Scaling properties of functionals and existence of constrained minimizers

Transcription

1 Journal of Functional Analysis 61 (11) Scaling properties of functionals and existence of constrained minimizers Jacopo Bellazzini a,, Gaetano Siciliano b a Università di Sassari, via Piandanna 4, 71 Sassari, Italy b Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 11, 558-9, São Paulo, Brazil Received 18 November 1; accepted June 11 Available online 5 July 11 Communicated by J. Coron Abstract In this paper we develop a new method to prove the existence of minimizers for a class of constrained minimization problems on Hilbert spaces that are invariant under translations. Our method permits to exclude the dichotomy of the minimizing sequences for a large class of functionals. We introduce family of maps, called scaling paths, that permits to show the strong subadditivity inequality. As byproduct the strong convergence of the minimizing sequences (up to translations) is proved. We give an application to the energy functional I associated to the Schrödinger Poisson equation in iψ t + ψ ( x 1 ψ ) ψ + ψ p ψ = when <p<3. In particular we prove that I achieves its minimum on the constraint {u H 1 ( ): u = ρ} for every sufficiently small ρ>. In this way we recover the case studied in Sanchez and Soler (4) [] for p = 8/3 and we complete the case studied by the authors for 3 <p<1/3 in Bellazzini and Siciliano (11) [4]. 11 Elsevier Inc. All rights reserved. Keywords: Constrained minimization; Subadditivity inequality; Schrödinger Poisson equations; Standing waves * Corresponding author. addresses: jbellazzini@uniss.it (J. Bellazzini), siciliano@dm.uniba.it (G. Siciliano). -136/$ see front matter 11 Elsevier Inc. All rights reserved. doi:1.116/j.jfa

2 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) Introduction The existence of minimizers for constrained functionals is an interesting problem either from a mathematical or from a physical point of view. Indeed in many applications often appears a C 1 functional whose critical points restricted to some constraint have a relevant physical meaning. For example, in Schrödiger-type equations the existence of standing wave solutions can be proved by finding minimizers for the energy functional on L constraint. In this paper, having in mind an application to a Schrödinger Poisson equation, we study the existence of minimizers for a class of functionals defined on a Hilbert space. We consider H, H 1 two Hilbert spaces of functions defined in R N, with norms H and H1 satisfying u( +a) H = u( ) H, u( +a) H1 = u( ) H1 for all a R N. Assume that H H 1, H L (R N ) with ( c 1 H1 + ( L (R )) N H c H1 + ) L (R N ) where L (R N ) is the usual Lebesgue space. Let I : H R be a functional of the following form I(u):= 1 u H 1 + T(u) (1.1) where the nonlinear operator T C 1 (H, R) satisfies some suitable assumptions. In particular we require that T is invariant for the noncompact group of translations in R N so that also the functional I is translation invariant, i.e. it satisfies I(u(x+ a)) = I(u(x)). We look at the constrained minimization problem I ρ := inf B ρ I(u) (we agree I = ) (1.) where B ρ ={u H: u = ρ} and I ρ > is assumed. The main difficulty for translation invariant functionals is due to the lack of compactness of the (bounded) minimizing sequences {u n } B ρ ; indeed the minimizing sequence {u n } could run off to spatial infinity and/or spread uniformly in space. So even up to translations two possible bad scenarios are possible: (vanishing) u n ; (dichotomy) u n ū and < ū <ρ. The general strategy in the applications (see for instance [13] in case of weakly lower semicontinuous functionals) is to prove that any minimizing sequence weakly converges, up to translation, to a function ū which is different from zero, excluding the vanishing case. Then one has to show that ū = ρ, which proves that dichotomy does not occur.

3 488 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) The subadditivity condition In [17], Lions proved that the invariance by translations of the problem implies in many cases an inequality that the infima I ρ have to satisfy and read as follows (weak subadditivity inequality) I ρ I μ + I ρ μ for all <μ<ρ. (1.3) However the necessary and sufficient condition in order that any minimizing sequence on B ρ is relatively compact is a stronger version of (1.3) and is given by the following inequality: I ρ <I μ + I ρ μ for all <μ<ρ. (1.4) In the literature it is referred as the strong subadditivity inequality. This condition is more difficult to prove and the classical approach to prove the strong subadditivity inequality (1.4) is to ensure that (MD) the function s I s s is monotone decreasing. Indeed, in case (MD) holds for μ (,ρ)we get μ I ρ ρ <I μ and ρ μ I ρ ρ <I ρ μ. Therefore I ρ = μ ρ I ρ + ρ μ ρ I ρ <I μ + I ρ μ μ (,ρ). The main problem when we try to apply the concentration-compactness principle to a specific functional is that also condition (MD) is not easy to prove. Indeed the function s I s can have s a fast oscillating behavior, even in a neighborhood of the origin, even if the function s I s is continuous and fulfills the weak subadditivity inequality (1.3); the reason is that this is a very weak condition in the sense that it is satisfied also by some pathological functions including for instance the Cantor function, see [11]. The main result of the paper is Theorem.1 which shows that condition (MD) can be recovered for a large class of functionals (including that involved in problem (.9)) provided that they satisfy some good scaling properties. In the theorem we give sufficient conditions that guarantee (MD) and thus the convergence, up to translation, of the minimizing sequences. The paper is organized as follows. In Section we state the main result, Theorem.1, which is proved in Section 3. In Section 4 we apply the abstract framework to the Schrödinger Poisson equation in, with p (, 3), H = H 1 ( ) and H 1 = D 1, ( ) (see below for the definition of these spaces), proving the existence of minimizers of the related functional with sufficiently small L -norm, see Theorem 4.1. As a natural consequence we get the orbital stability of the minimizers (Theorem 4.). 1.. Notations As a matter of notations, in the paper it is understood that all the functions, unless otherwise stated, are complex-valued, but for simplicity we will write L s (R N ), H 1 (R N )..., where N 3 and for any 1 s<+, L s (R N ) is the usual Lebesgue space endowed with the norm

