Stefan Le Coz

Standing waves in nonlinear Schrödinger equations

Keywords: Nonlinear Schrödinger equations, standing waves, orbital stability, instability, blow-up, variational methods. AMS classification. 35Q55, 35Q51, 35B35, 35A15

In the theory of nonlinear Schrödinger equations, it is expected that the solutions will either spread out because of the dispersive effect of the linear part of the equation or concentrate at one or several points because of nonlinear effects. In some remarkable cases, these behaviors counterbalance and special solutions that neither disperse nor focus appear, the so-called standing waves. For the physical applications as well as for the mathematical properties of the equation, a fundamental issue is the stability of waves with respect to perturbations. Our purpose in these notes is to present various methods developed to study the existence and stability of standing waves. We prove the existence of standing waves by using a variational approach. When stability holds, it is obtained by proving a coercivity property for a linearized operator. Another approach based on variational and compactness arguments is also presented. When instability holds, we show by a method combining a Virial identity and variational arguments that the standing waves are unstable by blow-up.

Introduction

In these notes, we consider the following nonlinear Schrödinger equation i∂ t u + ∆u + |u| p-1 u = 0.

(1.1)

Equation (1.1) arises in various physical and biological contexts, for example in nonlinear optics, for Bose-Einstein condensates, in the modelling of the DNA structure, etc. We refer the reader to [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation[END_REF] for more details on the physical background and references.

Here, the unknown u is a complex valued function of t ∈ R and x ∈ R N for N 1

u : R × R N → C.
In most of the applications, t stands for the time and x for the space variable, but sometimes it can be the converse, for example in nonlinear optics. The number i is the imaginary unit (i 2 = -1), ∂ t is the first derivative with respect to the time t, ∆ denotes the Laplacian with respect to the space variable x (∆ = N j=1 ∂ 2 ∂x 2 j ). Finally, p ∈ R is such that p > 1, which means that the equation is superlinear.

To simplify the exposition, we have restricted ourselves to the study of nonlinear Schrödinger equations with a power-type nonlinearity, but one can also consider more general versions of these equations, see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation[END_REF] and the references cited therein.

The purpose of theses notes is to study the properties of particular solutions of (1.1) of the form e iωt ϕ(x) with ω ∈ R and ϕ satisfying -∆ϕ + ωϕ -|ϕ| p-1 ϕ = 0.

(1.2)

Such solutions are called standing waves. They are part of a more general class of solutions arising for various nonlinear equations like Korteweg-de Vries, Klein-Gordon or Kadomtsev-Petviashvili equations. These equations enjoy special solutions whose profile remains unchanged under the evolution in time. These special solutions are called solitary waves or solitons (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Dauxois | Physics of Solitons[END_REF][START_REF] Drazin | Solitons: an introduction[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation[END_REF] for an overview of physical and mathematical questions around solitary waves). This kind of phenomena was discovered in 1834 by John Scott Russel. The Scottish engineer was supervising work in a canal near Edinburgh when he observed that the brutal stop of a boat in the canal was creating a wave that does not seem to vanish. Indeed, he was able to follow this wave horseback on a distance of several miles. His life long, he studied this surprising phenomenon but without being able to give it a theoretical justification. In fact, most of the scientists of that time did believe that such a wave, which does not disperse, could not exist. The first theoretical justification of the existence of solitary waves was given by Korteweg and de Vries in 1895. They derived an equation for the motion of water admitting solitary wave solutions. But one had to wait until the 1950's to see the beginning of an intensive research from mathematicians and physicists about solitary waves.

Heuristically, such solutions appear because of the balance of two contradictory effects: the dispersive effect of the linear part, which tends to flatten the solution as time goes on (see Figure 1), and the focusing effect of the nonlinearity, which tends to concentrate the solution, provided the initial datum is large enough (see Figure 2). For some special initial data, each effect compensates the other and the general profile of the initial data remains unchanged (see Figure 3).

In the study of standing waves, two types of questions arise naturally. is their decay at infinity, can we find variational characterizations for at least some of these solutions?

(ii) If standing waves exist, are they stable (in a sense to be made precise) as solutions of (1.1)? If they are unstable, what is the nature of instability?

The first type of questions is related to the study of existence of solutions for semilinear elliptic problems. The methods used in this context are often of variational nature and solutions are obtained by minimization under constraint or with min-max arguments. The second type of questions concerns the dynamics of the evolution equation. Nevertheless, both problems are intimately related. Indeed, informations of variational nature on the solutions of the stationary equation are essential to derive stability or instability results. The rest of these notes is divided into four sections and an appendix. We first give basic results on the Cauchy problem for (1.1) in Section 2. In Section 3, we study the existence of standing waves. Section 4 is devoted to stability whereas Section 5 deals with instability. The proof of a stability criterion is given in the appendix.

Notations. The space of complex measurable functions whose r-th power is integrable will be denoted by L r (R N ) and its standard norm by • L r (R N ) . When r = 2, the space L 2 (R N ) will be endowed with the real inner product

(u, v) 2 = Re R N u vdx for u, v ∈ L 2 (R N ).
The space of functions from L r (R N ) whose distribution derivatives of order less than or equal to m are elements of L r (R N ) will be denoted by W m,r (R N ). If m = 1 and r = 2, we shall denote W 1,2 (R N ) by H 1 (R N ), its usual norm by • H 1 (R N ) and the duality product between the dual space H -1 (R N ) and H 1 (R N ) by •, • . If m = r = 2, we denote W 2,2 (R N ) by H 2 (R N ).

For notational convenience, we shall sometimes identify a function and its value at some point. For example, we shall write e iωt ϕ(x) for the function (t, x) → e iωt ϕ(x). Similarly, we shall say that |x|v ∈ L 2 (R N ) if the function x → |x|v(x) belongs to L 2 (R N ). For a solution u of (1.1) we shall denote by u(t) the function x → u(t, x). Therefore, depending on the context, u may denote either a map from R to some functional space or a map from R × R N to C.

We make the convention that when we take a subsequence of a sequence (v n ) we denote it again by (v n ).

The letter C will denote various positive constants whose exact values may change from line to line but are not essential in the course of the analysis.

The Cauchy problem

For the physical properties of the model as well as for the mathematical study of the equation, it is interesting to look for quantities conserved along the time. At least formally, equation (1.1) admits three conserved quantities. The first one is the mass or charge:

If u satisfies (1.1) with initial datum u(0) = u 0 then Q(u(t)) := 1 2 u(t) 2 L 2 (R N ) = Q(u 0 ). (2.1)
The conservation of mass is obtained from multiplying (1.1) with ū, integrating over R N and taking the imaginary part. The second conserved quantity is the energy (multiply (1.1) by ∂ t ū, integrate over R N and take the real part),

E(u(t)) := 1 2 ∇u(t) 2 L 2 (R N ) - 1 p + 1 u(t) p+1 L p+1 (R N ) = E(u 0 ). (2.2)
Finally, multiplying (1.1) by ∇ ū, integrating over R N and taking the real part, we obtain the conservation of momentum,

M (u(t)) := Im R N u(t)∇ ū(t)dx = M (u 0 ).
Among these three conserved quantities, the mass and the energy are real-valued, but the momentum may be complex-valued, which makes it less convenient to use.

To benefit from the conserved quantities, it is natural to search solutions of (1.1) in function spaces where these quantities are well-defined. Consequently, we look for solutions of (1.1) in H 1 (R N ) and restrict the range of p to be subcritical for the Sobolev embedding of

H 1 (R N ) into L p+1 (R N ), i.e. 1 < p < 1 + 4 N -2 if N 3 and 1 < p < +∞ if N = 1, 2.
Then we have the following result concerning the wellposedness of the Cauchy problem for (1.1) (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and the references cited therein).

Proposition 2.1. Let 1 < p < 1 + 4 N -2 if N 3 and 1 < p < +∞ if N = 1, 2. For every u 0 ∈ H 1 (R N ) there exists a unique maximal solution u of (1.1), T min ∈ [-∞, 0), T max ∈ (0, +∞] such that u(0) = u 0 and u ∈ C((T min , T max ); H 1 (R N )) ∩ C 1 ((T min , T max ); H -1 (R N )).
Furthermore, we have the conservation of charge, energy and momentum: For all t ∈ (T min , T max ),

Q(u(t)) = Q(u 0 ), E(u(t)) = E(u 0 ), M (u(t)) = M (u 0 ). (2.3)
Finally, we have the blow-up alternative:

• If T min > -∞, then lim t↓T min ∇u(t) 2 L 2 (R N ) = +∞. • If T max < +∞, then lim t↑T max ∇u(t) 2 L 2 (R N ) = +∞.
From now on, it will be understood that 1

< p < 1+ 4 N -2 if N 3 and 1 < p < +∞ if N = 1, 2.
In view of Proposition 2.1, it is natural to ask under which circumstances global existence holds or blow-up occurs. The following proposition gives an answer to the question of global existence.

Proposition 2.2. If 1 < p < 1 + 4 N , then (1.1
) is globally well-posed, i.e. for any solution u given by Proposition 2.1, T min = -∞ and T max = +∞.

The proof of Proposition 2.2 relies on the Gagliardo-Nirenberg inequality (see [START_REF] Adams | Sobolev spaces[END_REF]): There exists a constant C > 0 such that for all v ∈ H 1 (R N )

v p+1 L p+1 (R N ) C ∇v N (p-1) 2 L 2 (R N ) v p+1- N (p-1) 2 L 2 (R N )
.

(2.4)

Proof of Proposition 2.2. Let 1 < p < 1 + 4 N and u be a solution of (1.1) as in Proposition 2.1. We prove the assertion by contradiction. Assume that T max < +∞ and therefore lim t↑T max ∇u(t) 2 L 2 (R N ) = +∞. By (2.4) we have

E(u(t)) ∇u(t) 2 L 2 (R N ) 1 2 -C ∇u(t) N (p-1) 2 -2 L 2 (R N ) u(t) p+1- N (p-1) 2 L 2 (R N )
.

In view of the conservation of charge and energy (see (2.3)), this implies

∇u(t) 2 L 2 (R N ) 1 -∇u(t) N (p-1) 2 -2 L 2 (R N )
< C for all t ∈ (T min , T max ).

(2.5)

Now, since p < 1 + 4 N , we have N (p-1)

2

-2 < 0 and thus letting ∇u(t) 2 L 2 (R N ) go to +∞ when t goes to T max leads to a contradiction in (2.5). Arguing in the same way if T min > -∞ leads to the same contradiction and completes the proof.

Concerning blow-up, we can give the following sufficient condition. Proposition 2.3. Assume that p 1 + 4

N and let u 0 ∈ H 1 (R N ) be such that

|x|u 0 ∈ L 2 (R N ) and E(u 0 ) < 0.
Then the solution u of (1.1) corresponding to u 0 blows up in finite time for positive and negative time, that is T min > -∞ and T max < +∞.

The proof of Proposition 2.3 relies on the Virial Theorem (the term "Virial Theorem" comes from the analogy to the Virial Theorem in classical mechanics).

Proposition 2.4 (Virial Theorem). Let u 0 ∈ H 1 (R N ) be such that |x|u 0 ∈ L 2 (R N )
and let u be the solution of (1.1) corresponding to u 0 . Then |x|u(t) ∈ L 2 (R N ) for all t ∈ (T min , T max ) and the function

f : t → xu(t) 2 L 2 (R N ) is of class C 2 and satisfies f ′ (t) = 4Im R N ū(t)x • ∇u(t)dx, f ′′ (t) = 8P (u(t)),
where P is given for v ∈ H 1 (R N ) by

P (v) := ∇v 2 L 2 (R N ) - N (p -1) 2(p + 1) v p+1 L p+1 (R N ) . (2.6)
The Virial Theorem comes from the work of Glassey [START_REF] Glassey | On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations[END_REF] in which the identities for f ′ and f ′′ were formally derived. For a rigorous proof, see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF].

