Study of a dissipative NLS equation

Study of a dissipative NLS equation

Tesi di Laurea Magistrale

CANDIDATO: Daniel Saint-Geours RELATORE: Prof. Vladimir Georgiev Anno Accademico 2019/2020

1 Basic Notions 2 1.1 Fourier transform . . . 2 1.1.1 Fourier transform on L1_(Rn) . . . 4 1.1.2 Fourier transform on L2_(Rn) . . . 4 2 Solitary waves NLS 5 2.1 The problem . . . 5 2.2 Main results . . . 6 2.3 Compactness principle . . . 7 2.4 Theorem (2.2) . . . 9 2.5 Theorem (2.3) . . . 15 2.6 N -dimensional case . . . . 18 2.7 Regularity . . . 20 2.8 Extensions . . . 21

3 More on the existence of Solitary waves 23 3.1 Concentrated Compactness method . . . 23

3.1.1 Translation independent case . . . 24

3.1.2 Translation dependent case . . . 27

3.2 Extensions . . . 28

4 Non-linear evolution problem 30 4.1 Stationary phase method . . . 34

4.2 Uni-dimensional case . . . 36 4.2.1 Strichartz estimates . . . 36 4.2.2 Local-existence of solutions . . . 39 4.2.3 Global-existence of solutions . . . 42 4.3 Multi-dimensional case . . . 48 4.3.1 Strichartz estimates . . . 49 4.3.2 Local-existence of solutions . . . 50 5 Stability 52 5.1 Ground state stability . . . 52 5.1.1 Introduction to the problem of ground state stability . 52

5.2 Orbital stability . . . 59 5.2.1 Introduction to the problem of orbital stability . . . . 59 5.2.2 Orbital stability for the problem . . . 60 5.2.3 Conclusions . . . 61

In this paper we introduce a new non-linear dispersive Schrödinger equation, which arises for non-linear fibre optic. The problem was taken from [1] and is part of a study of dispersive schockwaves in nematic fluids. The problem was then remodelled as:

(i∂t− A)φ := φ|φ|p−2V (x) φ : Rt× Rx → R, (1)

where A is the following operator:

A := −1

2∂ 2

x− iµ∂3xϕε(D), µ > 0,

where ϕε(D) is the convolution kernel such that ϕˆεu(ξ) = ˆϕε(ξ)ˆu(ξ) and we

defined D := ∂x

i , and ϕε is the Yoshida approximation of the unity:

ϕε:=

1 1 + εξ2.

p is an exponent such that p ∈ (2, 6) and V is constant. We can weaken the

assumptions on V , as will be done later in the paper.

The objectives of our work are mainly focused on the existence of ground states and the time evolution problem in dimension one. We found some interesting result regarding the existence of ground states using the concen-trated compactness lemma, afterward we were able to deduce results both for local and global existence of solutions. Throughout the paper some of the main results will be generalized in the multi-dimensional case, but we especially focused on the uni-dimensional case.

Lastly, we tried to introduce the problem of ground states’ stability, which, due to demanding it can be, will not be treated exhaustively. Even though the problem of stability is not formally treated, we tried to introduce it in order to hopefully further tackle such topic in the upcoming future.

Basic Notions

In this chapter we will briefly introduce some basic tools that will be used throughout this paper.

1.1

Fourier transform

Let us introduce the Schwartz space, the perfect place to introduce the Fourier transform.

Definition 1. The Schwartz space is the topological space of rapidly decaying

smooth function, that is f ∈ C∞_(Rn) such that

xα∂βf → 0 as |x| → ∞

for every multi-indices α, β ∈ Nn.

Let α, β ∈ Nnand f ∈ C∞_(Rn) we can define the norm k · k_α,β as follows: kf k_α,β = kxα∂βf k∞

and let

k · kS := sup

α,β∈Nn

k · kα,β.

Therefore, the Schwartz space is defined by

S(Rn) := {f ∈ C∞| kf kS < +∞} .

Observation 1.1.

C_c∞_(Rn_{) ⊂ S(R}n).

We are now able to define the Fourier transform for a Schwartz function:

Definition 2. Let f ∈ S(Rn), we define the Fourier transform of f as follows ˆ f (ξ) := Z Rn f (x)e−ihx,ξidx. 2

Observation 1.2. We can easily check that if f ∈ S(Rn), then ˆ_{f ∈ S(R}n). The above observation implies that the linear map

F : _S(Rn_{) → S(R}n)

f → ˆf

is well defined. But the map F enjoys more properties: indeed define ˇ f (x) := 1 (2π)n Z Rn f (ξ)eihx,ξidξ, then we have

Theorem 1.3. The map F is a one-to-one map, continuous map, with

inverse given by

F−1(f ) = ˇf .

The function ˇf is the inverse Fourier transform of f .

We can now list some of the basic properties of the Fourier transform.

Proposition 1.4. Let f ∈ S(Rn) then the followings hold:

1. ˆ f _L∞ ≤ kf kL1. 2. if τ_h(f )(x) = f (x − h), then τhˆ(f )(ξ) = e−iξhf .ˆ 3. eihxˆ_{f (ξ) = τ} h( ˆf ). 4. if σ_δ(f )(x) = _δ1nf (x_δ), then σδˆ(f )(ξ) = ˆf (δξ). 5. ∂xˆif = −iξif .ˆ 6. f ∗ g(ξ) = ˆˆ f (ξ)ˆg(ξ).

From property 5 of the previous proposition, it easily follows that for any differential operator

P (f ) := X

|α|≤m

cα∂αf

the composition with the map F works as a multiplication, in the sense of ˆ

P (f )(ξ) = P (ξ) ˆf (ξ),

where P (ξ) now denotes the polynomial

P (ξ) = X

|α|≤m

cα(−i)|α|ξα

and c_α are complex constants.

Theorem 1.5. For every f, g ∈ S(Rn) we have the following identity: (2π)nhf, gi = h ˆf , ˆgi,

where hf, gi =R

Rnf gdx.

In particular we obtain the Plancharel identity

(2π)nkf k2_L2 = ˆ f 2 L2. 1.1.1 Fourier transform on L1_(Rn)

Let f ∈ L1_(Rn), thus the integral that defines the Fourier transform is absolutely convergent. consequently, ˆf can be defined as the uniform limit

of any Schwartz’s approximation of f . This imply that ˆf is the uniform

limit of continuous functions vanishing at infinity, hence we obtain the Riemann-Lebesgue lemma.

Theorem 1.6. The Fourier transform is a bounded linear map F : L1_(Rn) → C0_(Rn_{) and} ˆ f _L∞ ≤ kf kL1 for every f ∈ L 1 (Rn),

Therefore, the norm of the map F : L1_(Rn) → C0_(Rn) is at most one.

1.1.2 Fourier transform on L2_(Rn₎

The Parseval’s formula implies that the Fourier transform preserves the L2 -inner product and norm, up to factors of 2π. It follows that we may extend the Fourier transform by density and continuity from S to an isomorphism of L2, with the same properties. Indeed given f ∈ L2, take any Schwartz’s approximation f_k of f in L2. Then by Plancharel’s Theorem ( ˆfk)k is a

Cauchy’s sequence in L2 and therefore, admits a L2-limit, which we define to be the Fourier transform of f . Then we have a similar Theorem as in the Schwartz space.

Theorem 1.7. The Fourier transform F : L2→ L2_{is a one-to-one, bounded,}

linear map. If f, g ∈ L2_{, then}

(2π)nhf, gi = h ˆf , ˆgi, where hf, gi =R

Rnf gdx.

