Delft University of Technology
Blow-up Dynamics and Orbital Stability for Inhomogeneous Dispersive Equations
Csobo, Elek DOI 10.4233/uuid:bd0b3a57-2f03-4bac-bf5e-1d5fac03df88 Publication date 2019 Citation (APA)
Csobo, E. (2019). Blow-up Dynamics and Orbital Stability for Inhomogeneous Dispersive Equations. https://doi.org/10.4233/uuid:bd0b3a57-2f03-4bac-bf5e-1d5fac03df88
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Blow-up Dynamics and Orbital Stability for
Inhomogeneous Dispersive Equations
Blow-up Dynamics and Orbital Stability for
Inhomogeneous Dispersive Equations
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.dr.ir. T.H.J.J. van der Hagen, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op vrijdag 29 november 2019 om 10.00 uur
door
Elek CSOBO
Master of Science in Applied Mathematics,
Budapest University of Technology and Economics, Hongarije geboren te Budapest, Hongarije
Dit proefschrift is goedgekeurd door de Promotor: Prof. dr. J.M.A.M. van Neerven Promotor: Dr. S. Le Coz
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. J.M.A.M. van Neerven Technische Universiteit Delft
Dr. S. Le Coz Universit´e Paul Sabatier, Toulouse
Onafhankelijke leden:
Dr. F. Genoud EPFL Lausanne, Switzerland
Dr. A. Geyer Technische Universiteit Delft
Prof. dr. T. Duyckaerts Universit´e Paris 13
Prof. dr. F.H.J. Redig Technische Universiteit Delft Prof. dr. ir. M.C. Veraar Technische Universiteit Delft
Prof. dr. ir. A.W. Heemink Technische Universiteit Delft, reservelid
Keywords: Schr¨odinger equation, Klein-Gordon equation, nonlinear partial differen-tial equation, orbital stability, singularity formation, Hamiltonian systems, ground states, standing waves equation
Contents
Chapter 1. Introduction 1
1. General context 2
2. Nonlinear Schr¨odinger equation with inverse square potential in dimension larger
than three 7
3. The Klein-Gordon equation with Delta potentials 10
4. The nonlinear Schr¨odinger equation with inverse square potential on the half-line 14
5. Open problems and research perspectives 16
Chapter 2. Minimal mass blow-up solutions for the L2 critical nonlinear Schr¨odinger
equation with inverse-square potential 23
1. Introduction 23
2. Ground states and the sharp global existence criterion 28
3. Construction of minimal mass blow up solutions 32
4. Classification of minimal mass blow-up solutions 33
Chapter 3. Stability of standing waves for a nonlinear Klein–Gordon equation with
delta potentials 49
1. Introduction 49
2. Local well-posedness of the Cauchy problem 56
3. Hamiltonian structure 64
4. Stability of standing waves 67
Chapter 4. Existence and orbital stability of standing waves to a nonlinear Schr¨odinger equation with inverse square potential on the half-line 85
1. Introduction 85
2. Local well-posedness of the Cauchy problem 87
3. Existence of bound states 93
4. Stability 100
CHAPTER 1
Introduction
In this thesis, we study the well-posedness, blow-up dynamics, and stability of non-linear Schr¨odinger and Klein-Gordon equations with space-dependent coefficients. These type of equations abound in various domains of physics in the modeling of nonlinear wave propagation in dispersive inhomogeneous media, such as surface waves in shallow water or electromagnetic waves in dielectric media.
For mathematicians, the focus so far has been mainly on equations with constant co-efficients, for which a fairly satisfactory description is now available, whereas the study of equations with space-dependent coefficients has only gained momentum recently and is still in its early stages. In real-life applications, however, it is crucial to consider equations with space-dependent coefficients, in order to account for the presence of an external field or inho-mogeneities of the propagating media. These equations are often termed as inhomogeneous and play a central role in areas such as nonlinear optics [38,53], cold quantum gases (e.g. Bose-Einstein condensates) [17,33,37], plasma physics [44,56] and water waves [48]. The presence of space-dependent coefficients typically breaks the space-translation and scaling invariance of homogeneous equations, giving rise to new physical phenomena and mathe-matical difficulties.
We are especially interested in the properties of the so-called solitary waves or standing waves to Schr¨odinger and Klein-Gordon equations with focusing power nonlinearities. Stand-ing waves are solutions of the form u(t, x) = eiωtϕ(x), where we suppose that ϕ is defined on RN, and ϕ(x) → 0 as |x| → ∞. This reflects the localized nature of standing wave solutions. Such a function is a solution of the Cauchy problem if and only if it solves the corresponding standing wave equation. For instance, in the case of the nonlinear Schr¨odinger equation
iut+ ∆u ± |u|p−1u = 0,
the standing wave equation reads as
Here the negative sign corresponds to the focusing nonlinearity, whereas the positive sign to the defocusing nonlinearity. Henceforth we will work only with equations with focusing non-linearities to investigate their blow-up behavior. Standing wave solutions play a pivotal role in the understanding of the global dynamics of nonlinear equations, and we will investigate their existence, variational characterization, and stability in various settings.
1. General context
Historical background. The study of nonlinear dispersive equations dates back to the 19th century. It starts with the observation of solitary waves in shallow water by John Scott Russell in 1834. These are waves which might undergo phase and spatial shifts, but preserve their original profile for a long time. Roughly speaking, the stability of the profile is the result of the balance of the competing dispersive and focusing effects, which correspond to the linear and nonlinear parts of the equation. The first theoretical justification of this phenomenon was given by Korteweg and de Vries in 1895. It is only in the 1970s that the rigorous mathematical analysis of solitary waves propagating in nonlinear media commenced. Since then, the literature on nonlinear dispersive equations has become abundant; it would be far too ambitious to give an exhaustive presentation of it. Thus, in this introduction, we will concentrate on the particular questions addressed in this thesis and attempt to situate them within the existing literature.
The existence of standing wave solutions to homogeneous equations was addressed in 1977 in the work by W. A. Strauss [51] in dimensions higher than three. In [8] H. Berestycki and P. L. Lions have established optimal conditions on a homogeneous nonlinearity to guarantee the existence of solutions in H1(RN), for N = 1 and N > 3. The two-dimensional case was
treated in the work by H. Berestycki, T. Gallouet, and O. Kavian [7]. In one dimension, the problem can be solved explicitly by ODE methods, whereas in higher dimensions critical point theory is used to obtain existence results in the Sobolev space H1(RN). In particular, one can define the action functional S on H1(RN),
S(u) = 1 2k∇uk 2 L2 + ω 2 kuk 2 L2− 1 p + 1kuk p+1 Lp+1,
whose critical points are the solutions of the standing wave equation. Therefore, to prove the existence of standing waves, it suffices to find a nontrivial critical point of the action functional. The functional S is unbounded from above and below, which prevents one from finding a nontrivial critical point by global minimization or maximization. One may overcome this difficulty by restricting S to the so-called Nehari manifold N = {u ∈ H1(RN) \ {0} :
S0(u)u = 0}. Clearly, the Nehari manifold contains all nontrivial critical points of the action functional. Solutions which minimize the action functional are called ground state solutions and are of particular interest. The advantage of these techniques is that it allows one to prove the existence of solutions on RN under rather general conditions on the nonlinearity.