Proof of Proposition 2.3. First, we remark that

P (u(t)) = 2E(u(t)) + 4 -N (p -1) 2(p + 1) u(t) p+1 L p+1 (R N ) .
Since p 1 + 4 N , we have in view of the conservation of energy (see (2.3)) P (u(t)) 2E(u(t)) = 2E(u 0 ) < 0 for all t ∈ (T min , T max ).

Therefore, by Proposition 2.4, we have

d 2 dt 2 xu(t) 2 L 2 (R N ) 16E(u 0 ) for all t ∈ (T min , T max ).

Integrating twice in time gives

xu(t) 2 L 2 (R N ) 8E(u 0 )t 2 + 4Im R N ū0 x • ∇u 0 dx t + xu 0 2 L 2 (R N ) .
(2.7)

The right member of (2.7) is a polynomial of order two with the main coefficient negative. Hence for |t| large the right-hand side of (2.7) becomes negative, which is in contradiction with xu(t) 2

L 2 (R N )
0. This implies T min > -∞ and T max < +∞ and finishes the proof.

Existence, uniqueness and properties of solitons

We will use the following definition of standing waves for (1.1). Definition 3.1. A standing wave or soliton of (1.1) is a solution of the form e iωt ϕ(x) with ω ∈ R and ϕ satisfying

-∆ϕ + ωϕ -|ϕ| p-1 ϕ = 0, ϕ ∈ H 1 (R N ) \ {0}.
(3.1)

Many techniques have been developed to study the existence of solutions to problems of type (3.1) (see e.g. [START_REF]Nonlinear analysis and semilinear elliptic problems[END_REF]). In this section, we look for solutions of (3.1) by using variational methods (see e.g. [START_REF] Struwe | Variational methods[END_REF] for a general overview).

For the study of solutions of (3.1), we define a functional S :

H 1 (R N ) → R by setting for v ∈ H 1 (R N ) S(v) := 1 2 ∇v 2 L 2 (R N ) + ω 2 v 2 L 2 (R N ) - 1 p + 1 v p+1 L p+1 (R N ) .
The functional S is often called action. It is standard that S is of class C 2 (see, for example, [START_REF] Willem | Minimax theorems[END_REF]) and for v ∈ H 1 (R N ) the Fréchet derivative of S at v is given by

S ′ (v) = -∆v + ωv -|v| p-1 v.
Therefore, ϕ is a solution of (3.1) if and only if ϕ ∈ H 1 (R N ) \ {0} and S ′ (ϕ) = 0. In other words, the nontrivial critical points of S are the solutions of (3.1). Therefore, to prove existence of solutions of (3.1) it is enough to find a nontrivial critical point of S. This section is divided as follows. First, we prove that if solutions to (3.1) exist then they are regular, exponentially decaying at infinity and satisfy some functional identities. Next, we prove the existence of a nontrivial critical point of S. Finally, we derive various variational characterizations for some special solutions of (3.1).

Preliminaries

Before studying existence of solutions to (3.1), it is convenient to prove that, if such solutions exist, they necessarily enjoy the following properties.

Proposition 3.2. Let ω > 0. If ϕ ∈ H 1 (R N ) satisfies (3.1)
, then ϕ is regular and exponentially decaying. More precisely

(i) ϕ ∈ W 3,r (R N ) for all r ∈ [2, +∞), in particular ϕ ∈ C 2 (R N ); (ii) there exists ε > 0 such that e ε|x| (|ϕ| + |∇ϕ|) ∈ L ∞ (R N ).
Sketch of proof. Point (i) follows from the usual elliptic regularity theory by a bootstrap argument (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]). We just indicate how to initiate the bootstrap and refer to [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] for a detailed proof. Let ϕ be a solution of (3.1). Suppose that ϕ ∈ L q (R N ) for some q > p. Since |ϕ| p-1 ϕ ∈ L q p (R N ) and ϕ satisfies

-∆ϕ + ωϕ = |ϕ| p-1 ϕ, (3.2)
it follows that ϕ ∈ W 2, q p (R N ). Choosing any q ∈ (p, +∞) if N = 1, 2 and q = 2N N -2

if N 3 and repeating the previous argument recursively, we obtain after a finite number of steps that ϕ ∈ W 2,r (R N ) for any r ∈ [2, +∞). This implies that |ϕ| p-1 ϕ ∈ W 1,r (R N ) for any r ∈ [2, +∞) and it follows from (3.2) that ϕ ∈ W 3,r (R N ) for all r ∈ [2, +∞), hence (i).

If ϕ is radial, i.e. ϕ(x) = ϕ(|x|), then ϕ satisfies

-ϕ ′′ - N -1 r ϕ ′ + ωϕ -|ϕ| p-1 ϕ = 0.
Here, we have employed that ∆ϕ = ϕ ′′ + N -1 r ϕ ′ if ϕ is radial. In this case, the exponential decay at infinity of ϕ follows from the classical theory of second order ordinary differential equations (see, for example, [START_REF] Hartman | Ordinary differential equations[END_REF]). See [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] for a proof of (ii) for non-radial solutions of (3.1).

Lemma 3.3. If ϕ ∈ H 1 (R N ) satisfies (3.1)
, then the following identities hold:

∇ϕ 2 L 2 (R N ) + ω ϕ 2 L 2 (R N ) -ϕ p+1 L p+1 (R N ) = 0, (3.3) ∇ϕ 2 L 2 (R N ) - N (p -1) 2(p + 1) ϕ p+1 L p+1 (R N ) = 0. (3.4) {v ∈ H 1 (R N ); v = 0, ∇v 2 L 2 (R N ) + ω v 2 L 2 (R N ) -v p+1 L p+1 (R N ) = 0} (3.5)
is called the Nehari manifold.

Proof of Lemma 3.3. Let ϕ ∈ H 1 (R N ) be a solution of (3.1).

To obtain (3.3), we simply multiply (3.1) by φ and integrate over R N .

For the proof of (3.4), recall first that we have defined in the beginning of this section a C 2 -functional S :

H 1 (R N ) → R by setting for v ∈ H 1 (R N ) S(v) := 1 2 ∇v 2 L 2 (R N ) + ω 2 v 2 L 2 (R N ) - 1 p + 1 v p+1 L p+1 (R N )
and that if ϕ is a solution of (3.1) then S ′ (ϕ) = 0.

For λ > 0, let ϕ λ (•) := λ N/2 ϕ(λ•). It follows from straightforward calculations that ∇ϕ λ 2 L 2 (R N ) = λ 2 ∇ϕ 2 L 2 (R N ) , ϕ λ 2 L 2 (R N ) = ϕ 2 L 2 (R N ) , and ϕ λ p+1 L p+1 (R N ) = λ N (p-1) 2 ϕ p+1 L p+1 (R N ) .
On the one hand, we have

S(ϕ λ ) = λ 2 2 ∇ϕ 2 L 2 (R N ) + ω 2 ϕ 2 L 2 (R N ) - λ N (p-1) 2 p + 1 ϕ p+1 L p+1 (R N ) and ∂ ∂λ S(ϕ λ ) λ=1 = ∇ϕ 2 L 2 (R N ) - N (p -1) 2(p + 1) ϕ p+1 L p+1 (R N ) . (3.6)
On the other hand, we have

∂ ∂λ S(ϕ λ ) λ=1 = S ′ (ϕ), ∂ϕ λ ∂λ λ=1 (3.7)
and, since ϕ is solution of (3.1), S ′ (ϕ) = 0, which implies

∂ ∂λ S(ϕ λ ) λ=1 = 0. (3.8)
Combining (3.6) and (3.8) proves (3.4). Note that the right-hand side of (3.7) is well-defined for ω > 0 since ∂ϕ λ ∂λ λ=1

= 1 2 ϕ + x • ∇ϕ is in H 1 (R N
) because of the regularity and exponential decay of ϕ stated in Proposition 3.2. If ω 0, the previous calculations are only formal. However, it is possible to give a rigorous proof of (3.4) also for ω 0, see [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF].

From Lemma 3.3 it is easy to derive a necessary condition for the existence of solutions to (3.1). Proof. We prove the assertion by contradiction. Let ω 0 and suppose that (3.1) has a solution ϕ ∈ H 1 (R N ). By (3.3) and (3.4), ϕ satisfies

(N -2)p -(N + 2) 2(p + 1) ϕ p+1 L p+1 (R N ) = -ω ϕ 2 L 2 (R N ) .
(3.9)

Since p < 1 + 4 N -2 if N 3 and p < +∞ if N = 1, 2
, the left-hand side of (3.9) is negative for any N , whereas the right-hand side is non negative by assumption, which is a contradiction.

In the rest of these notes, it will be understood that ω > 0.

Existence

Our main result in this section is the following. Theorem 3.5. There exists a nontrivial critical point ϕ ω of S, that is a solution of (3.1).

Several techniques are available to prove the existence of a nontrivial critical point of S. The functional S is clearly unbounded from above and below, and this prevents to find a critical point simply by global minimization or maximization. To overcome this difficulty, other techniques based on minimization under constraint were developed (see e.g. [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][START_REF] Coleman | Action minima among solutions to a class of Euclidean scalar field equations[END_REF][START_REF] Pohožaev | On the eigenfunctions of the equation ∆u + λf (u) = 0[END_REF][START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF] and the references cited therein). In what follows, we present a different approach based on the Mountain Pass Theorem of Ambrosetti and Rabinowitz [START_REF] Ambrosetti | Dual variational methods in critical point theory and applications[END_REF] and somehow inspired from [START_REF] Jeanjean | On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on R N[END_REF]. This approach was suggested to us by Louis Jeanjean. We first prove that the functional S has a Mountain Pass geometry. Thus there exists a Palais-Smale sequence at this level. It clearly converges to a critical point of S weakly in H 1 (R N ). Proving that this sequence is non-vanishing and taking advantage of the translation invariance of (3.1) we get the existence of a nontrivial critical point.

We define the Mountain Pass level c by setting

c := inf γ∈Γ max s∈[0,1] S(γ(s)) (3.10)
where Γ is the set of admissible paths: Proof.

Γ := {γ ∈ C([0, 1]; H 1 (R N )); γ(0) = 0, S(γ(1)) < 0}. ( 3
Let v ∈ H 1 (R N ) \ {0}.
Then, for any s > 0,

S(sv) = s 2 2 ( ∇v 2 L 2 (R N ) + ω v 2 L 2 (R N ) ) - s p+1 p + 1 v p+1 L p+1 (R N ) ,
and it is clear that if s is large enough then S(sv) < 0. Let C > 0 be such that S(Cv) < 0 and γ :

[0, 1] → H 1 (R N ) be defined by γ(s) := Csv. Then γ ∈ C([0, 1]; H 1 (R N )), γ (0)
= 0 and S(γ(1)) < 0 ; thus γ ∈ Γ and Γ is nonempty. Now, we clearly have

S(v) min{1, ω} 2 v 2 H 1 (R N ) - 1 p + 1 v p+1 L p+1 (R N )
and by the continuous embedding of

H 1 (R N ) into L p+1 (R N ) we find S(v) min{1, ω} 2 v 2 H 1 (R N ) - C p + 1 v p+1 H 1 (R N ) .
Let ε > 0 be small enough to have

δ := min{1, ω}ε 2 2 - Cε p+1 p + 1 > 0. Then for any v ∈ H 1 (R N ) with v H 1 (R N ) < ε, we have S(v) > 0. This implies that for any γ ∈ Γ, we have γ(1) H 1 (R N ) > ε,
and by continuity of γ there exists

s γ ∈ [0, 1] such that γ(s γ ) H 1 (R N ) = ε. Therefore, max s∈[0,1] S(γ(s)) S(γ(s γ )) δ > 0.
This implies for the Mountain Pass level c defined in (3.10) that

c δ > 0,
and thus S has a Mountain Pass geometry.

