UNIVERSIDADE FEDERAL DE MINAS GERAIS
INSTITUTO DE CIÊNCIAS EXATAS
PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA
PhD Thesis
Asymptotic Behavior for Inhomogeneous Nonlinear
Schrödinger Equation
MYKAEL DE ARAÚJO CARDOSO
Belo Horizonte 2020
UNIVERSIDADE FEDERAL DE MINAS GERAIS
INSTITUTO DE CIÊNCIAS EXATAS
PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA
MYKAEL DE ARAÚJO CARDOSO
Asymptotic Behavior for Inhomogeneous Nonlinear Schrödinger
Equation
Thesis presented to the Post-graduate Pro-gram in Mathematics at Universidade Fed-eral de Minas Gerais as partial fulfillment of the requirements for the degree of Doc-tor in Mathematics.
Adviser: Luiz Gustavo Farah Dias
Belo Horizonte 2020
Ficha catalográfica elaborada pela bibliotecária Irénquer Vismeg Lucas Cruz - CRB 6ª Região nº 819
Cardoso, Mykael de Araújo.
C268a Asymptotic behavior for inhomogeneous nonlinear Schrödinger Equation /Mykael de Araújo Cardoso — Belo Horizonte, 2020.
111 f. il.; 29 cm.
Tese (doutorado) - Universidade Federal de Minas Gerais – Departamento de Matemática.
Orientador: Luiz Gustavo Farah Dias.
1. Matemática - Teses. 2. Equações diferenciais parciais.– Teses. 3. Problemas de valor inicial – Teses. 4. Schrodinger, Equação de. – Teses I. Orientador. II. Título.
Agradecimentos
Primeiramente à Deus, por sempre me acompanhar durante todos os momentos em que passei e por aqueles que irei passar.
À minha esposa Maria de Fátima por todo amor, apoio e companherismo sempre e ao meu lindo filho Valentim Gael, que colocou nossos corações em uma comemoração constante.
Aos meus pais, Domingos e Salete, por terem infundido em mim todos os valores que achavam importantes e por me tornarem na pessoa que sou hoje. Agradeço também à todos os meus familiares que sempre acreditaram em meu potencial.
Aos professores da UFMG por terem colaborarado com minha formação.
Aos professores Ademir Pastor, Fábio Natali, Alex Ardila, Gastão Braga por terem aceito compor a banca de avaliação deste trabalho e pelas valiosas sugestões.
Ao professor Luiz Gustavo Farah por toda a disponibilidade e atenção dada desde o primeiro email em 2015. Agradeço pelos ensinamentos e conselhos durante esses anos.
Aos amigos Aldo e Mayara, Allan e Lays, Diego e Izabela, Franciele, Guido, Jeferson, Lázaro, Leandro, Lucas, Luccas, Marlon, Moacir, Natã e Dani, Rafael, Willer pelo companheirismo, churrascos e parcerias de estudo.
Às secretárias da pós-graduação, Kelli e Andréa, pela atenção e gentileza. À UFPI e ao Dep. de Matemática/CCN-UFPI.
O presente trabalho foi realizado com apoio da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - código de financiamento 001.
"Everything comes in time to those who can wait."
Resumo
Nesta tese investigamos algumas questões sobre o comportamento ao longo do tempo das soluções para o problema de valor inicial (PVI) associado à equação de Schrödinger não-linear não-homogênea (INLS)
iBtu ∆u κ|x|b|u|2σu 0,
onde κ 1 and σ, b ¡ 0. Dentre elas, (a) estabilidade de ondas viajantes da equação focusing L2-subcrítica INLS, para as quais damos uma prova alternativa ao resultado de
De Bouard and Fukuizumi [9]; (b) boa colocação local para a equação intercrítica INLS em 9HscpRNq X 9H1pRNq; (c) concentração da norma crítica para soluções em que o tempo máximo de existência é finito; (d) explosão da norma crítica para soluções com dado inicial radialmente simétrico em 9HscpRNq X 9H1pRNq, inspirado pelas ideias de Merle and Raphäel [52].
Palavras-chave: Boa-colocação, equação de Schrödinger não-linear, estabilidade, explosão da norma crítica.
Abstract
In this thesis we investigate some questions about the long-time behavior of the solutions for the initial value problem (IVP) associated to the inhomogeneous nonlinear Schrödinger (INLS) equation
iBtu ∆u κ|x|b|u|2σu 0,
where κ 1 and σ, b ¡ 0. Among them, (a) stability of standing waves for focusing L2-subcritical INLS equation for which we give an alternative proof for the result of
De Bouard and Fukuizumi [9]; (b) local well-posedness for the intercritical INLS equation in 9HscpRNq X 9H1pRNq; (c) critical norm concentration for finite-time blow up solutions; (d) blow-up of the critical norm for solutions with radially symmetric initial data in
9HscpRNq X 9H1pRNq, inspired by the idea of Merle and Raphäel [52].
Keyboards: Well posedness, nonlinear Schrödinger equation, stability, blow-up of the critical norm.
Agradecimentos iii
Introduction ix
1 Preliminaries 15
1.1 Notations . . . 15
1.2 Functional analysis . . . 16
1.3 The Banach spaces LppRNq . . . 17
1.4 The Fourier transform . . . 19
1.4.1 The Fourier transform in L1pRNq . . . . 19
1.4.2 The Fourier transform in Schwartz spaces . . . 20
1.4.3 The Fourier transform in LppRNq, 1 p ¤ 2 . . . . 22
1.4.4 The Fourier transform in the space of tempered distributions . . . . 23
1.5 Homogeneous Sobolev spaces . . . 24
1.6 Differential calculus for mappings in Banach spaces . . . 27
1.7 The Schrödinger equation . . . 30
2 Sobolev compact embedding 35 2.1 Introduction . . . 35
2.2 On the compactness . . . 37
2.3 Stability of standing waves . . . 39
3 Local well-posedness 47 3.1 Introduction . . . 47
3.2 A priori estimates . . . 48
3.3 Local well-posedness in 9HscX 9H1 . . . . 53 ii
4 Critical norm concentration 57
4.1 Introduction . . . 57
4.2 Gagliardo-Nirenberg inequality . . . 59
4.2.1 The ground states. . . 60
4.2.2 Gagliardo-Niremberg inequality . . . 62
4.3 Critical norm concentration . . . 65
4.3.1 Remark on another concentration for INLS . . . 66
5 Blow up critical norm 69 5.1 Introduction . . . 69
5.2 A radial interpolation estimate. . . 71
5.3 The main propositions . . . 75
5.3.1 Auxiliary Lemmas . . . 77
5.3.2 Upper bound on the blow-up rate for radial data in H1 . . . 86
5.3.3 Proof of Propositions 5.3.1 and 5.3.2 . . . 88
5.4 Blow up criteria . . . 93
5.5 Lower bound for the critical Lσc norm . . . . 94
A Complete metric space 101
B Cut-off function 103
Introduction
In this thesis, we consider the initial value problem (IVP) for the inhomogeneous nonlinear Schrödinger (INLS) equation
$ ' & ' % iBtu ∆u κ|x|b|u|2σu 0, x P RN, t¡ 0, up, 0q u0 P X, (1)
where κ 1, σ, b ¡ 0 are real numbers and X is a Sobolev space. The equation is called “focusing INLS” when κ 1 and “defocusing INLS” when κ 1.
