Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 39, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES FOR THE INHOMOGENEOUS FRACTIONAL SCHRÖDINGER EQUATION BINHUA FENG, ZHIQIAN HE, JIAYIN LIU Abstract. In this article, we study the blow-up and instability of standing waves for the inhomogeneous fractional Schrödinger equation i∂tu− (−∆)su + |x|−b|u|pu = 0, where s ∈ ( 1 2 , 1), 0 < b < min{2s,N} and 0 < p < 4s−2b N−2s . In the L2-critical and L2-supercritical cases, i.e., 4s−2b N ≤ p < 4s−2b N−2s , we establish general blow- up criteria for non-radial solutions by using localized virial estimates. Based on these blow-up criteria, we prove the strong instability of standing waves. 1. Introduction Over the past decade, there has been a great deal of interest in studying the fractional Schrödinger equation i∂tu = (−∆)su+ f(u), (1.1) where 0 < s < 1 and f(u) is the nonlinearity. The fractional differential oper- ator (−∆)s is defined by (−∆)su = F−1[|ξ|2sF(u)], where F and F−1 are the Fourier transform and inverse Fourier transform, respectively. Equation (1.1) was first deduced by Laskin in [24, 25] by extending the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths. The fractional Schrödinger equation also arises in the description of Bonson stars as well as in water wave dynamics (see [16]) and in the continuum limit of discrete models with long-range interactions (see [23]). In this article, we consider the blow-up criteria and instability of standing waves for the inhomogeneous fractional Schrödinger equation i∂tu− (−∆)su+ |x|−b|u|pu = 0, (t, x) ∈ [0, T ∗)× RN , u(0, x) = u0(x), (1.2) where u : [0, T ∗) × RN → C is the complex valued function, N ≥ 1, u0 ∈ Hs, 0 < s < 1, 0 < b < min{2s,N}, 0 < p < 4s−2b N−2s . 2010 Mathematics Subject Classification. 35B35, 35B40, 35K57, 35Q92, 92C17. Key words and phrases. Inhomogeneous fractional Schrödinger equation; blow-up criteria; strong instability. c©2021 Texas State University. Submitted November 28, 2020. Published May 7, 2021. 1 2 B. FENG, Z. HE, J. LIU EJDE-2021/39 This equation enjoys the scaling invariance. That is, if u(t, x) is a solution of (1.2), then uλ(t, x) = λ 2s−b p u(λ2st, λx) for all λ > 0, is also a solution of (1.2). By simple calculations, we have ‖uλ(t)‖Ḣs = λs+ 2s−b p −N 2 ‖u(λ2st)‖Ḣs . Thus, the critical Sobolev index is given by sc := N 2 − 2s− b p . (1.3) When sc < 0, equation (1.2) is L2-subcritical. The smallest power for which blow- up may occur is p = 4s−2b N , which is referred to L2-critical case corresponding to sc = 0. When 0 < sc < s, (1.2) is L2-supercritical and Hs-subcritical. When sc = s, (1.2) is Hs-critical. In this paper, we are interested in the L2-critical and L2-supercritical cases. Therefore, we restrict our attention to the case 0 ≤ sc < s. Rewriting this condition in terms of p, we obtain 4s− 2b N ≤ p < 4s− 2b N − 2s . If one considers initial data in Hs, then the equation enjoys mass and energy conservation laws: M(u(t)) := ‖u(t)‖L2 = ‖u0‖L2 , (1.4) and E(u(t)) := 1 2 ∫ RN |(−∆)s/2u(t, x)|2dx− 1 p+ 2 ∫ RN |x|−b|u(t, x)|p+2dx =E(u0). (1.5) Before entering some details of our results, let us recall known blow-up results . For the classical Shrödinger equation, i.e., s = 1, the Variance-Virial Law holds, that is 1 2 d dt ∫ RN |x|2|u(t, x)|2dx = 2 Im ∫ RN ū(t, x)x · ∇u(t, x)dx, (1.6) provided initial data u0 ∈ Σ := {u0 ∈ H1 and xu0 ∈ L2}. Combining (1.6) and the virial identity, one can obtain blow-up results for the classical Schrödinger equation with negative energy E(u0) < 0 and finite variance, see [2]. Ogawa and Tsutsumi [26] removed the assumption u0 ∈ Σ for the radial symmetry initial data. Applying similar ideas, when s = 1, Farah [10] and Dinh [7] established the blow-up criteria for equation (1.2) with initial