with positive potential

with positive potential

Dimitri Mugnai ^∗

Dipartimento di Matematica e Informatica Universit` a di Perugia

Via Vanvitelli 1, 06123 Perugia - Italy tel. +39 075 5855043, fax. +39 075 5855024,

e-mail: [email protected]

Abstract

We study the existence of radially symmetric solitary waves for a non- linear Schr¨ odinger-Poisson system. In contrast to all previous results, we consider the presence of a positive potential, of interest in physical appli- cations.

Keywords: Schr¨ odinger-Poisson system, positive potential, Lagrange multi- plier, nontrivial solutions

2000AMS Subject Classification: 35J50, 81T13

1 Introduction, motivations and main result

In recent years great attention has been paid to some classes of systems of partial differential equations in which a Schr¨ odinger equation is coupled to the Maxwell ones; the purpose is to describe the interaction of a nonlinear Schr¨odinger field with an electromagnetic field E − H. The gauge potentials A − Φ,

A : R ³ × R → R ³ , Φ : R ³ × R → R are related to E − H by the Maxwell equations

E = −



∇Φ + ∂A

∂t



, H = ∇ × A.

If we are interested in finding standing waves (solutions of a field equation whose energy travels as a localized packet preserving this localization in time) and we consider the electrostactic case (when A = 0), the Schr¨ odinger field is

∗

Research supported by the M.I.U.R. project Metodi Variazionali ed Equazioni Differen- ziali Nonlineari.

1

described by a real function u : R ³ → R, which represents the matter (see [4] or [8]). In this way we are led to the following stationary system of Schr¨ odinger–

Maxwell, or Schr¨ odinger–Poisson–Slater, type:

−∆u + ωu − λuΦ + W u (x, u) = 0 in R ³ , (1.1)

−∆Φ = u ² in R ³ , (1.2)

where ω, λ ∈ R and W : R ³ × R → R is a given potential. In particular, the second equation represents the repulsive character of the Coulomb force (the attractive case is described by the equation ∆Φ = u ² , see [26]).

Such a system was studied in [4] with W ≡ 0, and the existence of a sequence (ω n , u n , Φ n ) n of solutions was proved when the problem is settled in a bounded domain of R ³ . After that, in [8] the potential

W (x, s) = W (s) = − |s| ^γ

γ , γ > 2, (1.3)

was introduced and existence results in the whole physical space were proved for γ ∈ [4, 6), while non existence results for γ ∈ (0, 2] or γ ∈ [6, ∞) were proved in [7], where also more general potentials, behaving similarly to the one in (1.3), were considered. In all these papers a crucial assumption is that λ < 0 and ω > 0.

Since then a large number of generalizations and improvements have been produced; in particular a fundamental advance in the understanding of system (1.1)–(1.2) with W as in (1.3) was done in [2], [24] and [12]. But also more general problems, where the first equation is replaced by

−∆u + V (x)u + uΦ = K(x)|u| ^γ−2 u, (1.4) for some V, K : R ³ → R, were studied under different assumptions on V and K ([3], [19]); moreover, also some generalizations of the last cases, with K(x)|u| ^γ−2 u replaced by a more generic function f (x, u), were considered (for example in [30] and [31]).

In [27] a solution when W is as in (1.3) is found if γ = 8/3 and λ = −1;

however, it is not known if that solution is radial. The solution is found via a minimization argument and ω appears as a Lagrange multiplier.

Let us remark that the prototype potential defined in (1.3), as well as all the introduced generalizations, is not always positive, while for physical reasons a potential suitable to model physical phenomena should be nonnegative: indeed, the fact that W is nonnegative implies that the energy density of a solution (u, Φ) of system (1.1)–(1.2) is nonnegative as well, as one can see from the expression of the Lagrangian in equation (2.9) below.

Moreover, if we consider the autonomous electrostatic case, i.e. −∆u + W ^′ (u) = 0, calling “rest mass” of the particle u the quantity

Z

R

W (u) dx

(see [5]), the fact that W is positive implies that we are dealing with systems of particles having positive mass, which is, of course, of interest in physical applications.

Therefore, in this paper we are interested in system (1.1)–(1.2) with the main hypothesis that W is assumed nonnegative.

Before passing to the precise assumptions and the statement of our results, we recall that systems of this type were studied also in bounded domains both under Dirichlet ([25]) or Neumann boundary conditions ([23]), and that also similar systems ([9], [20], [21]) were treated with variational methods (as we are going to do), mainly by a suitable minimization process of the associated energy ([10], [13]), or by critical point theory (see [15] when W = 0 and ω > 0).

In particular we recall that in all the papers cited before, in equation (1.1) the authors assume λ < 0, while for W ≡ 0 the case λ > 0 is treated in [13] and [14], where actually (1.1) is replaced by

−∆u + ωu − 2u ² Φ = 0. (1.5)

Such an equation, also called Choquard equation, was introduced as an ap- proximation to the Hartree–Foch theory for a one component plasma, and in [13] the author proves the existence and uniqueness of a minimizing solution for the energy functional associated to a Schr¨odinger-Poisson problem, showing that such a minimizing solution satisfies a stationary equation of the form (1.5).

Moreover, it is proved that such a solution is radially decreasing by Schwarz symmetrization methods. See also the recent paper [18], where some powers are introduced in the nonlocal part.

Let us also note that equation (1.1) generalizes the equation

−∆u + ωu − u ² Φ + 5

3 u ^7/3 = 0, u > 0,

with ω > 0, introduced in [6] to describe a Hartree model for crystals. This equation seems to be the first example covered by our case, but in [6] the authors were interested in periodic periodic structures and did not look for solutions with finite global energy, as we are going to do.

Finally, we quote [11], where the author studies the existence and orbital stability of standing wave solutions for system (1.1)–(1.2) when W is given in (1.3).