Local well-posedness for nonlinear dispersive equations was rigorously studied in the works by J. Ginibre and G. Vello [25]. An essential tool to carry out a fixed point argu-ment in higher dimensions and to obtain local solutions are the Strichartz estimates for the Schr¨odinger group. In lower dimensions, however, these estimates are much more straight-forward, owing to the Sobolev embedding. A good overview of the local well-posedness theory for various dispersive equations can be found in [11], and an exhaustive study for homogeneous Schr¨odinger equations in [10].
Another essential problem is the global well-posedness and blow-up dynamics of nonlinear dispersive equations. In general, the asymptotic behavior of a nonlinear equation depends on the competition between the dispersive effect of the linear part of the equation and the focusing effect of the nonlinearity. If the focusing effect beats dispersion, then singularity formation occurs and the solution blows up in finite time, i.e., ku(t)kH1 → ∞ as t ↑ T for
some T < ∞. On the other hand, if there is a balance between focusing and dispersive effects, the solution exists for all times; hence, it is globally well-posed. For the focusing Schr¨odigner equation the so-called mass critical or L2 critical power nonlinearity, i.e. p = 1 + 4/N , plays
an essential role concerning the description of blow up. The Schr¨odinger equation with mass critical nonlinearity is invariant under the so-called L2 scaling
uλ(t, x) = λN/2u(λ2t, λx),
which leaves the L2 norm of u unchanged. Below the mass critical level solutions
corre-sponding to any initial condition are globally well-posed. This follows directly from the con-servation laws and the Gagliardo-Nirenberg inequality. For power nonlinearities above the mass critical exponent, however, global well-posedness does not hold, since the Gagliardo-Nirenberg inequality does not suffice to ensure global well-posedness. Instead, based on the purely Hamiltonian information that E(u0) < 0, one can prove that the corresponding
maximal solution blows up in finite time using the virial identities and a convexity argu-ment (see [4]). Hence, there is an open region in the energy space where blow up is a stable phenomenon. For the focusing Schr¨odinger equation with mass-critical nonlinearity, M. I. Weinstein has given a sharp characterization in [57] of global well-posedness. By es-tablishing the best constant in a Gagliardo-Nirenberg inequality in terms of the L2-norm of
the ground state solution to the homogeneous Schr¨odinger equation, he determined that the L2-norm of the ground state solution is the threshold for blow-up solutions, i.e., solutions
with initial condition below this threshold are globally well-posed. This threshold is indeed sharp since it is possible to construct minimal mass blow-up solutions from the ground state via the pseudo-conformal transformation. In his famous paper [39], F. Merle has shown that minimal-mass blow-up solutions admit the same profile, up to the symmetries of the equation. The original, rather complicated, proof, was later simplified by V. Banica [1], and by T. Hmidi and S. Keraani [30].
The blow-up dynamics above the minimal-mass level for the mass-critical homogeneous Schr¨odinger equations still poses difficult questions. Blow-up solutions can be constructed from a standing wave solution using the pseudo-conformal transformation. Blow-up solutions above the minimal mass were numerically investigated by Landman in [35]. Analytical results for the blow-up speed and the classification of blow-up solutions with L2 norm above
and close to the minimal mass level were obtained in the works of F. Merle and P. Rapha¨el [40–43]. This is known as the log-log blow up regime; solutions slightly above the minimal mass level have the blow-up rate
k∇u(t)kL2 ∼
r
log | log(T − t)|
T − t as t ↑ T. (1.1.1)
Hamiltonian formalism. Both Schr¨odinger and Klein-Gordon equations can be rewrit-ten as an infinite dimensional first-order Hamiltonian system. We briefly recall the basic notion of Hamiltonian formalism as it plays an important role in our investigations, an ex-haustive treatment of the subject can be found in the work by S. De Bi`evre, F. Genoud, S. Rota-Nodari [15] and C. A. Stuart [52]. LetH and L be Hilbert spaces with duals H ∗ and L∗, such that they form a variational triple
H ,→ L = L∗
,→H ∗, where we identifyL with L∗ via their Riesz isomorphism.
We also assume that there exist a mapping J from H to H∗ and a Hamiltonian func-tional E such that the system can be rewritten in the form
J d
dtU (t) = E
0
(U (t)).
The mapping J is called a symplector. It is a bounded, one-to-one, anti-symmetric map, and it serves as a suitable generalization of the notion of the symplectic form to infinite dimensional spaces. Indeed, ω(ξ, η) = hJ ξ, ηiH∗,H defines a symplectic form on H . It
{T (θ) : θ ∈ R} on H , which also preserves the symplectic structure of the system. Moreover, there is an operator A ∈ B(H , H ), which generates the group of isometries, i.e. T (θ) = eθA.
In this context, we can also define the charge Q(U ) = hJ AU, U iH∗,H.
Under certain conditions, the Cauchy problem is locally well-posed in H . Indeed, for any initial condition U0 ∈ H , there exists a maximal solution U ∈ C((−Tmin, Tmax),H ).
Additionally, the quantities E and Q remain constant along the flow of the solution, which is, in general, a consequence of the Hamiltonian structure of the system. Furthermore, a blow-up alternative holds, i.e. if Tmax(U0) < ∞, then limt→TmaxkU (t)kH = ∞, and if
Tmin(U0) < ∞, then limt→−TminkU (t)kH = ∞.
In this framework a standing wave solution is a solution of the following form
u(t, x) = T (ωt)ϕω(x) for all t ∈ R,
where ω ∈ R and ϕω ∈ H . Clearly, u(t) = T (ωt)ϕω(x) is a standing wave solution if and
only if (ω, ϕω) satisfies the standing wave equation
E0(ϕω) + ωQ0(ϕω) = 0.
Hence, standing wave solutions are the critical points of the so-called Lyapunov functional:
Lω(ϕω) = E(ϕω) + ωQ(ϕω).
Orbital stability. We will investigate the stability of standing waves of Schr¨odinger and Klein-Gordon equations. We wish to understand how the dynamical properties of partial differential equations change when a small perturbation of the system is introduced. One is typically interested in showing that a slight change in the initial conditions results only in a small perturbation of the long-time evolution of the system. However, owing to the symmetries of the equation, the usual Lyapunov stability is too restrictive, and no standing wave could be stable. To account for the symmetries of the equation, we need to define a weaker notion of stability, which is essentially the Lyapunov stability up to phase shifts and translations.
In order to do so, let us first present the notion of an orbit, which is auxiliary in the def-inition. The orbit of a stationary equation is determined by the symmetries of the equation. For instance, if we take a homogeneous Schr¨odinger equation
i∂tu + ∆u + |u|p−1u = 0 (1.1.2)
and ϕ is a solution of the stationary equation
then eiθϕ
ω(· + y) is also a solution to (1.1.3) for every θ ∈ R and y ∈ RN. Hence, the orbit
of a stationary solution u(t, x) = eiωtϕ
ω(x) is the set
O(ϕω) = {eiθϕω(· + y) : θ ∈ R, y ∈ RN}. (1.1.4)
We define the orbital stability of the stationary solution eiωtϕ
ω in H in the following way:
for every ε > 0 there exists a δ > 0 such that for all u0 ∈H satisfying kϕω− u(0)kH < δ,
it is true that
inf
v∈O(ϕω)
kv − u(t)kH < ε
for all t ∈ [0, +∞).
We will also investigate the linear instability of standing waves. Writing a solution U of the Cauchy problem in the form U (t) = eiω0t(Φ
ω0 + V (t)), we have that, at first order, V
satisfies the linearized equation
J d
dtV (t) = L
00
ω0(Φω0)V (t). (1.1.5)
The standing wave eiω0tΦ
ω0 is linearly unstable if V ≡ 0 is a linearly unstable solution
(in the sense of Lyapunov) of (1.1.5). Linear instability generally does not always imply orbital instability. For the homogeneous Schr¨odinger equation with power nonlinearity, this implication was shown by V. Georgiev and M. Ohta in [20].
