BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES FOR THE FRACTIONAL SCHR¨ODINGER POISSON EQUATION

. In this article, we consider blow-up criteria and instability of standing waves for the fractional Schr¨odinger-Poisson equation. By using the localized virial estimates, we establish the blow-up criteria for non-radial solutions in both mass-critical and mass-supercritical cases. Based on these blow-up criteria and three variational characterizations of the ground state, we prove that the standing waves are strongly unstable. These obtained re-sults extend the corresponding ones presented in the literature.


Introduction
In recent years, there has been a great deal of interest in using fractional Laplacians to model physical phenomena.By extending the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths, Laskin [23,24] used the theory of functionals over functional measure generated by the Lévy stochastic process to introduce the fractional nonlinear Schrödinger equation (NLS) where i 2 = −1, 0 < s < 1 and f (ψ) is the nonlinearity.The fractional differential operator (−∆) s is defined by (−∆) s ψ = F −1 [|ξ| 2s F(ψ)], where F and F −1 are the Fourier transform and inverse Fourier transform, respectively.The fractional NLS also appears in the continuum limit of discrete models with long-range interactions (see [22]) and in the description of Bonson stars as well as in water wave dynamics (see [15]).Recently, an optical realization of the fractional Schrödinger equation was proposed by Longhi [28].
In this article, we consider the blow-up criteria and the strong instability of standing waves for the fractional Schrödinger-Poisson equation where ψ : [0, T * ) × R 3 → C is the complex valued function, s, r ∈ (0, 1), and 0 < T * ≤ ∞, 0 < p < 4s 3−2s .Under this assumption, φ can be expressed as which is called the r-Riesz potential, where In (1.3), and in the sequel, in we often omit the constant c r for convenience of notation.Substituting φ into (1.2) leads to the fractional Schrödinger equation where ψ 0 ∈ H s .For the classical NLS, i.e., s = 1, we have the Variance-Virial Law 1 2 provided that ψ 0 ∈ Σ := {v ∈ H 1 : xv ∈ L 2 }, where Im denotes the imaginary part.By using (1.5) and the virial identity, we can obtain the blow-up results for the classical NLS with negative energy E(ψ 0 ) < 0 and finite variance [5].However, this argument breaks down for 0 < s < 1, since identity (1.5) fails in this case by the dimensional analysis.It turns out that the suitable generalization of the variance for the fractional NLS is (1.6) Given any sufficiently regular and spatially localized solution ψ(t) of the free fractional Schrödinger equation i∂ t ψ = (−∆) s ψ, a calculation yields 1 2 This idea has been successfully applied to prove the blow-up results for (1.1) with radial solutions and the Hartree-type nonlinearity (|x| −γ * |ψ| 2 )ψ with γ ≥ 1 in [6,43].But this method can not work due to the nontrivial error terms which seem very hard to control for the local nonlinearity |ψ| p ψ. Boulenger et al. [3] applied the Balakrishman's formula and obtained the differential estimate , where δ = 3p − 2s.With the help of this key estimate, they proved the existence of radial blow-up H s solutions by applying the comparison theory.
Inspired by the ideas in [9], we study the blow-up criteria for (1.4).The difficulty is the presence of the fractional order Laplacian (−∆) s .When s = 1, we have 1 2 Using this identity, Du et al. [9] derived an L 2 -estimate in the exterior ball.Thanks to this L 2 -estimate and the virial estimates, they established the blow-up criteria for the classical NLS.In the case s ∈ ( 1 2 , 1), the identity (1.9) does not hold.However, by exploiting the idea in [3] and the use of the Balakrishman's formula (1.8), we can obtain the time derivative of the virial action.Thus, we can obtain the blow-up criteria for (1.4).