Combined with Ekeland's variational principle (see e.g. [START_REF] Willem | Minimax theorems[END_REF]), Lemma 3.6 immediately implies the existence of a Palais-Smale sequence at the Mountain Pass level. More precisely: Corollary 3.7. There exists a Palais-Smale sequence (u n ) ⊂ H 1 (R N ) for S at the Mountain Pass level c, i.e. (u n ) satisfies, as n → +∞,

S(u n ) → c and S ′ (u n ) → 0. (3.12)
To find a critical point for S, we proceed now in two steps: First, we prove that the sequence (u n ) is, following the terminology of the concentration-compactness theory of Lions, non-vanishing; Second, we prove that, up to translations, (u n ) converges weakly in H 1 (R N ) to a nontrivial critical point of S.

Lemma 3.8. The Palais-Smale sequence (u n ) is non-vanishing: There exist ε > 0, R > 0 and a sequence (y n ) ∈ R N such that for all n ∈ N we have

B R (yn) |u n | 2 dx > ε, (3.13
)

where B R (y) := {z ∈ R N : |y -z| < R}.
To prove Lemma 3.8, we will use the following lemma, which is a kind of Sobolev embedding result (see [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF]). Lemma 3.9. Let R > 0. Then there exists α > 0 and C > 0 such that for any

v ∈ H 1 (R N ) we have v p+1 L p+1 (R N ) C sup y∈R N B R (y) |v| 2 dx α v 2 H 1 (R N ) . Proof. Let (Q k ) k∈N be a sequence of open cubes of R N of same volume such that Q k ∩ Q l = ∅ if k = l, ∪ k∈N Q k = R N and for each k there exists y k with Q k ⊂ B R (y k )
. By Hölder's inequality and the embedding of

H 1 (Q k ) into L p+1 (Q k ), there exists C > 0 independent of k such that for any v ∈ H 1 (R N ) we have Q k |v| p+1 dx C Q k |v| 2 dx α Q k |∇v| 2 + |v| 2 dx. with α = N p+1 -N -2 2 if N 3 and α = 1 if N = 1, 2. Summing over k ∈ N, this implies that v p+1 L p+1 (R N ) C sup k∈N Q k |v| 2 dx α v 2 H 1 (R N ) .
Now, since for any k there exists

y k with Q k ⊂ B R (y k ) the conclusion follows.
Proof of Lemma 3.8. We prove the result by contradiction. Assume that the Palais-Smale sequence (u n ) is vanishing, that is for all R > 0 we have

lim n→+∞ sup y∈R N B R (y) |u n | 2 dx = 0. (3.14)
Let ε > 0. For n large enough, we have sup y∈R N B R (y) |u n | 2 dx < ε thanks to (3.14). Then Lemma 3.9 implies that, still for n large enough,

S(u n ) 1 2 ∇u n 2 L 2 (R N ) + ω 2 u n 2 L 2 (R N ) -ε u n 2 H 1 (R N ) ,
and therefore

S(u n ) min{1, ω} 2 -ε u n 2 H 1 (R N ) .
Consequently, taking ε < min{1,ω} 2 , we infer that

(u n ) is bounded in H 1 (R N ), (3.15)
which allows to get

S ′ (u n ), u n → 0 as n → +∞. (3.16)
Combining (3.15) with (3.14) and Lemma 3.9 implies

lim n→+∞ u n p+1 L p+1 (R N ) = 0.
Consequently, we have

S(u n ) - 1 2 S ′ (u n ), u n = - p -1 2(p + 1) u n p+1 L p+1 (R N ) → 0 as n → +∞. (3.17)
On the other hand, we infer from (3.12) and (3.16) that

S(u n ) - 1 2 S ′ (u n ), u n → c as n → +∞,
which is a contradiction with (3.17) since c > 0. Hence, (u n ) is non-vanishing.

Proof of Theorem 3.5.

Let v n := u n (• + y n )
, where (u n ) is given by Corollary 3.7 and (y n ) by Lemma 3.8. Since the functional S is invariant under translations in space, (v n ) is still a Palais-Smale sequence for S:

S(v n ) → c and S ′ (v n ) → 0. (3.18) From (3.15) we infer that (v n ) is bounded in H 1 (R N ). Thus there exists v ∈ H 1 (R N )
such that, possibly for a subsequence only, v n ⇀ v weakly in H 1 (R N ). From (3.18), we infer that S ′ (v) = 0, i.e. v is a critical point for S. To show that v is nontrivial, we remark that, since the embedding

H 1 (B R (0)) ֒→ L 2 (B R (0)) is compact, (3.13) implies v ≡ 0. Setting ϕ ω := v finishes the proof.
The following corollary will be useful in the next subsection.

Corollary 3.10. The critical point ϕ ω is below the level c, i.e. S(ϕ ω ) c.

Proof. First, since S ′ (ϕ ω ) = 0, we have

S(ϕ ω ) = S(ϕ ω ) - 1 p + 1 S ′ (ϕ ω ), ϕ ω = p -1 2(p + 1) ∇ϕ ω 2 L 2 (R N ) + ω ϕ ω 2 L 2 (R N ) .
By virtue of weak convergence of

(v n ) towards ϕ ω in H 1 (R N ), this gives S(ϕ ω ) p -1 2(p + 1) lim inf n→+∞ ∇v n 2 L 2 (R N ) + ω v n 2 L 2 (R N ) . (3.19)
As in (3.16), we have

S ′ (v n ), v n → 0 as n → +∞
and combining with (3.18) and (3.19), we obtain

S(ϕ ω ) lim inf n→+∞ S(v n ) - 1 p + 1 S ′ (v n ), v n = c,
which completes the proof.

Variational characterizations

Among the solutions of (3.1), some are of particular interest.

Definition 3.11. A solution ϕ of (3.1) is said to be a ground state or least energy solution if S(ϕ) S(v) for any solution v of (3.1). We define the least energy level m by m := inf{S(v); v is a solution of (3.1)}.

(3.20)

The set of all least energy solutions is denoted by G:

G := {v ∈ H 1 (R N ); v is a solution of (3.1) and S(v) = m}. (3.21)
Ground states play an important role in the theory of nonlinear Schrödinger equations. In particular, in the critical case p = 1+ 4 N , they appear in an essential way in the derivation of global existence results (see [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF]) and in the description of the blow-up phenomenon (see [START_REF] Merle | On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation[END_REF][START_REF]The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF][START_REF]On one blow up point solutions to the critical nonlinear Schrödinger equation[END_REF][START_REF]On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation[END_REF] and the references cited therein).

Proposition 3.12. The solution ϕ ω of (3.1) found in Theorem 3.5 is a ground state and it is at the Mountain Pass level c (defined in (3.10)):

In addition, ϕ ω is a minimizer of S on the Nehari manifold (see (3.5)), i.e. it solves the following minimization problem

d := min{S(v); v ∈ H 1 (R N ) \ {0}, I(v) = 0} (3.22)
where

I(v) := ∇v 2 L 2 (R N ) + ω v 2 L 2 (R N ) -v p+1 L p+1 (R N ) .
The various variational characterizations of the critical point ϕ ω of S as a ground state or as a minimizer of S on the Nehari manifold will be useful when we will deal with the stability or instability issues.

Before proving Proposition 3.12, we need some preparation.

Lemma 3.13. The following inequality holds:

c d, (3.23)
where c is the Mountain Pass level defined in (3.10) and d is given by (3.22).

Proof.

Let v ∈ H 1 (R N ) \ {0} be such that I(v) = 0.
Following [START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF][START_REF]A remark on least energy solutions in R N[END_REF][START_REF] Rabinowitz | On a class of nonlinear Schrödinger equations[END_REF], the idea is to construct a path γ in Γ (recall that Γ was defined in (3.11)) such that S reaches its maximum on γ at v. From the proof of Lemma 3.6, we know that for C large enough the path γ :

[0, 1] → H 1 (R N ) defined by γ(s) := Csv belongs to Γ. It is easy to see that ∂ ∂s S(sv) = s( ∇v 2 L 2 (R N ) + ω v 2 L 2 (R N ) -s p-1 v p+1 L p+1 (R N ) )
. Therefore, we have at s = 1

∂ ∂s S(sv) s=1 = I(v) = 0,
and consequently,

∂ ∂s S(sv) > 0 if s ∈ (0, 1), ∂ ∂s S(sv) < 0 if s ∈ (1, +∞).
Thus S reaches its maximum on γ at v. This implies

c S(v) for all v ∈ H 1 (R N ) \ {0} such that I(v) = 0,
which finishes the proof.

Proof of Proposition 3.12. We first recall that, by Corollary 3.10, we have Remark 3.14. For general nonlinearities, the equality m = c between the Mountain Pass level and the least energy level always holds (see [START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF][START_REF]A remark on least energy solutions in R N[END_REF]).

S(ϕ ω ) c (3.

Uniqueness

It turns out that we are able to describe explicitly the set G of ground states (see (3.21)).

Theorem 3.15. There exists a real-valued, positive, spherically symmetric and decreasing function

Ψ ∈ H 1 (R N ) such that G = {e iθ Ψ(• -y); θ ∈ R, y ∈ R N }.
Therefore, the ground state is unique up to translations and phase shifts. It would exceed the scope of these notes to give a proof of Theorem 3.15 and we just indicate some references. First, a simple and general proof that any complex-valued ground state ϕ is of the form e iθ φ with θ ∈ R and φ a real positive ground state was recently given in [START_REF] Cingolani | Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions[END_REF] (see also [START_REF] Hajaiej | On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation[END_REF]). The fact that all real positive ground states of (3.1) are radial up to translations was first proved by Gidas, Ni and Nirenberg in 1979 (see [START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF]) by using the so-called moving planes method. Alternatively, the same result can be deduce by the method of Lopes [START_REF] Lopes | Radial symmetry of minimizers for some translation and rotation invariant functionals[END_REF] which relies on symmetrizations with respect to suitably chosen hyperplanes combined with a unique continuation theorem. Recently, Maris [START_REF] Mariş | On the symmetry of minimizers[END_REF] developed for the symmetry of minimizers for a large class of problems a new method that furnishes a simpler proof of radial symmetry of real ground states, see [START_REF] Byeon | Symmetry and monotonicity of least energy solutions[END_REF], without even assuming a priori that they are positive. Uniqueness follows from a result of Kwong [START_REF] Kwong | Uniqueness of positive solutions of ∆u -u + u p = 0 in R n[END_REF] in 1989. Remark 3.16. When N = 1, it turns out that the set of all solutions of (3.1) (not only those of least energy) is precisely the set of ground states

G = {e iθ Ψ(• -y); θ ∈ R, y ∈ R N } = {v; v is a solution of (3.1) }.
Moreover, we are able to give an explicit formula for Ψ:

Ψ(x) = (p + 1)ω 2 sech 2 (p -1) √ ω 2 x 1 p-1 .
For p = 3 and ω = 1, the shape of Ψ is given in Figure 4.