In general, there are two different methods to investigate the orbital stability of stationary wave solutions. One, which was developed by T. Cazenave and P. L. Lions [12] in 1982, relies on a variational characterization of the ground states subject to an L2 constraint. They have
shown, using Lions’ concentration-compactness principle, that the ground state solutions of the nonlinear Schr¨odinger equation are minimizers of the energy functional under an L2 constraint if 1 < p < 1 + 4/N . Orbital stability of the ground state follows directly from this characterization. This is an optimal result, in the sense that for 1 + 4/N 6 p < 1 + 4/(N − 2) the ground state is unstable by blow up, which was shown by H. Berestycki and T. Cazenave in [4]. At the same time, a different approach was initiated by J. Shatah [49], and J. Shatah and W. A. Strauss [50] to study the orbital stability of stationary wave solutions to Schr¨odinger and Klein-Gordon equations. Their approach was further developed in collaboration with M. Grillakis [28,29] to treat general Hamiltonian systems under symmetries. This general theory provides us with a framework to determine the orbital stability of eiωtϕ
ω(x), by determining the spectral properties of L00ω(ϕω), and the sign
of ∂ω∂ kϕωk 2
L2. Hence, this method is particularly advantageous when we are dealing with
2. Nonlinear Schr¨odinger equation with inverse square potential in dimension larger than three
In the first chapter of this thesis we investigate the global well-posedness and singularity formation for a nonlinear Schr¨odinger equation with an inverse square potential
iut+ ∆u + |x|c2u + |u|p−1u = 0, u(0) = u0 ∈ H1(RN), (1.2.1)
for the mass critical exponent, p = 1 + 4/N , with N > 3. We fix the coupling constant c ∈ (0, c∗), where c∗ = (N − 2)2/4 is the best constant in Hardy’s inequality.
Owing to the presence of the inverse square potential the Cauchy problem (1.2.1) is closely related to Hardy’s inequality
c∗ Z RN |u|2 |x|2dx 6 Z RN |∇u|2dx.
The Schr¨odinger equation with inverse-square potential plays an important role in physics, for instance in quantum field equations, or in the study of certain black hole solutions of the Einstein equations, see in [32]. The study of nonlinear Schr¨odinger equation with inverse-square potential has attracted substantial interest in the last years. Stricharz estimates for the Schr¨odinger equation with inverse square potential in dimensions N > 3 were estab-lished by N. Burq, F. Planchon and J. G. Stalker [9] and local well-posedness for c < c∗
was established by N. Okazawa, T. Suzuki, T. Yokota [46]. Local well-posedness for the Hardy critical case (c = c∗) has only been covered recently [45]. In this work, the uniqueness
of the ground state has also been established. In [16,34,58], the authors consider scatter-ing and global well-posedness of the Cauchy problem. In [2] the authors establish orbital stability of standing waves for mass subcritical nonlinearities and strong instability by blow-up for mass sblow-upercritical nonlinearities in the Hardy subcritical case. Orbital stability for mass subcritical nonlinearities in the Hardy critical case was established by G. P. Trachanas, N. B. Zographopoulos in [54]. In [2] and in [54] orbital stability is proved by showing the precompactness of minimizing sequences of the energy functional on an L2 constraint.
It has been established in [46] that along the flow of (1.2.1) we have conservation of the L2 norm, also known as the mass,
Z RN |u(t, x)|2dx = Z RN |u0(x)|2dx,
and of the energy
E(u(t)) = 1 2 Z RN |∇u(t, x)|2dx − c 2 Z RN |u(t, x)|2 |x|2 dx − 1 p + 1 Z RN |u(t, x)|p+1dx = E(u 0).
For our analysis it is convenient to introduce the Hardy functional, defined on H1(RN) by H(u) = Z RN |∇u|2dx − c Z RN |u|2 |x|2dx.
By Hardy’s inequality, for all c ∈ (0, c∗)
1 − c
c∗
k∇uk2L2 6 H(u) 6 k∇uk
2 L2.
Hence, for all c ∈ (0, c∗), the functional H(u) defines a seminorm on H1(RN), which is
equivalent to k∇uk2L2. Therefore, a solution u(t) blows up at time T > 0 if and only if
limt↑T H(u(t)) = +∞.
Since |x|−2 scales the same way as the Laplacian, the Schr¨odinger equation with p = 1 + 4/N is invariant under the so-called L2-scaling
uλ(t, x) = λN/2u(λ2t, λx), for λ ∈ R
which leaves the L2-norm invariant, as in the homogeneous case (c = 0). Hence, we call the
equation mass critical. Additionally, the equation is invariant under time translations
ut0(t, x) = u(t − t0, x) for t0 ∈ R,
and phase shifts
uγ0(t, x) = e
iγ0u(t, x) for γ
0 ∈ R.
Note that since c > 0, the equation is not invariant under space translations and Galilean transforms.
In a joint work with F. Genoud [13] we established the classification of minimal mass blow-up solutions and a sharp characterization of global well-posedness in terms of the L2
-norm of the ground state solutions for attractive inverse potential (c > 0). Thereby we extended the famous results of M. I. Weinstein [57] and F. Merle [39]. Our result was later revisited and extended by A. Bensouilah and D. V. Duong in [3] for repulsive inverse square potential (c < 0) for radially symmetric functions. More recently, D. Mukherjee, P. T. Nam, and P. Nguyen [45] showed that our global well-posedness and classification results also persist in the Hardy critical case. The Cauchy problem (1.2.1) admits standing wave solutions. Indeed, functions of the form eiωtϕ(x) are solutions of (1.2.1) if and only if ϕ ∈ H1(RN) is a solution of the nonlinear elliptic problem
∆ϕ + c
|x|2ϕ − ωϕ + |ϕ| p−1
ϕ = 0. (1.2.2)
In our work we used Weinstein’s variational approach to show that (1.2.2) admits a positive radial solution Q. We have defined ground states as positive radial solutions of
(1.2.2), which minimize the Weinstein functional Jp,N(u) = H(u) p−1 4 Nkuk2+ p−1 2 (2−N ) L2 kukp+1Lp+1 .
Moreover, ground state solutions have the same L2-norm, denoted by Mgs, which follows from
the Pohozaev identity. Additionally, we have shown that E(Q) = 0. We have established the following: Let u0 ∈ H1(RN), p = 1 + 4/N , and Mgs be the mass of a ground state solution
of the Schr¨odinger equation. Then a sufficient condition for the global well-posedness of the Cauchy problem is:
ku0kL2 < Mgs.
To prove the global well-posedness result, we rely on a sharp Gagliardo-Nirenberg inequal-ity adapted to the inverse-square potential. By minimizing Jp,N, we obtain the best constant
on terms of the unique mass of the ground state solutions. From the sharp Gagliardo-Nirenberg inequality follows that
E(u) 6 1 2 1 − kukL2 Mgs 4/N! , u ∈ H1(RN). The assumption ku0kL2 < Mgs yields an a priori bound on H(u(t)), namely
H(u(t)) 6 2E(u0) 1 −
ku0kL2
Mgs
4/N!−1 ,
which yields global well-posedness.
