On stability and instability of standing waves for the nonlinear Schrodinger equation with inverse-square potential

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On stability and instability of standing waves for the nonlinear Schrodinger equation with inverse-square

potential

Abdelwahab Bensouilah, van Duong Dinh, Shihui Zhu

To cite this version:

Abdelwahab Bensouilah, van Duong Dinh, Shihui Zhu. On stability and instability of standing waves

for the nonlinear Schrodinger equation with inverse-square potential. Journal of Mathematical Physics,

American Institute of Physics (AIP), 2018, 59 (101505). �hal-01936034�

NONLINEAR SCHR ¨ODINGER EQUATION WITH INVERSE-SQUARE POTENTIAL

ABDELWAHAB BENSOUILAH, VAN DUONG DINH, AND SHIHUI ZHU

Abstract. We consider the focusing nonlinear Schr¨odinger equation with inverse square poten- tial

i∂tu+ ∆u+c|x|⁻²u=−|u|^αu, u(0) =u0∈H¹, (t, x)∈R⁺×R^d, where d ≥3, c6= 0, c < λ(d) = ^d−2₂ 2

and 0 < α ≤ ⁴

d. Using the profile decomposition obtained recently by the first author [1], we show that in theL²-subcritical case, i.e. 0< α <_d⁴, the sets of ground state standing waves are orbitally stable. In theL²-critical case, i.e. α=_d⁴, we show that ground state standing waves are strongly unstable by blow-up.

1. Introduction

Consider the focusing nonlinear Schr¨odinger equation with inverse-square potential i∂_tu+ ∆u+c|x|⁻²u = −|u|^αu, (t, x)∈R⁺×R^d,

u(0) = u₀∈H¹, (1.1)

where d≥3,u:R⁺×R^d→C,u0:R^d →C, c6= 0 satisfiesc < λ(d) := ^d−2₂ ²

and 0< α≤⁴_d. The Schr¨odinger equation (1.1) appears in a variety of physical settings, such as quantum field equations or black hole solutions of the Einstein’s equations [8,7,16]. The mathematical interest in the nonlinear Schr¨odinger equation with inverse-square potential comes from the fact that the potential is homogeneous of degree −2 and thus scales exactly the same as the Laplacian.

LetP_c⁰denote the natural action of−∆−c|x|⁻²onC₀^∞(R^d\{0}). Whenc≤λ(d), the operator P_c⁰ is a positive semi-definite symmetric operator. Indeed, we have the following identity

ϕ, P_c⁰ϕ

= Z

|∇ϕ(x)|²−c|x|⁻²|ϕ(x)|²dx= Z

∇ϕ(x) +ρx|x|⁻²ϕ(x)

2dx≥0, for allϕ∈C₀^∞(R^d\{0}), where

ρ:= d−2

2 −

s d−2

2 2

−c.

Denote P_c the self-adjoint extension ofP_c⁰. It is known (see [16]) that in the rangeλ(d)−1< c <

λ(d), the extension is not unique. In this case, we do make a choice among possible extensions such as Friedrichs extension. Note also that the constantλ(d) is the sharp constant appearing in Hardy’s inequality

λ(d) Z

|x|⁻²|u(x)|²dx≤ Z

|∇u(x)|²dx, ∀u∈H¹. (1.2)

Key words and phrases. Nonlinear Schr¨odinger equation, inverse-square potential, Standing waves, Stability, Instability.

1

Throughout this paper, we denote the Hardy functional kuk²_H˙1

c

:=kp

Pcuk²_L2 = Z

|∇u(x)|²−c|x|⁻²|u(x)|²dx, (1.3) and define the homogeneous Sobolev space ˙H_c¹as the completion of C₀^∞(R^d\{0}) under the norm k · kH˙_c¹. It follows from (1.2) that forc < λ(d),

kukH˙_c¹∼ kukH˙¹. (1.4) This is an assertion of the isomorphism between the homogeneous space ˙H_c¹defined in terms ofP_c and the usual homogeneous space ˙H¹. We refer the interested reader to [18] for the sharp range of parameters for which such an equivalence holds.

The local well-posedness for (1.1) was established in [22]. More precisely, we have the following result.