Theorem 1.1.Let s ∈ (1/2, 1) and ψ 0 ∈ H s be the corresponding (not necessary radial) solution to (1.4) on the maximal time interval [0, T * ).If there exists δ > 0 such that where Q(ψ(t)) is defined by (1.14).Then one of the following statements is true: • ψ(t) blows up in finite time, i.e.T * < +∞; or • ψ(t) blows up in infinite time and there exists a time sequence (t n ) n≥1 such that t n → +∞ and Based on the blow-up criterion (1.10), we will study the strong instability of standing waves of (1.4).The standing waves of (1.4) are solutions of the form e iωt u, where ω ∈ R is a frequency and u ∈ H s \{0} is a nontrivial solution to the elliptic equation (1.12) Note that (1.12) can be written as S ω (u) = 0, where is the action functional.Then we define with u λ (x) := λ 3/2 u(λx) and where I ω (u) denotes the Pohozaev identity related to (1.12), see (2.3).
The usual strategy to show the strong instability of standing waves of the classical NLS (s=1) is to establish the finite time blow-up by using the variational characterization of ground states as minimizers of the action functional and the virial identity.More specifically, the variational characterization of ground states by the manifold , where u is the ground state solution.Then, it follows from the virial identity and the choice of initial data ψ 0 that for t ∈ [0, T * ).This implies that the solution ψ(t) blows up in a finite time.Thus, we can prove the strong instability of ground state standing waves [5,26,31,32,33,37].However, in many cases, it is hard to obtain the variational characterization of ground states by the manifold N .But we can obtain the variational characterization of ground states by the Nehari manifold and obtain the key estimate Q(ψ(t)) ≤ 2(S ω (ψ 0 ) − S ω (u)) [16,17,18,27,19,29,30,34,38,41].When s = r = 1 and p ∈ {2/3} ∪ (1, 4/3), Bellazzini and Siciliano [1] proved the existence of orbitally stable standing waves for (1.4).Kikuchi [25] showed the existence of standing waves for (1.4) with s = r = 1 and 0 < p < 4 and proved that the standing wave e iωt u is strongly unstable for all ω > 0 when 2 ≤ p < 4. When 4/3 < p < 2, it shows that there exists ω > 0 such that the standing wave e iωt u is strongly unstable for all ω > ω.In the L 2 -supercritical case, i.e., 4/3 < p < 4, Bellazzini et al. [2] improved the result of Kikuchi and proved that the standing wave e iωt u is strongly unstable for all ω > 0. When 4/3 ≤ p < 4, Feng et al. [13] proved that the standing wave e iωt u is strongly unstable for all ω > 0.
Equation (1.4) with p = 4s/3 is a class of nonlinear Schrödinger equations with combined L 2 -critical and L 2 -subcritical nonlinearities.When we try to study the variational characterization of ground states by the manifold N , it is hard to obtain S ω (u) = 0, see Lemma 5.4.Moreover, we find that the usual Nehari manifold is not a good choice in this case.Fortunately, we can obtain the variational characterization of ground states by the Nehari-Pohozaev manifold N 1 := {u ∈ H s \{0}, K ω (u) = 0}.Based on this variational characterization and a theoretical analysis, we can obtain the strong instability of standing waves for (1.4).
Theorem 1.2.Let ω > 0, s ∈ (1/2, 1), 2s + 2r > 3, 4s/3 ≤ p < 4s/(3 − 2s) and u be the ground state related to (1.12).Then the standing wave ψ(t, x) = e iωt u(x) is strongly unstable in the following sense: there exists {ψ 0,n } ⊂ H s such that ψ 0,n → u in H s as n → ∞ and the corresponding solution ψ n of (1.4) with initial data ψ 0,n blows up in finite time or infinite time for any n ≥ 1.This article is organized as follows.In Section 2, we present some useful lemmas such as the local well-posedness theory of (1.4), Brezis-Lieb's lemma, and the compactness lemma.In Section 3, we prove the localized virial estimates related to (1.4).In Sections 4 and 5, we prove Theorems 1.1 and 1.2, respectively.