4 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) u s s := R N u s dx, and H 1 (R N ) the usual Sobolev space endowed with the norm u H 1 := u dx + u dx. R N R N For our application, let us define the space D 1, (R N ). It is the completion of C (RN ) with respect to the norm u D 1, := R N u dx. Moreover the letter c will be used to denote a suitable positive constant, whose value may change also in the same line, and the symbol o(1) to denote a quantity which goes to zero. We also use O(1) to denote a bounded sequence.. The main result Before to state our main theorem, some preliminaries are in order. In the next lemma we give a class of functionals to which the Lions principle holds. The strong subadditivity condition is assumed as hypothesis; the novelty is that it is applicable to a large class of functionals. However we know that it is a version of the concentration-compactness principle of Lions [17] adapted to the problem we have in mind. It is, in some sense, the departure point of our main result. Lemma.1. (See [3,4].) Let T C 1 (H, R). Let ρ> and {u n } be a minimizing sequence for I ρ weakly convergent, up to translations, to a nonzero function ū. Assume that (1.4) holds and that T(u n ū) + T(ū) = T(u n ) + o(1); (.1a) T ( α n (u n ū) ) T(u n ū) = o(1) where α n = ρ ū ; (.1b) u n ū T (u n ), u n = O(1); (.1c) T (u n ) T (u m ), u n u m = o(1) as n, m +. (.1d) Then u n ū H. In particular it follows that ū B ρ and I(ū) = I ρ. The basic assumptions of this lemma are based on a Brezis Lieb splitting property of the nonlinear part T (condition (.1a), see [5,7]) and a sort of homogeneity (condition (.1b)) which together exclude dichotomy. We remark explicitly that the unique point where (1.4) is used in the proof of this lemma is only to exclude dichotomy, that is, only to ensure that the weak limit ū belongs to B ρ. Put in other way, even suppressing conditions (.1c) and (.1d) in the lemma we can conclude that ū = ρ. If in addition these last two conditions are fulfilled, we derive that {u n } strongly converges to ū in H.

5 49 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) We give now the following Definition.1. Let u H, u. A continuous path g u : θ R + g u (θ) H such that g u (1) = u is said to be a scaling path of u if Θ gu (θ) := g u (θ) u is differentiable and Θ g u (1) (.) where the prime denotes the derivative. We denote with G u the set of the scaling paths of u. The set G u is nonempty and indeed it contains a lot of elements: for example, g u (θ) = θu(x) G u, since Θ gu (θ) = θ.alsog u (θ) = u(x/θ) is an element of G u since Θ gu (θ) = θ N. As we will see in the application it is relevant to consider the family of scaling paths of u parametrized with β R given by G β u = { g u (θ) = θ 1 N β u ( x/θ β)} G u. (.3) Notice that all the paths of this family have as associated function Θ(θ) = θ. Moreover, fixed u, we define the following real valued function which is crucial for our purpose: h gu (θ) := I ( g u (θ) ) Θ gu (θ)i (u), θ. Definition.. Let u be fixed and g u G u. We say that the scaling path g u is admissible for the functional I if h gu is a differentiable function. In our application the function h gu will be obviously differentiable; this is due to the special form of the scaling path we choose; indeed we will work with the subfamily G β u. Our intent is to give some conditions which ensures the strong subadditivity condition for the functional I. It turns out to be simpler to give conditions in terms of h gu. Indeed under very mild assumptions on this auxiliary function (easily verified in the applications) (1.4) is achieved and so the minimizing sequences are strongly convergent; indeed our main result reads as follow. Theorem.1 (Avoiding dichotomy). Let T C 1 (H, R) satisfying the set of assumptions (.1). Assume that for every ρ>, all the minimizing sequences {u n } for I ρ have a weak limit, up to translations, different from zero. Assume finally (1.3) and the following conditions <I s < for all s> ( I() = ), (.4) s I s is continuous, (.5) lim s I s =. (.6) s Then for every ρ> the set M(ρ) = { u Bμ : I(u)= I μ } μ (,ρ] is nonempty.

