Existence of normalized peak solutions for a coupled nonlinear Schrödinger system

: In this article, we study the following nonlinear Schrödinger system


Abstract:
In this article, we study the following nonlinear Schrödinger system ( ) with the constraint ( ) 1

Introduction and main result
In this article, we consider the following Schrödinger system with coupled quadratic nonlinearities (1.1) under the constraint where > α 0 and > α β, ∈ μ is a chemical potential, and ( ) V x 1 and ( ) V x 2 are bounded functions.Such type of systems like (1.1) with quadratic interaction have wide applications in physics, such as Bose-Einstein condensates, plasma physics, and nonlinear optics.For example, the following coupled nonlinear Schrödinger equations for some ∈ μ j k , and = j k K , 1,2,…, , can describe the propagation of solitons with ( ) χ 3 nonlinear fiber couplers in nonlinear optic theory, where the complex function ω k denotes the kth component of the light beam, and 2 is the change in refractive index profile created by all the incoherent components in the light beam.This system has gotten a lot of attentions experimentally and theoretically.If we consider the standing wave solutions for (1.3) (1.4) for = k K 1, 2,…, .The existence and multiplicity of standing wave solutions to (1.4) and its relates problem have been explored by many authors in recent years, see previous studies by [2][3][4]6,9,10,13,22,24,26,30] and the references therein.However, when nonlinear optical effects such as second harmonic generation are investigated in the optical material that has a ( ) χ 2 nonlinear response, it is led to the system (1.1) (c.f.[1,7]).For the existence, multiplicity and asymptotic behavior of solutions for (1.1), we can refer to [27][28][29]31].
We note that the following Gross-Pitaevskii (GP) equations proposed by Gross [15] and Pitaevskii [25] in the 1960s (1.5) with the constraint has been investigated extensively due to some new experimental advances in quantum phenomena (c.f.[5,11,14]).Here, ≥ N 2, ( ) ≥ V x 0 is a real-valued potential and ∈ a is treated as an arbitrary dimensionless parameter.If we want to find a solution for (1.5) of the form ( ) ( ) = − ω x t e u x , iμt , where μ represents the chemical potential of the condensate and ( ) u x is a function independent of time, then the unknown pair ( ) μ u , satisfies the following nonlinear eigenvalue equation with the constraint For ground states of equation (1.6), one can refer to the studies by Guo et al. [18][19][20] and the references therein, where the main tools are the energy comparison under various assumptions of trapping potential ( ) V x , which can be described equivalently by positive L 2 minimizers of the following functional: Very recently the existence and the local uniqueness of excites states for a class of degenerated trapping potential with nonisolated critical points were given in the study by Luo et al. [23], where we call any eigenfunction of equations (1.6) and (1.7) whose energy is larger than that of the ground state the excited states in the physics literatures (c.f.[8]).As for two-component Bose-Einstein condensates with the mass constraint, the studies by Guo and Yang [16] showed the existence of the general exited states by using the reduction argument combined with the local Pohozaev identities, which generalize the existence of the ground state with trapping potentials given in the study by Guo et al. [17] to some extent.
To our best knowledge, there seems to be no results on the existence of peak solutions to (1.1) with the L 2 constraint (1.2).So in this article, we want to investigate this problem and suppose the following conditions hold.
To state our results, we first introduce some notations.Let U be the unique positive solution of the following problem: and we denote . From [21], we know that ( ) is the ground state of (1.8) provided that > α β with For any ∈ + t and ∈ y 4 , we define for , and for any The first result of our article is as follows.
Theorem 1.1.Assume that (K 1 ) holds and > α β and > α 0, then there exists a small constant > ε 0 such that for any , problems (1.1) and (1.2) have a peak solution ( ) u u μ , , depending on α and β with the form

Also as
Theorem 1.1 tells that we construct a synchronized solution for problems (1.1) and (1.2), where the synchronized solution means that the two components of the solutions for system (1.1) concentrate at the same set of concentrated points.Otherwise, we call it a segregated solution if the components concentrate at two different set of points.depending on α and β with the form μ j μ j , ,

Also as
1 and a and μ y y for l j , .
Remark 1.4.To our best knowledge, this seems to be the first time to consider the existence of the normalized peak (or bubbling) solutions for problems (1.1) and (1.2).Also we note that only one coefficient a in (1.6) need to be discussed with the constraint mass, while there are two parameters α and β in system (1.1) to be considered, corresponding to the interactions within and between the components, respectively, which makes the problems (1.1) and (1.2) more complicated than the single equation, and we have to analyze the mutual influence of the parameters and the constraint condition very carefully.
We will mainly use the finite dimensional reduction to prove our results, which is an effective way to construct solutions for perturbed elliptic problems.Since the singularly perturbed problem has a small parameter naturally, the approximate solutions can be constructed by the standard steps of the reduction argument.However, the constraint problems (1.1) and (1.2) itself do not provide any natural limiting process explicitly.So to apply the reduction method, we have to instigate the relationship between the constraint conditions and the parameters appeared in the system.This process involve some various Pohozaev identities, which play an important role and need some more accurate estimates.
The structure of this article is organized as follows.We carry out the finite dimensional reduction to study the corresponding problem without constraint in Section 2. In Section 3, we prove Theorems 1.1 and 1.3 is proved in Section 4.

