On the radiality of constrained minimizers to the Schrödinger–Poisson–Slater energy

Available online at www.sciencedirect.com

Ann. I. H. Poincaré – AN 29 (2012) 369–376

www.elsevier.com/locate/anihpc

On the radiality of constrained minimizers to the Schrödinger–Poisson–Slater energy

Vladimir Georgiev

, Francesca Prinari

, Nicola Visciglia

aUniversità di Pisa, Dipartimento di Matematica, Largo B. Pontecorvo 5, 56124, Pisa, Italy bUniversità di Ferrara, Dipartimento di Matematica, via Machiavelli 35, 44100, Ferrara, Italy Received 1 October 2011; received in revised form 7 December 2011; accepted 12 December 2011

Available online 14 December 2011

Abstract

We study the radial symmetry of minimizers to the Schrödinger–Poisson–Slater (S–P–S) energy:

inf u∈H¹(R³) u_L2(R3)=ρ

1 2 R³

|∇u|²+1 4 R³

R³

|u(x)|²|u(y)|²

|x−y| dx dy− 1 p R³

|u|^pdx

provided that 2< p <3 andρis small. The main result shows that minimizers are radially symmetric modulo suitable translation.

©2012 Elsevier Masson SAS. All rights reserved.

Résumé

On montre la radialité des minimiseurs de l’énergie de Schrödinger–Poisson–Slater

inf u∈H¹(R³) u_L2(R3)=ρ

1 2 R³

|∇u|²+1 4 R³

R³

|u(x)|²|u(y)|²

|x−y| dx dy− 1 p R³

|u|^pdx

pourvu que 2< p <3 etρest petit.

©2012 Elsevier Masson SAS. All rights reserved.

The following minimization problem associated to Schrödinger–Poisson–Slater (S–P–S) energy functional has been extensively studied in the literature (see for instance [4–6,16] and all the references therein):

I_ρ,p= inf

u∈H¹(R³) u_L2(R3)=ρ

Ep(u) (0.1)

* Corresponding author.

E-mail addresses:georgiev@dm.unipi.it (V. Georgiev), francesca.prinari@unife.it (F. Prinari), viscigli@dm.unipi.it (N. Visciglia).

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2011.12.001

where

Ep(u)=1 2

R³

|∇u|²+1 4

R³

R³

|u(x)|²|u(y)|²

|x−y| dx dy−1 p

R³

|u|^pdx.

The corresponding set of minimizers will be denoted since now on byMρ,p. It has been proved in [16] (based on the technique introduced in [6]) thatMρ,8/3= ∅provided that 0< ρ < ρ0for a suitableρ0>0 (i.e. under a smallness assumption on the charge). In [5] it is proved thatMρ,p= ∅provided thatρ >0 is small and 2< p <3. In [4] it is treated the case 3< p <¹⁰₃ andρsufficiently large.

The main aim to look at the minimization problem (0.1) is to construct (following the original argument by [7]) orbitally stable standing wave solutions to the following evolution problem

i∂tψ+ψ− 1

|x|∗ |ψ|²

ψ+ψ|ψ|^p−2=0 (t, x)∈R×R³.

For the sake of completeness we recall that standing waves are solutions of the following type ψ (t, x)=e^iωtv(x)

for a suitableω∈Randv(x)∈H¹(R³).

In this paper we study the radiality (up to translation) of the functions inMρ,pprovided thatρ >0 is small enough and 2< p <3.

There are different results on the symmetry of the minimizers. The basic result due to Gidas, Ni and Nirenberg [11]

implies the radial symmetry of the minimizers associated with the semilinear elliptic equation u+f (u)=0,

provided suitable assumptions on the function f (u) are satisfied and the scalar function u is positive. As in the previous result due to Serrin [17], the proof is based on the maximum principle and Hopf’s lemma.

The symmetry of the energy functional (even with constraint conditions) can not imply in general the radial sym- metry of the minimizers. This phenomena was discovered and studied in the works [8–10] in the scalar case.

Different techniques have been developed in the literature to prove the radiality of minimizers to suitable varia- tional problems. We quote some of them (see also all the references therein): [3] where it is proved a very general radiality result for nonnegative critical points of suitable variational problems (however Hartree type nonlinearity is not allowed), [13,15] where the case of nonlocal Hartree type nonlinearity is treated. However, as far as we can see, those techniques do not work in our context since the potential energy in SPS is defocusing on the nonlocal term (the Hartree nonlinearity) and focusing on the local term (theL^pnorm).

To underline the difficulty notice that it is not obvious to answer to the following weaker question:

Is there at least a radially symmetric function belonging toMρ,p?