In higher dimensions, it is known that there exist solutions of (3.1) that are not ground states (see [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF][START_REF]Nonlinear scalar field equations II[END_REF][START_REF]Solutions complexes d'équations elliptiques semilinéaires dans R N[END_REF]) and no explicit solution is available.

Stability

In this section, we will prove that for ϕ ∈ G the standing wave e iωt ϕ(x) is stable (in a sense to be made precise) if 1 < p < 1 + 4 N . In his report of 1844 [START_REF] Russell | Report on waves[END_REF], Russel was already mentioning the observed remarkable stability properties of solitary waves. From the mathematical point of view, the concept of stability that comes naturally in mind for the standing waves e iωt ϕ(x) of (1.1) is stability in the sense of Lyapunov, which means uniformly continuous dependence of the initial data: For all ε > 0 there exists δ > 0 such that for all u 0 ∈ H 1 (R N ),

u 0 -ϕ H 1 (R N ) < δ =⇒ u(t) -e iωt ϕ H 1 (R N ) < ε for all t,
where u is the maximal solution of (1.1) with u(0) = u 0 . However, with this definition, all standing waves would be unstable (see Remark 4.2), which is contradictory with what is observed for such phenomena. Therefore, we have to look for a different notion of stability.

The main reason for standing waves being unstable in the sense of Lyapunov is that (1.1) admits translation and phase shift invariance: If u = u(t, x) is a solution of (1.1) then for all θ ∈ R and y ∈ R N , e iθ u(•, •y) is still a solution of (1.1). In some sense, the equation does not prescribe the behavior of the solutions in the "direction" of translations and phase shifts. To take this fact into account, we define the concept of orbital stability, which is Lyapunov stability up to translations and phase shifts. Definition 4.1. Let ϕ be a solution of (3.1). The standing wave e iωt ϕ(x) is said to be orbitally stable in H 1 (R N ) if for all ε > 0 there exists δ > 0 such that if

u 0 ∈ H 1 (R N ) satisfies u 0 -ϕ H 1 (R N ) < δ, then the maximal solution u(t) of (1.1) with u(0) = u 0 exists for all t ∈ R and sup t∈R inf θ∈R inf y∈R N u(t) -e iθ ϕ(• -y) H 1 (R N ) < ε.
Otherwise, the standing wave is said to be unstable.

Remark 4.2. The concept of orbital stability is optimal in the following sense (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF]).

• Space translations are necessary: For a solution ϕ of (3.1), ε > 0 and y ∈ R N with |y| = 1, let ϕ ε (x) := e iεx•y ϕ(x). Then it is easy to check that the solution of (1.1) with initial datum ϕ ε is u ε (t, x) = e iε(x•y-εt) e iωt ϕ(x -2εty).

We clearly have

ϕ ε → ϕ in H 1 (R N ) as ε → 0, but for any ε > 0 sup t∈R inf θ∈R u ε (t) -e iθ ϕ H 1 (R N ) = 2 ϕ H 1 (R N ) .
Conversely, note that if we consider (1.1) in the subspace of H 1 (R N ) consisting of radial functions, then no space translation can occur and we can omit the translations in the definition of orbital stability.

• Phase shifts are necessary: For a solution ϕ of (3.1) and ε > 0, let ϕ ε (x) := (1 + ε)

1 p-1 ϕ((1 + ε) 1 2 x).
Then it is easy to check that the solution of (1.1) with initial datum ϕ ε is

u ε (t, x) = e iω(1+ε)t (1 + ε) 1 p-1 ϕ((1 + ε) 1 2 x).

We clearly have

ϕ ε → ϕ in H 1 (R N ) as ε → 0 but for any ε > 0 sup t∈R inf y∈R N u ε (t) -ϕ(• -y) H 1 (R N ) ϕ H 1 (R N ) .
Remark that ϕ ε is a solution of (3.1) with ω replaced by ω(1 + ε).

The main result of this section is the following.

Theorem 4.3. Let ϕ be a ground state of (3.1). If 1 < p < 1 + 4 N , then the standing wave e iωt ϕ(x) is orbitally stable.

Remark 4.4. The range of p is optimal: We will see in the next section that instability holds if p 1 + 4

N . Remark 4.5. Except for N = 1 where all solutions of (3.1) are ground states, Theorem 4.3 asserts only the stability of standing waves corresponding to ground states.

It is an open problem to describe the behavior of other standing waves of (1.1) under perturbations (see nevertheless [START_REF] Grillakis | Linearized instability for nonlinear Schrödinger and Klein-Gordon equations[END_REF][START_REF] Todorova | Dynamics of non-linear wave equations[END_REF] and the references cited therein).

The first rigorous stability study is due to Benjamin [START_REF] Benjamin | The stability of solitary waves[END_REF] for solitary waves of the Korteweg-de Vries equation. Roughly speaking, two different approaches are possible in the study of stability of standing waves. The first one was introduced by Cazenave [START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF] and then developed by Cazenave and Lions [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF]. It relies on variational and compactness arguments. The main step in this approach consists in characterizing the ground states as minimizers of the energy on a sphere of L 2 (R N ). To use this approach, it is essential to have a uniqueness result for ground states as Theorem 3.15, otherwise a weaker result is obtained: Stability of the set of ground states (see Remark 4.9). The second approach was introduced by Weinstein [START_REF]Modulational stability of ground states of nonlinear Schrödinger equations[END_REF][START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF] (see also [START_REF] Rose | On the bound states of the nonlinear Schrödinger equation with a linear potential[END_REF]) and then considerably generalized by Grillakis, Shatah and Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry I[END_REF][START_REF]Stability theory of solitary waves in the presence of symmetry II[END_REF]. A criterion based on a form of coercivity for S ′′ (ϕ) is derived in this work and allows to prove stability of standing waves. Sufficient conditions for instability are also given in [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry I[END_REF][START_REF]Stability theory of solitary waves in the presence of symmetry II[END_REF].

The rest of this section is devoted to the proofs of Theorem 4.3. In Subsection 4.1 we present the proof using Cazenave-Lions method and in Subsection 4.2 we present the proof using Grillakis-Shatah-Strauss method.

Cazenave-Lions method

The proof of Theorem 4.3 by Cazenave-Lions method relies on the following compactness result.

Proposition 4.6. Let 1 < p < 1 + 4

N . For any τ > 0, define

Σ τ := {v ∈ H 1 (R N ); v 2 L 2 (R N ) = τ }.
Consider the minimization problem

-ν τ := inf{E(v); v ∈ Σ τ },
where E is the functional defined in (2.2):

For v ∈ H 1 (R N ), E(v) = 1 2 ∇v 2 L 2 (R N ) - 1 p + 1 v p+1 L p+1 (R N ) .
Then ν τ < +∞, and if

(v n ) ⊂ H 1 (R N ) is such that v n 2 L 2 (R N ) → τ and E(v n ) → -ν τ as n → +∞,
then there exist a family (y n ) ⊂ R N and a function v ∈ H 1 (R N ) such that, possibly for a subsequence only,

v n (• -y n ) → v strongly in H 1 (R N ).
In particular, v ∈ Σ τ and

E(v) = -ν τ .
The proof of Proposition 4.6 is based on the concentration-compactness principle of Lions [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case. II[END_REF]. We refer to [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] for a detailed proof.

Remark 4.7.

For p > 1 + 4 N , it is easy to see that ν τ = +∞. Indeed, let v ∈ Σ τ . Using the scaled functions v λ (•) := λ N 2 v(λ•), we obtain v λ 2 L 2 (R N ) = v 2 L 2 (R N ) = τ , but lim λ→+∞ E(v λ ) = -∞.
Therefore Proposition 4.6 cannot hold in this case.

Next we characterize the ground states of (3.1) as minimizers of the energy on a sphere of L 2 (R N ).

4.8. Let 1 < p < 1 + 4 N . Then (i) there exists τ G > 0 such that ϕ 2 L 2 (R N ) = τ G for all ϕ ∈ G, (ii) ϕ ∈ G if and only if ϕ ∈ Σ τ G and E(ϕ) = -ν τ G .
Sketch of proof. Point (i) follows immediately from Theorem 3.15. We refer to [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] for the proof of (ii).

Proof of Theorem 4.3 by Cazenave-Lions method. The result is proved by contradiction. Assume that there exist ε > 0 and two sequences (u n,0

) ⊂ H 1 (R N ), (t n ) ⊂ R such that u n,0 -ϕ H 1 (R N ) → 0 as n → +∞, (4.1)
inf

θ∈R inf y∈R N u n (t n ) -e iθ ϕ(• -y) H 1 (R N ) > ε for all n ∈ N, (4.2)
where u n is the maximal solution of (1.1) with initial datum u n,0 . Set v n (x) := u n (t n , x). By (4.1) and the conservation of charge and energy (see (2.3)), we have, as n → +∞,

v n 2 L 2 (R N ) = u n (t n ) 2 L 2 (R N ) = u n,0 2 L 2 (R N ) → ϕ 2 L 2 (R N ) = τ G (4.3) E(v n ) = E(u n (t n )) = E(u n,0 ) → E(ϕ) = -ν τ G . (4.4)
By (4.3), (4.4) and Proposition 4.6, there exist Remembering that v n = u n (t n ) and substituting this in (4.5), we get

(y n ) ⊂ R N and v ∈ H 1 (R N ) such that v n (• -y n ) -v H 1 (R N ) → 0 as n → +∞, (4.5)
v ∈ Σ τ G and E(v) = -ν τ G . ( 4
u n (t n ) -e iθ ϕ(• -(y -y n )) H 1 (R N ) → 0 when n → +∞,
which is a contradiction with (4.2) and finishes the proof.

Remark 4.9. It is essential for the proof of Theorem 4.3 by Cazenave-Lions method that the set of ground states G can be explicitly described by {e iθ Ψ(•y); θ ∈ R, y ∈ R N } as in Theorem 3.15. This uniqueness result is far from being obvious even with our simple pure power nonlinearity. For general nonlinearities, such uniqueness results are usually not available. Without this uniqueness result, one obtains a result corresponding to a weaker notion of stability: stability of the set G. More precisely, the concept of stability would read as follows: The set of ground states G is said to be stable if for all ε > 0 there exists δ > 0 such that if

u 0 ∈ H 1 (R N ) satisfies u 0 -ϕ H 1 (R N ) < δ for some ϕ ∈ G then sup t∈R inf ψ∈G u(t) -ψ H 1 (R N ) < ε,
where u is the maximal solution of (1.1) with u(0) = u 0 .