Theorem 1.1 (Local well-posedness [22]). Let d≥3 andc 6= 0be such that c < λ(d). Then for any u0 ∈H¹, there exists T ∈ (0,+∞] and a maximal solution u∈ C([0, T), H¹) of (1.1). The maximal time of existence satisfies either T = +∞orT <+∞and

limt↑Tk∇u(t)kL² =∞.

Moreover, the solution enjoys the conservation of mass and energy, i.e.

M(u(t)) = Z

|u(t, x)|²dx=M(u0), E(u(t)) = 1

2 Z

|∇u(t, x)|²dx−c 2

Z

|x|⁻²|u(t, x)|²dx− 1 α+ 2

Z

|u(t, x)|^α+2dx, for any t∈[0, T). Finally, if0< α < ⁴_d, thenT = +∞, i.e. the solution exists globally in time.

We refer the reader to [22, Theorem 5.1] for the proof of this result. Note that the existence of solutions is based on a refined energy method and the uniqueness follows from Strichartz estimates.

Note also that Strichartz estimates for the linear NLS with inverse-square potential were first established in [5] except the endpoint case (2,2d/(d−2)). Recently, Bouclet-Mizutani [4] proved Strichartz estimates with the full set of admissible pairs for the linear NLS with critical potentials including the inverse-square potential. The local well-posedness for (1.1) can be proved using Strichartz estimates and the equivalence between Sobolev spaces defined byP_cand the usual ones via the Kato method. However, due to the appearance of inverse-square potential, the local well- posedness proved by Strichartz estimates requires a restriction on the validity ofc andd(see e.g.

[26,17,19,20]).

The main purpose of this paper is to study the stability and instability of standing waves for (1.1). In fact, the stability of standing waves for the nonlinear Schr¨odinger equations is widely pursued by physicists and mathematicians (see [28], for a review). For the classical nonlinear Schr¨odinger equation, Cazenave and Lions [3] were the first to prove the orbital stability of standing waves via the concentration-compactness principle. Then, a lot of results on the orbital stability were obtained. For the nonlinear Schr¨odinger equation with a harmonic potential, Zhang [27]

succeed in obtaining the orbital stability by the weighted compactness lemma. Recently, the stability phenomenon was proved for the fractional nonlinear Schr¨odinger equation by establishing the profile decomposition for bounded sequences in H^s(se [12,23,31]).

The first part of this paper concerns the stability of standing waves in the L²-subcritical case 0 < α < ⁴_d. Before stating our stability result, let us introduce some notations. For M > 0, we consider the following variational problems

• for 0< c < λ(d),

dM := inf

E(v) : v∈H¹,kvk²_L2 =M ; (1.5)

• forc <0,

dM,rad := inf

E(v) : v∈H_rad¹ ,kvk²_L2=M , (1.6) whereH_rad¹ is the space of radialH¹-functions. Note that in the casec <0, we are only interested in radial data. This is related to the fact that the sharp Gagliardo-Nirenberg inequality for non- radial data (see Section2) is never attained whenc <0. We will see later (Proposition3.1) that the above variational problems are well-defined. Moreover, the above infimums are attained. Let us denote

• for 0< c < λ(d),

SM :=

v∈H¹ : v is a minimizer of (1.5) ;

• forc <0,

SM,rad:=

v∈H_rad¹ : v is a minimizer of (1.6) .

By the Euler-Lagrange theorem (see Appendix), we see that if v ∈SM, then there existsω > 0 such that

−∆v−c|x|⁻²v+ωv=|v|^αv. (1.7)

Note also that ifvis a solution to (1.7), thenu(t, x) :=e^iωtv(x) is a solution to (1.1). One usually calls e^iωtv the orbit ofv. Moreover, if v∈SM, i.e. v is a minimizer of (1.5), thene^iωtv is also a minimizer of (1.5) ore^iωtv∈S_M. A similar remark goes forv∈S_M,rad.

We next define the following notion of orbital stability which is similar to the one in [24].

Definition 1.2. The setSM is said to be orbitally stableif, for any >0, there exists δ >0 such that for any initial data u0 satisfying

v∈SinfM

ku₀−vk_H1 < δ, the corresponding solution uto (1.1) satisfies

v∈SinfM

ku(t)−vk_H1 < , for allt≥0. A similar definition applies forSM,rad.

