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arXiv:1110.4213v2 [math.AP] 20 Oct 2011

SCHR ¨ ODINGER-NEWTON SYSTEM

SILVIA CINGOLANI, M ´ ONICA CLAPP, AND SIMONE SECCHI

Abstract. We study the problem

( (−εi∇ + A(x))

2

u + V (x)u = ε

−2



1

|x|

∗ |u|

2

 u, u ∈ L

2

(R

3

, C), ε∇u + iAu ∈ L

2

(R

3

, C

3

),

where A : R

3

→ R

3

is an exterior magnetic potential, V : R

3

→ R is an exte- rior electric potential, and ε is a small positive number. If A = 0 and ε = ~ is Planck’s constant this problem is equivalent to the Schr¨ odinger-Newton equa- tions proposed by Penrose in [23] to describe his view that quantum state re- duction occurs due to some gravitational effect. We assume that A and V are compatible with the action of a group G of linear isometries of R

3

. Then, for any given homomorphism τ : G → S

1

into the unit complex numbers, we show that there is a combined effect of the symmetries and the potential V on the number of semiclassical solutions u : R

3

→ C which satisfy u(gx) = τ (g)u(x) for all g ∈ G, x ∈ R

3

. We also study the concentration behavior of these solutions as ε → 0.

MSC2010: 35Q55, 35Q40, 35J20, 35B06.

Keywords: Schr¨ odinger-Newton system, nonlocal nonlinearity, electromag- netic potential, semiclassical solutions, intertwining solutions.

1. Introduction

The Schr¨ odinger-Newton equations were proposed by Penrose [23] to describe his view that quantum state reduction is a phenomenon that occurs because of some gravitational influence. They consist of a system of equations obtained by coupling together the linear Schr¨odinger equation of quantum mechanics with the Poisson equation from Newtonian mechanics. For a single particle of mass m this system has the form

(1.1)

( − 2m ~

2

∆ψ + V (x)ψ + U ψ = 0,

−∆U + 4πκ|ψ| 2 = 0,

where ψ is the complex wave function, U is the gravitational potential energy, V is a given potential, ~ is Planck’s constant, and κ := Gm 2 , G being Newton’s constant.

According to Penrose, the solutions ψ of this system are the basic stationary states into which a superposition of such states is to decay within a certain timescale, cf.

[22, 23, 18, 19, 24].

S. Cingolani is supported by the MIUR project Variational and topological methods in the study of nonlinear phenomena (PRIN 2007).

M. Clapp is supported by CONACYT grant 129847 and PAPIIT grant IN101209 (Mexico).

1

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After rescaling by ψ(x) = 1

~ ψ(x) ˆ

√ 2κm , V (x) = 1

2m V (x), ˆ U (x) = 1 2m U (x), ˆ system (1.1) can be written as

(1.2)

( −~ 2 ∆ ˆ ψ + ˆ V (x) ˆ ψ + ˆ U ˆ ψ = 0,

−~ 2 ∆ ˆ U + 4π | ˆ ψ | 2 = 0.

The second equation in (1.2) can be explicitly solved with respect to ˆ U , so this system is equivalent to the single nonlocal equation

(1.3) − ~ 2 ∆ ˆ ψ + ˆ V (x) ˆ ψ = 1

~ 2 Z

R

3

| ˆ ψ(ξ) | 2

|x − ξ| dξ

!

ψ ˆ in R 3 .

We shall consider a more general equation having a similar structure, namely (1.4) ( −εi∇ + A(x)) 2 u + V (x)u = 1

ε 2

 1

|x| ∗ |u| 2



u in R 3 ,

where A : R 3 → R 3 is an exterior magnetic potential, i is the imaginary unit and ∗ denotes the convolution operator. We are interested in semiclassical states, i.e. in solutions of this equation for ε → 0.

The existence of one solution can be traced back to Lions’ paper [15]. In the nonmagnetic case A = 0 equation (1.4) and related equations have been inves- tigated by many authors, see e.g. [2, 10, 11, 12, 13, 16, 17, 18, 20, 25, 26, 19]

and the references therein. Recently, Wei and Winter [27] showed the existence of positive multibump solutions which concentrate at local minima, local maxima or nondegenerate critical points of the potential V as ε → 0. The magnetic case A 6= 0 was recently studied in [6] where it was shown that equation (1.4) has a fam- ily of solutions having multiple concentration regions located around the (possibly degenerate) minima of V .

In this paper we consider the situation where A and V are symmetric and we look for semiclassical solutions of equation (1.4) having specific symmetries. The absolute value of the solutions we obtain concentrates at points which need not be local extrema, nor nondegenerate critical points of V (in fact, we do not even assume that V is differentiable). We state our main results in the following section and give some explicit examples.

2. Statement of results

2.1. The results. Let G be a closed subgroup of the group O(3) of linear isometries of R 3 , A : R 3 → R 3 be a C 1 -function, and V : R 3 → R be a bounded continuous function with inf R

3

V > 0, which satisfy

(2.1) A(gx) = gA(x) and V (gx) = V (x) for all g ∈ G, x ∈ R 3 .

Given a continuous homomorphism of groups τ : G → S 1 into the group S 1 of unit complex numbers, we look for solutions to the problem

(2.2)

 

 

 

( −εi∇ + A) 2 u + V (x)u = ε −2 

1

|x| ∗ |u| 2  u, u ∈ L 2 (R 3 , C),

ε ∇u + iAu ∈ L 2 (R 3 , C 3 ),

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which satisfy

(2.3) u(gx) = τ (g)u(x) for all g ∈ G, x ∈ R 3 , This implies that the absolute value |u| of u is G-invariant, i.e.

|u(gx)| = |u(x)| for all g ∈ G, x ∈ R 3 ,

whereas the phase of u(gx) is that of u(x) multiplied by τ (g). A concrete example is given in subsection 2.2 below.

Note that if u satisfies (2.2) and (2.3) then e u satisfies (2.2) and (2.3) for every θ ∈ R. We shall say that u and v are geometrically distinct if e u 6= v for all θ ∈ R.

We introduce some notation. For x ∈ R 3 , we denote by Gx the G-orbit of x and by G x the G-isotropy subgroup of x, i.e.

Gx := {gx: g ∈ G} , G x := {g ∈ G : gx = x}.

A subset X of R 3 is G-invariant if Gx ⊂ X for every x ∈ X. The G-orbit space of X is the set

X/G := {Gx : x ∈ X}

of G-orbits of X with the quotient topology.

Let #Gx denote the cardinality of Gx, and define ℓ G,V := inf

x∈R

3

(#Gx)V 3/2 (x), M τ := n

x ∈ R 3 : (#Gx)V 3/2 (x) = ℓ G,V , G x ⊂ ker τ o .