The finite dimensional reduction
In this section, we consider the following problem without constraint where > λ 0 is a large parameter.We want to find a peak solution of equation (2.1) with the form where ( ( for some ∈ y 4 .To this aim, we define L λ be the bounded linear operator from 4 to itself as follows: Then to obtain the solution ( of equation (2.1) with (2.2) is to solve the following problem: where , 2 To carry out the reduction argument, we first introduce the following result, which has been proved in [29].
4 in the sense that the kernel is given by Now we define the set and the norm ‖( Also we set the projection 4 to E λ as follows: , Proof.By contradiction, suppose that there exist and from this, it holds Now choosing R sufficiently large such that which gives that To obtain a contradiction, next we want to prove that To this end, we define . Then Thus, we can assume that as 4 , and 4 for = k 1, 2 and = j m 1,…, .Now we just need to prove that For any ( 2 , we can decompose ( ) ϕ ϕ , 1 2 as follows: for some constants b λ i j , , , , . Also, by (2.3), for some constants ϑ λ i j , , n .From this, we estimate ϑ λ i j , , n as follows: where > γ 0 is a small constant and n λn j n λn j n λn j n λn j n λn j n λn j n λn j n λn j It follows from the definition of E λn that Hence, we find n n λn j n λn j n λn j n λn j n , , , , which gives n and using (2.7), we obtain Existence of normalized peak solutions  7 Then, ( From this and the fact that ( ) U U , 1 2 is nondegenerate, we have (2.8) On the other hand, from n and (2.8), it holds By the property of ( ) U (2.9) Lemma 2.4.We have (2.10) and Using Hölder inequality, we find and . , Hence, (2.12) and (2.11) give (2.10).□ Lemma 2.5.It holds (2.13) . By the direct computations, we have where we use the following classical Gagliardo-Nirenberg inequality: (2.15) With the same argument, it holds (2.17) , where ( ) . Moreover, it holds Existence of normalized peak solutions  9 Proof.To obtain (2.17), it is equivalent to consider the following problem: (2.20) Now we define where > γ 0 is a fixed small constant.Next we will prove B is a contraction map from M to M .First, from Proposition 2.2 and Lemmas 2.4 and 2.5, we have On the other hand, for ( ) ( , it holds By applying the contraction mapping theorem, we find that for any depending on a k and λ solving (2.20) and then (2.17) follows.Finally, (2.18) . Now to obtain a true solution for (2.1), we just need to choose suitable y λ j , such that = = = i j m Γ 0, for 1, 2, 3, 4 and 1,…, .
To this end, we want to choose ( ) y y , …, , with = i 1, 2, 3, 4 and = j m 1,…, .Lemma 3.1.Assume that (K 1 ) holds.Then there exists some large λ 0 such that for any with the form ) Proof.From the aforementioned discussions, we have to choose suitable ∈ y λ j , , we find that (3.2) is equivalent to for some > θ 0. Then we have On the other hand, Similarly, we have By the assumption ( ) (3.9) Proof.Note that U satisfies (3.10) By direct computations, we have Also multiplying ( ) ⋅∇ x U on both sides of equation (3.10) and integrating on 4 , it holds So the aforementioned two equalities give (3.9).□ is a solution of (2.1) with the form (3.3) satisfying (3.4), then it holds Proof.Let ( ) u u , where 2Γ * 1 , . On the other hand, using (3.9), we find So summing (3.12) from = j 1 to = j m, we find (3.11).□ Now we are in a position to prove Theorem 1.1.
Proof of Theorem 1.1.Now we set ( ) Therefore, from Proposition 3.3, we obtain that Similar to (3.13), it holds which gives that there exists some λ ¯0 large enough such that , then from the fact that , which contradicts to (3.14).Now we define ( ) ( , then we have got the solution.Now we consider On the other hand, by (3.13), Thus, there exists some for small ε.
With the same argument as mentioned earlier, we can also obtain the existence of a concentrated solution to . □ Existence of normalized peak solutions  13 4 Proof of Theorem 1.3 In this section, we come to prove Theorem 1.3.First from Proposition 2.7, for any , there exist u i λ , and Γ λ i j , , such that ).Now we define , and Then to obtain a true solution of (2.1), we need to choose suitable y λ j , such that = = = i j m Γ 0, for 1, 2, 3, 4 and 1,…, .
To this aim, similar to Lemma 2.2.13 in [12], we will find the critical points of ( ) is the critical point of ( ) Q y , then (4.1) holds.
Proof.First, by the definition of I λ , we have where we use the fact that . Now, we find And from the assumption ( ) , we compute that , , On the other hand, it holds .So the aforementioned estimates give (4.2). □ Now we have the following energy expansion.
Existence of normalized peak solutions  15 By using the classical Gagliardo-Nirenberg inequality (2.15), similar to (2.14), we have On the other hand, from ( ) So, (4.4) gives that Then (4.where ε is a small fixed constant.Consider the following problem Suppose that it is achieved by ∈ y λ .To prove this y λ is a critical point of ( ) Q y , we just need to prove that y λ is the interior point of .
Let y ¯λ j , , = j m 1,…, , satisfy  for some > κ ˜0 0 , which gives a contradiction again.Thus, we have proved that y λ is the interior point of , and thus, it is a critical point of ( ) Q y .□ Finally, we conclude our second main result.
Proof of Theorem 1.3.Now we set ( ) satisfies that with ( )  Next, similar to the proofs of Theorem 1.1, we can finish the proof.□ Existence of normalized peak solutions  17

,
on both sides of the ith equation of equation (2.1) and integrating on ( )

(4. 7 )
Proof.First, it follows from Lemmas 4.1 and 4 fact that y λ is a maximum point of ( ) Q y in .Now suppose that there exist y λ j , 0 and y λ m , 0 such that | | 3) follows by the aforementioned estimates.□ Proposition 4.3.Assume that (K 2 ) holds.Then there exist some large λ 1 such that for any