The inhomogeneous nonlinear Schödinger equation (INLS) have been attracting attention from both physical (e.g. Gill [33], Liu and Tripathi [51] and Bergé [1]) and mathematical (e.g. De Bouard and Fukuizumi [9], Stuart [61], Genoud and Stuart [29], Genoud [28], Farah [18] and Farah and Guzmán [20]-[19]) point of view. The case b 0 is the classical Nonlinear Schrödinger equation (NLS), extensively studied in recent years (see Sulem and Sulem [62], Bourgain [3], Cazenave [6], Linares and Ponce [46], Fibich [22]
and the references therein).
Before reviewing some results about the IVP (1), we recall some facts for this equation. By Duhamel’s Principle, the solution of (1) is a solution uP X to the integral equation
up, tq eit∆u0 i
»t
0
eiptsq∆p|x|b|upsq|2σupsqq ds, (2) where eit∆ denotes the strongly continuous unitary group associated to the linear problem
iBtu ∆u 0
with initial data u0, defined by
eit∆u0 eit|ξ|2 u0 _ . ix
Note that, if u is a solution to (1), then uλ given by
uλpx, tq λ
2b
2σ upλx, λ2tq
is also solution to (1) for all λ¡ 0 with the homogeneous Sobolev norm of the initial data satisfying
}uλp0q}H9s λ
ssc}up0q}
9
Hs.
where sc N2 22σb is the critical Sobolev index. If sc 0 (alternatively σ 2Nb) the
problem is known as the mass-critical or L2-critical; if s
c 1 (alternatively σ N2b2) it
is called energy-critical or H1-critical, finally the problem is known as mass-supercritical and energy-subcritical if 0 sc 1 (alternatively 2Nb σ N2b2). On the other hand,
the inhomogeneous nonlinear Schrödinger equation conserves the mass Mrus and energy Erus which are defined by
Mruptqs » |uptq|2dx Mru 0s, Eruptqs 1 2 » |∇uptq|2dx 1 2σ 2 » |x|b|uptq|2σ 2dx Eru 0s. (3)
It is well know that for any ω¡ 0 the elliptic equation
∆φω |x|b|φω|2σφω ωφω. (4)
admits a unique positive radial solution φω,b where 0 σ N2b2 (see Stuart [60], Gidas
et al. [31], Li [44], Li and Ni [45] and Yanagida [67]). The global behavior of H1pRNq solutions to (1) is related to the existence of standing waves upx, tq eiωtφ
ω,bpxq. It is
known that a positive radial solution of (4) is a ground state. In [9] De Bouard and Fukuizumi proved that if N ¥ 3, 0 b 2 and 0 σ 2Nb, then the standing wave solution eiωtφω,bpxq is stable for any ω ¡ 0. On the other hand, it is shown that if N ¥ 3,
0 b 2 and 2Nb σ N2b2 then the standing wave solution eiωtφ
ω,bpxq is unstable for
any ω ¡ 0 (see Fukuizumi and Ohta [25]).
Genoud and Stuart [29] studied, by using the abstract theory of Cazenave [6] and without relying on Strichartz-type inequalities, the well-posedness for the INLS equation in the sense of distributions, that is, iBtu ∆u |x|b|u|2σu 0 in H1pRNq. They showed,
with 0 b 2, that the IVP (1) is well-posed • locally if 0 σ σb (sc 1);
xi
• globally1 for any initial data in H1pRNq if σ 2b
N and κ 1;
• globally for sufficiently small initial data if 2b
N ¤ σ σb and κ 1, where σb $ ' & ' % 2b N2, N ¥ 3 8, N ¤ 2. (5)
Recently, Guzmán [38] established local well-posedness of the INLS equation in HspRNq
based on Strichartz estimates. In particular, setting
˜ 2 $ ' & ' % N 3, N ¤ 3 2, N ¥ 4,
he proved that, for N ¥ 2, 0 σ σb and 0 b ˜2, the initial value problem (1) is locally well-posed in H1pRNq. Dinh [12] improved2 Guzmán’s result in dimension N 2,
for 0 b 1 and 0 σ σb. Cho and Lee [8] improved3, for N 3, 0 σ 2 b and
0 b 32. Note that, in all these results the ranges of b are more restricted than those in the results of Genoud and Stuart [29]. However, Guzmán, Dinh and Cho-Lee give more detailed information on the solutions, showing that there exists Tp}u0}H1q ¡ 0 such that
uP LqprT, T s; Lrq for any L2-admissible pair pq, rq satisfying
2 q N 2 N r , where $ ' ' ' & ' ' ' % 2¤ r ¤ N2N2 if N ¥ 3, 2¤ r 8 if N 2, 2¤ r ¤ 8 if N 1.
In the limiting case σ 2Nb (L2-critical case) Genoud [28] showed how small should
be the initial data to ensure global well posedness. Indeed, he proved global well-posedness in H1pRNq assuming
}u0}L2 }Q0}L2,
1It was also proved that for, κ 1, any local solution of (1) with u
0P H1pRNq extends globally in
time.
2Dinh also extends the range of b, for N 3 and 1 2 b
3
2, however with a extra assumption on σ,
σ 32b2b1.
3It is worth mentioning that Cho-Lee studied the INLS equation with a potential, i.e., iBu
t ∆u
where Q0 is the unique non-positive, radially symmetric, decreasing solution of the equation (4) with ω 1 and σ 2b N , i.e., ∆Q0 |x|b|Q0| 2p2bq N Q0 Q0.
Farah [18] extended this result for the intercritical case 0 sc 1 (alternatively 2Nb
σ σb) showing sufficient conditions on the initial data to obtain global and blow-up solutions in H1pRNq. For initial data u
0 P H1pRNq satisfying
Mru0s1scEru0ssc MrQs1scErQssc,
where Q is the unique non-negative, radially symmetric, decreasing solution to equation (4) with ω 1, that is,
∆Q |x|b|Q|2σQ Q,
he showed, inspired in Holmer and Roudenko [40] (see also Holmer and Roudenko [41] and Duyckaerts et al. [16]), that if
}u0}1sL2 c}∇u0}Lsc2 ¡ }Q}1sL2 c}∇Q}sLc2
and u0 has finite variance, i.e. |x|u0 P L2pRNq, then the solution u to (1) blows-up in
finite-time.
Merle and Tsutsumi [53] (in the radial case) and Weinstein [65] (in the general case), considered the L2-critical NLS and showed that all finite-time blow-up solutions must
concentrate the L2pRNq norm. However, without the radial assumption, the concentration
occurs around a traveling point xptq in space, for which there is no much information available - not even about the continuity of the map tÞÑ xptq. In [39] Hmidi and Keraani established a profile decomposition and provided another proof for this result. Applying the profile decomposition, Campos and Cardoso [5] proved mass concentration for blow-up solution to L2- critical INLS equation with xptq 0. Other authors, also using profile
decompositions, showed critical norm concentration (see Guo [37], Xie et al. [66], Dinh [14] and Farah and Pigott [21])) for finite blow-up solutions to the L2-supercritical NLS-type
equation with initial data in 9HscX 9H1.
This thesis is organized as follows. First, in Chapter 2 we prove the stability of standing waves eiωtφω for the focusing (κ 1) INLS equation (1), where ω ¡ 0 and φωpxq
is a ground state of the stationary problem (4), for L2-subcritical case 0 σ 2b
N
. Our
xiii
main tool is the compact embedding H1 ãÑ L2σ 2p|x|bq proved by Genoud and Stuart
[29]. This is an alternative proof for the same result firstly proved by De Bouard and Fukuizumi [9]. In addition, we generalize this compact embedding result and show that
9H1pRNq X LppRNq ãÑ L2σ 2p|x|bqpRNq is compact for 0 s c 1.