A general tool that could be useful to provide an answer to the question above is the Schwartz rearrangement map u→u^∗. The following properties are well known (see [14]):

∇u^∗

L²(R³)∇u²_L2(R³);

R³

R³

|u(x)|²|u(y)|²

|x−y| dx dy

R³

R³

|u^∗(x)|²|u^∗(y)|²

|x−y| dx dy; u^∗

L^q(R³)= uL^q(R³).

As a consequence there is a competition between the kinetic energy and the nonlocal energy which makes unclear whether or not the set Mρ,p is invariant under the map u→u^∗ (and hence it makes useless the rearrangement technique to provide an answer to the question raised above).

Next we state the main result of this paper.

Theorem 0.1.For every2< p <3there existsρ0=ρ0(p) >0such that

∀(v, ρ)∈Mρ,p×(0, ρ0)∃τ ∈R³ such thatv(x+τ )=v(|x| +τ ) ∀x∈R³.

Remark 0.1.Recall that in [5] it is proved thatMρ,p= ∅for 2< p <3 andρ >0 small.

Remark 0.2.Notice that in Theorem 0.1 the physically relevant casep=8/3 is allowed.

Next we fix some notations.

Notation.We shall denote byQ(x)the unique function such that:

(1) Q∈H¹(R³);

(2) Q(x)is radially symmetric;

(3) Q(x) >0 for everyx∈R³; (4) Q²_L2(R³)=1;

(5) Q(x)solves the following elliptic problem

−Q+ω0Q=Q|Q|^p−2 onR³ for a suitableω₀>0 (which is unique).

(We recall that the existence and uniqueness of a functionQthat satisfies the properties mentioned above follows by combining the results in [7,11,12] provided that 2< p <¹⁰₃.)

IfQ(x)is the radial function, satisfying the above relations, then we introduce G=

Q(x+τ )τ∈R³

and for everyτ∈R³we writeQ_τ =Q(x+τ ).

TQ(resp.TQ_τ) denotes the tangent space of the manifoldGat the pointQ(resp.Qτ). We also denote byT_Q^⊥

τ the intersection ofH¹(R³)with the orthogonal space (w.r.t. theL²scalar product) ofT_Qτ.

LetMbe a vector space thenπ_M denotes the orthogonal projection, with respect to theL²(R³)scalar product, on the vector spaceM.

H¹is the usual Sobolev space endowed with the following Hilbert norm u²_H1=

R³

|∇u|²dx+ω0

R³

|u|²dx, whereω₀is the constant introduced above.

H_rad¹ denotes the functions inH¹that are radially symmetric.

L^pwill denote the spaceL^p(R³).

In general . . . dxand

. . . dx dydenote _R3. . . dxand _R3 R³. . . dx dy.

Assume(H, (. , .))is a Hilbert space andF:H→Ris a differentiable functional,then∇uF is the gradient ofF at the pointu∈H.

Let(X,.)be a Banach space, thenB_X(x, r)denotes the ball of radiusr >0 centered inx∈X.

LetΦ be a differentiable map between two Banach spaces(X,.X)and(Y,.Y)thendΦ_x∈L(X, Y )denotes the differential ofΦat the pointx∈X.

1. An equivalent problem

By the rescalingu_ρ(x)=ρ^{4−3(p−2)4} u(ρ

2(p−2)

4−3(p−2)x)it is easy to check that the minimization problem (0.1) is equiva- lent to the following one:

J_ρ,p= inf

u∈H¹ u_L2=1

1 2

|∇u|²dx+ρ^α(p) |u(x)|²|u(y)|²

|x−y| dx dy−1 p

|u|^pdx

whereα(p)=⁸⁽³₁₀₋⁻_3p^p). Notice thatα(p) >0 provided that 2< p <3. Motivated by this fact we introduce the following minimization problem

K_ρ,p= inf

u∈H¹ u_L2=1

Eρ,p(u)

where

Eρ,p(u)=1 2

|∇u|²dx+ρ |u(x)|²|u(y)|²

|x−y| dx dy−1 p

|u|^pdx.

We also denote byNρ,pthe corresponding minimizers:

Nρ,p=

v∈H¹Eρ,p(v)=K_ρ,p .

It is easy to prove that Theorem 0.1 is equivalent to the following proposition.

Proposition 1.1.For every2< p <3there existsρ₀=ρ₀(p) >0such that any functionv∈Nρ,pis(up to transla- tion)radially symmetric provided that0< ρ < ρ₀.

The rest of the paper is devoted to the proof of Proposition 1.1.

In the sequel the functionQ(x)and the constantω0>0 are the ones defined in the introduction.

Next result will be useful in the sequel.

Proposition 1.2.Let2< p <3andv_k∈Nρ_k,pwherelimk→∞ρ_k=0. Then up to subsequence there existsτ_k∈R³ such that

v_k(x+τ_k)→Q inH¹.