6 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) If in addition u M(ρ) g u G u admissible, such that d dθ h g u (θ), (.7) θ=1 then (MD) holds. Moreover, if {u n } is a minimizing sequence weakly convergent to a certain ū (necessarily ) then u n ū H and I(ū) = I ρ. The novelty in Theorem.1 is given by some ingredients that, as far as we know, have never been introduced in the literature. First, the continuity of the function s I s s reveals a very important property of the infimum; it is shared by many different minimization problems. Second, that the monotonicity of I s can be proved by looking just at the scaling s properties of the minimizers found for the values of s that correspond to the global minima of I s in the interval [,ρ]. For this purpose we have introduced the scaling paths and the crucial s hypothesis (.7). Third, we notice that condition (.7) has to be checked on a specific minimizer u (so we can take advantage of the fact that it is a constrained minimum) and not on an abstract sequence. We underline that with our approach we can easily recover two well-known results concerning minimization problems where the strong subadditivity condition is proved with standard argument. The first one is related to the Choquard functional (see for instance [15]): E ρ := inf B ρ E(u) where E(u) := 1 u dx u(x) u(y) dxdy. x y The second one derives from the nonlinear Schrödinger equation, see [17] and [1] G ρ := inf B ρ G(u), G(u) := 1 u D 1, 1 p u p dx. (.8) In both cases it is not difficult to show that (MD) holds thanks to the scaling properties of the functionals. However, there are various examples in which condition (MD) is not clear if holds or not. This is the case of the following minimization problem in H 1 ( ) involving Coulombian nonlocal terms: { 1 I ρ = inf B ρ u dx u(x) u(y) dxdy 1 x y p } u p dx (.9)

7 49 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) where <p<3. This is exactly the problem we are going to study in Section 4 and, as we will see, the existence of minimizers for (.9) is related to the existence of stable standing waves for an equation that derives from the Hartree Fock equation, the so-called Schrödinger Poisson equation. Here the difficulty concerns the nonlocal repulsive Coulombian term u(x) u(y) x y dxdy that does not permit to show the strong subadditivity inequality with standard arguments, even for small ρ. Remark.1. Notice that to recover (.4), it is sufficient the weak subadditivity condition (1.3) in [, + ) and the fact that I s < only for s in a certain interval (, ρ]. Indeed, let ρ ( ρ, ρ]: then for every s ( ρ,ρ] we get I s I ρ + I s ρ < since s ρ < ρ. This shows that I s < fors in the larger interval (,ρ]. Iterating this procedure it follows that I s < for every s>. 3. Proof of Theorem.1 We first address the dichotomy case, i.e. when the minimizing sequences for I ρ weakly converge to a nonzero function ū which is not on the right constraint but satisfies ū = μ <ρ. The result is not surprising in view of the trichotomy of the Lions principle. Proposition 3.1 (Dichotomy). Let T C 1 (H, R) satisfying (.1a) and (.1b). Let ρ> and {u n } B ρ be a minimizing sequence for I ρ such that u n ū and assume that μ = ū (,ρ). Assume also that (1.3) holds. Then I ρ = I μ + I ρ μ (3.1) and I(ū) = I μ. This proposition shows that in the dichotomy case, in (1.3) the equality holds and the weak limit ū is a minimizer on the manifold given by the constraint u = μ. Although B μ is not the original constraint, we can take advantage of the fact that ū is a minimizer on u = μ as shown by Theorem.1. As far as we know, this simple result is new. Proof. Since u n ū, we get therefore u n ū + ū = u n + o(1) On the other hand, {u n } is a minimizing sequence for I ρ,so α n = ρ μ u n ū 1. (3.) 1 u n H 1 + T(u n ) = I ρ + o(1)

8 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) and by (.1a), we deduce also 1 u n ū H ū H 1 + T(u n ū) + T(ū) = I ρ + o(1). Hence using (3.) and (.1b) we infer that is, 1 α n (u n ū) H ū H 1 + T ( α n (u n ū) ) + T(ū) = I ρ + o(1) Then, since α n (u n ū) = ρ μ and (1.3) we get I ( α n (u n ū) ) + I(ū) = I ρ + o(1). (3.3) I ρ μ + I(ū) I( α n (u n ū) ) + I(ū) = I ρ + o(1) I ρ μ + I μ + o(1) which implies I(ū) = I μ and consequently (3.1). A crucial remark now for our purpose is in order. The strong subadditivity inequality (1.4) holds if the following condition is satisfied: (I) the function s I s s in the interval [,ρ] achieves its unique minimum in s = ρ. Indeed for μ (,ρ)we get μ I ρ ρ <I μ and ρ μ I ρ ρ <I ρ μ. Therefore I ρ = μ ρ I ρ + ρ μ ρ I ρ <I μ + I ρ μ μ (,ρ). We now show a lemma that asserts that the behavior of the function s I s near zero is sufficient to deduce almost (1.4). Lemma 3.1. Let us assume that condition (.4) is satisfied in a certain interval [,ρ] and that (.5) and (.6) hold. Then for every ρ> there exists ρ (,ρ] such that for every μ (,ρ ) I ρ <I μ + I ρ μ. Proof. Let us fix ρ>and define { ρ := min s [,ρ] s.t. I s s = I } ρ ρ which is strictly positive in virtue of (.5) and (.6). We claim that the function s I s in the interval [,ρ s ] achieves the minimum only in s = ρ. By the claim follows, as noticed before, that I ρ <I μ + I ρ μ for every μ (,ρ ).In

