Semiclassical limit for Schr¨ odinger equations with magnetic field and Hartree-type nonlinearities

Quaderni di Matematica

Semiclassical limit for Schr¨ odinger equations with magnetic field and Hartree-type nonlinearities

S. Cingolani, S. Secchi, M. Squassina

Quaderno n. 5/2009 (arxiv:math.AP/0903.5435)

Stampato nel mese di marzo 2009

presso il Dipartimento di Matematica e Applicazioni,

Universit` a degli Studi di Milano-Bicocca, via R. Cozzi 53, 20125 Milano, ITALIA .

Disponibile in formato elettronico sul sito www.matapp.unimib.it.

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Esemplare fuori commercio per il deposito legale agli effetti della Legge 15 aprile 2004

n.106 .

MAGNETIC FIELD AND HARTREE-TYPE NONLINEARITIES

SILVIA CINGOLANI, SIMONE SECCHI, AND MARCO SQUASSINA

Abstract. The semi-classical regime of standing wave solutions of a Schr¨ odinger equa- tion in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.

Contents

1. Introduction and main result 1

1.1. Introduction 1

1.2. Statement of the main result 3

1.3. A heuristic remark: multi-bump dynamics 5 2. Properties of the set of ground states 6

2.1. A Pohozaev type identity 7

2.2. Orbital stability property 8

2.3. Structure of least energy solutions 13 2.4. Compactness of the ground states set 14

3. The penalization argument 17

3.1. Notations and framework 18

3.2. Energy estimates and Palais-Smale condition 20 3.3. Critical points of the penalized functional 27

3.4. Proof for the main result 31

References 32

1. Introduction and main result

1.1. Introduction. Some years ago, Penrose derived in [26] a system of nonlinear equa- tions by coupling the linear Schr¨ odinger equation of quantum mechanics with Newton’s gravitational law. Roughly speaking, a point mass interacts with a density of matter described by the square of the wave function that solves the Schr¨ odinger equation. If m

Date: March 31, 2009.

2000 Mathematics Subject Classification. 35B40, 35K57, 35B35, 92D25.

Key words and phrases. Hartree equations, ground states, semiclassical limit, multipeak solutions, variational methods.

The research of the first and the third author was partially supported by the MIUR national research project Variational and Topological Methods in the Study of Nonlinear Phenomena, PRIN 2007. The research of the second author was partially supported by the MIUR national research project Variational methods and nonlinear differential equations, PRIN 2006.

1

is the mass of the point, this interaction leads to the system in R

(1.1)

(

2m

∆ψ − V (x)ψ + U ψ = 0,

∆U + 4πγ|ψ|

= 0,

where ψ is the wave function, U the gravitational potential energy, V a given Schr¨ odinger potential, ~ the Planck constant and γ = Gm

, G being Newton’s constant of gravitation.

Notice that, by means of the scaling ψ(x) = 1

~

ψ(x) ˆ

√ 8πγm , V (x) = 1 2m

V (x), ˆ U (x) = 1 2m

U (x) ˆ system (1.1) can be written, maintaining the original notations, as (1.2)

(

~

∆ψ − V (x)ψ + U ψ = 0,

~

∆U + |ψ|

= 0.

The second equation in (1.2) can be explicitly solved with respect to U , so that the system turns into the single nonlocal equation

(1.3) ~

∆ψ − V (x)ψ + 1 4π~

 Z

R³

|ψ(ξ)|

|x − ξ| dξ 

ψ = 0 in R

.

The Coulomb type convolution potential W (x) = |x|

in R

is also involved in vari- ous physical applications such as electromagnetic waves in a Kerr medium (in nonlinear optics), surface gravity waves (in hydrodynamics) as well as ground states solutions (in quantum mechanical systems). See for instance [1] for further details and [14] for the derivation of these equations from a many-body Coulomb system.

In the present paper, we will study the semiclassical regime (namely the existence and asymptotic behavior of solutions as ~ → 0) for a more general equation having a similar structure. Taking ε in place of ~, our model will be written as

(1.4)  ε

i ∇ − A(x) 

u + V (x)u = 1

ε

W ∗ |u|

u in R

,

in R

, where the convolution kernel W : R

\ {0} → (0, ∞) is an even smooth kernel, homogeneous of degree −1 and we denote by i the imaginary unit. The choice of W (x) =

|x|

recovers (1.3). Equation (1.4) is equivalent to

(1.5)  1

i ∇ − A

(x) 

u + V

(x)u = W ∗ |u|

u in R

, where we have set A

(x) = A(εx) and V

(x) = V (εx).

The vector-valued field A represents a given external magnetic potential, and forces the solutions to be, in general, complex-valued (see [11] and references therein). To the best of our knowledge, in this framework, no previous result involving the electromagnetic field can be found in the literature. On the other hand, when A ≡ 0, it is known that solutions have a constant phase, so that it is not a restriction to look for real-valued solutions.

In this simpler situation, we recall the results contained in [24, 25], stating that at fixed

~ = ε the system ( 1.2) can be uniquely solved by radially symmetric functions. Moreover

these solutions decay exponentially fast at infinity together with their first derivatives.

The mere existence of one solution can be traced backed to the paper [22].

Later on, Wei and Winter proposed in [28] a deeper study of the multi-bumps solutions to the same system, and proved an existence result that can be summarized as follows:

if k ≥ 1 and P

, . . . , P

∈ R

are given non-degenerate critical points of V (but local extrema are also included without any further requirements), then multi-bump solutions ψ

exist that concentrate at these points when ~ → 0. A similar equation is also studied in [23], where multi-bump solutions are found by some finite-dimensional reduction. The main result about existence leans on some non-degeneracy assumption on the solutions of a limiting problem, which was actually proved in [28, Theorem 3.1] only in the particular case W (x) = |x|

in R

. Moreover, the equation investigated in [23] cannot be deduced from a singularly perturbed problem like (1.3), because the terms do not scale coherently.