Grillakis-Shatah-Strauss method

The method of Grillakis-Shatah-Strauss is a powerful tool to derive stability or instability results. The results in [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry I[END_REF][START_REF]Stability theory of solitary waves in the presence of symmetry II[END_REF] state, roughly speaking, the following: If we consider a family (ϕ ω ) of solutions of the stationary equation, the standing wave e iωt ϕ ω (x) is stable if

∂ ∂ω ϕ ω 2 L 2 (R N ) > 0 (4.7)
and unstable if

∂ ∂ω ϕ ω 2 L 2 (R N ) < 0, (4.8)
provided some spectral assumptions on the linearized operator S ′′ (ϕ ω ) are satisfied. Slope conditions of the type (4.7)-(4.8) can easily be checked when the stationary equation admits some scaling invariance, typically when the nonlinearity is of power type. For example, in the case of (3.1), it is easy to see that if ϕ 1 is a solution of (3.1) with ω = 1, then the scaling ϕ ω (•) := ω

1 p-1 ϕ 1 (ω 1 2
•) provides a family of solutions of (3.1) for ω ∈ (0, +∞) such that

∂ ∂ω ϕ ω 2 L 2 (R N ) > 0 if p < 1 + 4 N , ∂ ∂ω ϕ ω 2 L 2 (R N ) < 0 if p > 1 + 4 N .
In such situations, the main difficulty is to handle the spectral conditions (see e.g. [START_REF] Gallay | Orbital stability of periodic waves for the nonlinear Schrödinger equation[END_REF][START_REF]Stability of small periodic waves for the nonlinear Schrödinger equation[END_REF][START_REF] Le Coz | Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential[END_REF]). If (3.1) has no scaling invariance, it becomes very difficult to obtain the slope conditions (4.7)-(4.8) (see nevertheless [START_REF] Genoud | Existence and orbital stability of standing waves for some nonlinear Schrödinger equations, perturbation of a model case[END_REF][START_REF] Genoud | Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves[END_REF][START_REF] Mcleod | Stability of standing waves for some nonlinear Schrödinger equations[END_REF]). An alternative is to use a stability criterion derived from the work [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry I[END_REF]. This stability criterion fits better the case of general nonlinearities, as in e.g. [START_REF] De Bouard | Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities[END_REF][START_REF]Stability of standing waves for nonlinear Schrödinger equations with potentials[END_REF][START_REF] Fukuizumi | Instability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities[END_REF][START_REF]An existence and stability result for standing waves of nonlinear Schrödinger equations[END_REF][START_REF] Kikuchi | Existence and stability of standing waves for Schrödinger-Poisson-Slater equation[END_REF][START_REF]Stability of standing waves for Klein-Gordon-Schrödinger equations and nonlinear Schrödinger equation with Yukawa potential[END_REF]. See also [START_REF] Stuart | Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation[END_REF] for a review of the different ways to obtain stability for general nonlinearities thanks to Grillakis-Shatah-Strauss' results.

Although theses notes are restricted to nonlinear Schrödinger equations with powertype nonlinearities, our goal is to provide the reader with methods applicable in rather general situations and this is why we choose to present the proof of Theorem 4.3 using the following stability criterion. Proposition 4.10 (Stability criterion). Let ϕ be a solution of (3.1). Suppose that there exists δ > 0 such that for all v ∈ H 1 (R N ) satisfying the following orthogonality conditions

(v, ϕ) 2 = (v, iϕ) 2 = v, ∂ϕ ∂x j 2
= 0 for all j = 1, ..., N (4.9)

we have

S ′′ (ϕ)v, v δ v 2 H 1 (R N ) . ( 4

.10)

Then the standing wave e iωt ϕ(x) is orbitally stable in H 1 (R N ).

Let us heuristically explain why the assumptions of Proposition 4.10 lead to stability. The idea comes from the theory of Lyapunov stability for an equilibrium in dynamical systems. A good candidate for a Lyapunov functional would be the functional S. Indeed, let us suppose for a moment that the coercivity condition (4.10) holds for any v ∈ H 1 (R N ) (this is not the case, as we will see in the sequel). Let u be a solution of (1.1) with initial datum u 0 close to ϕ in H 1 (R N ). A Taylor expansion gives

S(u(t)) -S(ϕ) = S ′ (ϕ), u(t) -ϕ + 1 2 S ′′ (ϕ)(u(t) -ϕ), u(t) -ϕ +o( u(t) -ϕ 2 H 1 (R N ) ).
Since ϕ is a solution of (3.1), S ′ (ϕ) = 0. Combined with (4.10), this would give, for some constant C > 0 independent of t, To avoid this kernel, we require the coercivity condition (4.10) only for v ∈ H 1 (R N ) satisfying

S(u(t)) -S(ϕ) C u(t) -ϕ 2 H 1 (R N ) . ( 4
(v, iϕ) 2 = v, ∂ϕ ∂x j 2
= 0 for all j = 1, ..., N, which allows phase shifts and translations in the right-hand side of (4.11). The other orthogonality condition (v, ϕ) 2 = 0 is related to the conservation of mass. Indeed, since the mass is conserved, the evolution takes place, in some sense, in the tangent space of the sphere of L 2 (R N ) at ϕ and therefore it is enough to ask for S to satisfy the coercivity condition (4.10) on this tangent space to get stability. The rigorous proof of Proposition 4.10 is involved and we have postponed it to the appendix. In view of Proposition 4.10, it is clear that Theorem 4.3 follows immediately from the following proposition. N and ϕ be a ground state of (3.1). Then there exists δ > 0 such that for all w ∈ H 1 (R N ) satisfying (w, ϕ) 2 = (w, iϕ) 2 = w, ∂ϕ ∂x j 2 = 0 for all j = 1, ..., N (4.12)

we have

S ′′ (ϕ)w, w δ w 2 H 1 (R N ) .
Before giving the proof of Proposition 4.11, some preparation is necessary. First, from Theorem 3.15, we can assume without loss of generality that the ground state ϕ is real, positive, and radial. Note that uniqueness is not involved here. It is convenient to split w in the real and imaginary part. We set w = u+iv for real-valued u, v ∈ H 1 (R N ). Then the operator S ′′ (ϕ) can be separated into a real and an imaginary part L 1 and L 2 such that S ′′ (ϕ)w, w = L 1 u, u + L 2 v, v . Here, L 1 and L 2 are two bounded operators defined on H 1 (R N ) restricted to real valued functions, with values in H -1 (R N ), and given by

L 1 u = -∆u + ωu -pϕ p-1 u, L 2 v = -∆v + ωv -ϕ p-1 v.
Until the end of this subsection, the functions considered will be real-valued. In particular, H 1 (R N ) and L 2 (R N ) will be restricted to real-valued functions.

Proposition 4.11 follows immediately from the two following lemmas.

Lemma 4.12. There exists

δ 1 > 0 such that for all u ∈ H 1 (R N ) satisfying (u, ϕ) 2 = u, ∂ϕ ∂x j 2 = 0 for all j = 1, ..., N
we have

L 1 u, u δ 1 u 2 H 1 (R N ) .
Lemma 4.13. There exists δ 2 > 0 such that for all v ∈ H 1 (R N ) satisfying

(v, ϕ) 2 = 0 we have L 2 v, v δ 2 v 2 H 1 (R N ) .
It is not hard to see that the operators L 1 and L 2 admit as Friedrich extensions (see e.g. [START_REF] Kato | Perturbation theory for linear operators[END_REF]) two unbounded self-adjoint operators L1 and L2 from

L 2 (R N ) to L 2 (R N ) with domain H 2 (R N ).
The proofs of Lemmas 4.12 and 4.13 rely on the analysis of the spectra of L1 and L2 . The following lemma gives the general structure of these spectra.

Lemma 4.14. The spectra of L1 and L2 consist of essential spectrum in [ω, +∞) and of a finite number of eigenvalues of finite multiplicity in (-∞, ω ′ ] for all ω ′ < ω.

Proof. We first remark that since L1 and L2 are self-adjoint operators, their spectra lie on the real line.

The spectra of L1 and L2 are bounded from below. Indeed, for all u ∈ H 1 (R N ) we have

L 2 u, u = ∇u 2 L 2 (R N ) + ω u 2 L 2 (R N ) - R N ϕ p-1 u 2 dx ∇u 2 L 2 (R N ) + ω u 2 L 2 (R N ) -p R N ϕ p-1 u 2 dx = L 1 u, u . Since ϕ ∈ L ∞ (R N ), there exists C > 0, independent of u, such that p R N ϕ p-1 u 2 dx C u 2 L 2 (R N )
and therefore

L 2 u, u L 1 u, u (ω -C) u 2 L 2 (R N )
. Now, since ϕ is exponentially decaying, L1 and L2 are compactly perturbed versions of

-∆ + ω : H 2 (R N ) ⊂ L 2 (R N ) → L 2 (R N ).
It is well-known that the essential spectrum of -∆ + ω is σ ess (-∆ + ω) = [ω, +∞), thus, by Weyl's Theorem (see e.g. [START_REF] Kato | Perturbation theory for linear operators[END_REF]),

σ ess ( L1 ) = σ ess ( L2 ) = [ω, +∞).
Since, for j = 1, 2, σ( Lj ) \ σ ess ( Lj ) consists of isolated eigenvalues of finite multiplicity and σ( Lj ) is bounded from below, this completes the proof of the Lemma.

From now on, we consider L 1 and L 2 separately. We begin with L 2 .

Lemma 4.15. There exists δ2 > 0 such that for all

v ∈ H 1 (R N ) with (v, ϕ) 2 = 0, we have L 2 v, v δ2 v 2 L 2 (R N ) .
Proof. We remark that L 2 ϕ = S ′ (ϕ) = 0 since ϕ is a solution of (3.1). This means that 0 is an eigenvalue of L 2 with ϕ as an eigenfunction. But ϕ > 0 and it is well-known (see e.g. [START_REF] Berezin | The Schrödinger equation[END_REF]) that this implies that 0 is the first simple eigenvalue of L2 . Let v ∈ H 1 (R N ) be such that (v, ϕ) 2 = 0. Then, by the min-max characterization of eigenvalues (see e.g. [START_REF] Berezin | The Schrödinger equation[END_REF]), there exists δ2 > 0 independent of v (in fact, δ2 is the second eigenvalue of L2 ) such that

L 2 v, v δ2 v 2 L 2 (R N ) .
Proof of Lemma 4.13. The proof is carried out by contradiction. Assume the existence of a sequence

(v n ) ⊂ H 1 (R N ) such that ∇v n 2 L 2 (R N ) + ω v n 2 L 2 (R N ) = 1, (v n , ϕ) 2 = 0 and L 2 v n , v n → 0 as n → +∞. Since v n 2 H 1 (R N ) ≤ max{ω, ω -1 }( ∇v n 2 L 2 (R N ) + ω v n 2 L 2 (R N ) ), (v n ) is bounded in H 1 (R N ) and there exists v ∈ H 1 (R N ) such that v n ⇀ v weakly in H 1 (R N ).
In particular, we have (v, ϕ) 2 = 0 and by Lemma 4.15

L 2 v, v 0. (4.13)
By the Sobolev embedding of H 1 (R N ) into L 2q (R N ), we have v 2 n ⇀ v 2 weakly in L q (R N ) for some q ∈ (1, 2 ⋆ /2). Since ϕ is exponentially decaying, we have ϕ p-1 ∈ L q ′ (R N ), where q ′ is the conjugate exponent of q, 1 q + 1 q ′ = 1. Therefore

R N ϕ p-1 v 2 n dx → R N ϕ p-1 v 2 dx as n → +∞. (4.14)
From (4.14) and by weak lower semi-continuity of the H 1 (R N )-norm, we infer that On the other hand,

L 2 v, v lim inf n→+∞ L 2 v n , v n = 0. ( 4
0 = lim inf n→+∞ L 2 v n , v n = 1 - R N ϕ p-1 v 2 dx.
Hence R N ϕ p-1 v 2 dx = 1, a contradiction with (4.16).

We now turn our attention to L 1 . The proof of Lemma 4.12 is more delicate, essentially because the spectrum of L1 contains nonpositive eigenvalues. Furthermore, ϕ is no longer an eigenfunction. We first deal with the negative eigenvalues of L1 .