In Chapter 3, we establish a local well-posedness result for the IVP (1) in the space 9HscpRNq X 9H1pRNq with 0 s
c 1.
Next, in Chapter 4, we investigate the phenomenon of critical norm concentration for H1pRNq solutions of (1) in the L2-supercritical and H1-subcritical case (0 s
c 1).
The main tool is the compact embedding introduced in Chapter 2. We also show a Gagliardo-Nirenberg type inequality in 9HscpRNq X 9H1pRNq, obtained by minimizing the associated Weinstein functional.
In Chapter 5, we investigate the behavior of the critical norm for some special solutions of the IVP (1) in the L2-supercritical and H1-subcritical case (0 s
c 1)
with initial data u0 P 9HscpRNq X 9H1pRNq. For finite-time blow up solutions to the INLS
equation with radially symmetric initial data we show that the critical norm also blows up. Moreover, we show a lower bound for the blow up rate of the critical norm. Our method relies on the ideas introduced by Merle and Raphäel [52] in their study of the intercritical nonlinear Schrödinger equation.
Chapter 1
Preliminaries
In this Chapter, we introduce some definitions and basics results that will be used throughout the work. For more details see, for instance, Brezis [4], Yosida [68], Rudin [57], Botelho et al. [2], Cazenave [6], Linares and Ponce [46], Grossinho and Tersian [36] and Kavian [43].
1.1
Notations
• We use c to denote various constants that may vary line by line.
• Let a, b be real positive numbers. By aÀ b (respectively a Á b) we mean that there is a constant c¡ 0 such that a ¤ cb (respectively a ¥ cb). We also denote a À b À a by a b.
• Cε denotes a constant depending on ε.
• a is a fixed number slightly smaller than a (a a ε with ε ¡ 0 small enough) and, in a similar way, we define a . Moreover, pa q1 is the positive number such that
1 a 1 pa q1 1 a .
• B denotes the unite ball in RN defined by Bp0, 1q tx P RN; |x| ¤ 1u.
• Let A RN. We denote by Ac RNzA the complement of A in RN.
• The inner product of x, yP RN on RN is denoted by x y.
• Given f a complex function, its real part will denoted by Re f and the imaginary part by Im f .
• We use MpRN
, Cq to denote the space of measurable functions from RN to C. • Let X, Y be Banach spaces, CkpX, Y q will denote the space of all maps from X to
Y with k continuous derivatives.
• We denote by X ãÑ Y if X Y with continuous injection.
• α pα1, ..., αNq with αi P N Y t0u and α1 ... αN k is a called multi-index of
order k. We denote the derivative of multi-index α by Bαf B kf Bα1x 1 ... BαNxN . • ∇f is given by ∇f pBx1f, ...,Bxnfq.
1.2
Functional analysis
In this section we recall some basic properties of functional analysis that we use throughout the text.
Lemma 1.2.1 (Young’s inequality). Let p, q ¡ 0 be real numbers such that 1p 1q 1. Then for all non-negative a, b real numbers and ε¡ 0 there exists Cε Cpε, p, qq ¡ 0 such
that
ab¤ εap Cεbq.
Proof. See, for instance, Brezis [4, Theorem 4.6, page 92].
Proposition 1.2.2. Let X ãÑ Y be two Banach spaces. Consider x P X and txnun18 X
a sequence. If xná x in X as n Ñ 8, then xná x in Y as n Ñ 8.
Proof. See, for instance, Brezis [4, Theorem 3.10, page 62]
Proposition 1.2.3. Let X be a reflexive space and txnun18 X a bounded sequence.
Then, there exists xP X such that xn á x as n Ñ 8, up to a subsequence.
1.3. THE BANACH SPACES LPpRNq 17
Consider two Banach spaces X1 and X2 which are subsets of a Hausdorff topological
vector space X . Let
X1X X2 tx P X ; x P X1 and xP X2u
and
X1 X2 tx P X ; Dx1 P X1, Dx2 P X2 such that x x1 x2u.
Proposition 1.2.4. X1X X2 pX1X X2,} }X1XX2q is a Banach space, where
} }X1XX2 } }X1 } }X2.
Proof. See Cazenave [6, Theorem 1.1.3, page 3].
The following Theorem is an important tool to study the existence of solutions. Theorem 1.2.5 (Banach’s fixed point). Let pX, dq be a complete metric space and F : X Ñ X. If there exists a positive constant L 1 such that dpF pxq, F pyqq ¤ Ldpx, yq for all x, y P X, then F has a unique fixed point x0 P X, i.e., there exists a unique x0 P X
such that Fpx0q x0.
Proof. See Cazenave [6, Theorem 1.1.1, page 1].
Theorem 1.2.6. Let X, Y be two Banach spaces. If T : DpT q X Ñ Y is a bounded linear operator, then there exists a unique T : DpT q X Ñ Y bounded linear operator such that T
DpT q T and }T } }T }.
1.3
The Banach spaces L
ppR
Nq
For each 1¤ p 8, let LppRNq be the class of measurable functions f : RN Ñ C such that the integral (of Lebesgue)
»
RN
|fpxq|pdx is finite. Also, let L8pRNq be the
collection of bounded functions at almost every point, i.e , at all points up to a null set. Equipped with the following norms
}f}Lp » RN |fpxq|pdx 1 p , 1¤ p 8 and }f}8 ess supt|fpxq|; x P RNu,
each LppRNq, 1 ¤ p ¤ 8 is a Banach space. Some elementary properties of LppRNq :
1. }f g}Lp ¤ }f}Lp }g}Lp ( Minkowski’s inequality); 2. }fg}L1 ¤ }f}Lp}g}Lq, where 1 1 p 1 q, if 1¤ p 8 and q 1 if p 8 (Hölder’s inequality). In particular, if p q 2, then }fg}L1 ¤ }f}L2}g}L2 (1.1)
which is known as Cauchy-Schwartz inequality.
In view of (1.1), we define the inner product in L2pRNq by
xf, gy »
RN
fpxqgpxqdx, f, g P L2pRNq.
With this inner product, L2pRNq is a Hilbert space. It is easy to see that xf, fy }f}2 2,
f P L2pRNq. It is possible to show that LppRNq is a Hilbert space if and only if p 2. We expose below two basic and essential results in the study of LppRNq spaces. Here,
when not specified, the parameter p is such that 1¤ p 8.
Proposition 1.3.1 (Generalized Hölder’s inequality). Let 1 ¤ p, q, r ¤ 8 such that
1 r 1 p 1 q. If f P L
p and g P Lq, then f g P LrpRNq and
}fg}r ¤ }f}p}g}q.
Proof. It’s an immediate consequence of Hölder’s inequality. For this end, consider the Hölder’s inequality with |f|r and |g|r.
Proposition 1.3.2 (Minkowski’s inequality for integrals). Let f : Rm RN Ñ R (or C) measurable such that f p, yq P LppRmq for almost all points y P RN and the function yÞÝÑ }fp, yq}LppRmq belongs to L1pRNq. Then the function x ÞÝÑ
» RN fpx, yqdy belongs to LppRmq and » RN fpx, yqdy LppRmq ¤ » RN }fp, yq}LppRmqdy. Proof. See Folland [23, Theorem 6.19, page 194].
