PARTIALLY CONCENTRATING STANDING WAVES FOR WEAKLY COUPLED SCHRÖDINGER SYSTEMS

A bstract . We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations


Introduction
The mathematical analysis of singularly perturbed semilinear elliptic equations and systems has been the object of a wide range of studies in the last decades.Among the many motivations, a big role is provided by models in Quantum Mechanics, and in particular by the semiclassical analysis of Schrödinger-type equations.In this context, one postulates that the classical Newtonian Mechanics should be recovered from the Quantum one by letting the Planck constant ℏ vanish.Accordingly, the wave function of the quantum particle should concentrate and collapse to one or more Dirac's deltas, which position should describe the sharp location of classical particles.When different quantum waves interact, e.g. in the case of weakly coupled NLS systems, the commonly investigated setting is the one in which all the waves concentrate in point particles.From the analytical point of view, this study lets different challenges arise.On the one hand, one may ask what is the limit concentrating profile at specific energy levels, for instance for ground states; this is typically done exploiting variational methods and blow-up analysis.On the other hand, solutions concentrating with prescribed shape and position can be constructed, mainly using the Lyapunov-Schmidt reduction approach.
A largely studied model is the case of a binary mixture of Bose-Einstein condensates, usually described by the Gross-Pitaevskii system, namely a systems of two weakly coupled nonlinear Schrödinger equations   for N = 1, 2, 3. Here, ψ 1 and ψ 2 are the order parameters of the two components of the mixture, m 1 and m 2 the corresponding masses, and V and W the external potentials, bounded from below.In general, the (trapping) potentials may be different, opening interesting possibilities concerning the geometrical configurations of the condensates.Finally, the interaction parameters µ 1 , µ 2 , β depend on the scattering lengths associated to the different states.These interaction parameters can be fine-tuned across a wide rage of values, profiting from the presence of a Feshbach resonance.For more details on this model we refer to the book by Pitaevskii and Stringari [21], in particular Chaps.5 and 21.
Looking for standing waves (ψ 1 (x, t), ψ 2 (x, t)) = (e iE 1 t/ℏ u(x), e iE 2 t/ℏ v(x)) of frequencies E i we have that (u, v) with V(x) = E 1 + V(x) and W(x) = E 2 + W(x), for E i such that inf R N V > 0 and inf R N W > 0.
The usually studied case is the one in which µ i , β are both very large, or ℏ is very small with respect to the other parameters, so that one is lead to consider the singularly perturbed elliptic system While the autonomous case, namely the case V i ≡ λ i with λ i positive constants, has been widely studied in the last two decades, (see the recent paper [25] for an exhaustive list of references), a few results concerning the non-autonomous situation are known.
The first result seems due to Lin and Wei [11] who studied the case of a binary mixture in a singularly perturbed regime and in presence of trapping potentials.They prove the existence of ground state solutions, derive their asymptotic behaviors as ε → 0 and show that each component has one maximum point (possibly the same), called spike, which is trapped at the minimum points of the potentials V i .The existence of a concentrating ground state solutions was also established by Montefusco, Pellacci and Squassina [15], Pomponio [22], Ikoma and Tanaka [7] and Byeon [4].All the previous papers are concerned with system (1.2) where both the equations are affected by the presence of the small parameter ε, so that every wave (given by the components of the vector solution) concentrates as ε approaches zero.
Here, we focus on another type of regime, as it may happen that only some of the waves act in a semiclassical way, while the others persist in a quantum behavior.To the best of our knowledge, this kind of analysis is not present in the PDEs literature yet, and this paper is a first contribution in this direction.
More precisely, in our study we consider 2m 1 = ℏ 2 and m 2 = 1 2 in (1.1), so that (u, v) solves the following elliptic weakly coupled system where ε 2 := ℏ 2 .According to the previous discussion, along this paper we deal with the system above in the singularly perturbed regime ε → 0.Moreover, our study will deal with the case of µ i > 0 and β < 0, corresponding to positive intraspecies and to a negative interspecies scattering length, describing a repulsive interaction between the condensates.Our analysis will include the class of potentials satisfying the following assumptions.
which is non-degenerate in the space H 1 e (R N ) of functions even with respect to each variables, i.e.
that is the only solutions to (W) W(x) is even with respect to all the variables, Our main result is stated as follows.
Theorem 1.1.Let N = 2, 3, and suppose that (V 1 ), (V 2 ) and (W) hold.Assume set ω 0 := ω(0) > 0 and assume that Then there exists ε 0 > 0 such that for every ε ∈ (0, ε 0 ) there exists a solution (u ε , v ε ) of system (1.3) even with respect to each variable and having the following asymptotic profile as ε → 0 where Υ solves (1.5), U is the positive radial solution of and the peaks P ε and −P ε collapse to the origin as (1.12) Theorem 1.1 states the existence of a solution whose first component looks like a genuine solution to (1.5), in particular it does not concentrate, and whose second component concentrates at two opposite points which collapse to the origin as ε goes to 0. As a consequence of the coupling in the equations, the first component of (1.3) plays the role of an additional potential in the singularly perturbed second equation, so that the concentration will be triggered by the modified potential W − βΥ 2 .Because of the assumption β < 0, we will obtain a solution in the repulsive regime, and, when the origin is a maximum point of W our solution exists for every β negative; while when ∂ 2  11 W(0) > 0 we obtain a solution for β < β 0 < 0 (see (1.14)).
