Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities

In this paper, we investigate the following fractional Sobolev critical nonlinear Schr\"{o}dinger (NLS) coupled systems: \begin{equation*} \left\{\begin{array}{lll} (-\Delta)^{s} u=\mu_{1} u+|u|^{2^{*}_{s}-2}u+\eta_{1}|u|^{p-2}u+\gamma\alpha|u|^{\alpha-2}u|v|^{\beta} ~ \text{in}~ \mathbb{R}^{N},\\ (-\Delta)^{s} v=\mu_{2} v+|v|^{2^{*}_{s}-2}v+\eta_{2}|v|^{q-2}v+\gamma\beta|u|^{\alpha}|v|^{\beta-2}v ~~\text{in}~ \mathbb{R}^{N},\\ \|u\|^{2}_{L^{2}}=m_{1}^{2} ~\text{and}~ \|v\|^{2}_{L^{2}}=m_{2}^{2}, \end{array}\right. \end{equation*} where $(-\Delta)^{s}$ is the fractional Laplacian, $N={3,4}$, $s\in(0,1)$, $\mu_{1}, \mu_{2}\in\mathbb{R}$ are unknown constants, which will appear as Lagrange multipliers, $2^{*}_{s}$ is the fractional Sobolev critical index, $\eta_{1}, \eta_{2}, \gamma, m_{1}, m_{2}>0$, $\alpha>1, \beta>1$, $p, q, \alpha+\beta\in(2+4s/N,2^{*}_{s}]$. Firstly, if $p, q, \alpha+\beta<2^{*}_{s}$, we obtain the existence of positive normalized solution when $\gamma$ is big enough. Secondly, if $p=q=\alpha+\beta=2^{*}_{s}$, we show that nonexistence of positive normalized solution. The main ideas and methods of this paper are scaling transformation, classification discussion and concentration-compactness principle.


Introduction and main result
The motivation for the problem studied in this article arises from finding stationary waves solutions of the following physical model: where i represents the imaginary unit and φ j = φ j (x, t) : R N × R + → C is the wave function of the jth ( j = 1, 2) component, the mass of them represents the number of particles of each component in the mean-field models for binary mixtures of Bose-Einstein condensation, see [1][2][3] and references therein, η j and γ denote the intraspecies and interspecies scattering lengths. The sign case for γ determines whether the interaction between the states is attractive or repulsive, i.e. the interaction is attractive if γ is positive, the interaction is repulsive if γ is negative. An important solution, known as travelling or standing wave, is characterized by ansat z φ 1 (x, t) = e iμ 1 t u(x), φ 2 (x, t) = e iμ 2 t v(x) (1.2) for two unknown functions u, v : R N → R, where μ 1 , μ 2 ∈ R. Because these solutions are very similar to each other and retain their mass over time, it makes sense to seek prescribed L 2 -norm solutions (normalized solutions). Therefore, combining (1.1) and (1.2), we arrive at the following fractional single or double Sobolev critical Schrödinger system: where N > 2s, s ∈ (0, 1), μ 1 , μ 2 ∈ R are unknown constants, which will appear as Lagrange multipliers, 2 * s is the fractional Sobolev critical index, η 1 , η 2 , γ, m 1 , m 2 > 0, α > 1, β > 1, p, q, α + β ∈ (2 + 4s/N , 2 * s ], and (− ) s is the fractional Laplacican defined by For more information about this type of operator we refer to [4]. With regard to the double Sobolev critical Schrödinger coupled systems, many experts and scholars have conducted extensive and in-depth research on this whether it is integer order or fractional order with fixed μ 1 and μ 2 . When s → 1, Zou et al. [5] considered the existence and symmetry of positive ground states for a double critical coupled systems. Moreover, they studied the limit behavior of positive ground states for another kind of double critical Schrödinger system when the interaction is repulsive in [6]. In the case of 0 < s < 1, Zou and Yin [7] proved the asymptotic behaviour and existence of the positive least energy solutions for k-coupled double critical systems driven by a fractional Laplace operator by means of the idea of induction. Yang [8] dealt with a class of fractional Laplacian doubly critical coupled systems, they gave sufficient conditions for the existence of weak solutions by establishing an embedding theorem. He et al. [9] investigated the existence of least energy solution with the help of the Nehari manifold.
However, as far as we know, few papers treat parameters μ 1 and μ 2 as Lagrange multipliers to study the normalized solution of double Sobolev critical problems.
In particular, when s → 1, problem (1.3) becomes the form: (1.4) Liu and Fang [10] studied the existence and nonexistence of positive normalized solution for equation (1.4) in the case of p, q, α + β < 2 * s and p = q = α + β = 2 * s respectively. Furthermore, if let N = 4, p = q, α = β = 2, the above problem reduces to the classical elliptic system: which was investigated by Zou et al. [11], they obtained the existence, nonexistence and asymptotic behavior of normalized ground state solutions for system (1.5) in different cases. Of course, a great deal of work has focused on the normalized solution of integerorder nonlinear Schrödinger systems [12][13][14][15][16][17][18] or fractional-order single Schrödinger equations [19][20][21][22][23][24][25][26]. In particular, we highlight that Jeanjean and Lu [21] obtained the existence and multiplicity of normalized solutions and the asymptotic behavior of the ground state solution for a class of mass supercritical problems. The method developed in the present paper is inspired by the techniques introduced in [21]. We also point out that Soave [27] established existence and stability properties of ground state solutions for a Schrödinger equation with Sobolev critical exponent under three different assumptions, respectively mass subcritical, mass critical and mass supercritical.
So far, we have found only one paper [28] dealing with normalized solution of fractional Schrödinger coupled systems but only the subcritical case is considered. Hence it is natural to inquire what difficulties will appear if we consider single critical nonlinearity or even double critical nonlinearities.
Motivated by the work above, we will consider the existence for single critical fractional Schrödinger coupled systems and the nonexistence of normalized solutions for double critical fractional Schrödinger coupled systems. The main features of this paper is the existence of nonlocal operator and Sobolev critical nonlinearities, which makes dealing with compactness conditions more complicated. Our main methods and tools for solving these problems are scaling transformation, classification discussion and concentration-compactness principle.
A classical method for studying the normalized solution of system (1.3) is to look for critical points of the following C 1 functional where H s (R N ) is the fractional Sobolev space defined by For convenience, we use u p to represent the norm of Lebesgue space L p (R N ) for p ∈ [1, ∞). We write We now present our main results: is a positive solution whose associated Lagrange multipliers μ 1 and μ 2 are negative.

