Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation

Over the past decade, there has been a great deal of interest in studying the fractional Schrödinger equation (NLS) i∂tψ = (−∆)sψ + f (ψ), (1.1) where 0 < s < 1 and f (ψ) is the nonlinearity. The fractional di erential operator (−∆)s is de ned by (−∆)sψ = F−1[|ξ |2sF(ψ)], where F and F−1 are the Fourier transform and inverse Fourier transform, respectively. The fractional NLS (1.1) was rst deduced by Laskin in [29, 30] by extending the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths. The fractional NLS also arises in the description of Bonson stars as well as inwater wave dynamics (see e.g. [22]) and in the continuum limit of discretemodels with long-range interactions (see e.g. [28]). In this paper, we consider blow-up criteria and instability of normalized standingwaves for the fractional nonlinear Schrödinger-Choquard equation { i∂tψ − (−∆)sψ + (Iα * |u|p)|ψ|p−2ψ = 0, (t, x) ∈ [0, T*) ×RN , ψ(0, x) = ψ0(x), (1.2)

Equation (1.2) enjoys the scaling invariance. That is, if ψ is a solution of ( . ) with initial data ψ , then ψµ(t, x) := µ α+ s s(p− ) ψ(µt, µ s x) for all µ > is also a solution of ( . ) with initial data µ α+ s s(p− ) ψ (µ s x). In particular, ψµ(t) Ḣsc = ψ(µt) Ḣsc , where Thus, sc is referred as the critical Sobolev exponent of ( . ). If the initial data ψ ∈ H s , then equation (1.2) enjoys mass and energy conservation laws: where the energy E is de ned by Before entering our main results, we rstly recall some known blow-up results for NLS. For the classical NLS, i.e., s = , when initial data ψ ∈ Σ := {ψ ∈ H and xψ ∈ L }, the following Variance-Virial Law holds d dt (1.5) By using (1.5) and the virial identity, ones can prove existence of blow-up solutions for the classical NLS with negative energy E(ψ ) < , see [7]. However, since identity (1.5) fails for s < , which readily checks by dimensional analysis, this argument cannot work. Rather, a possible generalization of the variance for the fractional NLS is given by the nonnegative quantity This method has been successfully applied to prove the existence of radial blow-up solutions of (1.1) with focusing Hartree-type nonlinearities, i.e., f (ψ) = −(|x| −γ * |ψ| )ψ with γ ≥ , see [8,9,48]. But this method can not work due to the nontrivial error terms which seem very hard to control for the local nonlinearities f (ψ) = −|ψ| p ψ, see [6]. In [6], Boulenger, Himmelsbach and Lenzmann applied the Balakrishman's formula and obtained the di erential inequality where δ = pN − s. Based on this key estimate, they proved the existence of radial blow-up solutions by applying a standard comparison ODE argument. For the fractional Schrödinger-Choquard equation (1.2), Saanouni in [39] proved the existence of radial blow-up solutions by using the method in [6]. In this paper, we will further study the existence of blow-up solutions of (1.2) for non-radial initial data by using the idea of Du, Wu and Zhang in [13]. The main di culty is the appearance of the fractional order Laplacian (−∆) s . When s = , the time derivative of the virial action can be easily obtained, that is d dt (1.9) Using this identity, Du, Wu and Zhang in [13] derived an L -estimate in the exterior ball. Combining this Lestimate and the virial estimates, they established blow-up criteria for the classical NLS. When s ∈ ( , ), the identity (1.9) does not hold. However, by exploiting the ideas in [6,12] and using the Balakrishman's formula (1.8), we can obtain the time derivative of the virial action, see Lemma 2.9. Thus, we can establish the blow-up criteria for (1.2).
where sc is de ned by (1.3) and u is a ground state of the following elliptic equation (1.11)  [8,9,25,33,37,40,41,47,48]. Here, we remove the assumption of radial solutions and extend these results to more general Choquard-type nonlinearity.
Based on blow-up criterion (1.10), we study the strong instability of normalized standing waves of (1.2). Firstly, we introduce some notations. Equation (1.2) enjoys a class of special solutions, which are called standing waves, namely solutions of the form e iωt uω, where ω ∈ R is a frequency and uω ∈ H s is a nontrivial solution to the elliptic equation (1.12) At this moment, our intention is reduced to study (1.12). To do this, there exist two substantially di erent choices in terms of the frequency ω. One is to x the frequency ω ∈ R. In this situation, every solution to (1.12) corresponds to a critical point of the action functional Sω(u) on H s , where Alternatively, it is interesting to study solutions of (1.12) having prescribed L -norm. That is, for any given c > , ones study solutions of (1.12) satisfying the L -norm constraint (1.14) Physically, such solutions are called normalized solutions of (1.12), which formally corresponds to critical points of the energy functional E(u) restricted on S(c), where E(u) is de ned by (1.4). In particular, in this situation, the frequency ω ∈ R is an unknown part, which corresponds to the associated Lagrange multiplier.