We have also exhibited that Mgs is the smallest L2-norm where blow-up occurs, which
shows that our global well-posedness condition is sharp. Indeed, we can construct blow-up solutions from the ground state by the pseudo-conformal transformation, which is given by the following expression:
uT(t, x) = e−i |x|2 4(T −t) (T − t)N/2u 1 T − t, x T − t .
We have shown that if u is a global solution of the Schr¨odinger equation that does not scatter to a linear solution, then uT is also a solution to it, which blows up in finite time T > 0.
Hence, from a ground state solution, which is global in time, we can construct a blow-up solution. Moreover, this transformation leaves the L2-norm of the solution invariant, which
is a specific feature of the L2-critical exponent.
Applying phase translation, L2-scaling, and the pseudo-conformal transformation to the solution e−itQ(x), we obtain a three-parameter family of blow-up solutions: for all T ∈ R,
λ0 > 0, and γ0 ∈ R the function ST ,λ0,γ0(t, x) = e iγ0ei λ20 T −te−i |x|2 4(T −t) λ0 T − t N/2 Q λ0x T − t
is a blow-up solution on [0, T ) of minimal mass, and Q is a ground state solution. Conversely, we have shown that all u minimal-mass blow-up solutions on [0, T ), can be constructed from a ground state solution by the pseudo-conformal transformation, i.e. there are T ∈ R, λ0 > 0,
γ0 ∈ R, and Q ground state solution such that
u(t) = ST ,λ0,γ0(t, x).
To establish our classification result, we employ the dynamical arguments by T. Hmidi and S. Keraani in [30], and a compactness result, which we obtain by the concentration-compactness principle. When applying the concentration-concentration-compactness principle, a difficulty arises owing to the loss of space-translation invariance of the problem.
3. The Klein-Gordon equation with Delta potentials
The second chapter of this thesis addresses the orbital stability and instability of the standing wave solutions to a Klein-Gordon equation
utt− uxx+ m2u + γδu + iαδut− |u|p−1u = 0,
(u(0), ∂tu(0)) = (u0, u1) ∈ H1(R) × L2(R).
(1.3.1)
where δ denotes the Dirac mass at the origin. We seek solutions u : R × R → C, m > 0, α ∈ R and γ ∈ R are parameters and p > 1 determines the strength of the nonlinearity.
The various properties of the nonlinear homogeneous Klein-Gordon equation are well understood. Local well-posedness can be established by a standard fixed point argument, a detailed presentation of which can be found in [11]. The orbital stability/instability was established in the seminal works of J. Shatah [49], and J. Shatah and W. Strauss [50]. They have shown, that in N space dimensions standing waves are unstable if p > 1 + 4/N. On the other hand, if 1 < p < 1+4/N , then there is a critical frequency ωc, such that standing waves
with frequency ω are orbitally stable if ωc< |ω| < m and orbitally unstable if 0 < |ω| < ωc.
Strong instability by blow up was established by M. Ohta and G. Todorova [47] under the condition 0 < 2(p + 1)ω2 < p − 1.
The effect of a delta potential on the dynamics of the nonlinear Schr¨odinger equation has also attracted substantial attention. The delta potential typically models the presence of an impurity or trapping potential localized at the origin and its interaction with a traveling wave. The delta potential at the origin has been efficiently modeled by imposing a jump
condition on the first derivative of the solution (see [18,19]). The orbital stability of ground state solutions of the Schr¨odinger equation with attractive delta potential (γ > 0) was first investigated by R. H. Goodman, P. J. Holmes and M. I. Weinstein in [27] for p = 3. A complete investigation of the orbital stability with attractive delta potential was provided by R. Fukuizumi, M. Ohta, and T. Ozawa in [19]. Instead of relying on the theory of [28,29], and investigating the spectral conditions of the linearized operator around the standing wave, they use the fact that the standing wave can be characterized as a minimizer of the action functional on the Nehari manifold. Since the Nehari manifold is of codimension one, one can deduce that the linearized operator has at most one negative eigenvalue (see [18]). In [18], the authors study the effect of a repulsive delta potential (γ < 0) on the dynamics with a similar method. However, the standing wave solution is not a minimizer of the action functional on the Nehari manifold, but one has to restrict the minimization problem to the subspace of even functions, where the minimum is attained. Hence one works on a manifold with codimension two, which leaves open the possibility that two negative eigenvalues exist. Hence the orbital stability of the ground state is studied only in the subspace of even functions, where it has a minimizing character. Several stability regimes were established in this context. However, this does not imply stability in the nonlinear setting. This problem was overcome by S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim, and Y. Sivan in [36], by applying Kato’s perturbation theory to obtain the exact number of negative eigenvalues. They use results of [28,29] to obtain a full characterization of the stability of the ground state solution for the nonlinear Schr¨odinger equation with repulsive delta potential. More recently, the dynamics of a one-dimensional Schr¨odinger equation with an attractive delta potential and a cubic-quintic combination of attractive and repulsive nonlinearities was investigated in [21]. They investigated the existence and stability of positive bound states, and establish the existence of a regime where two stable bound states coexist with the same propagation constant.
In a joint work with F. Genoud, M. Ohta, and J. Royer [14] we initiated a rigorous anal-ysis of the Klein-Gordon equation with delta potentials at the origin in one space dimension. We have established well-posedness of the Cauchy problem, the Hamiltonian framework of (1.3.1) and several stability/instability regimes of the unique standing wave solution. Our results were later extended to nonlinear Klein-Gordon equations on star graphs in [26] by N. Goloshchapova. To prove local well-posedness, we rewrite the equation as a first-order system by introducing the variables (u, v) = (u, ut). We seek solutions in the energy space
H = H1(R) × L2(R), equipped with the usual inner product
h(u1, v1), (u2, v2)iH = hu01, u 0
2iL2+ hu1, u2iL2 + hv1, v2iL2,
where the real L2 inner product is defined as hu, vi = Re
Z
R
u¯vdx.
To account for the presence of delta potentials at the origin, we impose a jump condition at the origin:
u0(0+) − u0(0−) = γu(0) + iαv(0).
Hence, the influence of the delta potentials will be accounted for in the domain of the free propagator A = 0 Id ∂x2− m2 0 ! , which is acting on
D= {(u, v) ∈ H1(R) ∩ H2(R \ {0}) × H1(R) : u0(0+) − u0(0−) = γu(0) + iαv(0)} ⊂ H1× L2.
This formulation allows us to rigorously investigate the local well-posedness and Hamiltonian structure of the nonlinear Klein-Gordon equation in the presence of Delta potentials. We have established that the pair (A, D) generates a C0 semi-group on H . Remarkably, this
might not be a unitary group. This is due to the fact that for positive values of γ the operator A is not skew-adjoint. Nevertheless, we were able to define an inner product onH , which is equivalent to the original one and in which A is skew-adjoint. Hence A generates a unitary group on H . However, for negative values of γ, the continuous group generated by A is possibly only exponentially bounded. Nevertheless, this is enough for our purpose; we were able to establish the local well-posedness of the Cauchy problem by using Duhamel’s formula and a fixed point argument.