For precise references to some classical works (well-posedness, regularity, long-term behaviour) related to the nonlinear Schr¨ odinger equation with Hartree nonlinearity for Coulomb potential and A = 0, we refer to [27, p.66]. We would also like to mention the work of Carles et al. [8].

1.2. Statement of the main result. We shall study equation (1.5) by exploiting a penalization technique which was recently developed in [12], whose main idea is searching for solutions in a suitable class of functions whose location and shape is the one expected for the solution itself. This approach seems appropriate, since it does not need very strong knowledge of the limiting problem (2.1) introduced in the next section. In particular, for a general convolution kernel W , we still do not know if its solutions are non-degenerate.

In order to state our main result (as well as the technical lemma contained in Section 2 and 3), the following conditions will be retained:

(A1): A : R

→ R

is of class C

.

(V1): V : R

→ R is a continuous function such that 0 ≤ V

= inf

x∈R³

V (x), lim inf

|x|→∞

V (x) > 0.

(V2): There exist bounded disjoint open sets O

, . . . , O

such that 0 < m

= inf

x∈Oⁱ

V (x) < min

x∈∂Oⁱ

V (x), i = 1, . . . , k.

(W): W : R

\ {0} → (0, ∞) is a function of class C

such that W (λx) = λ

W (x) for any λ > 0 and x 6= 0.

Convolution kernels such as W (x) = x

/|x|

, for x ∈ R

\ {0} or, more generally, W (x) = W

(x)/W

(x), for x ∈ R

\{0}, where W

, W

are positive, even and (respectively) homogeneous of degree m and m + 1 satisfy (W ).

For each i ∈ {1, . . . , k}, we define

M

= {x ∈ O

: V (x) = m

}

and Z = {x ∈ R

: V (x) = 0} and m = min

m

. By (V1) we can fix m > 0 with e

m < min e n

m, lim inf

|x|→∞

V (x) o

and define ˜ V

(x) = max{ m, V e

(x)}. Let H

be the Hilbert space defined by the completion of C

(R

, C) under the scalar product

hu, vi

= <

Z

R³

 1

i ∇u − A

(x)u  1

i ∇v − A

(x)v 

+ ˜ V

(x)uvdx and k · k

the associated norm.

The main result of the paper is the following

Theorem 1.1. Suppose that (A), (V1-2) and (W) hold. Then for any ε > 0 sufficiently small, there exists a solution u

∈ H

of equation (1.5) such that |u

| has k local maximum points x

∈ O

satisfying

lim

max

i=1,...,k

dist(εx

, M

) = 0, and for which

|u

(x)| ≤ C

exp  − C

min

i=1,...,k

|x − x

| ,

for some positive constants C

, C

. Moreover for any sequence (ε

) ⊂ (0, ε] with ε

→ 0 there exists a subsequence, still denoted by (ε

), such that for each i ∈ {1, . . . , k} there exist x

∈ M

with ε

x

→ x

, a constant w

∈ R and U

∈ H

(R

, R) a positive least energy solution of

(1.6) − ∆U

+ m

U

− (W ∗ |U

|

)U

= 0, U

∈ H

(R

, R);

for which one has

(1.7) u

(x) =

k

X

i=1

U

x − x

exp i w

+ A(x

)(x − x

)

+ K

(x)

where K

∈ H

satisfies kK

k

= o(1) as n → +∞.

The one and two dimensional cases would require a separate analysis in the construction of the penalization argument (see e.g. [5] for a detailed discussion). The study of the cases of dimensions larger than three is less interesting from the physical point of view.

Moreover, having in mind the soliton dynamics as a possible further development, in

dimensions N ≥ 4 the time dependent Schr¨ odinger equation with kernels, say, of the type

W (x) = |x|

does not have global existence in time for all H

initial data (see e.g. [9,

Remark 6.5.2, p.114]) as well as the heuristic discussion in the next section).

1.3. A heuristic remark: multi-bump dynamics. We could also think of Theo- rem 1.1 as the starting point in order to rigorously justify a multi-bump soliton dynamics for the full Schr¨ odinger equation with an external magnetic field

(1.8)

( iε∂

u +

∇ − A(x)

u + V (x)u =

(W ∗ |u|

) u in R

,

u(x, 0) = u

(x) in R

.

We describe in the following what we expect to hold (the question is open even for A = 0, see the discussion by J. Fr¨ ohlich et al. in [16]). Given k ≥ 1 positive numbers g

, . . . , g

, if E : H

(R

) → R is defined as

E(u) = 1 2

Z

R³

|∇u|

dx − 1 2

Z

R⁶

W (x − y)|u(x)|

|u(y)|

dxdy, let U

: R

→ R (j = 1, . . . , k) be the solutions to the minimum problems

E(U

) = min{E (u) : u ∈ H

(R

), kuk

= g

}, which solve the equations

− 1

2 ∆U

+ m

U

= W ∗ |U

|

U

in R

, for some m

∈ R. Consider now in ( 1.8) an initial datum of the form

u

(x) =

k

X

j=1

U

 x − x

ε



e

, x ∈ R

,

where x

∈ R

and ξ

∈ R

(j = 1, . . . , k) are initial position and velocity for the ODE

(1.9)



 

 



 

 