In the L -subcritical case, i.e., + α N < p < + s+α N , the energy E(u) is bounded from below on S(c). Feng and Zhang in [21] studied existence of normalized ground states to (1.12) by using the pro le decomposition theory in H s . On the contrary, in the L -supercritical case, the energy E(u) restricted on S(c) becomes unbounded from below for any c > . For this reason, it is unlikely to obtain a solution to (1.12)-(1.14) by developing a global minimizing problem. Motivated by minimizing method on Pohozaev manifold, we try to construct a submanifold of S(c), on which E(u) is bounded from below and coercive, and then we look for minimizers of E(u) on such a submanifold. Precisely, we introduce the following minimizing problem 16) and the functional K(u) is de ned by Indeed, the identity K(u) = is the Pohozaev identity related to (1.12). The constraint V(c) is the so-called Pohozaev manifold related to (1.12)-(1.14). In the following theorem, we can prove the existence of minimizers of (1.15).
Remark. This theorem can be proved by using the method in [18]. Here, we will use the pro le decomposition of bounded sequences in H s to prove this theorem. The pro le decomposition theory has been extensively applied to study existence of normalized standing waves in the L -subcritical case, see, e.g., [20,21,49].
Here, we successfully apply it to study existence of normalized standing waves in the L -supercritical case. Therefore, our approach is of particular interest. Next, we denote the set of minimizers of E on V(c) as In the following theorem, we can show any minimizer to (1.15) is a ground state to (1.12)-(1.14).
Then for any uc ∈ Mc, there exists ωc > such that (uc , ωc) ∈ H s ×R is a weak solution to problem (1.12). Furthermore, uc is a ground state solution to problem (1.12) with ω = ωc.
Finally, we consider the strong instability of normalized standing waves. The usual strategy to study the strong instability of standing waves for the classical NLS (s=1) is to use the variational characterization of the ground states as minimizers of the action functional and obtain the key estimate K(ψ(t)) ≤ (Sω(ψ )−Sω(uω)). Then, it follows from the virial identity that where K(ψ(t)) is de ned by (1.17) with s = . This implies that the solution ψ(t) of (1.1) with s = blows up in nite time. Thus, one can prove the strong instability of ground state standing waves, see [7, 11, 16, 17, 23, 24, 31, 34-36, 38, 43, 44].
Here, we only need to use the blow-up criterion (1.10) to study the strong instability of normalized standing waves.
Then for any uc ∈ Mc, the standing wave ψ(t, x) = e iωc t uc(x) is strongly unstable in the following sense: there exists {ψ ,n } ⊂ H s such that ψ ,n → uc in H s as n → ∞ and the corresponding solution ψn of (1.2) with initial data ψ ,n blows up in nite or in nite time for any n ≥ .

Remark 1.
In previous results, in order to construct blow-up solutions around the ground state solution, one need to assume that the ground state solution uω is radial or uω ∈ Σ := {v ∈ H and xv ∈ L }. Here, we remove these assumptions, so our result greatly improve some previous results.
Remark 2. When p = and N − α = s, i.e., in the L -critical case, Zhang and Zhu in [46] proved the strong instability of radial ground state standing waves of (1.2). Here, we remove this radial assumption and extend this result to the L -supercritical case and more general Choquard-type nonlinearity.
This paper is organized as follows: in Section 2, we will recall and prove some lemmas such as the local well-posedness theory of (1.2), a sharp Gagliardo-Nirenberg type inequality and the localized virial estimate related to (1.2). In section 3, we will establish blow-up criteria for (1.2). In section 4, we will prove the existence and strong instability of normalized standing waves.
Notations. In this paper, we use the following notations. For any s ∈ ( , ), the fractional Sobolev space where up to a multiplicative constant is the so-called Gagliardo semi-norm of u. In this paper, we often use the abbreviations with the usual modi cation when either q or r are in nity. In the case q = r, we shall use L q (J × R N ) instead of L q (J, L r ).

