Stefan Le Coz

A NOTE ON BERESTYCKI-CAZENAVE'S CLASSICAL INSTABILITY RESULT FOR NONLINEAR SCHR ÖDINGER EQUATIONS

Keywords: Mathematics Subject Classification. 35Q55, (35B35, 35A15). 1

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.

Introduction

In 1981, in a celebrated note [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF], Berestycki and Cazenave studied the instability of standing waves for the nonlinear Schrödinger equation [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] iu t + ∆u + |u| p-1 u = 0 where u = u(t, x) ∈ C, t ∈ R, x ∈ R N and p > 1. A standing wave is a solution of (1) of the form e iωt ϕ(x) with ϕ ∈ H 1 (R N ) and ω > 0. Thus ϕ is solution of (2) -∆ϕ + ωϕ = |ϕ| p-1 ϕ, ϕ ∈ H 1 (R N ).

We say that ϕ ∈ H 1 (R N ) is a ground state solution of (2) if it satisfies S(ϕ) = inf{ S(v); v ∈ H 1 (R N ) \ {0} is a solution of (2)

}
where S is defined for v ∈ H 1 (R N ) by

S(v) := 1 2 ∇v 2 2 + ω 2 v 2 2 - 1 p + 1 R N |v| p+1 dx.
In [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] it is shown that if 1+ 4 N < p < 1+ 4 N -2 when N 3 and 1+ 4 N < p < +∞ when N = 1, 2, then any standing wave associated with a ground state solution ϕ of ( 2) is unstable by blow up. More precisely, there exists (ϕ n ) ⊂ H 1 (R N ) such that ϕ n → ϕ in H 1 (R N ) and the corresponding maximal solution u n of (1) with u n (0) = ϕ n blows up in finite time.

The result and perhaps more the methods introduced in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] still have a deep influence on the field of instability for nonlinear Schrödinger and related equations. In particular the idea of defining appropriate invariant sets and how to use them to establish the blow-up. We should mention that in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] more general nonlinearities were considered. The paper [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] is only a short note which contains the main ideas but no proofs. For the special nonlinearity |u| p-1 u these proofs can be found in [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]. For the general case it seems that the extended version [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] of [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] has remained unpublished so far.

The aim of the present note is to present an alternative, shorter proof of the result of [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] for general nonlinearities. Also the instability of the standing waves is proved under slightly weaker assumptions. Before stating our result we need to introduce some notations. Let g : R → R be an odd function extended to C by setting g(z) = g(|z|)z/|z| for z ∈ C \ {0}. Equation (1) now becomes [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF] iu t + ∆u + g(u) = 0 and correspondingly for the ground states we have (4) -∆ϕ + ωϕ = g(ϕ).

∈ [0, 4 N -2 ) if N 3, α ∈ [0, ∞) if N = 2, such that |g(s) -g(t)| C(1 + |s| α + |t| α )|t
-s| for all s, t ∈ R. If N = 1 we assume that for every M > 0, there exists

L(M) > 0 such that |g(s) -g(t)| L(M)|s -t| for all s, t ∈ R such that |s| + |t| M. Finally we define for v ∈ H 1 (R N ) the functional S(v) := 1 2 ∇v 2 2 + ω 2 v 2 2 - R N G(v)dx
and set

m := inf{S(v); v ∈ H 1 (R N ) \ {0} is a solution of (4) }.
Our main result is Theorem 1. Assume that (A 0 )-(A 2 ) hold and let ϕ be a ground state solution of (4), i.e. a solution of (4) such that S(ϕ) = m. Then for every ε > 0 there exists