9 494 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) order to prove the claim we notice that if there exists ρ <ρ such that I ρ ρ continuity a ρ <ρ such that I ρ = I ρ which contradicts the definition of ρ ρ ρ. With this result in hands we can give now the < I ρ ρ it will exist by Proof of Theorem.1. To prove that M(ρ) let us fix ρ>. By Lemma 3.1 there exists ρ (,ρ] such that for every μ (,ρ ) I ρ <I ρ μ + I μ. Then by Lemma.1 we get {u B μ : I(u)= I μ }. To get (MD) it is sufficient to prove condition (I) on every interval [,ρ]. So let us fix ρ> I and call c := min s [,ρ] <, by (.4). Let s We have to prove that ρ = ρ. Thanks to (.5) and (.6), ρ > and { ρ := min s [,ρ] s.t. I } s s = c. s [,ρ ): I ρ ρ < I s s (3.4) namely, the function [,ρ ] s I s R s achieves the minimum only in s = ρ, by definition of ρ. Since condition (I) is satisfied in [,ρ ] we have the strong subadditivity inequality I ρ <I μ + I ρ μ μ (,ρ ). Therefore we can apply Lemma.1 to the minimization problem I ρ = inf B ρ I(u) and we deduce the existence of ū B ρ such that I(ū) = I ρ. In particular ū M(ρ).Now we argue by contradiction by assuming that ρ <ρ. Then fixed gū Gū with its associated Θ, by (3.4) and the definition of ρ : I ρ ρ I Θ(θ)ρ Θ(θ)ρ for all θ (1 ε, 1 + ε). Therefore we have I Θ(θ)ρ I(gū(θ)) Θ(θ)ρ Θ(θ)ρ I ρ ρ = I(ū) ρ for every θ (1 ε, 1 + ε).

10 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) This means that the map h gū(θ) = I(gū(θ)) Θ(θ)I(ū), defined in a neighborhood of θ = 1, is nonnegative and has a global minimum in θ = 1 with h gū(1) =. Then we get h gū(1) =. Since gū is arbitrary this relation has to be true for every map gū, so we have found a ū M(ρ) such that for every gū Gū, h gū(1) = : this clearly contradicts (.7) and so ρ = ρ. This implies condition (I) on every interval of type [,ρ] and so (MD), that is, s I s /s is monotone decreasing in [, + ). To prove the final part, let {u n } be a minimizing sequence for I ρ weakly convergent to a certain ū. We already know that ū. Since we have just shown that in (,ρ)the strong subadditivity condition is satisfied we can apply Lemma.1 and conclude the proof. 4. Application to a Schrödinger Poisson equation We apply the aforementioned results to a concrete minimization problem for which the dichotomy of minimizing sequence cannot a priori be excluded. We consider the following Schrödinger Poisson type equation iψ t + ψ ( x 1 ψ ) ψ + ψ p ψ = in, (4.1) where ψ(x,t): [,T) C is the wave function, denotes the convolution and <p<3. Eq. (4.1) has a very important physical meaning in case p = 8/3 due to the fact that it derives as a simplification, due to Slater, of the Hartree Fock equation. Eq. (4.1) describes a quantum mechanical system of many particles, and it has been used to describe a wide variety of physical phenomena in Quantum Chemistry and Solid State Physics. We refer to [16] and [18] for a detailed study of equations which model physical phenomena with nonlocal terms. From a mathematical point of view however the Cauchy problem associated to Eq. (4.1) is globally well posed for <p<1/3, and for this reason we will restrict to this range on p. We are interested to the existence of particular class of solutions of the Schrödinger Poisson equation. By a solitary wave we mean a solution of (4.1) whose energy travels as a localized packet; if a solitary wave exhibits orbital stability it is called soliton. We are looking for standing waves, that is solitary waves of the form ψ(x,t)= e iωt u(x), ω R, u(x) C. Plugging in a solitary wave ψ into (4.1), we reduce to study the following semilinear elliptic equation with a nonlocal nonlinearity u + φ u u u p u = ωu in, (4.) where we have set φ u (x) = u(y) x y dy.