Proof. First step:Kρ_k,p→K0,pask→ ∞. First notice that

K_0,pK_ρk,p

due to the positivity ofρk. Hence it is sufficient to prove lim sup_k→∞Kρ_k,pK0,p. This fact follows from K_ρk,pEρk,p(Q)=Ep(Q)+ρ_k |Q(x)|²|Q(y)|²

|x−y| dx dy=K_0,p+o(1).

Second step:v_kconverge toQup to subsequence and translation.

By the previous step we deduce that{v_k}is a minimizing sequence forK_0,p. As a consequence of the results proved in [7,11,12] we deduce that{v_k}converge strongly (up to translation) toQ(x). 2

In next result we get a qualitative information on the Lagrange multipliers associated to the constrained minimizers belonging toNρ,pwhenρ >0 is small enough.

Proposition 1.3.Let2< p <3be fixed. For every >0there existsρ() >0such that sup

ω∈Aρ

|ω−ω0|< ∀0< ρ < ρ()

where Aρ=

ω∈R−v+ωv+ρ

|v|²∗ 1

|x|

v−v|v|^p−2=0, v∈Nρ,p

.

Proof. By looking at the equation satisfied byv∈Nρ,pwe deduce

ω=v^p_L^p− ∇v²_L2−ρ _|v(x)|²_|v(y)|²

|x−y| dx dy v²_L2

.

The proof can be concluded since by Proposition 1.2 we get that the r.h.s. converges to Q^p_Lp− ∇Q²_L2

Q²_L2

=ω₀

forρ→0. 2

2. The implicit function argument

In this section we present some results strictly related to the implicit function theorem (see [1]).

Proposition 2.1.There exist₀,₁>0such that

∀u∈B_H1(Q, ₀)∃!τ (u)∈R³, R(u)∈T_Q^⊥

τ (u) s.t.

maxτ (u)

R³,R(u)

H¹

< 1 and u=Qτ (u)+R(u).

Moreoverlimu→Qτ (u)_R³=limu→0R(u)H¹=0and the nonlinear operators P :B_H1(Q, ₀)→G,

R:B_H1(Q, ₀)→H¹

(whereP (u)=Q_{τ (u)}andR(u)is defined as above)are smooth.

Remark 2.1. Every radially symmetric function u ∈H_rad¹ can be written as u=Q+(u−Q) and moreover u−Q∈T_Q^⊥(this follows by noticing thatT_Q=span{∂_xiQ|i=1, . . . , n}). In particularP u=Qfor everyu∈H_rad¹ . Remark 2.2.Notice thatT_Qτ = {v(x+τ )|v∈T_Q}. As a consequence it is easy to proveP (u(x+τ ))=P (u)(x+τ ) and hence (sinceP (Q)=Q)P (Qτ)=Qτ.

Proof. It is sufficient to apply the implicit function theorem to the map Φ:G×H¹(Q_τ, h)→

Q_τ+h,

h, v₁(x+τ ) ,

h, v₂(x+τ ) ,

h, v₃(x+τ )

∈H¹×R³ wherespan{v1, v2, v3} =TQand( , )denotes the usualL²scalar product.

Next we shall prove that dΦ(Q,0)∈L

TQ×H¹, H¹×R³ is invertible. By explicit computation we get

dΦ_(Q,0):T_Q×H¹(w, k)→

w+k, (k, v₁), (k, v₂), (k, v₃)

∈H¹×R³ and hence:

dΦ_(Q,0)(−h, h)h∈T_Q

= {0} ×R³; dΦ_(Q,0)(0, h)h∈T_Q^⊥

=T_Q^⊥× {0};

dΦ_(Q,0)(h,0)h∈T_Q

=T_Q× {0}.

As a consequence we deduce thatdΦ(Q,0)is surjective.

Next we prove thatdΦ_(Q,0)is injective. Assume that(w, k)∈T_Q×H¹satisfies dΦ_(Q,0)(w, k)=(0,0)∈H¹×R³

which in turn (by looking at the explicit structure ofdΦ_(Q,0)) is equivalent to k∈T_Q^⊥ and w= −k,

hencew∈T_Q^⊥. By combining this fact with the hypothesisw∈T_Qwe getw=0 and alsok= −w=0. 2 Proposition 2.2.There exists₂>0such that the equation

−w+ωw+ρ

|w|²∗ 1

|x|

w−w|w|^p−2=0

has a solutionw(ρ, ω)∈H_rad¹ for every(ρ, ω)∈(0, 2)×(ω0−2, ω0+2). Moreover

(ω,ρ)lim→(ω₀,0)w(ω, ρ)=Q inH¹.

Proof. It follows by an application of the implicit function theorem at the following operator:

Φ:R×R⁺×H_rad¹ (ω, ρ, u)→ ∇uFρ,ω,p∈H_rad¹ where

Fρ,ω,p(u)=Eρ,p(u)+ω 2u²_L2. Notice that∇QF0,ω₀,p=0 and moreover

dΦ_(0,ω0,Q)[h] =h+Kh ∀h∈H_rad¹ (2.1)

where

(p−1)⁻¹K=(−+ω₀)⁻¹◦ |Q|^p−2.