˙x

(t) = ξ

(t),

ξ ˙

(t) = −∇V (x

(t)) − ε

k

P

i6=j

m

∇W (x

(t) − x

(t)) − ξ

(t) × B(x

(t)), x

(0) = x

, ξ

(0) = ξ

, j = 1, . . . , k,

with B = ∇ × A. The systems can be considered as a mechanical system of k interacting particles of mass m

subjected to an external potential as well as a mutual Newtonian type interaction. Therefore, the conjecture it that, under suitable assumptions, the following representation formula might hold

(1.10) u

(x, t) =

k

X

j=1

U

 x − x

(t) ε



e

+ ω

,

locally in time, for certain phases θ

: R

→ [0, 2π), where ω

is small (in a suitable sense) as ε → 0, provided that the centers x

in the initial data are chosen sufficiently far from each other. Now, neglecting as ε → 0 the interaction term (ε-dependent)

ε

k

X

i6=j

m

∇W (x

(t) − x

(t))

in the Newtonian system (1.9) and taking

x

, . . . , x

∈ R

: ∇V (x

) = 0 and ξ

= 0, for all j = 1, . . . , k, then the solution of (1.9) is

x

(t) = x

, ξ

(t) = 0, for all t ∈ [0, ∞) and j = 1, . . . , k, so that the representation formula (1.10) reduces, for ε = ε

→ 0,

u

(x, t) =

k

X

j=1

U

 x − x

ε



e

+ ω

,

namely to formula (1.7) up to a change in the phase terms and up to replacing x with ε

x and x

with ε

x

for all j = 1, . . . , k.

Plan of the paper.

In Section 2 we obtain several results about the structure of the solutions of the limiting problem (1.6). In particular, we study the compactness of the set of real ground states solutions and we achieve a result about the orbital stability property of these solutions for the Pekar-Choquard equation. In Section 3 we perform the penalization scheme. In particular we obtain various energy estimates in the semiclassical regime ε → 0 and we get a Palais-Smale condition for the penalized functional which allows to find suitable critical points inside the concentration set. Finally we conclude the proof of Theorem 1.1.

Main notations.

(1) i is the imaginary unit.

(2) The complex conjugate of any number z ∈ C is denoted by ¯ z.

(3) The real part of a number z ∈ C is denoted by <z.

(4) The imaginary part of a number z ∈ C is denoted by =z.

(5) The symbol R

(resp. R

) means the positive real line [0, ∞) (resp. (−∞, 0]).

(6) The ordinary inner product between two vectors a, b ∈ R

is denoted by ha | bi.

(7) The standard L

norm of a function u is denoted by kuk

. (8) The standard L

norm of a function u is denoted by kuk

. (9) The symbol ∆ means D

1

+ D

2

+ D

3

.

(10) The convolution u ∗ v means (u ∗ v)(x) = R u(x − y)v(y)dy.

2. Properties of the set of ground states

For any positive real number a, the limiting equation for the Hartree problem (1.4) is

(2.1) − ∆u + au = W ∗ |u|

u in R

.

2.1. A Pohozaev type identity. We now give the statement of a useful identity satisfied by solutions to problem (2.1).

Lemma 2.1. Let u ∈ H

(R

, C) be a solution to ( 2.1). Then

(2.2) 1

2 Z

R³

|∇u|

dx + 3 2 a

Z

R³

|u|

dx = 5 4

Z

R³×R³

W (x − y)|u(x)|

|u(y)|

dxdy.

Proof. The proof is straightforward, and we include it just for the sake of completeness.

We multiply equation (2.1) by hx | ∇ui. Notice that

∆uhx | ∇ui = div



hx | ∇ui∇u − 1

2 |∇u|

x

 , (2.3)

−auhx | ∇ui = −a div  1 2 u

x

 + 3

2 au

, (2.4)

ϕ(x)uhx | ∇ui = div  1

2 u

ϕ(x)x 

− 1

2 u

div (ϕ(x)x) , (2.5)

where ϕ(x) = R

R³

W (x − y)|u(y)|

dy. We can easily obtain that div(ϕ(x)x) =

3

X

i=1

∂

∂x

 x

Z

R³

W (x − y)|u(y)|

dy 

= N Z

R³

W (x − y)|u(y)|

dy + Z

R³

h∇W (x − y) | xi|u(y)|

dy.

Summing up (2.3), (2.4) and (2.5) and integrating by parts, we reach the identity (2.6) 1

2 Z

R³

|∇u|

dx + 3 2 a

Z

R³

u

dx − 3 2

Z

R³×R³

W (x − y)|u(x)|

|u(y)|

dx dy

− 1 2

Z

R³×R³

h∇W (x − y) | xi|u(x)|

|u(y)|

dxdy = 0.

By exchanging x with y, we find that Z

R³×R³

h∇W (x − y) | xi|u(x)|

|u(y)|

dx dy =

− Z

R³×R³

h∇W (x − y) | yi|u(x)|

|u(y)|

dx dy.

Therefore, Z

R³×R³

h∇W (x − y) | xi|u(x)|

|u(y)|

dx dy

= 1 2

Z

R³×R³

h∇W (x − y) | x − yi|u(x)|

|u(y)|

dx dy

= − 1 2

Z

R³×R³

W (x − y)|u(x)|

|u(y)|

dx dy

via Euler’s identity for homogeneous functions. Plugging this into (2.6) yields (2.2). 

2.2. Orbital stability property. In this section, we consider the Schr¨ odinger equation

(2.7)

( i

+ ∆u + W ∗ |u|

u = 0 in R

× (0, ∞), u(x, 0) = u

(x) in R

.