Assumption (2.1) implies that M τ is G-invariant. Observe that the points of M τ

need not be neither local minima nor local maxima of V .

Given ρ > 0 we set B ρ M τ := {x ∈ R 3 : dist(x, M τ ) ≤ ρ}, and write cat B

ρ

M

τ

/G (M τ /G)

for the Lusternik-Schnirelmann category of M τ /G in B ρ M τ /G.

Finally, we denote by E 1 the least energy of a nontrivial solution to problem (2.4)

( −∆u + u = ( |x| 1 ∗ u 2 )u, u ∈ H 1 (R 3 , R).

We shall prove the following results.

Theorem 2.1. Assume there exists α > 0 such that the set

(2.5) n

x ∈ R 3 : (#Gx)V 3/2 (x) ≤ ℓ G,V + α o

is compact. Then, given ρ, δ > 0, there exists ε > 0 such that, for every ε b ∈ (0, b ε), problem (2.2) has at least

cat B

ρ

M

τ

/G (M τ /G) geometrically distinct solutions u which satisfy (2.3) and

(2.6)

1

4 Z

R

3

 1

|x| ∗ |u| 2



|u| 2 − ε 5 ℓ G,V E 1

< ε 5 δ.

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The last inequality says that the energy of the solutions is arbitrarily close to ε 3 ℓ G,V E 1 for ε small enough. So considering different groups G and G for which ℓ G,V 6= ℓ G

,V will lead to solutions with energy in disjoint ranges.

For u ∈ H 1 (R 3 , R) set

kuk 2 ε :=

Z

R

3

2 |∇u| 2 + u 2 ).

The following theorem describes the module of the solutions given by Theorem 2.1 as ε → 0.

Theorem 2.2. Let u n be a solution to problem (2.2) which satisfies (2.3) and (2.6) for ε = ε n > 0, δ = δ n > 0. Assume ε n → 0 and δ n → 0. Then, after passing to a subsequence, there exists a sequence (ξ n ) in R 3 such that ξ n → ξ ∈ M τ , G ξ

n

= G ξ , and

ε −3 n

|u n | − P

n

∈Gξ

n

ω ξ

 · − gξ n ε n



2

ε

n

→ 0, where ω ξ is the unique ground state of problem

−∆u + V (ξ)u =

 1

|x| ∗ u 2



u, u ∈ H 1 (R 3 , R), which is positive and radially symmetric with respect to the origin.

Next, we give an example which illustrates our results.

2.2. Rotationally invariant potentials. Let S 1 act on R 3 ≡ C×R by e (z, t) :=

(e z, t), and let A and V satisfy assumption (2.1) for the cyclic group G m gener- ated by e 2πi/m , for some m ∈ N. For example, the standard magnetic poten- tial A(x 1 , x 2 , x 3 ) := ( −x 2 , x 1 , 0) associated to the constant magnetic field B(x) = (0, 0, 2) has this property for every m.

For each j = 0, 1, . . . , m − 1 we look for solutions to problem (2.2) which satisfy (2.7) u(e 2πi/m z, t) = e 2πij/m u(z, t) for all (z, t) ∈ C × R.

Solutions of this type arise in a natural way in some problems where the magnetic potential is singular and the topology of the domain produces an Aharonov-Bohm type effect, cf. [1, 8]. Taking τ j (g) := g j we see that these are solutions of the type furnished by Theorem 2.1.

If V satisfies (2.8) V 0 := inf

x∈R

3

V < lim inf

|x|→∞ V (x) and mV 0 3/2 < inf

t∈R V 3/2 (0, t),

then assumption (2.5) in Theorem 2.1 is satisfied, ℓ G

m

,V = mV 0 3/2 and M τ is simply the set of minima of V,

M = 

x ∈ R 3 : V (x) = V 0 .

Thus, for each j = 0, 1, . . . , m − 1 and ρ, δ > 0, Theorem 2.1 yields at least cat B

ρ

M/G

m

(M/G m ) geometrically distinct solutions to problem (2.2) satisfying (2.7) and (2.6), for ε small enough.

For each k dividing m the potentials A and V satify assumption (2.1) for G k

and V satisfies (2.8) with k instead of m. Property (2.6) implies that the solutions

obtained for G k are different from those for G m if k 6= m and ε is small enough.

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This paper is organized as follows. In section 3 we discuss the variational problem related to the existence of solutions to problem (2.2) satisfying (2.3). We also outline the strategy for proving Theorem 2.1. Sections 4 and 5 are devoted to the construction of an entrance map and a local baryorbit map which will help us estimate the Lusternik-Schnirelmann category of a suitable sublevel set of the variational functional for ε small enough. Finally, in section 6 we prove Theorems 2.1 and 2.2.

3. The variational problem Set ∇ ε,A u := ε ∇u + iAu and consider the real Hilbert space

H ε,A 1 (R 3 , C) := {u ∈ L 2 (R 3 , C) : ∇ ε,A u ∈ L 2 (R 3 , C 3 ) } with the scalar product

(3.1) hu, vi ε,A,V := Re Z

R

3

ε,A u · ∇ ε,A v + V (x)uv  . We write

kuk ε,A,V :=

Z

R

3

 |∇ ε,A u | 2 + V (x) |u| 2  1/2

for the corresponding norm.

If u ∈ H ε,A 1 (R 3 , C), then |u| ∈ H 1 (R 3 , R) and

(3.2) ε |∇|u(x)|| ≤ |ε∇u(x) + iA(x)u(x)| for a.e. x ∈ R 3 . This is called the diamagnetic inequality [14]. Set

D (u) :=

Z

R

3

Z

R

3

|u(x)| 2 |u(y)| 2

|x − y| dxdy.

The standard Hardy–Littlewood–Sobolev inequality [14, Theorem 4.3] yields (3.3)

Z

R

3

Z

R

3

f (x)h(y)

|x − y| dx dy

≤ Ckfk L

6/5

(R

3

) khk L

6/5

(R

3

)

for all f, h ∈ L 6/5 (R 3 ), where C is a positive constant independent of f and h. In particular,

(3.4) D (u) ≤ Ckuk 4 L

12/5

(R

3

)

for every u ∈ H ε,A 1 (R 3 , C).

The energy functional J ε,A,V : H ε,A 1 (R 3 , C) → R associated to problem (2.2), defined by

J ε,A,V (u) := 1

2 kuk 2 ε,A,V − 1 4ε 2 D (u), is of class C 2 , and its derivative is given by

J ε,A,V (u)v := hu, vi ε,A,V − 1 ε 2 Re

Z

R

3

 1

|x| ∗ |u| 2

 uv.

Therefore, the solutions to problem (2.2) are the critical points of J ε,A,V . We write

ε J ε,A,V (u) for the gradient of J ε,A,V at u with respect to the scalar product (3.1).