Theorem 1.3.3 (Riesz-Thorin). Let pX; ΣX; µq and pY ; ΣY; νq be measure spaces and
p0, p1, q0, q1 P r1, 8s, p0 p1, q0 q1. For 0 θ 1, let pθ and qθ be such that
1 pθ 1 θ p0 θ p1 , 1 qθ 1 θ q0 θ qθ .
1.4. THE FOURIER TRANSFORM 19
If T : Lp0pµq Lp1pµq ÝÑ Lq0pνq Lq1pνq is a linear operator such that,
}T f}q0 ¤ M0}f}p0, for f P L p0pµq and }T f} q1 ¤ M1}f}p1, for f P L p1pµq, then }T f}qθ ¤ M 1θ 0 M θ 1}f}pθ,@ f P L pθpµq and 0 θ 1;
that is, denoting by Mθ the norm of T : Lpθ ÝÑ Lqθ we have, Mθ ¤ M01θM1θ.
Proof. See Linares and Ponce [46, Theorem 2.1, page 27].
Definition 1.3.4. For 1¤ p, q 8 we define the Banach space LqTLpx tf : R r0, T s Ñ C measurable : }f}Lq TL p x 8u, where }f}LqTLpx »T 0 » 8 8 |fpx, tq| pdx q p dt 1 q . The notation LqtLp
x (observe that now t is in lowercase letter) means that the integral
in the variable t is from8 to 8. If p 8 or q 8, }f}Lq TL
p
x is defined in the natural way.
Remark 1.3.5. Let B Bp0, 1q tx P RN;|x| ¤ 1u then |x|bLγpBq 8 if N γ b ¡ 0, and |x|bLγpBCq 8 if N γ b 0.
1.4
The Fourier transform
In this section we review the key properties of the Fourier transform theory in LppRNq, 1 ¤ p ¤ 2, in Schwartz space and in the space of tempered distributions.
1.4.1
The Fourier transform in L
1pR
Nq
Proposition 1.4.1. The Fourier transform F : L1pRNq Ñ L8pRNq, defined by Fpfqpξq :
»
RN
e2πipxξqfpxqdx, is a continuous linear operator with }F} 1.
Proof. See Linares and Ponce [46, Theorem 1.1, page 1] and Folland [23].
In addition to the notation Fpfqpξq, we also use pfpξq to denote the Fourier transform of a function f P L1pRNq.
Below we present some basic properties of the Fourier transform in L1pRNq.
Theorem 1.4.2. The Fourier transform in L1pRNq satisfies the following properties 1. f P L1pRNq, then pf : RN Ñ C is continuous. 2. lim |ξ|Ñ 8fppξq 0 (Riemann-Lebesgue lemma). 3. pτhfqppξq e2πihξfppξq, where τhfpxq fpx hq. 4. Fpe2iπxhfqpξq τhfppξq. 5. For a¡ 0, Fpfpaxqqpξq aNfppa1ξq
6. pf gqppξq pfpξqpgpξq, where the convolution product is defined by f gpyq
»
RN
fpxqgpy xqdx.
Proof. See, for instance, Linares and Ponce [46, Theorem 1.1, page 1] and Folland [23].
1.4.2
The Fourier transform in Schwartz spaces
We introduce now a functional space where the Fourier transform has an inverse and, due to its regularity and density in LppRNq, 1 ¤ p 8, we can use it for a broader
study of the Fourier transform.
Definition 1.4.3. We define the Schwartz space SpRNq, as the space of the C8-functions
whose derivative decay polynomially fast at infinity, i.e., SpRNq tϕ P C8pRNq; }ϕ}
α,β sup xPRN|x
αBβϕ| 8, @ multi index α, β P ZNu.
Definition 1.4.4. We say that a sequence pϕjqjPN of functions in SpRNq converges to a
function ϕP SpRNq, when lim
jÑ 8}ϕj ϕ}α,β 0, for any multi-indices α, β.
Proposition 1.4.5. The metric space SpRNq with the topology generated by the semi-norms }ϕ}α,β supxPRN|xαBβϕ|, x P RN and α, β multi-indices is complete.
1.4. THE FOURIER TRANSFORM 21
Proof. See Friedlander and Joshi [24, page 93].
Theorem 1.4.6. The Schwartz space has the following basics properties:
1. Let ϕ P SpRNq, then P pxqϕ P SpRNq and P pBqϕ P SpRNq, for any polynomial P pxq.
In particular, if Ppxq, Qpxq are polynomials and ϕ P SpRNq, we have P pxqQpBqϕ P SpRNq.
2. S ãÑ Lp and is dense in Lp, @ 1 ¤ p 8, with the norm of Lp.
3. If ϕ, ψ P SpRNq, then ϕ ψ P SpRNq, that is, the convolution product ϕ ψ is a
closed operation in SpRNq.
Proof. See Friedlander and Joshi [24, Theorem 8.2.1, page 94].
Corollary 1.4.7. All properties in Theorem 1.4.2 for the Fourier transform in L1pRNq
are valid for the Schwartz space SpRNq.
Proof. We have from item 2 of Theorem 1.4.6 that SpRNq ãÑ L1. Thus, the result follows.
Theorem 1.4.8. Let ϕP SpRNq. Then
1. pBα
xϕqppξq p2πiξqαϕppξq, for all α multi-index.
2. pp2πixqαϕpxqq ppξq Bα
ξϕppξq, for all α multi-index
3. ϕpP SpRNq, i.e., F : SpRNq Ñ SpRNq.
Proof. See Friedlander and Joshi [24, Theorem 8.2.1, page 94]. Theorem 1.4.9 (Parseval’s identity). Let ϕ, ψP SpRNq, then
» RN ψϕdx » RN p ψϕdx.p Proof. See Linares and Ponce [46, Theorem 1.1, page 2]. Theorem 1.4.10 (Plancherel). If ϕP SpRNq, then }ϕ}
L2 }pϕ}L2.
Proof. See, for example, Friedlander and Joshi [24, Theorem 9.2.2, page 118] and Duoandikoetxea and Zuazo [15].
Theorem 1.4.11 (Fourier). For ϕP SpRNq, ϕpxq » RN p ϕpξqe2πipxξqdξ. Proof. See Friedlander and Joshi [24, Theorem 8.2.2, page 95].
The Fourier’s Theorem gives us an inverse formula, that allows us to define the inverse Fourier transform of a function ϕP SpRNq by
F1pϕpxqq qϕpxq »
RN
ϕpξqe2πipxξqdξ. (1.2) Theorem 1.4.12. The Fourier transform F : SpRNq Ñ SpRNq is a continuous isomor-phism and its inverse F1 : SpRNq Ñ SpRNq also is a continuous isomorphism.
Proof. See Friedlander and Joshi [24, Theorem 8.2.3, page. 96].
Remark 1.4.13. One can define the inverse Fourier transform F1 in L1 using formula (1.2). Moreover, all properties listed previously for the Fourier transform F are also valid
for its inverse F1. In what follows the results for F also are true for F1.
1.4.3
The Fourier transform in L
ppR
Nq, 1 p ¤ 2
Having already defined the Fourier transform in L1pRNq, we extend the definition
to functions belonging to L2pRNq, and using the Riesz-Thorin interpolation Theorem, we
also define the Fourier transform in LppRNq, 1 p 2.