u 0 ∈ H 1 (R N ) such that u 0 -ϕ H 1 (R N ) < ε and the solution u of (3) with u(0) = u 0 satisfies lim t→Tu 0 ∇u(t) 2 = +∞ with T u 0 < +∞.
From [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF][START_REF] Berestycki | Nonlinear scalar field equations I[END_REF] we know that assumption (A 0 ) is almost necessary to guarantee the existence of a solution for (4). Assumption (A 1 ) is a weaker version of the assumption (H.1) in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]. An assumption of this type, on the growth of g, is necessary since it is known from [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] that when g(u) = |u| p-1 u with 1 < p < 1 + 4 N the standing waves associated with the ground states are, on the contrary, orbitally stable. Assumption (A 2 ) is purely technical and is aimed at ensuring the local well-posedness of the Cauchy problem for (3). It replaces assumption (H.2) in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]. Indeed, in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] the authors were using the results of Ginibre and Velo [START_REF] Ginibre | On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case[END_REF] for that purpose. Since [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] has been published, advances have been done in the study of the Cauchy problem (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Cazenave | The Cauchy problem for the nonlinear Schrödinger equation in H 1[END_REF] and the references therein). In particular, under our condition (A 2 ), for all 3) such that lim t→Tu 0 ∇u(t) 2 = +∞ if T u 0 < +∞. Furthermore, the following conservation properties hold : for all t ∈ [0, T u 0 ) we have S(u(t)) = S(u 0 ), ( 5)

u 0 ∈ H 1 (R N ) there exist T u 0 > 0 and a unique solution u ∈ C([0, T u 0 ), H 1 (R N )) ∩ C 1 ([0, T u 0 ), H -1 (R N )) of (
u(t) 2 = u 0 2 . (6)
Finally, the function f : t → xu(t) 2 2 is C 2 and we have the virial identity ( 7)

∂ tt f (t) = 8Q(u(t)),
where

Q is defined for v ∈ H 1 (R N ) by Q(v) := ∇v 2 2 - N 2 R N (g(|v|)|v| -2G(v))dx.
The proofs of instability in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] and here share some elements, in particular the introduction of sets invariant under the flow. The main difference lies in the variational characterization of the ground states which is used to define the invariant sets and how to derive this characterization.

In [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] it is shown that a ground state of (4) can be characterized as a minimizer of S on the constraint

M := {v ∈ H 1 (R N ) \ {0}, Q(v) = 0}.
To show this characterization, S is directly minimized on M. Additional assumptions (see (H.1) in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]) are necessary at this step to insure that the minimizing sequences are bounded. Once the existence of a minimizer for S on M has been established, one has to get rid of the Lagrange multiplier, namely to prove that it is zero. There, a stronger version of (A 0 ), requiring in particular g ∈ C 1 (R, R) and a control on g ′ (s) at infinity, is necessary along with tedious calculations.

Having established that the ground states of (4) minimize S on M, Berestycki and Cazenave show that the set

K := {v ∈ H 1 (R N ), 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 on

M := {v ∈ H 1 (R N ) \ {0}; Q(v) = 0, I(v) 0}, where I(v) is defined for v ∈ H 1 (R N ) by I(v) := ∇v 2 2 + ω v 2 2 - R N g(|v|)|v|dx
and correspondingly our invariant set is

{v ∈ H 1 (R N ), S(v) < m, Q(v) < 0 and I(v) < 0}.
The dominant feature of our approach, which also explains why our assumptions on g are weaker than in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] is that we never explicitly solve a minimization problem. At the heart of our approach is an additional characterization of the ground states as being at a mountain pass level for S. This characterization was derived in [START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF] for N 2 and in [START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF] for N = 1. We also strongly benefit from recent techniques developed by several authors [START_REF] Le Coz | Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential[END_REF][START_REF] Liu | Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation[END_REF][START_REF] Liu | Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions[END_REF][START_REF] Liu | Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity[END_REF][START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF][START_REF] Zhang | Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential[END_REF] where minimization approches using two constraints have been introduced.

Proof of Theorem 1

We first prove the existence of ground states and the fact that they correspond to minimizers of S on the Nehari manifold.

Lemma 1. Assume that (A 0 ) and (A 1 ) hold. Then (4) admits a ground state solution. Furthermore, the ground states solutions of (4) are minimizers for

d(ω) := inf S(v); v ∈ H 1 (R N ) \ {0}, I(v) = 0 .
Before proving Lemma 1, we prove a technical result.

