution of the minimization prob-lems ( elledueIelledueI , ( elledueEelledueE . Proof.
As observed in Remark elledueIEelledueIE elledueIelledueI elledueEelledueE elledueEelledueE fm [8] equation (9)), we get that thefollowing inequality holds for any U ∈ M γ E ( U ) ≥ k∇ U k n ( p − " k∇ U k − n ( p − − C ω,β p γ p − ( p − n / ,
10o that E is bounded from below if and only if ( pzeropzero c γ in( elledueEelledueE λ n = µ so that, for any U ∈ M γ , U µ,λ = (cid:16) u µ,λ , u µ,λ (cid:17) still belongsto M γ . By ( scalingscaling h defined by h ( λ ) = E ( U µ,λ ) = λ k∇ U k − λ n ( p − F ( U ) , and from condition ( pzeropzero λ = λ ( U ) > λ ∈ (0 , λ ) h ( λ ) is negative and this shows the claim. Then, Theorem criticicritici m ∈ K I suchthat c γ = T ( m ). Finally, since in mmp [15] it is proved that m N is achieved, using Corollary equigroundequiground gamma Remark 3.9
It is easy to see that every U in G satisfies k U k ,ω = m N pp − − n ! , thanks to the regularity properties of U and to Pohozaev identity. muguali Theorem 3.10
Assume ( pzeropzero and let γ be fixed as defgamma (3.11) γ = m N pp − − n ! . Then m N = m γ . Proof.
From the definition of γ immediately follows that m N ≥ m γ . In order to show that theequality is achieved, we only have to observe that m γ = c γ + γ / = T ( m N ) + γ /
2. Using thedefinition of T joint with ( defgammadefgamma Remark 3.11
Consider the minimization problems ( cnecne cner (3.12) m N ’ : = inf {I ( W ) : W ∈ H ( ’ n , ’ ) \ { } : hI ′ ( W ) , W i = } , it results that m N = m N ’ . Indeed, m N ≤ m N ’ . Moreover, if U ∈ ˆ , U = ( u , u ) is a solution of( cnecne k U k ,ω = γ , where γ is defined in ( defgammadefgamma U ∈ M γ and Theorem mugualimuguali E ( U ) = c γ . Then, from Remark realireali θ , θ such that U = ( u , u ) = ( e i θ | u | , e i θ | u | ) , E ( | u | , | u | ) = c γ . Finally, Theorem mugualimuguali I ( | u | , | u | ) = m N and, since |∇ u i | = |∇| u i || it results that ( | u | , | u | ) ∈N is a solution of ( cnercner m N is achieved on a vector with real valued components. Further-more, we have shown that G = { ( e i θ u , e i θ u ) , θ j ∈ ’ , ( u , u ) ∈ H ( ’ n , ’ ) solves ( cnercner } . In order to prove the instability result Theorem instapinstap erre (3.13) R ( U ) = k∇ U k − n ( p − F ( U )and the infimum m P = inf P I where P = { U ∈ ˆ : R ( U ) = } . The following results hold. 11 nfs
Proposition 3.12
Assume that p > + / n, then the following conclusions hold:a ) P is a natural constraint for I ;b ) m P = m N . Proof.