Our first result is the following orbital stability of standing waves for the L²-subcritical (1.1).

Theorem 1.3 (Orbital stability). Letd≥3,0< α < ⁴_d andM >0.

(1) If 0< c < λ(d), thenSM is orbitally stable.

(2) If c <0, thenS_M,rad is orbitally stable.

Let us mention that the stability of standing waves in the case 0 < c < λ(d) was studied in [24]. However, they only considered radial standing waves in this case. Here, we remove the radially symmetric assumption and prove the stability for non-radial standing waves in the case 0< c < λ(d). Moreover, our approach is based on the profile decomposition which is of particular interest. We also study the stability of radial standing waves in the case c < 0, which to our knowledge is new.

The proof of our stability result is based on the profile decomposition related to (1.1). Note that this type of profile decomposition was recently established by the first author in [1]. The main difficulty is the lack of space translation invariance due to the inverse-square potential. A careful analysis is thus needed to overcome the difficulty. We refer the reader to Section3for more details.

The second part of this paper is devoted to the strong instability result in the L²-critical case α=⁴_d. There are two main difficulties in studying this problem. The first difficulty is the lack of regularity of solutions to the elliptic equation

−∆Q−c|x|⁻²Q+Q=|Q|^4dQ. (1.8)

More precisely, we do not know whetherQ∈L²(|x|²dx) for any solutionQof (1.8). This is a strong contrast with the classical NLS (c= 0) where solutions to (1.8) are known to have an exponential decay at infinity. Another difficulty is that the uniqueness (up to symmetries) of positive radial solutions to (1.8) is not yet known. To overcome these difficulties, we need to define properly the notion of ground states. To do this, we follow the idea of Csobo-Genoud [10] and define the set of ground states Gand the set of radial ground states Grad(see Section 4for more details). Using this notion of ground states, we are able to show that any ground stateQsatisfiesQ∈L²(|x|²dx), similarly any radial ground stateQ_radsatisfiesQ_rad∈L²(|x|²dx). Thanks to this fact, the standard virial identity yields the following instability in theL²-critical case.

Theorem 1.4. (1) Let d≥ 3,0 < c < λ(d) and Q∈ G. Then the standing wave e^itQ(x) is unstable in the following sense: there exists(u_0,n)_n≥1⊂H¹ such that

u_0,n→Qstrongly in H¹,

as n→ ∞and the corresponding solution u_n to the L²-critical(1.1)with initial data u_0,n blows up in finite time for any n≥1.

(2) Let d≥3, c <0 andQ_rad∈ G_rad. Then the radial standing wave e^itQ_rad(x)is unstable in the following sense: there exists (u0,n)_n≥1⊂H¹ such that

u0,n→Qrad strongly in H¹,

as n→ ∞and the corresponding solution un to the L²-critical(1.1)with initial data u0,n

blows up in finite time for any n≥1.

If we are interested in radialH¹-solutions to (1.8), then we can show another version of instability of standing waves. The interest of this instability is that it allows radialH¹-solutions of (1.8) whose L²-norms are greater than the L²-norms of ground states. We refer the reader to Section 4 for more details.

The paper is organized as follows. In Section2, we recall sharp Gagliardo-Nirenberg inequalities and the profile decomposition related to (1.1). In Section3, we give the proof of the stability result stated in Theorem1.3. Finally, we study the strong instability of standing waves in Section4.

2. Preliminaries

2.1. Sharp Gagliardo-Nirenberg inequalities. In this section, we recall the sharp Gagliardo- Nirenberg associated to (1.1), namely

kuk^α+2_Lα+2≤C_GN(c)kuk

4−(d−2)α 2

L² kuk_H^dα_˙²₁

c

, (2.1)

for allu∈H¹. The sharp constantCGN(c) is defined by C_GN(c) := sup

J_c^α(u) : u∈H¹\{0} , where J_c^α(u) is the Weinstein functional

J_c^α(u) := kuk^α+2_Lα+2

kuk

4−(d−2)α 2

L² kuk_H^dα_˙²₁

c

. (2.2)

We also recall the sharp radial Gagliardo-Nirenberg inequality, namely kuk^α+2_Lα+2≤C_GN(c,rad)kuk

4−(d−2)α 2

L² kuk_H^dα_˙²₁

c

, (2.3)

for allu∈H_rad¹ . The sharp constantC_GN(c,rad) is defined by CGN(c,rad) := sup

J_c^α(u) : u∈H_rad¹ \{0} .