Let us make some comments.
Remark 1.2.We point out that, in case Υ satisfies then assumption (1.9) is satisfied as long as On the other hand, assumption (1.13) is verified in case V is radially non-decresing near 0 and Υ is radial and has a (local) strict maximum at 0, as one can verify applying Hopf's Lemma.Indeed, first we observe that, in such a case, for any i the function ∂ i Υ solves By Hopf's Lemma we deduce ∂ ii Υ(0) < 0 and (1.13) follows.
Remark 1.3.It is useful to recall the classical results concerning the case of constant potential, i.e. −∆U (1.15) It is well known that (1.15) has an unique positive solution which is radially symmetric and also that the set of solution of the corresponding linearized equation is spanned by the N partial derivatives ∂U ∂x i which are odd in each variable.In addition, U is radially decreasing and it satisfies the following exponential decay (see [2,3,9]) Remark 1.4.We observe that a class of potentials V which satisfy hypotheses (V 1 ) and (V 2 ) includes both the constant potentials and, at least in dimension N = 3, the trapping ones, like V(x) = λ + |x| m for some λ > 0 and m > 0. Indeed, (V 1 ) follows from Remarks 1. we believe that a version of such results should hold also in lower dimension, but this is far beyond the aim of this paper).
Remark 1.5.Our result relies on the simmetry of the potentials V and W which allows to build symmetric solutions with symmetric peaks P ε and −P ε collapsing to the origin.We strongly believe that a similar construction could be carried out in a more general setting in the spirit of Kang and Wei [8], when the radial solution of the first equation (1.5) is non-degenerate in the whole space H 1 (R N ) (i.e.V is a trapping potential as in [5,23]).In that case, it should be possible to build a solutions whose first component resembles the radial solution Υ of (1.5) and the second component has two different peaks collapsing to a maximum point of the modified potential W − βΥ 2 .
Remark 1.6.We will prove Theorem 1.1 using a classical Lyapunov-Schmidt reduction.This will allow us to build each component with a prescribed profile: the first component will look as one bump solution for ε sufficiently small, while the second will develop two spikes collapsing at the origin and will be exponentially small far from them.In performing this classical procedure, we faced some new difficulties.First of all, in view of the square growth of the coupling term, we need to correct the ansatzs of both the components to detect the suitably reduced problem.Moreover, the use of the regularity theory will be crucial in order to make suitable expansion of all the terms involved in the construction.Our existence result does not cover the case N = 1, as in this case the size of the error term does not produce the suitable smallness of the reminders terms in the ansatz despite of the presence of the correction term.We think that this point could be managed introducing further correction terms, again in both the equations, which at the prices of heavy technicality should allow to construct a remaining term sufficiently small.
Remark 1.7.Our result deals with the case of a binary mixture and it is natural to ask if our construction can be extended to the case of a larger number of equations, i.e.
In particular, we wonder if it is possible to build a solution whose components v j concentrate at different pairs of points (P ε j , −P ε j ) collapsing to the origin as ε goes to zero.Remark 1.8.The unperturbed version of system (1.2) (let us say ε = 1) was firstly studied by Peng and Wang [19] who (in presence of radial potentials) constructed an unbounded sequence of non-radial solutions exhibiting an arbitrarily large number of peaks.Their result has been successively extended to the case of more than two equations by Pistoia and Vaira [20] and very recently by Li, Wei and Wu [10].We wonder if it is possible, by combining the ideas used in the above papers, to produce a solution to system (1.3) with ε = 1 whose first component looks like the solution to (1.5) and second component concentrates at an arbitrary large number of points approaching infinity as ε goes to zero.Remark 1.9.Let us finally observe that the existence of solutions to the system (1.2) is closely related to the study of the normalized solutions for nonlinear Schrödinger systems.We refer the reader to the recent papers by Lu [14], Liu and Yang [13], Guo and Xie [6] and Liu and Tian [12].In particular, it would be interesting to produce normalized solutions using as a parameter their L 2 -norms, in the spirit of the results obtained by Pellacci, Pistoia, Vaira and Verzini [18].
The paper is organized as follows.In the next section we set the problem, by introducing the the main blocks of our construction and by reformulating problem (1.3) as a system of two equations, one set in an infinite dimensional set, the other, called the reduced problem, in a finite dimensional one.In Section 3 we solve the the infinite dimensional equation.Finally, in Section 4 we study the reduced problem and we complete the proof of Theorem 1.1.
Acknowledgments.The authors warmly thank the anonymous referee for her/his precious comments, and in particular for having pointed out a gap in the proof of Lemma 2.1 in a previous version of this manuscript.