In this paper we are concerned with system (1.1)–(1.2). In all the papers just cited above in which λ > 0, the potential W always has the form of a pure power W (s) = |s| ^γ /γ for some γ > 2; hence, in order to include the previous cases, in this paper we make the following assumptions:

W 1 ) W : R ³ × R → [0, ∞) is such that the derivative W u : R ³ × R → R is a Carath´eodory function, W (x, s) = W (|x|, s) for a.e. x ∈ R ³ and for every s ∈ R, and W (x, 0) = W u (x, 0) = 0 for a.e. x ∈ R ³ ;

W 2 ) ∃ C 1 , C 2 > 0 and 1 < q < p < 5 such that |W u (x, s)| ≤ C 1 |s| ^q + C 2 |s| ^p for

every s ∈ R and a.e. x ∈ R ³ ;

W 3 ) ∃ k ≥ 2 such that 0 ≤ sW u (x, s) ≤ kW (x, s) for every s ∈ R and a.e.

x ∈ R ³ .

Remark 1.1. As it will be clear from the proof, the requirement that W de- pends radially on the space variable is a technical assumption which lets us reduce the problem to a radial setting. Of course, if W is independent of x, the pure power case is included in our setting and the requirement is obviously satisfied.

From W 1 ) we get in particular that s = 0 is an absolute minimum point for W for a.e. x ∈ R ³ , and that the system always admits the trivial solution u = Φ = 0. Moreover, we remark that, as in case of (1.3), by W 2 ) the potential W is more than quadratic both at 0 and infinity. Let us also note that by direct integration of W 3 ) we obtain that there exist a, b : R ³ → [0, ∞] such that

W (x, s) ≤ a(x)|s| ^k + b(x) for every s ∈ R and a.e. x ∈ R ³ ,

so that W may have superquadratic growth at infinity, which is in agreement with W 2 ). On the other hand, in W 3 ) we exclude the case k < 2, since in this latter case we would obtain that W (x, s)/s ^k is decreasing if s > 0, so that W (x, s) ≤ W (x, ε)

ε ^k s ^k for s > ε > 0, and passing to the limit as ε → 0 ⁺ , by W 1 ) we would get W ≤ 0, so that W ≡ 0, as it was already considered in the literature.

Remark 1.2. Note that our assumptions cover the case W ≡ 0, but even the more intriguing case in which W (x, s) = 0 for x belonging to a proper subset Ω of R ³ . An interesting case is when Ω is a ball, so that the potential W is active only in an exterior domain, or when Ω is an annulus. All these situations have not been considered up to now, since also in the generalizations mentioned above the coefficients V and K in (1.4) were assumed to be strictly positive in R ³ . To our best knowledge, this paper is the first one to cover the case of potentials possibly vanishing somewhere.

As usual, for physical reasons, we look for solutions that have finite energy, i.e. (u, Φ) ∈ H ¹ × D ¹ , where H ¹ = H ¹ (R ³ ) is the usual Sobolev space endowed with the scalar product

hu, vi H

:=

Z

R

Du · Dv + uv
dx

and norm kuk = (R |Du| ² + R u ² ) ^1/2 , and D ¹ = D ¹ (R ³ ) is the completion of C ₀ ^∞ (R ³ ) with respect to the norm

kuk ² _D

:=

Z

R

|Du| ² dx, induced by the scalar product hu, vi D

:= R Du · Dv dx.

Before stating our main result, let us note that if u = 0 in (1.1), then Φ = 0

in (1.2), and if u 6= 0, then also Φ 6= 0. Therefore, we can say that a solution

(u, Φ) of system (1.1)–(1.2) is nontrivial if both u and Φ are different from the trivial function, and a sufficient condition for this occurrence is that u 6= 0.

Our first result is the following easy one.

Proposition 1.1. Assume that W satisfies W 1 ) − W 3 ). Then for any ω > 0 there exist λ > 0 and nontrivial radial functions (u, Φ) ∈ H ¹ (R ³ ) × D ¹ (R ³ ) which solve the related system (1.1)–(1.2).

Analogously to the previous literature, in which ω was found as a Lagrange multiplier, got by a minimization process on a suitable manifold, here we find the value of λ in this way. However, in our case the manifold is different from the usual one used in the papers cited above, which adopted similar procedures considering the unit sphere in L ² (R ³ ). Indeed, we introduce a new manifold which in the end turns out to be a good choice. On the other hand, our manifold is defined as R u ² Φ =constant, so that no rearrangement approach can de done to prove radial decreasing properties of the solutions, as done in [13] and [26], where the equality R |u| ^p = R (u ^⋆ ) ^p is fundamental. In fact, one could proceed as follows:

1. prove that u, Φ ≥ 0;

2. substitute u by its radial decreasing rearrangement u ^⋆ (i.e. its Schwartz symmetrization);

3. prove that Φ is radially decreasing, so that it coincides with its decreasing rearrangement Φ ^⋆ .

Actually the last step is true for any radial function u, since Φ is radial as well (by (ii) of Proposition 2.2 below) and superharmonic. However, in this way the integral R uΦ increases by the classical Hardy–Littlewood inequality

Z

R

uΦ dx ≤ Z

R

u ^⋆ Φ ^⋆ dx,

so that a symmetrization process pushes out of the manifold under consideration.

Let us note that under assumption W 3 ) above, system (1.1)–(1.2) admits only the trivial solution when λ ≤ 0, independently of its possible symmetry:

Proposition 1.2. Assume W 3 ), and let (u, Φ) ∈ H ¹ (R ³ )×D ¹ (R ³ ) be a solution of system (1.1)–(1.2) with λ ≤ 0. Then u = Φ = 0

Proof. Multiplying (1.1) by u and integrating by parts, W 3 ) gives 0 =

Z

R

|Du| ² dx + ω Z

R

u ² dx − λ Z

R

u ² Φ dx + Z

R

W u (x, u)u dx

≥ Z

R

|Du| ² dx + ω Z

R

u ² dx;

since ω > 0. This implies u ≡ 0, as claimed. By (1.2) we immediately get

Φ = 0.

Thus, if W 3 ) holds, it is natural to look for solutions when λ > 0, as stated in Theorem 1.1.

On the other hand, the importance of radial functions, and in particular of positive ones, has been widely used in open quantum problems (see [26]), and in addition they provide a more friendly functional setting from a mathematical point of view.

On the other hand, from a physical viewpoint, positive solutions are the most natural ones, and in fact we can prove

Proposition 1.3. Under the assumptions of Proposition 1.1, the solution u of (1.1) is strictly positive in R ³ .