To investigate the Hamiltonian formalism of the equation we relied on the work of De Bi`evre et al [15]. The Hamiltonian setting is rather intriguing, since the corresponding symplector J : H → H∗ depends on the coupling constant α. Moreover, we established that the energy
E(u, v) = 1 2ku 0k2 L2 + m2 2 kuk 2 L2+ 1 2kvk 2 L2 + γ 2|u(0)| 2− 1 p + 1kuk p+1 Lp+1 (1.3.2)
and the charge
Q(u, v) = Im Z R u¯vdx − α 2|u(0)| 2 (1.3.3)
are constant along the flow of the solution. Additionally, we were able to rewrite the Cauchy problem in a Hamiltonian formalism:
J d
dtU (t) = E
0
(U (t)),
where U = (u, v). The Klein-Gordon equation (1.3.1) possesses standing wave solutions of the form u(t, x) = eiωtϕ(x). The corresponding standing wave equation reads as
−ϕ00ω+ (m2− ω2)ϕω+ (γ − αω)δϕω− |ϕω|p−1ϕω = 0. (1.3.4)
It has been shown in [11,12], that there exists a unique solution to (1.3.4) if and only if (γ − αω)2 4 < (m 2− ω2). It is given explicitly by ϕω(x) = (p + 1)(m2− ω2) 2 sech 2 (p − 1) √ m2− ω2 2 |x| + tanh −1 −(γ − αω) 2√m2− ω2 1 p−1 .
This solution is constructed from the solution of unperturbed standing wave equation on each side of the delta potential.
By introducing Uω(t, x) = eiωt(ϕω(x), iωϕω(x)), the standing wave equation in the
Hamil-tonian setting becomes:
E0(Uω) + ωQ0(Uω) = 0.
Hence standing wave solutions are exactly the critical points of the Lyapunov functional
Lω(u, v) = E(u, v) + ωQ(u, v).
To investigate the orbital stability and instability regimes of the equation, we relied on the general theory by Grillakis, Shatah, and Strauss developed in [28] and [29]. According to this, under general assumptions, such as local well-posedness of the Cauchy problem, Hamiltonian structure, and existence of a smooth curve of standing wave solutions ω → ϕω, to determine orbital stability of the standing waves, it suffices to examine the spectral
properties of the linear operator
L00(ϕω, ωϕω) :H → H∗,
and the sign of the derivative
d dω ω=ω 0 Q(Uω)
where where Q is the charge
Q(Uω) = −ωkϕωk2L2 −
α
2|ϕω(0)|
This last condition is called the Vakhitov-Kolokolov condition in the physics literature. Since we know solution ϕω explicitly, we are in a position to investigate the slope condition by
explicit calculations.
4. The nonlinear Schr¨odinger equation with inverse square potential on the half-line
In the last chapter of this thesis, we investigate the local well-posedness and orbital stability of a nonlinear Schr¨odinger equation with inverse square potential on the positive half-line iut+ u00+ cxu2 + |u|p−1u = 0, u(0) = u0 ∈ H01(R+), (1.4.1) where u(t, x) : R × R+ → C, u
0 : R+ → C, 1 < p < ∞, and 0 < c < 1/4. We show that the
Cauchy problem is locally well-posed in H1
0(R+) by using a fixed point argument. In this case
we do not need Strichartz estimates, but we benefit from the embedding H1
0(R+) ,→ L∞(R+).
In particular, we have established that to all initial value u0 ∈ H01(R+), there exists a unique
maximal solution u ∈ C((−Tmin, Tmax), H01(R+)), which satisfies the conservation laws for
the mass
ku(t)kL2 = ku0kL2
and the energy
E(u(t)) = 1 2ku 0 (t)k2L2 − c 2 u x 2 L2 − 1 p + 1kuk p+1 Lp+1.
along the flow of the solution to (1.4.1).
We have also investigated the properties and the existence of standing wave solutions to (1.4.1). By introducing the Ansatz u(t, x) = eiωtϕ(x) we obtain the standing wave equation
ϕ00+ c
x2ϕ − ωϕ + |ϕ|
p−1ϕ = 0, ϕ ∈ H1 0(R
+). (1.4.2)
First we have established regularity and exponential decay of |ϕ| and |ϕ0| at infinity of solutions to (1.4.2). Moreover, we have shown that solutions of (1.4.2) satisfy the following identities kϕ0k2L2 − c ϕ x 2 L2 + ω kϕk 2 L2 − kϕk p+1 Lp+1 = 0, (1.4.3) kϕ0k2L2 − c ϕ x 2 L2 − p − 1 2(p + 1)kϕk p+1 Lp+1 = 0. (1.4.4)
To establish existence of ground states we solve the following constrained minimization problem
where for u ∈ H1
0(R+) the actions functional is given by
S(u) = 1 2ku 0k2 L2 − c 2 u x 2 L2 + ω 2 kuk 2 L2 − 1 p + 1kuk p+1 Lp+1,
and the Nehari manifold is defined by
N = u ∈ H01(R+) : ku0k2L2 − c u x 2 L2 + ω kuk 2 L2 − kuk p+1 Lp+1 = 0 .
Since the compactness provided by Strauss’ lemma is absent in H1
0(R+), we cannot use
standard techniques to obtain minimizers of (1.4.5). Instead, we use Palais-Smale sequences to obtain a critical point to the minimization problem. The behavior of bounded Palais-Smale sequences is described by a profile decomposition lemma, which we obtain by adapting the arguments of [55]. To show that the bounded Palais-Smale sequence has a strongly convergent subsequence we compared the problem on the half-line to the problem at ’infinity’, i.e. (c = 0), which is the classical case on R. In particular, we have defined
m∞ = minnS∞(u) : u ∈ H1(R) and ku0k2L2+ ω kuk
2
L2 − kuk
p+1 Lp+1 = 0
o
and we have shown that if 0 < c < 1/4, then m < m∞. This implies that the Palais-Smale sequences are strongly convergent and the minimum is attained. After this, by relying on the identities (1.4.3) and (1.4.4), we show that all ground states admit the same L2 norm.
We then prove that the ground state solutions are orbitally stable for mass subcritical exponents by giving a variational characterization of the solutions on L2 constraints using
Lions’ concentration-compactness principle and adopting the ideas of Cazenave and Lions [10]. In particular, we show that the following minimization problem has a solution
I = inf{S(u) : u ∈ H01(R+) and kuk2L2 = µ}, (1.4.6)
and that all minimizing sequences are convergent. This compactness result will directly imply orbital stability. As in the higher dimensional case, a difficulty here arises owing to the loss of translational invariance of the problem when we apply the concentration-compactness principle. We overcome this difficulty by again comparing the problem on the half-line with the problem at ’infinity’
I∞= min{S∞(u) : u ∈ H1(R) and kuk2L2 = µ}.
We have established that if 0 < c < 1/4, then I < I∞, which will imply the convergence of all minimizing sequences. On the other hand, if c < 0 then the infimum (1.4.6) is not attained. Hence we cannot expect that ground states are orbitally stable for a repulsive inverse square potential. This is in contrast with the result of [2], in which orbital stability
for radially symmetric functions is proved for c < 0. This negative result is due to the lack of compact embedding provided by Strauss’ lemma.
Finally, we show that ground state solutions are unstable by blow up for mass supercritical exponents. In particular, in any neighborhood of a ground state solution there exists an initial condition such that the corresponding maximal solution blows up in finite time.
5. Open problems and research perspectives
We would like to present several open problems and researcher perspectives related to the results of this thesis.