Preliminary lemmas
In this section, we recall some preliminary results that will be used later.Firstly, let us recall the local theory for the Cauchy problem (1.4).The local well-posedness for the fractional NLS in the energy space H s was studied by Hong and Sire in [20].The proof is based on Strichartz's estimates and the contraction mapping argument.Note that for non-radial data, Strichartz's estimates have a loss of derivatives.Fortunately, this loss of derivatives can be compensated by using the Sobolev embedding.However, it leads to a weak local well-posedness in the energy space compared to the classical nonlinear Schrödinger equation.We refer the reader to [7,20] for more details.We 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 3 5 ≤ s < 1.
The following compactness lemma is vital in our discussions [8,10].
Lemma 2.4.Let 0 < p < 4s 3−2s and {u n } be a bounded sequence in H s such that lim sup Then there exist a sequence (x n ) n≥1 in R 3 and u ∈ H s \ {0} such that up to a subsequence, Finally, we recall the Pohozaev identity related to (1.12) [40].

Localized virial estimates
In this section, we prove some localized virial estimates related to (1.4).Let us recall some useful results in [3].
for some constant C > 0.
To study the localized virial estimates for (1.4), we introduce the auxiliary function where for some constant C > 0 dependent only on s.
We refer the reader to [3, Appendix A] for the proof of Lemmas 3.1 and 3.2.Given that sin πs π Plancherel's and Fubini's theorems imply that for any u ∈ Ḣs .
for some constant C > 0 dependent only on s.
Proof.The idea is essentially similar to [3,Lemma A.2].For the reader's convenience, we just present the outline of our proof.Splitting m-integral into ρ 0 . . .and ∞ ρ . . .with ρ > 0 to be chosen later.For the first term, we use integration by parts and Hölder's inequality to have Here we use the fact ∇u m L 2 ≤ Cm −1/2 u L 2 and u m L 2 ≤ Cm −1 u L 2 which follows from the definition of u m .For the second term, we have for arbitrary ρ > 0. Minimizing the right hand side with respect to ρ, i.e. choosing ρ 2 , we obtain the desired result.
By the same argument as in Lemma 3.3 and Lemma 3.1, we obtain the following result.
Let 1/2 < s < 1 and ϕ : R 3 → R be such that ϕ ∈ W 2,∞ .Assume that ψ ∈ C([0, T * ), H s ) is a solution to (1.4).We define the localized virial action of ψ associated to ϕ by where Proof.It suffices to prove Lemma 3.5 for ψ(t) ∈ C ∞ 0 (R 3 ).The general case follows by an approximation argument.We write where [X, Y ] = XY − Y X is the commutator of X and Y .To study [(−∆) s , ϕ], we recall the Balakrishman's formula Using the fact that for operators A ≥ 0, B with m > 0 being any positive real number and letting A = (−∆) s , B = ϕ and using the Balakrishman's formula, we have Thus we obtain A direct consequence of Lemmas 3.3 and 3.4 is the following estimate.
We next define the localized Morawetz action of ψ associated to ϕ by By Lemma 3.1, we obtain the bound Hence the quantity Lemma 3.7 (Morawetz identity).Let s ∈ (1/2, 1) and ϕ : R 3 → R be such that ∇ϕ ∈ W 3,∞ .Assume that ψ ∈ C([0, T * ), H s ) is a solution to (1.4).Then for each t ∈ [0, T * ) we have where Proof.Integration by parts yields The rest proof is similar to [3, Lemma 2.1], so we omit the details.
4. Blow-up criteria for (1.4) Lemma 4.1.Let η > 0, R > 1 and the solution ψ(t) of (1.4) satisfy Then there exists a constant C > 0 independent of R and C 1 such that Proof.Let us now introduce θ : [0, ∞) → [0, 1] a smooth function satisfying For R > 1, we denote the radial function We have In particular, we have We define the localized virial potential as We have By Lemma 3.6, (4.1) and (4.2), we obtain for some constant C > 0 independent of R and C 1 .Therefore, , for all t ≥ 0. By the choice of θ and the conservation of mass, we have Combing the above estimates, we arrive at the desired result.