Setting of the problem
Let us introduce the Banach spaces equipped with the norms and Henceforth, we omit the subscript ε in u, v and we agree that a b means |a| ≤ c|b| for some constant c which does not depend on a and b.
Performing a change of variable in the second equation, we are lead to seek a solution (u, v) of −∆u in the space ))}.In the next subsection, we introduce the main building blocks in the construction of our solution.

The ansatz and the correction terms. In view of assumptions
and U be the solution of where, since β < 0 We look for a solution (u, v) of (2.2) of the form where and the concentration points satisfy (see (1.12)) The function Φ ε and Ψ ε , are suitable correction terms, whose existence and properties are established in Lemma 2.1, 2.3.The reimander terms ϕ and ψ belong to the space where solves the linear equation Moreover, it is worthwhile to point out that all the functions Φ ε , U ε and Z ε are even functions.
In the following we introduce the two correction terms we need in our construction of the solution.Let us start from the term in the first component.

Lemma 2.1. There exists a unique even
(2.9) Proof.By exploiting assumptions (V 1 ) − (V 2 ) we deduce that for any even function , for some constant c which does not depend on f .Now we point that the function ε is an even function with Indeed, by scaling x = εy we immediately deduce As a direct consequence, we infer . Now, we write (2.9) as in R N and we observe that, reasoning as in (2.10), we deduce that , for any m ≥ 2; then, assumptions (V 2 ) implies, Remark 2.2.It is useful to remark that by Lemma (2.1) since ∇Φ ε (0) = 0 we deduce Moreover, since ∇Υ(0) = 0 we also have (2.12)