Lemma 4.16. The operator L1 has only one negative eigenvalue -λ 1 with a corresponding eigenfunction

e 1 ∈ H 2 (R N ) such that e 1 L 2 (R N ) = 1.
Proof. Let S : H 1 (R N ) → R be the restriction of S to real-valued functions. It is not hard to see that ϕ is also a Mountain Pass critical point of S. Then it is well-known (see, for example, [START_REF]Nonlinear analysis and semilinear elliptic problems[END_REF]) that the Morse index of S at ϕ is at most 1 (recall that the Morse index is the number of negative eigenvalues of S′′ (ϕ)). We remark that L 1 = S′′ (ϕ). Therefore L1 has at most one negative eigenvalue. On the other hand,

L 1 ϕ, ϕ = ∇ϕ 2 L 2 (R N ) + ω ϕ 2 L 2 (R N ) -p ϕ p+1 L p+1 (R N ) = -(p -1) ϕ p+1 L p+1 (R N ) < 0
where the second equality follows from the Nehari identity (3.3). Therefore, L1 has exactly one negative eigenvalue -λ 1 and we can pick up an eigenfunction

e 1 ∈ H 2 (R N ) such that e 1 L 2 (R N ) = 1.
The following result originates from the work of Weinstein [START_REF]Modulational stability of ground states of nonlinear Schrödinger equations[END_REF].

Lemma 4.17. The second eigenvalue of L1 is 0 and

Z := ker L1 = span ∂ϕ ∂x j ; j = 1, ..., N .
It is out of reach in these notes to give a complete proof of Lemma 4.17, therefore we only give a partial proof and refer to [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] for a complete proof.

Partial proof of lemma 4.17. We remember that ϕ satisfies -∆ϕ + ωϕϕ p = 0.

(4.17)

Differentiating (4.17) with respect to x j gives

-∆ ∂ϕ ∂x j + ω ∂ϕ ∂x j -pϕ p-1 ∂ϕ ∂x j = 0. This is allowed because ϕ ∈ W 3,2 (R N ) implies ∂ϕ ∂xj ∈ H 2 (R N ).
Hence 0 is an eigenvalue of L1 and span ∂ϕ ∂x j ; j = 1, ..., N ⊂ ker L1 .

We admit that the reverse inclusion also holds.

Lemma 4.18. The space

H 1 (R N ) can be decomposed into H 1 (R N ) = E 1 ⊕ Z ⊕ E +
where E 1 = span{e 1 } and E + is the image of the spectral projection corresponding to the positive part of the spectrum of L1 .

Note that in the direct sum E 1 ⊕ Z ⊕ E + the spaces are mutually orthogonal with respect to the inner product of L 2 (R N ).

Proof of Lemma 4.18. The assertion follows immediately from Lemmas 4.14, 4.16, 4.17 and the spectral decomposition theorem (see e.g. [43, p. 177]).

Lemma 4.19. For all

u ∈ H 1 (R N ) \ {0} satisfying (u, ϕ) 2 = u, ∂ϕ ∂x j 2 = 0 for all j = 1, ..., N (4.18)
we have

L 1 u, u > 0. ( 4.19)
Proof. Let u ∈ H 1 (R N ) satisfying (4.18). We first look for a function ψ such that

L 1 ψ = -ωϕ.
To do this, we use the scaled functions ϕ λ defined by ϕ λ (•) := λ

1 p-1 ϕ(λ 1 2 •) for λ > 0. The functions ϕ λ satisfy -∆ϕ λ + ωλϕ λ -ϕ p λ = 0. ( 4.20)
Differentiating (4.20) with respect to λ gives at λ = 1

-∆ψ + ωψ -pϕ p-1 ψ = -ωϕ for ψ := ∂ϕ λ ∂λ λ=1 = 1 p-1 ϕ + 1 2 x • ∇ϕ. Note that ψ ∈ H 1 (R N ) since x • ∇ϕ ∈ H 1 (R N
) by Proposition 3.2. Furthermore, since ϕ is radial, ϕ is even in each x j . Therefore, ψ is also even in each x j whereas ∂ϕ ∂xj is odd in x j . This implies ψ, ∂ϕ ∂x j 2 = 0 for all j = 1, ..., N.

We decompose u and ψ with respect to

H 1 (R N ) = E 1 ⊕ Z ⊕ E + : There exist α, β ∈ R and ξ, η ∈ E + such that u = αe 1 + ξ, ψ = βe 1 + η.
If α = 0, then ξ ≡ 0 since u ≡ 0, and we have

L 1 u, u = L 1 ξ, ξ > 0.
Therefore, u satisfies (4.19). Now, we suppose α = 0. We easily find:

L 1 ψ, ψ = -ω ϕ, 1 p -1 ϕ + 1 2 x • ∇ϕ 2 = -ω 1 p -1 ϕ 2 L 2 (R N ) + 1 2 R N ϕx • ∇ϕdx .
With integration by parts, it is easy to see that

R N ϕx • ∇ϕdx = -N ϕ 2 L 2 (R N ) - R N ϕx • ∇ϕdx. Therefore, R N ϕx • ∇ϕdx = -N 2 ϕ 2 L 2 (R N ) and L 1 ψ, ψ = -ω 1 p -1 - N 4 ϕ 2 L 2 (R N ) .
Since ω > 0 and p < 1 + 4 N this implies L 1 ψ, ψ < 0 and thus β = 0. On E + , L 1 defines a positive definite quadratic form. Thus we have the Cauchy-Schwarz inequality for ξ, η

∈ E + L 1 ξ, η 2 L 1 ξ, ξ L 1 η, η . Now, L 1 u, u = -α 2 λ 1 + L 1 ξ, ξ -α 2 λ 1 + L 1 ξ, η 2 L 1 η, η . (4.21) But 0 = -ω (ϕ, u) 2 = L 1 ψ, u = -αβλ 1 + L 1 ξ, η .
Thus L 1 ξ, η = αβλ 1 . This gives

-α 2 λ 1 + L 1 ξ, η 2 L 1 η, η = -α 2 λ 1 + α 2 β 2 λ 2 1 L 1 η, η = -α 2 λ 1 + α 2 β 2 λ 2 1 β 2 λ 1 + L 1 ψ, ψ = -α 2 λ 1 L 1 ψ, ψ L 1 η, η > 0.
Combined with (4.21) this finishes the proof.

Proof of Lemma 4.12. The proof of Lemma 4.12 follows the same lines as the proof of Lemma 4.13. We omit the details.

Instability

In this section, we will prove that if 1

+ 4 N p < 1 + 4 N -2
then the standing waves are unstable by blow-up in finite time. More precisely, for any ground state ϕ ∈ G, we will find initial data, as close to ϕ as we want, such that the solutions of (1.1) corresponding to these initial data will all blow up in finite time. As in Proposition 2.3, our basic tool will be the Virial Theorem (Proposition 2.4). However, since the energy of the ground state is nonnegative, it is not possible to argue directly as in Proposition 2.3. To overcome this difficulty, we follow the approach of [START_REF] Coz | A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations[END_REF], which is a recent improvement of the method introduced by Berestycki and Cazenave in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF][START_REF]Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]. At the heart of this method is a new variational characterization of the ground states as minimizers of the action S on constraints related to the Pohozaev identity (3.4). Compare to [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF][START_REF]Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF], the mean feature of the approach of [START_REF] Coz | A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations[END_REF] is that we do not need to solve directly a new minimization problem; instead, we take advantage of the variational characterizations of the ground states obtained in Section 3.

Before stating our results, we give a precise definition of instability by blow-up.

Definition 5.1. Let ϕ be a solution of (3.1). We say that the standing wave e iωt ϕ(x) is unstable by blow-up in finite time if for all ε > 0 there exists

u ε,0 ∈ H 1 (R N ) such that u ε,0 -ϕ H 1 (R N ) < ε but the corresponding maximal solution u ε of (1.1) in the interval (T ε min , T max ε ) satisfies T ε min > -∞, T max ε < +∞ and thus lim t↓T ε min u ε (t) H 1 (R N ) = +∞ and lim t↑T max ε u ε (t) H 1 (R N ) = +∞.
We start with the simplest case: p = 1+ 4 N . The following result is due to Weinstein [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF].

Theorem 5.2. Let p = 1 + 4

N . Then for every solution ϕ of (3.1) the standing wave e iωt ϕ(x) is unstable by blow-up in finite time.

Proof. First, we remark that since p = 1 + 4

N , E(v) = P (v) (recall that P was defined in (2.6)) for all v ∈ H 1 (R N ). From Pohozaev identity (3.4), we have

E(ϕ) = P (ϕ) = 0.
Let u ε,0 be defined by u ε,0 := (1 + ε)ϕ. Then it is easy to see that E(u ε,0 ) < 0. In view of the exponential decay of ϕ (see Proposition 3.2), we have |x|u ε,0 ∈ L 2 (R N ). The conclusion follows from Proposition 2.3.

We consider now the general case. N (see [START_REF] Todorova | Dynamics of non-linear wave equations[END_REF] for a review of related results and open problems).

For the proof of Theorem 5.3 it is not possible to mimic the proof of Theorem 5.2. Indeed, for p > 1 + 4 N , the identity E(v) = P (v) does not hold any more. Moreover, for ϕ ∈ G, E(ϕ) > 0, which prevents to use Proposition 2.3.

The scaling v λ (•) := λ N/2 v(λ•) will play an important role in the proof. In the following lemma, we investigate the behavior of different functionals under the scaling. Lemma 5.5. Let v ∈ H 1 (R N )\{0} be such that P (v) 0. Then there exists λ 0 ∈ (0, 1] such that

(i) P (v λ 0 ) = 0, (ii) λ 0 = 1 if and only if P (v) = 0, (iii) ∂ ∂λ S(v λ ) = 1 λ P (v λ ), (iv) ∂ ∂λ S(v λ ) > 0 for λ ∈ (0, λ 0 ) and ∂ ∂λ S(v λ ) < 0 for λ ∈ (λ 0 , +∞), (v) λ → S(v λ ) is concave on (λ 0 , +∞).
Proof. A simple calculation leads to

P (v λ ) = λ 2 ∇v 2 H 1 (R N ) -λ N (p-1) 2 N (p -1) 2(p + 1) v p+1 L p+1 (R N ) .
Recalling that N (p-1)

2

> 2 (because of p > 1 + 4 N ) we infer that P (v λ ) > 0 for λ small enough. Thus, by continuity of P , there must exist λ 0 ∈ (0, 1] such that P (v λ 0 ) = 0. Hence (i) is proved. If λ 0 = 1, it is clear that P (v) = 0. Conversely, suppose that P (v) = 0. Then

P (v λ ) = λ 2 P (v) + λ 2 -λ N (p-1) 2 N (p -1) 2(p + 1) v p+1 L p+1 (R N ) = λ 2 -λ N (p-1) 2 N (p -1) 2(p + 1) v p+1 L p+1 (R N ) ,
and, since N (p-1)

2

> 2, this implies that P (v λ ) > 0 for all λ ∈ (0, 1). Hence (ii) follows. From a simple calculation, we obtain

∂ ∂λ S(v λ ) = λ ∇v 2 H 1 (R N ) -λ N (p-1) 2 -1 N (p -1) 2(p + 1) v p+1 L p+1 (R N ) = λ -1 P (v λ ).
Hence (iii) is shown. To see (iv), we remark that

P (v λ ) = λ 2 λ -2 0 P (v λ 0 ) + λ 2 λ N (p-1) 2 -2 0 -λ N (p-1) 2 N (p -1) 2(p + 1) v p+1 L p+1 (R N ) = λ 2 λ N (p-1) 2 -2 0 -λ N (p-1) 2 -2 N (p -1) 2(p + 1) v p+1 L p+1 (R N ) .
Therefore, since N (p-1)

2

-2 > 0, P (v λ ) > 0 for λ < λ 0 and P (v λ ) < 0 for λ > λ 0 . Combined with (iii), this gives (iv). Finally,

∂ 2 ∂λ 2 S(v λ ) = ∇v 2 L 2 (R N ) -λ N (p-1) 2 -2 N (p -1) 2 -1 N (p -1) 2(p + 1) v p+1 L p+1 (R N ) = λ -2 P (v λ ) -λ N (p-1) 2 -2 N (p -1) 2 -2 N (p -1) 2(p + 1) v p+1 L p+1 (R N )
which implies that ∂ 2 ∂λ 2 S(v λ ) < 0 for λ > λ 0 since N (p-1)

2

> 2, hence (v) follows.