The action of G on H ε,A 1 (R 3 , C) defined by (g, u) 7→ u g , where

(u g )(x) := τ (g)u(g −1 x),

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satisfies

hu g , v g i ε,A,V = hu, vi ε,A,V and D (u g ) = D(u)

for all g ∈ G, u, v ∈ H ε,A 1 (R 3 , C). Hence, J ε,A,V is G-invariant. By the principle of symmetric criticality [21, 28], the critical points of the restriction of J ε,A,V to the fixed point space of this G-action, denoted by

H ε,A 1 (R 3 , C) τ = 

u ∈ H ε,A 1 (R 3 , C) : u g = u

= 

u ∈ H ε,A 1 (R 3 , C) : u(gx) = τ (g)u(x) ∀x ∈ R 3 , g ∈ G , are the solutions to problem (2.2) which satisfy (2.3). Those which are nontrivial lie on the Nehari manifold

N ε,A,V τ := n

u ∈ H ε,A 1 (R 3 , C) τ : u 6= 0, ε 2 kuk 2 ε,A,V = D(u) o ,

which is a C 2 -manifold radially diffeomorphic to the unit sphere in H ε,A 1 (R 3 , C) τ . The critical points of the restriction of J ε,A,V to N ε,A,V τ are precisely the nontrivial solutions to (2.2) which satisfy (2.3).

The radial projection π ε,A,V : H ε,A 1 (R 3 , C) τ \ {0} → N ε,A,V τ is given by (3.5) π ε,A,V (u) := ε kuk ε,A,V

p D (u) u.

Note that

(3.6) J ε,A,V (π ε,A,V (u)) = ε 2 kuk 4 ε,A,V

4D(u) for all u ∈ H ε,A 1 (R 3 , C) τ \ {0}.

Recall that J ε,A,V : N ε,A,V τ → R is said to satisfy the Palais-Smale condition (P S) c at the level c if every sequence (u n ) such that

u n ∈ N ε,A,V τ , J ε,A,V (u n ) → c, ∇ N

ε,A,Vτ

J ε,A,V (u n ) → 0,

contains a convergent subsequence. Here ∇ N

ε,A,Vτ

J ε,A,V (u) denotes the orthogonal projection of ∇ ε J ε,A,V (u) onto the tangent space to N ε,A,V τ at u. The following holds.

Proposition 3.1. For every ε > 0, the functional J ε,A,V : N ε,A,V τ → R satisfies (P S) c at each level

c < ε 3 min

x∈R

3

\{0} (#Gx)V 3/2 E 1 , where V ∞ := lim inf |x|→∞ V (x).

Proof. This was proved in [5] for ε = 1. For ε > 0 the assertion follows after per- forming the change of variable u ε (x) := u(εx) since a straightforward computation shows that

ε −3 J ε,A,V (u) = J 1,A

ε

,V

ε

(u ε ) and ε −3/2N

ε,A,Vτ

J ε,A,V (u) = ∇ N

1τ,Aε ,Vε

J 1,A

ε

,V

ε

(u ε ),

where A ε (x) := A(εx) and V ε (x) := V (εx). 

S 1 acts on H ε,A 1 (R 3 , C) τ by scalar multiplication: (e , u) 7→ e u. The Nehari

manifold N ε,A,V τ and the functional J ε,A,V are invariant under this action. Two

solutions of (2.2) are geometrically distinct iff they lie on different S 1 -orbits. Equi-

variant Lusternik-Schnirelmann theory yields the following result, see e.g. [7].

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Proposition 3.2. If J ε,A,V : N ε,A,V τ → R satisfies (P S) c at each level c ≤ c, then J ε,A,V has at least

cat 

N ε,A,V τ ∩ J ε,A,V c

 /S 1  critical S 1 -orbits in N ε,A,V τ ∩ J ε,A,V c .

Here ( N ε,A,V τ ∩ J ε,A,V c )/S 1 denotes the S 1 -orbit space of N ε,A,V τ ∩ J ε,A,V c , where, as usual, J ε,A,V c := 

u ∈ H ε,A 1 (R 3 , C) : J ε,A,V (u) ≤ c . To prove Theorem 2.1 we will show that

(3.7) cat B

ρ

M

τ

/G M τ /G ≤ cat 

N ε,A,V τ ∩ J ε,A,V d

 /S 1 

for some d = d(ε) ∈ (c τ ε,A,V , ε 3 min x∈R

3

\{0} (#Gx)V ∞ 3/2 E 1 ), where

(3.8) c τ ε,A,V := inf

N

ε,A,Vτ

J ε,A,V . To obtain inequality (3.7) we shall construct maps

M τ /G −→ C/S ι

ε

1 β −→ B

ε

ρ M τ /G,

whose composition is the inclusion M τ /G ֒ → B ρ M τ /G, where C is a union of connected components of N ε,A,V τ ∩ J ε,A,V d . A standard argument then yields

cat B

ρ

M

τ

/G M τ /G ≤ cat C/S 1 

≤ cat 

N ε,A,V τ ∩ J ε,A,V d

 /S 1  .

The main ingredients for defining these maps are contained in the following two sections.

4. The entrance map For any positive real number λ we consider the problem (4.1)

( −∆u + λu = ( |x| 1 ∗ u 2 )u, u ∈ H 1 (R 3 , R).

Its associated energy functional J λ : H 1 (R 3 , R) → R is given by J λ (u) = 1

2 kuk 2 λ − 1

4 D (u), with kuk 2 λ :=

Z

R

3

 |∇u| 2 + λu 2  . Its Nehari manifold will be denoted by

M λ : = n

u ∈ H 1 (R 3 , R) : u 6= 0, kuk 2 λ = D(u) o . We set

E λ := inf

u∈M

λ

J λ (u).

The critical points of J λ on M λ are the nontrivial solutions to (4.1). Note that u solves (2.4) if and only if u λ (x) := λu( √

λx) solves (4.1). Therefore, E λ = λ 3/2 E 1 .

Minimizers of J λ on M λ are called ground states. Lieb established in [13] the

existence and uniqueness of ground states up to sign and translations. Recently

Ma and Zhao [17] showed that every positive solution to problem (4.1) is radially

symmetric, and they concluded from this fact that the positive solution to this

problem is unique up to translations. We denote by ω λ the positive solution to

problem (4.1) which is radially symmetric with respect to the origin.

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Fix a radial function ̺ ∈ C (R 3 , R) such that ̺(x) = 1 if |x| ≤ 1 2 and ̺(x) = 0 if |x| ≥ 1. For ε > 0 set ̺ ε (x) := ̺( √ εx), ω λ,ε := ̺ ε ω λ and

(4.2) υ λ,ε = kω λ,ε k λ

p D (ω λ,ε ) ω λ,ε .