In particular we obtain the Plancharel identity

(2π)nkf k2_L2 = ˆ f 2 L2.

Solitary waves NLS

We will introduce and give the main results for the 1 dimensional case (x ∈ R), afterward we will discuss how it is possible to modify them for the general case of dimension N ≥ 2.

2.1

The problem

In this chapter the main focus is the research of solitary waves for our NLS problem, which we recall to be:

(i∂t− A)φ := φ|φ|p−2V (x) φ : Rt× Rx → R, (2.1)

where A is the following operator:

A := −1

2∂ 2

x− iµ∂3xϕε(D), µ > 0,

where ϕ_ε(D) is the convolution kernel such that ϕˆεu(ξ) = ˆϕε(ξ)ˆu(ξ) and we

defined D := ∂x

i , and ϕε is the Yoshida approximation of the unity:

ϕε:=

1 1 + εξ2.

p is an exponent such that p ∈ (2, 6) and V is a function satisfying the

following properties: • V ∈ Cb(R),

• V ≤ 0 _{∀x ∈ R,}

• V → V∞< 0 as |x| → ∞.

Finally µ and ε are positive constant arbitrarily small, which will be fixed later. We will now explain how to prove the existence of solitary waves, but first:

Definition 3. A solitary wave φ(t, x) is a solution of NLS of the form

φ(t, x) = eiωtu(x),

where ω ∈ R and u is a real function such that u(x) → 0 as x → ∞ in some weak sense.

Observation 2.1. Let φ be a solution to (2.1) then observe that

t → kφ(t, x)k_L2

x

is constant: indeed multiplying (2.1) by ¯φ and taking the imaginary part we

get

∂tkφ(t, x)kL2

x = 0.

Moreover if a solitary wave φ = eitωu(x) is a solution to 2.1, than u

satisfies the differential equation

Au + ωu = −V (x)|u|p−2u (2.2) Then it is natural to check, for the existence of solitary waves for (2.1), the following minimization problem:

Iλ := inf

S(λ)E(u), (2.3)

where E is the functional given by

E(u) := 1 2hAu, ui + 1 p Z R V (x)|u|pdx (2.4) and S(λ) is the following subset of H1_(R),

S(λ) = {u ∈ H1(R)| kukL2 = λ}.

We will use the abuse of notation hAu, ui even though the principle part of the operator has to be intented in the weak formulation as the problem is studied in H1_(R).

2.2

Main results

Before checking the well posedness of the problem (2.3), let us enunciate our main results.

Theorem 2.2. Suppose V ≡ V∞ < 0 is constant and p ∈ (2, 6). Then every minimizing sequence of (2.3) is relatively compact in H1_{(R) up to} translations. In particular (2.3) admits a minimum.

Theorem 2.3. Suppose V (x) < V∞ a.e and p ∈ (2, 6). Then every mini-mizing sequence of (2.3) is relatively compact in H1_{(R). In particular (2.3)} admits a minimum.

The strategy to prove the two Theorems is to exploit the well-known

2.3

Compactness principle

Lemma 2.4. Let (ρ_n) ⊂ L1_(RN) such that, ρ_n ≥ 0 and kρ_nk_L1 = λ ∀n ∈ N. Then, up to subsequences, (ρn)n satisfies one of the three following

possibilities: (i) Compactness

There exists yn ∈ RN such that the family (ρn(· + yn))_n is tight in

L1_(RN_{), that is ∀ε > 0 ∃R < ∞ such that}

Z yn+BR ρn≥ λ − ε ∀n ∈ N; (ii) Vanishing lim n sup y∈RN Z y+BR ρn= 0 ∀R < ∞; (iii) Dichotomy

There exists α ∈ (0, λ) such that ∀ε > 0 there exist k0 ∈ N and

ρ1_n, ρ2_n∈ L1_(RN_{) positive, satisfying for n ≥ k}

0:                ρ1_{n L}1 − α  < ε   ρ2_{n L}1 − (λ − α)  < ε ρ_n− ρ1_n− ρ2_{n L}1 < ε

dist supp(ρ1_n), supp(ρ2_n)

→

n ∞

Proof. Let us introduce the concentration function given by Qn(t) := sup

y∈RN

Z

y+Bt

ρn(ξ)dξ.

We observe that (Qn)nis a sequence of nondecreasing, nonnegative, uniformly

bounded functions such that limnQn(t) = λ, which means that, up to a

subsequence, there exists a nondecreasing, nonnegative bounded function

Q(t) which is the pointwise limit of the extracted subsequence (observe that

(Qn)n is uniformly bounded in BV (0, T ) ∀T > 0).

For the sake of clean notation we will not rename said subsequence. Let α = lim

t→∞Q(t) and observe that α ∈ [0, λ].

• α = 0

Clearly, by definition of Q, (ii) occurs. • α = λ

In this case (i) occurs.

Take µ ∈ (λ₂, λ), then there exists R = R(µ) such that for all n ≥ 1 Qn(R) > µ.

For this it is sufficient to observe that we can choose R big enough such that Q(R) > µ then the above statement is true for all k ≥ k1, up to choosing R0 > R the statement holds for every k ≥ 1. For the

definition of Qn we can find yn= yn(µ) such that

Z

yn+BR

ρn(ξ)dξ > µ.

Let R(λ₂) and (y_n)_n be, respectevely, the radius and the sequence of centers constructed as above for µ = λ₂ and observe that we have

|y_n(µ) − y_n| ≤ R(µ) + R(λ 2), otherwise kρnkL1 ≥ Z yn(µ)+BR(µ) ρn(ξ)dξ + Z yn+BR(λ₂) ρn(ξ)dξ > λ.

Then by choosing R0(µ) = R(λ₂) + 2R(µ) we obtain the claim. • α ∈ (0, λ)

Finally we have to show that (iii) holds. Let ε > 0 and choose R such that Q(R) > α − ε.

Then definitely we have Qn(R) ∈ (α − ε, α + ε), moreover we can find

a sequence R_n% ∞ such that Q_n(R_n) ≤ α + ε.

Finally we can find y_n, by the definition of Q_n, for which we have:

Z

yn+BR

ρn(ξ)dξ ∈ (α − ε, α + ε).

Then we set ρ1_n= ρ_n1_yn+BR, ρ2_n= ρ_n1_yn+Bc

Rn and all that is left is to

check the properties.

Clearly dist supp(ρ1_n), supp(ρ2_n)

→

n ∞.

The choice of R, Rn and yn implies

   ρ 1 n _L1− α   < ε and    ρ 2 n _L1 − (λ − α)   < ε. Finally ρn− ρ 1 n− ρ2n _L1 = Z R≤|ξ−yn|≤Rn ρn(ξ)dξ ≤ Q_n(R_n) − Q_n(R) − 2ε ≤ 4ε

This is the most known version of the concentrated compactness lemma regarding distributions, but for our purposes we will give a less general variant.

Lemma 2.5. Let (u_n)_n be a boundend sequence in H1_(RN) such that ku_nk2_L2 = λ. Then ρn= u2n satisfies the Concentrated compactness Lemma.

As a result, up to subsequences, either compactness (i) occurs, vanishing (ii) occurs or dichotomy occurs as follows: there exists α ∈ (0, λ) such that for all ε > 0, there exist k₀ ≥ 1, u1

n, u2n bounded in H1(RN) satisfying for all

n ≥ k0 all the properties already listed in the classical version of the lemma

and the following:

lim n hAun, uni ≥ lim n hAu1_n, u1_ni + lim n hAu2_n, u2_ni.

