2 Necessity of the positivity of the quadratic form

GROUND STATE SOLUTIONS TO NONLINEAR

SCHR ¨ODINGER EQUATIONS WITH MULTISINGULAR

INVERSE-SQUARE ANISOTROPIC POTENTIALS

By

VERONICAFELLI^∗

Abstract. A class of nonlinear Schr¨odinger equations with critical power- nonlinearities and potentials exhibiting multiple anisotropic inverse square sin- gularities is investigated. Conditions on strength, location, and orientation of singularities are given for the minimum of the associated Rayleigh quotient to be achieved, both in^RNand in bounded domains.

1 Introduction and statement of the main results

This paper is concerned with the following class of nonlinear Schr ¨odinger equations with a critical power-nonlinearity and a potential exhibiting multiple anisotropic inverse square singularities:

(1)







−∆v −Pk i=1

hi x−ai

|x−ai|

|x−ai|² v = v^2∗−1, v > 0 in^RN _{\ {a}1, . . . , ak},

where_{N ≥ 3},_{k ∈ N},hi∈ C¹(S^{N −1}),(a1, a2, . . . , ak) ∈ R^kN,ai 6= a^jfor_{i 6= j}, and 2^∗= _{N −2}^2N is the critical Sobolev exponent.

The interest in such a class of equations arises in nonrelativistic molecular physics. Inverse square potentials with anisotropic coupling terms turn out to describe the interaction between electric charges and dipole moments of molecules;

see [16]. In crystalline matter, the presence of many dipoles leads one to consider multisingular Schr ¨odinger operators of the form

(2) _{−∆ −}

k

X

i=1

λi(x − ai) · di

|x − ai|³ ,

∗Supported by Italy MIUR, national project “Variational Methods and Nonlinear Differential Equa- tions”.

189

whereλi > 0,i = 1, . . . , k, is proportional to the magnitude of thei-th dipole and d_i,i = 1, . . . , k, is the unit vector giving the orientation of thei-th dipole.

Schr ¨odinger equations and operators with isotropic inverse-square singular potentials have been largely investigated in the literature, both for the case of a single pole [1, 13, 15, 19, 21] and for multiple singularities [2, 5, 6, 8, 9, 12]. The anisotropic case was first considered in [21], where the problem of existence of ground state solutions to (1) was discussed fork = 1. In [10], an asymptotic formula for solutions to an equation associated with dipole-type Schr ¨odinger operators near the singularity was established. We also mention that positivity, localization of binding and essential self-adjointness properties of a class of Schr ¨odinger operators with many anisotropic inverse-square singularities were investigated in [11].

Ground state solutions to (1), i.e., solutions with the smallest energy, can be obtained through minimization of the associated Rayleigh quotient

(3) S(h1, h2, . . . , hk) = inf

u∈D^1,2(R^N)\{0}

Q(u) R

R^N|u|^2∗dx
2/2^∗, where_D^1,2(R^N)denotes the closure ofC₀^∞(R^N)with respect to the norm

kukD^1,2(R^N):=

 Z

R^N

|∇u|²dx

1/2

,

and_{Q : D}^1,2(R^N) → Ris the quadratic form associated to the left-hand side of equation (1), viz.,

(4) _{Q(u) :=}

Z

R^N

|∇u|²dx −

k

X

i=1

Z

R^N

hi x−ai

|x−ai|

|x − ai|² u²(x) dx.

Positive minimizers of (3) suitably rescaled give rise to weak_D^1,2(R^N)-solutions to (1); by the Brezis–Kato Theorem [3] and standard elliptic regularity theory, these turn out to be classical solutions in^RN _{\ {a}1, . . . , ak}.

The present paper means to extend to problems (1) and (3) the analysis per- formed in [12] in the case of locally isotropic inverse square potentials (i.e., for allhi’s constant), proving conditions on the strength, location and orientation of singularities for their solvability.

A necessary condition for the existence of positive classical solutions to (1) in R^N \ {a¹, . . . , ak}is that_Qbe positive semidefinite in_D^1,2(R^N).

Proposition 1.1. A necessary condition for the solvability of problem (1) is that the quadratic form_Q(u)defined in (4) be positive semidefinite, i.e.,

Q(u) ≥ 0 for all_{u ∈ D}^1,2(R^N).

A necessary condition on the angular coefficientshi for the positive semidefi- niteness of the quadratic form can be expressed in terms of the first eigenvalues of the associated Schr ¨odinger operators on the sphere. Indeed, for any_{h ∈ C}^{1 SN −1}
, letµ1(h)be the first eigenvalue of the operator_−∆S^N−1− h(θ)on^{SN −1}, i.e.,

µ1(h) = min

ψ∈H¹(S^N−1)\{0}

R

S^N−1|∇^SN−1ψ(θ)|²dV (θ) −R

S^N−1h(θ)ψ²(θ) dV (θ) R

S^N−1ψ²(θ) dV (θ) . A necessary (but not sufficient) condition for the quadratic form defined in (4) to be positive semidefinite is that

(5)

µ1(hi) ≥ − N − 2 2

2

, for alli = 1, . . . , k, and µ1

Pk i=1hi

≥ − N − 2 2

2

;

see [10].

In particular, condition (5) is necessary for solvability of problem (1). In this paper, we consider multisingular anisotropic potentials with angular terms satisfying the stronger assumption

(6)

µ1(hi) > − N − 2 2

²

, for alli = 1, . . . , k, and µ1

Pk i=1hi

> − N − 2 2

² .

In [11, Proposition 1.2], condition (6) was shown to be necessary for the quadratic form_Qto be positive definite, i.e., for

(7) µ(h1, . . . , hk, a1, . . . , ak) := inf

D^1,2(R^N)\{0}

Q(u) kuk²D^1,2(R^N)

> 0.

On the other hand, (6) is not sufficient for the validity of (7), see [11, Example 1.5]. However, if (6) holds, then (7) turns out to be necessary for the solvability of (1).

Proposition 1.2. If (6) holds and (1) admits a positive _D^1,2(R^N)-solution, then (7) is necessarily satisfied.

Due to the above proposition, in order to look for solutions to (1), we assume that the quadratic form_Q is positive definite. The dependence of positivity of the quadratic form on the location and orientation of dipoles has been deeply investigated in [11], where conditions on thehi’s andai’s ensuring the validity of (7) can be found. If _Q(u) is positive definite, then Sobolev’s inequality implies that

S(h1, h2, . . . , hk) ≥ µ(h1, . . . , hk, a1, . . . , ak) S > 0,

whereSis the best constant in the classical Sobolev inequality, i.e., S = inf

D^1,2(R^N)\{0}

kuk²D^1,2(R^N)

kuk²_L^2∗_(RN)

.

Problems (1) and (3) have been treated by Terracini in [21] in the one-dipole case k = 1. For_{h ∈ C}¹(S^{N −1}), let

(8) S(h) := inf

u∈D^1,2(R^N)\{0}

R

RN|∇u(x)|²−^h(x/|x|)_|x|² u²(x) dx R

R^N|u|^2∗
2/2^∗ .

Let us recall from [21] the following existence result for the one-dipole type problem.

Theorem 1.3 ([21, