In this note we give an alternative, shorter proof of the classical result of
Berestycki and Cazenave on the instability by blow-up for the standing waves of
some nonlinear Schr\"odinger equations

Coz, Stefan Le

English

arXiv

International audienceIn this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schrödinger equations

Le Coz, Stefan

HAL - Université de Franche-Comté

hal-00182752, version 1 - 27 Oct 2007A NOTE ON BERESTYCKI-CAZENAVE’S CLASSICALINSTABILITY RESULT FOR NONLINEAR SCHRO¨DINGEREQUATIONSSTEFAN LE COZAbstract. In this note we give an alternative, shorter proof of the classicalresult of Berestycki and Cazenave on the instability by blow-up for thestanding waves of some nonlinear Schro¨dinger equations.1. IntroductionIn 1981, in a celebrated note [1], Berestycki and Cazenave studied the insta-bility of standing waves for the nonlinear Schro¨dinger equation(1) iut +∆u+ |u|p−1u = 0where u = u(t, x) ∈ C, t ∈ R, x ∈ RN and p > 1. A standing wave is a solutionof (1) of the form eiωtϕ(x) with ϕ ∈ H1(RN) and ω > 0. Thus ϕ is solution of(2) −∆ϕ + ωϕ = |ϕ|p−1ϕ, ϕ ∈ H1(RN).We say that ϕ ∈ H1(RN) is a ground state solution of (2) if it satisfiesS˜(ϕ) = inf{S˜(v); v ∈ H1(RN) \ {0} is a solution of (2) }where S˜ is defined for v ∈ H1(RN ) byS˜(v) :=12‖∇v‖22 +ω2‖v‖22 −1p+ 1∫RN|v|p+1dx.In [1] it is shown that if 1+ 4N< p < 1+ 4N−2when N > 3 and 1+ 4N< p < +∞when N = 1, 2, then any standing wave associated with a ground state solutionϕ of (2) is unstable by blow up. More precisely, there exists (ϕn) ⊂ H1(RN)such that ϕn → ϕ in H1(RN) and the corresponding maximal solution un of(1) with un(0) = ϕn blows up in finite time.The result and perhaps more the methods introduced in [1] still have adeep influence on the field of instability for nonlinear Schro¨dinger and relatedequations. In particular the idea of defining appropriate invariant sets andDate: October 28, 2007.2000 Mathematics Subject Classification. 35Q55,(35B35,35A15).12 STEFAN LE COZhow to use them to establish the blow-up. We should mention that in [1]more general nonlinearities were considered. The paper [1] is only a shortnote which contains the main ideas but no proofs. For the special nonlinearity|u|p−1u these proofs can be found in [5]. For the general case it seems that theextended version [2] of [1] has remained unpublished so far.The aim of the present note is to present an alternative, shorter proof ofthe result of [1] for general nonlinearities. Also the instability of the standingwaves is proved under slightly weaker assumptions. Before stating our resultwe need to introduce some notations. Let g : R 7→ R be an odd functionextended to C by setting g(z) = g(|z|)z/|z| for z ∈ C \ {0}. Equation (1) nowbecomes(3) iut +∆u+ g(u) = 0and correspondingly for the ground states we have(4) −∆ϕ + ωϕ = g(ϕ).For z ∈ C let G(z) :=∫ |z|0g(s)ds. We assume(A0) The function g satisfies(a) g ∈ C(R,R).(b) lims→0g(s)s= 0.(c) when N > 3, lims→+∞ g(s)s−N+2N−2 = 0;when N = 2, for any α > 0, there exists Cα > 0 such that|g(s)| 6 Cαeαs2 for all s > 0.(A1) The function h(s) := (sg(s)− 2G(s))s−(2+4/N) is strictly increasing on(0,+∞) and lims→0 h(s) = 0.