data u0 ∈ Σ := {v ∈ H1 and xv ∈ L2} and radial symmetry initial data. However, when s < 1, identity (1.6) fails and these arguments cannot work. However, a generalization of the variance for the fractional Schrödinger equation is given by V(s)[u(t)] := ∫ RN ū(t, x)x · (−∆)1−sxu(t, x)dx = ‖x(−∆) 1−s 2 u(t)‖2L2 . (1.7) Let u(t) be the solution of equation i∂tu = (−∆)su, a formal calculation yields 1 2 d dt V(s)[u(t)] := 2 Im ∫ RN ū(t, x)x · ∇u(t, x)dx. (1.8) Based on this identity, the authors in [3, 4, 33] successfully obtained blow-up results for (1.1) with radial initial data and Hartree-type nonlinearity, i.e., f(u) = −(|x|−γ∗ EJDE-2021/39 BLOW-UP CRITERIA FOR SCHRÖDINGER EQUATIONS 3 |u|2)u with γ ≥ 1. Because it is very hard to control the nontrivial error terms, this method fails to work for the local nonlinearities f(u) = −|u|pu, see [1]. Using the Balakrishman’s formula (−∆)s = sinπs π ∫ ∞ 0 ms−1 −∆ −∆ +m dm, (1.9) Boulenger, Himmelsbach and Lenzmann [1] established the differential estimate d dt ( Im ∫ RN ū(t)∇ϕR · ∇u(t)dx ) ≤ 4pNE(u0)− 2δ‖(−∆)s/2u(t)‖2L2 + ◦R(1)(1 + ‖(−∆)s/2u(t)‖p/s+L2 ), where δ = pN − 2s. Based on this key estimate and a standard comparison ODE argument, they proved the existence of radial blow-up Hs solutions. For the inhomogeneous fractional Schrödinger equation (1.2), Peng and Zhao in [27] obtained the existence of radial blow-up solutions. In this paper, by using localized virial estimates and the ideas of Du, Wu and Zhang [9], we remove this assumption, and establish general blow-up criteria for non-radial solutions in the L2-critical and L2-supercritical cases. The main difficulty is the appearance of the fractional order Laplacian (−∆)s and the singular potential |x|−b. When s = 1, it easily follows that the time derivative of the virial action 1 2 d dt ∫ RN ϕ(x)|u(t, x)|2dx = 2 Im ∫ RN ū(t, x)∇ϕ(x) · ∇u(t, x)dx. (1.10) By applying this identity, Du, Wu and Zhang [9] established an L2-estimate in the exterior ball. Based on this L2-estimate and the virial estimates, they established blow-up criteria for the classical Schrödinger equation. In the case s ∈ ( 1 2 , 1), the identity (1.10) does not hold. However, by using the Balakrishman’s formula (1.9) and exploiting the ideas in [1], we can obtain the time derivative of the virial action. We consequently obtain the following general blow-up criteria for non-radial solutions in both L2-critical and L2-supercritical cases. Theorem 1.1. Let N ≥ 1, s ∈ ( 1 2 , 1), 4s−2b N ≤ p < 4s−2b N−2s , and u0 ∈ Hs. Assume that u ∈ C([0, T ∗), Hs) is a solution of (1.2). Furthermore, we assume that either E(u0) < 0, or, if E(u0) ≥ 0 and E(u0)sc‖u0‖2(s−sc)L2 < E(Q)sc‖Q‖2(s−sc)L2 , ‖(−∆)s/2u0‖scL2‖u0‖s−scL2 > ‖(−∆)s/2Q‖scL2‖Q‖s−scL2 , . (1.11) where sc is defined by (1.3) and Q is the ground state of the elliptic equation (−∆)sQ+Q− |x|−b|Q|pQ = 0. (1.12) Then one of the following statements holds: • u(t) blows up in finite time, i.e. T ∗ < +∞; • u(t) blows up infinite time, i.e., there exists (tn)n≥1 such that tn → +∞ and lim n→∞ ‖(−∆)s/2u(tn)‖L2 =∞. Our blow-up criteria also hold for (1.2) with s = 1, which to our knowledge is new. When s = 1, similar blow-up criteria for (1.2) with radial solutions or initial data u0 ∈ Σ := {v ∈ H1 and xv ∈ L2} have been established in [7, 10]. Here, we remove the assumption of radial solutions and u0 ∈ Σ := {v ∈ H1 and xv ∈ L2}. 