We remark that the fact that Φ is positive is a free result, since by direct integration of (1.2) we get

Φ(x) = Z

R

1

4π|x − y| u ² (y)dy, (1.6)

so that Φ > 0, as soon as u 6= 0.

Variational techniques can improve the existence result above when W sat- isfies some symmetry conditions. In this way we get the two main results of this paper, collected in the theorems below.

Theorem 1.1. Under the assumptions of Proposition 1.1, let us also assume that W (x, s) = W (x, −s) for a.e. x ∈ R ³ and every s ∈ R. Then for every ω > 0 there exist infinitely many triples (λ, u, Φ) ∈ R ⁺ × H ¹ (R ³ ) × D ¹ (R ³ ) which solve the related system (1.1)–(1.2).

The main fact in the previous results is that λ is not given a priori, but is found as a Lagrange multiplier. However, if we let p vary in a smaller set, then λ can be fixed from the very beginning, thus improving the result of Proposition 1.1 in the following way:

Theorem 1.2. Assume that W satisfies W 1 ), W 2 ) with p ∈ (1, 3) and W 3 ) with k ≤ 4. Then for any λ, ω > 0 there exist nontrivial radial functions (u, Φ) ∈ H ¹ (R ³ ) × D ¹ (R ³ ) which solve (1.1)–(1.2).

The fact that in Theorem 1.2 p is not too large lets us employ classical variational techniques, such as the Mountain Pass Theorem, which could not be applied to prove Theorem 1.1, since the geometric conditions of the latter theorem fail if p ≥ 3.

Open problems and final comments: In the case of the model potential

(1.3), it is still an open problem to prove that the Palais–Smale condition holds

if γ ∈ (2, 3), while the geometrical structure of a mountain pass was proved

to hold in [8]. Here, on the contrary, it is possible to prove the Palais–Smale

condition and the right mountain pass geometry if the growth at infinity of the

potential is not too fast.

2 Preliminary setting

First we recall the standard notation L ^p ≡ L ^p (R ³ ) for the usual Lebesgue space endowed with the norm

kuk ^p _p :=

Z

R

|u| ^p dx, 1 ≤ p < ∞.

We also recall the continuous embeddings

H ¹ (R ³ ) ֒→ L ^p (R ³ ) ∀ p ∈ [2, 6] and D ¹ (R ³ ) ֒→ L ⁶ (R ³ ), (2.7) being 6 the critical exponent for the Sobolev embedding H ¹ (R ³ ) ֒→ L ^p (R ³ ).

Now, for any λ > 0 let us consider the functional F : H ¹ × D ¹ → R defined as

F ω (u, Φ) = J(u) − λA(u, Φ), (2.8) where

J(u) = 1 2

Z

R

|Du| ² dx + ω 2

Z

R

u ² dx + Z

R

W (x, u) dx, (2.9) and

A(u, Φ) = − 1 4

Z

R

|DΦ| ² dx + 1 2

Z

R

u ² Φ dx.

It is easily seen that under the growth assumptions on W we have the fol- lowing

Proposition 2.1. The functional F ω belongs to C ¹ (H ¹ × D ¹ ) and its critical points solve (1.1) − (1.2).

Moreover, a by–now standard approach shows the following result, which appeared for the first time in [8].

Proposition 2.2. For every u ∈ H ¹ (R ³ ), there exists a unique Φ = Φ[u] ∈ D ¹ (R ³ ) which solves (1.2). Furthermore

(i) Φ[u] ≥ 0;

(ii) if u is radially symmetric, then Φ[u] is radial, too;

(iii) for any t ∈ R and any u ∈ H ¹ (R ³ ) there holds Φ[tu] = t ² Φ[u].

As a consequence of Proposition 2.2, we can define the map Φ : H ¹ (R ³ ) −→ D ¹ (R ³ )

which maps each u ∈ H ¹ (R ³ ) in the unique solution of (1.2). By the Implicit Function Theorem, Φ ∈ C ¹ (H ¹ , D ¹ ) and from the very definition of Φ we get

∂F ω

∂Φ (u, Φ[u]) = 0 ∀ u ∈ H ¹ , or equivalently ∂A

∂Φ (u, Φ[u]) = 0. (2.10)

We have the following result, which will be used later in the stronger context

of radial functions.

Lemma 2.1. The functional Ξ : H ¹ (R ³ ) → R defined as Ξ(u) := A(u, Φ[u]) = − 1

4 Z

R

|DΦ[u]| ² dx + 1 2 Z

R

u ² Φ[u] dx is of class C ¹ and

Ξ ^′ (u)(v) = Z

R

uΦ[u]v dx for any u, v ∈ H ¹ (R ³ ).

Proof. The regularity of all characters appearing in its definition, imply that Ξ is of class C ¹ , as claimed.

Moreover, Ξ(u) = A(u, Φ[u]), so that Ξ ^′ (u) = ∂A

∂u (u, Φ[u]) + ∂A

∂Φ (u, Φ[u])Φ ^′ [u] = ∂A

∂u (u, Φ[u]) by (2.10), and thus for any u, v ∈ H ¹ (R ³ ) there holds

Ξ ^′ (u)(v) = Z

R

uΦ[u]v dx, as claimed.

Now let us consider the functional I : H ¹ (R ³ ) −→ R, given by I(u) :=

F ω (u, Φ[u]). By definition of F ω , we obtain I(u) = 1

2 Z

R

|Du| ² dx + ω 2

Z

R

u ² dx + Z

R

W (x, u) dx + λ

4 Z

R

|DΦ[u]| ² dx − λ 2 Z

R

u ² Φ[u] dx.

(2.11)

By Proposition 2.1 and Lemma 2.1, I ∈ C ¹ (H ¹ , R) and by (2.10) we get I ^′ (u) = ∂F ω

∂u (u, Φ[u]) − λ ∂A

∂Φ (u, Φ[u]Φ ^′ [u] = ∂F ω

∂u (u, Φ[u]) ∀ u ∈ H ¹ (R ³ ).

Standard calculations (for example, see [8] or [24]) give the following result.