Lemma 2.3. There exists a unique
Proof.As U is radial, there exists a unique Ψ radial solution to then the function Ψ ε defined as ).The regularity properties of Ψ are consequence of the regularity properties of U, while the bound from above of the H 2 (R N ) norm follows from the upper bound on the The exponential decay of Ψ ε will follow from the analogous decay of Ψ.In order to prove this property, let us first show that there exists R > 1 such that Ψ(r) ≥ 0 for every r > R. By contradiction there exists a sequence r n → +∞ of minimum point at a negative level for Ψ.Then as soon as r n is sufficiently large so that the parenthesis is positive .
Remark 2.4.Let us point out that assumptions (V 1 ) and (V 2 ) are satisfied by constant potentials as well as by potentials of the type V(x) = |x| m with m > 0 as shown in [5, 23].
Remark 2.5.As a consequence of Sobolev embedding, the bounds from above stated in Lemma 2.1 hold for Ψ ε as well.
Finally, the nonlinear term N = (N 1 , N 2 ) is defined by

The linear theory
Let us start the study of the second equation in (2.15) by proving the following crucial result.

Lemma 3.1.
There exists c > 0, and ε 0 > 0 such that for every ε ∈ (0, ε 0 ) and for every β < 0 it results Proof.We argue by contradiction and suppose that there exist Step 1.Let us first show that ) and almost everywhere in R N .Taking into account that By applying Lemma 2.1 one deduces that Moreover, by applying Lemma 2.3 we obtain Arguing analogously on the other terms on the right hand side, it follows that ϕ solves Since ϕ is an even function, by the assumtpion (V 1 ) we get ϕ ≡ 0.
In order to show that ϕ n → 0 strongly in H 2 V (R N ), it is enough to exploit Lemma 2.1 and 2.3 to verify that the L 2 (R N )−norm of the right hand side (R.H.S. for short) of the first equation goes to zero, and then apply hypothesis (V 1 ).Indeed, as (ϕ n , where we have also taken into account that Υ decays exponentially and ϕ → 0 strongly in L 2 loc (R N ), so that we also have Step 2. We now study the second equation in (3.1) and we prove that t n → 0. We test with Z ε n and we remind that Z ε n solves =o (1).
Indeed, we use the exponential decay of U and of its derivatives.Since Moreover a direct computation and Lemma A.1 shows that It is possible to show that all the other integral terms on the left hand side tend to zero by applying Lemma 2.1 and 2.3, Step 1. and Sobolev embeddings.
Finally, since it is immediate to check that for some C > 0, we deduce that t n = o(1).
Step 3. Let us now introduce the sequences We will show that (up to subsequences) ψ ±P n ⇀ 0 weakly in H 1 (R N ) and strongly in L 2 loc (R N ).Both these sequences are bounded in H 2 W εn (R N ), so that, up to subsequences, ψ ±P n ⇀ ψ ± weakly in H 1 (R N ) and strongly in L 2 loc (R N ).Let us first show that ψ + ≡ 0, then an analogous argument will yield that ψ + ≡ 0. In the following we will use the notation ψ +P n (x) = ψ P n (x).Recalling (3.1), the function ψ P n satisfies the equation Arguing as in the first step, applying Lemma 2.1 and Lemma 2.3, and taking into account that ϕ n L ∞ (R N ) → 0, we obtain that the limit function ψ+ solves the limit problem On the other hand, the function ψ + inherits the symmetry properties of the function ψ ±P n , namely it is even in the last two variables and it satisfies the orthogonality condition This, together with (3.3), yields ψ + ≡ 0.
Step 4. Let us prove that a contradiction arises.First, let us prove that . By testing the second equation with ψ n , and recalling that β < 0 in view of (1.14), we deduce that where we have repeatedly applied Lemma 2.1, 2.3, that φ n L ∞ (R N ) = o(1) and that g n → 0 strongly in L 2 (R N ).Concerning the last term, we have that because ψ ±P n → 0 strongly in L 2 loc (R N ) (as shown in the previous step) and U decays exponentially.This implies that ψ n → 0 strongly in H 1 (R N ), thanks to (1.7).Finally, let us prove that a contradiction arises by showing that also In order to show this, it is enough to use hypothesis (V 2 ) and to check that the L 2 (R N )−norm of the right hand side of the second equation in (3.1) goes to zero.Indeed, by Lemma 2.1, taking into account that Remark 3.2.Let us observe that the hypothesis β < 0 is needed only in the proof of Step 4.Moreover, it is not needed in the case of a constant potential so that the sequence W 0 − βΥ 2 (ε n x) ψ 2 n ≥ 0 for every x and the final contradiction can be obtained applying Fatou Lemma.However, even if at this point we can manage β ≥ 0 (in the case of W constant), the study of the finite dimensional problem will require β < 0 as shown in hypothesis (1.14).
3.1.The size of the error term.In this subsection we compute the L 2 (R N ) of E which will determine the norm of the remainder term (ϕ, ψ).