The first one is related to nonlinear Schr¨odinger equation with inverse square potential on the half-line. We have established orbital stability and instability respectively for the mass subcritical and supercritical case, thereby extending result already known in the higher dimensional case. However, a full characterization of the mass critical case still needs to be established, similar to the one that is presented in the second chapter. In particular, an optimal description of global well-posedness is missing. This is due to the difficulty of establishing a sharp interpolation constant in the Gagliardo-Nirenberg inequality, which problem arises from the loss of compact embedding by Strauss lemma in the non-radial setting. On the other hand, assuming that a sharp interpolation constant exists, we expect that a similar argument would yield a classification of minimal mass blow-up solutions for 0 < c < 1/4, as stated in the second chapter, thereby we extending our results.
Another open problem is the existence or non-existence of standing wave solutions to (1.4.2) when c < 0. Again this is a problem arising from the loss of compactness in H01(R+). For 0 < c < 1/4, we have managed to establish the existence of a minimizer on the Nehari by comparing the perturbed problem 0 < c < 1/4 to the homogeneous case c = 0. This method, however, fails when c < 0. Moreover, we do not expect that the orbital stability and the classification of minimal mass blow-up solutions will persist when c < 0. In the higher dimensional case, this is stated for radial functions, hence it cannot hold on the half-line. In general, we expect that any argument based on the concentration-compactness principle will fail when c < 0, an important tool to obtain both orbital stability and classification of minimal mass blow-up solutions.
Additionally, we would like to propose the study of blow-up solutions for the mass crit-ical Schr¨odinger equation with inverse square potential above the minimal mass level. In particular, we would like to study the nature of blow-up for the inhomogeneous Schr¨odinger equation above and close to the threshold level, thereby investigating whether the log-log
blow-up regime (1.1.1) established by F. Merle and P. Rapha¨el in [40–43] persists in the presence of an inverse square potential.
We also propose the study of a defocusing Schr¨odinger equation with an inverse square potential
iut+ ∆u +
c
|x|2u + (1 − |u|
2)u = 0, (1.5.1)
with the boundary condition
|u(t, x)| → 1, as |x| → +∞. (1.5.2)
The defocusing Schr¨odinger equation with nonstandard boundary condition (1.5.2) is also called the Gross-Pitaevskii equation. The homogeneous Gross-Pitaevskii equation has been addressed in several works [5,6,22–24]. In particular, it is known that there exist stationary solutions of the homogeneous Schr¨odinger equation with boundary condition (1.5.2), called dark solitons. We are interested in the Cauchy problem for the above equation and in the existence and orbital stability of stationary solutions. Since we assume that the modulus of the solution is one at infinity, the natural energy space corresponding to the equation is not a vector space. Additionally, owing to the presence of the inverse square potential, we need to use Strichartz estimates adopted to the potential [9] in higher dimensions.
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CHAPTER 2
Minimal mass blow-up solutions for the L
2critical nonlinear
Schr¨
odinger equation with inverse-square potential
Abstract. We study minimal mass blow-up solutions of the focusing L2critical nonlinear Schr¨odinger equation with inverse-square potential,i∂tu + ∆u +
c
|x|2u + |u|
4 Nu = 0,
with N ≥ 3 and 0 < c < (N −2)4 2. We first prove a sharp global well-posedness result: all H1 solutions with a mass (i.e. L2norm) strictly below that of the ground states are global.
Note that, unlike the equation in free space, we do not know if the ground state is unique in the presence of the inverse-square potential. Nevertheless, all ground states have the same, minimal, mass. We then construct and classify finite time blow-up solutions at the minimal mass threshold. Up to the symmetries of the equation, every such solution is a pseudo-conformal transformation of a ground state solution.
1. Introduction
The purpose of this study is to classify the time blow-up dynamics of a focusing nonlinear Schr¨odinger equation (NLS) with an attractive inverse-square potential,
i∂tu + ∆u +
c
|x|2u + |u|
p−1u = 0, u(0, ·) = u
0 ∈ H1(RN), (2.1.1)
in the L2 critical case, p = 1 +N4, with N ≥ 3. We shall fix the coupling constant c ∈ (0, c∗),
where c∗ = (N − 2)2/4 is the best constant in Hardy’s inequality:
c∗ Z RN |u|2 |x|2dx ≤ Z RN |∇u|2dx, u ∈ H1 (RN). (2.1.2)
The NLS equation with inverse-square potential has received substantial attention re-cently, see e.g. [12–14,21,30] for various results of local/global well-posedness, scattering, and harmonic analysis issues related to the operator −∆ − c|x|−2. All these recent contribu-tions rely on the Strichartz estimates for this operator, which were established by Burq et al. in [3]. A scattering/blow-up dichotomy result `a la Duyckaerts–Holmer–Roudenko [7,10] was proved by Killip et al. in [14] for the cubic nonlinearity in dimension N = 3, and re-cently extended by Lu et al. [17] to all L2 supercritical, energy subcritical nonlinearities,
in dimensions 3 ≤ N ≤ 6. To the best of our knowledge, apart from these contributions, blow-up solutions of (2.1.1) are mostly virgin territory.
The present work is a first step in this direction, and we shall focus here on the L2
criti-cal power p = 1 + N4, which is the smallest power for which finite time blow-up occurs. L2
criticality (discussed in more detail below) follows from the fact that the potential |x|−2 is homogeneous of degree −2, like the Laplace operator. On the other hand, the presence of a space-dependent coefficient in (2.1.1) breaks the translation invariance, which is a funda-mental feature of the classical NLS (i.e. the case c = 0). Mathematically, the inverse-square potential, with its remarkable scaling property, yields a fairly tractable instance of NLS without translation invariance. It also plays an important role in various areas of physics, for instance in quantum field equations, or in the study of certain black hole solutions of the Einstein equations; see the references in [3,11].
Let us now describe the main results of our work, and their relations to the literature. We consider solutions u ∈ C([0, T ), H1(RN)), where T > 0 is the maximal time of existence.
Along the flow of the solution, the L2-norm is conserved, also known as the mass:
ku(t)kL2
x ≡ ku0kL2x, (2.1.3)
and of the energy:
E(u(t)) = 1 2 Z RN |∇u(t, x)|2dx − c 2 Z RN |u(t, x)|2 |x|2 dx − 1 p + 1 Z RN |u(t, x)|p+1dx ≡ E(u 0). (2.1.4) We call a solution global if T = +∞. The local well-posedness of (2.1.1) with c ∈ (0, c∗) is
ensured by the following result. It can be proved using Strichartz estimates as in the case of the wave equation with inverse-square potential considered by Planchon et al. [23], although another proof is given in [21]. We will comment further on this point and on the case c = c∗
in Subsection 1.1.
Theorem 2.1 (Theorem 5.1 of [21]). Let c ∈ (0, c∗) and 1 < p < 1 +N −24 . For any initial
value u0 ∈ H1(RN), there exist T ∈ (0, +∞] and a maximal solution u ∈ Ct0Hx1([0, T ) × RN)
of (2.1.1), satisfying (2.1.3)–(2.1.4) for all t ∈ (0, T ). Moreover, the blow-up alternative holds: if T < +∞ then limt↑T k∇u(t)kL2 = +∞. Finally, if 1 < p < 1 + 4
N, then the solution
is global.
The constants of the motion (2.1.3)–(2.1.4) are related to the symmetries of (2.1.1) in H1(RN). More precisely, if u(t, x) is a solution of (2.1.1), then so are:
(2) uγ0(t, x) = e
iγ0u(t, x), for all γ
0 ∈ R (phase shift);
(3) uλ0(t, x) = λ
2/(p−1)
0 u(λ20t, λ0x), for all λ0 > 0 (scaling invariance).
Note that (2.1.1) with c > 0 does not admit space translations invariance. Owing to the symmetries (1) and (2) of the mass and the energy are conserved along the flow of (2.1.1). We will describe the scaling invariance in more detail below.