Note that supp(υ λ,ε ) ⊂ B(0, 1/ √ ε) := {x ∈ R 3 : |x| ≤ 1/ √ ε } and υ λ,ε ∈ M λ . An easy computation shows that

(4.3) lim

ε→0 J λ (υ λ,ε ) = λ 3/2 E 1 . Observe that

ℓ G,V := inf

x∈R

3

(#Gx)V 3/2 (x) < V 3/2 (0) < ∞.

We assume from now on that there exists α > 0 such that the set n y ∈ R 3 : (#Gy)V 3/2 (y) ≤ ℓ G,V + α o

is compact. Then

M G,V := n

y ∈ R 3 : (#Gy)V 3/2 (y) = ℓ G,V

o

is a compact G-invariant set and all G-orbits in M G,V are finite. We split M G,V ac- cording to the orbit type of its elements as follows: we choose subgroups G 1 , . . . , G m

of G such that the isotropy subgroup G x of every point x ∈ M G,V is conjugate to precisely one of the G i ’s, and we set

M i := 

y ∈ M G,V : G y = gG i g −1 for some g ∈ G .

Since isotropy subgroups satisfy G gx = gG x g −1 , the sets M i are G-invariant and, since V is continuous, they are closed and pairwise disjoint, and

M G,V = M 1 ∪ · · · ∪ M m . Moreover, since

|G/G i | V 3/2 (y) = (#Gy)V 3/2 (y) = ℓ G,V for all y ∈ M i ,

the potential V is constant on each M i . Here |G/G i | denotes the index of G i in G.

We denote by V i the value of V on M i .

Let υ i,ε := υ V

i

,ε be defined as in (4.2) with λ := V i . For ξ ∈ M i set φ ε,ξ (x) := υ i,ε

 x − ξ ε

 exp



−iA(ξ) ·

 x − ξ ε



.

The proofs of the following two lemmas are similar to those of Lemmas 1 and 2 in [3], so we shall omit them.

Lemma 4.1. Uniformly in ξ ∈ M i , we have that

ε→0 lim ε −3 J ε,A,V [π ε,A,V (φ ε,ξ )] = V i 3/2 E 1 , where π ε,A,V is as in (3.5).

It is well known that the map G/G ξ → Gξ given by gG ξ 7→ gξ is a homeomor- phism, see e.g. [9]. So, if G i ⊂ ker τ and ξ ∈ M i , then the map

Gξ → S 1 , gξ 7→ τ(g),

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is well defined and continuous. Set (4.4) ψ ε,ξ (x) := X

gξ∈Gξ

τ (g)υ i,ε

 x − gξ ε



e −iA(gξ)· (

x−gξε

).

Lemma 4.2. Assume that G i ⊂ ker τ. Then, the following hold:

(a) For every ξ ∈ M i and ε > 0, one has that

ψ ε,ξ (gx) = τ (g)ψ ε,ξ (x) ∀g ∈ G, x ∈ R 3 . (b) For every ξ ∈ M i and ε > 0, one has that

τ (g)ψ ε,gξ (x) = ψ ε,ξ (x) ∀g ∈ G, x ∈ R 3 . (c) One has that

ε→0 lim ε −3 J ε,A,V [π ε,A,V (ψ ε,ξ )] = ℓ G,V E 1 . uniformly in ξ ∈ M i .

Let

M τ := {y ∈ M G,V : G y ⊂ ker τ} = [

G

i

⊂ker τ

M i .

Proposition 4.3. The mapε : M τ → N ε,A,V τ given byε (ξ) := π ε,A,V (ψ ε,ξ ) is well defined and continuous, and satisfies

τ (g) bι ε (gξ) = bι ε (ξ) ∀ξ ∈ M τ , g ∈ G.

Moreover, given d > ℓ G E 1 , there exists ε d > 0 such that

ε −3 J ε,A,V ( bι ε (ξ)) ≤ d ∀ξ ∈ M τ , ε ∈ (0, ε d ).

Proof. This follows immediately from Lemma 4.2. 

5. A local baryorbit map

Let W : R 3 → R be a bounded, uniformly continuous function with inf R

3

W > 0 and such that W (gx) = W (x) for all g ∈ G, x ∈ R 3 . We assume that the set

(5.1) n

y ∈ R 3 : (#Gy)W 3/2 (y) ≤ ℓ G,W + α o

is compact, where ℓ G,W := inf x∈R

3

(#Gx)W 3/2 (x), and consider the real-valued problem

(5.2)

 

 

−ε 2 ∆v + W (x)v = ε 1

2



1

|x| ∗ u 2  u, v ∈ H 1 (R 3 , R),

v(gx) = v(x) ∀x ∈ R 3 , g ∈ G.

We write hv, wi ε,W :=

Z

R

3

ε 2 ∇v · ∇w + W (x)vw 

, kvk 2 ε,W :=

Z

R

3

 |ε∇v| 2 + W (x)v 2  , and set

H 1 (R 3 , R) G := {v ∈ H 1 (R 3 , R) : v(gx) = v(x) ∀x ∈ R 3 , g ∈ G}.

The nontrivial solutions of (5.2) are the critical points of the energy functional J ε,W (v) = 1

2 kvk 2 ε,W − 1

2 D (v)

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on the Nehari manifold

M G ε,W := {v ∈ H 1 (R 3 , R) G : v 6= 0, kvk 2 ε,W = ε −2 D (v) }.

Set

(5.3) c G ε,W := inf

M

Gε,W

J ε,W = inf

v∈H

1

(R

3

,R)

G

v6=0

ε 2 kvk 4 ε,W

4D(v) .

We wish to study the behavior of ”minimizing sequences” for the family of prob- lems (5.2), parametrized by ε, as ε → 0. This is described in Proposition 5.4 below.

We start with some lemmas.

Lemma 5.1. 0 < (inf R

3

W ) 3/2 E 1 ≤ ε −3 c G ε,W for every ε > 0, and lim sup

ε→0

ε −3 c G ε,W ≤ ℓ G,W E 1 ,

Proof. Set W 0 := inf R

3

W and write v ε (x) := v(εx). Then kv ε k 2 W

0

= ε −3 kvk 2 ε,W

0

and D(v ε ) = ε −5 D (v). If follows immediately from (5.3) that W 0 3/2 E 1 ≤ c G 1,W

0

= ε −3 c G ε,W

0

≤ ε −3 c G ε,W .

To prove the second inequality, take ξ ∈ R 3 such that (#Gξ)W 3/2 (ξ) = ℓ G,W E 1 . Write Gξ := {ξ 1 , ..., ξ m }. Fix 0 < ρ < 1 2 min i6=j |ξ i − ξ j | , and let W ρ := sup B(ξ

1

,ρ) W . Let υ ρ,ε := υ W

ρ

,ε be defined as in (4.2) with λ := W ρ . Set

w ρ,ε (x) :=

P m i=1

υ ρ,ε

 x − ξ i ε

 .