Lemma 2.2. The following statements are equivalent:

i) (u, Φ) ∈ H ¹ × D ¹ is a critical point of F , ii) u is a critical point of I and Φ = Φ[u].

Then, in order to get solutions of (1.1)–(1.2), we could look for critical points

of I. It is readily seen that the functional I is strongly indefinite, in the sense

that it is unbounded both from above and below. Moreover, a more delicate

problem in getting the existence of critical points is the fact that I presents a

lack of compactness due to its invariance under the translation group, given by

the set of transformations having the form u(x) = u(x + x 0 ) for any x 0 ∈ R ³ .

In order to avoid the latter problem, it is standard to consider only the set of radial functions, so that we take

H _r ¹ = H _r ¹ (R ³ ) := u ∈ H ¹ (R ³ ) : u(x) = u(|x|) ,

and the functional I |H

: H _r ¹ (R ³ ) → R. In this way, if u ∈ H _r ¹ (R ³ ), then also Φ[u] is radial by (ii) of Proposition 2.2, and we can write

Φ[u] ∈ D _r ¹ = D ¹ _r (R ³ ) := φ ∈ D ¹ (R ³ ) : φ(x) = φ(|x|) .

Such a choice is justified by the fundamental fact that H _r ¹ (R ³ ) is compactly embedded in the Lebesgue space of radial functions L ^s _r (R ³ ) for any s ∈ (2, 6) by Strauss’ Lemma (see [28]).

Remark 2.1. The introduction of I |H

is very useful also in an abstract way, since H _r ¹ is a natural constraint for I, in the sense that a function u ∈ H _r ¹ is a critical point for I |H

if and only if u is a critical point for I, by the principle of symmetric criticality of Palais (see [22]).

Thus, from now on, we could look for critical points of I |H

in H _r ¹ (R ³ ).

But also this strategy seems hard in general and will be applied, in fact, to prove Theorem 1.2 with the restrictions on the parameters required there. In the general case, looking for nontrivial triples (u, Φ[u], ω) which solve system (1.1)–(1.2), we proceed as follows: fix σ > 0 and introduce the set

M _σ :=



u ∈ H _r ¹ (R ³ ) : Ξ(u) = − 1 4 Z

R

|DΦ[u]| ² dx + 1 2 Z

R

u ² Φ[u] dx = σ

 . (2.12) For further use, we note that by (1.2) we get

Z

R

|DΦ[u]| ² dx = Z

R

u ² Φ[u] dx, (2.13)

so that an easy computation gives Ξ(u) = A(u, Φ[u]) = 1

4 Z

R

u ² Φ[u] dx = 1 4

Z

R

|DΦ[u]| ² dx, (2.14) which guarantees that in the definition of M σ the quantity Ξ(u), equal to σ, must be, in effect, nonnegative.

Now our approach will be to look for minimizers of J constrained on M σ for a given σ > 0, and in this way λ will be found as the Lagrange multiplier.

We conclude this section noting that the following uniform estimate holds true applying the Sobolev inequality to (2.13):

∃ C > 0 such that kΦ[u]k ≤ Ckuk ² _12/5 ∀u ∈ H ¹ (R ³ ).

3 Proof of Propositions 1.1 and 1.3

The first step is to prove than M σ is a good set on which we can minimize J.

Indeed:

Lemma 3.1. For any σ > 0 the set M σ defined in (2.12) is a non empty manifold of codimension 1.

Proof. Take any u ∈ H _r ¹ (R ³ ), u 6= 0. By (iii) of Proposition 2.2, we know that Φ[tu] = t ² Φ[u] for any t ∈ R. Thus, taken t > 0, by (2.14) we have

κ(t) := Ξ(tu) = t ³ 4

Z

R

u ² Φ[u] dx.

Hence κ(0 ⁺ ) = 0 and by the continuity and monotonicity of κ, it is readily seen that there exists a unique t 0 > 0 such that κ(t 0 ) = Ξ(t 0 u) = σ, so that M σ is non empty.

Now, M σ = {u ∈ H _r ¹ : Ξ(u) = σ}, and by Lemma 2.1 we have Ξ ^′ (u) = 0 ⇔ uΦ[u] = 0.

In this case also u ² Φ[u] ≡ 0, so that Z

R

u ² Φ[u] dx = 0,

and thus u cannot belong to M σ by (2.14). The claim follows.

Remark 3.1. As clear from the proof of Lemma (3.1), any line through 0 intersect M σ . More precisely, if kuk = 1 the half–line R ⁺ u intersect M σ exactly once, a fact we will use later.

Now, let us prove Proposition 1.1, showing that J |M

has a minimum. Let (u n ) n be a minimizing sequence for J |M

, so that

J(u n ) = 1 2

Z

R

|Du n | ² dx + ω 2

Z

R

u ² _n dx + Z

R

W (x, u n ) dx → inf

M

J. (3.15) First step: The sequence (u n ) n is bounded in H _r ¹ (R ³ ) and the sequence (Φ n ) n

is bounded in D _r ¹ (R ³ ).

Indeed, since W ≥ 0 and ω > 0, by (3.15) it is immediately seen that the sequence (u n ) n is bounded in H ¹ (R ³ ). Moreover, since u n ∈ M σ , from (2.12) and (2.14), setting Φ n := Φ[u n ], we immediately get kΦ n k ² = 4σ for all n ∈ N, so that (Φ n ) n is bounded in D ¹ _r (R ³ ).

Now, (u n ) n and (Φ n ) n are bounded in H _r ¹ (R ³ ) and D ¹ _r (R ³ ) respectively, so

that there exists a subsequence (still labelled (u n ) n ) such that u n ⇀ u in H _r ¹ (R ³ )

and a.e.; the associated (sub)sequence (Φ n ) n admits a subsequence converging

weakly in D ¹ _r (R ³ ) and a.e.. Therefore we can assume, after relabelling, that

u n ⇀ u in H _r ¹ (R ³ ), Φ n ⇀ Φ in D ¹ _r (R ³ ) and a.e. in R ³ . Moreover, we can

also assume that (u n ) n converges strongly to u in L ^s (R ³ ) for any s ∈ (2, 6) by Strauss’ Lemma.