11 496 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) Evidently, φ u satisfies φ u = 4π u, is uniquely determined by u and is usually interpreted as the scalar potential of the electrostatic field generated by the charge density u. At this point, there are two different ways to approach Eq. (4.) according to the role of ω: (a) the frequency ω is a fixed and assigned parameter, (b) the frequency ω is an unknown of the problem. In the first case, it is easy to see that the critical points of the following functional defined in H 1 ( ) J(u)= 1 u dx ω u dx + 1 φ u u dx 1 u p dx (4.3) 4 p give rise to solutions of (4.). This case has been extensively studied by many authors in these last years, see e.g. [1,1,19] and the references therein. On the other hand, the second case has been less investigated. Thanks to our abstract framework developed in the previous section, we can give a contribution in this direction. Indeed the solutions of (4.1) with ω unknown, can be seen as the critical points of a functional restricted to the constraint of functions with fixed L -norm. Note also that the critical points of the Schrödinger Poisson functional on the manifold of fixed charge (L -norm), are physically relevant since the charge is a quantity which is conserved during the evolution in time of the standing waves. So the natural way to find the solutions of Eq. (4.) with fixed L -norm is to look for the constrained critical points of the functional I(u)= 1 u dx φ u u dx 1 p u p dx (4.4) on the L -spheres in H 1 ( ) B ρ = { u H 1( ) : u = ρ }. Recalling that in this case ω is not a parameter but an unknown of the problem, by a solution of (4.) we mean a couple (ω ρ,u ρ ) R H 1 ( ), where ω ρ is the Lagrange multiplier associated to the critical point u ρ on B ρ and is given explicitly by ω ρ = 1 ρ ( u ρ + φ uρ u ρ dx u ρ ) p dx. Actually we are interested in the existence of solutions of (4.) with minimal energy (constrained to the spheres), therefore we are reduced to study the minimization problem we have considered in the abstract framework, i.e. I ρ = inf B ρ I(u). (4.5) Note also that this problem makes sense for <p<1/3; indeed it is well known (see e.g. [4]) the following

12 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) Proposition 4.1. For every ρ> and p (, 1/3) the functional I is bounded from below and coercive on B ρ. As a consequence of this proposition, whenever ρ is fixed and {u n } is a minimizing sequence for I ρ, we implicitly assume that {u n } is bounded in H 1 ( ), so weakly convergent up to subsequences. We recall that the energy and the charge associated to the wave function ψ(x,t) evolving according to (4.1) are given by E ( ψ(x,t) ) := 1 ψ dx + 1 ( x 1 ψ ) ψ dx 1 4 p = E ( ψ(x,) ) ψ p dx and Q ( ψ(x,t) ) := 1 ψ dx = Q ( ψ(x,) ). So our action functional I is exactly the energy of the standing wave and the charge is the L -norm. We underline that in a recent paper by Benci and Fortunato [6] the relevance of the energy/charge ratio for the existence of standing waves in field theories has been discussed under a general framework. In our context, the function s I s has the physical interpretation of the s ratio between the infimum of the energy of the standing waves with fixed charge and the charge itself. So conditions (.5) and (.6) seem less abstract and concern the properties of the above mentioned ratio. In spite of the case in which the frequency ω is fixed, the problem with fixed charge has been less investigated: there is just a result by Sanchez and Soler [] in the case p = 8/3 and by the authors in [4] in the case 3 <p<1/3. Moreover for a nonhomogeneous nonlinearity of the form u 1/3 1/3 u 8/3 8/3 we quote [8]. For p = 8/3, the so-called Schrödinger Poisson Slater equation, the existence of minimizers is proved in [] only for ρ small, that is for small values of the charge. The difficulty, in considering all ρ>, concerns the possibility of dichotomy for an arbitrary minimizing sequence. On the other hand, in [4] it is proved that for p (3, 1/3) the functional I ρ has a minimum on B ρ provided that ρ is greater than a certain ρ 1. In particular it is proved that I ρ < forρ (ρ 1, ) and that condition (MD) holds by means of scaling arguments. In case p (, 3) the standard scaling arguments do not permit to show that (MD) holds and the possibility of dichotomy for an arbitrary minimizing sequence cannot be excluded. With our abstract frameworks we are able to prove that (MD) holds for p in the above range at least for small value of ρ and the compactness of every minimizing sequence up translations is proved. Indeed our result is the following. Theorem 4.1. Let p (, 3). Then there exists ρ 1 (depending on p) such that all the minimizing sequences for (4.5) are precompact in H 1 ( ) up to translations provided that <ρ<ρ 1.

