Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 24, pp. 1–23. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES FOR THE FRACTIONAL SCHRÖDINGER POISSON EQUATION YICHUN MO, MIN ZHU, BINHUA FENG Abstract. In this article, we consider blow-up criteria and instability of standing waves for the fractional Schrödinger-Poisson equation. By using the localized virial estimates, we establish the blow-up criteria for non-radial so- lutions 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. 1. Introduction In recent years, there has been a great deal of interest in using fractional Lapla- cians 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) i∂tψ − (−∆)sψ + f(ψ) = 0, (1.1) where i2 = −1, 0 < s < 1 and f(ψ) is the nonlinearity. The fractional differential operator (−∆)s is defined by (−∆)sψ = F−1[|ξ|2sF(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 i∂tψ − (−∆)sψ − φψ + |ψ|pψ = 0, (t, x) ∈ [0, T ∗)× R3, (−∆)rφ = |ψ|2, (1.2) 2020 Mathematics Subject Classification. 35J60, 35Q55, 35J20. Key words and phrases. Schrödinger-Poisson equation; blow-up criteria; strong instability; standing waves; well-posedness. ©2023. This work is licensed under a CC BY 4.0 license. Submitted October 28, 2022. Published March 6, 2023. 1 2 Y. MO, M. ZHU, B. FENG EJDE-2023/24 where ψ : [0, T ∗) × R3 → 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 φ(x) = cr ∫ R3 |ψ(y)|2 |x− y|3−2r dy, (1.3) which is called the r-Riesz potential, where cr = π−3/22−2r Γ( 3 2 − 2r) Γ(r) . In (1.3), and in the sequel, in we often omit the constant cr for convenience of notation. Substituting φ into (1.2) leads to the fractional Schrödinger equation i∂tψ − (−∆)sψ − (|x|−(3−2r) ∗ |ψ|2)ψ + |ψ|pψ = 0, (t, x) ∈ [0, T ∗)× R3, ψ(0, x) = ψ0(x), (1.4) where ψ0 ∈ Hs. For the classical NLS, i.e., s = 1, we have the Variance-Virial Law 1 2 d dt ∫ R3 |x|2|ψ(t, x)|2 dx = 2 Im ∫ R3 ψ̄(t, x)x · ∇ψ(t, x) dx, (1.5) provided that ψ0 ∈ Σ := {v ∈ H1 : xv ∈ L2}, 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(ψ0) < 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 V(s)[ψ(t)] := ∫ R3 ψ̄(t, x)x · (−∆)1−sxψ(t, x) dx = ‖x(−∆) 1−s 2 ψ(t)‖2L2 . (1.6) Given any sufficiently regular and spatially localized solution ψ(t) of the free frac- tional Schrödinger equation i∂tψ = (−∆)sψ, a calculation yields 1 2 d dt V(s)[ψ(t)] := 2 Im ∫ R3 ψ̄(t, x)x · ∇ψ(t, x) dx. (1.7) 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|pψ. Boulenger et al. [3] applied the Balakrishman’s formula (−∆)s = sinπs π ∫ ∞ 0 ms−1 −∆ −∆ +m dm, (1.8) and obtained the differential estimate d dt ( Im ∫ R3 ψ̄(t)∇ϕR · ∇ψ(t) dx ) ≤ 12pE(ψ0)− 2δ‖(−∆)s/2ψ(t)‖2L2 + ◦R(1)(1 + ‖(−∆)s/2ψ(t)‖p/s+L2 ), where δ = 3p − 2s. With the help of this key estimate, they proved the existence of radial blow-up Hs solutions by applying the comparison theory. However, to the best of our knowledge, there are no any blow-up results for (1.4) so far. In particular, equation (1.4) includes two classical nonlinearities, i.e., power-type |ψ|pψ and Hartree-type (|x|−(3−2r) ∗ |ψ|2)ψ. The study of blow-up EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 3 solutions for (1.4) is of particular challenge, because the methods for proving blow- up results of (1.1) with power-type |ψ|pψ or Hartree-type (|x|−(3−2r) ∗ |ψ|2)ψ are usually different, so we should develop a new method when both nonlinearities appear simultaneously. 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 d dt ∫ R3 ϕ(x)|ψ(t, x)|2 dx = 2 Im ∫ R3 ψ̄(t, x)∇ϕ(x) · ∇ψ(t, x) dx. (1.9) Using this identity, Du et al. [9] derived an L2-estimate in the exterior ball. Thanks to this L2-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 ψ0 ∈ Hs be the corresponding (not necessary radial) solution to (1.4) on the maximal time interval [0, T ∗). If there exists δ > 0 such that sup t∈[0,T∗) Q(ψ(t)) ≤ −δ < 0, (1.10) 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 (tn)n≥1 such that tn → +∞ and lim n→∞ ‖(−∆)s/2ψ(tn)‖L2 =∞. (1.11) 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 eiωtu, where ω ∈ R is a frequency and u ∈ Hs\{0} is a nontrivial solution to the elliptic equation (−∆)su+ ωu+ (|x|−(3−2r) ∗ |u|2)u− |u|pu = 0. (1.12) Note that (1.12) can be written as S′ω(u) = 0, where Sω(u) := 1 2 ‖u‖2 Ḣs + ω 2 ‖u‖2L2 + 1 4 ∫ R3 (|x|−(3−2r) ∗ |u|2)|u|2dx − 1 p+ 2 ‖u‖p+2 Lp+2 (1.13) is the action functional. Then we define Q(u) := ∂λSω(uλ)|λ=1 = s‖u‖2 Ḣs + 