Lemma 2. Assume that (A 0 ) and (A 1 ) hold. Then the nonlinearity g satisfies g(s)

s is increasing for s > 0. ( 8)

g(s) s → +∞ as s → +∞. ( 9
)
Proof of Lemma 2. From the definition of h(s) we have [START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF] g(s)

s = s 4/N h(s) + 2G(s) s 2 . Furthermore, for s > 0 (11) ∂ ∂s G(s) s 2 = s(sg(s) -2G(s)) s 4 > 0
where the last inequality follows from (A 1 ). Thus, combining ( 10), ( 11) and (A 1 ) we get ( 8) and ( 9).

Proof of Lemma 1. It follows from Lemma 2 that (P) There exists

s 0 > 0 such that -if N 2, then 1 2 ωs 2 0 < G(s 0 ); -if N = 1, then 1 2 ωs 2 > G(s) for s ∈ (0, s 0 ), 1 2 ωs 2 0 = G(s 0
) and ωs 0 < g(s 0 ). Now, from [3, Théorème 1] and [4, Theorem 1] we know that the conditions (A 0 ) 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 the lemma it is enough to show that d(ω) m. From [START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF][START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF] we know that under (A 0 ) and (P) the functional S has a mountain pass geometry. More precisely, if we set

Γ := {χ ∈ C([0, 1], H 1 (R N )); χ(0) = 0, S(χ(1)) < 0}, then Γ = ∅ and c := inf χ∈Γ max t∈[0,1] S(χ(t)) > 0.
In addition it is shown1 in [START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF][START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF] that

m = 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 ∈ H 1 (R N ) satisfies I(v) = 0 then t → S(tv) achieves its unique maximum on [0, +∞) at t = 1. Also [START_REF] Jeanjean | Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments[END_REF] shows that lim t→+∞ S(tv) = -∞. From the definition of c, it implies that c S(v) for all v ∈ H 1 (R N ) such that I(v) = 0. Hence we have

d(ω) c,
and combined with the fact that m = c it ends the proof. Now we investigate the behavior of the functionals under some rescaling Lemma 3. Assume that (A 0 ) and

(A 1 ) hold. For λ > 0 and v ∈ H 1 (R N ), we define v λ ( • ) := λ N 2 v(λ • ). We suppose Q(v) 0. Then there exists λ 0 1 such that (i) Q(v λ 0 ) = 0, (ii) λ 0 = 1 if and only if Q(v) = 0, (iii) ∂ ∂λ S(v λ ) > 0 for λ ∈ (0, λ 0 ) and ∂ ∂λ S(v λ ) < 0 for λ ∈ (λ 0 , +∞), (iv) λ → S(v λ ) is concave on (λ 0 , +∞), (v) ∂ ∂λ S(v λ ) = 1 λ Q(v λ ). Proof of Lemma 3. Easy computations lead to ∂ ∂λ S(v λ ) = 1 λ Q(v λ ) = λ ∇v 2 2 - N 2 R N λ -(N +2) λ N 2 g(λ N 2 |v|)|v| -2G(λ N 2 v) dx ,
and recalling from (A 1 ) that the function h(s) := (sg(s) -2G(s))s -(2+4/N ) is strictly increasing on [0, +∞), (i),(ii),(iii) and (v) follow easily. To see (iv), we remark that since

∇v 2 2 - N 2 R N λ -(N +2) λ N 2 g(λ N 2 |v|)|v| -2G(λ N 2 v) dx < 0
on (λ 0 , +∞), we infer from (A 1 ) that ∂ ∂λ S(v λ ) is strictly decreasing on (λ 0 , +∞), which implies (iv).

Proof of Theorem 1. We recall that

M = {v ∈ H 1 (R N ) \ {0}; Q(v) = 0, I(v) 0},
and define

d M := inf{S(v); v ∈ M }.
We proceed in three steps.