Proof. Let R, R_n and y_n be the same as the previous proof, ϕ ∈ C_c∞_{(R) a} cutoff function such that ϕ ≡ 1 if |x| ≤ 1 and ϕ ≡ 0 if |x| ≥ 2 and set

ξ = 1 − ϕ. We define ϕR= ϕ(_R·), and similarly ξR.

Finally we set u1

n= unϕR(· − yn) and un2 = unξRn(· − yn).

The first properties follows as in the previous proof. Now for the last one, we first suppose that the function u → hAu, ui is well defined on H1(RN) (which will be proved later).

From what we have just proved we can write un = u1n+ u2n+ vn, where

u1_nu2_n= u_n1vn= u2nvn= 0 a.e.

Then

hAun, uni = hAu1n, u1ni + hAu2n, u2ni + hAvn, vni

≥ hAu1

n, u1ni + hAu2n, u2ni,

where the last inequality follows from the fact that u → hAu, ui is a positive real function (as will be proved later).

2.4

Theorem (2.2)

First of all we will prove the well-posedness of the problem (2.3) (when µ is small), that is

Iλ > −∞ ∀λ,

but before that let us make some observations on the bilinear form hAu, vi.

Observation 2.6. It is easy to check that the bilinear form hAu, vi is in

fact a scalar product, thanks to the fact that A is a skew-adjoint operator. Moreover hAu, ui ≥ 0, indeed from Parseval formula follows that

where ˆu is the Fourier transform of u. By well known properties of the

Fourier transform we get

hAu, ui = c

Z

g(ξ)|ˆu(ξ)|2dξ,

where g(ξ) = ξ₂2 + µξ3ϕε(ξ).

The main point is to choose µ and ε such that there exist c1, c2 > 0 costants satisfying c₁ξ2 ≤ g(ξ) ≤ c₂ξ2 and therefore,

0 ≤ ˆc1k ˙uk2L2 = ˜c1 Z ξ2|ˆu|2 ≤ c Z g(ξ)|ˆu|2 ≤ ˜c2 Z ξ2|ˆu|2 = ˆc2k ˙uk2L2. Observe that if |ξ| >> 1, then ξ3ϕε(ξ) ∼ ξ_ε, thus we can find Rε >> 1 such

that g(ξ) ≥ ξ₄2 for |ξ| ≥ Rε. Up to choosing R0ε> Rε the inequalty g(ξ) ≤ ξ2

holds.

If |ξ| << 1 observe that ξ3ϕε(ξ) ∼ ξ3, then similarly we can find rε<< 1

such that for |ξ| ≤ r_ε the same inequalities hold.

Lastly, for |ξ| ∈ [rε, Rε] we can find c1, c2 constants to get the desired inequalities, since we are working on a compact subset this can be easily achieved by choosing µ = µ(ε) small enough.

Observation 2.7. The above series of inequalities prove that E(u) < ∞ for

all u ∈ H1_{(R). Indeed we have}

E(u) ≤ c2kuk2_H1 + kV k_L∞kukp_Lp ≤ c2kuk2_H1 + ˜C kukp_H1 < +∞. Clearly enough hAu, ui = 0 if and only if u = 0, which means that

u → hAu, ui12 is a norm on H1(R) which we will denote as k·k

A.

What is more interesting is that the norm k·k_A is equivalent to the usual norm of H1_{(R), as proved by the inequalities of (2.6).}

We are now ready to prove the well-posedness of (2.3)

Proposition 2.8. The problem (2.3) is well-posed for every λ > 0, that is

Iλ > −∞ ∀λ > 0.

Proof. Let us recall the Gagliardo-Niremberg inequality in dimension one:

kuk_Lp ≤ cpkuk1−θL2 k ˙ukθ_L2, (2.5) for p > 2 and θ = p−2_2p . Recalling the equivalence of the two norms, we get

kuk_Lp ≤ ˜cpkuk1−θL2 hAu, ui

θ

Therefore, using (2.6), we get, for every u ∈ S(λ),     Z V (x)|u|p     ≤ c(p, λ) kV k_L∞hAu, ui pθ 2, (2.7) but if p < 6, then pθ₂ < 1. Finally using (2.7) E(u) ≥ 1 2kuk 2 A− c kuk pθ A

and the RHS is bounded from below, which proves the proposition.

Thanks to this proposition we can show that every minimizing sequence is bounded in H1_(R).

Proposition 2.9. Let (un)n be a minimizing sequence of (2.3) for λ > 0

fixed, then (u_n)_n is bounded in H1_(R).

Proof. Since (un) is minimizing and thanks to (2.7) we can find C1, C2∈ R such that

C1≤ E(un) ≤ C2. From which follows that

C2 ≥ E(un) ≥ 1 2kunk 2 A− c kunk p−2 2 A , which implies ku_nk_A≤ C.

Then from (2.6) we get k ˙unkL2 ≤ ˜C, recalling that for every minimizing sequence of (2.3) holds ku_nk2_L2 = λ we get the thesis.

Observation 2.10. Iλ < 0 for all λ > 0.

Proof. Let u ∈ C_c∞_{(R) such that kuk}2_L2 = λ and set uθ :=

√ θu(θ·). Then we get E(uθ) = kuθk2A+ θp2−1 p Z V (x)|u|p.

Using the observation made before, it is easy to check that ku_θk_A= o(θ2_{) as}

θ ∼ 0, while p₂ − 1 < 2, which is enough to conclude.

Observation 2.11. For every θ > 1 and λ > 0 we have I_θλ< θIλ.

Proof. Let u ∈ C_c∞_{(R) such that kuk}2_L2 = λ and set uθ :=

√ θu. Accordingly, Iθλ≤ E(uθ) = kuθk2A+ θp2 p Z V (x)|u|p ≤ θ kuk2_A+θ p 2 p Z V (x)|u|p ≤ θp2E(u).

Recalling that I_λ < 0, taking a minimizing sequence in C_c∞(R), we can conclude since p₂ > 1.

Finally we can prove the fundamental property for the application of the concentration compactness principle: the strict subadditivity.

Proposition 2.12. Given λ > 0 for every α ∈ (0, λ) the following holds

Iλ < Iα+ Iλ−α. Proof. Suppose α ≥ λ − α, Iλ = Iλ αα < λ αIα = Iα+ λ − α α Iα≤ Iα+ Iλ−α.

Before getting to the proof of Theorem 1 we need the following Lemma, which will be crucial to prove Theorem (2.2).

Lemma 2.13. Let u ∈ H1_{(R) and I}

k= [k, k + 1] then • kukp_Lp ≤ cp sup k∈Z kuk_L2_(I k) !p−2 kuk2_H1 if 2 < p ≤ 6, • kukp_Lp ≤ cp sup k∈Z kuk_L2_(I k) !p+2 2 kuk p−2 2 H1 if p ≥ 6.

Proof. Recall that from the Gagliardo-Niremberg-Sobolev inequality we have

kukp_Lp_(I k) ≤ cpkuk p+2 2 L2_(I k)kuk p−2 2 H1_(I k) (2.8) - Case p ≥ 6

Then p−2₂ ≥ 2 and we can write k ˙uk

p−2 2 L2 = k ˙uk 2 L2k ˙uk p−6 2 L2 . So in (2.8) we have the product of the bounded sequence

ak:= kuk p+2 2 L2_(I k)k ˙uk p−6 2 L2_(I k) kakk∞≤ sup k∈Z kuk_L2_(I k) !p+2₂ kuk p−6 2 H1

and the summable sequence

bk:= cpkuk2L2_(I

k)

X

bk= cpkuk2L2 ≤ cpkuk2H1 from which easily follows the claim.