The proof of Theorem 5.3 runs in three steps. First, we derive a new variational characterization for the ground states of (3.1). Then we use this characterization to define a set I, invariant under the flow of (1.1), such that any initial datum in I gives rise to a blowing up solution of (1.1). Finally, we prove that the ground states can be approximated by sequences of elements of I.

We consider the following minimization problem:

d M := inf v∈M S(v),
where the constraint M is given by

M := {v ∈ H 1 (R N ) \ {0}; P (v) = 0, I(v) 0}.
Recall that I was defined in Proposition 3.12.

Lemma 5.6. The following equality holds:

m = d M ,
where m is the least energy level defined in (3.20).

Proof. Let ϕ ∈ G. Since from Lemma 3.3 we have I(ϕ) = P (ϕ) = 0, it is clear that ϕ ∈ M. Therefore S(ϕ) d M and thus m d M .

(5.1)

Now, let v ∈ M. If I(v) = 0 then S(v)
m by Proposition 3.12. Suppose that I(v) < 0. We use the scaling v λ (•) := λ N/2 v(λ•). Then

I(v λ ) = λ 2 ∇v 2 L 2 (R N ) + ω v 2 L 2 (R N ) -λ N (p-1) 2 v p+1 L p+1 (R N ) implies lim λ→0 I(v λ ) = ω v 2 L 2 (R N ) > 0.
By continuity of I, there exists λ 1 < 1 such that I(v λ 1 ) = 0. Therefore, by Proposition 3.12, m S(v λ 1 ).

(

From P (v) = 0 and Lemma 5.5, we deduce that (5.4)

S(v λ 1 ) < S(v). (5.3)
The conclusion follows from (5.1) and (5.4) Now, we define the set I by:

I := {v ∈ H 1 (R N ); I(v) < 0, P (v) < 0, S(v) < m},
and use Lemma 5.6 to prove that I is invariant under the flow of (1.1).

Lemma 5.7. If u 0 ∈ I then the corresponding maximal solution u in (T min , T max ) of (1.1) satisfies u(t) ∈ I for all t ∈ (T min , T max ).

Proof. Let u 0 ∈ I and u be the corresponding maximal solution of (1.1) in (T min , T max ).

Since S is a conserved quantity for (1.1), S(u(t)) = S(u 0 ) < m for all t ∈ (T min , T max ).

(5.5)

The assertion is proved by contradiction. Suppose that there exists t such that

I(u(t)) 0.
Then by the continuity of I and u there exists t 0 such that I(u(t 0 )) = 0.

By Proposition 3.12, this implies

S(u(t 0 )) m,
which is a contradiction with (5.5). Therefore I(u(t)) < 0 for all t ∈ (T min , T max ).

(5.6)

Finally, suppose that for some t ∈ (T min , T max ) P (u(t)) 0.

Still by continuity, there exists t 1 such that P (u(t 1 )) = 0.

From (5.6), we also have I(u(t 1 )) < 0, and thus by Lemma 5.6

S(u(t 1 )) m,
which is another contradiction. Therefore P (u(t)) < 0 for all t ∈ (T min , T max ) and this completes the proof.

Lemma 5.8. Let u 0 ∈ I and u be the corresponding maximal solution of (1.1) in (T min , T max ). Then there exists δ > 0 independent of t such that P (u(t)) < -δ for all t ∈ (T min , T max ).

Proof. Let t ∈ (T min , T max ) and define v := u(t) and v λ (•) := λ N/2 v(λ•). By Lemma 5.5, there exists λ 0 < 1 such that P (v λ 0 ) = 0. If I(v λ 0 ) 0, we keep λ 0 , otherwise if I(v λ 0 ) > 0 there exists λ0 ∈ (λ 0 , 1) such that I(v λ0 ) = 0 and we replace λ 0 by λ0 . In any case, by Proposition 3.12 or Lemma 5.6,

S(v λ 0 ) m. (5.7)
Now, by (v) in Lemma 5.5,

S(v) -S(v λ 0 ) (1 -λ 0 ) ∂ ∂λ S(v λ ) λ=1 .
(5.8)

From (iii) in Lemma 5.5, we obtain that

∂ ∂λ S(v λ ) λ=1 = P (v).
(5.9) Furthermore, P (v) < 0 and λ 0 ∈ (0, 1) implies

(1λ 0 )P (v) > P (v).

(5.10) Combining (5.7)-(5.10) gives

S(v) -m > P (v).
Let -δ := S(v)m. Then δ > 0 since v ∈ I and δ is independent of t since S is a conserved quantity. In conclusion, for any t ∈ (T min , T max ),

P (u(t)) < -δ
and this ends the proof.

Lemma 5.9. Let u 0 ∈ I be such that |x|u 0 ∈ L 2 (R N ). Then the corresponding maximal solution u of (1.1) in (T min , T max ) blows up in finite time, i.e. T min > -∞, T max < +∞ and

lim t↓T min u(t) H 1 (R N ) = +∞ and lim t↑T max u(t) H 1 (R N ) = +∞.
Proof. By Lemma 5.8, there exists δ > 0 such that P (u(t)) < -δ for all t ∈ (T min , T max ).

Remembering from Proposition 2.4 that ∂ 2 ∂t 2 xu(t) 2 L 2 (R N ) = 8P (u(t)) we get by integrating twice in time

xu(t) 2 L 2 (R N ) -4δt 2 + C(t + 1).
As in the proof of Proposition 2.3, this leads to a contradiction for large |t|. Therefore T min > -∞, T max < +∞ and by Proposition 2.1 lim

t↓T min u(t) H 1 (R N ) = +∞ and lim t↑T max u(t) H 1 (R N ) = +∞.
Proof of Theorem 5.3. In view of Lemma 5.9, all that remains to do is to find a sequence in I converging to ϕ in H 1 (R N ). We define ϕ λ (•) := λ N/2 ϕ(λ•). Then, by Lemma 5.5, I(ϕ λ ) < 0, P (ϕ λ ) < 0, S(ϕ λ ) < m, thus ϕ λ ∈ I for all 0 < λ < 1. Furthermore, by Proposition 3.2, ϕ is exponentially decaying and so is ϕ λ . Therefore, |x|ϕ λ ∈ L 2 (R N ) for all 0 < λ < 1. It is clear that ϕ λ → ϕ when λ → 1 and from Lemma 5.9 ϕ λ gives rise to a blowing-up solution of (1.1) for any 0 < λ < 1. This achieves the proof.

Remark 5.10. When the nonlinearity is more general, it may be impossible to use the Virial Theorem to obtain a result of instability by blow-up. In such cases, a good alternative to prove instability would be to follow the method introduced by Shatah and Strauss in [START_REF] Shatah | Instability of nonlinear bound states[END_REF] and then developed in [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry I[END_REF][START_REF]Stability theory of solitary waves in the presence of symmetry II[END_REF]. Other methods, based on modifications of the original idea of Shatah and Strauss, are also available, see for example [START_REF] Fukuizumi | Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential[END_REF][START_REF] Fukuizumi | Instability of standing waves for nonlinear Schrödinger equations with potentials[END_REF][START_REF] Gonçalves-Ribeiro | Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field[END_REF][START_REF] Kikuchi | Instability of standing waves for the Klein-Gordon-Schrödinger system[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. Note that proving instability by these methods does not necessarily provide informations on the long time behavior (blow-up or global existence) of solutions starting near a standing wave.

Remark 5.11. The type of method employed here to prove instability by blow-up of standing waves is not restricted to nonlinear Schrödinger equations. See, for example, [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF][START_REF] Jeanjean | Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments[END_REF][START_REF] Liu | Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations[END_REF][START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF][START_REF]Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system[END_REF][START_REF] Shatah | Unstable ground state of nonlinear Klein-Gordon equations[END_REF] and the references cited therein for results on the instability by blow-up for standing waves of nonlinear Klein-Gordon equations.

Appendix

The appendix is devoted to the proof of Proposition 4.10.

For ϕ being a solution of (3.1), we define a tubular neighborhood of ϕ of size

ε > 0 in H 1 (R N ) by U ε (ϕ) := {v ∈ H 1 (R N ); inf θ∈R,y∈R N e iθ v(• -y) -ϕ H 1 (R N ) < ε}.
Before proving Proposition 4.10, some preliminaries are needed. Lemma 6.1. Let ϕ be a solution of (3.1). For all δ > 0 there exists ε > 0 such that if for θ ∈ R and y ∈ R N we have

e iθ ϕ(• -y) -ϕ L 2 (R N ) < ε then |(θ, y)| < δ. Here, | • | denotes the norm in (R/2πZ) × R N .
Proof. The proof is carried out by contradiction. Let δ > 0 and assume that for all n ∈ N there exists (θ n , y n ) such that

e iθn ϕ(• -y n ) -ϕ L 2 (R N ) < 1 n and |(θ n , y n )| > δ.
Possibly for a subsequence only, we have e iθn ϕ(• -y n ) → ϕ almost everywhere as n → +∞, and in fact everywhere by regularity of ϕ (see Proposition 3.2). Suppose that (y n ) is unbounded. Then, possibly for a subsequence only, |y n | → +∞, but by exponential decay of ϕ (see Proposition 3.2), this implies that for all x ∈ R N ,

e iθn ϕ(x + y n ) → 0,
which is a contradiction with ϕ ≡ 0. Therefore (y n ) is bounded and converges up to a subsequence to some y ∈ R N . Since each θ n can be chosen in [0, 2π), we also have θ n → θ for some θ ∈ [0, 2π). By hypothesis, we have

|(θ, y)| δ, (6.1)
but for all x ∈ R N e iθ ϕ(x + y) = ϕ(x). (6.2)

We claim that (6.2) can hold true only if y = θ = 0. Indeed, let x 0 be such that ϕ(x 0 ) = 0. If y = 0, then lim n→+∞ e inθ ϕ(x 0 + ny) = ϕ(x 0 ) = 0, what is in contradiction with the decay of ϕ at infinity. Therefore, y = 0 and this immediately implies θ = 0. This is in contradiction with (6.1) and finishes the proof. Lemma 6.2. Let ϕ be a solution of (3.1). There exist ε > 0 and two functions σ :

U ε (ϕ) → R and Y : U ε (ϕ) → R N such that for all v ∈ U ε (ϕ) e iσ(v) v(• -Y (v)) -ϕ L 2 (R N ) = inf θ∈R,y∈R N e iθ v(• -y) -ϕ L 2 (R N ) .
Furthermore, the function

w := e iσ(v) v(• -Y (v)) satisfies (w, iϕ) 2 = w, ∂ϕ ∂x j 2 = 0 for all j = 1, ..., N. (6.3)
Proof. For the sake of simplicity, we assume that ϕ is real-valued and radial. Let

Φ : R × R N × H 1 (R N ) → R be defined by Φ(θ, y, v) := 1 2 e iθ v(• -y) -ϕ 2 L 2 (R N ) = 1 2 v -e -iθ ϕ(• + y) 2 L 2 (R N )
and let F : R × R N × H 1 (R N ) → R N +1 be the derivative of Φ with respect to (θ, We have F (0, 0, ϕ) = 0 and

D θ,y F (0, 0, ϕ) =        ϕ 2 L 2 (R N ) (0) ∂ϕ ∂x 1 2 L 2 (R N )
. . .