If √ ε ≤ ρ, then supp(w ρ,ε ) ⊂ ∪ m i=1 B(ξ i , ρ). Therefore w ρ,ε ∈ M G ε,W

ρ

and ε −3 c G ε,W ≤ ε −3 J ε,W (w ρ,ε ) ≤ ε −3 J ε,W ρ (w ρ,ε ) = mJ W

ρ

(υ ρ,ε ).

It follows from (4.3) that

lim sup

ε→0

ε −3 c G ε,W ≤ mW ρ 3/2 E 1 . Letting ρ → 0, we conclude that

lim sup

ε→0 ε −3 c G ε,W ≤ (#Gξ)W 3/2 (ξ)E 1 = ℓ G,W E 1 ,

as claimed. 

Lemma 5.2. Let ε n > 0 and ξ n ∈ R 3 such that ε n → 0 and (W (ξ n )) converges.

Set c W n (x) := W (ε n x + ξ n ) and c W := lim n→∞ W (ξ n ). Then, for every sequence (u n ) in H 1 (R 3 , R) such that u n ⇀ u weakly in H 1 (R 3 , R) and every w ∈ H 1 (R 3 , R), the following hold:

n→∞ lim

 hu n , w i 1,c W

n

− hu n − u, wi 1,c W

n

 = hu, wi 1,c W

and

n→∞ lim

 ku n k 2 1,c W

n

− ku n − uk 2 1,c W

n

 = kuk 2 1,c W .

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Proof. The argument is similar for both equalities. We prove the second one. Since (u n ) is bounded in L 2 (R 3 ) there exists C > 2 kuk L

2

(R

3

) such that

ku n k 2 1,c W

n

− ku n − uk 2 1,c W

n

− kuk 2 1,c W

≤ ku n k 2 1,c W − ku n − uk 2 1,c W − kuk 2 1,c W

+

Z

R

3

(c W n − c W )(2u n u − u 2 )

≤ o(1) + C (c W n − c W )u

L

2

(R

3

) . Given ε > 0 we fix R > 0 such that

Z

|x|≥R

(c W n − c W ) 2 u 2 ≤ (2 sup

x∈R

3

W ) 2 Z

|x|≥R

u 2 < ε 2 . Since W is uniformly continuous, there exists δ > 0 such that

|W (ε n x + ξ n ) − W (ξ n ) | < ε

C if |x| < δ ε n

.

Fix n 0 ∈ N such that |W (ξ n ) − c W | < C ε and ε δ

n

> R if n ≥ n 0 . Then, Z

|x|≤R

(c W n − c W ) 2 u 2 < ε 2 for all n ≥ n 0 . Therefore,

n→∞ lim

(c W n − c W )u

L

2

(R

3

) = 0.

This concludes the proof. 

Lemma 5.3. Let (z n ) be a sequence in R N . Then, after passing to a subsequence, there exist a closed subgroup Γ of G and a sequence (ζ n ) in R N such that

(a) (dist(Gz n , ζ n )) is bounded, (b) G ζ

n

= Γ,

(c) if |G/Γ| < ∞ then |gζ n − ˜gζ n | → ∞ for all g, ˜g ∈ G with ˜gg −1 ∈ Γ, /

(d) if |G/Γ| = ∞, there exists a closed subgroup Γ of G such that Γ ⊂ Γ , |G/Γ | =

∞ and |gζ n − ˜gζ n | → ∞ for all g, ˜g ∈ G with ˜gg −1 ∈ Γ / .

Proof. See Lemma 3.2 in [5]. 

Set

M G,W := n

y ∈ R 3 : (#Gy)W 3/2 (y) = ℓ G,W

o

Abusing notation we write again G i and M i for the groups and the sets defined as in Section 4 but now for W instead of V . So the value of W on M i is constant and we denote it by W i . We fix ρ > 0 such that b

(5.4) |y − gy| > 2b ρ if gy 6= y ∈ M G,W , dist(M i , M j ) > 2 ρ b if i 6= j,

For ρ ∈ (0, b ρ), let

M i ρ := {y ∈ R 3 : dist(y, M i ) ≤ ρ, G y = gG i g −1 for some g ∈ G}, and for each ξ ∈ M i ρ and ε > 0, define

θ ε,ξ (x) := P

gξ∈Gξ

ω i

 x − gξ ε



,

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where ω i is unique positive ground state of problem (4.1) with λ := W i which is radially symmetric with respect to the origin. Set

Θ ρ,ε := {θ ε,ξ : ξ ∈ M 1 ρ ∪ · · · ∪ M m ρ }.

The following holds.

Proposition 5.4. Let ε n > 0 and v n ∈ H 1 (R 3 , R) G be such that (5.5) ε n → 0, ε −3 n J ε

n

,W (v n ) → bc, ε −3 n k∇ ε

n

J ε

n

,W (v n ) k 2 ε

n

,W → 0,

where bc := lim inf ε→0 ε −3 c G ε,W andε

n

J ε

n

,W is the gradient of J ε

n

,W with respect to the scalar product h·, ·i ε

n

,W . Then, passing to a subsequence, there exist an i{1, ..., m} and a sequence (ξ n ) in R 3 such that

(i) G ξ

n

= G i , (ii) ξ n → ξ ∈ M i ,

(iii) ε −3 n k|v n | − θ ε

n

n

k 2 ε

n

,W → 0, (iv) bc = lim ε→0 ε −3 c G ε,W = ℓ G,W E 1 .

Proof. A standard argument shows that the sequence (ε −3 n kv n k 2 ε

n

,W ) is bounded and that

n→∞ lim ε −3 n kv n k 2 ε

n

,W = lim

n→∞ ε −5 n D (v n ) = 4 bc =: c > 0.

Let ev n ∈ H 1 (R 3 , R) G be given by ev n (z) := v n (ε n z). Then,

kev n k 2 1,W

n

= ε −3 kv n k 2 ε

n

,W and D ( ev n ) = ε −5 n D (v n ), where W n (z) := W (ε n z). Set

δ := lim sup

n→∞ sup

y∈R

3

Z

B(y,1) |ev n | 2 .

Since c > 0, Lions’ lemma [28, Lemma 1.21], together with inequality (3.4), yields that δ > 0. Choose z n ∈ R 3 such that

Z

B(z

n

,1) |ev n | 2 ≥ δ 2

and replace (z n ) by a sequence (ζ n ) having the properties stated in Lemma 5.3. Set bv n (z) := ev n (z + ζ n ). After passing to a subsequence, we may assume that bv n ⇀ bv weakly in H 1 (R 3 , R), bv n (x) → bv(x) a.e. on R 3 and bv n → bv in L 2 loc (R 3 , R). Choosing C ≥ dist(ζ n , Gz n ) for all n, we obtain

Z

B(0,C+1) |bv n | 2 = Z

B(ζ

n

,C+1) |ev n | 2 ≥ Z

B(z

n

,1) |ev n | 2 ≥ δ 2 . Therefore, bv 6= 0.