- Case 2 < p ≤ 6

Then p−2₂ ≤ 2 and 6−p₂ = 2 −p−2₂ < 2 and we get

kuk p+2 2 L2_(I k)k ˙uk p−2 2 L2_(I k)≤ kuk p+2 2 − 6−p 2 L2_(I k) kuk p−2 2 + 6−p 2 H1_(I k) = kuk p−2 L2_(I k)kuk 2 H1_(I k)

We are now ready to proceed with the proof of Theorem (2.2).

Proof. The proof relies on the use of the variation of the concentrated

compctness Lemma given in (2.5) and the goal is to prove that vanishing (ii) and dichotomy (iii) cannot occur. Let (u_n)_n be a minimizing sequence for (2.3), thanks to (2.9) the hypothesis of (2.5) are satisfied. We will not rename the extracted subsequence given by the Lemma.

- Step 1: vanishing does not occur. If (ii) occurs then

lim

n sup_k∈ZkunkL2(Ik)= 0,

where we have denoted with Ik the interval [k, k + 1]. Then, using

Lemma (2.13), un→ 0 strongly in Lp, indeed we have

ku_nkp_Lp ≤ cp sup k∈Z ku_nk_L2_(I k) !p−2 ku_nk2_H1,

taking the limit on n we get the claim, recalling that (un)nis bounded

in H1_(R).

But this means that

Z

V (x)|un|p → 0,

implying I_λ ≥ 0 which is a contradiction since we have proved in (2.10) that I_λ < 0 for every λ > 0.

- Step 2: dichotomy does not occur.

Suppose there exists α ∈ (0, λ) such that for ε > 0 fixed we can find

u1

n, u2n and vn satisfying the dichotomy properties, where vnis defined

as vn:= un− u1n− u2n. In particular we know that

    u 1 n 2 L2 − α    < ε     u 2 n 2 L2− (λ − α)    < ε,

then, up to subsequences, there exist αε ∈ (α − ε, α + ε) and βε ∈

(λ − α − ε, λ − α + ε) such that u 1 n 2 L2 → αε u 2 n 2 L2 → βε.

Since the supports of u1_n, u2_n and vn are a.e disjoint we can write

Z V (x)|un|p= Z V (x)|u1_n|p₊Z _{V (x)|u}2 n|p+ Z V (x)|vn|p,

observing that from GN     Z V (x)|vn|p     = |V∞| kvnkpLp≤ cpkvnk p+2 2 L2 kvnk p−2 2 H1 ≤ δ_ε−ε→0−−−→ 0,+

where the last inequality follows from the fact that (v_n)_nis bounded in

H1(R) (due to the fact that vn= un− u1n− u2n) and that kvnk2L2 < ε. Putting everything together:

Iλ= lim n E(un ) ≥ lim n E(u 1 n)+ lim n E(u 2 n) − δε≥ Iαε + Iβε − δε, letting ε → 0, we get Iλ≥ Iα+ Iλ−α,

which is in contradiction with the strict subadditivity. - Step 3: conclusion.

Now we know there exists a sequence (yn)n⊂ R such that the family

(u_n(· + y_n))_n is tight in L2, let ¯un= un(· + yn) and observe that (¯un)n

is bounded in H1_{(R), then, up to subsequences not renamed, it admits} a weak limit which will be denoted as ¯u.

Given ε > 0 there exists R < ∞ such that R

BRu¯

2

n ≥ λ − ε for all n,

then testing the weak convergence with the function1BR we get

k¯uk2_L2 ≥ Z BR ¯ u2= lim n Z BR ¯ u2_n≥ λ − ε. (2.9)

This means k¯uk_L2 ≥ lim_nk¯unkL2 and recalling that the L2 norm is weak-SCI, we obtain the equality, which implies the convergence is strong in L2.

From this we get that (¯un) converges strongly to ¯u also in Lp, indeed

using the usual GN inequality

k¯u − ¯umkpLp≤ cpk¯u − ¯umk p+2 2 L2 u˙¯_n− ˙¯u_m p−2 2 L2 ,

meaning that (¯un) is of Cauchy in Lp, assuring its convergence as

stated.

Our next, claim is that ¯u is a minimum of (2.3).

Suppose this is true then lim

This is holds because the function u →R

V |u|p is strongly continuous in L2 (thanks to GN). Now we prove that if this holds than the conver-gence to ¯u must be strong in H1(R):

u˙¯n− ˙¯u 2 L2 ≤ chA¯un− ¯u, ¯un− ¯ui = c   hA¯un, ¯uni + hA¯u, ¯ui | {z } I − 2hA¯un, ¯ui | {z } II   ,

the first term obviously tends to 2hA¯u, ¯ui, so does the second term due

to the weak convergence in H1_(R).

In fact, let v_n* v in H1_{(R) then we have} hA(v_n− v), wi ≤ ˜c

Z

( ˙v_n− ˙v)w → 0,

where the first inequality is obtained through Parseval’s formula as already seen different times.

All that is left to prove is that ¯u is a minimum point for (2.3). Since

we already proved the strong continuity of the function u →R

V |u|p, it is sufficient to observe that thanks to the weak-SCI of the norm it holds

lim

n

hA¯un, ¯uni ≥ hA¯u, ¯ui

and the proof is completed.

2.5

Theorem (2.3)

Let us now focus on the proof of Theorem (2.3), although very similar to the previous there are some essential differences to point out.

First we need to introduce the problem at "infinity":

I_λ∞:= inf

u∈S(λ)E

∞_{(u), (2.10)}

where E∞ is the functional defined as

E∞(u) := hAu, ui +1

pV

Observation 2.14. Let (un)n be a minimizing sequence for (2.3) and

con-sider ¯un= un(· + yn), where ynis a sequence such that |yn| % +∞.

Then it follows that

Iλ = lim n E(¯un ) = lim n E ∞_(u n) ≥ Iλ∞.

Moreover from the assumption V (x) < V∞a.e we obtain the strict inequality. Observe that the arguments used in observation (2.10) work for I_λ∞ proving that for all λ > 0 we have I_λ∞< 0.

In the case translation-dependent the main difference is in the subaddi-tivity condition, in fact it reformulates as

Iλ< Iα+ Iλ−α∞ for all α ∈ [0, λ) , (2.11)

where we define I₀ := 0.

Proposition 2.15. The condition of subadditivity (2.11) holds for (2.10)

for all λ > 0

Proof. From (2.12) and the observation (2.14) follows:

Iλ< Iα+ Iλ−α ≤ Iα+ Iλ−α∞ for all α ∈ (0, λ).

We still need to prove that the condition holds for α = 0, but this is the case

Iλ < Iλ∞,

which has already been proved to be true.

We are now ready to prove Theorem (2.3).

Proof. We will proceed as in Theorem 1. Let (un)nbe a minimizing sequence

of (2.3).