Proof of Theorem 1.1.If T * < +∞, then the proof is done.If T * = +∞, then we need to show (1.11).Let ϕ : R 3 → R be such that ∇ϕ ∈ W 3,∞ .In addition, we assume that ϕ = ϕ(r) is radial and satisfies for r ≤ 1, const.for r ≥ 10, and ϕ (r) ≤ 1 for r ≥ 0. Given R > 0 , we define the rescaled function ϕ R : R 3 We readily verify the inequalities for all r ≥ 0 and all x ∈ R 3 .It is easy to see that and Applying Lemma 3.7, we have where ψ m (t) = ψ m (t, x) is defined in (3.1).Since supt(∆ 2 ϕ R ) ⊂ {|x| ≥ R}, by Lemma 3.2, we have Using (3.2) leads to Thus, we have L 2 (|x|≥R) .Thus we obtain We denote the last term in (4.4) by T .We have

Thus, we obtain
To estimate this term, we deduce from the Sobolev embedding that Thus, it follows from the Hardy-Littlewood-Sobolev inequality and the conservation of mass that L 2 (|x|≥R) .We can derive an estimate in the region {|y| ≥ R} too.Similarly, we can obtain L 2 (|x|≥R) .(4.9) By using (4.5)-(4.9),we obtain We first choose η > 0 small enough so that We next choose R > 1 large enough so that . Note that η > 0 is fixed, so we can choose R > 1 large enough such that T 0 is as large as we want.From (4.11) it follows that for all t ∈ [t 0 , T 0 ] with some sufficiently large t 0 ∈ [0, T 0 ].On the other hand, by Lemma 3.1 and the conservation of mass, we have for any t ∈ [0, +∞), By interpolating between L 2 and Ḣs , we obtain for any t for all t ∈ [t 1 , T 0 ] with some sufficiently large Taking R > 1 sufficiently large, we have a contradiction with (4.1).The proof is complete.

Strong instability of standing waves
In this section, we prove Theorem 1.2.Let us start with the following characterization of the ground state related to (1.12).Proposition 5.1.Let ω > 0, 2s + 2r > 3 and 4s 3 ≤ p < 4s 3−2s .Then u is the ground state related to (1.12) if and only if u solves the minimization problem (5.1) To solve this minimization problem, we consider the minimization problem where (5.3) for sufficiently small λ > 0. Thus, there exists λ 0 ∈ (0, 1) such that K ω (λ 0 v) = 0.It follows that This implies that (5.4) In following lemma, we will solve the minimizing problem (5.2).
Furthermore, we deduce from the definition of d(ω) and the weak lower semicontinuity of norm that This yields S ω (u) = d(ω).
Proof.We firstly prove K ω (u) = 0.If K ω (u) = 0, then we have (5.8) where The first equation comes from the fact that S ω (u) = d(ω).The second one holds since K ω (u) = 0.The third one follows by multiplying (5.8) by u and integrating both sides.The fourth one is derived by applying the Pohozaev equality to (5.8).
After a direct calculations, we have Because of 2s + 2r > 3 and 4s 3 ≤ p < 4s 3−2s , it follows that (p − 2)s + pr > 0. We will show that µ must be equal to zero by excluding the other possibilities: (1) If µ = 0, µ = − We now denote the set of all minimizers of (5.1) by Proof.Let u ∈ M ω .It follows from Lemma 5.4 that S ω (u) = 0.In particular, we have u ∈ A ω .To prove u ∈ G ω , it remains to show that S ω (u) ≤ S ω (v) for all v ∈ A ω .To see this, we notice that for all v ∈ A ω , where I ω (v) is defined by (2.3).By definition of d(ω), we have S ω (u) ≤ S ω (v).Thus, u ∈ G ω .
Thus, applying Theorem 1.1, we obtain that the solution ψ n (t) of (1.4) with initial data ψ 0,n blows up in finite or infinite time for any n ≥ 1.