4 B. FENG, Z. HE, J. LIU EJDE-2021/39 So our results improve some previous results. Based on blow-up criterion (1.11), we can prove the strong instability of standing waves of (1.2). Firstly, we introduce some notation. Throughout this paper, we call a standing wave solution of (1.2) of the form eiωtQω, where ω ∈ R is a frequency and Qω ∈ Hs is a nontrivial solution to the elliptic equation (−∆)sQω + ωQω − |x|−b|Qω|pQω = 0. (1.13) LetQω(x) = ω 2s−b 2sp Q(ω 1 2sx) in (1.13), thenQ satisfies equation (1.12). In particular, by some basic calculations, we have E(Qω)sc‖Qω‖2(s−sc)L2 = E(Q)sc‖Q‖2(s−sc)L2 , (1.14) ‖(−∆)s/2Qω‖scL2‖Qω‖s−scL2 = ‖(−∆)s/2Q‖scL2‖Q‖s−scL2 . (1.15) In fact, these two quantities are scaling invariant of (1.2). Definition 1.2. A function Q ∈ Hs\{0} is called a ground state for (1.12) if it is a minimizer of the Weinstein’s functional J(v) := ‖v‖ Np+2b 2s Ḣs ‖v‖p+2−Np+2b 2s L2∫ RN |x|−b|v(x)|p+2dx , (1.16) that is, J(Q) = inf{J(v) : v ∈ Hs\{0}}. (1.17) The existence of ground states related to (1.12) has been established in Lemma 2.2. In addition, a direct computation shows that ‖Qω‖L2 = ω 4s−2b−Np 4sp ‖Q‖L2 , ‖Qω‖Ḣs = ω 2sp−Np+4s−2b 4sp ‖Q‖Ḣs ,∫ RN |x|−b|Qω(x)|p+2dx = ω 2sp−Np+4s−2b 2sp ∫ RN |x|−b|Q(x)|p+2dx. These imply that J(Qω) = J(Q). That is, Qω is also a minimizer of the Weinstein’s functional. Thus, we can define the ground states related to (1.13) as follows: A function Qω ∈ Hs\{0} is called a ground state solution of (1.13) if it is a minimizer of the Weinstein’s functional (1.16). We can derive ground states of (1.13) from ground states related to (1.12). This implies the existence of ground states related to (1.13) when ω > 0. In addition, the uniqueness of ground states related to (1.12) is an open problem. Note also that (1.13) can be written as S′ω(Qω) = 0, where Sω(Q) :=E(Q) + ω 2 ‖Q‖2L2 = 1 2 ‖Q‖2 Ḣs + ω 2 ‖Q‖2L2 − 1 p+ 2 ∫ RN |x|−b|Q(x)|p+2dx, (1.18) is the action functional. We also define the following functional K(Q) := ∂λSω(Qλ)|λ=1 = s‖Q‖2 Ḣs − Np+ 2b 2p+ 4 ∫ RN |x|−b|Q(x)|p+2dx, (1.19) where Qλ(x) := λN/2Q(λx). (1.20) To the best of our knowledge, the general method to investigate the strong in- stability of standing waves for the classical Schrödinger equation is to apply the EJDE-2021/39 BLOW-UP CRITERIA FOR SCHRÖDINGER EQUATIONS 5 variational characterization of the ground states as minimizers of the action func- tional and derive the key estimate K(u(t)) ≤ 2(Sω(u0)−Sω(Qω)). Then, it follows from the virial identity that d2 dt2 ‖xu(t)‖2L2 = 8K(u(t)) ≤ 16(Sω(u0)− Sω(Qω)), where K(u(t)) is defined by (1.19) with s = 1. Finally, one can choose the initial data u0 such that Sω(u0)−Sω(Qω) < 0. This implies that the solution u(t) of (1.1) with s = 1 blows up in finite time. Thus, one can prove the strong instability of ground state standing waves, see [2, 6, 12, 13, 14, 15, 22, 29, 30]. Here, we present a simpler method to study the strong instability of standing waves, which is based on the blow-up criterion (1.11). Theorem 1.3. Let N ≥ 1, s ∈ ( 1 2 , 1), 4s−2b N ≤ p < 4s−2b N−2s , ω > 0, Qω be the ground state related to (1.13). Then, the standing wave u(t, x) = eiωtQω(x) is strongly unstable in the following sense: there exists {u0,n} ⊂ Hs such that u0,n → Qω in Hs as n → ∞ and the corresponding solution un of (1.2) with initial data u0,n blows up in finite or infinite time for any n ≥ 1. In previous results, to construct blow-up solutions around the ground state solution, one needs to assume that the ground state solution Qω is radial or Qω ∈ Σ := {v ∈ H1 and xv ∈ L2}. Here, we remove these assumptions, so our result greatly improve some previous results. This article is organized as follows: in Section 2, we recall and prove some lemmas such as the local well-posedness theory of (1.2), Brezis-Lieb’s lemma, the sharp Gagliardo-Nirenberg type inequality (2.1) and the localized virial estimate related to (1.2). In section 3, we establish blow-up criteria for (1.2). In section 4, we prove the strong instability of standing waves. Throughout this article, we use the following notation. C > 0 stands for a constant that may be different from line to line when it does not cause any confusion. For any s ∈ (0, 1), the fractional Sobolev space Hs(RN ) is defined by Hs(RN ) = { u ∈ L2(RN ) : ∫ RN (1 + |ξ|2s)|û(ξ)|2dξ <∞ } , endowed with the norm ‖u‖Hs(RN ) = ‖u‖L2(RN ) + ‖u‖Ḣs(RN ), where up to a multiplicative constant, ‖u‖Ḣs(RN ) = {∫∫ RN×RN |u(x)− u(y)|2 |x− y|N+2α dx dy }1/2 is the so-called Gagliardo semi-norm of u. In this paper, we often use the abbrevi- ations Lr = Lr(RN ), Hs = Hs(RN ). 