Remark 1.1
The proof of Theorem 1.1 faces some difficulties and challenges. Firstly, strong convergence of sequences in are not compact. Secondly, the lack of compactness caused by Sobolev critical index makes verifying the Palais-Smale condition more complicated. Thirdly, the idea of classification discussion is going to be used since we don't infer which of the three indices p, q, θ is big and which is small. [29] in some aspects. Compared with the local case of our result, this kind of system is studied by Mederski and Schino [29] under more generalized assumptions on the nonlinear terms. The paper is organized as follows. Section 2 introduces relative results of scalar equations and some preliminaries, which play an important role in the proof of Palais-Smale condition. Section 3 proves Theorem 1.1 by using the methods of scaling transformation, classification discussion and concentration-compactness principle. Section 4 gives the proof of Theorem 1.2 with the help of Pohozaev identity.

Relevant results for scalar equations and preliminaries
In order to study the fractional critical Schrodinger coupling system, we first need to review related results of following scalar equations, i.e. γ = 0 in (1.3): which has been investigated in [25] by constraining on the Pohozaev manifold

2)
A standard way to get normalized solutions of (2.1) is to look for critical points for C 1 functional As we all know, P m 1 ,η 1 contains every critical point of I η 1 (u)| S m 1 , due to the Pohozaev identity (see [30,Proposition 4.1]).
It follows from [31] that there exists a best fractional critical Sobolev constant S > 0 such that which is famous fractional Sobolev inequality. In order to prove our result in Sect. 3, we need to obtain the following monotonicity result of scalar equations, which is necessary in the proof of Lemma 3.7.
where ξ p is defined by (2.2).
The above fractional Gagliardo-Nirenberg inequality plays an key role in the next series of proofs.
Proof On account of u 0 and fulfills thus we infer that (− ) s u ≥ 0 if μ 1 ≥ 0. It follows from [32, Proposition 2.17] that u ≡ 0. This contradicts to the condition u 0, which means that μ 1 < 0. Similarly, we also can get μ 2 < 0 from v 0.

Proof of theorem 1.1
We will do a scaling transformation make the functional I (ρ u, ρ v) satisfy the mountain pass geometry. For (u, v) ∈ W and ρ ∈ R, we let which comes from the inspiration of Jeanjean [33]. The results show that the original functional I (u, v) and the transformed functional I = I (ρ u, ρ v) have the same mountain pass geometry and mountain pass level. Lemma 3.1 Suppose that (u, v) ∈ S m 1 × S m 2 is arbitrary but fixed. Then we have the following conclusions: Proof Through simple calculations, we have 3), ζ ≥ 2, and θ > 2, we get Thus, we have , which also means that (2) holds.