13 498 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) With standard arguments, following [9] and [], the compactness of minimizers on H 1 ( ) and the conservation laws of energy and charge give rise to the orbital stability of the standing waves ψ ρ = e iω ρt u ρ without further efforts; so we get the following result, whose details are given in the final subsection. Theorem 4.. Let p (, 3). Then the set for ρ small, is orbitally stable. S ρ = { e iθ u(x): θ [, π), u = ρ, I(u)= I ρ }, We mention [14] where the orbital stability of standing waves for (4.3) is achieved by following the original approach of [1]. Using Mountain Pass arguments, Kikuchi in [14] proved that for p (, 3) there exist orbitally stable standing waves u ω (x)e iωt for ω (,ω 1 ). However, by studying the functional (4.3), nothing can be said a priori on the L -norm of the solutions. Before the proof of Theorem 4.1, we define, for short, the following quantities: A(u) := u dx, B(u) := φ u u dx, C(u) := u p dx so that I(u)= 1 A(u) B(u) + 1 p C(u). Note that if we set u λ ( ) = λ δ u(λ γ ( )), δ,γ R, λ>, then φ uλ (x) = λ δ+γ u(λ γ y) λ γ x λ γ y dy = λ (δ γ) u(y) λ γ x y dy = λ(δ γ) φ u ( λ γ x ). To prove our theorem, we have to verify all the hypotheses of Theorem.1. In particular, the crucial hypothesis concerning the behavior of I ρ near zero (condition (.6)), is obtained by a comparison argument with the simpler constrained minimization problem related to the nonlinear energy functional (.8) (see Step 4 below). It corresponds to the standing wave of the Schrödinger equation without the contribution of the nonlocal term. For this reason in a brief Appendix A it is proved that lim s G s /s =. Now we can give the Proof of Theorem 4.1. The proof is now divided in some steps where we verify all the hypotheses of Theorem.1. Step 1. Condition (1.3) holds and the functional T satisfies (.1). These are proved in [, Proposition.3] and [4, Proposition 3.1] respectively. Step. If <p<3, then condition (.4) is satisfied.

14 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) We already know that I s > for all s>,seee.g.[4],sowejusthavetoprovethati s < for every s>. Let u H 1 ( ) and choose the family of scaling paths given in (.3) g u (θ) = θ 1 3 β u ( x/θ β) such that Θ(θ) = θ and g u (θ) = θ. We easily find the following scaling laws: For β = we get A ( g u (θ) ) = θ β A(u), B ( g u (θ) ) = θ 4 β B(u), C ( g u (θ) ) = θ (1 3 β)p+3β C(u). I ( g u (θ) ) = θ 6 A(u) + θ 6 θ 4p 6 B(u) + 4 p C(u) for θ, since 4p 6 < 6 and C(u) <. This proves that there exists a small θ such that I s < s (,θ ]. Then by Step 1 and Remark.1 we conclude that I s < for every s>. Step 3. For every ρ>, all the minimizing sequences {v n } for I ρ have a weak limit, up to translations, different from zero. Furthermore the weak limit is in M(ρ). Let {v n } be a minimizing sequence in B ρ for I ρ. Notice that for any sequence {y n } we have that v n (. + y n ) is still a minimizing sequence for I ρ. Then the proof of this step can be concluded provided that we show the existence of a sequence {y n } such that the weak limit of v n (. + y n ) is different from zero. By the well-known Lions lemma [17], it follows that if ( lim sup n y B(y,1) ) v n dx =, then v n inl q ( ) for any q (, ) and so C(v n ). Here B(a,r) ={x : x a r}. On the other hand, by Step, I ρ < so we have necessarily that sup y B(y,1) v n dx δ>. In this case we can choose y n such that vn (. + y n ) dx δ> B(,1)

15 5 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) and hence, due to the compactness of the embedding H 1 (B(, 1)) L (B(, 1)), we deduce that the weak limit of the sequence v n (. + y n ), let us call it v, is not the trivial function. It follows that v M(ρ) (if v = ρ it is trivial, otherwise use Lemma 3.1). Step 4. The function s I s satisfies (.5) and (.6). We first prove that if ρ n ρ then lim n I ρ n = I ρ. For every n N, letw n B ρn such that I(w n )<I ρ n + n 1 < n 1. Therefore, by using the interpolation and the Sobolev inequality, we get 1 w n 6 p 3(p ) Cρ n w n 1 w n 1 p w n p p I(w n )< 1 n. Since 3(p ) < and {ρ n } is bounded, we deduce that {w n } is bounded in H 1( ). In particular {A(w n )} and {C(w n )} are bounded sequences, and also {B(w n )} since in general, see e.g. [19]. So we easily find u H 1( ) : B(u) = ( ) ρ I ρ I w n = 1 ( ρ ρ n ρ n = I(w n ) + o(1)<i ρ n + o(1). ) A(w n ) + 1 ( ρ 4 ρ n φ u u dx C u 4 H 1 ( ), ) 4 B(w n ) + 1 ( ) ρ p C(w n ) p ρ n On the other hand, given a minimizing sequence {v n } B ρ for I ρ,wehave ( ) ρn I ρ n I ρ v n = I(v n ) + o(1) = I ρ + o(1) which, join to the previous computation, gives lim n I ρ n = I ρ. I In order to show that lim ρ ρ =, we notice that ρ G ρ ρ I ρ ρ < where { 1 G ρ = inf u D 1, 1 p } u p dx. Since G ρ /ρ (see Lemma A.1 of Appendix A) we easily conclude.