In order to prove a ) let us consider U a constrained critical point of I on P , then there exists λ ∈ ’ such that the following identities are satisfied uno (3.14) (1 − λ ) k∇ U k + k U k ,ω = p [ λ n (1 − p ) + F ( U ) , due (3.15) (1 − λ ) (cid:18) n − (cid:19) k∇ U k + n k U k ,ω = n [ λ n (1 − p ) + F ( U ) , tre (3.16) k∇ U k = n ( p − F ( U ) . Hence, using ( tretre unouno k U k ,ω = " λ (1 − p ) + pn ( p − − k∇ U k , and using this and ( tretre duedue p > + / n , we obtain that λ = U is a free critical point of I .In order to prove b ) take a minimum point U of I in P ; from a ) it follows that then U belongs to N so that m P ≥ m N ; viceversa if V is a minimum point of I in N then V is a free critical point of I and Pohozaev identity implies that V ∈ P so that m P ≤ m N , yielding the conclusion. glambda Lemma 3.13
Assume that p > + / n. For any U , (0 , let us consider U λ n / ,λ = (cid:16) u λ n / ,λ , u λ n / ,λ (cid:17) ,for u µ,λ defined in Lemma ulamuulamu ) there exists a unique λ ∗ = λ ∗ ( U ) such that U λ n / ∗ ,λ ∗ belong to P ,b ) the function g ( λ ) = I (cid:16) U λ n / ,λ (cid:17) has its unique maximum point in λ = λ ∗ ,c ) λ ∗ < if and only if R ( U ) < and λ ∗ = if and only if R ( U ) = ,d ) the function g ( λ ) is concave on ( λ ∗ , + ∞ ) . Proof. a ): for any λ > R (cid:16) U λ n / ,λ (cid:17) = λ k∇ U k − λ n ( p − n ( p − F ( U ) , then there is a unique λ ∗ ( U ) = λ ∗ = k∇ U k n ( p − F ( U ) / [ n ( p − − such that R (cid:16) U λ n / ∗ ,λ ∗ (cid:17) =
0. Computing the first derivative of g ( λ ) b ) is proved. Since p > + / n , c )easily follows. d ) immediately follows from writing the second derivative of the function g . RI Lemma 3.14
For any U ∈ ˆ with R ( U ) < it results R ( U ) ≤ I ( U ) − m N . roof. Conclusion d) in Lemma glambdaglambda g (1) ≥ g ( λ ∗ ) + g ′ (1)(1 − λ ∗ ) . Direct computation yields I ( U ) = g (1) ≥ g ( λ ∗ ) + g ′ (1)(1 − λ ∗ ) = g ( λ ∗ ) + R ( U )(1 − λ ∗ ) ≥ I (cid:18) U λ n / ∗ ,λ ∗ (cid:19) + R ( U ) , where in the last inequality we have used that R ( U ) <
0. Recalling that U λ n / ∗ ,λ ∗ ∈ P and applyingconclusion b ) of Proposition infsinfs It is well known (see Section 8 in caz [5]) that z ωβ defined in ( defzbodefzbo miniequa (3.17) E ( u ) = c δ = min M δ E where M δ = { u ∈ H ( ’ n , ’ ) : k u k = δ } ;where the functional E : H ( ’ n ) → ’ is defined by defE1 (3.18) E ( u ) = k∇ u k − β + p k u k p p . and when we prescribe δ = δ ( ω ) = ω p − − n ( β + p − k z k . Otherwise, z ωβ can be equivalently obtained as the solution of the minimization problem nehari1 (3.19) I ( u ) = m = min N I where N = n u ∈ H ( ’ n , ’ ) , u . hI ′ ( u ) , u i = o ;where the functional I : H ( ’ n , ’ ) → ’ is defined by I ( u ) = k∇ u k + ω k u k − β + p k u k p p . The following result is the starting point in proving Theorem caraboundcarabound moduliN
Proposition 3.15
Assume that ω = ω = ω . If U solves ( cneduecnedue it results | u | = | u | almost every-where. Proof.
Consider the variational characterization z ωβ as the solution of ( nehari1nehari1 Z = ( z ωβ , z ωβ )belongs to N , so that 2 m = I ( z ) = I ( Z ) ≥ m . Let now U be a solution of ( cneduecnedue m = I ( U ) ≥ I ( u ) + I ( u ) ≥ m ≥ m equaI (3.20) I ( U ) = I ( u ) + I ( u ) , and m = m . Writing down this equality we get β p Z | u u | p = β p Z (cid:16) | u | p + | u | p (cid:17) , that is Z ( | u | p − | u | p ) = , giving the conclusion. Proof of Theorem caraboundcarabound