Theorem 1.4.14 (Plancherel). If f P L2pRNq, then pf P L2pRNq e }f}L2pRNq } pf}L2pRNq.
In other words, F is a unitary operator (an isometry) in L2pRNq.
Proof. See Linares and Ponce [46, Theorem 1.3, page 7].
In Proposition1.4.1 and Theorem 1.4.14, we have seen that the Fourier transform is a linear continuous operator of the strong type p1, 8q and p2, 2q, respectively. Thus, by the Riesz-Thorin interpolation Theorem we have the following result.
Theorem 1.4.15 (Hausdorff-Young inequality). If f P LppRNq, 1 ¤ p ¤ 2, then pf P LqpRNq, with 1
p 1 q 1,
} pf}Lq ¤ }f}Lp. Proof. See Folland [23, Theorem 8.21, page 248].
1.4. THE FOURIER TRANSFORM 23
1.4.4
The Fourier transform in the space of tempered
distribu-tions
Definition 1.4.16. Let F : SpRNq Ñ C be a functional. We say that F is a tempered
distribution if it is both linear and continuous, i.e., for all tϕnun18 such that ϕnÑ 0 in
SpRNq as n Ñ 8, we have F pϕ
nq Ñ 0 as n Ñ 8.
We know that SpRNq is a topological vector space, and thus the space of all continuous
linear functionals on SpRNq is closed under the standard operations of addition and scalar
multiplication. We adopt the notation S1pRNq for the space of tempered distributions.
In general the set of linear and continuous functionals on a vector space is called the continuous dual of the space.
Remark 1.4.17. Let f be a function such that f ϕ is integrable on RN for all ϕP SpRNq.
We define the tempered distribution induced by f as Ff : SpRNq Ñ R and define it as
Ffpϕq
»
RN
fpxqϕpxqdx, @ g P SpRNq.
In particular, by Hölder’s inequality, any functions in Lp can be identified with a tempered
distribution.
Definition 1.4.18. For F P S1pRNq, we define the Fourier transform by
x pF , ϕy pFpϕq xF, pϕy F ppϕq, @ϕ P SpRNq.
Observe that if f P L1pRNq, then pf coincides with the Fourier transform in S1pRNq.
As in SpRNq, we have that the following statement holds in S1pRNq.
Theorem 1.4.19. F : S1pRNq Ñ S1pRNq is an isomorphism and both F and F1 are
continuous.
Proof. See Friedlander and Joshi [24, Theorem 8.3.2, page 99] and Duoandikoetxea and Zuazo [15].
Definition 1.4.20. Considering F P S1pRNq and ψ P SpRNq, we define the convolution
of F and ψ as
Proposition 1.4.21. Let F ψ be a C8-function of polynomial growth, then F ψ defines a tempered distribution by the formula
xF ψ, φy »
RN
pF ψqpxqφpxqdx, @φ P SpRNq.
Proof. See Friedlander and Joshi [24, Theorem 5.1.1, page 51]. With these tools, we can prove the following result. Theorem 1.4.22. If F P S1pRNq, then
z
F ψ pF pψ, @ψ P SpRNq,
where pF pψ P S1pRNq is defined as
x pF pψ, φy pF pψpφq pFp pψφq x pF , pψφy, @φ P SpRNq.
Proof. By the previous properties and the fact that rψ ppψ, where rψpxq ψpxq, it follows that
x zF ψ, φy xF ψ, pφy xF, pφ ppψy xF, zφ pψy x pF , φ pψy x pF pψ, φy, thus obtaining the result.
Lemma 1.4.23. For α P p0, Nq y1
|x|αpξq CN,s
1 |ξ|Nα
as a tempered distribution, i.e., for all ϕP SpRNq
» 1 |x|αϕppxq dx CN,s » 1 |ξ|Nαϕpξq dξ where CN,s πα N 2Γ N 2 α 2 {Γ α 2 . Proof. See, for instance, Guzmán [38].
1.5
Homogeneous Sobolev spaces
The Sobolev spaces 9Hs,rpRNq are formed by the distributions whose derivatives up
to order s are in LrpRNq. In the case r 2, besides being Hilbert spaces, the Fourier transform is a unitary operator in these spaces.
1.5. HOMOGENEOUS SOBOLEV SPACES 25
Definition 1.5.1. Let sP R, 1 ¤ r 8. The homogeneous Sobolev space 9Hs,rpRNq is
defined by
9Hs,rpRNq !f P SpRNq : Dsf P LrpRNq where Dsf |ξ|sfp _)
}} Hs,r9
with the norm
}f}H9s,r p|ξ|sfpq_
Lr }D
sf} Lr.
Remark 1.5.2. For more details about the definition and proprieties of the homogeneous Sobolev spaces 9Hs,rpRNq see, for instance, Wang et al. [64, page 19], Grafakos [34, page
16] and Triebel and Schmeisser [63, page 237].
We often use 9Hs,r to denote 9Hs,rpRNq. In the case r 2, we write 9Hs instead of
9Hs,2. Note that, when r 2 the norm comes from the inner product
xf, gys » RN Dsf Dsgdx » RN |ξ|2sfppξqpgpξqdξ.
Theorem 1.5.3. Let s P p0,Nrq and 1 r 8. Then 9Hs,rpRNq is continuously embedded in LppRNq with s N r N p. Moreover, for f P 9H s,rpRNq, we have }f}Lp ¤ CN,s,r}Dsf}Lr.
Proof. First, assume that g Dsf P SpRNq. Then,
Dsf g or f Dsg 1 |ξ|spg _ CN,s |x|Ns g,
where we have used Lemma 1.4.23. Thus by Hardy-Littlewood-Sobolev estimate (see Linares and Ponce [46] Theorem 2.6, page 35) it follows that
}f}Lp }Dsg}Lp CN,s |x|Ns g Lp ¤ CN,s,r}g}Lr CN,s,r}Dsf}Lr.
The general case follows from the density of SpRNq in Lr1.
Proposition 1.5.4. 9Hs,rpRNq satisfies the following properties:
(i) SpRNq is dense in 9Hs,r, with the norm of 9Hs,r, for all sP R and 1 r 8. (ii) 9H0 L2 and } }
9
H0 } }L2.
Proof. Item (i) immediately follows from the definition of 9Hs. Items (ii) and (iii) do not
present difficult.
In view of Proposition1.2.4, for s1, s2 ¡ 0 the space 9Hs1 X 9Hs2 with the norm
}f}H9s1X 9Hs2 }f}H9s1 }f}H9s2 is Banach. For ¡ 0, we use the notation Hs : L2 X 9Hs and
} }Hs } }L2 } }H9s.
Proposition 1.5.5. Let s¡ 0. We have the following embeddings (i) If s N2, then HspRNq ãÑ LrpRNq for every 2 ¤ r 2N
N2s.
(ii) If s¥ N2, then HspRNq ãÑ LrpRNq for every 2 ¤ r 8.
(iii) If s¡ N2, then HspRNq ãÑ L8pRNq.
Proof. The proof is inspired in Demengel et al. [11]. (i) For each f P HspRNq, s N
2, from Theorem1.5.3we already have that f P L
2XLN2N2s.