This equation is also known as Pekar-Choquard equation (see e.g. [10, 19, 21]). Consider the functionals

E(u) = 1

2 k∇uk

− 1

4 D(u), J (u) = 1

2 k∇uk

+ a

2 kuk

− 1 4 D(u), where

(2.8) D(u) =

Z

R⁶

W (x − y)|u(x)|

|u(y)|

dxdy, and let us set

M = u ∈ H

(R

, C) : kuk

= ρ ,

N = u ∈ H

(R

, C) : u 6= 0 and J

(u)(u) = 0}, for some positive number ρ > 0.

Definition 2.2. We denote by G the set of ground state solutions of (2.1), that is solutions to the minimization problem

(2.9) Λ = min

u∈N

J (u).

In Lemma 2.5 we will prove that a ground state solution of (2.1) can be obtained as scaling of a solution to the minimization problem

(2.10) Λ = min

u∈M

E(u),

which is a quite useful characterization for the stability issue. We now recall two global existence results for problem (2.7) (see e.g. [9, Remark 6.5.2, p.114]).

Proposition 2.3. Let u

∈ H

(R

). Then problem (2.7) admits a unique global solution u ∈ C

([0, ∞), H

(R

, C)). Moreover, the charge and the energy are conserved in time, namely

(2.11) ku(t)k

= ku

k

, E(u(t)) = E(u

), for all t ∈ [0, ∞).

Definition 2.4. The set G of ground state solutions of (2.1) is said to be orbitally stable for the Pekar-Choquard equation (2.7) if for every ε > 0 there exists δ > 0 such that

∀u

∈ H

(R

, C): inf

φ∈G

ku

− φk

< δ implies that sup

t≥0

inf

ψ∈G

ku(t, ·) − ψk

< ε,

where u(t, ·) is the solution of (2.7) corresponding to the initial datum u

.

Roughly speaking, the ground states are orbitally stable if any orbit starting from an initial datum u

close to G remains close to G, uniformly in time.

In the classical orbital stability of Cazenave and Lions (see e.g. [10]) the ground states set G is meant as the set of minima of the functional E constrained to a sphere of L

(R

).

In this section we just aim to show that orbital stability holds with respect to G as defined in Definition 2.2.

Consider the following sets:

K

= {m ∈ R : there is w ∈ N with J

(w) = 0 and J (w) = m}, K

= {c ∈ R

: there is u ∈ M with E

|

(u) = 0 and E (u) = c}.

In the next result we establish the equivalence between minimization problems (2.9) and (2.10), namely that a suitable scaling of a solution of the first problem corresponds to a solution of the second problem with a mapping between the critical values.

Lemma 2.5. The following minimization problems are equivalent

(2.12) Λ = min

u∈M

E(u), Γ = min

u∈N

J (u), for Λ < 0 and Λ = Ψ(Γ), where Ψ : K

→ K

is defined by

Ψ(m) = − 1 2

 3 aρ



m

, m ∈ K

.

Proof. Observe that if u ∈ M is a critical point of E |

with E (u) = c < 0, then there exists a Lagrange multiplier γ ∈ R such that E

(u)(u) = −γρ, that is k∇uk

− D(u) = −γρ.

By combining this identity with D(u) = 2k∇uk

− 4c, we obtain −k∇uk

+ 4c = −γρ, which implies that γ > 0. The equation satisfied by u is

−∆u + γu = W ∗ |u|

u in R

. After trivial computations one shows that the scaling

(2.13) w(x) = T

u(x) := λ

u(λx), λ := r a γ

is a solution of equation (2.1). On the contrary, if w is a nontrivial critical point of J , then choosing

(2.14) λ = ρ

kwk

,

the function u = T

w belongs to M and it is a critical point of E

. Now, Let m be the value of the free functional J on w, m = J (w). Then

m = 1

2 k∇wk

+ a

2 kwk

− 1 4 D(w) (2.15)

= 1

2 λ

k∇uk

+ a

2 λkuk

− 1

4 λ

D(u)

= λ

E(u) + a 2 λρ

= cλ

+ a

2 λρ.

Observe that, since of course D(w) = k∇wk

+ akwk

and w satisfies the Pohozaev identity (2.2), we have the system

1

2 k∇wk

+ 3

2 akwk

= 5

4 k∇wk

+ akwk

, 1

4 k∇wk

+ a

4 kwk

= m, namely

3k∇wk

− akwk

= 0, k∇wk

+ akwk

= 4m, which, finally, entails

(2.16) k∇wk

= m, kwk

= 3 a m.

As a consequence a simple rescaling yields the value of λ, that is ρλ = kwk

= 3

a m.

Replacing this value of λ back into formula (2.15), one obtains m = c  3m

aρ



+ 3 2 m, namely

− 1

2 m = c  3m aρ



. In conclusion, we get

(2.17) c = Ψ(m)

= − 1

2

 3 aρ



m

,

where the function Ψ : R

→ R

is injective. In order to prove that Ψ

is surjective, let m be a free critical value for J , namely m = J (w), with w solution of equation (2.1).

Then, if we consider u = T

w(x) = λ

w(λ

x) with λ given by (2.14), it follows that u ∈ M is a critical point of E |

with lagrange multiplier γ = aλ

. By using

λ =  aρ 3m



, in light of (2.16) we have

4c = k∇uk

− γρ = λ

k∇wk

− aρλ

=  aρ 3m



m − aρ  3m aρ



,

which yields m = Ψ

(c), after a few computations. We are now ready to prove the assertion. Notice that by formula (2.17) we have

Λ = min

u∈M

E(u) = min

u∈M

c

= min

v∈N

Ψ(m

) = − max

v∈N

1 2

 3 aρ



m

= − 1 2

 3 aρ





min

m



= − 1 2

 3 aρ



Γ

= Ψ(Γ).