- Step 1: vanishing does not occur. Suppose vanishing (ii) occurs, then

Z |V (x)−V∞||u_n|p ≤ kV (x)−V∞k∞ Z |x|≤R |u_n|p+ Z |x|≥R |V (x)−V∞||u_n|p,

the first term is arbitrarily small choosing n big enough, since ku_nkp_Lp ≤ cpkunk p+2 2 L2 kunk p−2 2 H1

and the RHS vanishes as n → ∞ for hypothesis; the second term goes to 0 as R → ∞ for dominated convergence.

Putting everything together, we obtain

Iλ≥ Iλ∞,

which is a contradiction.

- Step 2: dichotomy does not occur.

Suppose there exists α as in the Lemma (2.5) and fix ε > 0. Let (yn)n, R, (Rn)n as in the proof of (2.5), then we distinguish two cases:

(i) (yn) is bounded. Z |V (x) − V∞||u2_n|p= Z yn+BRn |V (x) − V∞||u_n|p → 0. (ii) (y_n) is not bounded.

Then, up to subsequences, we can assume |yn| → ∞ and therefore,

Z

|V (x) − V∞||u1_n|p =

Z

yn+BR

|V (x) − V∞||un|p → 0.

In either cases we proceed as in the proof of Theorem (2.2) and we end up with one of the following

Iλ≥ Iα+ Iλ−α∞ ,

Iλ ≥ Iα∞+ Iλ−α.

Both of those contradict the subadditivity condition (2.11). - Step 3: conclusion.

Now we know there exists a sequence (y_n)_{n⊂ R such that the family} (un(· + yn))n is tight in L2, let ¯un = un(· + yn).Now that we have

observed this, we can apply the same reasoning as in Theorem (2.2) to conclude, up to subsequences, that ¯un→ ¯u strongly in L2.

Using GN we obtain the strong convergence of ¯un in Lp and as a result,

the convergence of the second term of the operator E due to his strong-continuity. Applying the same logic used in the proof of the previous Theorem we would end up showing that ¯unconverges strongly in H1(R),

but our claim is that, up to subsequences, un converges strongly in

H1(R). To prove this is sufficient to show that yn is bounded.

Suppose it is not true, then, up to subsequences, |yn| → ∞. Since the

sequence ¯un converges strongly in Lp, we can see that

Z

|V (x) − V∞||u_n|p₌Z _{|V (x + y}

which again contradicts the subadditivity condition since it implies that

Iλ≥ Iλ∞.

This ends the proof.

2.6

N -dimensional case

Let us now introduce the N -dimensional case and point out the few modifi-cations needed to apply the results obtained previously.

The NLS-equation is

(i∂t− A)φ := φ|φ|p−2V (x) φ : Rt× RNx → R, (2.12)

where A is the following operator:

A := −1

2∆x− iµP ϕε(D), µ > 0,

ϕε(D) denotes once again a convolution kernel such that ˆϕεu(ξ) = ˆϕε(|ξ|)ˆu(ξ)

and ϕε is once again the smooth function

ϕε(ξ) :=

1 1 + ε|ξ|2,

P is a differential operator of the form P := X

|α|=3

∂_xα and α ∈ NN multiindex

Finally p ∈ (2, l), where l = 4+2N_N and V is a function satisfying the following properties:

• V ∈ Cb(RN),

• V ≤ 0 _{∀x ∈ R}N_,

• V → V∞_{< 0 as |x| → ∞.}

The research of solitary waves proceed as for the one dimensional case, studying the already introduced problem:

Iλ := inf S(λ)E(u),

(2.13) where E is the functional given by

E(u) := 1 2hAu, ui + 1 p Z RN V (x)|u|pdx (2.14)

and S(λ) is the following subset of H1_(RN),

S(λ) = {u ∈ H1(RN)| kuk_L2 = λ}.

The main difference is given by the symbol denoted as g(ξ), in fact using Parseval formula

hAu, ui = ch ˆAu, ˆui =

Z

RN

h(ξ)|ˆu|2dξ,

where the symbol h is given by:

h(ξ) = |ξ|

2

2 + µP (ξ)ϕε(|ξ|).

Observation 2.16. The following holds true

0 ≤ ˆc1k∇uk2L2 ≤ c

Z

h(ξ)|ˆu|2≤ ˆc2k∇uk2L2.

The only difference left is that the GN inequality (2.5) does not hold true in dimension N , at least not in that formulation:

Proposition 2.17. There exist Kp such that for every u ∈ H1(RN):

kuk_Lp≤ Kpk∇ukθL2kuk1−θ_L2 ,

where θ = N (p−2)_2p for N > 2 and 2 < p < 2∗ = _{N −2}2N or N = 2 and p > 2.

Thanks to GN we conclude that for every u ∈ S(λ) we have E(u) > −∞. Indeed observe that

    Z RN V (x)|u|p     ≤ c(λ, p) k∇ukpθ_L2 ≤ ˜c(λ, p)hAu, ui pθ 2 ,

now p ∈ (2, l) implies that pθ₂ < 1 assuring the boundedness from below as

in the 1-dimensional case.

Almost all of the properties given in the 1-dimensional case and those used for proving (2.2) and (2.3) are dimensional independent. The few differences (the properties following the uni-dimensional GN) can be resolved thanks to (2.17).

There is one Lemma heavily used for the proves of the two Theorems which cannot be easily translated in the multi-dimensional case, that is Lemma (2.13). Using this Lemma proving that vanishing could not occur was easy. In the multi-dimensional case we can still get a similar Lemma, but the proof has to be adjusted.

Lemma 2.18. Let u ∈ H1_(RN) then there exist a countable family of disjoint

balls of fixed radius R < ∞ (Bj)_j∈J such that RN ⊂S j

5Bj and moreover the

following holds true:

kukp_Lp ≤ cp sup j∈J kuk_L2_(B j) !p−2 kuk2_H1 if 2 < p ≤ l,

where 5B is the ball of same center as the ball B and radius 5 five times the one of B.

The proof of the Lemma proceeds as in (2.13) given the existence of such family of balls. This property follows from Vitali’s Covering Lemma.

Now the result holds true even in the case of dimension N and the proof follows the same steps as in the uni-dimensional case, having care of using the above Propositions and Lemmas instead of their uni-dimensional counterpart.

2.7

Regularity

We have proved that there exists u ∈ S(λ) weak solution of (2.13), that means that

δE(u, v) − ρu = 0 for every v ∈ C_c∞_(RN),

where δE(u, v) denotes the first variation of E in v and ρ is the langrangian multiplier.

We ask ourselves whether it is possible to obtain more regularity for u, in particular we question if u ∈ H2_(RN).

Definition 4. Let u ∈ H1_(RN) and g ∈ L2_(RN). We say that g is the laplacian of u in distributional sense if

Z RN h∇u, ∇ϕidx = − Z RN gϕdx for every ϕ ∈ C_c∞_(RN). We will write ∆u = g.

Theorem 2.19. Let u ∈ H1_(RN) with distributional laplacian ∆u ∈ L2_(RN).

Then u ∈ H2_(RN) and the following estimate holds: kuk2 2,2,RN ≤ 3 2 h kuk2 L2_(RN₎+ k∆uk2_L2_(RN₎ i .

We shall observe that

δE(u, v) = hAu, vi + 1 p

Z

Therefore, we obtain that 1

2∆u = λu − V (x)|u|

p−2_{u + iµP (D)ϕ(D)u}

in weak sense.