In the case c = 0, it is well known (see [25]) that the sharp constant C_GN(0) is attained by the functionQ₀ which is the unique (up to symmetries) positive radial solution of

−∆Q0+Q0=|Q0|^αQ0. (2.4)

In the case c6= 0 andc < λ(d), we have the following result (see [17] and also [11]).

Theorem 2.1 (Sharp Gagliardo-Nirenberg inequality). Letd≥3,0< α < _d−2⁴ andc6= 0be such that c < λ(d). ThenCGN(c)∈(0,∞)and

• if 0 < c < λ(d), then the equality in (2.1) is attained by a function Q_c ∈ H¹ which is a positive radial solution to the elliptic equation

−∆Qc−c|x|⁻²Qc+Qc=|Qc|^αQc. (2.5)

• if c <0, thenC_GN(c) =C_GN(0) and the equality in (2.3) is never attained. However, the constant C_GN(c,rad) is attained by a functionQ_c,rad ∈H¹ which is a positive solution to the elliptic equation

−∆Qc,rad−c|x|⁻²Qc,rad+Qc,rad=|Qc,rad|^αQc,rad. (2.6) We refer the reader to [17, Theorem 3.1] for the proof in the cased= 3 andα= 2 and to [11, Theorem 4.1] for the proof in the general case.

2.2. Profile decomposition. We next recall the profile decomposition related to the nonlinear Schr¨odinger equation with inverse-square potential.

Theorem 2.2 (Profile decomposition [1]). Letd≥3 andc6= 0be such thatc < λ(d). Let(vn)n≥1

be a bounded sequence in H¹. Then there exist a subsequence still denoted by (v_n)n≥1, a family (x^j_n)_n≥1 of sequences inR^d and a sequence (V^j)_j≥1 of H¹-functions such that

(1) for every j6=k,

|x^j_n−x^k_n| → ∞, (2.7)

as n→ ∞;

(2) for every l≥1and every x∈R^d, we have vn(x) =

l

X

j=1

V^j(x−x^j_n) +v_n^l(x), with

lim sup

n→∞

kv_n^lk_Lp→0, (2.8)

as l→ ∞ for any2< p < _d−2^2d .

Moreover, for every l≥1, we have

kvnk²_L2=

l

X

j=1

kV^jk²_L2+kv_n^lk²_L2+on(1), (2.9)

k∇v_nk²_L2=

l

X

j=1

k∇V^jk²_L2+k∇v_n^lk²_L2+o_n(1), (2.10)

kvnk²_H˙1

c

=

l

X

j=1

kV^j(· −x^j_n)k²_H˙1

c

+kv_n^lk²_H˙1

c

+on(1), (2.11)

kvnk^α+2_Lα+2=

l

X

j=1

kV^jk^α+2_Lα+2+kv^l_nk^α+2_Lα+2+o_n(1), (2.12)

for any 0< α < _d−2⁴ .

We refer the reader to [1, Theorem 4] for the proof of Theorem 2.2. We note here that the profile decomposition argument was first proposed by G´erard in [13]. Later, Hmidi and Keraani [15] gave a refined version and used it to give a simple proof of some dynamical properties of blow-up solutions for the classical nonlinear Schr¨odinger equation. Using the same idea as in [15], the profile decomposition of bounded sequences inH²andH^s(0< s <1) were then established in [29,30] to study dynamical aspects of blow-up solutions for the fourth-order nonlinear Schr¨odinger equation and the fractional nonlinear Schr¨odinger equation. The profile decomposition was also successfully used to study the stability of standing waves for the fractional nonlinear Schr¨odinger equation (see e.g. [31,12,23]).

3. Orbital stability of standing waves

In this section, we will give the proof of the stability result stated in Theorem1.3. Let us firstly study the variational problems (1.5) and (1.6) by using the profile decomposition of bounded sequences in H¹.