Set ξ n := ε n ζ n and c W n (x) := W (ε n x + ξ n ). Since W is bounded, a subsequence of W (ξ n ) converges. We set c W := lim n→∞ W (ξ n ). The weak continuity of D [2, Lemma 3.5], together with Lemma 5.2 and assumption (5.5) imply that bv is a so- lution to problem (4.1) with λ := c W .

Since v n and W are G-invariant we have that bv n (g −1 x) = v n (ε n x+gξ n ), c W n (g −1 x) = W (ε n x + gξ n ), and c W := lim n→∞ W (gξ n ) for each g ∈ G. Fix g 1 , ..., g k ∈ G such that |g i ζ n − g j ζ n | → ∞ if i 6= j. Then,

(5.6) bv n g j −1 − P k

i=j+1 bvg i −1 ( · − g i ζ n + g j ζ n ) ⇀ bvg −1 j

(13)

weakly in H 1 (R 3 , R). Applying Lemma 5.2 we obtain

bv n g j −1 − P k

i=j+1 bvg i −1 ( · − g i ζ n + g j ζ n )

2

1,c W

n

g

−1j

=

bv n g −1 j − bvg −1 j − P k

i=j+1 bvg i −1 ( · − g i ζ n + g j ζ n )

2

1,c W

n

g

−1j

+ bvg j −1

2

1,c W + o(1), and performing the change of variable y = ε n x + g j ξ n we conclude that

ε −3 n v n −

P k i=j+1 bvg i −1

 · − g i ξ n

ε n



2

ε

n

,W

= ε −3 n

v n − P k

i=j bvg i −1

 · − g i ξ n

ε n



2

ε

n

,W

+ kbvk 2 1,c W + o(1).

Iterating these equalities we conclude that

4 bc = lim n→∞ ε −3 n kv n k 2 ε

n

,W = lim

n→∞ ε −3 n v n −

X k i=1

bvg i −1

 · − g i ξ n

ε n



2

ε

n

,W

+ k kbvk 2 1,c W .

This implies that 4 bc ≥ k kbvk 2 1,c W which, together with property (d) in Lemma 5.3, implies |G/Γ| < ∞. Property (c) allows us to take k := |G/Γ|. Then, property (b) and Lemma 5.1 yield

ℓ G,W E 1 ≤ lim n→∞ (#Gξ n )W 3/2 (ξ n )E 1 = |G/Γ| c W 3/2 E 1

≤ |G/Γ| 1

4 kbvk 2 1,c W ≤ bc ≤ lim sup

ε→0 ε −3 c G ε,W ≤ ℓ G,W E 1 . This proves (iv) and gives also

(5.7) lim

n→∞ ε −3 n v n

P k i=1 bvg i −1

 · − g i ξ n

ε n



2

ε

n

,W

= 0.

Moreover, (#Gξ n )W 3/2 (ξ n ) ≤ ℓ G,W +α for n large enough. Thus, assumption (2.5) implies, after passing to a subsequence, that ξ n → ξ. Hence, W (ξ) = c W and

ℓ G,W E 1 ≤ (#Gξ)W (ξ)E 1 ≤ |G/Γ| c W 3/2 E 1 ≤ |G/Γ| 1

4 kbvk 2 1,c W ≤ ℓ G,W E 1 . We conclude that ξ ∈ M i for some i = 1, ..., m, as claimed in (ii). Then, c W = W i , Γ = G ξ = gG i g −1 for some g ∈ G, and bv is a ground state of problem (4.1) with λ = W i .

Since the ground state is unique up to sign and translation we must have that bv(z) = ±ω i (z − z 0 ) for some z 0 ∈ R 3 . Observe that bv is Γ-invariant. So, if Γ is nontrivial, then z 0 = 0 and, since ω i is radial, equation (5.7) becomes (iii). If, on the other hand, Γ is the trivial group, we replace ξ n by ξ n := ξ n + ε n z 0 . Since Gξ n ∼ = G and ε n → 0, ξ n has the same properties as ξ n for n large enough. Moreover, since ω i is radially symmetric,

bv

 g −1 z − ξ n ε n



= ±ω i

 g −1 z − ξ n

ε n



= ±ω i

 z − gξ n

ε n



(14)

and, again, equation (5.7) yields (iii). This completes the proof.  Proposition 5.5. Given ρ ∈ (0, b ρ) there exist d ρ > ℓ G,W E 1 and ε ρ > 0 with the following property: For every ε ∈ (0, ε ρ ) and every v ∈ M G ε,W with J ε,W (v) ≤ ε 3 d ρ

there exists precisely one G-orbit Gξ ε,v with ξ ε,v ∈ M 1 ρ ∪ · · · ∪ M m ρ such that ε −3

|v| − θ ε,ξ

ε,v

2

ε,W = min

θ∈Θ

ρ,ε

k|v| − θk 2 ε,W .

Proof. The proof is analogous to that of Proposition 5.3 in [4]. We omit the details.

 Fix ρ ∈ (0, b ρ) and ε ∈ (0, ε ρ ). Proposition 5.5 allows us to define a map

(5.8) β b ρ,ε,0 : M G ε,W ∩ J ε

3

d

ρ

ε,W −→ (M 1 ρ ∪ · · · ∪ M m ρ ) /G by taking

β b ρ,ε,0 (v) := Gξ ε,v .

Here, as usual, J ε,W c := {v ∈ H 1 (R 3 , R) : J ε,W (v) ≤ c}. The map b β ρ,ε,0 is the G- equivariant analogon to the usual baricenter map. It is only defined for functions in M G ε,W with small enough energy. We call it the local baryorbit map. It reflects the fact that such functions concentrate at a unique G-orbit with minimal cardinality as ε → 0.

6. Proofs of the main results Let V ∞ := lim inf |x|→∞ V (x). Assumption (2.5) implies that

ℓ G,V < min

x∈R

3

\{0} (#Gx)V 3/2 . We fix δ 0 > 0 and λ ∈ (0, V ∞ ) such that

(6.1) ℓ G,V E 1 + δ 0 < min

x∈R

3

\{0} (#Gx)λ 3/2 E 1 < min

x∈R

3

\{0} (#Gx)V 3/2 E 1 ,

and define W (x) := min {V (x), λ}. This W has all properties stated in section 5, in particular, it is uniformly continuous. Moreover, ℓ G,W = ℓ G,V and M G,W = M G,V . Let π ε,W : H 1 (R 3 , R) G \ {0} → M G ε,W denote the radial projection onto the Nehari manifold, which is given by

(6.2) π ε,W (v) := ε kuk ε,W

p D (u) v.