Preliminaries
In this section, we recall some preliminary results that will be used later. Firstly, we recall the local wellposedness for the Cauchy problem (1.2). Hong and Sire in [27] rst studied the local well-posedness of the fractional NLS in H s by using Strichartz's estimates and the contraction mapping argument. Since Strichartz's estimates for non-radial data have a loss of derivatives, a weak local well-posedness holds in the energy space compared to the classical nonlinear Schrödinger equation, see [10,27] for more details. One can remove the loss of derivatives in Strichartz's estimates by considering radially symmetric data. However, it needs a restriction on the validity of s, namely N N− ≤ s < .
Then for all ψ ∈ H s , there exist T * ∈ ( , +∞] and a unique solution , for some q > max{ p − , } when N = and some q > p − when N ≥ . Moreover, the following properties hold: • The solution enjoys conservation of mass and energy, i.e. ψ(t) L Therefore, we guess that these results also hold for + α N < p < . However, we cannot prove these results since the nonlinearity (Iα*|ψ| p )|ψ| p− ψ is singular when + α N < p < , see [19]. Next, we recall a sharp Gagliardo-Nirenberg type inequality established in [21].

2)
where the optimal constant C opt is given by where Q is the ground state of the elliptic equation (1.11). In particular, in the L -critical case, i.e., p = + s+α N , Moreover, the following Pohozaev's identities hold true: Next, we recall the pro le decomposition of bounded sequences in H s , which has been established in [48].
Finally, we recall and prove some virial estimates related to (1.2) which is the main ingredient in the proof of Theorem 1.1.

Lemma 2.5 ([6]
). Let N ≥ and suppose φ : R N → R is such that ∇φ ∈ W ,∞ (R N ). Then, for all u ∈ H , it holds that for some constant C > that depends only on N.
In order to study localized virial estimates for ( . ), we need to introduce the auxiliary function where cs := sin πs π .
Lemma 2.6 ([6]). Let N ≥ , s ∈ ( , ) and suppose φ : R N → R with ∆φ ∈ W ,∞ (R N ). Then, for all u ∈ L , it holds that for some constant C > that depends only on s and N.
We refer the reader to [ for any u ∈Ḣ s . Lemma 2.7. [12,Lemma 4.2] Let N ≥ , s ∈ ( / , ) and φ : R N → R be such that ∇φ ∈ W ,∞ . Then for any u ∈ L , it holds that for some constant C > that depends only on s and N.
By the same argument as in Lemma . and using in addition Lemma . , we obtain the following estimate. Let N ≥ , / < s < and φ : R N → R be such that φ ∈ W ,∞ . Assume that ψ ∈ C([ , T * ), H s ) is a solution to ( . ). We de ne the localized virial action of ψ associated to φ by

Proof. It follows from an integration by parts that
Therefore, following the method used in [6], we prove Lemma . .