Step thus we suppose I(v) < 0. We use the rescaling defined in Lemma 3 : for λ > 0 we have

I(v λ ) = λ 2 ∇v 2 2 + ω v 2 2 - R N λ -N/2 g(λ N/2 |v|)|v|dx.
It follows from (A 0 )-(b) that lim λ→0 I(v λ ) = ω v 2 2 and thus by continuity there exists λ 1 < 1 such that I(v λ 1 ) = 0. Thus S(v λ 1 ) d(ω). Now, from Q(v) = 0 and (iii) in Lemma 3 we have

S(v) S(v λ 1 ) d(ω), hence d M = d(ω).
Step 2. For λ > 0, we set u λ := ϕ λ . For λ > 1 close to 1, we have S(u λ ) < S(ϕ) and Q(u λ ) < 0, ( 12)

I(u λ ) < 0. ( 13
)
Indeed, [START_REF] Le Coz | Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential[END_REF] follows from (iii) and (v) in Lemma 3. For (13), we write

I(u λ ) = 2S(u λ ) + 2 N Q(u λ ) - 2 N ∇u λ 2 2 2S(ϕ) + 2 N Q(ϕ) -I(ϕ) - 2λ 2 N ∇ϕ 2 2 2(1 -λ 2 ) N ∇ϕ 2 2 < 0.
Let u(t) be the solution of (3) with u(0) = u λ . We claim that the properties described in [START_REF] Le Coz | Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential[END_REF], [START_REF] Liu | Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation[END_REF] are invariant under the flow of (3). Indeed, since from [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] we have for all t > 0 (14) S(u(t)) = S(u λ ) < S(ϕ), we infer that I(u(t)) = 0 for any t 0, and by continuity we have I(u(t)) < 0 for all t 0. It follows that Q(u(t)) = 0 for any t 0 (if not u(t) ∈ M and thus S(u(t)) S(ϕ) which contradicts ( 14)), and by continuity we have Q(u(t)) < 0 for all t 0. Thus for all t > 0 we have S(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 v β (x) := β N 2 v(βx). From Step 2 we have Q(v) < 0, thus from Lemma 3 there exists β 0 < 1 such that Q(v β 0 ) = 0. If I(v β 0 ) 0, we keep β 0 , otherwise we replace it by β0 ∈ (β 0 , 1) such that I(v β0 ) = 0. Thus in any case we have [START_REF] Liu | Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity[END_REF] S(v β 0 ) d(ω)

and Q(v β 0 ) 0. Now from (iv) in Lemma 3, we have

S(v) -S(v β 0 ) (1 -β 0 ) ∂ ∂β S(v β ) |β=1 .
Thus, from (v) in Lemma 3, Q(v) < 0 and β 0 < 1, we get

S(v) -S(v β 0 ) Q(v).
Combined with [START_REF] Liu | Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity[END_REF], this gives ( 16)

Q(v) S(v) -d(ω) := -δ < 0
where 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 -δt 2 + Ct + xu λ 2 2 , and since the right hand side of (17) becomes negative when t grows up, we easily deduce that T u λ < +∞ and lim t→T u λ ∇u(t) 2 = +∞.

For

  z ∈ C let G(z) := |z| 0 g(s)ds. We assume (A 0 ) The function g satisfies (a) g ∈ C(R, R). (b) lim s→0 g(s) s = 0. (c) when N 3, lim s→+∞ g(s)s -N+2 N-2 = 0; when N = 2, for any α > 0, there exists C α > 0 such that |g(s)| C α e αs 2 for all s > 0. (A 1 ) The function h(s) := (sg(s) -2G(s))s -(2+4/N ) is strictly increasing on (0, +∞) and lim s→0 h(s) = 0. (A 2 ) There exist C > 0 and α

1 .

 1 Let us prove d(ω) = d M . Since the ground states ϕ satisfy Q(ϕ) = I(ϕ) = 0, we have ϕ ∈ M . Combined with S(ϕ) = d(ω), this implies d M d(ω). Conversely, let v ∈ M . If I(v) = 0, then trivially S(v) d(ω),

In fact, the results of[START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF][START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF] are proved only for real valued functions; however, it is not hard to see that they can be extended to the complex case (see[START_REF] Jeanjean | Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments[END_REF] Lemma 14]).

Acknowledgements. The author is grateful to Louis Jeanjean for helpful advice during the writing of this paper. He also wishes to thank Thierry Cazenave for sharing with him a digitalized version of the unpublished paper [2].