For our analysis it is convenient to introduce the Hardy functional, defined on H1(RN) by H(u) = Z RN |∇u|2dx − c Z RN |u|2 |x|2dx.
Using H, the energy can be rewritten as
E(u) = 1 2H(u) − 1 p + 1kuk p+1 Lp+1.
Moreover by Hardy’s inequality, for all c ∈ (0, c∗],
1 − c c∗ Z RN |∇u|2dx ≤ H(u) ≤ Z RN |∇u|2dx. (2.1.5)
In particular, for c ∈ (0, c∗), H(u) defines on H1(RN) a seminorm equivalent to k∇ukL2.
A solution u(t) therefore blows up at time T > 0 if and only if limt↑T H(u(t)) = +∞.
Furthermore, H(u) scales as k∇uk2
L2 under space dilations. More precisely, the self-adjoint
operator −∆ − c|x|−2 associated with the positive semi-definite quadratic form H(u) is homogeneous of degree −2.1 The scaling symmetry (c) above is a crucial consequence of this fact. Now, p = 1 + N4 yields 2/(p − 1) = N/2 and, as in the classical case c = 0, (2.1.1) is invariant under the L2 scaling
u(t, x) → uλ(t, x) = λN/2u(λ2t, λx) (λ > 0).
This transformation preserves the L2 norm and (2.1.1) is called L2 critical.
The Cauchy problem (2.1.1) also admits standing waves. Indeed, u(t, x) = eitϕ(x) solves (2.1.1) if and only if ϕ ∈ H1(RN) is a solution of the nonlinear equation
∆ϕ + c
|x|2ϕ − ϕ + |ϕ|
4
Nϕ = 0. (2.1.6)
In Section 2, we will use Weinstein’s variational approach [27] to prove the existence in H1(RN) of a positive radial solution Q of (2.1.6), called ground state. Ground states will
be defined as positive radial solutions of (2.1.6) which minimize a suitable functional, see
1Note that the self-adjoint operator associated with H(u) is unique when c ≤ c
∗− 1, as −∆ − c|x|−2
is essentially self-adjoint on C0∞(RN \ {0}) in this case. For c
∗− 1 < c ≤ c∗, this operator has deficiency
indices (1, 1) and so admits a one-parameter family of self-adjoint extensions in L2(RN). See [11,12] for more details.
Proposition 2.4. We shall see that all ground states Q have the same mass kQkL2 =: Mgs > 0
and satisfy E(Q) = 0. We denote the set of ground states by G.
For c = 0, it is well known that the ground state is unique, up to the symmetries of (2.1.6). More precisely, there exists a unique positive radial solution Q ∈ H1(RN) of (2.1.6),
and G = {eiγ0Q(· − x
0) : γ0 ∈ R, x0 ∈ RN}. Unfortunately, for c > 0, we are not aware of
any uniqueness result for (2.1.6) on RN. Uniqueness results for radial solutions of nonlinear elliptic PDEs are typically based on an intricate analysis of the corresponding ODEs in the radial variable r = |x|, see e.g. [5,15,22,29]. We shall not consider this problem here.
Our first result shows that ground states play a pivotal role in the global dynamics of (2.1.1).
Theorem 2.2. Let c ∈ (0, c∗) and p = 1 + N4. If u0 ∈ H1(RN) satisfies
ku0kL2 < Mgs, (2.1.7)
then the corresponding solution of (2.1.1) given by Theorem 2.1 is global.
The proof of this theorem relies on the inequality
E(u) ≥ 1 2H(u) 1 − kukL2 Mgs N4! , u ∈ H1(RN), (2.1.8)
which follows from a sharp Gagliardo–Nirenberg inequality established in Section 2. Indeed, owing to the conservation of the energy and the L2 norm, the inequality (2.1.8) yields an a
priori bound on H(u(t)) — and hence on k∇u(t)kL2 — when ku0kL2 < Mgs. Namely, we
have H(u(t)) ≤ 2E(u0) 1 − kukL2 Mgs N4!−1 , (2.1.9)
which implies global existence.
We shall next exhibit blow-up solutions at the mass threshold ku0kL2 = Mgs, which is
thus the minimal mass where blow-up can occur. This shows that the global well-posedness condition (2.1.7) is sharp. The minimal mass blow-up solutions are constructed explicitly by applying the pseudo-conformal transformation (defined in Lemma 2.8) to the standing wave eitQ. Taking into account the above symmetries of (2.1.1) we obtain, for each ground state Q ∈ G, a set (SQ,T ,λ0,γ0)T ∈R,λ0>0,γ0∈R of minimal mass blow-up solutions of (2.1.1) given by
SQ,T ,λ0,γ0(t, x) = e iγ0ei λ20 T −te−i |x|2 4(T −t) λ0 T − t N/2 Q λ0x T − t . (2.1.10)
Let us remark that minimal mass blow-up solutions are self-similar: for t ∈ [0, T ), there exists λ(t) > 0 for which |SQ,T ,λ0,γ0(t, x)| = λ(t)
SQ,T ,λ0,γ0 preserves its profile as Q during its blow-up dynamics up to the scaling symmetry
of the equation.
The striking fact is that all finite time blow-up solutions at the minimal mass threshold are of this form. Indeed, we have the following classification result.
Theorem 2.3. Let T > 0 and u ∈ C [0, T ), H1(RN) be a minimal mass solution of (2.1.1) with p = 1 +N4, which blows up at time T , i.e. ku0kL2 = Mgs and limt↑T H(u(t)) =
+∞. Then there exist a ground state Q ∈ G, λ0 > 0 and γ0 ∈ R such that, for all t ∈ [0, T ),
u(t) = SQ,T ,λ0,γ0(t).
Singularity formation of the mass-critical, homogeneous Schr¨odinger equations have been widely studied in the past. A comprehensive review of the theory for the classical focusing NLS can be found in [24]. Theorems 2.2 and 2.3 respectively extend the famous results of Weinstein [27] and Merle [18], from the case c = 0 to the case c ∈ (0, c∗). We are indebted
to these authors for the fundamental ideas supporting our proofs. In fact, our approach here is based closely on Hmidi and Keraani [9], where the arguments of [18] have been simplified, using a Cauchy–Schwarz inequality due to Banica [1].
The paper [9] deals with the classical NLS with L2 critical nonlinearity and constant coefficients. To adapt it to space-dependent coefficients, the main difficulties lie in a crucial compactness result. In the present work, this is Proposition 2.14, which relies on a subtle combination of Hardy’s inequality and the sharp Gagliardo–Nirenberg inequality (2.2.6). Various authors have also considered blow-up solutions for focusing NLS equations with space-dependent coefficients, notably [2,19,25]. These references are discussed in some detail in the introduction of the paper [6] by Combet and Genoud, where the classification of minimal mass blow-up solutions is obtained for the L2 critical equation
i∂tu + ∆u + |x|−b|u|
4−2b
N u = 0, with 0 < b < min{2, N }, N ≥ 1.