(0) ∂ϕ ∂x N 2 L 2 (R N )       
, where (0) means that all the off-diagonal entries are 0. Since ϕ is a solution of (3.1), ϕ is non zero and therefore ϕ 2 L 2 (R N ) > 0 and ∂ϕ ∂xj 2 L 2 (R N ) > 0 for all j = 1, ..., N . Consequently, the matrix D θ,y F (0, 0, ϕ) is positive definite and by the Implicit Function Theorem there exists ε > 0, δ > 0 and (σ, Y ) : V ε → Ω, where For v ∈ V ε , we have

e i θϕ(•-ỹ)-ϕ L 2 (R N ) < e i θ(ϕ(•-ỹ)-v(•-ỹ)) L 2 (R N ) + e i θv(•-ỹ)-ϕ L 2 (R N ) < 2ε.
If ε is small enough, this implies by Lemma 6.1 that |( θ, ỹ)| < δ which is a contradiction. Therefore, we have for all v ∈ V ε

e iσ(v) v(• -Y (v)) -ϕ L 2 (R N ) = inf (θ,y)∈R×R N e iθ v(• -y) -ϕ L 2 (R N ) .
It remains to extend (σ, Y ) to U ε (ϕ). Let v ∈ U ε (ϕ). Then there exists (θ ⋆ , y ⋆ ) such that e iθ ⋆ v(• -y ⋆ )ϕ L 2 (R N ) < ε. Define σ(v) := σ(e iθ ⋆ v(•y ⋆ )) + θ ⋆ and Y (v) := Y (e iθ ⋆ v(•y ⋆ )) + y ⋆ . Then it is not hard to see that this definition is independent of the choice of (θ ⋆ , y ⋆ ) and allows us to extend (σ, Y ) to U ε (ϕ). The orthogonality relations in (6.3) follow from the relation (6.4) Recall that Q was defined in (2.1) and E in (2.2). Lemma 6.3. Under the hypothesis of Proposition 4.10, there exists ε > 0 and C > 0 such that for all v ∈ U ε (ϕ) satisfying Q(v) = Q(ϕ) we have

E(v) -E(ϕ) C inf (θ,y)∈R×R N e iθ v(• -y) -ϕ 2 H 1 (R N ) .
Proof. For the sake of simplicity, we assume that ϕ is radial. Let ε > 0 and v ∈ U ε (ϕ).

For ε small enough, let (σ, Y ) be as in Lemma 6.2 and define w := e iσ(v) v(• -Y (v)). (6.6)

Then by Lemma 6.2 the function w satisfies (w, iϕ) 2 = w, ∂ϕ ∂x j 2 = 0 for all j = 1, ..., N. (6.7)

Let λ ∈ R and z ∈ H 1 (R N ) be such that (z, ϕ) 2 = 0 (6.8) and wϕ = λϕ + z. (6.9)

Since ϕ is radial up to translations, we have ϕ, ∂ϕ ∂x j 2 = 0 for j = 1, ..., N. (6.10)

Combining (6.7), (6.9) and (6.10) we get z, ∂ϕ ∂x j 2 = 0 for j = 1, ..., N. (6.11)

Moreover, we have (ϕ, iϕ) 2 = Re(-i ϕ 2 L 2 (R N ) ) = 0, and therefore (z, iϕ) 2 = 0. (6.12)

From (6.8), (6.11) and (6.12) we see that z satisfies the orthogonality conditions in (4.9) and therefore

S ′′ (ϕ)z, z δ z 2 H 1 (R N ) . (6.13)
By a Taylor expansion, we obtain

Q(ϕ) = Q(v) = Q(w) = Q(ϕ) + Q ′ (ϕ), w -ϕ + O( w -ϕ 2 H 1 (R N )
). But Q ′ (ϕ) = ϕ and therefore Q ′ (ϕ), wϕ = (ϕ, w -ϕ) 2 = (ϕ, λϕ + z) 2 = λ ϕ 2 L 2 (R N ) , where the last equality follows from (6.8). Consequently, Furthermore, from (6.9) we get But since Q(v) = Q(ϕ), we have

λ = O( w -ϕ 2 H 1 (R N ) ). ( 6
E(v) -E(ϕ) = S(v) -S(ϕ)
and with (6.13), (6. [START_REF] De Bouard | Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities[END_REF]) and (6.21), we obtain

E(v) -E(ϕ) δ 2 w -ϕ 2 H 1 (R N ) + o( w -ϕ 2 H 1 (R N ) ).
Therefore, we have for ε small enough

E(v) -E(ϕ) δ 4 w -ϕ 2 H 1 (R N ) .
Setting C := δ/4 and remembering how w was defined in (6.6) gives

E(v) -E(ϕ) C inf (θ,y)∈R×R N e iθ v(• -y) -ϕ 2 H 1 (R N ) ,
which finishes the proof.

Proof of Proposition 4.10. The assertion is proved by contradiction. Assume that there exists u n,0 and ε > 0 such that u n,0 -ϕ H 1 (R N ) → 0 as n → +∞ (6. where u n is the maximal solution of (1.1) in (T n min , T max n ) corresponding to u 0,n . By continuity, we can pick up the first time t n such that inf (θ,y)∈R×R N e iθ u n (t n , •y)ϕ H 1 (R N ) = ε In view of (6.22) and the conservation of charge and energy (see (2.3)), it is clear that

E(u n (t n )) = E(u n,0 ) → E(ϕ) Q(u n (t n )) = Q(u n,0 ) → Q(ϕ)
as n → +∞.

Let v n := un(tn) un(tn) L 2 (R N ) ϕ L 2 (R N ) . Then Q(v n ) = Q(ϕ), E(v n ) → E(ϕ) and v n -u n (t n ) H 1 (R N ) → 0.
Choosing ε small enough, we can apply Lemma 6. C(E(v n ) -E(ϕ)) → 0.

On the other hand, we have

ε = inf (θ,y)∈R×R N e iθ u n (t n , • -y) -ϕ H 1 (R N ) inf (θ,y)∈R×R N e iθ v n (• -y) -ϕ H 1 (R N ) + u n -v n H 1 (R N ) ,
which yields a contradiction for n large.

Figure 1 .

 1 Figure 1. Dispersive effect.

Figure 2 .

 2 Figure 2. Focusing effect.

Figure 3 .

 3 Figure 3. Balance between dispersion and focusing.

Corollary 3 . 4 .

 34 If ω 0 then (3.1) has no solution.

. 11 ) 3 . 6 .

 1136 LemmaThe functional S has a Mountain Pass geometry, i.e. Γ = ∅ and c > 0.

  [START_REF] Fukuizumi | Instability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities[END_REF] and byLemma 3.13, c d. (3.25) Now, we remark that, since by (3.3) any solution v of (3.1) satisfies I(v) = 0, we have d m. (3.26) Since ϕ ω is a solution of (3.1), it follows from the definition of the least energy level m (see (3.20)) that m S(ϕ ω ). (3.27) Combining (3.24)-(3.27) gives S(ϕ ω ) = c = d = m and completes the proof.

Figure 4 .

 4 Figure 4. The function Ψ for N = 1, p = 3 and ω = 1.

Proposition

. 6 )

 6 By (4.6) and Proposition 4.8 we have v ∈ G. Therefore, by Theorem 3.15, there exist θ ∈ R and y ∈ R N such that v = e iθ ϕ(•y).

Proposition 4 . 11 .

 411 Let 1 < p < 1 + 4

Theorem 5 . 3 .

 53 Let p > 1 + 4 N . For all ϕ ∈ G the standing wave e iωt ϕ(x) is unstable by blow-up in finite time. Remark 5.4. Theorem 5.2 asserts the instability of standing waves corresponding to any solution of (3.1) whereas Theorem 5.3 concerns only standing waves corresponding to ground states. It is still an open problem to prove instability by blow-up for any solution of (3.1) if p > 1 + 4

Combining ( 5 . 2 )

 52 and (5.3) gives m S(v), hence m d M .

Fe 2 - 2 

 22 (θ, y, v) := D θ,y Φ(θ, y, v) = iθ v(•y), iϕ e iθ v(•y), ∂ϕ ∂x 1 2 . . . e iθ v(•y), ∂ϕ ∂x N

V

  ε := {v ∈ H 1 (R N ); vϕ H 1 (R N ) < ε} and Ω := {(θ, y) ∈ R × R N ; |(θ, y)| < δ}, such that for all v ∈ V ε F (σ(v), Y (v), v) = 0 (6.4) and e iσ(v) v(• -Y (v))ϕ L 2 (R N ) = inf (θ,y)∈Ω e iθ v(•y)ϕ L 2 (R N ) .(6.5)Now we extend (6.5) to all (θ, y) ∈ R × R N . By contradiction, assume that there exists ( θ, ỹ) ∈ R × R N such that |( θ, ỹ)| > δ and e i θv(• -ỹ)ϕ L 2 (R N ) < e iσ(v) v(• -Y (v))ϕ L 2 (R N ) .

. 14 )

 14 Now, another Taylor expansion givesS(v) -S(ϕ) = S(w) -S(ϕ) = S ′ (ϕ), wϕ + 1 2 S ′′ (ϕ)(wϕ), wϕ + o( wϕ 2 H 1 (R N ) ).

(6. 15 )

 15 Since ϕ is solution of (3.1), we have S ′ (ϕ) = 0 and thereforeS ′ (ϕ), wϕ = 0. (6.16)

22 )

 22 but for alln ∈ N sup t∈(T n min ,T max n ) inf (θ,y)∈R×R N e iθ u n (t, •y)ϕ H 1 (R N ) > ε

  3 to get inf (θ,y)∈R×R N e iθ v n (•y)ϕ 2 H 1 (R N )

  ′′ (ϕ)ϕ, ϕ +2λRe S ′′ (ϕ)ϕ, z + S ′′ (ϕ)z, z .(6.17) From (6.14) we haveλ 2 S ′′ (ϕ)ϕ, ϕ = o( wϕ 2 H 1 (R N ) ).H 1 (R N ) 2 wϕ 2 H 1 (R N ) + 2λ 2 ϕ 2 H 1 (R N ), we have by (6.14) thatz 2 H 1 (R N ) = O( wϕ 2 H 1 (R N ) ),(6.19)and therefore 2λReS ′′ (ϕ)ϕ, z = o( wϕ 2 H 1 (R N ) ). (6.20)Combining (6.17), (6.18) and (6.20), we getS ′′ (ϕ)(wϕ), wϕ = S ′′ (ϕ)z, z + o( wϕ 2 H 1 (R N )). Together with (6.15)-(6.16), this givesS(v) -S(ϕ) 1 2 S ′′ (ϕ)z, z + o( wϕ 2 H 1 (R N ) ). (6.21)

		(6.18)
	Since	z 2

S ′′ (ϕ)(wϕ), wϕ = λ 2 S

S(ϕω ) = m = c.

Acknowledgments. The material presented in these notes is based on two lecture series given at the TU Berlin and at SISSA in 2008. The author would like to thank these two institutions for their hospitality. He is also grateful to Antonio Ambrosetti, Etienne Emmrich, and Petra Wittbold for giving him the opportunity to teach these courses. Finally, he wishes to thank Louis Jeanjean for helpful discussions regarding Section 3 and useful comments on a preliminary version of these notes, and Petra Wittbold and Etienne Emmrich for their careful reading of the manuscript and valuable suggestions.

Author information

Stefan Le Coz, SISSA, via Beirut 2-4, 34014 Trieste, Italy. E-mail: lecoz@sissa.it