The question that arise is: does the RHS belong to L2? Certainly u ∈ L2 and so does the last term; indeed we have

kP (D)ϕ(D)uk_L2 = c kP (ξ)ϕ(ξ)ˆuk_L2 ≤ ˜c |ξ| 2_{uˆ L}2 = c1k∇ukL2 < +∞. If we show that |u|p−2u ∈ L2 _{we can conclude that ∆u ∈ L}2 _{and use Theorem} (2.19). That is equivalent to prove that u ∈ L2(p−1).

Consequently, for the cases N = 1, 2 the claim holds true, in fact thanks to Sobolev embeddings we know that u ∈ Lq for every q ≥ 2.

In the general case n ≥ 3 if 2(p − 1) ≤ 2∗ the Sobolev embeddings, once again, prove that the claim, where 2∗ is the Sobolev conjugate exponent. In those cases we can conclude that the weak solutions u lays in H2_(RN).

2.8

Extensions

In this section we will try to weaken the hypothesis on the potential, given it is not constant, allowing a more general environment for the study of the problem.

First of all let us point out that the condition on the sign of V (x), although clearly sufficient, it is not necessary.

Observation 2.20. Suppose V ∈ Cb(RN) such that V (x) → V∞as |x| → ∞.

If there exists x0 ∈ RN such that V (x0) < 0 and V (x) ≤ V∞, with the inequality being strict over a set of positive measure, then the Theorem (2.3) still hold.

The second condition is sufficient to prove that we still have the property

Iλ < Iλ∞ for every λ > 0.

The local condition of the sign of the potential is what assures that the infimum is negative. Consequently, following the same approach to the problem as in the less general case already treated, we can prove the existence of solitary wave.

To prove that I_λ < 0 take u = c1J, where J is a small neighborhood of x0 such that V (x) < 0 on J and c is a positive constant such that the constraint is satisfied. Therefore, Iλ≤ E(u) = cp p Z J V (x) < 0.

Observe the necessarity of the condition V (x0) < 0 for some x0 ∈ RN. In fact V ≥ 0 would imply that I_λ = 0 for every λ > 0 and the problem admits no minimum.

We clearly have Iλ ≥ 0, let un= cn1Jn, where (Jn) is a sequence of measurable

sets, with positive measure, and cn is a constant such the costraint satisfied.

Observe that ku_nk2_L2 = λ implies L(Jn) = _cλ2

n, hence,

E(un) ∼ cp−2n .

Choosing c_n→ 0 and J_nproperly it is obvious that such minimizing sequence proves Iλ= 0.

Finally we claim that the hypothesis V∞ < 0 is not necessary in the

translation-dependent case, whereas it is in the translation-independent case as we have just shown. This will be treated in the next section, as it is important to define a different problem, which gives the same minimizers.

More on the existence of

Solitary waves

We will focus our attention once again to the case N = 1, but before we begin our study we want to underline the fact that the logic used in this section can be easily adjusted to the case N ≥ 2.

Up until now we have proved the existence of weak solutions for the case

p < 6, we now want to study the other case, p ≥ 6. In the general case of

dimension N ≥ 2 we will restrain the interval of p to [l, 2∗) . Let us recall the problem:

(A + ω)u = −V (x)|u|p−2u. (3.1) First thing is to observe that ω has to be nonnegative, this follows from the same argument one can use in the classical NLS equation.

3.1

Concentrated Compactness method

As already done in Chapter 2, we will study the following family of problems:

Iλ:= inf

T (λ)E(u) λ > 0, (3.2)

where the functional E : H1_(RN_{) → R is given by}

E(u) = 1 2hAu, ui + Z RN ω 2|u| 2_dx and T (λ) is the following proper subset of H1_(RN)

T (λ) :=  u ∈ H1(RN)| 1 p Z |V (x)||u|pdx = λ  .

Next we need to define the problem at "infinity"

I_λ∞:= inf

T∞_(λ)E(u) λ > 0, (3.3) 23

where T∞(λ) :=  u ∈ H1_(RN)| 1 p Z |V∞||u|pdx = λ  .

Before proceeding we need to point out that minimum points of (3.2) produce weak solutions of (3.1).

Let u be a minimum point of (3.1), then ther exists ν such that

δE(u, v) − νδG(u, v) = 0 for every v ∈ C_c∞_(RN), where G(u) is the constraint of T (λ).

Observation 3.1. ν > 0.

Indeed, for a density argument, the above equation holds for v ∈ H1_(RN). Therefore, using v = u we obtain

hAu, ui + Z RN ω|u|2dx = ν Z |V ||u|p_dx

And the RHS is positive, which implies ν > 0.

Now suppose u is a weak solution, let ρ > 0 and consider ¯u = ρu .

|V ||¯u|p−2u = ρ¯ p−1|V ||u|p−2_{u =} ρp−2

ν [A¯u + ω ¯u] .

Accordingly, it is sufficient to choose ρ such that ρp−2_ν = 1.

This means that studying (3.2) could lead us to finding solutions to (3.1), as wanted.

3.1.1 Translation independent case

In this subsection we will have V (x) = V∞ constant.

Clearly the problem is well posed since E ≥ 0, but if restricted on T (λ) we can sharpened the lower bound of E.

Proposition 3.2. There exists r = r(λ) > 0 such that I_λ > r. Moreover we can find c₁, c2> 0 such that for every minimizing sequence (un)n

c1 ≤ kunk_H1 ≤ c2.

Proof. Let (u_n)_n be a minimizing sequence, from Sobolev embedding and the constraint, we have

kunk_H1 ≥ cpkunkLp = ˜cpλ.

Recalling the observation (2.6) and that ω ≥ 0 the function u →p

E(u) is a

lower bound.

Let (un)n be a minimizing sequence, therefore,

r ≤ E(un) ≤ c.

Using once again the equivalence of the two norms we conclude that the sequence is bounded in H1 as stated.

As in the prevoius chapter the condition of subbaditivity is needed to prove the existence of minimizer using the Concentration-Compactness principle. So we have

Lemma 3.3. For every α ∈ (0, λ) the following holds :

Iλ < Iα+ Iλ−α.

Proof. The proof follows from the identity: Iλ = λ

2

p_I

1.

Indeed, since p ≥ 6, the concavity of the function x → x2p _{concludes the}

proof.

Let (un)n be a minimizing sequence for I1 and consider vn = λ

1 p_u n. Then we have Iλ≤ lim n E(vn) = λ 2 p_lim n E(un) = λ 2 p_I 1.

At the same way the other inequality is proved and hence the Lemma. We are now ready for the main Theorem of the subsection

Theorem 3.4. Every minimizing sequence of (3.1) is relatively compact in

H1(RN), up to transaltions. In particular I_λ admits a minimum.

Proof. Let (u_n)_nbe a minimizing sequence. We could use the Concentration-Compactness by considering ρ_n= 1_p|V ||u_n|p _{as already done in the previous}

chapter, but we want to give an idea of different approaches that could be taken.

Let ρ_n= ω|u_n|2_{and observe that from (3.2) we know that, defined ˜λ}

n:=R ρn,

there exist c1, c2 constants such that

0 < c₁≤ ˜λn≤ c2,

therefore, up to subsequences, the sequence (˜λn) converges. Up to considering

˜

- Step 1: No vanishing.

Using Lemma (2.18), we can find a suitable family (B_k)_k∈K of disjoint ball of fixed radius, such that

kunkpLp ≤ cp sup k∈K kunkL2_(B k) !p+2 2 kunk p−2 2 H1 .

If vanishing is supposed, then the LHS tends to zero, contradicting the costraint.