If ˆ u ∈ M is a minimizer for Λ, that is Λ = E (ˆ u) = min

E, the function ˆ w = T

u is ˆ a critical point of J with J ( ˆ w) = Ψ

(Λ) = Γ, so that w is a minimizer for Γ, that is

J (w) = min

J . This concludes the proof. 

Corollary 2.6. Any ground state solution u to equation (2.1) satisfies

(2.18) kuk

= ρ, ρ = 3Γ

a ,

where Γ is defined in (2.12). Moreover, for this precise value of the radius ρ, we have

(2.19) min

u∈M

J (u) = min

u∈N

J (u), where M = M

.

Proof. The first conclusion is an immediate consequence of the previous proof. Let us now prove that the second conclusion holds, with ρ as in (2.18). We have

min

u∈M

J (u) = min

u∈M

E(u) + a

2 kuk

= Λ + aρ 2

= − 1 2

 3 aρ



Γ

+ aρ 2

= Γ = min

u∈N

J (u),

by the definition of ρ. 

The following is the main result of the section.

Theorem 2.7. Then the set G of the ground state solutions to (2.1) is orbitally stable for (2.7).

Proof. Assume by contradiction that the assertion is false. Then we can find ε

> 0, a sequence of times (t

) ⊂ (0, ∞) and of initial data (u

) ⊂ H

(R

, C) such that

(2.20) lim

n→∞

inf

φ∈G

ku

− φk

= 0 and inf

ψ∈G

ku

(t

, ·) − ψk

≥ ε

,

where u

(t, ·) is the solution of (2.7) corresponding to the initial datum u

. Taking into account (2.18) and (2.19) of Corollary 2.6, for any φ ∈ G, we have

kφk

= ρ

, J (φ) = min

u∈M_ρ0

J (u), ρ

= 3Γ

a .

Therefore, considering the sequence Υ

(x) := u

(t

, x), which is bounded in H

(R

, C), and recalling the conservation of charge, as n → ∞, from (2.20) it follows that

kΥ

k

= ku

(t

, ·)k

= ku

k

= ρ

+ o(1).

Hence, there exists a sequence (ω

) ⊂ R

with ω

→ 1 as n → ∞ such that (2.21) kω

Υ

k

= ρ

, for all n ≥ 1.

Moreover, by the conservation of energy (2.11) and the continuity of E , as n → ∞, J (ω

Υ

) = E (ω

Υ

) + a

2 kω

Υ

k

= E (Υ

) + a

2 kΥ

k

+ o(1) (2.22)

= E (u

(t

, ·)) + a

2 ku

k

+ o(1) = E (u

) + a

2 ku

k

+ o(1)

= J (u

) + o(1) = min

u∈M_ρ0

J (u) + o(1).

Combining (2.21)-(2.22), it follows that (ω

Υ

) ⊂ H

(R

, C) is a minimizing sequence for the functional J (and also for E ) over M

. By taking into account the homogeneity property of W , following the lines of the proof of [10, Theorem IV.1], we learn that, up to a subsequence, (ω

Υ

) converges to some function Υ

, which thus belongs to the set G, since by (2.21)-(2.22) and equality (2.19)

J (Υ

) = min

u∈M_ρ0

J (u) = min

u∈N

J (u) Evidently, this is a contradiction with (2.20), as we would have

ε

≤ lim

n→∞

inf

ψ∈G

kΥ

− ψk

≤ lim

n→∞

kΥ

− Υ

k

= 0.

This concludes the proof. 

In the particular case W (x) = |x|

, due to the uniqueness of ground states up to translations and phase changes (cf. [24]), Theorem 2.7 strengthens as follows.

Corollary 2.8. Assume that w is the unique real ground state of

−∆w + aw = |x|

∗ w

w, in R

. Then for all ε > 0 there exists δ > 0 such that

u

∈ H

(R

, C) and inf

y∈R3

θ∈[0,2π)

ku

− e

w(· − y)k

< δ

implies that

sup

t≥0

inf

y∈R3

θ∈[0,2π)

ku(t, ·) − e

w(· − y)k

< ε.

2.3. Structure of least energy solutions. We can now state the following Lemma 2.9. Any complex ground state solution u to (2.1) has the form

u(x) = e

|u(x)|, for some θ ∈ [0, 2π).

Proof. In view of Lemma 2.5 (see also Corollary 2.6), searching for ground state solutions of (2.1) is equivalent to consider the constrained minimization problem min

E(u) for a suitable value of ρ > 0. Then the proof is quite standard; we include a proof here for the sake of selfcontainedness. Consider

σ

= inf E(u) : u ∈ H

(R

, C), kuk

= ρ , σ

= inf E(u) : u ∈ H

(R

, R), kuk

= ρ .

It holds σ

= σ

. Indeed, trivially one has σ

≤ σ

. Moreover, if u ∈ H

(R

, C), due to the well-known inequality |∇|u(x)|| ≤ |∇u(x)| for a.e. x ∈ R

, it holds

Z

|∇|u(x)||

dx ≤ Z

|∇u(x)|

dx,

so that E (|u|) ≤ E (u). In particular σ

≤ σ

, yielding σ

= σ

. Let now u be a solution to σ

and assume by contradiction that µ({x ∈ R

: |∇|u|(x)| < |∇u(x)|}) > 0, where µ denotes the Lebesgue measure in R

. Then k|u|k

= kuk

= 1, and

σ

≤ 1 2

Z

|∇|u||

dx − 1

4 D(|u|) < 1 2

Z

|∇u|

dx − 1

4 D(u) = σ

,

which is a contradiction, being σ

= σ

. Hence, we have |∇|u(x)|| = |∇u(x)| for a.e.

x ∈ R

. This is true if and only if < u∇(= u) = = u∇(< u). In turn, if this last condition holds, we get

¯

u∇u = < u∇(< u) + = u∇(= u), a.e. in R

,

which implies that < (i¯ u(x)∇u(x)) = 0 a.e. in R

. From the last identity one finds θ ∈ [0, 2π) such that u = e

|u|, concluding the proof.  We then get the following result about least-energy levels for the limiting problem (2.1).