Let us first show that the weak limits u and Φ are related by the fact that Φ = Φ[u]: indeed for any ψ ∈ D ¹ (R ³ ) there holds

Z

R

DΦ n · Dψ dx = Z

R

u ² _n Φ n ψ dx;

but u ² _n → u ² in L ^6/5 (R ³ ), while, as for R u ² _n Φ n ψ, we have that Φ n ⇀ Φ in L ⁶ (R ³ ), ψ ∈ L ⁶ (R ³ ) and u ² _n → u ² in L ^3/2 (R ³ ). Thus, passing to the limit,

Z

R

DΦ · Dψ dx = Z

R

u ² Φψ dx, (3.16)

i.e. Φ = Φ[u], as claimed.

Now let us prove that u minimizes J on M σ . First we show that J(u) ≤ lim inf J(u n ). In fact, the first part of J is essentially the norm in H ¹ , which is weakly lower semicontinuous. Moreover, since W is a nonnegative Carath´eodory function, the sequence W (|x|, u n (x)) converges a.e. to W (|x|, u(x)), so that Fatou’s Lemma gives the lower semicontinuity of R W and the claim follows. We remark that radial symmetry is not essential here, since the full space H ¹ (R ³ ) is compactly embedded in L ² _loc and this is sufficient for the convergence a.e. of a subsequence.

Now we show that u ∈ M σ . Indeed, since u n converges strongly to u in L ^12/5 , then u ² _n converges strongly to u ² in L ^6/5 . Moreover, Φ n converges weakly to Φ in L ⁶ . Thus R u ² _n Φ n converges to R u ² Φ . As a consequence, u ∈ M σ , and so it minimizes J on M σ . Therefore, by Lemma 3.1, there exists a Lagrange multiplier λ such that

J ^′ (u) − λΞ ^′ (u) = 0 in H ⁻¹ := (H _r ¹ ) ^′ ,

i.e. u is a nontrivial solution of (1.1). As already remarked in Proposition 1.2, this implies that λ > 0.

Proposition 1.1 is thus completely proved.

Proof of Proposition 1.3. Let us note that all the calculations above can be repeated word by word, replacing J with the functional

J + (u) = 1 2 Z

R

|Du ⁺ | ² dx + ω 2

Z

R

(u ⁺ ) ² dx + Z

R

W (x, u ⁺ ) dx,

where u ⁺ = max{u, 0} is the positive part of u. In this way we find a nonnegative solution u ∈ H _r ¹ (R ³ ) of the equation

−∆u ⁺ + ωu ⁺ − λu ⁺ Φ[u ⁺ ] + W u (x, u ⁺ ) = 0,

that is u is a nonnegative solution of (1.1). An application of the strong maxi-

mum principle gives u > 0 in R ³ .

4 Proof of Theorem 1.2 and Theorem 1.1

While it is quite standard to prove the result stated in Proposition 1.1 with the usual techniques of the calculus of variations, subtler considerations are necessary in this cases.

We start from the functional F ω defined in (2.8), and recalling the calcu- lations made in Section 2, we know that radial solutions of system (1.1)–(1.2) can be characterized as critical points of the functional I : H _r ¹ → R defined in (2.11), which can be re–written as

I(u) = 1 2

Z

R

|Du| ² dx + ω 2

Z

R

u ² dx + Z

R

W (x, u) dx − λ 4 Z

R

u ² Φ[u] dx by (2.13). We now show that under the assumptions of Theorem 1.2, I satisfies the assumptions of the Mountain Pass Theorem (see [1]). Indeed, I is of class C ¹ on H _r ¹ (R ³ ) and I(0) = 0. Moreover, by W 1 ) and (2.13), we easily get the existence of a universal positive constant c such that

I(u) ≥ min{1, ω}

2 kuk ² − cλkuk ⁴ , (4.17)

which is bounded away from 0 if kuk is sufficiently small.

Finally, taken u 6= 0 and t > 0, by (iii) of Proposition 2.2 we have I(tu) = t ²

2

Z

R

|Du| ² dx + ω Z

R

u ² dx

 +

Z

R

W (x, tu) dx − λt ⁴ 4

Z

R

u ² Φ[u] dx, and by W 2 )

I(tu) ≤ t ² 2

Z

R

|Du| ² dx + ω Z

R

u ² dx



+ C 1 t ^q+1 q + 1

Z

R

|u| ^q+1 dx

+ C 2 t ^p+1 p + 1

Z

R

|u| ^p+1 dx − λt ⁴ 4

Z

R

u ² Φ[u] dx,

(4.18)

and the l.h.s. of the previous inequality goes to −∞ as t → ∞, since p < 3.

At this point we can apply the Mountain Pass Theorem, provided that we show

Lemma 4.1. Under the assumptions of Theorem 1.2, I : H _r ¹ → R satisfies the Palais–Smale condition, i.e. if (u n ) n in H _r ¹ (R ³ ) is such that I(u n ) is bounded and I ^′ (u n ) → 0 as n → ∞, then there exists a subsequence (u n

) k which con- verges strongly in H _r ¹ (R ³ ).

Proof. Take (u n ) n as in the statement; then there exist M, N > 0 such that 4I(u n ) − I ^′ (u n )(u n ) ≤ M + N ku n |.

On the other hand, the left hand side of the previous inequality equals Z

R

|Du n | ² dx + ω Z

R

u ² dx + Z

R

[4W (u n ) − W u (x, u n )u n ] dx,

and by W 3 ) this is greater than R |Du n | ² + ω R u ² , since k ≤ 4. Thus (u n ) n is bounded, and we can assume that it converges to u weakly in H _r ¹ (R ³ ) and a.e.

in R ³ .

Now, W u (x, u n ) → W u (x, u) a.e., since W u is a Carath´eodory function by W 1 ), and by W 2 )

|W u (x, u n )(u n − u)| ≤ C 1 |u n | ^q |u n − u| + C 2 |u n | ^p |u n − u|.