- Step 2: No dichotomy. Let α ∈ (0, λ) and u1_n, u2

n, vn as in Lemma (2.5). Then

E(un) ≥ E(u1n) + E(u2n)

and observe that u1_{n L}p, u2_{n L}p are bounded. As a result, up to subsequences, Z |V ||u1 n|p→ λ1, Z |V ||u2 n|p→ λ2. From dichotomy u_n− u1_n− u2_{n L}2 ≤ δε ε→0+

−−−−→ 0 and, by the usual GN inequality un− u 1 n− u2n _Lp ≤ ˜δε ε→0+ −−−−→ 0, therefore, |λ − λ₁− λ₂| ≤ c(ε).

Observe that λ1, λ2 ≥ 0. If λ1 = 0, then u1n → 0 strongly in Lp(RN)

and since it is bounded in H1, using Hölder and Sobolev

u 1 n _L2 ≤ u 1 n _Lp u 1 n _Lp0 ≤ cp u 1 n _Lp u 1 n _H1 → 0, which is a contradiction since

   u 1 n _L2 − α   < ε.

Thus, λ1, λ2> 0 and the usual argument can be used to prove

Iλ ≥ Iα+ Iλ−α

showing dichotomy cannot occur. - Step 3: Conclusions.

There exists (yn) such that ¯un= un(· + yn) is tight in L2(RN). Let ¯u

be the weak limit of (¯un) in H1, recall it is a bounded sequence. Since

it is tight we know that, ¯un→ ¯u strongly in L2(RN).

¯

u satisfy the costraint. Using the weak-SCI of the norm k·k_A, ¯u is a

minimum of I_λ which implies

hA¯u, ¯ui = lim

n

hA¯u, ¯ui.

This, as already seen multiple times, proves that the convergence is strong in H1.

3.1.2 Translation dependent case

First of all observe that all of the preliminary results of the previous case are still valid. Indeed it is sufficient to observe that V (x) satisfies the condition |V∞_{| < |V (x)| < kV k}

L∞ and with little effort it can be proved that this is enough to prove all of the mentioned results.

The only result which is not obvious is the one concerning the subadditivity. So let us give the brief but not evident proof of this fact.

Proposition 3.5. For every α ∈ [0, λ), the following holds true

Iλ < Iα+ Iλ−α∞ .

Proof. Since Iα≤ Iα∞ we easily get the strict inequality for α ∈ (0, λ). We

need to prove

Iλ < Iλ∞.

To prove this observe that thanks to Theorem (3.4) we know there exists u. Let θ such that

θ1 p

Z

|V (x)||u|pdx = λ.

Observe that, since |V (x)| > |V∞|, θ < 1. Now we can conclude, indeed considering v = θu

Iλ ≤ E(v) = θ

2

p_{E(u) < E(u) = I}∞

λ .

We shall point out that the importance of knowing before hand the existence of a minimizer for I_λ∞ as stated in Theorem (3.4). In fact working with a minimizing sequence and considering (θn) as in the proof we would

have the strict inequality for every fixed n, but taking the liminf we would los the strict inequality and therefore, this proof would not work.

Theorem 3.6. Every minimizing sequence of (3.2) is relatively compact in

Proof. Let (un)n be a minimizing sequence and observe that it is bounded

in H1. We will now proceed as usual, considering ρn= ω|un|2.

- Step 1: no vanishing.

We procced in the same way as in Theorem (3.4) and get to a contra-diction since vanishing implies that u_n has to violates the constraint. - Step 2: no dichotomy.

Let α ∈ (0, λ) and u1_n, u2_n, vn as in the proof of Theorem (3.4).

Con-sidering the two distinct cases in which y_n is either bounded or not (where yn is the usual sequence given by Lemma (2.5)) and following

the blueprint of the previous proof, we get:

Iλ≥ Iα+ Iλ−α∞ if yn is bounded

Iλ≥ Iα∞+ Iλ−α if yn is unbounded.

Which is in contradiction with the subadditivity. - Step 3: conclusions.

We apply the usual arguments. Let (yn)n be the sequence of the

compactness in Lemma (2.5) and ¯un the usual translation of un. We

know that ¯un converge s strongly in H1, up to subsequences, to some

¯

u. The proof is concluded if we show that (yn)n has to be bounded.

But this can be done as already seen in the proof of Theorem (2.3).

Before concluding the section let us point out that the arguments used in Section 2.7 are still valid, proving that every weak soutions does indeed lay in H2_(RN).

This concludes the study of problem (3.1) using the classical arguments of the Concentration Compactness Principle.

3.2

Extensions

Even in the critical and super-critical case we can weaken the hypothesis on the potential. Indeed observe that, in the same way as we did in Section

2.8, the hypothesis V (x) < 0 a.e it is not necessary. Let us focus our

attention first in the translation-independent case: here we need not to assume V∞< 0.Indeed it is sufficient to consider a different constraint for

the problem,

Z

V∞|u|p= λ, where λ > 0

translation-dependent case and observe that the condition ∃x0 ∈ RN such that V (x0) < 0 is still sufficient to conclude I_λ < 0, where now the infimum is taken over

T0(λ) :=  u ∈ H1(RN)| Z −V (x)|u|pdx = λ  .

This modification let us treat the case V∞ > 0. In fact if V∞ is positive then we set

I_λ∞= +∞ and the subadditivity condition still holds.

To prove the Theorem, though, a small modification, in the Step proving dichotomy is impossible, has to be done; in fact it lied on the fact that the constraint of the functions given by the dichotmy were at most zero, but if

V∞> 0 the constraint could very well be negative.

Suppose, in the notations of the proof of Theorem (3.4), without loss of generality, that λ2 < 0. Observe that

E(u2_n) ≥ ω u 2 n _L2 ≥ c > 0,

where the constant c depends only on α. Therefore, we obtain

Iλ = lim

n E(un) ≥ limn E(u

1

n) + lim n E(u

2

n) ≥ Iλ1+ c.

Recalling that λ₁ ≥ λ − c(ε), since |λ − λ₁ − λ₂| ≤ c(ε) −ε→0−−−→ 0, letting+

ε → 0 we get the following:

Iλ ≥ I_λ¯+ c > I¯_λ,

where ¯λ ≥ λ, which is a contradiction since for every λ > 0 Iλ = λ

2

p_I

Non-linear evolution problem

Now that the existence of ground states (see Definition 5.1) for the NLS equation (2.1) has been proved, it is natural to study the related evolutionary problem, that is the Cauchy problem

(

(i∂t− A) u = F (x, u) (t, x) ∈ RN +1,

u(0, x) = u0(x) x ∈ RN,

(4.1) where the initial datum u0 is in a suitable functional space X and the non-linearity F is the function F (x, u) = V (x)|u|p−2u.

The main goal is to prove existence results, either locally or globally, which means we are looking for a solution

u ∈ C1([−T, T ]; X) such that u(0, x) = u₀(x), that satisfies the differential equation pointwise.

That is at least the classical definition of solution, since regularity is hard to treat, we will initially look for solution u ∈ Lq_tLr

x, where we

de-note by Lq_tLr

x the functional spaces Lq



R; Lrx(RN)



, and the local spaces Lq_t,TLr

x := Lq



[−T, T ] ; Lr_x(RN). The definition of solution in this space will be given shortly.

Let us, for now, work with smooth solutions, in order to formally introduce the problem. Consider the homogeneous problem given by

(

(i∂_t− A) u = 0 _{(t, x) ∈ R}N +1_,

u(0, x) = u0(x) x ∈ RN.