(A2) There exist C > 0 and α ∈ [0,4N−2) if N > 3, α ∈ [0,∞) if N = 2, suchthat|g(s)− g(t)| 6 C(1 + |s|α + |t|α)|t− s|for all s, t ∈ R. If N = 1 we assume that for every M > 0, there existsL(M) > 0 such that|g(s)− g(t)| 6 L(M)|s− t|for all s, t ∈ R such that |s|+ |t| 6M.Finally we define for v ∈ H1(RN) the functionalS(v) :=12‖∇v‖22 +ω2‖v‖22 −∫RNG(v)dxand setm := inf{S(v); v ∈ H1(RN) \ {0} is a solution of (4) }.ON BERESTYCKI-CAZENAVE’S INSTABILITY RESULT FOR NLS 3Our main result isTheorem 1. Assume that (A0)−(A2) hold and let ϕ be a ground state solutionof (4), i.e. a solution of (4) such that S(ϕ) = m. Then for every ε > 0 thereexists u0 ∈ H1(RN) such that ‖u0 − ϕ‖H1(RN ) < ε and the solution u of (3)with u(0) = u0 satisfieslimt→Tu0‖∇u(t)‖2 = +∞ with Tu0 < +∞.From [3, 4] we know that assumption (A0) is almost necessary to guaranteethe existence of a solution for (4). Assumption (A1) is a weaker version of theassumption (H.1) in [1]. An assumption of this type, on the growth of g, is nec-essary since it is known from [6] that when g(u) = |u|p−1u with 1 < p < 1+ 4Nthe standing waves associated with the ground states are, on the contrary, or-bitally stable. Assumption (A2) is purely technical and is aimed at ensuringthe local well-posedness of the Cauchy problem for (3). It replaces assumption(H.2) in [1]. Indeed, in [1] the authors were using the results of Ginibre andVelo [8] for that purpose. Since [1] has been published, advances have beendone in the study of the Cauchy problem (see [5, 7] and the references therein).In particular, under our condition (A2), for all u0 ∈ H1(RN) there exist Tu0 > 0and a unique solution u ∈ C([0, Tu0), H1(RN )) ∩ C1([0, Tu0), H−1(RN)) of (3)such that limt→Tu0 ‖∇u(t)‖2 = +∞ if Tu0 < +∞. Furthermore, the followingconservation properties hold : for all t ∈ [0, Tu0) we haveS(u(t)) = S(u0),(5)‖u(t)‖2 = ‖u0‖2.(6)Finally, the function f : t 7→ ‖xu(t)‖22 is C2 and we have the virial identity(7) ∂ttf(t) = 8Q(u(t)),where Q is defined for v ∈ H1(RN) byQ(v) := ‖∇v‖22 −N2∫RN(g(|v|)|v| − 2G(v))dx.The proofs of instability in [1] and here share some elements, in particularthe introduction of sets invariant under the flow. The main difference lies inthe variational characterization of the ground states which is used to definethe invariant sets and how to derive this characterization.In [1] it is shown that a ground state of (4) can be characterized as a mini-mizer of S on the constraintM := {v ∈ H1(RN) \ {0}, Q(v) = 0}.4 STEFAN LE COZTo show this characterization, S is directly minimized on M . Additional as-sumptions (see (H.1) in [1]) are necessary at this step to insure that the mini-mizing sequences are bounded. Once the existence of a minimizer for S on Mhas been established, one has to get rid of the Lagrange multiplier, namely toprove that it is zero. There, a stronger version of (A0), requiring in particularg ∈ C1(R,R) and a control on g′(s) at infinity, is necessary along with tediouscalculations.Having established that the ground states of (4) minimize S on M , Beresty-cki and Cazenave show that the setK := {v ∈ H1(RN), S(v) < m and Q(v) < 0}is invariant under the flow of (3) and that one can choose