Observe that

(6.3) J ε,W (π ε,W (v)) = ε 2 kvk 4 ε,W

4D(v) for all v ∈ H 1 (R 3 , R) G \ {0}.

Let bι ε be the map defined in Proposition 4.3 and b β ρ,ε,0 be as in (5.8). Then, for d ρ > ℓ G,V E 1 and ε ρ > 0 as in Proposition 5.5 the following holds.

Proposition 6.1. For each ρ ∈ (0, b ρ) and ε ∈ (0, ε ρ ), the map β b ρ,ε : N ε,A,V τ ∩ J ε

3

d

ρ

ε,A,V → (M 1 ρ ∪ · · · ∪ M m ρ ) /G, β b ρ,ε (u) := b β ρ,ε,0 (π ε,W ( |u|)), is well defined and continuous and satisfies

(i) b β ρ,ε (γu) = b β ρ,ε (u) for all γ ∈ S 1 ,

(ii) b β ρ,ε ( bι ε (ξ)) = Gξ for all ξ ∈ M τ with J ε,A,V ( bι ε (ξ)) ≤ ε 3 d ρ .

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Proof. If u ∈ N ε,A,V τ then |u| ∈ H 1 (R 3 , R) G \ {0} and, since W ≤ V, formulas (6.3) and (3.6), together with the diamagnetic inequality (3.2) yield

(6.4) J ε,W (π ε,W ( |u|)) ≤ J ε,V (π ε,V ( |u|)) ≤ J ε,A,V (u).

So J ε,W (π ε,W ( |u|)) ≤ ε 3 d ρ if J ε,A,V (u) ≤ ε 3 d ρ . Therefore, b β ρ,ε is well defined. It is straightforward to verify that it has the desired properties. 

Let

M τ ρ := S

G

i

⊂ker τ

M i ρ .

Propositions 4.3 and 6.1 allow us to estimate the Lusternik-Schnirelmann category of low energy sublevel sets as follows.

Corollary 6.2. For every ρ ∈ (0, b ρ) and d ∈ (ℓ G,V E 1 , d ρ ) there exists ε ρ,d > 0 such that

cat M

τρ

/G M τ /G ≤ cat 

( N ε,A,V τ ∩ J ε,A,V ε

3

d )/S 1  for every ε ∈ (0, ε ρ,d ).

Proof. Set ε ρ,d := min {ε d , ε ρ } where ε d is as in Proposition 4.3. Fix ε ∈ (0, ε ρ,d ).

Then,

J ε,A,V ( bι ε (ξ)) ≤ ε 3 d and b β ρ,ε ( bι ε (ξ)) = ξ for all ξ ∈ M τ . Since M 1 ρ , ..., M m ρ are G-invariant and pairwise disjoint, the set

C := {u ∈ N ε,A,V τ ∩ J ε,A,V ε

3

d : b β ρ,ε (u) ∈ M τ ρ /G } is a union of connected components of N ε,A,V τ ∩ J ε,A,V ε

3

d . Therefore,

cat C/S 1 

≤ cat 

( N ε,A,V τ ∩ J ε,A,V ε

3

d )/S 1  . By Propositions 4.3 and 6.1, the maps

M τ /G −→ C/S ι

ε

1 β −→ M

ρ,ε

τ ρ /G,

given by ι ε (Gξ) := bι ε (ξ) and β ρ,ε (S 1 u) := b β ρ,ε (u), are well defined and satisfy β ρ,ε (ι ε (ξ)) = ξ for all ξ ∈ M τ . Therefore,

cat M

τρ

/G M τ /G ≤ cat C/S 1  .

This finishes the proof. 

Another consequence of our previous results is the following.

Corollary 6.3. If there exists ξ ∈ R 3 such that (#Gξ)V 3/2 (ξ) = ℓ G,V and G ξ ⊂ ker τ , then

ε→∞ lim ε −3 c τ ε,A,V = ℓ G,V E 1 , where c τ ε,A,V := inf N

ε,A,Vτ

J ε,A,V .

Proof. Inequality (6.4) yields c G ε,W := inf M

Gε,W

J ε,W ≤ inf N

ε,A,Vτ

J ε,A,V =: c τ ε,A,V . By Proposition 5.4 and Lemma 4.2(c),

ℓ G,W E 1 = lim

ε→∞ ε −3 c G ε,W ≤ lim inf ε→0 ε −3 c τ ε,A,V ≤ lim sup

ε→∞

ε −3 c τ ε,A,V ≤ ℓ G,V E 1 .

Since ℓ G,W E 1 = ℓ G,V E 1 , our claim follows. 

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Proof of Theorem 2.1. Let ρ, δ > 0 be given. We may assume that ρ ∈ (0, b ρ) with ρ as in (5.4) and that δ b ∈ (0, δ 0 ) with δ 0 as in (6.1). By Corollary 6.3 there exists ε δ > 0 such that

ℓ G,V E 1 − δ < ε −3 c τ ε,A,V for all ε ∈ (0, ε δ ).

Fix d ∈ (ℓ G,V E 1 , min {d ρ , ℓ G,V E 1 + δ }) and set b ε := min {ε δ , ε ρ,d } with ε ρ,d as in Corollary 6.2. Since (6.1) holds, Proposition 3.1 asserts that J ε,A,V : N ε,A,V τ → R satisfies (P S) c for every c ≤ ε 3 d. Applying Proposition 3.2 and Corollary 6.2 we conclude that J ε,A,V has at least

cat M

τρ

/G M τ /G geometrically distinct solutions u ∈ N ε,A,V τ satisfying

ε 3 ℓ G,V E 1 − ε 3 δ < J ε,A,V (u) = 1

2 D (u) ≤ ε 3 d < ε 3 ℓ G,V E 1 + ε 3 δ,

for each ε ∈ (0, b ε), as claimed. 

Proof of Theorem 2.2. After passing to a subsequence, we may assume that ℓ G,V E 1 − 2n 1 ≤ ε −3 n c G ε

n

,W and δ n ≤ 2n 1 . Then, inequality (6.4) yields

(6.5) c G ε

n

,W ≤ J ε

n

,W (π ε

n

,W ( |u n |)) ≤ J ε

n

,A,V (u n ) ≤ ε 3 n (ℓ G E 1 +δ n ) ≤ c G ε

n

,W +ε 3 n /n.

By Ekeland’s variational principle [28, Theorem 8.5] we may choose v n ∈ M G ε

n

,W

such that

(6.6) ε −3 nε

n

,W ( |u n |) − v n k 2 ε

n

,W → 0,

ε −3 n J ε

n

,W (v n ) → ℓ G E 1 and ε −3 n k∇ ε

n

J ε

n

,W (v n ) k 2 ε

n

,W → 0.