Proposition 3.1. Letd≥3,0< α < ⁴_d andM >0.

(1) If 0< c < λ(d), then the variational problem(1.5) is well-defined and there existsC₁>0 such that

d_M ≤ −C1<0. (3.1)

Moreover, there exists v∈H¹ such thatE(v) =d_M.

(2) If c <0, then the variational problem (1.6) is well-defined and there exists C₂ >0 such that

d_M,rad≤ −C2<0. (3.2)

Moreover, there exists v∈H_rad¹ such thatE(v) =d_M,rad.

Proof. (1) Let us firstly consider the case 0< c < λ(d). Letv∈H¹be such thatkvk²_L2 =M. By the sharp Gagliardo-Nirenberg inequality (2.1), we have

E(v) = 1 2kvk²_H˙1

c − 1

α+ 2kvk^α+2_Lα+2

≥ 1 2kvk²_H˙1

c −C_GN(c) α+ 2 kvk

4−(d−2)α 2

L² kvk_H^dα_˙²₁

c

= 1 2kvk²_H˙1

c

−C_GN(c)

α+ 2 M^{4−(d−2)α4} kvk^dα_˙²

H_c¹.

Since 0 < ^dα₂ <2, we apply the Young’s inequality, that is for any a, b >0, any > 0 and any 1< p, q <∞satisfying _p¹+¹_q = 1, there existsC(, p, q)>0 such that

ab≤a^p+C(, p, q)b^q, to have

C_GN(c)

α+ 2 M^{4−(d−2)α4} kvk_H^dα_˙²₁

c

≤kvk²_H˙1

c +C(, d, α, M).

We thus get

E(v)≥ 1

2 −

kvk²_H˙1

c −C(, d, α, M). (3.3)

By choosing 0 < < ¹₂, we see that E(v) ≥ −C(, d, α, M). This shows that the variational problem (1.5) is well-defined.

Forλ >0, we definevλ(x) :=λ^d2v(λx). It is easy to check thatkvλk²_L2 =kvk²_L2=M and E(v_λ) = λ²

2 kv_λk²_H˙1

c

− λ^dα2

α+ 2kvk^α+2_Lα+2.

Since 0 < ^dα₂ < 2, one can find a value λ1 > 0 sufficiently small so that E(vλ₁) < 0. Taking C1:=−E(vλ₁)>0, one gets (3.1).

Let (vn)n≥1be a minimizing sequence ofdM, that iskvnk²_L2 =Mfor alln≥1 and limn→∞E(vn) = dM. There existsC >0 such that

E(vn)≤dM +C, for alln≥1. For 0< <¹₂ fixed, we use (3.3) to infer

1 2 −

kvnk²_H˙1

c ≤d_M+C+C(, d, α, M).

This shows that kvnkH˙¹_c (hence kvnkH˙¹) is bounded for all n ≥ 1. In particular (vn)n≥1 is a bounded sequence inH¹. Therefore, the profile decomposition given in Theorem2.2 implies that up to a subsequence, we can write for everyl≥1,

vn(x) =

l

X

j=1

V^j(x−x^j_n) +v_n^l(x), (3.4) and (2.7)−(2.12) hold. In particular, we have

E(vn) =

l

X

j=1

E(V^j(· −x^j_n)) +E(v_n^l) +on(1), (3.5) as n→ ∞. Denote ˜V_n^j(x) :=λ_jV^j(x−x^j_n) and ˜v_n^l(x) =λ^l_nv^l_n(x), where

λ_j:=

√M kV^jkL²

≥1, λ^l_n :=

√M kv_n^lkL²

≥1.

We readily check that

kV˜_n^jk²_L2 =λ²_jkV^j(· −x^j_n)k²_L2 =M, k˜v_n^lk²_L2 = (λ^l_n)²kv_n^lk²_L2 =M.