in K an initial data,arbitrarily close to the ground state, for which the blow-up occurs.In our approach we characterize the ground states as minimizers of S onM := {v ∈ H1(RN) \ {0};Q(v) = 0, I(v) 6 0},where I(v) is defined for v ∈ H1(RN) byI(v) := ‖∇v‖22 + ω‖v‖22 −∫RNg(|v|)|v|dxand correspondingly our invariant set is{v ∈ H1(RN ), S(v) < m,Q(v) < 0 and I(v) < 0}.The dominant feature of our approach, which also explains why our assump-tions on g are weaker than in [1] is that we never explicitly solve a minimizationproblem. At the heart of our approach is an additional characterization of theground states as being at a mountain pass level for S. This characterizationwas derived in [10] for N > 2 and in [11] for N = 1. We also strongly benefitfrom recent techniques developed by several authors [12, 13, 14, 15, 16, 17]where minimization approches using two constraints have been introduced.2. Proof of Theorem 1We first prove the existence of ground states and the fact that they corre-spond to minimizers of S on the Nehari manifold.Lemma 1. Assume that (A0) and (A1) hold. Then (4) admits a ground statesolution. Furthermore, the ground states solutions of (4) are minimizers ford(ω) := inf{S(v); v ∈ H1(RN) \ {0}, I(v) = 0}.Before proving Lemma 1, we prove a technical result.ON BERESTYCKI-CAZENAVE’S INSTABILITY RESULT FOR NLS 5Lemma 2. Assume that (A0) and (A1) hold. Then the nonlinearity g satisfiesg(s)sis increasing for s > 0.(8)g(s)s→ +∞ as s→ +∞.(9)Proof of Lemma 2. From the definition of h(s) we have(10)g(s)s= s4/Nh(s) +2G(s)s2.Furthermore, for s > 0(11)∂∂s(G(s)s2)=s(sg(s)− 2G(s))s4> 0where the last inequality follows from (A1). Thus, combining (10), (11) and(A1) we get (8) and (9). Proof of Lemma 1. It follows from Lemma 2 that(P) There exists s0 > 0 such that– if N > 2, then 12ωs20 < G(s0);– if N = 1, then 12ωs2 > G(s) for s ∈ (0, s0),12ωs20 = G(s0) andωs0 < g(s0).Now, from [3, The´ore`me 1] and [4, Theorem 1] we know that the conditions(A0) and (P) are sufficient to insure the existence of a ground state.If v is a solution of (4), then S ′(v)v = I(v) = 0; therefore, to prove thelemma it is enough to show that d(ω) > m. From [10, 11] we know that under(A0) and (P) the functional S has a mountain pass geometry. More precisely,if we setΓ := {χ ∈ C([0, 1], H1(RN));χ(0) = 0, S(χ(1)) < 0},then Γ 6= ∅ andc := infχ∈Γmaxt∈[0,1]S(χ(t)) > 0.In addition it is shown1 in [10, 11] thatm = c.Namely the mountain pass level c corresponds to the ground state level m.Now it is well-known that (8) ensure that if v ∈ H1(RN ) satisfies I(v) = 0then t 7→ S(tv) achieves its unique maximum on [0,+∞) at t = 1. Also (9)1In fact, the results of [10, 11] are proved only for real valued functions; however, it is nothard to see that they can be extended to the complex case (see [9, Lemma 14]).6 STEFAN LE COZshows that limt→+∞ S(tv) = −∞. From the definition of c, it implies thatc 6 S(v) for all v ∈ H1(RN ) such that I(v) = 0. Hence we haved(ω) > c,and combined with the fact that m = c it ends the proof. Now we investigate the behavior of the functionals under some rescalingLemma 3. Assume that (A0) and (A1) hold. For λ > 0 and v ∈ H1(RN ), wedefine vλ( · ) := λN2 v(λ · ). We suppose Q(v) 6 0. Then there exists λ0 6 1such that(i) Q(vλ0) = 0,(ii) λ0 = 1 if and only if Q(v) = 0,(iii) ∂∂λS(vλ) > 0 for λ ∈ (0, λ0) and∂∂λS(vλ) < 0 for λ ∈ (λ0,+∞),(iv) λ 7→ S(vλ) is concave on (λ0,+∞),(v) ∂∂λS(vλ) = 1λQ(vλ).Proof of Lemma 3. Easy computations lead to∂∂λS(vλ) =1λQ(vλ)= λ(‖∇v‖22 −N2∫RNλ−(N+2)(λN2 g(λN2 |v|)|v| − 2G(λN2 v))dx),and recalling from (A1) that the function h(s) := (sg(s) − 2G(s))s−(2+4/N) isstrictly increasing on [0,+∞), (i),(ii),(iii) and (v) follow easily. To see (iv), weremark that since(‖∇v‖22 −N2∫RNλ−(N+2)(λN2 g(λN2 |v|)|v| − 2G(λN2 v))dx)< 0on (λ0,+∞), we infer from (A1) that∂∂λS(vλ) is strictly decreasing on(λ0,+∞), which implies (iv). Proof of Theorem 1. We recall thatM = {v ∈ H1(RN) \ {0};Q(v) = 0, I(v) 6 0},and definedM := inf{S(v); v ∈ M }.We proceed in three steps.Step 1. Let us prove d(ω) = dM . Since the ground states ϕ satisfyQ(ϕ) = I(ϕ) = 0, we have ϕ ∈ M . Combined with S(ϕ) = d(ω), this impliesdM 6 d(ω). Conversely, let v ∈ M . If I(v) = 0, then trivially S(v) > d(ω),ON BERESTYCKI-CAZENAVE’S INSTABILITY RESULT FOR NLS 7thus we suppose I(v) < 0. We use the rescaling defined in Lemma 3 : for λ > 0we haveI(vλ) = λ2‖∇v‖22 + ω‖v‖22 −∫RNλ−N/2g(λN/2|v|)|v|dx.It follows from (A0)-(b) that limλ→0 I(vλ) = ω‖v‖22 and thus by continuitythere exists λ1 < 1 such that I(vλ1) = 0. Thus S(vλ1) > d(ω). Now, fromQ(v) = 0 and (iii) in Lemma 3 we haveS(v) > S(vλ1) > d(ω),hence dM = d(ω).Step 2. For λ > 0, we set uλ := ϕλ. For λ > 1 close to 1, we haveS(uλ) < S(ϕ) and Q(uλ) < 0,(12)I(uλ) < 0.(13)Indeed, (12) follows from (iii) and (v) in Lemma 3. For (13), we writeI(uλ) = 2S(uλ) +2NQ(uλ)−2N‖∇uλ‖226 2S(ϕ) +2NQ(ϕ)− I(ϕ)−2λ2N‖∇ϕ‖2262(1− λ2)N‖∇ϕ‖22 < 0.Let u(t) be the solution of (3) with u(0) = uλ. We claim that the propertiesdescribed in (12), (13) are invariant under the flow of (3). Indeed, since from(5) we have for all t > 0(14) S(u(t)) = S(uλ) < S(ϕ),we infer that I(u(t)) 6= 0 for any t > 0, and by continuity we have I(u(t)) < 0for all t > 0. It follows that Q(u(t)) 6= 0 for any t > 0 (if not u(t) ∈ Mand thus S(u(t)) > S(ϕ) which contradicts (14)), and by continuity we haveQ(u(t)) < 0 for all t > 0. Thus for all t > 0 we haveS(u(t)) < S(ϕ), I(u(t)) < 0 and Q(u(t)) < 0.Step 3. We fix t > 0 and define v := u(t). For β > 0, let vβ(x) := βN2 v(βx).From Step 2 we have Q(v) < 0, thus from Lemma 3 there exists β0 < 1such that Q(vβ0) = 0. If I(vβ0) 6 0, we keep β0, otherwise we replace it byβ˜0 ∈ (β0, 1) such that I(vβ˜0) = 0. Thus in any case we have(15) S(vβ0) > d(ω)8 STEFAN LE COZand Q(vβ0) 6 0. Now from (iv) in Lemma 3, we haveS(v)− S(vβ0) > (1− β0)∂∂βS(vβ)|β=1.Thus, from (v) in Lemma 3, Q(v) < 0 and β0 < 1, we getS(v)− S(vβ0) > Q(v).Combined with (15), this gives(16) Q(v) 6 S(v)− d(ω) := −δ < 0where δ is independent of t since S is a conserved quantity.To conclude, it suffices to observe that thanks to (7) and (16) we have(17) ‖xu(t)‖22 6 −δt2 + Ct+ ‖xuλ‖22,and since the right hand side of (17) becomes negative when t grows up, weeasily deduce that Tuλ < +∞ and limt→Tuλ‖∇u(t)‖2 = +∞. Acknowledgements. The author is grateful to Louis Jeanjean for helpfuladvice during the writing of this paper. He also wishes to thank ThierryCazenave for sharing with him a digitalized version of the unpublished paper[2].References[1] H. Berestycki and T. Cazenave, Instabilite´ des e´tats stationnaires dans lese´quations de Schro¨dinger et de Klein-Gordon non line´aires, C. R. Acad. Sci. Paris293, (1981), 489–492.