Blow-up criteria
In this section, we will prove Theorem 1.1. To this end, we will establish the following blow-up criterion for (1.2). Proof. If T * < +∞, then the proof is completed. If T * = +∞, then we show ( . ). Assume by contradiction that the solution ψ(t) exists globally and there exists C > such that Combining this and the conservation of mass, we have Next, we introduce a smooth function θ : [ , ∞) → [ , ] and satisfy For R > , we de ne the radial function After some simple calculations, we can obtain These imply Thus, we can de ne the localized virial function It easily follows that Combining Corollary . , ( . ) and ( . ), we can obtain for some constant C > independent of R and C . We consequently obtain for all t ≥ . We infer from the de nition of θ that Collecting the above estimates, we can obtain the following control about the L -norm of the solution ψ(t) outside a large ball.
Lemma 3.2. Let η > , R > and C be as in ( . ). Then there exists a constant C > independent of R and C such that for any t ∈ [ , T ] with T := ηR CC , Next, we introduce a radial function φ(x) = φ(r) which satis es φ(r) = r for r ≤ , const. for r ≥ , and φ (r) ≤ for r ≥ . For any R > , we de ne the rescaled function φ R : It easily follows that for all r ≥ and all x ∈ R N . It is easy to see that Applying Lemma . , we can obtain Since φ R is a radial function, applying Since φ R ≤ , we deduce from Cauchy-Schwarz inequality that ∞ m s Thus, we can obtain By the Hardy-Littlewood-Sobolev inequality and the conservation of mass, we can estimate as follows We consequently obtain Denote the last term in ( . ) by I. We can obtain In the region {|x| ≥ R}, it follows that This implies that We have a similar control in the region {|y| ≥ R}. By a similar argument as above, we can obtain Applying Lemma . , for any η > and any R > , there exists C > independent of R and C such that for any t ∈ [ , T ] with T = ηR CC , We rst choose η > small enough so that We next choose R > large enough so that for any t ∈ [ , T ] with T = ηR CC . Note that η > is xed, so we can choose R > large enough so that T is as large as we want. By ( . ), it follows that By interpolating between L andḢ s , we get for any t ∈ [t , T ], This implies that for all t ∈ [t , T ] with some su ciently large t ∈ [t , T ]. Taking t close to T = ηR CC , we see that (−∆) s/ ψ(t) L → ∞ as R → ∞. Taking R > su ciently large, we have a contradiction with ( . ). The proof is complete.
Proof of Theorem 1.1. We need only to check that (3.1) follows. In the L -critical case, i.e., sc = , we infer from (1.10) that ψ L < u L and ψ L > u L , which is an contradiction. Thus, when sc = , we have E(ψ ) < . Applying the conservation of energy and + s+α N ≤ p < N+α N− s , it follows that for all t ∈ [ , T * ). Hence, (3.1) follows with δ = − sE(ψ ). Next, we consider the case E(ψ ) ≥ . We deduce from the assumption (1.10) that By a direct calculation, we also have Multiplying both sides of E(ψ(t)) by ψ(t) σ L and use the sharp Gagliardo-Nirenberg inequality (2.2), we obtain It easily follows that f is increasing on ( , x ) and decreasing on (x , ∞), where where the last equality follows from (3.17). It follows from (3.17) and (3.18) that Thus the conservation of mass and energy together with the rst condition in (1.10) imply for all t ∈ [ , T * ). Using the second condition (1.10), the continuity argument shows that for any t ∈ [ , T * ). On the other hand, since E(ψ ) ψ σ L < E(u) u σ L , we pick η > small enough so that Thus, by the conservation of energy, (3.18) and (3.19), we have for all t ∈ [ , T * ). This implies (3.1) with δ = ηθE(u) u σ L . Thus, the solution ψ(t) of (1.2) blows up in nite or in nite time. This completes the proof.

Existence and instability of normalized standing waves
In this section, we will prove the existence and instability of normalized standing waves of (1.2). Firstly, we prove Theorem 1.2.
Proof of Theorem 1.2. We rst show m(c) > . By K(v) = and the inequality (2.2), we have where θ = Np − N − α, which implies that there exists C > such that v Ḣs ≥ C > . Thus, it follows from there exists µ ∈ ( , ) such that Vµ L = c. We consequently estimate as follows: which is a contraction. Finally, we claim that there exists only one term U j ≠ . Indeed, if there exist two terms U j ≠ and U j ≠ , it follows from (4.7) that K(U j ) = , K(U j ) = and Next, we set U j µ = µ s+α p− U j (µx), U j µ = µ s+α p− U j (µx). (4.14) It follows from that K(U j µ ) = K(U j ) = , and K(U j µ ) = K(U j ) = for all µ > . By R N |U j | dx < c and R N |U j | dx < c, we obtain that there exist µ , µ ∈ ( , ) such that (4.15) Thus, we can estimate as follows: which is a contradiction. Therefore, there exists only one term U j ≠ in the decomposition (4.2) and K(U j ) = , which together with (2.5) implies that the in mum of the variational problem (1.15) is attained at U j . This completes the proof. It is easy to see that the equation ∂ λ E(u λ c ) = has a unique non-zero solution sp uc Ḣs θ R N (Iα * |uc| p )(x)|uc(x)| p dx θ− s = .