By definition ofd_M, we have

E( ˜V_n^j)≥dM, E(˜v^l_n)≥dM. (3.6) Moreover, a direct computation shows that

E( ˜V_n^j) = λ²_j

2 kV^j(· −x^j_n)k²H˙_c¹− λ^α+2_j

α+ 2kV^jk^α+2_Lα+2. So,

E(V^j(· −x^j_n)) = E( ˜V_n^j)

λ²_j +λ^α_j −1

α+ 2 kV^jk^α+2_Lα+2. (3.7) Similarly,

E(v_n^l) = E(˜v^l_n)

(λ^l_n)² +(λ^l_n)^α−1

α+ 2 kv^l_nk^α+2_Lα+2≥ E(˜v^l_n)

(λ^l_n)². (3.8)

Inserting (3.7), (3.8) to (3.5) and using (3.6), we obtain E(vn)≥

l

X

j=1

E( ˜V_n^j)

λ²_j +λ^α_j −1

α+ 2 kV^jk^α+2_Lα+2

!

+E(˜v_n^l)

(λ^l_n)² +on(1)

≥

l

X

j=1

kV^jk² M dM +

j≥1inf λ^α_j −1

α+ 2 ^l

X

j=1

kV^jk^α+2_Lα+2+kv_n^lk²_L2

M dM +on(1)

= dM

M kvnk²_L2+on(1) +

j≥1inf λ^α_j −1

α+ 2

kvnk^α+2_Lα+2− kv_n^lk^α+2_Lα+2+on(1)

+on(1).

Since P∞

j=1kV^jk²_L2 is convergent, there exists j₀≥1 such that kV^j0k²_L2 = sup

j≥1

kV^jk²_L2. We thus have

j≥1infλ^α_j = inf

j≥1

√M kV^jkL²

!^α

=

√M kV^j0kL²

!^α

. (3.9)

On the other hand, by the definition of energy, kv_nk^α+2_Lα+2

α+ 2 ≥ −E(vn)% −dM. Using (3.1), we get

kvnk^α+2_Lα+2

α+ 2 ≥ −E(v_n)≥C₁

2 , (3.10)

fornsufficiently large. By (3.9) and (3.10), we get E(vn)≥dM+on(1) +

" √ M kV^j0k_L2

!^α

−1

# C1

2 −kv_n^lk^α+2_Lα+2

α+ 2

!

+on(1).

Taking the limits n→ ∞andl→ ∞and using (2.8), we obtain dM ≥dM +

" √ M kV^j0k_L2

!^α

−1

#C1

2 .

Therefore, kV^j0k²_L2 ≥ M. The almost orthogonality (2.9) and the factkvnk²_L2 =M then imply that there is only one termV^j0 6= 0 in the profile decomposition (3.4) andkV^j0k²_L2 =M.

The identity (3.4) implies for everyl≥j0,

vn(x) =V^j0(x−x^j_n⁰) +v^l_n(x).

Fixl=j0. Using

kv_nk²_L2=kV^j0k²_L2+kv_n^j0k²_L2+o_n(1), as n→ ∞, andkvnk²_L2=kV^j0k²_L2 =M, we get (up to a subsequence)

n→∞lim kv^j_n⁰k_L2 = 0.

This shows that the sequence (v^j_n⁰)_n≥1 converges to zero weakly in H¹ and strongly inL². The boundedness of (v_n^j0)_n≥1 inH¹along with the strong convergence inL² to zero yield that

n→∞lim kv_n^j0k^α+2_Lα+2= 0.

The lower semi-continuity of Hardy’s functional then gives 0≤lim inf

n→∞ E(v^j_n⁰), thus

lim inf

n→∞ E(V^j0(· −x^j_n⁰))≤lim inf

n→∞ E(V^j0(· −x^j_n⁰)) + lim inf

n→∞ E(v_n^j0)

≤lim inf

n→∞ E(V^j0(· −x^j_n⁰)) +E(v_n^j0)

= lim inf

n→∞ E(vn) =dM.

On the other hand, sincekV^j0(· −x^j_n⁰)k²_L2 =kV^j0k²_L2=M for alln≥1, we haveE(V^j0(· −x^j_n⁰))≥ dM for alln≥1. Therefore,

lim inf

n→∞ E(V^j0(· −x^j_n⁰)) =dM, or equivalently,

1

2k∇V^j0k²_L2− 1

α+ 2kV^j0k^α+2_Lα+2−c

2lim sup

n→∞

Z

|x|⁻²|V^j0(x−x^j_n⁰)|²dx=dM. (3.11) We next prove that the sequence (x^j_n⁰)_n≥1 is bounded. Indeed, if it is not true, then up to a subsequence, we assume that |x^j_n⁰| → ∞as n → ∞. Without loss of generality, we assume that V^j0 is continuous and compactly suported. We have

Z

|x|⁻²|V^j0(x−x^j_n⁰)|²dx= Z

supp(V^j0)

|x+x^j_n⁰|⁻²|V^j0(x)|²dx.