3− 2r 4 ∫ R3 (|x|−(3−2r) ∗ |u|2)|u|2dx− 3p 2(p+ 2) ‖u‖p+2 Lp+2 (1.14) 4 Y. MO, M. ZHU, B. FENG EJDE-2023/24 with uλ(x) := λ3/2u(λx) and Kω(u) := (s+ r)〈S′ω(u), u〉 − Iω(u) = 4s+ 2r − 3 2 ‖u‖2 Ḣs + ω(2s+ 2r − 3) 2 ‖u‖2L2 + 4s+ 2r − 3 4 ∫ R3 (|x|−(3−2r) ∗ |u|2)|u|2dx − (s+ r)(p+ 2)− 3 p+ 2 ‖u‖p+2 Lp+2 , (1.15) where Iω(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 clas- sical 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 N := {v ∈ H1\{0}, Q(v) = 0} can imply the key estimate Q(ψ(t)) ≤ 2(Sω(ψ0) − Sω(u)), where u is the ground state solution. Then, it follows from the virial identity and the choice of initial data ψ0 that d2 dt2 ‖xψ(t)‖2L2 = 8Q(ψ(t)) ≤ 16(Sω(ψ0)− Sω(u)) < 0, 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 manifoldN . But we can obtain the variational characterization of ground states by the Nehari manifold and obtain the key estimate Q(ψ(t)) ≤ 2(Sω(ψ0)− Sω(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 eiωtu is strongly unstable for all ω > 0 when 2 ≤ p < 4. When 4/3 < p < 2, it shows that there exists ω̄ > 0 such that the standing wave eiωtu is strongly unstable for all ω > ω̄. In the L2-supercritical case, i.e., 4/3 < p < 4, Bellazzini et al. [2] improved the result of Kikuchi and proved that the standing wave eiωtu is strongly unstable for all ω > 0. When 4/3 ≤ p < 4, Feng et al. [13] proved that the standing wave eiωtu is strongly unstable for all ω > 0. Equation (1.4) with p = 4s/3 is a class of nonlinear Schrödinger equations with combined L2-critical and L2-subcritical nonlinearities. When we try to study the variational characterization of ground states by the manifold N , it is hard to ob- tain S′ω(u) = 0, see Lemma 5.4. Moreover, we find that the usual Nehari mani- fold is not a good choice in this case. Fortunately, we can obtain the variational characterization of ground states by the Nehari-Pohozaev manifold N1 := {u ∈ Hs\{0}, Kω(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) = eiωtu(x) is strongly unstable in the following sense: there exists {ψ0,n} ⊂ Hs such that ψ0,n → u in Hs as n→∞ and the corresponding solution ψn of (1.4) with initial data ψ0,n blows up in finite time or infinite time for any n ≥ 1. EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 5 This article is organized as follows. In Section 2, we present some useful lem- mas 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. 2. 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 Hs 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. Proposition 2.1. Let 3/5 ≤ s < 1, 0 < p < 4s 3−2s and ψ0 ∈ Hs be radial. Then there exists T = T (‖ψ0‖Hs) such that (1.4) admits a unique solution ψ ∈ C([0, T ], Hs). Let [0, T ∗) be the maximal time interval on which the solution ψ is well-defined. If T ∗ < ∞, then ‖ψ(t)‖Ḣs → ∞ as t ↑ T ∗. Moreover, for all 0 ≤ t < T ∗, the solution ψ(t) satisfies the following conservation of mass and energy ‖ψ(t)‖L2 = ‖ψ0‖L2 , E(ψ(t)) = E(ψ0), where E(ψ(t)) = 1 2 ‖ψ(t)‖2 Ḣs + 1 4 ∫ R3 (|x|−(3−2r) ∗ |ψ(t)|2)(x)|ψ(t, x)|2dx − 1 p+ 2 ‖ψ(t)‖p+2 Lp+2 . (2.1) In this article, we use the so called Brezis-Lieb’s lemma [4]. Lemma 2.2. Let 0 < p < ∞. Suppose that un → u almost everywhere and {un} is a bounded sequence in Lp. Then lim n→∞ (‖un‖pLp − ‖un − u‖pLp − ‖u‖pLp) = 0. Lemma 2.3 ([40]). Let u ∈ Hs and 2s+ 2r > 3. Suppose that un ⇀ u in Hs and un → u a.e. in R3. Then∫ R3 (|x|−(3−2r) ∗ |un|2)|un|2dx = ∫ R3 (|x|−(3−2r) ∗ |un − u|2)|un − u|2dx + ∫ R3 (|x|−(3−2r) ∗ |u|2)|u|2dx+ ◦(1). The following compactness lemma is vital in our discussions [8, 10]. 6 Y. MO, M. ZHU, B. FENG EJDE-2023/24 Lemma 2.4. Let 0 < p < 4s 3−2s and {un} be a bounded sequence in Hs such that lim sup n→∞ ‖un‖Ḣs ≤M, lim inf n→∞ ‖un‖Lp+2 ≥ m. Then there exist a sequence (xn)n≥1 in R3 and u ∈ Hs \ {0} such that up to a subsequence, un(·+ xn) ⇀ u weakly in Hs. Finally, we recall the Pohozaev identity related to (1.12) [40]. Lemma 2.5. If u ∈ Hs satisfies equation (1.12), then it holds ‖u‖2 Ḣs + ω‖u‖2L2 + ∫ R3 (|x|−(3−2r) ∗ |u|2)|u|2dx− ‖u‖p+2 Lp+2 = 0 (2.2) and Iω(u) := 3− 2s 2 ‖u‖2 Ḣs + 3ω 2 ‖u‖2L2 + 2r + 3 4 ∫ R3 (|x|−(3−2r) ∗ |u|2)|u|2dx − 3 p+ 2 ‖u‖p+2 Lp+2 = 0. (2.3) 3. Localized virial estimates In this section, we prove some localized virial estimates related to (1.4). Let us recall some useful results in [3]. Lemma 3.1 ([3]). Suppose ϕ : R3 → R is such that ∇ϕ ∈ W 1,∞. Then for all u ∈ H1/2, it holds∣∣ ∫ R3 u(x)∇ϕ(x) · ∇u(x)dx ∣∣ ≤ C‖∇ϕ‖W 1,∞ ( ‖|∇|1/2u‖2L2 + ‖u‖L2‖|∇|1/2u‖L2 ) for some constant C > 0. To study the localized virial estimates for (1.4), we introduce the auxiliary func- tion um(x) := cs 1 −∆ +m u(x) = csF−1 ( û(ξ) |ξ|2 +m ) , m > 0, (3.1) where cs := √ sinπs π . Lemma 3.2 ([3]). Let s ∈ (0, 1) and ϕ : R3 → R with ∆ϕ ∈ W 2,∞. Then for all u ∈ L2 it holds∣∣ ∫ ∞ 0 ms ∫ R3 (∆2ϕ)|um|2 dx dm ∣∣ ≤ C‖∆2ϕ‖sL∞‖∆ϕ‖1−sL∞ ‖u‖ 2 L2 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 sinπs π ∫ ∞ 0 ms (|ξ|2 +m)2 dm = s|ξ|2s−2, EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 7 Plancherel’s and Fubini’s theorems imply that∫ ∞ 0 ms ∫ R3 |∇um|2 dx dm = ∫ R3 ( sinπs π ∫ ∞ 0 ms dm (|ξ|2 +m)2 ) |ξ|2|û(ξ)|2dξ = ∫ R3 (s|ξ|2s−2)|ξ|2|û(ξ)|2dξ = s‖(−∆)s/2u‖2L2 (3.2) for any u ∈ Ḣs. Lemma 3.3. Let s ∈ (1/2, 1) and ϕ : R3 → R be such that ∇ϕ ∈W 1,∞. Then for any u ∈ L2 it holds∣∣∣ ∫ ∞ 0 ms ∫ R3 (∆ϕ)|um|2 dx dm ∣∣∣ ≤ C‖∆ϕ‖2s−1 L∞ ‖∇ϕ‖ 2−2s L∞ ‖u‖ 2 L2 for some constant C > 0 dependent only on s. Proof. The idea is essentially similar to [3, Lemma A.2]. For the reader’s conve- nience, 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∣∣∣ ∫ ρ 0 ms ∫ R3 (∆ϕ)|um|2 dx dm ∣∣∣ = ∣∣∣ ∫ ρ 0 ms ∫ R3 ∇ϕ · (∇umum + um∇um) dx dm ∣∣∣ = ‖∇ϕ‖L∞ ∫ ρ 0 ms‖∇um‖L2‖um‖L2 dm = ‖∇ϕ‖L∞‖u‖2L2 (∫ ρ 0 ms−3/2 dm ) ≤ Cρs−1/2‖∇ϕ‖L∞‖u‖2L2 . Here we use the fact ‖∇um‖L2 ≤ Cm−1/2‖u‖L2 and ‖um‖L2 ≤ Cm−1‖u‖L2 which follows from the definition of um. For the second term, we have∣∣∣ ∫ ∞ ρ ms ∫ R3 (∆ϕ)|um|2 dx dm ∣∣∣ ≤ C‖∆ϕ‖L∞(∫ ∞ ρ ms‖um‖2L2 dm ) ≤ C‖∆ϕ‖L∞‖u‖2L2 (∫ ∞ ρ ms−2 dm ) ≤ Cρs−1‖∆ϕ‖L∞‖u‖2L2 . Combining two terms yields∣∣∣ ∫ ∞ 0 ∫ R3 (∆ϕ)|um|2 dx dm ∣∣∣ ≤ C(ρs−1/2‖∇ϕ‖L∞ + ρs−1‖∆ϕ‖L∞ ) ‖u‖2L2 for arbitrary ρ > 0. Minimizing the right hand side with respect to ρ, i.e. choosing ρ = ( (1−s)‖∆ϕ‖L∞ (s−1/2)‖∇ϕ‖L∞ )2 , we obtain the desired result. � By the same argument as in Lemma 3.3 and Lemma 3.1, we obtain the following result. Lemma 3.4. Let s ∈ (1/2, 1) and ϕ : R3 → R be such that ∇ϕ ∈W 1,∞. Then for any u ∈ H1/2 we have∣∣∣ ∫ ∞ 0 ∫ R3 um∇ϕ · ∇um dx dm ∣∣∣ ≤ C‖∇ϕ‖W 1,∞‖u‖2H1/2 for some constant C > 0. 8 Y. MO, M. ZHU, B. FENG EJDE-2023/24 Let 1/2 < s < 1 and ϕ : R3 → R be such that ϕ ∈ W 2,∞. Assume that ψ ∈ C([0, T ∗), Hs) is a solution to (1.4). We define the localized virial action of ψ associated to ϕ by Vϕ[ψ(t)] := ∫ R3 ϕ(x)|ψ(t, x)|2 dx. Lemma 3.5 (Virial identity). Let s ∈ (1/2, 1) and ϕ : R3 → R be such that ϕ ∈ W 2,∞. Assume that ψ ∈ C([0, T ∗), Hs) is a solution to (1.4). Then for any t ∈ [0, T ∗) we have d dt Vϕ[ψ(t)] = −i ∫ ∞ 0 ms ∫ R3 (∆ϕ)|ψm(t)|2 dx dm− 2i ∫ ∞ 0 ms ∫ R3 ψm(t)∇ϕ · ∇ψm(t) dx dm, where ψm(t) = cs(−∆ +m)−1ψ(t). Proof. It suffices to prove Lemma 3.5 for ψ(t) ∈ C∞0 (R3). The general case follows by an approximation argument. We write Vϕ[ψ(t)] = 〈ψ(t), ϕψ(t)〉, where 〈·, ·〉 is the scalar product in L2. Since ψ(t) satisfies (1.4), it is easy to see that d dt Vϕ[ψ(t)] = i〈ψ(t), [(−∆)s, ϕ]ψ(t)〉, where [X,Y ] = XY −Y X is the commutator of X and Y . To study [(−∆)s, ϕ], we recall the Balakrishman’s formula (−∆)s = sinπs π ∫ ∞ 0 ms−1 −∆ −∆ +m dm. Using the fact that for operators A ≥ 0, B with m > 0 being any positive real number[ A A+m ,B ] = [ 1− m A+m ,B ] = −m [ 1 A+m ,B ] = m 1 A+m [A,B] 1 A+m . and letting A = (−∆)s, B = ϕ and using the Balakrishman’s formula, we have [(−∆)s, ϕ] = sinπs π ∫ ∞ 0 ms [ −∆ −∆ +m ,ϕ ] dm = sinπs π ∫ ∞ 0 ms 1 −∆ +m [−∆, ϕ] 1 −∆ +m dm. Thus we obtain 〈ψ(t), [(−∆)s, ϕ]ψ(t)〉 = 〈ψ(t), ( sinπs π ∫ ∞ 0 ms 1 −∆ +m [−∆, ϕ] 1 −∆ +m dm ) ψ(t)〉 = c2s ∫ ∞ 0 ms〈ψ(t), 1 −∆ +m [−∆, ϕ] 1 −∆ +m ψ(t)〉 dm = ∫ ∞ 0 ms〈cs(−∆ +m)−1ψ(t), [−∆, ϕ]cs(−∆ +m)−1ψ(t)〉 dm = ∫ ∞ 0 ms ∫ R3 ψm(t) ( −∆ϕψm(t)− 2∇ϕ · ∇ψm(t) ) dx dm EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 9 = ∫ ∞ 0 ms ∫ R3 ( (−∆ϕ)|ψm(t)|2 − 2ψm(t)∇ϕ · ∇um(t) ) dx dm. � A direct consequence of Lemmas 3.3 and 3.4 is the following estimate. Lemma 3.6. Let s ∈ (1/2, 1) and ϕ : R3 → R be such that ϕ ∈ W 2,∞. Assume that ψ ∈ C([0, T ∗), Hs) is a solution to (1.4). Then for any t ∈ [0, T ∗) we have∣∣ d dt Vϕ[ψ(t)] ∣∣ ≤ C‖∇ϕ‖W 1,∞‖ψ(t)‖2Hs for some constant C > 0 dependent only on s. We next define the localized Morawetz action of ψ associated to ϕ by Mϕ[ψ(t)] := 2 Im ∫ R3 ψ̄(t, x)∇ϕ(x) · ∇ψ(t, x)dx. (3.3) By Lemma 3.1, we obtain the bound |Mϕ[ψ(t)]| ≤ C (‖∇ϕ‖L∞ , ‖∆ϕ‖L∞) ‖ψ(t)‖2H1/2 . Hence the quantity Mϕ[ψ(t)] is well-defined, given ψ(t) ∈ Hs(R3) with s > 1/2. Lemma 3.7 (Morawetz identity). Let s ∈ (1/2, 1) and ϕ : R3 → R be such that ∇ϕ ∈W 3,∞. Assume that ψ ∈ C([0, T ∗), Hs) is a solution to (1.4). Then for each t ∈ [0, T ∗) we have d dt Mϕ[ψ(t)] = ∫ ∞ 0 ms ∫ R3 { 4∂kψm(t)(∂2 klϕ)∂lψm(t)− (∆2ϕ)|ψm(t)|2 } dx dm + (3− 2r) ∫ R3 ∫ R3 |u(t, x)|2|u(t, y)|2(x− y) · (∇ϕ(x)−∇ϕ(y)) |x− y|5−2r dx dy − 2p p+ 2 ∫ R3 ∆ϕ|ψ(t)|p+2 dx, (3.4) where ψm(t) = ψm(t, x) is defined in (3.1). Proof. Integration by parts yields 〈u(t), [−(|x|−(3−2r) ∗ |u(t)|2), iΓϕ]u(t)〉 = −〈u(t), [(|x|−(3−2r) ∗ |u(t)|2),∇ϕ · ∇+∇ · ∇ϕ]u(t)〉 = 2 ∫ R3 ∇ϕ · ∇(|x|−(3−2r) ∗ |u(t)|2)|u(t)|2dx = −(3− 2r) ∫ R3 ∫ R3 |u(t, x)|2|u(t, y)|2(x− y) · (∇ϕ(x)−∇ϕ(y)) |x− y|5−2r dx dy. The rest proof is similar to [3, Lemma 2.1], so we omit the details. � 10 Y. MO, M. ZHU, B. FENG EJDE-2023/24 4. Blow-up criteria for (1.4) Lemma 4.1. Let η > 0, R > 1 and the solution ψ(t) of (1.4) satisfy C1 := sup t∈[0,+∞) ‖ψ(t)‖Hs <∞. (4.1) Then there exists a constant C > 0 independent of R and C1 such that∫ |x|≥R |ψ(t, x)|2 dx ≤ η + oR(1) for all t ∈ [0, T0] with T0 := ηR CC2 1 . Proof. Let us now introduce θ : [0,∞)→ [0, 1] a smooth function satisfying θ(r) = { 0 if 0 ≤ r ≤ 1/2, 1 if r ≥ 1. For R > 1, we denote the radial function φR(x) := θ(r/R), r = |x|. We have ∇φR(x) = x rR θ′(r/R), ∆φR(x) = 1 R2 θ′′(r/R) + 2 rR θ′(r/R). In particular, we have ‖∇φR‖W 1,∞ ∼ ‖∇φR‖L∞ + ‖∆φR‖L∞ ≤ CR−1. (4.2) We define the localized virial potential as VφR [ψ(t)] := ∫ R3 φR(x)|ψ(t, x)|2 dx. We have VφR [ψ(t)] = VφR [ψ0] + ∫ t 0 d dτ VφR [ψ(τ)]dτ ≤ VφR [ψ0] + ( sup τ∈[0,t] ∣∣ d dτ VφR [ψ(τ)] ∣∣)t. By Lemma 3.6, (4.1) and (4.2), we obtain sup τ∈[0,t] ∣∣ d dτ VφR [ψ(τ)] ∣∣ ≤ C‖∇φR‖W 1,∞ sup τ∈[0,t] ‖ψ(τ)‖2Hs ≤ CC2 1R −1, for some constant C > 0 independent of R and C1. Therefore, VφR [ψ(t)] ≤ VφR [ψ0] + CC2 1R −1t, for all t ≥ 0. By the choice of θ and the conservation of mass, we have VφR [ψ0] = ∫ R3 φR(x)|ψ0(x)|2 dx ≤ ∫ |x|>R/2 |ψ0(x)|2 dx→ 0, as R→∞ or VφR [ψ0] = oR(1). On the other hand,∫ |x|≥R |ψ(t, x)|2 dx ≤ VφR [ψ(t)]. Combing the above estimates, we arrive at the desired result. � EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 11 Proof of Theorem 1.1. If T ∗ < +∞, then the proof is done. If T ∗ = +∞, then we need to show (1.11). Let ϕ : R3 → R be such that ∇ϕ ∈ W 3,∞. In addition, we assume that ϕ = ϕ(r) is radial and satisfies ϕ(r) = { r2 2 for r ≤ 1, const. for r ≥ 10, and ϕ′′(r) ≤ 1 for r ≥ 0. Given R > 0 , we define the rescaled function ϕR : R3 → R by ϕR(r) := R2ϕ ( x R ) . (4.3) We readily verify the inequalities 1− ϕ′′R(r) ≥ 0, 1− ϕ′R(r) r ≥ 0, 3−∆ϕR(x) ≥ 0, for all r ≥ 0 and all x ∈ R3. It is easy to see that ‖∇kϕR‖L∞ ≤ R2−k, k = 0, . . . , 4, and supt(∇kϕR) ⊂ { {|x| ≤ 10R} for k = 1, 2, {R ≤ |x| ≤ 10R} for k = 3, 4. Applying Lemma 3.7, we have d dt MϕR [ψ(t)] = ∫ ∞ 0 ms ∫ R3 { 4∂kψm(t)(∂2 klϕR)∂lψm(t)− (∆2ϕR)|ψm(t)|2 } dx dm − 2p p+ 2 ∫ R3 ∆ϕR|ψ(t)|p+2 dx + (3− 2r) ∫ R3 ∫ R3 |ψ(t, x)|2|ψ(t, y)|2(x− y) · (∇ϕR(x)−∇ϕR(y)) |x− y|5−2r dx dy, (4.4) where ψm(t) = ψm(t, x) is defined in (3.1). Since supt(∆2ϕR) ⊂ {|x| ≥ R}, by Lemma 3.2, we have∣∣∣ ∫ ∞ 0 ms ∫ R3 (∆2ϕR)|ψm(t)|2 dx dm ∣∣∣ ≤ C‖∆2ϕR‖sL∞‖∆ϕR‖1−sL∞ ‖ψ(t)‖2L2(|x|≥R) ≤ CR−2s‖ψ(t)‖2L2(|x|≥R). (4.5) Since ϕR is radial, we use ∂2 jk = (δjk r − xjxk r3 ) ∂r + xjxk r2 ∂2 r to deduce ∫ ∞ 0 ms ∫ R3 ∂kψm(t)(∂2 jkϕR)∂lψm(t) dx dm = ∫ ∞ 0 ms ∫ R3 ϕ′R r |∇ψm(t)|2 dx dm + ∫ ∞ 0 ms ∫ R3 (ϕ′′R r2 − ϕ′R r3 ) |x · ∇ψm(t)|2 dx dm. 12 Y. MO, M. ZHU, B. FENG EJDE-2023/24 Using (3.2) leads to∫ ∞ 0 ms ∫ R3 ϕ′R r |∇ψm(t)|2 dx dm = s‖(−∆)s/2ψ(t)‖2L2 + ∫ ∞ 0 ms ∫ R3 (ϕ′R r − 1 ) |∇ψm(t)|2 dx dm. Since ϕ′′R ≤ 1, the Cauchy-Schwarz inequality implies∫ ∞ 0 ms ∫ R3 (ϕ′R r − 1 ) |∇ψm(t)|2 dx dm + ∫ ∞ 0 ms ∫ R3 ( ϕ′′R − ϕ′R r ) |x · ∇ψm(t)|2 r2 dx dm ≤ 0. Thus, we have 4 ∫ ∞ 0 ms ∫ R3 ∂kψm(t)(∂2 jkϕR)∂lψm(t) dx dm ≤ 4s‖(−∆)s/2ψ(t)‖2L2 . (4.6) Note that − 2p p+ 2 ∫ R3 ∆ϕR|ψ(t)|p+2 dx = − 6p p+ 2 ‖ψ(t)‖p+2 Lp+2+ 2p p+ 2 ∫ R3 (3−∆ϕR)|ψ(t)|p+2 dx. Since supt(3−∆ϕR) ⊂ {|x| ≥ R} and ‖3−∆ϕR‖L∞ ≤ C, we have∫ R3 (3−∆ϕR)|ψ(t)|p+2 dx ≤ C ∫ |x|≥R |ψ(t)|p+2 dx ≤ C‖ψ(t)‖ 3p 2s L 6 3−2s (|x|≥R) ‖ψ‖ 4s−(3−2s)p 2s L2(|x|≥R) ≤ C‖ψ(t)‖ 3p 2s Hs‖ψ(t)‖ 4s−(3−2s)p 2s L2(|x|≥R) ≤ CC 3p 2s 1 ‖ψ(t)‖ 4s−(3−2s)p 2s L2(|x|≥R) . Thus we obtain − 2p p+ 2 ∫ R3 ∆ϕR|ψ(t)|p+2 dx ≤ − 6p p+ 2 ‖ψ(t)‖p+2 Lp+2 + CC 3p 2s 1 ‖ψ(t)‖ 4s−(3−2s)p 2s L2(|x|≥R) . (4.7) We denote the last term in (4.4) by T . We have T = (3− 2r) ∫ R3 ∫ R3 (x− y) · (∇ϕR(x)−∇ϕR(y)) |ψ(t, x)|2|ψ(t, y)|2 |x− y|5−2r dx dy. By using supt(|x− y|2 − (x− y) · (∇ϕR(x)−∇ϕR(y))) ⊂ {|x| ≥ R} ∪ {|y| ≥ R}, in the region {|x| ≥ R} we obtain∣∣ |x− y|2 − (x− y) · (∇ϕR(x)−∇ϕR(y)) ∣∣ ≤ C|x− y|2. Thus, we obtain∣∣∣ ∫ |x|≥R ∫ R3 [|x− y|2 − (x− y) · (∇ϕR(x)−∇ϕR(y))] |u(t, x)|2|ψ(t, y)|2p2 |x− y|5−2r dx dy ∣∣∣ ≤ C ∫ |x|≥R (|x|−(3−2r) ∗ |ψ(t)|2)|ψ(t)|2 dx. EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 13 To estimate this term, we deduce from the Sobolev embedding that ‖ψ(t)‖2 L 12 3+2r ≤ C‖ψ(t)‖ 4s+2r−3 2s L2 ‖ψ(t)‖ 3−2r 2s L 6 3−2s ≤ C‖(−∆)s/2ψ(t)‖ 3−2r 2s L2 . (4.8) Thus, it follows from the Hardy-Littlewood-Sobolev inequality and the conservation of mass that∫ |x|≥R (|x|−(3−2r) ∗ |ψ(t)|2)|ψ(t)|2 dx ≤ C‖|x|−(3−2r) ∗ |ψ(t)|2‖ L 6 3−2r (|x|≥R) ‖|ψ(t)|2‖ L 6 3+2r (|x|≥R) ≤ C‖ψ(t)‖2 L 12 3+2r ‖ψ(t)‖2 L 12 3+2r (|x|≥R) ≤ C‖ψ(t)‖ 3−2r s Hs ‖ψ(t)‖ 4s+2r−3 2s L2(|x|≥R) ≤ CC 3−2r s 1 ‖ψ(t)‖ 4s+2r−3 2s L2(|x|≥R). We can derive an estimate in the region {|y| ≥ R} too. Similarly, we can obtain T ≤ (3− 2r) ∫ R3 (|x|−(3−2r) ∗ |ψ(t)|2)|ψ(t)|2 dx+ CC 3−2r s 1 ‖ψ(t)‖ 4s+2r−3 2s L2(|x|≥R). (4.9) By using (4.5)–(4.9), we obtain d dt MϕR [ψ(t)] ≤ 4s‖(−∆)s/2ψ(t)‖2L2 + CR−2s‖ψ(t)‖2L2(|x|≥R) + (3− 2r) ∫ R3 (|x|−(3−2r) ∗ |ψ(t)|2)|u(t)|2 dx− 6p p+ 2 ‖ψ(t)‖p+2 Lp+2 + CC 3p 2s 1 ‖ψ(t)‖ 4s−(3−2s)p 2s L2(|x|≥R) + CC 3−2r s 1 ‖ψ(t)‖ 4s+2r−3 2s L2(|x|≥R) ≤ 4Q(ψ(t)) + CR−2s‖ψ(t)‖2L2(|x|≥R) + CC 3p 2s 1 ‖ψ(t)‖ 4s−(3−2s)p 2s L2(|x|≥R) + CC 3−2r s 1 ‖ψ(t)‖ 4s+2r−3 2s L2(|x|≥R). (4.10) By Lemma 4.1, we see that for any η > 0 and any R > 1, there exists C > 0 independent of R and C1 such that for any t ∈ [0, T0] with T0 = ηR CC2 1 , we have d dt MϕR [ψ(t)] ≤ 4Q(ψ(t)) + CR−2s(η + oR(1))2 + CC 3p 2s 1 (η + oR(1)) 4s−p(3−2s) 2s + CC 3−2r s 1 (η + oR(1)) 4s+2r−3 2s ≤ −4δ + CR−2s(η2 + oR(1)) + CC 3p 2s 1 (η 4s−p(3−2s) 2s + oR(1)) + CC 3−2r s 1 (η 4s+2r−3 2s + oR(1)). We first choose η > 0 small enough so that CC 3p 2s 1 η 4s−p(3−2s) 2s + CC 3−2r s 1 η 4s+2r−3 2s = −3δ > 0. We next choose R > 1 large enough so that d dt MϕR [ψ(t)] ≤ −δ < 0 (4.11) 14 Y. MO, M. ZHU, B. FENG EJDE-2023/24 for any t ∈ [0, T0] with T0 = ηR CC2 1 . Note that η > 0 is fixed, so we can choose R > 1 large enough such that T0 is as large as we want. From (4.11) it follows that MϕR [ψ(t)] ≤ −δt, for all t ∈ [t0, T0] with some sufficiently large t0 ∈ [0, T0]. On the other hand, by Lemma 3.1 and the conservation of mass, we have for any t ∈ [0,+∞), |MϕR [ψ(t)]| ≤ CC(ϕR) ( ‖|∇|1/2ψ(t)‖2L2 + ‖ψ(t)‖L2‖|∇|1/2ψ(t)‖L2 ) ≤ CC(ϕR) ( ‖|∇|1/2ψ(t)‖2L2 + ‖ψ(t)‖2L2 ) ≤ CC(ϕR) ( ‖|∇|1/2ψ(t)‖2L2 + 1 ) . By interpolating between L2 and Ḣs, we obtain for any t ∈ [t0, T0] δt ≤ −MϕR [ψ(t)] = |MϕR [ψ(t)]| ≤ CC(ϕR) ( ‖(−∆)s/2ψ(t)‖ 1 s L2 + 1 ) . This implies that ‖(−∆)s/2ψ(t)‖L2 ≥ Cts (4.12) for all t ∈ [t1, T0] with some sufficiently large t1 ∈ [t0, T0]. Taking t close to T0 = ηR CC2 1 , we see that ‖(−∆)s/2ψ(t)‖L2 →∞ as R→∞. Taking R > 1 sufficiently large, we have a contradiction with (4.1). The proof is complete. � 5. Strong instability of standing waves In this section, we prove Theorem 1.2. Let us start with the following charac- terization 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 d(ω) = inf{Sω(v) : v ∈ Hs\{0},Kω(v) = 0}. (5.1) To solve this minimization problem, we consider the minimization problem d̃(ω) = inf{S̃ω(v) : v ∈ Hs\{0},Kω(v) ≤ 0}, (5.2) where S̃ω(v) := Sω(v)− Kω(v) 4s+ 2r − 3 = ωs 4s+ 2r − 3 ‖v‖2L2 + p(s+ r)− 2s (p+ 2)(4s+ 2r − 3) ‖v‖p+2 Lp+2 . (5.3) If Kω(v) < 0, then Kω(λv) = 4s+ 2r − 3 2 λ2‖v‖2 Ḣs + 4s+ 2r − 3 4 λ4 ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx + ω(2s+ 2r − 3) 2 λ2‖v‖2L2 − (s+ r)(p+ 2)− 3 p+ 2 λp+2‖v‖p+2 Lp+2 > 0, for sufficiently small λ > 0. Thus, there exists λ0 ∈ (0, 1) such that Kω(λ0v) = 0. It follows that S̃ω(λ0v) = ωs 4s+ 2r − 3 λ2 0‖v‖2L2 + p(s+ r)− 2s (p+ 2)(4s+ 2r − 3) λp+2 0 ‖v‖p+2 Lp+2 < S̃ω(v), EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 15 This implies that d̃(ω) = inf{S̃ω(v) : v ∈ Hs\{0},Kω(v) = 0}. (5.4) In following lemma, we will solve the minimizing problem (5.2). Lemma 5.2. Let ω > 0, 2s + 2r > 3 and 4s 3 ≤ p < 4s 3−2s . Then there exists u ∈ Hs\{0}, such that Kω(u) = 0 and S̃ω(u) = d̃(ω). Proof. We first show that d̃(ω) > 0. From Kω(v) ≤ 0, we have 4s+ 2r − 3 2 ‖v‖2 Ḣs + ω(2s+ 2r − 3) 2 ‖v‖2L2 + 4s+ 2r − 3 4 ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx ≤ (s+ r)(p+ 2)− 3 