Let U = ( u , u ) ∈ B , from Proposition moduliNmoduliN k∇ u k + ω k u k = k u k p p + β k u u k pp = ( β + k u k p p , that is u ∈ N , so that disI (3.21) I ( u ) ≥ I ( z ωβ ) . On the other hand, from Proposition moduliNmoduliN equaIequaI I ( U ) = I ( u ) and recallingthat Z = ( z ωβ , z ωβ ) ∈ N we derive2 I ( u ) = I ( U ) ≤ I ( Z ) = I ( z ωβ ) . This, ( disIdisI nehari1nehari1 u = e i θ z ωβ ( · − y ) for some θ ∈ ’ and y ∈ ’ n . The sameargument for u gives u = e i θ z ωβ ( · − y ), and Proposition moduliNmoduliN y = y .As we did for ground states we want to investigate the connection of Problem ( cneduecnedue E under suitable constraints. Since we are now considering vectors with both nontrivialcomponents we are naturally lead to study the following problem for E . cmdue (3.22) E ( U ) : = c ( δ ,δ ) = inf M ( δ ,δ E where M ( δ ,δ ) = { ( u , u ) ∈ ˆ : k u k = δ , k u k = δ } where δ , δ are positive real numbers.If δ , δ we do not know how to solve problem ( cmduecmdue cmduecmdue δ = δ = δ . In this case we have the followingminimization problem ohta (3.23) E ( U ) = c ( δ,δ ) = inf M ( δ,δ ) E where M ( δ,δ ) = n ( u , u ) ∈ ˆ : k u k = k u k = δ o . As we did for the ground state solutions, investigating the relation between Problems ( ohtaohta cneduecnedue ω = ω = ω naturally lead us to choose δ = δ ( ω ) such that every solution of ( ohtaohta ellitticoellittico B (given in Theorem caraboundcarabound A ( δ ( ω ) ,δ ( ω )) set of solutions of ( ohtaohta oh [20] for the case p = β = araA Proposition 3.16
It results A ( δ ( ω ) ,δ ( ω )) = n ( e i θ z ωβ ( · − y ) , e i θ z ωβ ( · − y )) , θ i ∈ ’ , y ∈ ’ n o . Proof.
Taking U = ( u , u ) ∈ A ( δ ( ω ) ,δ ( ω )) , using the variational characterization z ωβ as the solutionof ( miniequaminiequa moduliNmoduliN moduli (3.24) | u | = | u | , E ( U ) = E ( u ) + E ( u ) , c ( δ,δ ) = c δ Moreover, k u i k = δ ( ω ), so that u i ∈ M δ and it results E ( u i ) ≥ c δ . Then ( modulimoduli E ( u i ) = c δ .Thus, u i are solutions of the minimization problem ( miniequaminiequa k [11] and Theorem II.1 in cl [6] to deduce that there exist θ , θ , y , y such that u = e i θ z ωβ ( · − y ), u = e i θ z ωβ ( · − y ). Moreover, ( modulimoduli y = y . proofs In this section we will prove the main results concerning the stability (or instability) of the standingwaves. In particular in the following subsections we show, in the subcritical case 1 < p < + / n ,the orbital stability of the sets G and S , and also of the set B for ω = ω . Finally in subsection subcritsubcrit p > + / n the sets G , S and B (for ω = ω ) are unstable and for p = + / n the instability holds for every bound state. The proofs of Theorems mainstabmainstab mainstab2mainstab2 instapinstap instapBinstapB cl,caz [6, 5] for the single equation, while the proof of Theorem mainstab3mainstab3 oh [20]. Proof of Theorem mainstabmainstab
Let us argue by contradiction, and suppose that there exist ε > { t k } ⊂ ’ and a sequence of initial data { Φ k } ⊂ ˆ such that ipodati (4.1) lim k →∞ inf U ∈G k Φ k − U k ˆ = { Φ k } of Problem ( schrschr iposol (4.2) inf U = ( u , u ) ∈G k Φ k ( · , t k ) − ( u , u ) k ≥ ε . Condition ( ipodatiipodati groundground defgammadefgamma mugualimuguali I yield I ( Φ k ) → m γ k Φ k k ,ω = γ + o (1) . Let us denote Ψ k ( x ) = Φ k ( x , t k ), then, conservation laws ( massmass energyenergy I ( Ψ k ) → m γ , k Ψ k k ,ω = γ + o (1) . Following the arguments of mmp [15], we use the Ekeland variational principle to obtain a new minimizingsequence ˜ Ψ k which is also a Palais-Smale sequence for I and we find ˜ Ψ such that I ( ˜ Ψ ) = m γ , I ′ ( ˜ Ψ ) = . − p ! k ˜ Ψ k ˆ = m γ = I ( ˜ Ψ k ) − p hI ′ ( ˜ Ψ k ) , ˜ Ψ k i + o (1) = − p ! k ˜ Ψ k k ˆ + o (1)showing that k ˜ Ψ k k ˆ → k ˜ Ψ k ˆ and, since ˆ is an Hilbert space, we have the strong convergence in ˆ . By the choice of ˜ Ψ k wederive that also Ψ k → ˜ Ψ strongly in ˆ , which is an evident contradiction with ( iposoliposol Remark 4.1