Due to the decay properties of the functionQ(x)and the Rellich Compactness Theorem (see [2]) the operatorH_rad¹ v→ |Q|^p−2v∈L²_radis compact. Moreover(−+ω0)⁻¹∈L(L²_rad, H_rad¹ )and hence the operatorK∈L(H_rad¹ , H_rad¹ ) is a compact operator. By combining this fact with (2.1) we deduce thatdΦ(0,ω₀,Q)∈L(H_rad¹ , H_rad¹ )is a Fredholm operator with index zero (see [2]).

Moreover by the work [18] it is easy to deduce that ker_H¹

raddΦ_(0,ω0,Q)=

h∈H_rad¹

R³ h+Kh=0

= {0}

and hencedΦ_(0,ω0,Q)is invertible (sincedΦ_(0,ω0,Q)is injective and has Fredholm index zero). 2

In the next proposition (and along its proof) the operators P (u), R(u) and the number0>0 are the ones in Proposition 2.1.

Proposition 2.3.There exist₃, ₄>0such that:

∀(ω, ρ)∈(ω0−3, ω0+3)×(0, 3)∃!u=u(ρ, ω)∈H¹ s.t.

u(ρ, ω)−Q

H¹< ₄, P (u)=Q and π_T⊥

Q(∇uFρ,ω,p)=0 (2.2)

where

Fρ,ω,p(u)=Eρ,p(u)+ω

2u²_L2. (2.3)

Proof. It is sufficient to apply the implicit function theorem to the map Φ:R×R×B_H1(Q, 0)(ρ, ω, u)→

π_T⊥

Q(∇uFρ,ω,p), P (u)

∈T_Q^⊥×G.

Hence we have to show thatdΦ_(0,ω0,Q)∈L(H¹, T_Q^⊥×T_Q)is invertible. Recall that by Remark 2.2 we getP (Q_τ)= Q_τ and henceΦ(0, ω0, Q_τ)=(0, Qτ)which in turn implies

dΦ_(0,ω0,Q)[v] =(0, v) ∀v∈T_Q. (2.4)

Arguing as in Proposition 2.2 we deduce that the operator d(∇uFρ,ω,p)(0,ω₀,Q)∈L

H¹, H¹

is a Fredholm operator of index zero inH¹. (2.5) Moreover by the work [18] we get

TQ=kerd(∇uF)(0,ω₀,Q) (2.6)

and by the self-adjointness (w.r.t. to theL²scalar product) of the operatord(∇uFρ,ω,p)_(0,ω0,Q)we get d(∇uFρ,ω,p)_(0,ω0,Q)

T_Q^⊥

⊂T_Q^⊥. (2.7)

By combining (2.5), (2.6) and (2.7) we conclude that dΦ(0,ω₀,Q)∈L

T_Q^⊥, T_Q^⊥

is invertible. (2.8)

By using (2.4) and (2.8) it is easy to deduce thatdΦ_(0,ω0,Q)∈L(H¹, T_Q^⊥×T_Q)is invertible. 2 3. Proof of Proposition 1.1

Recall that the operatorsP (u),R(u)are the ones introduced along Proposition 2.1.

Let v∈Nρ,p. Due to Proposition 1.2 for every >0 there existsρ₁() >0 such that (up-to translation) v∈ B_H1(Q, )provided thatρ < ρ₁(). Moreovervsolves the problem

−v+ωv+ρ

|v|²∗ 1

|x|

v−v|v|^p−2=0 or equivalently

∇vFρ,ω,p=0 (3.1)

(see (2.3) for definition ofFρ,ω,p) for a suitableωsuch that|ω−ω0|< provided thatρ < ρ2()(see Proposition 1.3). Notice that by Proposition 2.1 there exists δ() >0 such that we can write in a unique way (provided that >0 is small enough)v(x)=Q_τ +r(x)withrH¹ < δandr∈T_Q^⊥

τ, and hencev(x−τ )=Q+r(x−τ )with r(x−τ )∈T_Q^⊥. By combining this fact with (3.1) (recall the translation invariance of the functionalFρ,ω,p) we get

P

v(x−τ )

=Q and π_T⊥

Q(∇v(x−τ )Fρ,ω,p)=0. (3.2)

On the other hand by combining Remark 2.1 with Proposition 2.2 we deduce thatw(ρ, ω)∈H_rad¹ (given in Propo- sition 2.2) satisfies the same properties ofv(x−τ )in (3.2). By the uniqueness property included in Proposition 2.3 (see (2.2)) we getv(x−τ )=w(ρ, ω)and hencev(x−τ )is radially symmetric.

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