[2] H. Berestycki and T. Cazenave, Instabilite´ des e´tats stationnaires dans lese´quations de Schro¨dinger et de Klein-Gordon non line´aires, Publications du Labora-toire d’Analyse Nume´rique, Universite´ de Paris VI.[3] H. Berestycki, T. Gallouet and O. Kavian, E´quations de champs scalaires eu-clidiens non line´aires dans le plan, C. R. Acad. Sci. Paris 297, (1983), 307-310.[4] H. Berestycki and P.L. Lions, Nonlinear scalar field equations I, Arch. Ration.Mech. Anal., 82, (1983), 313–346.[5] T. Cazenave, Semilinear Schro¨dinger equations, Courant Lecture Notes in Mathemat-ics, 10, (2003).[6] T. Cazenave and P.L. Lions, Orbital stability of standing waves for some nonlinearSchro¨dinger equations, Comm. Math. Phys., 85, 4, (1982), 549–561.[7] T. Cazenave and F.B. Weissler, The Cauchy problem for the nonlinear Schro¨dingerequation in H1, Manuscripta Math. 61, (1988), 477-494.[8] J. Ginibre and G. Velo, On a class of nonlinear Schro¨dinger equations. I. TheCauchy problem, general case, J. Func. Anal. 32, 1, (1979), 1–32.[9] L. Jeanjean and S. Le Coz, Instability for standing waves of nonlinear Klein-Gordonequations via mountain-pass arguments, preprint.[10] L. Jeanjean and K. Tanaka, A note on a mountain pass characterization of leastenergy solutions, Adv. Nonlinear Stud. 3, 4, (2003), 445–455.ON BERESTYCKI-CAZENAVE’S INSTABILITY RESULT FOR NLS 9[11] L. Jeanjean and K. Tanaka, A remark on least energy solutions in RN , Proc. Amer.Math. Soc. 131, 8, (2003), 2399–2408.[12] S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability ofbound states of a nonlinear Schro¨dinger equation with a Dirac potential, preprint.[13] Y. Liu, Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation, Trans. Amer. Math. Soc. 353, 1, (2001), 191–208.[14] Y. Liu, Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equa-tion in three dimensions, J. Differential Equations 180, 1, (2002), 153–170.[15] Y. Liu, X.-P. Wang and K. Wang, Instability of standing waves of the Schro¨dingerequation with inhomogeneous nonlinearity, Trans. Amer. Math. Soc. 358, (2006), 2105–2122.[16] M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst. 12, 2, (2005), 315–322.[17] J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schro¨dingerequations under a harmonic potential, Comm. Partial Differential Equations, 30, 10-12,(2005), 1429–1443.(Stefan LE COZ) Laboratoire de Mathe´matiques,Universite´ de Franche-Comte´, 25030 Besanc¸on, FRANCEE-mail address : slecoz@univ-fcomte.fr

A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations

Abstract
                In this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schrödinger equations.</jats:p

A note on Berestycki-Cazenave's classical instability result for nonlinear Schr\"odinger equations

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