In the proof of the previous result it was crucial to show that the sequence ˜ Ψ k stronglyconverges in ˆ , to get the desired contradiction. In other words, in proving orbital stability re-sults we made use of conservation laws, minimization property and compactness of the minimizingsequence. Proof of Theorem mainstab2mainstab2
Let us argue for the set of scalar solution with the second component equalto zero, the other case can be handled analogously. The conclusion can be obtained arguing as in theprevious Theorem assuming that there exist ε > { t k } ⊂ ’ and a sequence of initial data { Φ k } ⊂ ˆ such that lim k →∞ inf θ ∈ ’ , y ∈ ’ n k Φ k − ( e i θ z ω ( · − y ) , k ˆ = { Φ k } of Problem ( schrschr absurd (4.3) inf θ ∈ ’ , y ∈ ’ n k Φ k ( · , t k ) − ( e i θ z ω ( · − y ) , k ≥ ε As before, via conservation laws, Ψ k ( x ) = Φ k ( x , t k ) satisfies consescal (4.4) E ( Ψ k ( t )) → E ( z ω , = c ( δ , , k ψ k , k = k z ω k + o (1) , k ψ k , k = o (1) , where E ( z ω , = c ( δ , = min M ( δ , E , M ( δ , = { U ∈ ˆ : k u k = k z ω k = δ , k u k = } . ¿From Gagliardo-Nirenberg inequality we deduce that Ψ k is bounded in ˆ , then interpolation in-equality, joint with ( consescalconsescal convforte (4.5) ψ k , → L p . Therefore, Holder inequality yields Z | ψ k , | p | ψ k , | p ≤ k ψ k , k p p k ψ k , k p p → . This and ( convforteconvforte gradient (4.6) E ( Ψ k ) = E ( ψ k , ) + k∇ ψ k , k + o (1) , E ( u ) = k∇ u k − p k u k p p . So that we are lead to E ( ψ k , ) ≤ E ( Ψ k ) + o (1) = c ( δ , + o (1) . Consider the scalar minimization problem E ( z ω ) = c δ = min M δ E M δ = { u ∈ H ( ’ n , ’ ) : k u k = k z ω k } . It is easy to verify that equac (4.7) c δ = c ( δ , , so that scalarc (4.8) E ( ψ k , ) ≤ E ( z ω ) + o (1) = c δ + o (1) , k ψ k , k → k z ω k . By using Gagliardo-Nirenberg type inequality and arguing as in the proof of Theorem groundstategroundstate c ( δ , <
0, so that for k large, E ( Ψ k ) ≤ c ( δ , /
2. This and ( convforteconvforte k ψ k , k p p > σ , σ > . This, ( convforteconvforte scalarcscalarc cl [6] (see also caz [5]), and by means of concentration com-pactness technique, get the strong convergence (up to a subsequence) of ψ k , . Then, there exists ψ such that k ψ k = k z ω k , so that E ( ψ ) ≥ c δ , but, passing to the limit in ( scalarcscalarc E ( ψ ) = c δ . Therefore, there exist θ ∈ ’ and y ∈ ’ n such that ψ = e i θ z ω ( · − y ). Moreover, ( consescalconsescal gradientgradient equacequac ψ k , → H ,giving a contradiction with ( absurdabsurd Proof of Theorem mainstab3mainstab3
Arguing by contradiction, as in the proofs of the other stability results,and using Theorem caraboundcarabound caraAcaraA ε and a sequence Ψ k suchthat ultima (4.9) inf θ ∈ ’ , y ∈ ’ n k Ψ k − ( e i θ z ωβ ( · − y ) , e i θ z ωβ ( · − y )) k ≥ ε , k ψ k , i k → k z ωβ k , E ( Ψ k ) → c ( δ,δ ) Following the proof of Lemma 2.3 in oh [20], it can be proved that the sub-additivity condition holdsfor Problem ( ohtaohta lions [13]) we obtainthat Ψ k is compact and the conclusion follows passing to the limit in ( ultimaultima subcrit In this subsection we will prove Theorem instapinstap instapBinstapB
Proof of Theorem instapinstap
In order to prove conclusion a ) let us assume 1 + / n < p < n / ( n −
2) andconsider first the set G . Let U ∈ G so that U ∈ P . When we fix U s = U s n / , s with s > λ ∗ ( U s ) < c ) of Lemma glambdaglambda R ( U s ) < b) of Lemma glambdaglambda questa (4.10) I ( U s ) = g (1) < g ( λ ∗ ) = I ( U ) = m N . Φ s the solution generated by U s . By ( energyenergy I ( Φ s ) = I ( U s ), when the solution exists. Bycontinuity R ( Φ s ( t )) < t small; moreover, Lemma RIRI questaquesta R ( Φ s ( t )) ≤ I ( Φ s ( t )) − m N = − σ < , showing that R ( Φ s ( t )) < V ( t ) = k| x | Φ s ( t ) k , it follows that V ′′ ( t ) = R ( Φ s ( t )) ≤ − σ . Thus, there exists T ∗ such that V ( T ∗ ) = Φ s blows up in T ∗ (see fm [8]), which gives conclusion a) for the set G .Let now U ∈ S then U = ( u ,