16 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) Step 5. For small ρ the functional I satisfies (.7). Since / M(ρ), A(u), B(u) and C(u) are all different from zero whenever u M(ρ). We claim now that u M(ρ): A(u) p B(u) + C(u) =. (4.6) p Indeed, for u M(ρ) (i.e. u = μ (,ρ] and I(u)= I μ ) we define v(θ,u) = θ 3 u( x θ ) so that v(θ,u) = u. It follows that A ( v(θ,u) ) = θ A(u), B ( v(θ,u) ) = θ 1 B(u), C ( v(θ,u) ) = θ 3 3 p C(u). Since the map θ I(v(θ,u))is differentiable and u achieves the minimum on B μ, we get d dθ I( v(θ,u) ) = θ=1 which is exactly our claim (4.6). Now, for u we compute explicitly h gu (θ) by choosing the family of scaling paths of u parametrized with β R given by G β u = { g u (θ) = θ 1 3 β u ( x/θ β)} G u. (4.7) All the paths of this family have as associated function Θ(θ) = θ. We get h gu (θ) = 1 ( θ β θ ) A(u) + 1 ( θ 4 β θ ) B(u) + 1 ( θ (1 3 β)p+3β θ ) C(u), 4 p which shows that the paths in G β u are admissible, i.e. h gu is differentiable for every g u G β u.we have also, for g u G β u : h g u (1) = βa(u) + β B(u) + (1 3 β)p + 3β C(u). 4 p We will show that the admissible scaling path satisfying d dθ h g u (θ) θ=1 can be chosen in G β u. For future reference we compute also I(g u (θ)) θ u = h g u (θ) θ u = 1 u + I(u) u ( 1 θ β A(u) θ β B(u) + 1 p θ (1 3 β)p+3β C(u) ). (4.8)

17 5 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) To prove Step 5 we argue by contradiction. Assume that there exists a sequence {u n } M(ρ) with ρ u n = ρ n such that for all β R (that is: for all g un G β u n ) h g un (1) = βa(u n ) + β B(u n ) + (1 3 β)p + 3β C(u n ) = 4 p then, by using (4.6) we get and hence (again by (4.6)) 1 B(u n) + p p C(u n) = B(u n ) = A(u n ), C(u n ) = p p A(u n), I(u n ) = A(u n) + B(u n) 4 + C(u n) p = 3 p p A(u n). (4.9) The contradiction is achieved by showing that relations (4.9) are impossible for p (, 3) for small ρ. We know that { I(un ) = I ρ n (by continuity), A(u n ), B(u n ), C(u n ) (by (4.9)). (4.1) Because of the following Hardy Littlewood Sobolev inequality B(u n ) = u n (x) u n (y) dxdy c u n 4 1/5 x y that we will frequently use, it is convenient to consider some cases. Case (a): <p<1/5. Then B(u n ) c u n 4 1/5 c u n 4α p u n 4(1 α) 6, α= 3p (6 p). We get, thanks to (4.9) and the Sobolev inequality u n 6 SA(u n), B(u n ) cb(u n ) 4α p B(u n ) 4(1 α). (4.11) We notice that 4α p + 4(1 α) > 1 since p<3. This is in contradiction with (4.1).

18 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) Case (b): p = 1/5. This case is simpler: thanks to (4.9) we get which contradicts (4.1). Case (c): 1/5 <p<8/3. Interpolating L 1/5 between L and L p we get u n 1/5 1/5 = cb(u n) c u n 4 1/5 u n p p = cb(u n ) c u n 4 1/5 c u n 4α u n p 4(1 α) 5p 1, α= 6(p ) i.e. u n p p ρ 4α n u n 4(1 α) p. Since p<4(1 α),i.e.p<8/3, we get a contradiction with (4.1). Case (d): p = 8/3. Again by interpolation we get B(u n ) c u n 4 1/5 cρ4/3 n u n 8/3 8/3, and again, using that B(u n ) = u n 8/3 8/3 we get a contradiction. Case (e): 8/3 <p<3. In this case for u satisfying (4.9), with u = ρ we get I θ ρ θ ρ I(g u (θ)) θ ρ = 1 ρ ( 1 θ β A(u ) + 1 θ β A(u ) + A(u ) ) p θ (1 3 β)p+3β. Now let us choose β = ( p) 1 3p so that < β = (1 3 ) β p + 3β < β. Hence we obtain I θ ρ θ ρ I(g u (θ)) θ ρ = p 3 p I(u ) ρ = A(u [ ) 4 p ρ ( p) θ 4(p ) 1 3p + 1 ] θ 4(4 p) 1 3p [ 4 p ( p) θ 4(p ) 1 3p + 1 ] θ 4(4 p) 1 3p