After passing again to a subsequence, Proposition 5.4 gives a sequence (ξ n ) in R 3 such that ξ n → ξ ∈ M τ , G ξ

n

= G ξ , and

(6.7) ε −3 n

v n − P

n

∈Gξ

n

ω ξ

 · − gξ n ε n



2

ε

n

,W

→ 0.

Since u n ∈ N ε τ

n

,A,V , multiplying inequality (6.5) by 4ε 2 n [D( |u n |)] −1 = 4ε 2 n [D(u n )] −1 = [J ε

n

,A,V (u n )] −1 , using (6.3) and (3.6), and observing that ε −3 n J ε

n

,A,V (u n ) → ℓ G E 1 , we get

1 − ε n k |u n | k ε

n

,W p D ( |u n |)

! 4 =

ε 2 n ku n k 2 ε

n

,A,V

D (u n )

! 2

− ε 2 n k |u n | k 2 ε

n

,W

D ( |u n |)

! 2

≤ ε −3 n J ε

n

,A,V (u n )  −1 1 n → 0.

Recalling (6.2) and using the diamagnetic inequality (3.2), we then obtain

ε −3 n k |u n | − π ε

n

,W ( |u n |) k 2 ε

n

,W =

1 − ε n k |u n | k ε

n

,W

p D ( |u n |)

2

ε −3 n k |u n | k 2 ε

n

,W

1 − ε n k |u n | k ε

n

,W

p D ( |u n |)

2

−3 n J ε

n

,A,V (u n ) → 0.

(6.8)

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Finally, combining (6.6), (6.7), and (6.8) we conclude that

ε −3 n

|u n | − P

n

∈Gξ

n

ω ξ

 · − gξ n ε n



2

ε

n

,W

→ 0.

Since kvk 2 ε ≤ C kvk 2 ε,W for some constant independent of ε, our claim follows.  References

[1] L. Abatangelo, S. Terracini, Solutions to nonlinear Schr¨ odinger equations with singular elec- tromagnetic potential and critical exponent, preprint 2010.

[2] N. Ackermann, On a periodic Schr¨ odinger equation with nonlocar superlinear part, Math. Z.

248 (2004), 423-443.

[3] S. Cingolani, M. Clapp, Intertwining semiclassical bound states to a nonlinear magnetic equation, Nonlinearity 22 (2009), 2309-2331.

[4] S. Cingolani, M. Clapp, Symmetric semiclassical states to a magnetic nonlinear Schr¨ odinger equation via Equivariant Morse Theory, Commun. Pure Appl. Anal. 9 (2010), 1263–1281.

[5] S. Cingolani, M. Clapp, S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys. at press.

[6] S. Cingolani, S. Secchi, M. Squassina, Semiclassical limit for Schr¨ odinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh, 140 A (2010), 973–1009.

[7] M. Clapp, D. Puppe, Critical point theory with symmetries, J. reine angew. Math. 418 (1991), 1-29.

[8] M. Clapp, A. Szulkin, Multiple solutions to a nonlinear Schr¨ odinger equation with Aharonov- Bohm magnetic potential, Nonlinear Differ. Equ. Appl. (NoDea) 17 (2010), 229-248.

[9] T. tom Dieck, ”Transformation groups”, Walter de Gruyter, Berlin-New York 1987.

[10] J. Fr¨ ohlich, E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, in S´ eminaire: ´ Equations aux D´ eriv´ ees Partielles 2003–2004, Exp. No. XIX, 26 pp., S´ emin. ´ Equ. D´ eriv. Partielles, ´ Ecole Polytech., Palaiseau, 2004.

[11] J. Fr¨ ohlich, T.-P. Tsai, H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys. 225 (2002), 223–274.

[12] R. Harrison, I. Moroz, K.P. Tod, A numerical study of the Schr¨ odinger-Newton equations, Nonlinearity 16 (2003), 101–122.

[13] E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93-105.

[14] E.H. Lieb, M. Loss, ”Analysis”, Graduate Studies in Math. 14, Amer. Math. Soc. 1997.

[15] P.-L. Lions, The concentration-compacteness principle in the calculus of variations. The locally compact case, Ann. Inst. Henry Poincar´ e, Analyse Non Lin´ eaire 1 (1984), 109-145 and 223-283.

[16] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal. T.M.A. 4 (1980), 1063–1073.

[17] L. Ma, L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equa- tion, Arch. Rational Mech. Anal. 195 (2010), 455-467.

[18] I.M. Moroz, R. Penrose, P. Tod, Spherically-symmetric solutions of the Schr¨ odinger-Newton equations, Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity 15 (1998), 2733–2742.

[19] I.M. Moroz, P. Tod, An analytical approach to the Schr¨ odinger-Newton equations, Nonlin- earity 12 (1999), 201–216.

[20] M. Nolasco, Breathing modes for the Schr¨ odinger–Poisson system with a multiple–well ex- ternal potential, Commun. Pure Appl. Anal. 9, No. 5 (2010), 1411–1419.

[21] R. Palais, The principle of symmetric criticallity, Comm. Math. Phys. 69 (1979), 19-30.

[22] R. Penrose, On gravity’s role in quantum state reduction, Gen. Rel. Grav. 28 (1996), 581–600.

[23] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos.

Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), 1927–1939.

[24] R. Penrose, ”The road to reality. A complete guide to the laws of the universe”, Alfred A.

Knopf Inc., New York (2005).

(18)

[25] S. Secchi, A note on Schr¨ odinger–Newton systems with decaying electric potential, Nonlinear Analysis 72 (2010), 3842–3856.

[26] P. Tod, The ground state energy of the Schr¨ odinger-Newton equation, Physics Letters A 280 (2001), 173–176.

[27] J. Wei, M. Winter, Strongly interacting bumps for the Schr¨ odinger–Newton equation, J.

Math. Phys. 50 (2009), 012905.

[28] M. Willem, ”Minimax theorems”, PNLDE 24, Birkh¨ auser, Boston-Basel-Berlin 1996.

Dipartimento di Matematica, Politecnico di Bari, via Orabona 4, 70125 Bari, Italy.

E-mail address: s.cingolani@poliba.it

Instituto de Matem´ aticas, Universidad Nacional Aut´ onoma de M´ exico, Circuito Ex- terior, C.U., 04510 M´ exico D.F., Mexico.

E-mail address: mclapp@matem.unam.mx

Dipartimento di Matematica ed Applicazioni, Universit` a di Milano-Bicocca, via Cozzi 53, 20125 Milano, Italy.

E-mail address: Simone.Secchi@unimib.it

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