0) (for example) with u = e i θ z ω . u is unstable for the singleequation(with β =
0) then Theorem 8.2.2 in caz [5] implies that there exists u ε, such that k u ε, − u k ≤ ε and the solution generated by u ε, , φ ε, blows up in finite time. Now if we choose U ε = ( u ε, ,
0) weget k U ε − U k ≤ ε and the solution generated by U ε is (by the well posedness of the Cauchy problem) Φ ε = ( φ ε, ,
0) blows up in finite time.Now consider p = + / n . From Proposition infsinfs U solution of ( ellitticoellittico tretre R ( U ) = R ( λ U ) < λ >
1. Let U λ = λ U be the initial datum of ( schrschr Φ λ the corresponding solution. ( energyenergy > R ( U λ ) = E ( U λ ) = E ( Φ λ ) = R ( Φ λ ) , so that, also in this case, the variance is concave and the solution Φ λ blows up in finite time. Proof of Theorem instapBinstapB
Let U ∈ B then U = ( u , u ) with u = e i θ z ωβ and u = e i θ z ωβ , again z ωβ is unstable because it is a ground state for the single equation with coe ffi cient β + caz [5] to obtain an initial datum u ε such that k u ε − z ωβ k ≤ ε and the solution that starts from u ε φ ε blows up in finite time. If we choose U ε = ( u ε , u ε ) we have (by the well posedness of the Cauchy problem) that the solution generatingfrom U ε is Φ ε = ( φ ε , φ ε ) and it blows up in finite time. In summary, we have studied the problem of the orbital stability of standing waves in two weaklycoupled nonlinear Schr¨odinger equations. In analogy of what happens for the single equation casewe have that least action solutions give rise to orbitally stable standing waves. But, the system admitsalso other families of orbitally stable standing waves, for example the set of solutions with one trivialcomponent whose elements are not ground states (and have Morse index greater than one) for β large.Moreover, for ω = ω least action solution with both nontrivial components also generate orbitallystable solutions of ( schrschr ellitticoellittico I , equal to one is not a necessary property to gain orbitalstability. In our opinion this is linked to the facts that the L norms of the components are conservedseparately.We remark that it remains open the question of the stability for the set of minima of the energywhose components have di ff erent L norms, at least for β small. Moreover, it is an interesting openproblem to find conditions, maybe related to the geometrical properties of I , on a solution in orderto produce instability. More precisely, it would be interesting to understand how to extend the resultof gss [9] for this kind of system. 18 cknowledgement. Part of this work was developed while the second autor was visiting the
Uni-versidade de Bras´ılia : he is very grateful to all the
Departmento de Matem´atica for the warm hos-pitality. Moreover he wishes to thank also professor Gustavo Gilardoni for many stimulating andfriendly discussions.The authors wish to thank Louis Jeanjean and Hichem Hajaiej for some very useful and stimulatingobservations.
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Eugenio Montefusco,Dipartimento di Matematica,SapienzaUniversit`a di Roma,piazzale A. Moro 5, 00185 Roma, Italy.E-mail address: [email protected]
Benedetta Pellacci,Dipartimento di Scienze Applicate, 20niversit`a degli Studi di Napoli Parthenope,Centro Direzionale, Isola C4 80143 Napoli, Italy.E-mail address: [email protected]@uniparthenope.it