We claim that there exists r ∈ (2, 6), such that 2 ≤ qr ^′ ≤ 6 and Z

R

|u n | ^q |u n − u| dx ≤ k|u n | ^q k r

ku n − uk r → 0, (4.19) since u n → u in L ^r and (u n ) n is bounded in L ^qr

. Indeed: if q ∈ [2, 3), then 2 < 2r ^′ ≤ qr ^′ < 3r ^′ < 6 for any r ∈ (2, 6); if q ∈ (1, 2), then qr ^′ ∈ (2, 6) by choosing r ∈ 

2, min n

2 2−q , 6 o

. Analogous calculations can be done with p, so that Lebesgue’s Theorem gives

Z

R

W u (x, u n )(u n − u) dx → 0. (4.20) In conclusion, we know that I ^′ (u n )(u n − u) → 0, i.e.

Z

R

Du n · D(u n − u) dx + ω Z

R

u n (u n − u) dx + Z

R

W u (x, u n )(u n − u) dx → 0, which easily implies that u n → u in H ¹ (R ³ ) by (4.20).

Remark 4.1. As it is clear from the proof above, the geometrical structure of a mountain pass fails when p ≥ 3, while the Palas–Smale condition fails for k > 4; this is why such restrictions are made in Theorem 1.2.

Although we assume some symmetries on W , such as W (x, −s) = W (x, s), in general we are not able to apply the Z 2 –version of the Mountain Pass ([1]) in order to obtain infinitely many solutions. In fact, from (4.18) we cannot infer that I(u) < 0 when kuk is large enough in any finite dimensional subspace of H _r ¹ . However, this is not surprising, since the same problem could arise in presence of the model potential W (s) = −|s| ^γ /γ, γ ∈ (2, 6) (see [2], [3], [8], [24],. . . ), where different situations appeared depending on the values of γ. On the other hand, the approach based on Krasnoselskii genus and constrained critical points, in this case reveals fruitful thanks to the introduction of the manifold M σ made in Section 3. For this purpose we recall the following Theorem (see [29, Theorem 5.7]), where γ(A) denotes the Krasnoselskii genus of a symmetric set A not containing 0. For the reader’s convenience we only recall the definition of genus, referring to [29] for an introduction to the genus and related results.

Definition 1. Let A be a closed and symmetric subset of a Banach space B.

We set γ(∅) = 0 and when A 6= ∅ γ(A) =

 inf{m ∈ N : ∃ h ∈ C ⁰ (A, R ^m \ {0}) odd},

∞ if {m ∈ N : ∃ h ∈ C ⁰ (A, R ^m \ {0}) odd} = ∅.

In particular γ(A) = ∞ for any symmetric set containing 0.

Theorem 4.1. Suppose that J is an even C ¹ functional on a complete sym- metric C ^1,1 manifold M contained in a Banach space B, and suppose that J satisfies (P S) and is bounded from below. Let

˜

γ(M) = sup{γ(K) : K ⊂ M is compact and symmetric} ≤ ∞.

Then J admits at least ˜ γ(M) pairs of critical points on M.

In our case J is the functional defined in (2.9) and M = M σ , and the fact that J is bounded from below is clear form its very definition. Moreover, M σ is a symmetric manifold (not containing 0), since by (1.2) it is clear that Φ[−u] = Φ[u], so that if u ∈ M σ then also −u ∈ M σ . In addition, the regularity of M σ follows from the proof of Lemma 3.1.

Now we prove

Lemma 4.2. ˜ γ(M σ ) = ∞.

Proof. Take any N ∈ N and consider a subspace H N of H _r ¹ (R ³ ) of dimension N . We first prove that M N := M σ ∩ H N , which is non empty by Remark 3.1, is bounded. Indeed, assume by contradiction that there exists an unbounded sequence (u n ) n in M N . Up to subsequences, we can assume that _ku ^u

_k → u ∈ H N

with kuk = 1. If H N = Span(e 1 , . . . , e N ), then by (1.6) we get that Φ n := Φ[u n ] ∈ Span

Z

R

e i (y)e j (y)

| · −y| dy, 1 ≤ i ≤ j ≤ N

 .

Thus, by (2.14) we get kΦ n k ² = 4σ, so that we may assume that Φ n → Φ, where, as in (3.16), Φ = Φ[u].

From (2.14) we know that R u ² _n Φ n = 4σ; dividing by ku n k ² and passing to the limit we get

Z

u ² Φ = 0.

By (1.6) we also get that Φ is positive, so that the previous integral implies u ² = 0 in R ³ , a contradiction.

Now, take N ∈ N and a subspace H N of H _r ¹ (R ³ ) of dimension N ; then consider the set K N := H N ∩M σ , which is non empty by Remark 3.1. Moreover, by the previous considerations, K N is a compact subset of H N . We finally prove that γ(K N ) ≥ N : indeed, consider the map from the unit sphere S ^{N −1} of H N , π : S ^{N −1} → M σ , defined as π(u) = u σ , where u σ is the unique point of R ⁺ u intersecting M σ ; we remark that the definition of π is well posed by Remark 3.1. Clearly π is an odd continuous map, and thus, by [29, Proposition 5.4 (4 ^◦ )]

we get γ(K N ) ≥ γ(S ^{N −1} ), and γ(S ^{N −1} ) = N by [29, Proposition 5.2]. Since N is arbitrary, we get ˜ γ(M σ ) = ∞, and the lemma is proved.

Thus, in order to prove Theorem 1.1, it is enough to prove that J |M

satisfies

(P S), and then to apply Theorem 4.1, noting that u is a critical point for J on

M σ if and only if there exists λ ∈ R such that J ^′ (u)v − λ R uΦ[u]v = 0 for every

v ∈ M. The fact that λ > 0 follows from Proposition 1.2.

Lemma 4.3. J |M

satisfies (P S).

Proof. Let (u n ) n be a sequence in M σ such that (J(u n )) n is bounded and J _|M ^′

(u n ) → 0. By Lemma 2.1, taken v ∈ H _r ¹ , we have Ξ ^′ (u n )(v) = R u n Φ[u n ]v for any n ∈ N, so that, by Lemma 3.1, there exist real numbers λ n such that for all v ∈ M σ

Z

R

Du n · Dv dx + ω Z

R

u n v dx + Z

R

W u (x, u n )v dx − λ n

Z

R

u n Φ n v → 0 (4.21) as n → ∞, where, as usual, Φ n = Φ[u n ].