Proposition 3.3. There exists ε
Proof.Let us start studying E 1 given in (2.17).We have Lemma 2.1 and Sobolev embedding imply that (see Remark 2.2) Then we deduce and Moreover by applying Lemma 2.3 we obtain As far as concern the last three terms, similar computations show that Let us now study the L 2 (R N ) norm of the terms in (2.18).In view of (1.7) and (1.12), recalling that x 0 = 0 is a critical point of ω by symmetry, On the other hand Lemma 2.3 allows us to deduce that so that, we obtain Moreover, conclusion (ii) of Lemma A.1 and (1.12) yield as, by using (1.12) it follows that ρ ε ∼ 1 √ ω 0 ε| ln ε| and hence ε ).Let us finally study the last terms in (2.18) , concluding the proof.(2.15).Lemma 3.1 and Proposition 3.3 yields the following result Proposition 3.4.There exists ε > 0 such that for every ε ∈ (0, ε 0 ) there exists a unique solution (ϕ, ψ) ∈ K ⊥ of the equation

Solving the second equation in
Proof.We will obtain the result by applying the contraction principle to the continuous map where is well defined thanks to Lemma 3.1, and A is a suitable positive constant to be chosen.In order to find A, it is sufficient to prove that In addition, this, together with (3.7), implies (3.6).Then, the claim follows by the contraction mapping theorem.

Solving the reduced problem
In this section, we are going to study the first equation in (2.15).Let (ϕ, ψ) the solution of the second equation in (2.15), then Our goal will be to prove that c 0 = 0. From now on, we fix (ϕ, ψ) given in Proposition 3.4.
Proof.Arguing as in the proof of Proposition 3.4 it is easy to obtain that Now by using (2.16) and (2.8) we get that Let us study the right hand side.First of all, arguing as in (3.4) and taking into account (3.5), we have where we have applied Lemma A.1.Similarly In addition, (3.5) and Lemma 2.1, 2.3 yield and the terms in (4.2) can be handled analogously.Let us focus on the terms in (4.3).Recalling that we obtain Taking into account Remark 3.5 (choosing α = 1 2 for N = 2), we infer Moreover, (1.12) and (3.5) yield The last two terms in (4.5) can be studied similarly, by applying Lemma A.1.It results The last term in (4.3) can be easier studied as concluding the proof.
We are now in position to study the relevant term in (4.1).We claim that the other terms on the right hand side of (4.7) are of higher order with respect to ερ ε .Indeed, Lemma A.2 and (1.12) yield The last term in (4.7) can be managed, by applying Lemma 2.3.Indeed, we have Let us now study the second term on the right hand side of (4.6).It holds In view of Remark 2.2, we obtain Let us study the cubic and square term in U P ε in (4.6).We have  Applying again Lemma A.2-(ii) with s = t = 2 we infer 3µ 2 Hence