(4.2) Consider the equation given by

i∂tu = Au. (4.3)

Taking the spatial Fourier transform we obtain the ordinary equation

i∂tu(t)(ξ) = g(ξ) ˆˆ u(t)(ξ),

with intial datum ˆu0 and g(ξ) is the usual simbol related to the operator A. The unique solution to this problem is given by the following function

ˆ

u(t)(ξ) = e−itg(ξ)uˆ0(ξ).

Formally we can apply the inverse Fourier transform and obtain the solution

u(t, x) :=

Z

RN

eihx,ξi−itg(ξ)uˆ0(ξ)dξ. We shall introduce the following operator:

etL : S → S

u →

Z

RN

eihx,ξi−itg(ξ)u(ξ)dξ,ˆ

where S denotes the Schwartz space, so that the solution to the linear equation is given by

ulin:= etLu0.

Observe that the operator etL, defined on the Schwartz space, can be extended by density arguments to spaces like Lq_tLr

x. Recall that using Duhamel’s

formula one can easily check that the solution to (4.1) can be written as

u(t, x) = ulin+ i

Z t

0

e(t−s)LF (x, u(s))ds, (4.4) for any s ∈ I, where I is an interval containing 0 and we use the notations

Z t s = − Z s t if t < s.

Now that we have introduced the problem and how to approach we can give the definition of solution in Lq_tLr

x and L q t,TLrx.

By time reversal-simmetry we may assume t > 0 and work in the upper time-axis [0, ∞).

Definition 5. We say that u(t, x) is a strong L2_xsolution to (4.1) with initial datum in L2_x if it solves the integral form of the problem:

u(t, x) = ulin+ i

Z t

0

e(t−s)LF (x, u(s))ds.

To prove local existence, for every suitable initial datum, we will be using contraction Theorem-like results. In fact we have the following:

Proposition 4.1. Let N , S be two Banach spaces.

Let M : N → S be a linear operator such that there exists C0 ∈ R+ such that kM F kS ≤ C0kF kN ∀F ∈ N .

Suppose we are given a non-linear operator N : S → N , such that N (0) = 0 and N obeys the Lipschitz bound

kN (u) − N (v)kN ≤ 1

2C₀ku − vkS,

for all u, v ∈ Bε:= {u ∈ S|kukS≤ ε} for some ε > 0.

Then for all u_lin∈ Bε

2, there exists a unique solution u ∈ Bε to

u = ulin+ M N (u).

The map u_lin→ u is Lipschitz, with constant at most 2.

Proof. The claim will follow if we show that the proposition implies the

contraction Theorem, applied to the map

T (u) := ulin+ M N (u).

Let X := Bε, we need to prove that X is stable under the action of T , that is

kT (u)k_S ≤ ε _{∀u ∈ X.}

Using the triangular inequality and the hypothesis on M , N we obtain the following chain of inequalities

kT (u)k_S ≤ kulinkS+ kM N (u)kS ≤

ε 2 + C0kN (u)kN ≤ ε 2+ kuk_S 2 ≤ ε. To conclude the proof we need to show that T is a contraction, the last claim will follow easily.

Let u, v ∈ X, then we have

kT (u) − T (v)k_S = kM (N (u) − N (v)k_S ≤ C₀kN (u) − N (v)k_N ≤ 1

2ku − vkS.

Therefore, to prove local existence we need to be able to use the above proposition, considering N as the non-linearity F , M as the Duhamel’s operator and for some suitable spaces N , S, which will be chosen later. Before choosing the spaces N , S where we will be working, let us explain the tools that we will need to proceed, afterward the choices of N , S will be obvious.

What we need is some sort of Strichartz estimates for our problem. Suppose that for each time t ∈ R we have an operator

U (t) : H → L2(X), which obeys the energy estimate

• ∀t ∈ R, ∀f ∈ H

kU (t)f k_L2

x . kf kH (4.5)

and for some σ > 0, the following decay estimate • ∀t 6= s, ∀g ∈ L1

kU (s)U∗(t)gk_L∞ . |t − s|−σkgk_L1. (4.6)

Definition 6. We say that a pair of exponent (q, r) is σ-admissible, if

q, r ≥ 2, (q, r, σ) 6= (2, ∞, 1) and 1 q + σ r ≤ σ 2. (4.7)

If equality holds in (4.7) we say that (q, r) is sharp σ-admissible, otherwise (q, r) is non-sharp σ-admissible.

Note that when σ > 1 the endpoint P = (2,_σ−12σ ) is sharp σ-admissible. The Theorem that we need is given in the paper of M. Keel, T. Tao [7] and it is the following

Theorem 4.2. If U (t) obeys (4.5)-(4.6) then the estimates

kU (t)f k_Lq tLrx . kf kH, (4.8) Z (U (s))∗F (s)ds _H. kF kL q0 t Lr0x (4.9) k Z s<t U (t)(U (s))∗F (s)dskLq_tLr x . kF kLq0_t˜Lr0˜ x (4.10)

hold for all sharp σ-admissible exponent pairs (q, r), (˜q, ˜r).

In the setting of Schrödinger equation we will set X = RN and H = L2_(RN_).

Therefore, our objective is to prove the estimates (4.5)-(4.6) in order to use Theorem (4.2) in order to fullfill the hypothesis of proposition (4.1), in a way that will be explained once the estimates are achieved.

Observe that since A is a skew-adjoint operator, the symbol g(ξ) is a real function, as already seen multiple times. As a result, the operator

etL: L2_x→ L2_x is unitary, up to a constant, thus we have

ketLf kL2

x . kf kH ∀t ∈ R, ∀f ∈ L

2

satisfying the energy estimate (4.5).

In order to get a decay estimates it is sufficient to prove the following estimate kK_t(x)k_L∞ . |t|−σ

for some σ > 0, where Kt(x) is the kernel associated to the operator etL.

Indeed if this estimate holds, we have

kU (s)U∗(t)gk_L∞ = kK_t−s∗ uk_L∞ ≤ kK_t−sk_L∞kgk_L1 . |t − s|−σkgk_L1. The kernel of the operator etL has not an explicit formula, like the kernel of et∆ associated to the classical Schrödinger equation, due to the presence of the dissipative cubic term. Hence, the study of the sup-norm of such kernel is not as immediate as in the classical case.

To study kKtkL∞, we need to introduce the stationary method phase.

4.1

Stationary phase method

Let R > 0 and consider an oscillatory integral of type

I(R) :=

Z

RN

eiRφ(x)f (x)dx. (4.11)

Here φ(x), f (x) are smooth functions defined on RN, with φ(x) being real valued.

First we consider the case where φ(x) has no critical points.

More precisely, we consider the case, when there exist δ ∈ (0, 1] and C > 0 such that

|∇φ(x)| ≥ C−1hxiδ, hxi2= 1 + |x|2, (4.12) |∂_xαφ(x)| ≤ Chxiδ−|α| (4.13) For all x ∈ supp(f );

Then we have

Proposition 4.3. Suppose the assumptions (4.12)-(4.13) are fullfilled and

f is a smooth function with compact support. Then for every n ≥ 0 and for every ε > 0 |I(R)| ≤ C Rn X |α|≤n hxi −nδ−n+|α|+N 2+ε∂αf _L2. (4.14) It is important to notice that in the previous proposition the assumption that f has to have compact support is limiting, but it can be weakened. Indeed, by usual density argument, the proposition can be extended to regular function such that the RHS of (4.14) is finite.