19 54 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) and so renaming θ ρ = s we get I s 4(p ) cs 1 3p + o ( s 4(p ) ) 1 3p (4.1) s for sufficiently small s. On the other hand for u n satisfying (4.9) that is u n p p = cb(u n ) c u n 4 1/5 c u n 4α u n p 4(1 α) 5p 1, α= 6(p ), u n p p cρ 4α n u n 4(1 α) p. (4.13) Since now 8/3 <p(that is 4(1 α) < p) we cannot argue as in Case (c) to get the contradiction. (5p 1) 3p 8 But we deduce from (4.13) that u n p p cρn, and hence using (4.9), I ρ n ρ n 4(p ) 3p 8 cρn. (4.14) Combining (4.14) with (4.1) we find 4(p ) 3p 8 cρn I ρn ρn This drives to a contradiction for ρ n since 4(p ) 3p 8 4(p ) 1 3p cρn + o ( ρ > 4(p ) 1 3p. 4(p ) 1 3p n Summing up, we have verified all the hypotheses of Theorem.1 so u n ū in H 1 ( ) and this finishes the proof The orbital stability In this subsection we prove Theorem 4. following the ideas of [9]. First of all we recall the definition of orbital stability. We define S ρ = { e iθ u(x): θ [, π), u = ρ, I(u)= I ρ }. We say that S ρ is orbitally stable if for every ε> there exists δ> such that for any ψ H 1 ( ) with inf v Sρ v ψ H 1 ( ) <δwe have ). t > inf v S ρ ψ(t,.) v H 1 ( ) <ε, where ψ(t,.) is the solution of (4.1) with initial datum ψ. We notice explicitly that S ρ is invariant by translation, i.e. if v S ρ then also v(. y) S ρ for any y.

20 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) In order to prove Theorem 4. we argue by contradiction assuming that there exists a ρ such that S ρ is not orbitally stable. This means that there exists ε> and a sequence of initial data {ψ n, } H 1 ( ) and {t n } R such that the maximal solution ψ n, which is global and ψ n (,.)= ψ n,, satisfies lim inf ψ n, v n + H v S 1 ( ) = and inf ψ n (t n,.) v ρ v S H 1 ( ε. ) ρ Then there exists u ρ H 1 ( ) minimizer of I ρ and θ R such that v = e iθ u ρ and ψ n, v = ρ and I(ψ n, ) I(v)= I ρ. Actually we can assume that ψ n, B ρ (there exist α n = ρ/ ψ n, 1 so that α n ψ n, B ρ and I(α n ψ n, ) I ρ, i.e. we can replace ψ n, with α n ψ n, ). So {ψ n, } is a minimizing sequence for I ρ, and since I ( ψ n (., t n ) ) = I(ψ n, ), also {ψ n (., t n )} is a minimizing sequence for I ρ. Since we have proved that every minimizing sequence has a subsequence converging (up to translation) in H 1 -norm to a minimum on the sphere B ρ, we readily have a contradiction. Finally notice that, writing ψ(x,t)= ψ(x,t) e is(x,t) we get I ( ψ(x,t) ) = I ( ψ(x,t) ) + ψ(x,t) S(x,t) dx, so we easily conclude that the minimizer u ρ has to be real valued. Note added in proof In a private comunication, O. Sanchez informed the authors that in collaboration with I. Catto and J. Soler they are obtaining results similar to our Theorem 4.1. Acknowledgments The authors would like to thank V. Benci, L. Jeanjean and O. Sanchez for useful and stimulating discussions on the subject and the referee for his suggestions. The authors are partially supported by M.I.U.R. project PRIN7 Variational and topological methods in the study of nonlinear phenomena. The second author is also supported by J. Andalucía (FQM 116) and FAPESP, São Paulo, Grant 11/ Appendix A As already anticipated, we prove here that lim s G s /s = where G ρ = inf B ρ G(u) (A.1)

21 56 J. Bellazzini, G. Siciliano / Journal of Functional Analysis 61 (11) and G is the functional associated to the pure Schrödinger equation. It is defined in (.8), that is G(u) = 1 u D 1, 1 p u p dx. Here we can even allow <p<1/3. It is known that, for every ρ>, u ρ B ρ such that G ρ = G(u ρ )<, u B ρ : G(u) 1 u D 1, b p ρ 6 p 3(p ) u. D 1, For these facts the reader is referred to [] and [1]. As a consequence we get ( 1 >G(u ρ ) b pρ 6 p p ) 3p 1 u ρ u D 1, ρ D 1, (A.) which implies, since p<1/3, that the sequence {u ρ } ρ> is bounded in D 1, for ρ. (A.3) Lemma A.1. We have lim ρ G ρ ρ =. Proof. Since the minimizer u ρ for G ρ satisfies u ρ u ρ p u ρ = ω ρ u ρ, (A.4) we get, taking into account (A.), ω ρ = u ρ D 1, u ρ p dx u ρ dx 1 u ρ D 1, 1 p u ρ dx u ρ p dx = G(u ρ) ρ < (A.5) where ω ρ is the Lagrange multiplier associated to the minimizer. Actually we prove that lim ρ ω ρ =, so by comparison in (A.5) we get the lemma. To show that lim ρ ω ρ = we argue by contradiction by assuming that there exists a sequence ρ n such that ω ρn < c for some c (, 1). Since the minimizers u n := u ρn satisfy Eq. (A.4), we get c u n H 1 u n dx + c u n dx u n dx ω ρn u n dx = u n p dx C u n p, H 1

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