Since|x^j_n⁰| → ∞asn→ ∞, we see that|x+x^j_n⁰| ≥ |x^j_n⁰| − |x| → ∞asn→ ∞for allx∈supp(V^j0).

This shows that

Z

|x|⁻²|V^j0(x−x^j_n⁰)|²dx→0, as n→ ∞. This yields

1

2k∇V^j0k²_L2− 1

α+ 2kV^j0k^α+2_Lα+2 =dM.

By the definition ofE(V^j0), we obtain E(V^j0) +c

2 Z

|x|⁻²|V^j0(x)|²dx=dM.

Since 0 < c < λ(d), we get E(V^j0) < dM, which is absurd (note that E(V^j0) ≥ dM due to kV^j0k²_L2 =M). Therefore, the sequence (x^j_n⁰)_n≥1is bounded and up to a subsequence, we assume that x^j_n⁰ →x^j0 as n→ ∞.

We now write

vn(x) = ˜V^j0(x) + ˜v^j_n⁰(x),

where ˜V^j0(x) =V^j0(x−x^j0) and ˜v^j_n⁰(x) :=V^j0(x−x^j_n⁰)−V^j0(x−x^j0) +v_n^j0(x). Using the fact kvnk²_L2 =kV^j0k²_L2 =M, it is easy to see that

˜

v_n^j0*0 weakly in H¹ and lim

n→∞k˜v_n^j0k_L2= 0.

The first observation on ˜v_n^j0 allows us to write

E(vn) =E( ˜V^j0) +E(˜v^j_n⁰) +on(1).

Again, the lower semi-continuity of Hardy’s functional and the fact lim_n→∞k˜v^j_n⁰k^α+2_Lα+2 = 0, we get that lim infn→∞E(˜v^j_n⁰)≥0. Hence, using the fact thatkV˜^j0k²_L2 =M, we infer that

d_M = lim inf

n→∞ E(v_n)≥lim inf

n→∞ E( ˜V^j0) +E(˜v^j_n⁰)

≥E( ˜V^j0) + lim inf

n→∞ E(˜v^j_n⁰)

≥E( ˜V^j0)≥dM. Therefore,E( ˜V^j0) =dM which completes the proof of Item (1).

(2) We now consider the case c < 0. By the same argument (with CGN(c,rad) in place of CGN(c)), the variational problem (1.6) is well-defined and there exists C2 > 0 such that (3.2) holds. It remains to show that there existsv∈H¹radial such thatE(v) =dM,rad. Let (vn)_n≥1be a minimizing sequence ofdM,rad, that isvn∈H_rad¹ ,kvnk²_L2 =M for alln≥1 and lim_n→∞E(vn) = dM,rad. Arguing as in the first case, we see that (vn)_n≥1is a bounded radial sequence inH¹. Thanks to the fact that

H_rad¹ ,→L^p compactly, (3.12)

for any 2< p < _d−2^2d , there existsV ∈H_rad¹ (see Appendix) such that v_n* V weakly inH¹andv_n→V strongly inL^α+2. We write

vn(x) =V(x) +rn(x),

withr_n*0 weakly inH¹ (note thatr_n can be taken radially symmetric). We have the following expansions

kvnk²_L2=kVk²_L2+krnk²_L2+on(1), (3.13) E(vn) =E(V) +E(rn) +on(1), (3.14) as n→ ∞. Denote ˜V =λV and ˜r_n=λ_nr_n, where

λ:=

√M kVkL²

≥1, λ_n :=

√M krnkL²

≥1.

It is obvious thatkV˜k²_L2 =k˜r_nk²_L2 =M, hence

E( ˜V)≥d_M,rad, E(˜r_n)≥d_M,rad.