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 (1.2). By applying Strichartz’s estimates and the contraction mapping argument, Hong and Sire in [21] first studied the local well-posedness for the fractional Schrödinger equation in Hs. Because Strichartz’s estimates have a loss of derivatives in the non-radial symmetry case, a weak local well-posedness follows in the energy space compared to the classical 6 B. FENG, Z. HE, J. LIU EJDE-2021/39 Schrödinger equation, see [5, 21] for more details. In the radial symmetry case, one can remove the loss of derivatives in Strichartz’s estimates. But it needs a restriction on the validity of s, namely N 2N−1 ≤ s < 1. For the inhomogeneous Schrödinger equation (1.2) with s = 1, Genoud and Stuart [17] first studied the well-posedness by using the argument of Cazenave [2]. By using Strichartz’s estimates and the contraction mapping argument, Guz- man [19] also established the local well-posedness as well as the small data global well-posedness in Sobolev spaces. By using radial Strichartz’s estimates and the contraction mapping argument, we can obtain the following local well-posedness for (1.2) with radial Hs initial data. The proof is standard, see [5, 19, 21]. So we omit it. Theorem 2.1. Let N ≥ 2, N 2N−1 ≤ s < 1, 0 < p < 4s−2b N−2s and 0 < b < min{2s,N}. If u0 ∈ Hs is radial, then there exists T = T (‖u0‖Hs) such that (1.2) admits a unique solution u ∈ C([0, T ], Hs). Let [0, T ∗) be the maximal time interval on which the solution u is well-defined, if T ∗ < ∞, then ‖u(t)‖Ḣs → ∞ as t ↑ T ∗. Moreover, for all 0 ≤ t < T ∗, the solution u(t) satisfies the conservations of mass and energy. Next, we recall the following sharp Gagliardo-Nirenberg inequality, which has been established in [27]. Lemma 2.2 ([27]). Let 0 < s < 1, 0 < p < 4s−2b N−2s and 0 < b < min{2s,N}. Then, for all u ∈ Hs, ∫ RN |x|−b|u(x)|p+2dx ≤ Copt‖u‖ Np+2b 2s Ḣs ‖u‖p+2−Np+2b 2s L2 , (2.1) where the best constant Copt is given by Copt = ( Np+ 2b 2s(p+ 2)− (Np+ 2b) ) 4s−(Np+2b) 4s 2s(p+ 2) (Np+ 2b)‖Q‖pL2 , where Q is the ground state of (1.12). Moreover, the following Pohozaev’s identities hold ‖Q‖2 Ḣs = Np+ 2b 2s(p+ 2) ∫ RN |x|−b|Q|p+2dx = Np+ 2b 2s(p+ 2)− (Np+ 2b) ‖Q‖2L2 . (2.2) Lemma 2.3 ([1]). Let N ≥ 1, ϕ : RN → R and ∇ϕ ∈ W 1,∞(RN ). Then, for all u ∈ H1/2, it follows that∣∣ ∫ RN u(x)∇ϕ(x) · ∇u(x)dx ∣∣ ≤ C‖∇ϕ‖W 1,∞ ( ‖|∇|1/2u‖2L2 + ‖u‖L2‖|∇|1/2u‖L2 ) , where C > 0 depends only on N . To study localized virial estimates for (1.2), we introduce an auxiliary function um(x) := cs 1 −∆ +m u(x) = csF−1 ( û(ξ) |ξ|2 +m ) , m > 0, (2.3) where cs := √ sin(πs)/π. Lemma 2.4 ([1]). Let N ≥ 1, s ∈ (0, 1), ϕ : RN → R and ∆ϕ ∈ W 2,∞(RN ). Then, for all u ∈ L2,∣∣ ∫ ∞ 0 ms ∫ RN (∆2ϕ)|um|2 dx dm ∣∣ ≤ C‖∆2ϕ‖sL∞‖∆ϕ‖1−sL∞ ‖u‖ 2 L2 , EJDE-2021/39 BLOW-UP CRITERIA FOR SCHRÖDINGER EQUATIONS 7 where C > 0 depends only on s and N . Applying the identity sinπs π ∫ ∞ 0 ms (|ξ|2 +m)2 dm = s|ξ|2s−2, we deduce form the Plancherel’s and Fubini’s theorems that∫ ∞ 0 ms ∫ RN |∇um|2 dx dm = ∫ RN ( sinπs π ∫ ∞ 0 msdm (|ξ|2 +m)2 ) |ξ|2|û(ξ)|2dξ = ∫ RN (s|ξ|2s−2)|ξ|2|û(ξ)|2dξ = s‖(−∆)s/2u‖2L2 , (2.4) for any u ∈ Ḣs. Lemma 2.5 ([8]). Let N ≥ 1, s ∈ (1/2, 1), ϕ : RN → R and ∇ϕ ∈ W 1,∞. Then for any u ∈ L2,∣∣ ∫ ∞ 0 ms ∫ RN (∆ϕ)|um|2 dx dm ∣∣ ≤ C‖∆ϕ‖2s−1L∞ ‖∇ϕ‖ 2−2s L∞ ‖u‖ 2 L2 , where C > 0 depends only on s and N . Lemma 2.6 ([8]). Let N ≥ 1, s ∈ (1/2, 1), ϕ : RN → R and ∇ϕ ∈ W 1,∞. Then for any u ∈ H1/2,∣∣ ∫ ∞ 0 ms ∫ RN um∇ϕ · ∇um dx dm ∣∣ ≤ C‖∇ϕ‖W 1,∞‖u‖2H1/2 , where C > 0 depends only on N . Lemma 2.7 (Virial identity). Let N ≥ 1, s ∈ (1/2, 1) and ϕ : RN → R be such that ϕ ∈W 2,∞. Assume that u ∈ C([0, T ∗), Hs) is a solution to (1.2). Then d dt Vϕ[u(t)] =− i ∫ ∞ 0 ms ∫ RN (∆ϕ)|um(t)|2 dx dm − 2i ∫ ∞ 0 ms ∫ RN um(t)∇ϕ · ∇um(t) dx dm (2.5) for any t ∈ [0, T ∗), where Vϕ[u(t)] := ∫ RN ϕ(x)|u(t, x)|2dx is the localized virial action of u associated to ϕ and um(t) = cs(−∆ +m)−1u(t). Proof. Because the general case follows by an approximation argument, we only prove (2.5) for u ∈ C∞0 (RN ). Since u(t) satisfies (1.2), it easily follows that d dt Vϕ[u(t)] = d dt 〈u(t), ϕu(t)〉 = i〈u(t), [(−∆)s, ϕ]u(t)〉, where [X,Y ] = XY −Y X is the commutator of X and Y . To study [(−∆)s, ϕ], we use the fact that for operators A ≥ 0, B and m > 0 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 , see [1]. Using this identity with A = (−∆)s and B = ϕ, by the Balakrishman’s formula we have [(−∆)s, ϕ] = sinπs π ∫ ∞ 0 ms [ −∆ −∆ +m ,ϕ ] dm 8 B. FENG, Z. HE, J. LIU EJDE-2021/39 = sinπs π ∫ ∞ 0 ms 1 −∆ +m [−∆, ϕ] 1 −∆ +m dm. Thus, 〈u(t), [(−∆)s, ϕ]u(t)〉 = 〈 u(t), ( sinπs π ∫ ∞ 0 ms 1 −∆ +m [−∆, ϕ] 1 −∆ +m dm ) u(t) 〉 = c2s ∫ ∞ 0 ms〈u(t), 1 −∆ +m [−∆, ϕ] 1 −∆ +m u(t)〉dm = ∫ ∞ 0 ms〈cs(−∆ +m)−1u(t), [−∆, ϕ]cs(−∆ +m)−1u(t)〉dm = ∫ ∞ 0 ms ∫ RN um(t) (−∆ϕum(t)− 2∇ϕ · ∇um(t)) dx dm = ∫ ∞ 0 ms ∫ RN ( (−∆ϕ)|um(t)|2 − 2um(t)∇ϕ · ∇um(t) ) dx dm. The proof is complete. � The following estimate is a direct consequence of Lemmas 2.5, 2.6 and 2.7. Corollary 2.8. Let N ≥ 1, s ∈ (1/2, 1) and ϕ : RN → R be such that ϕ ∈ W 2,∞. Assume that u ∈ C([0, T ∗), Hs) is a solution to (1.2). Then for any t ∈ [0, T ∗), | d dt Vϕ[u(t)]| ≤ C‖∇ϕ‖W 1,∞‖u(t)‖2Hs , for some constant C > 0 depending only on s and N . Now we define the localized Morawetz action of u associated to ϕ by Mϕ[u(t)] := 2 Im ∫ RN ū(t, x)∇ϕ(x) · ∇u(t, x)dx. (2.6) By Lemma 2.3, we obtain the bound |Mϕ[u(t)]| ≤ C ( ‖∇ϕ‖L∞ , ‖∆ϕ‖L∞ ) ‖u(t)‖2H1/2 . Hence the quantityMϕ[u(t)] is well-defined, since u(t) ∈ Hs with some s > 1/2 by assumption. By a similar argument as that in [1, Lemma 2.1], we have the following time evolution of Mϕ[u(t)]. Lemma 2.9 (Morawetz identity). Let N ≥ 1, s ∈ (1/2, 1) and ϕ : RN → R be such that ∇ϕ ∈ W 3,∞. Assume that u ∈ C([0, T ∗), Hs) is a solution to (1.2). Then for any t ∈ [0, T ∗), it holds that d dt Mϕ[u(t)] = ∫ ∞ 0 ms ∫ RN { 4∂kum(t)(∂2klϕ)∂lum(t)− (∆2ϕ)|um(t)|2 } dx dm − 2p p+ 2 ∫ RN ∆ϕ|x|−b|u(t, x)|p+2dx − 4b p+ 2 ∫ RN |x|−b−2x · ∇ϕ|u(t, x)|p+2dx, (2.7) where um(t) = um(t, x) is defined by (2.3). EJDE-2021/39 BLOW-UP CRITERIA FOR SCHRÖDINGER EQUATIONS 9 Proof. It follows from an integration by parts that 〈u(t), [−|x|−b|u(t)|p, iΓϕ]u(t)〉 = −〈u(t), [|x|−b|u(t)|p,∇ϕ · ∇+∇ · ∇ϕ]u(t)〉 = 2 ∫ RN |x|−b|u(t, x)|2∇ϕ · ∇|u(t, x)|pdx+ 2 ∫ RN |u(t, x)|p+2∇ϕ · ∇|x|−bdx = − 2p p+ 2 ∫ RN ∆ϕ|x|−b|u(t, x)|p+2dx− 4b p+ 2 ∫ RN |x|−b−2x · ∇ϕ|u(t, x)|p+2dx, where we used the identities ∇|x|−b = −b|x|−b−2x and ∇|u|p+2 = p+ 2 p ∇|u|p|u|2. Following the method used in [1], we complete the proof. � 3. Blow-up criteria In this section, we will prove Theorem 1.1. Firstly, we establish the following blow-up criteria for (1.2). Lemma 3.1. Let N ≥ 1, s ∈ ( 1 2 , 1), 4s−2b N ≤ p < 4s−2b N−2s . Assume that u0 ∈ Hs and u ∈ C([0, T ∗), Hs) is the corresponding solution of (1.2). If there exists δ > 0 such that K(u(t)) ≤ −δ (3.1) for all t ∈ [0, T ∗), then one of the following two statements holds: • u(t) blows up in finite time, i.e. T ∗ < +∞; • u(t) blows up infinite time and there exists (tn)n≥1 such that tn → +∞ and lim n→∞ ‖(−∆)s/2u(tn)‖L2 =∞. (3.2) Proof. If T ∗ < +∞, then the proof is done. If T ∗ = +∞, we prove (1.1) by contradiction. If