Sublemma 4.1. There exists λ ∈ [0, _4σ ^k sup _n J(u n )] such that λ n → λ as n →

∞ (up to subsequences).

Remark 4.2. Actually at the end we will get that λ > 0, as before, but for the moment this preliminary result is enough to get the strong convergence of (u n ) n .

Proof of Sublemma 4.1. First, let us start noting that, by (2.9), we immediately get that (u n ) n is bounded, since W is nonnegative. Thus, from (4.21) we have

hJ ^′ (u n ), u n i − λ n

Z

R

u ² _n Φ n dx := ε n → 0 in R. (4.22) On the other hand, (2.14) ensures that

Z

R

u ² _n Φ n dx = 4σ ∀ n ∈ N,

so that from (4.22) and the form of J ^′ (u n )u n , we get that the sequence (λ n ) n

is bounded. Thus we can assume that there exists λ ∈ R such that, up to subsequences (which we assume to consider),

λ n → λ.

By (4.22) we also get λ n = 1

4σ (hJ ^′ (u n ), u n i − ε n )

= 1 4σ

Z

R

|Du n | ² dx + ω Z

R

u ² _n dx + Z

R

W u (x, u n )u n dx



− ε n

4σ .

(4.23)

Thus by W 3 ), since k ≥ 2, λ n ≤ 1

4σ

Z

R

|Du n | ² dx + ω Z

R

u ² _n dx + k Z

R

W (x, u n ) dx



− ε n

4σ

≤ k 4σ

 1 2

Z

R

|Du n | ² dx + ω 2

Z

R

u ² _n dx + Z

R

W (x, u n ) dx



− ε n

4σ

= k

4σ J(u n ) − ε n

4σ .

On the other hand, again by W 3 ), hJ ^′ (u n ), u n i =

Z

R

|Du n | ² dx + ω Z

R

u ² _n dx + Z

R

W u (x, u n )u n dx ≥ 0, so that by (4.23) we finally get

− ε n

4σ ≤ λ n ≤ k

4σ J(u n ) − ε n

4σ . Thus, up to subsequences, λ n → λ, where we now know that

0 ≤ λ ≤ k 4σ sup

n

J(u n ). (4.24)

As in the proof of Proposition 1.1 we get that, up to a subsequence, (u n ) n

converges weakly in H _r ¹ (R ³ ) and Φ n converges weakly to Φ in D ¹ _r (R ³ ). But actually we have the following result:

Sublemma 4.2. (u n ) n converges strongly in H _r ¹ (R ³ ).

Proof. We already know that u n ⇀ u in H _r ¹ (R ³ ), so that we need to show that u n → u in L ² (R ³ ) and Du n → Du in (L ² (R ³ )) ³ . Of course we have the following two possibilities: either u n 6→ u or u n → u in L ² (R ³ ), and we want to exclude the first possibility.

By the Concentration–Compactness Principle (see Appendix), in the former case only vanishing can occur, since we are treating radial functions and then dichotomy cannot take place. If this is the case, by the final statement of Lemma A.1, u n → 0 in L ^s (R ³ ) for any s ∈ (2, 6), in particular for s = 12/5. We also recall that Φ n ⇀ Φ = Φ[u] in D _r ¹ (R ³ ), and so in L ⁶ (R ³ ); then by (2.14) we would have

4σ = Z

R

u ² _n Φ n dx → 0, which is absurd.

Then we can conclude that u n → u in L ² (R ³ ); from this very last information we obtain that u n → u in H ¹ (R ³ ). Indeed,

Z

R

Du n · D(u n − u) dx + ω Z

R

u n (u n − u) dx − Z

R

W u (x, u n )(u n − u) dx

− λ n

Z

R

u n Φ n (u n − u) dx → 0 (4.25) by (4.21), since (u n − u) n is a bounded sequence. On the other hand, by W 2 ) we get

|W u (x, u n )(u n − u)| ≤ C 1 |u n | ^q |u n − u| + C 2 |u n | ^p |u n − u|. (4.26)

If p ∈ (1, 3], then Z

R

|u n | ^p |u n − u| dx ≤ ku n k 2p ku n − uk 2 → 0,

since 2 < 2p ≤ 6; if p ∈ (3, 5) take ε ∈ (0, 1) such that µ := 

6−ε p

 ′

∈ (2, 6), so

that Z

R

|u n | ^p |u n − u| dx ≤ ku n k ^p _6−ε ku n − uk µ → 0,

since now u n → u in L ^s (R ³ ) for any s ∈ [2, 6), and since (u n ) n is bounded in L ^6−ε . The addendum with q in (4.26) can be obviously treated in the same way.

Moreover, Z

R

u n Φ n (u n − u) dx = Z

R

u ² _n Φ n dx − Z

R

u n Φ n u dx;

now, in the r.h.s. of the previous identity Φ n ⇀ Φ in L ⁶ (R ³ ); in addition, in the first term u ² _n → u ² in L ^6/5 (R ³ ), while in the second term u n → u in L ³ (R ³ ) and u ∈ L ² (R ³ ). In conclusion (4.25) gives kDu n k 2 → kDuk 2 , so that, summing up, u n → u in H ¹ (R ³ ), as claimed.

In conclusion, Sublemma 4.2 gives that J |M

satisfies (P S), and thus Lemma 4.3 is proved, leading to the validity of Theorem 1.1.