Corollary 2.10. Consider the two problems

− ∆u + au = W ∗ |u|

u, u ∈ H

(R

, R), (2.23)

−∆u + au = W ∗ |u|

u, u ∈ H

(R

, C), . (2.24)

Let E

and E

denote their least-energy levels. Then

(2.25) E

= E

.

Moreover any least energy solution of (2.23) has the form e

U where U is a positive least

energy solution of (2.24 ) and τ ∈ R.

2.4. Compactness of the ground states set. In light of assumption (W), there exist two positive constants C

, C

such that

(2.26) C

|x| ≤ W (x) ≤ C

|x| , for all x ∈ R

\ {0}.

We recall two Hardy-Littlewood-Sobolev type inequality (see e.g. [20 ]) in R

:

∀u ∈ L

(R

) :

|x|

∗ u

≤ Ckuk

L

6q 3+2q

, (2.27)

∀u ∈ L

(R

) : Z

R⁶

|x − y|

u

(y)u

(x)dydx ≤ Ckuk

L¹²⁵

. (2.28)

Notice that, taking the limit q → ∞ in (2.27) yields

∀u ∈ L

(R

) :

|x|

∗ u

≤ Ckuk

. We have the following

Lemma 2.11. There exists a positive constant C such that

∀u ∈ H

(R

) : D(u) ≤ Ckuk

kuk

.

Proof. By combining (2.26), (2.28) and the Gagliardo-Nirenberg inequality, we obtain D(u) ≤ C

Z

R⁶

|x − y|

u

(y)u

(x)dydx ≤ Ckuk

L¹²⁵

≤ Ckuk

kuk

,

which proves the assertion. 

Let S

denote the set of (complex) least energy solutions u to equation (2.1) such that

|u(0)| = max

x∈R³

|u(x)|.

By Lemma 2.9, up to a constant phase change, we can assume that u is real valued.

Proposition 2.12. For any a > 0 the set S

is compact in H

(R

, R) and there exist positive constants C, σ such that u(x) ≤ C exp(−σ|x|) for any x ∈ R

and all u ∈ S

. Proof. If J

: H

(R

) → R denotes the functional associated with ( 2.1),

J

(u) = J (u) = 1 2

Z

R³

|∇u|

dx + a 2

Z

R³

u

dx − 1 4 D(u), since D(u) = k∇uk

+ akuk

for all u ∈ M

, we have

m

= L

(u) = 1

4 k∇uk

+ akuk

,

where m

= min{L

(u) : u 6= 0 solves (2.1)}. Hence, it follows that the set M

is bounded

in H

(R

, R). Moreover, S

is also bounded in L

(R

) and u(x) → 0 as |x| → ∞ for any

u ∈ S

. Indeed, from the Hardy-Littlewood-Sobolev inequality (2.27), for any q ≥ 3

(2.29) kW ∗ u

k

≤ Ck|x|

∗ u

k

≤ Ckuk

≤ Ckuk

it follows that |x|

∗ u

∈ L

(R

) for any q ≥ 3. Then, by H¨ older inequality, we get for all m such that 3/2 < m < 6,

kW ∗ u

uk

≤ Ck|x|

∗ u

uk

≤ Ck|x|

∗ u

k

L

6m

6−m

kuk

.

From equation (2.1) it follows that u ∈ W

(R

) for all 2 ≤ m < 6. Hence, it follows that u is a bounded function which vanishes at infinity. Actually u has further regularity.

Indeed, using again the equation, the boundedness of u as well as the Hardy-Littlewood- Sobolev inequality, we have

kuk

≤ Ck∆uk

≤ Ckuk

+ Ck(|x|

∗ u

)uk

≤ C + Ck|x|

∗ u

k

≤ C.

It follows, that u ∈ W

(R

) for every m > 3. Also these summability properties implies that S

is uniformly bounded in C

(R

). Let us now show that the limit u(x) → 0 as

|x| → ∞ holds uniformly for u ∈ S

. Assuming by contradiction that u

(x

) ≥ σ > 0 along some sequences (u

) ⊂ S

and (x

) ⊂ R

with |x

| → ∞, shifting u

as v

(x) = u

(x + x

) it follows that (v

) is bounded in H

(R

) ∩ L

(R

) and it converges, up to a subsequence to a function v, weakly in H

(R

) and locally uniformly in C(R

). If u denotes the weak limit of u

, we also claim that u, v are both solutions to equation (2.1), which are nontrivial as follows from (local) uniform convergence and u

(0) ≥ u

(x

) ≥ σ (since 0 is a global maximum for u

) and v

(0) = u

(x

) ≥ σ. To see that u, v are solutions to (2.1), set

ϕ

(x) = Z

R³

W (x − y)u

(y)dy, ϕ(x) = Z

R³

W (x − y)u

(y)dy.