not, the solution u(t) exists globally and there exists C0 > 0 such that C0 := sup t∈[0,+∞) ‖(−∆)s/2u(t)‖L2 <∞. (3.3) This, together with the conservation of mass, implies that C1 := sup t∈[0,+∞) ‖u(t)‖Hs <∞. (3.4) Now, we claim that for every η > 0, R > 1, there exists a constant C > 0 indepen- dent of R and C1 such that for any t ∈ [0, ηR CC2 1 ],∫ |x|≥R |u(t, x)|2dx ≤ η + oR(1). (3.5) To this end, we define a smooth function θ : [0,∞)→ [0, 1] that satisfies θ(r) = { 0 if 0 ≤ r ≤ 1/2, 1 if r ≥ 1. For R > 1, we define the radial function φR(x) = φR(r) := θ(r/R), r = |x|. 10 B. FENG, Z. HE, J. LIU EJDE-2021/39 It easily follows that ∇φR(x) = x rR θ′(r/R), ∆φR(x) = 1 R2 θ′′(r/R) + (N − 1) rR θ′(r/R). In particular, we have ‖∇φR‖W 1,∞ ∼ ‖∇φR‖L∞ + ‖∆φR‖L∞ ≤ CR−1. (3.6) Now, we can define the localized virial potential VφR [u(t)] := ∫ RN φR(x)|u(t, x)|2dx. We have VφR [u(t)] = VφR [u0] + ∫ t 0 d dτ VφR [u(τ)]dτ ≤ VφR [u0] + ( sup τ∈[0,t] ∣∣ d dτ VφR [u(τ)] ∣∣)t. By Corollary 2.8, (3.4) and (3.6), we obtain sup τ∈[0,t] ∣∣ d dτ VφR [u(τ)] ∣∣ ≤ C‖∇φR‖W 1,∞ sup τ∈[0,t] ‖u(τ)‖2Hs ≤ CC2 1R −1, for some constant C > 0 independent of R and C1. We thus obtain VφR [u(t)] ≤ VφR [u0] + CC2 1R −1t, for all t ≥ 0. By the choice of θ and the conservation of mass, we have VφR [u0] = ∫ RN φR(x)|u0(x)|2dx ≤ ∫ |x|>R/2 |u0(x)|2dx→ 0, as R→∞ or VφR [u0] = oR(1). On the other hand, we have∫ |x|≥R |u(t, x)|2dx ≤ VφR [u(t)]. Collecting the above estimates, we can obtain the control on the L2-norm of the solution outside a large ball, i.e., claim (3.5). Next, we assume that ϕ(x) = ϕ(r) is radial and satisfies ϕ(r) = { r2/2 for r ≤ 1, const. for r ≥ 10, and ϕ′′(r) ≤ 1 for r ≥ 0. GivenR > 0 , we define the rescaled function ϕR : RN → R by ϕR(x) := R2ϕ ( x R ) . (3.7) We readily verify the inequalities 1− ϕ′′R(r) ≥ 0, 1− ϕ′R(r) r ≥ 0, N −∆ϕR(x) ≥ 0, for all r ≥ 0 and all x ∈ RN . It is easy to see that ‖∇kϕR‖L∞ ≤ CR2−k, k = 0, · · · , 4, and supp(∇kϕR) ⊂ { {x : |x| ≤ 10R} for k = 1, 2, {x : R ≤ |x| ≤ 10R} for k = 3, 4. EJDE-2021/39 BLOW-UP CRITERIA FOR SCHRÖDINGER EQUATIONS 11 By Lemma 2.9, we have d dt MϕR [u(t)] = ∫ ∞ 0 ms ∫ RN { 4∂kum(t)(∂2klϕR)∂lum(t)− (∆2ϕR)|um(t)|2 } dx dm − 2p p+ 2 ∫ RN ∆ϕR|x|−b|u(t, x)|p+2dx − 4b p+ 2 ∫ RN |x|−b−2x · ∇ϕR|u(t, x)|p+2dx (3.8) where um(t) = um(t, x) is defined in (2.3). Since supp(∆2ϕR) ⊂ {|x| ≥ R}, by Lemma 2.4, we have∣∣ ∫ ∞ 0 ms ∫ RN (∆2ϕR)|um(t)|2 dx dm ∣∣ ≤ C‖∆2ϕR‖sL∞‖∆ϕR‖1−sL∞ ‖u(t)‖2L2(|x|≥R) ≤ CR−2s‖u(t)‖2L2(|x|≥R). (3.9) Since ϕR is radial, we use ∂2jk = (δjk r − xjxk r3 ) ∂r + xjxk r2 ∂2r to write ∫ ∞ 0 ms ∫ RN ∂kum(t)(∂2jkϕR)∂lum(t) dx dm = ∫ ∞ 0 ms ∫ RN ϕ′R r |∇um(t)|2 dx dm + ∫ ∞ 0 ms ∫ RN (ϕ′′R r2 − ϕ′R r3 ) |x · ∇um(t)|2 dx dm. Using (2.4), we write∫ ∞ 0 ms ∫ RN ϕ′R r |∇um(t)|2 dx dm = s‖(−∆)s/2u(t)‖2L2 + ∫ ∞ 0 ms ∫ RN (ϕ′R r − 1 ) |∇um(t)|2 dx dm. Since ϕ′′R ≤ 1, the Cauchy-Schwarz inequality implies∫ ∞ 0 ms ∫ RN (ϕ′R r − 1 ) |∇um(t)|2 dx dm + ∫ ∞ 0 ms ∫ RN ( ϕ′′R − ϕ′R r ) |x · ∇um(t)|2 r2 dx dm ≤ 0. Therefore, 4 ∫ ∞ 0 ms ∫ RN ∂kum(t)(∂2jkϕR)∂lum(t) dx dm ≤ 4s‖(−∆)s/2u(t)‖2L2 . (3.10) 12 B. FENG, Z. HE, J. LIU EJDE-2021/39 We next write − 2p p+ 2 ∫ RN |x|−b|u(t, x)|p+24ϕRdx = − 2pN p+ 2 ∫ RN |x|−b|u(t, x)|p+2dx − 2p p+ 2 ∫ RN |x|−b|u(t, x)|p+2(4ϕR −N)dx. (3.11) The second term can be estimated as follows:∣∣− 2p p+ 2 ∫ RN (∆ϕR −N)|x|−b|u(t, x)|p+2dx ∣∣ ≤ C ∫ |x|≥R |x|−b|u(t, x)|p+2dx ≤ CR−b ∫ |x|≥R |u(t, x)|p+2dx ≤ CR−b‖u(t)‖p+2−Np 2s L2(|x|≥R)‖u(t)‖ Np 2s L 2N N−2s (|x|≥R) ≤ CR−b‖u(t)‖p+2−Np 2s L2(|x|≥R)‖u(t)‖ Np 2s Hs ≤ CC Np 2s 1 R−b‖u(t)‖p+2−Np 2s L2(|x|≥R). (3.12) Thus, we have − 2p p+ 2 ∫ RN |x|−b|u(t, x)|p+24ϕRdx ≤ − 2pN p+ 2 ∫ RN |x|−b|u(t, x)|p+2dx+ CC Np 2s 1 R−b‖u(t)‖p+2−Np 2s L2(|x|≥R). (3.13) For the last term in (3.8), we have − 4b p+ 2 ∫ RN (x · ∇ϕR)|x|−b−2|u(t, x)|p+2dx = − 4b p+ 2 ∫ RN |x|−b|u(t, x)|p+2dx − 4b p+ 2 ∫ |x|≥R ( x · ∇ϕR(r) |x|2 − 1)|x|−b|u(t, x)|p+2dx . By the similar method as (3.12), we deduce∣∣− 4b p+ 2 ∫ |x|≥R ( x · ∇ϕR(r) |x|2 − 1)|x|−b|u(t, x)|p+2dx ∣∣ ≤ CC Np 2s 1 R−b‖u(t)‖p+2−Np 2s L2(|x|≥R). (3.14) Collecting (3.9)-(3.14), we obtain d dt MϕR [u(t)] ≤ 4s‖(−∆)s/2u(t)‖2L2 − 2Np+ 4b p+ 2 ∫ RN |x|−b|u(t, x)|p+2dx + CR−2s‖u(t)‖2L2(|x|≥R) + CC Np 2s 1 R−b‖u(t)‖p+2−Np 2s L2(|x|≥R). (3.15) EJDE-2021/39 BLOW-UP CRITERIA FOR SCHRÖDINGER EQUATIONS 13 By (3.5), 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 , d dt MϕR [u(t)] ≤ 4K(u(t)) + CR−2s(η + oR(1))2 + CC Np 2s 1 R−b(η + oR(1))p+2−Np 2s ≤ −4δ + CR−2s(η2 + oR(1)) + CC Np 2s 1 R−b(ηp+2−Np 2s + oR(1)). We first choose η > 0 small enough and R > 1 large enough so that d dt MϕR [u(t)] ≤ −δ < 0, (3.16) for any t ∈ [0, T0] with T0 = ηR CC2 1 . Note that η > 0 is fixed, so we can choose R > 1 large enough so that T0 is as large as we want. By (3.16), it follows that MϕR [u(t)] ≤ −ct, for all t ∈ [t0, T0] with some sufficiently large t0 ∈ [0, T0]. The constant c > 0 depends only on δ. On the other hand, we deduce from Lemma 2.3 and the conservation of mass that |MϕR [u(t)]| ≤ C(ϕR) ( ‖|∇|1/2u(t)‖2L2 + ‖u(t)‖L2‖|∇|1/2u(t)‖L2 ) ≤ C(ϕR) ( ‖|∇|1/2u(t)‖2L2 + ‖u(t)‖2L2 ) ≤ C(ϕR) ( ‖|∇|1/2u(t)‖2L2 + 1 ) , for every t ∈ [0,+∞). By interpolating between L2 and Ḣs, we obtain ct ≤ −MϕR [u(t)] = |MϕR [u(t)]| ≤ C(ϕR) ( ‖(−∆)s/2u(t)‖ 1 s L2 + 1 ) , for any t ∈ [t0, T0]. This implies that ‖(−∆)s/2u(t)‖L2 ≥ Cts, (3.17) for all t ∈ [t1, T0] with some sufficiently large t1 ∈ [t0, T0]. Taking t close to T0 = ηR CC2 1 , we see that ‖(−∆)s/2u(t)‖L2 →∞ as R →∞, which contradicts (3.4). The proof is complete. � Applying Lemma 3.1, we can prove blow-up criteria for (1.2). Proof of Theorem 1.1. We only check that (3.1) holds under the assumptions of this Theorem. In the L2-critical case, i.e., sc = 0. The blow-up condition (1.11) implies that ‖u0‖L2 < ‖Q‖L2 and ‖u0‖L2 > ‖Q‖L2 , which is impossible. Thus, for sc = 0 the only admissible condition is E(u0) < 0. It follows from the conservation of energy and p = 4s−2b N that K(u(t)) =s‖u(t)‖2 Ḣs − Np+ 2b 2p+ 4 ∫ RN |x|−b|u(t, x)|p+2dx =2sE(u(t)) + 4s−Np− 2b 2p+ 4 ∫ RN |x|−b|u(t, x)|p+2dx =2sE(u0), for all t ∈ [0, T ∗). Hence, when E(u0) < 0, (3.1) follows with δ = −2sE(u0). 14 B. FENG, Z. HE, J. LIU EJDE-2021/39 Next, we consider the case E(u0) > 0. The assumption (1.11) implies E(u0)‖u0‖2σL2 < E(Q)‖Q‖2σL2 , ‖(−∆)s/2u0‖L2‖u0‖σL2 > ‖(−∆)s/2Q‖L2‖Q‖σL2 , (3.18) where σ := s− sc sc = 2sp−Np+ 4s− 2b Np+ 2b− 4s . We notice that the sharp constant in Gagliardo-Nirenberg inequality (2.1) can be written as Copt = ∫ RN |x|−b|Q(x)|p+2dx ‖Q‖ Np+2b 2s Ḣs ‖Q‖p+2−Np+2b 2s L2 , (3.19) which, by (2.2), can be rewritten as Copt = 2s(p+ 2) Np+ 2b 1 (‖Q‖Ḣs‖Q‖σL2) Np+2b−4s 2s . (3.20) It easily follows that E(Q)‖Q‖2σL2 = Np+ 2b− 4s 2(Np+ 2b) (‖Q‖Ḣs‖Q‖σL2)2. (3.21) Multiplying both sides of E(u(t)) by ‖u(t)‖2σL2 , we deduce from the sharp Gagliardo- Nirenberg inequality (2.1) that E(u(t))‖u(t)‖2σL2 = 1 2 ‖u(t)‖2 Ḣs‖u(t)‖2σL2 − 1 p+ 2 ∫ RN |x|−b|u(t, x)|p+2dx‖u(t)‖2σL2 ≥ 1 2 (‖u(t)‖Ḣs‖u(t)‖σL2)2 − Copt p+ 2 (‖u(t)‖Ḣs‖u(t)‖σL2) Np+2b 2s = f(‖u(t)‖Ḣs‖u(t)‖σL2), where f(x) := 1 2x 2 − Copt p+2 x Np+2b 2s . It is easy to see that f is increasing on (0, x0) and decreasing on (x0,∞), where x0 = ( 2sp+ 4s Copt(Np+ 2b) ) 2s Np+2b−4s = ‖Q‖Ḣs‖Q‖σL2 , where the last equality follows from (3.20). It follows from (3.20) and (3.21) that f(‖Q‖Ḣs‖Q‖σL2) = E(Q)‖Q‖2σL2 . Thus the conservation of mass and energy together with the first condition in (1.11) imply f(‖u(t)‖Ḣs‖u(t)‖σL2) ≤ E(u(t))‖u(t)‖2σL2 = E(u0)‖u0‖2σL2 < E(Q)‖Q‖2σL2 = f(‖Q‖Ḣs‖Q‖σL2), for all t ∈ [0, T ∗). Using the second condition (1.11), the continuity argument shows that ‖u(t)‖Ḣs‖u(t)‖σL2 > ‖Q‖Ḣs‖Q‖σL2 (3.22) for any t ∈ [0, T ∗). On the other hand, since E(u0)‖u0‖2σL2 < E(Q)‖Q‖2σL2 , we pick η > 0 small enough so that E(u0)‖u0‖2σL2 ≤ (1− η)E(Q)‖Q‖2σL2 . EJDE-2021/39 BLOW-UP CRITERIA FOR SCHRÖDINGER EQUATIONS 15 Thus, by the conservation of energy, (3.21) and (3.22), we have K(u(t))‖u(t)‖2σL2 = Np+ 2b 2 E(u(t))‖u(t)‖2σL2 − Np+ 2b− 4s 4 ‖u(t)‖2 Ḣs‖u(t)‖2σL2 = Np+ 2b 2 E(u0)‖u0‖2σL2 − Np+ 2b− 4s 4 (‖u(t)‖Ḣs‖u(t)‖σL2)2 ≤ Np+ 2b 2 (1− η)E(Q)‖Q‖2σL2 − Np+ 2b− 4s 4 (‖Q‖Ḣs‖Q‖σL2)2 = −ηNp+ 2b 2 E(Q)‖Q‖2σL2 , for all t ∈ [0, T ∗). This implies (3.1) with δ = ηNp+2b 2 E(Q)‖Q‖2σL2 . Thus, the solution u(t) of (1.2) blows up in finite or infinite time. This completes the proof. � 4. Strong instability In this section, we apply the blow-up criteria in Theorem 1.1 to prove Theorem 1.3. Proof of Theorem 1.3. We divide the proof into two cases: (1) p = 4s−2b N and (2) 4s−2b N < p < 4s−2b N−2s . Case (1) p = 4s−2b N . Firstly, we deduce from Pohozaev’s identities (2.2) that E(Qω) = 0, where Qω is the ground state solution of (1.13). Thus, if we can construct initial data u0,n such that E(u0,n) < 0 and u0,n → Qω in Hs, as n→∞, then the corresponding solution un blows up in finite or infinite time by applying Theorem 1.1. This implies that the standing wave u(t, x) = eiωtQω(x) is unstable. Let {cn} ⊆ C be such that |cn| > 1 and limn→∞ |cn| = 1, and {λn} ⊆ R+ be such that limn→∞ λn = 1. We take the initial data u0,n(x) := cnλ N/2 n Qω(λnx). Then, we have lim n→∞ ‖u0,n‖L2 = lim n→∞ |cn|‖Qω‖L2 = ‖Qω‖L2 , lim n→∞ ‖u0,n‖Ḣs = lim n→∞ |cn|λsn‖Qω‖Ḣs = ‖Qω‖Ḣs . Thus, from Brezis-Lieb’s lemma we deduce that u0,n → Qω in Hs as n→∞. On the other hand, from Pohozaev’s identities (2.2) we deduce that E(u0,n) = 1 2 ‖u0,n‖2Ḣs − 1 p+ 2 ∫ RN |x|−b|u0,n(x)|p+2dx = |cn|2λ2sn 2 ‖Qω‖2Ḣs − |cn|p+2λ b+Np 2 n p+ 2 ∫ RN |x|−b|Qω(x)|p+2dx = (|cn|2 − |cn|p+2)λ2sn 2 ‖Qω‖2Ḣs < 0. Applying Theorem 1.1, the solution un of (1.2) with initial data u0,n blows up in finite time. 16 B. FENG, Z. HE, J. LIU EJDE-2021/39 Case (2) 4s−2b N < p < 4s−2b N−2s . Let Qω be the ground state related to (1.13), a direct computation shows Sω(Qλω) = 1 2 λ2s‖Qω‖2Ḣs + ω 2 ‖Qω‖2L2 − λ Np 2 +b p+ 2 ∫ RN |x|−b|Qω(x)|p+2dx, and ∂λSω(Qλω) = sλ2s−1‖Qω‖2Ḣs − (Np+ 2b)λ Np 2 +b−1 2p+ 4 ∫ RN |x|−b|Qω(x)|p+2dx = K(Qλω) λ . It is easy to see that the equation ∂λSω(Qλω) = 0 has a unique non-zero solution,( s(2p+ 4)‖Qω‖2Ḣs (Np+ 2b) ∫ RN |x|−b|Qω(x)|p+2dx ) 2 Np+2b−4s = 1. The last inequality comes from the fact that K(Qω) = 0, which follows from Po- hozaev’s identities (2.2). We thus obtain ∂λSω(Qλω) { > 0 if λ ∈ (0, 1), < 0 if λ ∈ (1,∞). This implies that Sω(Qλω) < Sω(Qω) for any λ > 0 and λ 6= 1. This, together with ‖Qλω‖L2 = ‖Qω‖L2 , implies that for any λ > 1, E(Qλω) < E(Qω). (4.1) Let λn > 1 such that limn→∞ λn = 1. We take the initial data u0,n(x) = Qλn ω (x) = λN/2n Qω(λnx). By Brezis-Lieb’s lemma, we have u0,n → Qω in Hs as n → ∞. We deduce from (4.1) that E(u0,n) < E(Qω), and ‖(−∆)s/2u0,n‖L2 = λsn‖(−∆)s/2Qω‖L2 > ‖(−∆)s/2Qω‖L2 . Thus, by ‖u0,n‖L2 = ‖Qω‖L2 , (1.14) and (1.15), we have E(u0,n)sc‖u0,n‖2(s−sc)L2 < E(Qω)sc‖Qω‖2(s−sc)L2 = E(Q)sc‖Q‖2(s−sc)L2 , and ‖(−∆)s/2u0,n‖scL2‖u0,n‖s−scL2 > ‖(−∆)s/2Qω‖scL2‖Qω‖s−scL2 = ‖(−∆)s/2Q‖scL2‖Q‖s−scL2 , where sc = N 2 − 2s−b p . Applying Theorem 1.1, the solution un of (1.2) and initial data u0,n blows up in finite time. This completes the proof. � Acknowledgments. B. Feng was supported by the Outstanding Youth Science Fund of Gansu Province (No. 20JR10RA111), and by the program NWNU-LKQN2019- 7. Z. He was supported by the Natural Science Foundation of Qinghai Province (No. 2021-ZJ-957Q). J. 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Introduction 2. Preliminary lemmas 3. Blow-up criteria 4. Strong instability Acknowledgments References