Since 2 r 2N
N2s there exists θP p0, 1q such that
1 r 1 θ 2 θpN 2sq 2N . By Proposition 1.3.1 we have » |fpxq|rdx 1 r » p|fpxq|1θ|fpxq|θqrdx 1 r ¤ }f}1θ L2 }f} θ LN2N2s, and thus, f P LrpRNq.
(ii) We can use that if s1 s, then Hs admits an embedding into Hs1. To see this, it
suffices to use the definition of the norms. Using the previous result with s1 N2, we obtain the desired result.
1.6. DIFFERENTIAL CALCULUS FOR MAPPINGS IN BANACH SPACES 27
Remark 1.5.6. The Sobolev space 9H1pRNq coincides with the space of function
9
W1,2 tf P LN2N2; ∇f P L2pRNqu.
Indeed, if f P 9H1, then by the Theorem 1.5.3f P LN2N2 and D1f2L2 1 4π2 N ¸ k1 » |p2πiqξkfppξq|2dξ 1 4π2 N ¸ k1 » |Bxkfpxq| 2 dx 1 4π2}∇f} 2 L2
In the case sP Z we have the following important inequality.
Theorem 1.5.7. [Gagliardo-Nirenberg’s inequality] Consider 1¤ p, q, r ¤ 8 and let j, m be two integers, 0¤ j m. If 1 q j N θ 1 r m N 1 θ p ,
for some θ P mj, 1 (θ<1 if r ¡ 1 and m j Nr 0), then there exists a constant c cpj, m, p, q, rq such that ¸ |α|j }Dαf} Lq ¤ c ¸ |β|m }Dβf} Lr θ }f}1θ Lp for all f P SpRNq.
Proof. See Cazenave [6, Theorem 1.3.7, page 9] and Nirenberg [55] .
Remark 1.5.8. An immediate consequence of above theorem is the following embedding 9H1pRNq X LppRNq ãÑ LqpRNq (1.3)
for all p q 2 with
2 $ ' & ' % 2N N2, if N ¥ 3 8, if N ¤ 2. (1.4)
1.6
Differential calculus for mappings in Banach spaces
Let X and Y be Banach spaces with norms } }X and} }Y respectively. Let U X
Definition 1.6.1. Let x be a point of the open subset U X. The mapping F : U Ñ Y is Fréchet-differentiable at x P U if there exists a linear operator (unique) A P LpX, Y q such that
Fpx hq F pxq Ah rphq, where limhÑ0 }rprq}}h} Y
X 0.
The operator A is said to be the Fréchet derivative of the mapping F at x and can be denoted as F1pxq. Let F : U Ñ Y be differentiable at every point of U. The mapping F1 : UÑLpX, Y q is called the Fréchet-derivative of F .
Remark 1.6.2. 1. Let H be a Hilbert space with inner product x , y and norm } }. The functional F : H Ñ C such that
Fpxq 1 2}x} 2 1 2xx, xy is Fréchet-differentiable and F1pxqϕ Re xx, ϕy. 2. Let F : Lp Ñ C be the functional given by
Fpxq 1 p}x}
p Lp. Then, F is Fréchet-differentiable and
F1pxqϕ Re x|x|p2x, ϕy. Lemma 1.6.3. Let Fpuq 1
p »
|x|b|u|pdx for p¡ 2. Then, F P C2pH1; Cq with
F1puqϕ »
|x|b|u|p2uϕ dx for all u, ϕP H1pRq.
Proof. See Genoud and Stuart [29, Lemma 2.1].
Proposition 1.6.4. (Chain rule) Let F : U Ñ Y , G : V Ñ Z be mappings with U and V open subsets of X and Y such that V F pUq. Let G F : U Ñ Z be the composition mapping. If F is Fréchet-differentiable at xP U and G : V Ñ Z is Fréchet-differentiable at y F pxq P V , then G F is Fréchet-differentiable at x and
1.6. DIFFERENTIAL CALCULUS FOR MAPPINGS IN BANACH SPACES 29
Remark 1.6.5. Let J : U Ñ R be a mapping Fréchet-differentiable with U open subset of X, 0Q I an open interval and v, ϕ P U. Thus, Jpv tϕq : I Ñ R is a real C1function. Suppose that the minimum value of Jpv tϕq in U is attained at t 0. From the theory of critical points for real functions and from the chain rule, we have
0 d dt t0rJpv tϕqs J 1pvqϕ.
Now, consider the problem
∆u fpx, uq in RN (1.5)
where f : Rn C Ñ C is a continuous function.
Definition 1.6.6. A classical solution of (1.5) is a function u P C2pRNq satisfying (1.5)
(in the usual sense). A weak solution of (1.5) is a function uP 9H1pRNq satisfying
»
∇u ∇ϕ dx »
f ϕ dx, @ϕ P 9H1 where ∇u ∇ϕ °Nk1BxkuBxiϕ.
Remark 1.6.7. Let X H be a Hilbert space with inner product x , y. By the Riesz representation theorem there exists a unique element ∇Fpxq P H such that
F1pxqh x∇F pxq, hy, @h P H.
The equation F1pxq 0 is said to be the Euler-Lagrange equation of the functional F : H Ñ R. Its solutions are assumed in the weak sense, that is,
x∇F pxq, hy 0, @h P H
and are considered as critical points of the functional F : H Ñ R.
Theorem 1.6.8 (Lagrange multipliers). Let X be a Banach space, let F, J P C1pX, Rq and the set
M tv P X; F pvq 0u. Let S M, S H, and suppose u0 P S satisfies
Jpu0q inf
vPSJpvq.
If F1pu0q 0 and if M X tx P X; }x u0}X ¤ ηu S for some η ¡ 0, then there exists a
Lagrange multiplier λP R such that J1pu0q λF1pu0q.
1.7
The Schrödinger equation
Let u0 P SpRNq and let u P C8pR, SpRNqq be defined by
y
uptqpξq e4π2i|ξ|2tup0pξq for all t P R and ξ P RN.
We have iBtpuptq 4π2|ξ|2puptq in R RN, and so iBtu ∆u 0 in R RN. In other
words, upx, tq e4π2i|ξ|2tpu0 _ : eit∆u0
(this notation is motivated by Stone’s Theorem, for more details see Yosida [68]) is a solution to following IVP
$ ' & ' % iBtu ∆u 0 upx, 0q u0pxq.
Note that for all sP R and t P R,
eit∆u0H9s }u0}H9s.
Since SpRNq is dense in 9Hs for all sP R, we deduce that for any s P R, teit∆u
tPR can be
extended to a group of isometries in 9Hs (see Theorem 1.2.6), which we still denote by
teit∆u
tPR. Now we establish the properties of the groupteit∆ut88 in the spaces LppRNq.
Lemma 1.7.1. If t 0, 1p p11 1 and p1 P r1, 2s, then eit∆ : Lp1pRNq Ñ LppRNq is
continuous and }eit∆f} Lp ¤ c|t| N 2 1 p1 1 p }f}Lp1.
Proof. See Linares and Ponce [46, Theorem 4.1, page 66].
Let I be a bounded, open interval of R with 0 P I. Let s P R, σ, b ¡ 0, u0 P SpRNq
and uP C1 I; SpRNq. We can deduce1 that u satisfies
uptq eit∆u0 iκ
»t
0
eiptsq∆|x|b|upsq|2σupsq ds (1.6) for tP I if and only if
$ ' & ' % iBtu ∆u κ|x|b|u|2σu 0 up, 0q u0. (1.7) 1due to regularity of u
1.7. THE SCHRÖDINGER EQUATION 31
In general, by Duhamel’s principle the solutions to (1.7) are given by (1.6). The difference between the equation (1.6) and the IVP (1.7) is that (1.6) does not require any differen-tiability of the solution. In what follows, we define that uP X is a solution to (1.7) with initial data u0 if u satisfies the integral equation (1.6).