Let us show that ϕ

(x) → ϕ(x), as m → ∞, for any fixed x ∈ R

. Indeed, we can write ϕ

(x) − ϕ(x) = I

(ρ) + I

(ρ) for any m ≥ 1 and any ρ > 0, where we set

I

(ρ) = Z

Bρ(0)

W (x − y)(u

(y) − u

(y))dy, I

(ρ) =

Z

R³\B_ρ(0)

W (x − y)(u

(y) − u

(y))dy.

Fix x ∈ R

and let ε > 0. Choose ρ

> 0 sufficiently large that (2.30) I

(ρ

) ≤

Z

R³\B_ρ0(0)

C

|y| − |x| |u

(y) − u

(y)|dy ≤ C

ρ

− |x| < ε 2 .

On the other hand, by the uniform local convergence of u

to u as m → ∞ and H¨ older inequality, for some 1 < r < 3

I

(ρ

) ≤ Cku

− uk

 Z

B_ρ0(0)

1

|x − y|

|u

(y) + u(y)|

dy 

(2.31)

≤ Cku

− uk

 Z

B_ρ0(x)

1

|y|

dy 

≤ Cku

− uk

< ε

2

for all m sufficiently large, where r

denotes the conjugate exponent of r. The bound r < 3 ensures that the singular integral which appears in the second inequality is finite.

Combining (2.30) with (2.31) concludes the proof of the pointwise convergence of ϕ

to ϕ. Notice that (ϕ

) is uniformly bounded by inequality (2.27), since kW ∗ u

k

≤ Cku

k

≤ C. Hence, ϕ

(x)u

(x)η(x) → ϕ(x)u(x)η(x) as m → ∞, for all η ∈ C

(R

) with compact support K and any x ∈ R

fixed. In turn, by the Dominated Convergence Theorem (recall that |ϕ

u

η| ≤ C ∈ L

(K)), we get

m→+∞

lim Z

K

ϕ

u

η dx = Z

K

ϕuη dx,

for all η ∈ C

(R

) with compact support K. This concludes the proof that u, v are nontrivial solutions to (2.1). It follows that, for any m and k,

J

(u

) = J

(u

) = 1 4

Z

R³

(|∇u

|

+ au

)dx, J

(u) ≥ J

(z) = m

,

J

(v) ≥ J

(z) = m

,

for all z ∈ S

. On the other hand, for any R > 0 and m ≥ 1 with 2R ≤ |x

|, m

= J

(u

) ≥ 1

4 lim inf

m

Z

BR(0)

(|∇u

|

+ au

)dx + 1

4 lim inf

m

Z

BR(0)

(|∇v

|

+ av

)dx

≥ J

(u) + J

(v) − ε ≥ 2m

− o(1)

as R → ∞, which yields a contradiction for R large enough. Hence the conclusion follows.

Let us now prove that

(2.32) lim

|x|→∞

ϕ(x) = 0, ϕ(x) = Z

R³

W (x − y)u

(y) dy.

Taken ρ > 0, write ϕ(x) =

Z

Bρ(0)

W (x − y)u

(y) dy + Z

R³\B_ρ(0)

W (x − y)u

(y) dy.

Chosen ε > 0, there exists ρ

> 0 such that u

(y) ≤ ε for all y in R

\ B

(0). Hence, for any x ∈ R

, we obtain

ϕ(x) ≤ kuk

η

(x) + εη

(x).

where

η

(x) = Z

B_ρ0(0)

W (x − y) dy, η

(x) = Z

R³\B_ρ0(0)

u

(y)

|x − y| dy.

Notice that η

is bounded, since by Hardy-Littlewood-Sobolev inequality (2.27) it follows kη

k

≤

|x|

∗ u

≤ Ckuk

,

since u ∈ L

(R

). Moreover, since |x − y| ≥ |x| − |y| ≥ |x| − ρ

for any x, y ∈ R

with

|y| ≤ ρ

and |x| > ρ

, then we have (µ is the Lebesgue measure in R

) 0 < η

(x) ≤ C

Z

B_ρ0(0)

1

|x| − ρ

dy ≤ C

µ(B

(0))

|x| − ρ

→ 0, as |x| → ∞.

Then, in conclusion, there exists R(ε) ≥ ρ

such that η

(x) ≤ Cε, as |x| ≥ R(ε), which yields ϕ(x) ≤ Cε as |x| ≥ R(ε), so that assertion (2.32) follows. In light of (2.32), let R

> 0 such that ϕ(x) ≤

, for any |x| ≥ R

. As a consequence,

−∆u(x) + a

2 u(x) ≤ 0, for |x| ≥ R

.

It is thus standard to see that this yields the exponential decay ou u, with uniform decay constants in S

. We can finally conclude the proof. Let (u

) be any sequence in S

. Up to a subsequence it follows that (u

) converges weakly to a function u which is also a solution to equation (2.1 ). If D is the function defined in ( 2.8), we immediately get

(2.33) lim

n→+∞

Z

R³

|∇u

|

+ au

− D(u

) = 0 = Z

R³

|∇u|

+ au

− D(u).

Hence the desired strong convergence of (u

) to u in H

(R

) follows once we prove that D(u

) → D(u), as n → ∞. In view of the uniform exponential decay of u

it follows that u

→ u strongly in L

(R

), as n → ∞. Taking into account that (u

) is bounded in H

(R

) and that W is even, we easily get the inequality

|D(u

) − D(u)| ≤ q

D(||u

|

− |u|

|

) q

D((|u

|

+ |u|

)

), n ∈ N.

By Hardy-Littlewood-Sobolev inequality and H¨ older’s inequality, it follows that

|D(u

) − D(u)|

≤ Ck ||u

|

− |u|

|

k

L¹²⁵

k (|u

|

+ |u|

)

k

L¹²⁵

≤ Cku

− uk

L¹²⁵

. As a consequence

D(u

) = D(u) + o(1), as n → ∞,

which concludes the proof in light of formula (2.33). 