We also have

E( ˜V) = λ² 2 kVk²_H˙1

c− λ^α+2

α+ 2kVk^α+2_Lα+2. So,

E(V) = E( ˜V)

λ² +λ^α−1

α+ 2 kVk^α+2_Lα+2. Similarly,

E(rn) =E(˜r_n)

λ²_n +λ^α_n−1

α+ 2 krnk^α+2_Lα+2 ≥E(˜r_n) λ²_n . Pluging above estimates to (3.14), we have

E(v_n)≥E( ˜V)

λ² +λ^α−1

α+ 2 kVk^α+2_Lα+2+E(˜r_n)

α²_n +o_n(1)

=kVk²_L2dM,rad

M +λ^α−1

α+ 2 kVk^α+2_Lα+2+krnk²_L2dM,rad

M +on(1)

=dM,rad

M kVk²_L2+krnk²_L2

+λ^α−1

α+ 2 kvnk^α+2_Lα+2− krnk^α+2_Lα+2

+on(1).

Since

kvnk^α+2_Lα+2

α+ 2 ≥ −E(v_n)% −d_M,rad. Using the upper bound (3.2), we see that

kvnk^α+2_Lα+2

α+ 2 ≥ −E(vn)≥C₂ 2 ,

fornsufficiently large. Takingn→ ∞, this combined with the fact lim_n→∞krnk^α+2_Lα+2= 0 yield dM,rad≥dM,rad+

" √ M kVk_L2

!^α

−1

#C2

2 .

We thus obtain kVk²_L2 ≥M. Sincekvnk²_L2 =M, we have from (3.13) thatkVk²_L2 =kvnk²_L2 =M. In particular, we have lim_n→∞krnk²_L2 = 0 and E(V)≥dM,rad. Sincern *0 weakly inH¹ and strongly in L². The lower semi-continuity of Hardy’s functional implies

lim inf

n→∞ E(rn)≥0.

Therefore,

dM,rad= lim inf

n→∞ E(vn)≥lim inf

n→∞ (E(V) +E(rn))≥E(V) + lim inf

n→∞ E(rn)

≥E(V)≥d_M,rad.

We thus obtain E(V) =dM,rad, which implies that the variational problem (1.6) is attained. The

proof is complete.

Remark 3.2. (1) The proof Proposition3.1 still holds

• in the case 0< c < λ(d) if in place of

kvnk²_L2=M for alln≥1 and lim

n→∞E(vn) =dM, we assume that

n→∞lim kvnk²_L2 =M and lim

n→∞E(vn) =dM.

• in the casec <0 if in place of

kvnk²_L2=M for alln≥1 and lim

n→∞E(v_n) =d_M,rad, we assume that

n→∞lim kvnk²_L2 =M and lim

n→∞E(vn) =dM,rad. (2) It follows from the proof of Proposition3.1that

• in the case 0< c < λ(d),

E(vn) =E( ˜V^j0) +E(˜v^j_n⁰) +on(1), as n→ ∞, and

n→∞lim E(v_n) =d_M, lim

n→∞kv˜_n^j0k^α+2_Lα+2= 0, E( ˜V^j0) =d_M. We thus have that up to a subsequence,

n→∞lim k˜v_n^j0k²_H˙1

c

= 0.

Since k˜v_n^j0k²_˙

H¹_c ∼ k˜v^j_n⁰k²_˙

H¹, we conclude that

n→∞lim k∇˜v^j_n⁰k_L2= 0.

So,

n→∞lim k∇vnkL²=k∇V˜^j0kL², which along with lim_n→∞kvnkL² =kV˜^j0kL² yield

vn→V˜^j0 strongly inH¹, as n→ ∞.

• in the casec <0,

E(vn) =E(V) +E(rn) +on(1), as n→ ∞, and

n→∞lim E(vn) =dM,rad, lim

n→∞krnk^α+2_Lα+2 = 0, E(V) =dM,rad. We thus have

n→∞lim krnk²_H˙1 c = 0, which together withkrnk²_H˙1

c

∼ krnk²_H˙1 yield

n→∞lim k∇r_nk_L2 = 0.

This implies

n→∞lim k∇vnkL² =k∇VkL², which together with limn→∞kvnkL² =kVkL² imply

vn→V strongly in H¹, as n→ ∞.