Lemma 3.2 There exists
Proof According to Lemma 2.2 and the Hölder inequality, for any (u, v) ∈ S m 1 × S m 2 we get that for A small enough, the scaling of inequality above takes advantage of this fact that pξ p , qξ q , θξ θ > 2. On the other hand, for any ( for A sufficiently small. So, we can pick A small enough for the inequality in Lemma 3.2 to be true. Now we have obtained that the geometry of mountain pass, then we give the minimax picture: Because Zhang [25] had proved the u := u m 1 ,η 1 is a ground state of (2.1) involving parameters p, η 1 , m 1 and also v := u m 2 ,η 2 is a ground state of (2.1) involving parameters p, η 2 , m 2 . Therefore, we fix (u, v) ∈ W m,r , according to Lemma

and Lemma 3.2, there exist two numbers ρ
We define the path then is not empty. In fact, let χ 0 (t) clearly, c γ (m 1 , m 2 ) > 0, then we have the following asymptotic behavior of critical values: we discuss the maximum value of the following function , for any l ≥ 0. Letting we can obtain the maximum point of g(l), that is As a result, there exists C > 0 that don't depend on γ > 0 such that thanks to θ > 2 + 4s/N . We complete the proof of Lemma 3.3 now.
To further prove our main results, we need to define Pohozaev manifold P m 1 ,m 2 ,μ 1 ,μ 2 of vector equations (1.3) in the same way as scalar equations (2.1).

Lemma 3.4 Suppose that {(u n , v n )} ⊂ W m,r is a Palais-Smale sequence for I (u, v).
Then lim n→∞ P(u n , v n ) = 0.

Proof Let
(ρ n u n , ρ n v n ) = (x n , y n ).
Proof Observing that ξ p p, ξ q q, ξ θ θ > 2 due to the fact that p, q, θ > 2 + 4s/N . It follows from Lemma 3.4 that In view of Lemma 3.5, there exists a nonnegative ( u, v) ∈ W r such that, up to a subsequence, as n → ∞. Since {(u n , v n )} ⊂ S m 1 × S m 2 is a Palais-Smale sequence for I (u, v), on basis of the Lagrange multipliers rule, there exists a sequence {(μ n 1 , μ n 2 )} ⊂ R × R such that Next, we take (u n , 0) and (0, v n ) as test functions in (3.7), it follows from the proof of [25, Proposition 2.2] that {(μ n 1 , μ n 2 )} is bounded in R. Therefore up to a subsequence (μ n 1 , μ n 2 ) → (μ 1 , μ 1 ) ∈ R × R.

Lemma 3.6
There exists γ * = γ * (m 1 , m 2 ) > 0 big enough, such that for any γ ≥ γ * , Proof We first claim that u n → u in L 2 * s (R N ). In fact, according to the concentrationcompactness principle in [34], we know that there exist two nonnegative measures ω, ν and a (at most countable) index set J such that |(− ) where x j is the different point in R N , δ x j denotes the Dirac measure at x j , S is best Sobolev constant in (2.3). Furthermore, it is possible to lose mass at infinity. i.e., where where B (x j ) represents the small ball with radius and center x j . By Lemma 3.5, we note that {ψ (x)u n } is bounded in H s (R N ). Next, again take {(ψ (x)u n , 0)} as a test function in (3.8) and letting → 0, we obtain that From (3.7), the Hölder inequality and the absolute continuity of the Lebesgue integral, we have (3.13) According to (3.12) and (3.13), we deduce that (3.14) Therefore it follows from (3.9) and (3.14) that ν j ≥ ω j , thereby which means that J is a finite set.
For any R, let ϕ R (x) = ϕ( x R ), by Lemma 3.5, we also have that {ϕ R (x)u n } is bounded in H s (R N ). Similarly, take {(ϕ R (x)u n , 0)} as a test function in (3.8) and letting R → ∞, we also obtain that Similarly to the proof of [35, Lemma 3.3], we can also get Again by (3.10), we have For the rest of the proof, we will only prove Case 1, because Case 2 and Case 1 are almost exactly the same.
If for any j ∈ J , ω j = 0, then we have that ν j = 0 since (3.9), thereby |u n | 2 * s → | u| 2 * s . By the Brézis-Lieb lemma [36], we conclude that u n → u in L 2 * s (R N , as claimed. If instead ω j ≥ S N 2s for some j ∈ J . In view of I (u n , v n ) → c γ (m 1 , m 2 ), Lemma 3.4 and (3.11), we adopt the method of categorical discussion: (1) If θ = min{ p, q, θ}. It follows from Lemma 3.3 that there exists a positive constant for any γ ≥ γ . By (3.9), we have which is a contradiction. (2) If p = min{ p, q, θ}. Similar to (1), there is also a constant γ big enough, such that c γ (m 1 , for any γ ≥ γ . We also obtain which contradicts our hypothesis. (3) If q = min{ p, q, θ}. Analogously as (1) and (2), we can also get our conclusion, which we omit here.
To sum up, there exists a bigger positive constant γ * (m 1 , m 2 ) such that ω j = 0 = ν j for any j ∈ J and γ ≥ γ * . Therefore, for γ ≥ γ * , we have u n → u in L 2 * s (R N ). The proof for v n → v in L 2 * s (R N ) is similar. Finally, we claim that u, v = 0. In fact, if not, we have ( u, v) = (0, 0). According to (3.6)

Conflict of interests
The declare that the authors have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.
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