1.1. The threshold case c = c∗. Our initial motivation for studying (2.1.1) came from
the paper [26] by Trachanas and Zographopoulos, where the orbital stability of standing waves of (2.1.1) is considered, with a special focus on the threshold value c = c∗. From
a functional analytic perspective, an interesting difficulty arises in this case due to the sharpness of Hardy’s inequality at c = c∗. The natural energy space associated with (2.1.1),
H = {u ∈ L2(RN) : H(u) < +∞}, then satisfies H1(RN) ( H. Indeed, the ground states
of (2.1.1) with c = c∗ have a singularity of order |x|−(N −2)/2 at the origin, and thus lie in
H \ H1
analysis of orbital stability, and our initial hope was to be able to extend our analysis to the case c = c∗ using this functional framework. However, it is not clear to us that local
well-posedness holds in this case. Trachanas and Zographopoulos [26, Theorem 3.1] claim that (in the radial case) it follows by adapting a proof by Cazenave, but we were not able to carry this through. Let us briefly explain why.
For c ∈ (0, c∗), Theorem 2.1 above was proved in [21], by adapting to −∆ − c|x|−2
Cazenave’s proof of [4, Theorems 3.3.9], originally developed to deal with −∆ in bounded domains, where dispersive estimates are not available. This approach allows one to obtain existence of local (in time) solutions [4, Theorems 3.3.5], but an additional uniqueness result is required to obtain the full well-posedness result [4, Theorems 3.3.9]. In [21], uniqueness relies on the Strichartz estimates for −∆ − c|x|−2, which were established in [3]. Unfortu-nately, as pointed out on p. 521 of [3], these estimates break down at the threshold value c = c∗. Hence, the existence of local in time solutions is ensured by [21], but it is not clear if
and how uniqueness can be proved. As uniqueness is essential in our proof of Theorem 2.3, we shall only consider c ∈ (0, c∗) here. Note that, for c = c∗, inequality (2.4.6) also breaks
down, and we do not know how to prove the crucial Proposition 2.14.
2. Ground states and the sharp global existence criterion
In this section, we will prove Theorem 2.2. We start by solving a minimization problem, the minimum of which is attained at the ground states of the stationary equation. A crucial consequence will be the sharp Gagliardo–Nirenberg inequality leading to (2.1.8). As these results may be useful in other problems, we will state them for any 1 < p < 1 + N −24 . A similar problem was considered in [14] for the specific case of three space dimensions and p = 3. Consider the Weinstein functional
Jp,N(u) := H(u) p−1 4 Nkuk2+ p−1 2 (2−N ) L2 kukp+1Lp+1 . Proposition 2.4. For 1 < p < 1 + N −24 , αp,N := inf u∈H1(RN)\{0}J p,N(u)
is attained at a positive radial function Q ∈ H1(RN), solution of the Euler–Lagrange equation
Np − 1 4 ∆Q + c Q |x|2 − 1 + p − 1 4 (2 − N ) Q + Qp = 0. (2.2.1) Furthermore, αp,N = 2 kQkp−1L2 p + 1 . (2.2.2)
In the case p = 1 + 4
N, any minimizer of J
p,N can be rescaled into a solution of (2.1.6).
Proof. First note that the functional Jp,N is invariant under the scaling
u(x) → uλ,µ(x) := µu(λx) (λ, µ > 0). Indeed, we have H(uλ,µ) = λ2−Nµ2H(u), kuλ,µk2L2 = λ−Nµ2kuk2L2, kuλ,µkp+1Lp+1 = λ −N µp+1kukp+1Lp+1 and so Jp,N(uλ,µ) = Jp,N(u).
Let {un} ⊂ H1(RN) be a minimizing sequence, αp,N = limn→∞Jp,N(un) ≥ 0. Since J (|u|) ≤
J (u), we can suppose that un ≥ 0 for all n ∈ N. Furthermore, denoting by u∗n the Schwarz
symmetrization of un (see e.g. [16, pp. 80-83]), we have that
Z RN (u∗n)2 |x|2 dx ≥ Z RN u2n |x|2dx,
k∇u∗nkL2 ≤ k∇unkL2 and ku∗nkL2 ≤ kunkL2.
We can therefore suppose that un = u∗n and, in particular, that each un is positive, radial
and radially decreasing. Thanks to the scaling invariance of Jp,N, we can further rescale the minimizing sequence by choosing λn = kunkL2/H(un) and µn = kunkN/2−1
L2 /HN/2(un). We
thus obtain a minimizing sequence ψn= uλnn,µn with the following properties:
ψn≥ 0, ψn= ψn(|x|), kψnkL2 = 1, H(ψn) = 1, lim n→∞J p,N(ψ n) = αp,N.
In particular, {ψn} is bounded in H1(RN). Thus, up to a subsequence, we can suppose that
{ψn} has a weak limit ψ∗ ∈ H1(RN). Since {ψn} ⊂ Hrad1 (RN) and p + 1 ∈ (2, 2
∗), we can
also suppose that ψn converges strongly to ψ∗ in Lp+1(RN). By weak lower semi-continuity
of k · kL2 and H (see [20]), we obtain that kψ∗kL2 ≤ 1 and H(ψ∗) ≤ 1. Hence,
αp,N ≤ Jp,N(ψ∗) ≤ 1 kψ∗kp+1 Lp+1 = lim n→∞J p,N(ψ n) = αp,N. It follows that H(ψ∗)p−14 Nkψ∗k2+(p−1)(2−N )/2 L2 = 1 and, therefore, H(ψ∗) = kψ∗kL2 = 1.
Now, ψ∗ must satisfy the Euler–Lagrange equation DJp,N(ψ∗) = 0. We first remark that,
for any u ∈ H1(RN), DJp,N(u) = 0 reads
−Np − 1 4 ∆u + c |x|2u +1 + p − 1 4 (2 − N ) H(u) kuk2 L2 u −p + 1 2 H(u) kukp+1Lp+1 |u|p−1u = 0. (2.2.3) Taking into consideration H(ψ∗) = 1, kψ∗kL2 = 1, kψ∗kp+1
Lp+1 = α −1 p,N and ψ ∗ ≥ 0, we get Np − 1 4 ∆ψ∗+ c ψ ∗ |x|2 − 1 + p − 1 4 (2 − N ) ψ∗+ αp,N p + 1 2 (ψ∗)p = 0.
Then Q defined by ψ∗ = (αp,N(p + 1)/2)−1/(p−1)Q readily satisfies (2.2.1) and (2.2.2).
Finally, in view of Remark 2.5 and Remark 2.6 below, if p = 1 +N4 and u is a minimizer of J1+N4,N, the Euler–Lagrange equation (2.2.3) reads
∆u + c |x|2u − 2 N H(u) kuk2 L2 u + |u|p−1u = 0, (2.2.4)
and any solution u can be rescaled into a solution Q of (2.1.6) by letting
u(x) = λN/2Q(λx) with λ = r 2 N pH(u) kukL2 .
The proof is complete.
Remark 2.5. (a) Note that, by phase invariance of Jp,N, if u is a minimizer so is eiθu, θ ∈ R.
(b) For p = 1 + N4, the Pohozaev identity for (2.1.6) reads as
H(u) = 1 1 + N2 kuk 2+N4 L2+ 4N , which implies so J1+N4,N(u) = H(u)kuk 4 N L2 kuk2+N4 L2+ 4N = kuk 4 N L2 1 + 2 N . (2.2.5)
Therefore, all minimizers have the same mass.
(c) The proof of Proposition 2.12 below shows that all minimizers are positive and radial, up to a global phase factor.
Definition 1. In the case p = 1 + N4, we call ground states the minimizers of J1+N4,N
that are positive radial solutions of (2.1.6). We denote the set of ground states by G. In view of Remark 2.5 (b), there exists Mgs > 0 such that kQkL2 = Mgs for all Q ∈ G. We call