p+ 2 ‖v‖p+2 Lp+2 , which implies 1 2 Hω(v) ≤ (s+ r)(p+ 2)− 3 (p+ 2)(2s+ 2r − 3) Hω(v) p 2 +1, where Hω(v) = ‖v‖2 Ḣs + ω‖v‖2L2 . Thus, there exists C0 > 0 such that Hω(v) > C0 for all Kω(v) ≤ 0. This implies that there exists C1 > 0 such that S̃ω(v) ≥ p(s+ r)− 2s 2(4s+ 2r − 3) ‖v‖p+2 Lp+2 ≥ p(s+ r)− 2s (p+ 2)(4s+ 2r − 3) (2s+ 2r − 3) (s+ r)(p+ 2)− 3 Hω(v) ≥ C1. Taking the infimum over v, we obtain d̃(ω) > 0. We now show that the minimizing problem (5.2) attains its minimum. Let {vn} be a minimizing sequence for (5.2), i.e., {vn} ⊆ Hs\{0}, Kω(vn) ≤ 0 and S̃ω(vn)→ d̃(ω) as n→∞. Thus, there exists C > 0 such that ‖vn‖2L2 + ‖vn‖p+2 Lp+2 ≤ C. (5.5) This, together with Kω(vn) ≤ 0 implies that {vn} is bounded in Hs. It follows from d̃(ω) > 0 that lim infn→∞ ‖vn‖p+2 Lp+2 > 0. Therefore, applying Lemma 2.4, there exists a subsequence, still denoted by {vn} and u ∈ Hs\{0} such that un := τxn vn ⇀ u 6= 0 weakly in Hs for some {xn} ⊆ R3. We deduce from Brezis-Lieb’s lemma (Lemma 2.2) and Lemma 2.3 that Kω(un)−Kω(un − u)−Kω(u)→ 0, (5.6) S̃ω(un)− S̃ω(un − u)− S̃ω(u)→ 0. (5.7) Now, we claim that Kω(u) ≤ 0. If not, it follows from (5.6) and Kω(un) ≤ 0 that Kω(un − u) ≤ 0 for sufficiently large n. Thus, by the definition of d̃(ω), it follows that S̃ω(un − u) ≥ d̃(ω), which, together with S̃ω(un)→ d̃(ω), implies that S̃ω(u) ≤ 0, which is a contradic- tion with S̃ω(u) > 0. We thus obtain Kω(u) ≤ 0. 16 Y. MO, M. ZHU, B. FENG EJDE-2023/24 Furthermore, we deduce from the definition of d̃(ω) and the weak lower semi- continuity of norm that d̃(ω) ≤ S̃ω(u) ≤ lim inf n→∞ S̃ω(un) = d̃(ω). This yields S̃ω(u) = d̃(ω). Finally, we show that Kω(u) = 0. Suppose that Kω(u) < 0 and set Kω(uλ) = 4s+ 2r − 3 2 λ2s‖u‖2 Ḣs + ω(2s+ 2r − 3) 2 ‖u‖2L2 + 4s+ 2r − 3 4 λ3−2r ∫ R3 (|x|−(3−2r) ∗ |u|2)|u|2dx − (s+ r)(p+ 2)− 3 p+ 2 λ 3p 2 ‖u‖p+2 Lp+2 > 0 for sufficiently small λ > 0. Then there exists λ0 ∈ (0, 1) such that Kω(uλ0) = 0. It follows that S̃ω(uλ0) = ωs 4s+ 2r − 3 ‖u‖2L2 + p(s+ r)− 2s (p+ 2)(4s+ 2r − 3) λ 3p 2 0 ‖u‖ p+2 Lp+2 < S̃ω(u) = d̃(ω), which contradicts the definition of d̃(ω). Hence, we have Kω(u) = 0. � From d(ω) = d̃(ω) and the above lemma, we can obtain the existence of mini- mization problem (5.1). Lemma 5.3. Let ω > 0, 2s + 2r > 3 and 4s 3 ≤ p < 4s 3−2s . Then there exists u ∈ Hs\{0} such that Kω(u) = 0 and Sω(u) = d(ω). Lemma 5.4. Let ω > 0, 2s + 2r > 3, and 4s 3 ≤ p < 4s 3−2s . Assume that u ∈ Hs\{0} is a solution of the minimizing problem (5.1), i.e., such that Kω(u) = 0 and Sω(u) = d(ω). Then S′ω(u) = 0. Proof. We firstly prove K ′ω(u) 6= 0. If K ′ω(u) = 0, then we have (4s+ 2r − 3)(−∆)su+ ω(2s+ 2r − 3)u+ (4s+ 2r − 3)(|x|−(3−2r) ∗ |u|2)u − ((s+ r)(p+ 2)− 3)|u|pu = 0. (5.8) Then A+B + C −D = d(ω), (4s+ 2r − 3)A+ (2s+ 2r − 3)B + (4s+ 2r − 3)C − ((s+ r)(p+ 2)− 3)D = 0, 2(4s+ 2r − 3)A+ 2(2s+ 2r − 3)B + 4(4s+ 2r − 3)C − (p+ 2)((s+ r)(p+ 2)− 3)D = 0, (3− 2s)(4s+ 2r − 3)A+ 3(2s+ 2r − 3)B + (4s+ 2r − 3)(3 + 2r)C − 3((s+ r)(p+ 2)− 3)D = 0, (5.9) where A = 1 2 ‖u‖2 Ḣs , B = ω 2 ‖u‖2L2 , C = 1 4 ∫ R3 (|x|−(3−2r) ∗ |u|2)(x)|u(x)|2dx, D = 1 p+ 2 ‖u‖p+2 Lp+2 . EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 17 The first equation comes from the fact that Sω(u) = d(ω). The second one holds since Kω(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 sA = tC, C = p((s+ r)(p+ 2)− 3)D 2(4s+ 2r − 3) , (2s+ 2r − 3)B + (p(s+ r)− 2s)((s+ r)(p+ 2)− 3) 2s D = 0. These A = B = C = D = 0 which is a contradiction with A,B,C,D > 0. Thus, K ′ω(u) 6= 0. Next, applying the Lagrange multiplier rule, there exists µ ∈ R such that S′ω(u)+ µK ′ω(u) = 0. We claim that µ = 0. As above, the equation S′ω(u) + µK ′ω(u) = 0 can be written as (−∆)su+ ωu+ (|x|−(3−2r) ∗ |u|2)u− |u|pu + µ [ (4s+ 2r − 3)(|x|−(3−2r) ∗ |u|2)u+ ω(2s+ 2r − 3)u + (4s+ 2r − 3)(−∆)su− ((s+ r)(p+ 2)− 3)|u|pu ] = 0. (5.10) By the same argument as in (5.9), we have A+B + C −D = d(ω), (4s+ 2r − 3)A+ (2s+ 2r − 3)B + (4s+ 2r − 3)C − ((s+ r)(p+ 2)− 3)D = 0, 2(µ(4s+ 2r − 3) + 1)A+ 2(µ(2s+ 2r − 3) + 1)B +4(µ(4s+ 2r − 3) + 1)C − (p+ 2)(µ((s+ r)(p+ 2)− 3) + 1)D = 0, (3− 2s)(µ(4s+ 2r − 3) + 1)A+ 3(µ(2s+ 2r − 3) + 1)B + (µ(4s+ 2r − 3) + 1)(3 + 2r)C − 3(µ((s+ r)(p+ 2)− 3) + 1)D = 0. We now deal with the above system. Consider A,B,C,D as unknown quantities, and denote the coefficient matrix by M . Computing its determinant, we have detM = −4sµp(s+ r)(1 + µ(4s+ 2r − 3))((p− 2)s+ pr). Note that detM = 0⇐⇒ µ = 0, p = 0, µ = − 1 4s+ 2r − 3 , (p− 2)s+ pr = 0. 