Theorem 1.7.2. Let u0 P H1pRNq. Then there exists T ¡ 0 and unique solution u P
CppT, T q; H1pRNqq to (1.7). Furthermore, there is conservation of mass and energy, i.e.,
}u0}L2 }uptq}L2 and Eru0s Eruptqs
for all tP pT, T q.
Proof. See Genoud and Stuart [29] and Cazenave [6]. Definition 1.7.3. We call the pair pq, pq 9Hs-admissible if
2 q N 2 N p s, where $ ' ' ' & ' ' ' % 2N N2s ¤ p 2N N2 if N ¥ 3, 2 1s ¤ p ¤ p 2 1sq 1 if N 2, 2 12s ¤ p ¤ 8 if N 1.
We also define that pq, pq is 9Hs-admissible if 2 q N 2 N p s, where $ ' ' ' & ' ' ' % 2N N2s ¤ p 2N N2 if N ¥ 3, 2 1s ¤ p ¤ p 2 1 sq 1 if N 2, 2 12s ¤ p ¤ 8 if N 1.
Given s P R, let As tpq, pq; pq, pq is 9Hs admissibleu and As tpq1, p1q; 1q q11
1 and 1p p11 1 for pq, pq P Asu. We define the following Strichartz norm
}u}Sp 9Hsq sup
pq,pqPAs
}u}LqtLpx
and the dual Strichartz norm
}u}S1p 9Hsq infpq,pqPA
s}u}L
q1 t L
p1 x.
The main tools to show local and global well-posedness of the IVP (1) are the well-known Strichartz estimates. We recall some Strichartz type estimates associated to the linear Schrödinger propagator. See, for instance, Linares-Ponce [46] and Kato [42] (see also Holmer and Roudenko [41] and the references therein).
Lemma 1.7.4. The following statements hold. (i) (Linear estimates).
}eit∆f}
SpL2q¤ c}f}L2,
}eit∆f}
Sp 9Hsq¤ c}f}H9s. (ii) (Inhomogeneous estimates).
» R eiptt1q∆gp., t1qdt1 SpL2q »t 0 eiptt1q∆gp., t1qdt1 SpL2q ¤ c}g}S1pL2q, »t 0 eiptt1q∆gp., t1qdt1 Sp 9Hsq ¤ c}g}S1p 9Hsq.
If u upx, tq is a solution to the PDE in (1.7), then the following functions are also solutions:
(i) uλpx, tq λ
2b
2σ upλx, λ2tq; (scaling invariance)
(ii) uθpx, tq eiθupx, tq; (phase invariance)
(iii) ut0px, tq upx, t t0q, t0 P R (time-translation)
(iv) vpx, tq upx, tq. (time-reversal)
We closed this section recalling the following sharp Gagliardo-Nirenberg type in-equality.
Theorem 1.7.5. Let 0 b 2 and 0 σ σb (see (5)). Then, » |x|b|u|2σ 2dx¤ C GN}∇u}2σsL2 c 2}u} 2σp1scq L2 @u P H1pRNq. (1.8) with CGN ¡ 0 given by CGN 2σp1 scq 2σsc 2 σsc 2σ 2 p2σsc 2q}Q}2σL2 ,
1.7. THE SCHRÖDINGER EQUATION 33
where Q is the unique non-negative, radially-symmetric positive, decreasing solution of the elliptic problem
∆Q |x|b|Q|2σQ Q.
Proof. For the intercritical case, 2bN σ σb, see Farah [18] and for σ 2bN see Genoud [28]. Following the same steps of the proof in [18] we can prove the Theorem if 0 σ 2bN .
Chapter 2
Sobolev compact embedding in a
weighted L
p
space
2.1
Introduction
In the study of the PDEs the compactness is an important tool in the minimization of variational problems and another themes (see e.g. Cazenave [6]). For a bounded domain Ω RN it is well known the Sobolev compact embedding 9H0mpΩq ãÑ LppΩq with p N2N2m (where 9Hs
0pΩq is the closing of the C8-functions f from Ω to C such that
Bαf P L2 for|α| m and fpxq 0, @x P BΩ). A great and well celebrated contribution to
the theory of elliptic PDEs was given by Lions [47], [48], [49] and [50], who introduced the concentration-compactness method, which immediately turned out to be a standard tool. After the work by Lions, Solimini [58] and Gérard [30] independently, and with different proofs, were able to describe in a precise way the lack of compactness of the Sobolev embedding 9HspRNq ãÑ LppsqpRNq (see Proposition 1.5.3) where ppsq 2N
N2s. Recently,
Hmidi and Keraani [39] showed, inspired by the work of Gérard [30], the following result which is known as the profile decomposition.
Theorem 2.1.1 (Hmidi and Keraani [39]). Let tvnun18 be a bounded sequence in H1pRNq.
Then, there exists a subsequence of tvnun18 (also denoted tvnun18), a family tx j nu
8
n1 of
sequences in RN and a sequence tVj nu
8
n1 of H
1pRNq functions such that
(i) for all k j,
|xk n x
j
n| ÝÝÝÝÑnÑ 8 0
(ii) for all l¥ 1 and x P RN, vnpxq l ¸ j1 Vjpx xjnq vlnpxq, with lim sup nÑ 8 }v l n}Lp ÝÝÝÝÑ lÑ 8 0, for all pP p2, 2q1. Moreover, as nÑ 8, }vn}2L2 l ¸ j1 }Vj}2 L2 }vnl}2L2 op1q, }∇vn}2L2 l ¸ j1 }∇Vj}2 L2 }∇vnl}2L2 op1q.
In the sequel, we consider the following definitions.
Definition 2.1.2. Let X be a Banach space. We say that a function f : X Ñ C is weakly sequentially continuous if for all sequence txnun18 in X such that xn á x weakly in X we
have fpxnq Ñ fpxq as n Ñ 8.
Definition 2.1.3. Let X and Y be two normed vector spaces with norms } }X and } }Y
respectively, and suppose that X Y . We say that X is compactly embedded in Y if X is continuously embedded in Y , i.e., there is a constant c such that}x}Y ¤ c}x}X for all x in
X and the embedding of X into Y is a compact operator, i.e. every sequence in such a bounded set has a subsequence that is Cauchy in the norm } }Y.
More recently, the profile decomposition has been used as alternative to study of the behavior solutions for dispersive equations. However, if we consider a weighted Lp space
we have compactness. Indeed, it is known the following result.
Proposition 2.1.4. Let N ¥ 1, 0 b mint2, Nu and 0 σ σb. Then the functional f ÞÑ
»
|x|b|fpxq|2σ 2dx
is weakly sequentially continuous in H1pRNq.
2.2. ON THE COMPACTNESS 37
Proof. See Genoud and Stuart [29, Section 2.1] and Genoud [27, Section 1.1] for N ¥ 2, Genoud [26, Section 2] for N 1.
In this chapter, we first show a generalization for the previous proposition. In sequel, we use it to study the stability of standing waves for the focusing L2-subcritical INLS
equation.