3. The penalization argument

Throughout this and the following sections we shall mainly used the arguments of [12]

highlighting the technical steps where the Hartree nonlinearity is involved in place of the local one. For the sake of self-containedness and for the reader’s convenience we develop the arguments with all the detail.

For any set Ω ⊂ R

and ε > 0, let Ω

= {x ∈ R

: εx ∈ Ω}.

3.1. Notations and framework. The following lemmas, taken from [12] show that the norm in H

is locally equivalent to the standard H

norm.

Lemma 3.1. Let K ⊂ R

be an arbitrary, fixed, bounded domain. Assume that A is bounded on K and 0 < α ≤ V ≤ β on K for some α, β > 0. Then, for any fixed ε ∈ [0, 1], the norm

kuk

= Z

Kε

 1

i ∇ − A

(y)

 u

2

+ V

(y)|u|

dy

is equivalent to the usual norm on H

(K

, C). Moreover these equivalences are uniform, namely there exist constants c

, c

> 0 independent of ε ∈ [0, 1] such that

c

kuk

≤ kuk

≤ c

kuk

.

Corollary 3.2. Retain the setting of Lemma 3.1. Then the following facts hold.

(i) If K is compact, for any ε ∈ (0, 1] the norm kuk

:=

Z

K

 1

i ∇ − A

(y)

 u

2

+ V

(y)|u|

dy is uniformly equivalent to the usual norm on H

(K, C).

(ii) For A

∈ R

and b > 0 fixed, the norm kuk

:=

Z

R³

 1

i ∇ − A

 u

2

+ b|u|

dy is equivalent to the usual norm on H

(R

, C).

(iii) If (u

) ⊂ H

(R

, C) satisfies u

= 0 on R

\ K

for any n ∈ N and u

→ u in H

(R

, C) then ku

− uk

→ 0 as n → ∞.

For future reference we recall the following Diamagnetic inequality: for every u ∈ H

, (3.1)

 ∇ i − A

 u

≥  ∇|u| 

, a.e. in R

.

See [15] for a proof. As a consequence of (3.1), |u| ∈ H

(R

, R) for any u ∈ H

. For any u ∈ H

, let us set

(3.2) F

(u) = 1 2

Z

R³

|D

u|

+ V

(x)|u|

dx − 1 4

Z

R³×R³

W (x − y)|u(x)|

|u(y)|

dxdy where we set D

= (

− A

). Define, for all ε > 0,

χ

(y) =

( 0 if y ∈ O

,

ε

if y / ∈ O

, χ

(y) =

( 0 if y ∈ (O

)

, ε

if y / ∈ (O

)

, and

(3.3) Q

(u) =  Z

R³

χ

|u|

dx − 1 

+

, Q

(u) =  Z

R³

χ

|u|

dx − 1 

+

.

The functional Q

will act as a penalization to force the concentration phenomena of the solution to occur inside O. In particular, we remark that the penalization terms vanish on elements whose corresponding L

-norm is sufficiently small. This device was firstly introduced in [7]. Finally we define the functionals Γ

, Γ

, . . . , Γ

: H

→ R by setting (3.4) Γ

(u) = F

(u) + Q

(u), Γ

(u) = F

(u) + Q

(u), i = 1, . . . , k.

It is easy to check, under our assumptions, and using the Diamagnetic inequality (3.1), that the functionals Γ

and Γ

are of class C

over H

. Hence, a critical point of F

corresponds to a solution of (1.5). To find solutions of (1.5) which concentrate in O as ε → 0, we shall look for a critical point of Γ

for which Q

is zero.

Let

M =

k

[

i=1

M

, O =

k

[

i=1

O

and for any set B ⊂ R

and α > 0, B

= {x ∈ R

: dist(x, B) ≤ δ} and set δ = 1

10 min



dist(M, R

\ O), min

i6=j

dist(O

, O

), dist(O, Z )

 .

We fix a β ∈ (0, δ) and a cutoff ϕ ∈ C

(R

) such that 0 ≤ ϕ ≤ 1, ϕ(y) = 1 for |y| ≤ β and ϕ(y) = 0 for |y| ≥ 2β. Also, setting ϕ

(y) = ϕ(εy) for each x

∈ (M

)

and U

∈ S

, we define

U

(y) =

k

X

i=1

e

ϕ

 y − x

ε

 U



y − x

ε

 . We will find a solution, for sufficiently small ε > 0, near the set

X

= {U

(y) : x

∈ (M

)

and U

∈ S

for each i = 1, . . . , k}.

For each i ∈ {1, . . . , k} we fix an arbitrary x

∈ M

and an arbitrary U

∈ S

and we define

W

(y) = e

ϕ

 y − x

ε

 U



y − x

ε

 . Setting

W

(y) = e

ϕ

 y − x

ε

 U

 y

t − x

εt

 , we see that

lim

kW

k

= 0, Γ

(W

) = F

(W

), t ≥ 0.

In the next Proposition we shall show that there exists T

> 0 such that Γ

(W

) < −2 for any ε > 0 sufficiently small. Assuming this holds true, let γ

(s) = W

for s > 0 and γ

(0) = 0. For s = (s

, . . . , s

) ∈ T = [0, T

] × . . . × [0, T

] we define

γ

(s) =

k

X

i=1

W

and D

= max

s∈T

Γ

(γ

(s)).

Finally for each i ∈ {1, . . . , k}, let E

= L

(U ) for U ∈ S

.