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 6= 0, µ 6= − 1 4s+2r−3 , then detM 6= 0, and hence the linear system has a unique solution (depending on the parameters µ, p, d(ω)). Applying Cramer’s rule, we obtain D = −d(ω)(4s+ 2r − 3)(2s+ 2r − 3) p(s+ r)((p− 2)s+ pr) < 0, which contradicts D > 0. (2) If µ = − 1 4s+2r−3 , then the third equation reads 4sB + (p+ 2)(s(p− 2) + pr)D = 0, which contradicts B,D > 0. Thus, µ = 0 and S′ω(u) = 0. � 18 Y. MO, M. ZHU, B. FENG EJDE-2023/24 We now denote the set of all minimizers of (5.1) by Mω = {u ∈ Hs\{0} : Sω(u) = d(ω), Kω(u) = 0}. Lemma 5.5. Mω ⊆ Gω. Proof. Let u ∈ Mω. It follows from Lemma 5.4 that S′ω(u) = 0. In particular, we have u ∈ Aω. To prove u ∈ Gω, it remains to show that Sω(u) ≤ Sω(v) for all v ∈ Aω. To see this, we notice that Kω(v) = (s+ r)〈S′ω(v), v〉 − Iω(v) = 0 for all v ∈ Aω, where Iω(v) is defined by (2.3). By definition of d(ω), we have Sω(u) ≤ Sω(v). Thus, u ∈ Gω. � Lemma 5.6. Gω ⊂Mω. Proof. Let u ∈ Gω. Since Mω is not empty, we take v ∈ Mω. By Lemma 5.5, v ∈ Gω. In particular, Sω(u) = Sω(v). Since v ∈Mω, we obtain Sω(u) = Sω(v) = d(ω). It remains to show that Kω(u) = 0. Since u ∈ Aω, we have S′ω(u) = 0 and Iω(u) = 0, hence Kω(u) = (s+ r)〈S′ω(u), u〉 − Iω(u) = 0. Therefore, u ∈Mω. � Proof of Proposition 5.1. It follows immediately from Lemmas 5.3, 5.5, and 5.6. � When 4s 3 ≤ p < 4s 3−2s , to study the strong instability of standing waves for (1.4), we need to establish the following characterization of the ground state related to (1.12). Lemma 5.7. Let ω > 0, 2s + 2r > 3, 4s 3 ≤ p < 4s 3−2s , and u be the ground state related to (1.12). Then Sω(u) = inf{Sω(v) : v ∈ Hs\{0}, Q(v) = 0}. (5.11) Proof. Firstly, we claim that the minimizing problem in (5.11) is well-defined. Let v ∈ Hs\{0} and Q(v) = 0. If p = 4s 3 , then Sω(v) = Sω(v)− 1 2s Q(v) = ω 2 ‖v‖2L2 + 2s+ 2r − 3 8s ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx > 0. (5.12) And if 4s 3 < p < 4s 3−2s , then Sω(v) = Sω(v)− 2 3p Q(v) = 3p− 4s 6p ‖v‖2 Ḣs + ω 2 ‖v‖2L2 + 3p+ 4t− 6 12p ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx > 0. (5.13) Thus we denote d := inf{Sω(v) : v ∈ Hs\{0}, Q(v) = 0}. Firstly, we deduce from (2.2) and (2.3) that Kω(u) = Q(u) = 0. By the definition of d, we have Sω(u) ≥ d. EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 19 Let v ∈ Hs\{0} be such that Q(v) = 0. If Kω(v) = 0, then it follows from Proposition 5.1 that Sω(v) ≥ Sω(u). If Kω(v) 6= 0, we notice that Kω(vλ) = 4s+ 2r − 3 2 λ2s‖v‖2 Ḣs + ω(2s+ 2r − 3) 2 ‖v‖2L2 + 4s+ 2r − 3 4 λ3−2r ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx − (s+ r)(p+ 2)− 3 p+ 2 λ 3p 2 ‖v‖p+2 Lp+2 , where vλ(x) := λ3/2v(λx). When 4s 3 < p < 4s 3−2s , we have lim λ→0 Kω(vλ) = ω(2s+ 2r − 3) 2 ‖v‖2L2 > 0, and lim λ→∞ Kω(vλ) < 0. (5.14) When p = 4s/3, it follows from Q(v) = 0 that s‖v‖2 Ḣs < 3p 2(p+ 2) ‖v‖p+2 Lp+2 , which implies that (5.14) holds. Thus, there exists λ0 > 0 such that Kω(vλ0) = 0. This implies that Sω(vλ0) ≥ Sω(u). On the other hand, by some basic calculations, we have ∂λSω(vλ) = sλ2s−1‖v‖2 Ḣs + 3− 2r 4 λ2−2r ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx − λ 3p 2 −1 p+ 2 3p 2 ‖v‖p+2 Lp+2 = Q(vλ) λ . Next, we define f(λ) := Q(vλ) = sλ2s‖v‖2 Ḣs + 3− 2r 4 λ3−2r ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx− λ 3p 2 p+ 2 3p 2 ‖v‖p+2 Lp+2 . When p = 4s 3 , it follows from Q(v) = 0 that s‖v‖2 Ḣs < 3p 2(p+2)‖v‖ p+2 Lp+2 . Thus, it is easy to see that the equation f(λ) = 0 admits a unique positive solution λ = 1. When 4s 3 < p < 4s 3−2s , assume that there exists λ1 6= 1 such that f(λ1) = 0. It easily follows that 3− 2r 4 ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx(λ2s 1 − λ3−2r 1 ) = ‖v‖p+2 Lp+2 p+ 2 3p 2 (λ2s 1 − λ 3p 2 1 ). If λ1 > 1, then λ2s 1 − λ3−2r 1 > 0 and λ2s 1 − λ 3p 2 1 < 0, which is a contradiction. If λ1 < 1, then λ2s 1 −λ3−2r 1 < 0 and λ2s 1 −λ 3p 2 1 > 0, which is a contradiction. Therefore, the equation f(λ) = 0 admits a unique positive solution λ = 1. Therefore, ∂λSω(vλ) > 0, for all λ ∈ (0, 1), ∂λSω(vλ) < 0, for all λ ∈ (1,∞). 