A Appendix

We recall the properties of Lions’s Concentration–Compactness Principle that we use in the form of the following lemma, which is the collection of [16, Lemma I.1] and [17, Lemma I.1]

Lemma A.1 (Concentration–Compactness Principle). Let (ρ n ) n be a sequence in L ¹ (R ^N ), N ≥ 1, such that ρ n ≥ 0 and R ρ n → β > 0. Then there exists a subsequence (ρ n

) k satisfying one of the three alternatives below:

(compactness) there exists a sequence of points (y k ) k in R ^N such that ∀ δ > 0

∃ R > 0 with the property that Z

B

(y

)

ρ n

dx ≥ β − δ;

(vanishing)

k→∞ lim sup

y∈R

Z

B

(y)

ρ n

dx = 0 for any R > 0;

(dichotomy) there exists α ∈ (0, β) such that for every δ > 0 there exists k 0 ≥ 1 and two sequences of nonnegative functions (ρ ¹ _k ) k , (ρ ² _k ) k in L ¹ (R ^N ) such that for any k ≥ k 0

kρ n

− (ρ ¹ _k + ρ ² _k )k L

(R

) < δ,     Z

R

ρ ¹ _k dx − α    

≤ δ, 

   Z

R

ρ ² _k dx − (β − α)    

≤ δ, and

dist Supp(ρ ¹ _k ), Supp(ρ ² _k )
→ ∞ as k → ∞, where Supp(ρ) denotes the support of ρ.

Moreover, if 1 < p < N , 1 ≤ q < _{N −p} ^pN , (u n ) n is bounded in L ^q (R ^N ) and (Du n ) n is bounded in (L ^p (R ^N )) ^N , and

sup

y∈R

Z

B

(y)

|u n | ^q dx → 0 as n → ∞ for some R > 0,

then u n → 0 in L ^s (R ^N ) for any s ∈ q, _{N −p} ^pN
.

References

[1] Ambrosetti, A., Rabinowitz, P.H. (1973). Dual variational methods in crit- ical point theory and applications. J. Funct. Analysis 14:349–381.

[2] Ambrosetti, A., Ruiz, D. (2008). Multiple bound states for the Schr¨ odinger–

Poisson problem. Commun. Contemp. Math 10:391–404.

[3] Azzollini, A., Pomponio, A. (2008). Ground state solutions for the nonlinear Schr¨ odinger–Maxwell equations. J. Math. Anal. Appl. 345:90–108.

[4] Benci, V., Fortunato, D. (1998). An eigenvalue problem for the Schr¨ odinger- Maxwell equations. Topol. Methods Nonlinear Anal. 11:283–293.

[5] Benci, V., Fortunato, D. (2004). Towards a Unified Field Theory for Clas- sical Electrodynamics. Arch. Rational Mech. Anal. 173:379-414.

[6] Catto, I., Le Bris, C., Lions, P.-L. (2002). On some periodic Hartree–type models for crystals. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 19:143–190.

[7] D’Aprile, T., Mugnai, D. (2004). Non–existence results for the coupled Klein–Gordon–Maxwell equations. Adv. Nonlinear Stud. 4:307–322.

[8] D’Aprile, T., Mugnai, D. (2004). Solitary Waves for nonlinear Klein–

Gordon–Maxwell and Schr¨ odinger–Maxwell equations. Proc. R. Soc. Edinb.

Sect. A 134:1–14.

[9] Esteban, M.J., Georgiev, V., S´er´e, E. (1996). Stationary waves of the Maxwell–Dirac and the Klein–Gordon–Dirac equations. Calc. Var. Partial Differential Equations 4:265–281.

[10] Guo, Y., Rein, G. (1999). Stable Steady States in Stellar Dynamics. Arch.

Rational Mech. Anal. 147:225–243.

[11] Kikuchi, H. (2007). Existence and stability of standing waves for Schr¨ odinger–Poisson–Slater equation. Adv. Nonlinear Stud. 7:403-437.

[12] Kikuchi, H. (2007). On the existence of a solution for elliptic system related to the Maxwell–Schr¨ odinger equations. Nonlinear Anal. 67:1445–1456.

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

[14] Lieb, E.H., Thomas, L.E. (1997). Exact Ground State Energy of the Strong–Coupling Polaron. Commun. Math. Phys. 183:511–519.

[15] Lions, P.-L. (1980). The Choquard equation and related questions. Nonlin- ear Anal. 4:1063-1072.

[16] Lions, P.-L. (1984). The concentration–compactness principle in the calcu- lus of variations. The locally compact case. Part I. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 1:109–145.

[17] Lions, P.-L. (1984). The concentration–compactness principle in the calcu- lus of variations. The locally compact case. Part II. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 1:223–283.

[18] Ma, L., Zhao, L. (2010). Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195:455-467.

[19] Mercuri, C. (2008). Positive solutions of nonlinear Schr¨ odinger–Poisson sys- tems with radial potentials vanishing at infinity. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19:211–227.

[20] Mugnai, D. (2004). Coupled Klein–Gordon and Born–Infeld type equations:

looking for solitons. R. Soc. Lond. Proc. Ser. A 460:1519–1528.

[21] Mugnai, D. (2010). Solitary waves in Abelian Gauge Theories with strongly nonlinear potentials. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 27:1055–

1071.

[22] Palais, R.S. (1979). The principle of symmetric criticality. Comm. Math.

Phys. 69:19-30.

[23] Pisani, L., Siciliano, G. (2007). Neumann condition in the Schr¨ odinger-

Maxwell system. Topol. Methods Nonlinear Anal. 29:251–264.

[24] Ruiz, D. (2006). The Schr¨ odinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237:655–674.

[25] Ruiz, D., Siciliano, G. (2008). A note on the Schr¨ odinger–Poisson–Slater equation on bounded domains. Adv. Nonlinear Stud. 8:179–190.

[26] Ruiz Arriola, E., Soler, J. (2001). A variational approach to the Schr¨ odinger-Poisson system: asymptotic behaviour, breathers and stability.

J. Stat. Phys. 103:1069-1106.

[27] S´ anchez, O., Soler, J. (2004). Long–time dynamics of the Schr¨ odinger- Poisson-Slater system. J. Statist. Phys. 114:179-204.

[28] Strauss, W.A. (1977). Existence of solitary waves in higher dimensions.

Comm. Math. Phys. 55:149–162.

[29] Struwe, M. (2008). Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, Springer-Verlag, Berlin, 4th edition.

[30] Wang, Z., Zhou, H.-S. (2007). Positive solution for a nonlinear stationary Schr¨ odinger–Poisson system in R ³ . Discrete Contin. Dyn. Syst. 18:809-816.

[31] Zhao, L., Zhao, F. (2008). On the existence of solutions for the Schr¨ odinger–

Poisson equations. J. Math. Anal. Appl. 346:155–169.