2.2
On the compactness
In this section, we present a generalization of Proposition 2.1.4. For 1 ¤ p 8, consider the functional
} }L2σb 2 : 9H
1X Lp Ñ R such that }u}
L2σb 2
»
|x|b|u|2σ 2dx.
Here, we prove that the functional f ÞÑ
»
|x|b|fpxq|2σ 2dx
is weakly sequentially continuous in 9H1XLp (see Corollary2.2.2). More than that, defining
L2σ 2b pRNq tf P MpRN; Cq; }f }L2σ 2
b 8u, we show that the embedding
9H1pRNq X LppRNq ãÑ L2σ 2
b pR Nq
is compact.
Proposition 2.2.1. Let N ¥ 1, 0 b 2, 0 σ σb and 2¤ p p2σ 2qNNb . The Sobolev embedding
9H1pRNq X LppRNq ãÑ L2σ 2
b pR Nq
is compact.
Proof. For the case p 2 see Genoud [28]. Let tunun18 be a bounded sequence in 9H1X Lp.
Then, there exists u P 9H1 X Lp such that u
n á u in 9H1 X Lp as n Ñ 8. Defining
wn un u, we will show that
»
|x|b|w
as nÑ 8. First, from the weak convergence, twnun18 is uniformly bounded in 9H1X Lp,
and thus, using the Sobolev embedding (1.3), we get that twnun18 is uniformly bounded in L
qpRNq for all p q 2. (2.1)
Moreover, for all R¡ 0 and α ¡ N, we have that »
RNzBp0;Rq
|x|αdx¤ Cpαq
RαN. (2.2)
Recalling the condition σ N2b2, we get pσ 1qp2 bq σpN bq ¡ 0. Let 0 ε pσ 1qp2 bq σpN bq, and choose γ11 such that γ11
1
b
N ε (i.e., γ11 ¡
N b) and
p p2σ 2qγ1 2 (where γ1 is such that γ11 γ11
1 1). Thus, by Hölder’s inequality, we
have » RNzBp0;Rq |x|b|w n|2σ 2 dx¤ » RNzBp0,Rq |x|bγ1 1dx 1 γ1 1 » RNzBp0;Rq |wn|p2σ 2qγ1dx 1 γ1 (2.3)
for all R¡ 0. Given ε ¡ 0, from (2.1), (2.2) and (2.3), we can choose R¡ 0 such that » RNzBp0,Rq |x|b|w n|2σ 2dx ε 2. (2.4)
Now, we are going to estimate the integral over the ball Bp0; Rq. For R ¡ 0 chosen in (2.4), we have wn
Bp0;Rq P H
1pBp0; Rqq and w
n á 0 in H1pBp0; Rqq. By compactness of
the Sobolev embedding H1pBp0; Rqq ãÑ LqpBp0; Rqq (see Evans [17, Theorem 1, page
286]), we have
the convergence wnÑ 0 strong in LqpBp0; Rqq for 2 q 2. (2.5)
Again, since σ N2b2, we obtain NNb pσ 1qNN2 ¡ 0. Then, we can choose γ11 2
b N ε,
then γ21 Nb and p p2σ 2qγ2 2 (where γ2 is such that γ12 γ11
2 1q. Hence, by
Hölder’s inequality we have » Bp0;Rq |x|b|w n| 2σ 2 dx¤ » Bp0;Rq |x|bγ1 2dx 1 γ1 2 » Bp0;Rq |wn|p2σ 2qγ2 dx 1 γ2 ,
and thus, together with (2.5), there exists n0 such that for any n¥ n0
» Bp0;Rq |x|b|w n|2σ 2 dx ε 2, which yields the proof of Proposition 2.2.1.
2.3. STABILITY OF STANDING WAVES 39
Corollary 2.2.2. Let N ¥ 1, 0 b mint2, Nu and 0 σ σb. Then the functional f ÞÑ
»
|x|b|fpxq|2σ 2
dx
is weakly sequentially continuous in 9H1pRNq X LppRNq for 2 ¤ p p2σ 2qN Nb .
Proof. Lettfnun18 be a sequence in 9H1XLp and f P 9H1XLp such that fná f in 9H1XLp.
Note that
}fn}L2σb 2 }f}Lb2σ 2 ¤ }fn f}L2σb 2
and thus, by the arguments of the proof of Proposition 2.2.1 we have the result. Remark 2.2.3. If p 2 and 0 σ σb, then we have Proposition 2.1.4.
2.3
Stability of standing waves for focusing L
2-subcritical
INLS
For ω ¡ 0, consider the following elliptic equation
∆ψ |x|b|ψ|2σψ ωψ x P RN. (2.6) Let ψω,b P H1pRNq be unique positive radial solution to the equation (2.6). Now, for δ ¡ 0,
define the set
Uδpψω,bq : ! v P H1; inf θPR}v e iθ} H1 δ ) .
Using the criteria of Grillakis et al. [35], De Bouard and Fukuizumi [10] studied the stability of the standing waves eiωtψ
ω,b. They showed that for any ω ¡ 0, the standing
wave eiωtψω,b is stable, i.e., given ε¡ 0 there exists δ ¡ 0 such that for any initial data
u0 P Uδpψω,bq, the corresponding solution uptq satisfies uptq P Uεpφωq of for any t ¥ 0. The
approach is based on studying properties on the linearized operator.
The main purpose of this section is to present an alternative proof for the stability of standing waves for focusing (κ 1) L2-subcritical INLS equation by variational methods.
Our approach uses as main tool the compact embedding of H1pRNq 9H1pRNq X L2pRNq
Remark 2.3.1. It is possible to show that our result is equivalent to the result obtained by De Bouard and Fukuizumi [10], using a similar argument to of Cazenave [6, Corollary 8.3.8, page 277].
Let N ¥ 1, 0 b 2 and 0 σ 2b
N . We consider the following minimization
problem
dM : inftErvs, v P H1 and }v}2L2 Mu (2.7)
where Ervs is the energy functional defined in (3) We will see later (Proposition 2.3.2) that the above minimization problem is well defined (i.e., dM is a real positive number),
and moreover, the infimum is attained. We denote by SM the set
SM : tv P H1; v is a minimizer of dMu.
We now are going to study the variational problems (2.7) using Proposition 2.2.1. Proposition 2.3.2. Let N ¥ 1, 0 b 2 and 0 σ 2Nb. Then dM is well defined and
there exists C1 ¡ 0 such that
dM ¤ C1 0. (2.8)
Moreover, there exists vP H1pRNq such that Ervs dM and }v}2L2 M.
Proof. First, by the Gagliardo-Nirenberg inequality (1.8), if v P H1pRNq and }v}2
L2 M, then Ervs 1 2}∇v} 2 L2 1 2σ 2 » |x|b|v|2σ 2 dx (2.9) ¥ 1 2}∇v} 2 L2 2σp1 scq 2σsc 2 σsc 1 p2σsc 2q}Q}2σL2 M2σ 2pNσ bq2 }∇v}N σ b L2 .
Since 0 Nσ b 2, we have that 2
N σ b ¡ 1, and thus, using Young’s inequality (Lemma
1.2.1), for any ε¡ 0 2σp1 scq 2σsc 2 σsc M2σ 2pNσ bq2 p2σsc 2q}Q}2σL2 }∇v}N σ b L2 ¤ ε}∇v}2L2 Cpε, N, σ, Mq. (2.10)
So, replacing (2.10) in (2.9), we have Ervs ¥ 1
2}∇v}
2