20 Y. MO, M. ZHU, B. FENG EJDE-2023/24 We thus obtain that Sω(vλ) < Sω(v) for any λ > 0 and λ 6= 1. In particular, we have Sω(vλ0) ≤ Sω(v). Thus, Sω(u) ≤ Sω(vλ0) ≤ Sω(v) for all v ∈ Hs\{0} and Q(v) = 0. Taking the infimum over v, we have Sω(u) ≤ d. This completes the proof. � To obtain the key estimate (5.19), we need establish the following variational characterization of the ground states to (1.12). Firstly, when p = 4s 3 , we define S1 ω(v) := Sω(v)− 1 2s Q(v) = ω 2 ‖v‖2L2 + 2s+ 2r − 3 8s ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx. (5.15) When 4s 3 < p < 4s 3−2s , we define S2 ω(v) := Sω(v)− 2 3p Q(v) = 3p− 4s 6p ‖v‖2 Ḣs + ω 2 ‖v‖2L2 + 3p+ 4t− 6 12p ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx. (5.16) Lemma 5.8. Let ω > 0, 2s + 2r > 3, 4s 3 ≤ p < 4s 3−2s , and u be the ground state related to (1.12). Then for k = 1, 2 we have Sω(u) = Skω(u) = inf{Skω(v) : v ∈ Hs\{0}, Q(v) ≤ 0}. (5.17) Proof. We only prove the case k = 1. The proof of the case k = 2 is similar. We denote d1(ω) = inf{Skω(v) : v ∈ Hs\{0}, Q(v) ≤ 0}. Since u is the ground state related to (1.12), Q(u) = 0. It follows from the definition of d1(ω) that S1 ω(u) ≥ d1(ω). (5.18) Let v ∈ Hs\{0} and Q(v) ≤ 0. If Q(v) = 0, then from Lemma 5.7 it follows that S1 ω(v) = Sω(v)− 1 2s Q(v) = Sω(v) ≥ Sω(u) = S1 ω(u). If Q(v) < 0, we note that Q(vλ) = λ2s‖v‖2 Ḣs + λ3−2r 4 ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx− λ 3p 2 p+ 2 3p 2 ‖v‖p+2 Lp+2 > 0 for sufficiently small λ > 0, so there exists λ0 ∈ (0, 1) such that Q(vλ0) = 0. We thus have S1 ω(v) = ω 2 ‖v‖2L2 + 2s+ 2r − 3 8s ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx ≥ ω 2 ‖v‖2L2 + 2s+ 2r − 3 8s λ3−2r 0 ∫ R3 (|x|−(3−2r) ∗ |v|2)|v|2dx = S1 ω(vλ0) = Sω(vλ0) ≥ Sω(u) = S1 ω(u). This implies that d1(ω) ≥ S1 ω(u). This, together with (5.18) implies that S1 ω(u) = d1(ω). � EJDE-2023/24 BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES 21 Let u be the ground state related to (1.12). We define Bω = {v ∈ Hs \ {0} : Sω(v) < Sω(u), Q(v) < 0}. Lemma 5.9. Let ω > 0, 2s+ 2r > 3, and u be the ground state related to (1.12). If 4s 3 ≤ p < 4s 3−2s , then the set Bω is invariant under the flow of (1.4). That is, if ψ0 ∈ Bω, then the solution ψ(t) to (1.4) with initial data ψ0 belongs to Bω and Q(ψ(t)) ≤ 2s(S(ψ0)− S(u)) (5.19) for any t ∈ [0, T ∗). Proof. Let ψ0 ∈ Bω, by Proposition 2.1, we see that there exists a unique solution ψ ∈ C([0, T ∗), Hs) with initial data ψ0. We deduce from the conservations of mass and energy that Sω(ψ(t)) = Sω(ψ0) < Sω(u) (5.20) for any t ∈ [0, T ∗). In addition, by the continuity of the function t 7→ Q(ψ(t)) and Lemma 5.7, if there exists t0 ∈ [0, T ∗) such that Q(ψ(t0)) = 0, then Sω(ψ(t0)) ≥ Sω(u), which contradicts (5.20). Therefore, we have Q(ψ(t)) < 0 for any t ∈ [0, T ∗). This, together with Lemma 5.8 implies that Sω(u) ≤ S1 ω(ψ(t)) = Sω(ψ(t))− 1 2s Q(ψ(t)) = Sω(ψ0)− Q(ψ(t)) 2s , Sω(u) ≤ S2 ω(ψ(t)) = Sω(ψ(t))− 2 3p Q(ψ(t)) < Sω(ψ0)− Q(ψ(t)) 2s for all t ∈ [0, T ∗). This completes the proof. � Proof of Theorem 1.2. Let u be the ground state related to (1.12) and {λn} ⊆ R+ be such that λn > 1 and limn→∞ λn = 1. We take the initial data ψ0,n(x) := λ3/2 n u(λnx). Therefore, lim n→∞ ‖ψ0,n‖L2 = lim n→∞ ‖u‖L2 = ‖u‖L2 , lim n→∞ ‖ψ0,n‖Ḣs = lim n→∞ λsn‖u‖Ḣs = ‖u‖Ḣs . Thus, we deduce from Brezis-Lieb’s lemma (Lemma 2.2) that ψ0,n → u in Hs as n→∞. By Lemma 5.7, we have Sω(ψ0,n) < Sω(u), Q(ψ0,n) < 0 for all n ≥ 1. Thus, ψ0,n ∈ Bω. Let ψn be the maximal solution of (1.4) with the initial data ψ0,n. We deduce from Lemma 5.9 that ψn(t) ∈ Bω for all t ∈ [0, T ∗) and Q(ψn(t)) ≤ 2s(S(ψ0,n)− S(u)) < 0. Thus, applying Theorem 1.1, we obtain that the solution ψn(t) of (1.4) with initial data ψ0,n blows up in finite or infinite time for any n ≥ 1. � Acknowledgments. This research was supported by the Outstanding Youth Sci- ence Fund of Gansu Province (No. 20JR10R A111), by the Department of Educa- tion of Gansu Province: Youth Doctoral Fund Project (No. 2022QB-031), and by NWNU-LKZD2022-03 and NWNU-LKQN2019-7. 22 Y. MO, M. ZHU, B. FENG EJDE-2023/24 References [1] J. Bellazzini, G. 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Yichun Mo Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Email address: sg25888@163.com Min Zhu (corresponding author) Department of Mathematics, Nanjing Forestry University, Nanjing, Jiangsu 210037, China Email address: zhumin@njfu.edu.cn Binhua Feng (corresponding author) Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Email address: binhuaf@nwnu.edu.cn 1. Introduction 2. Preliminary lemmas 3. Localized virial estimates 4. Blow-up criteria for (??) 5. Strong instability of standing waves Acknowledgments References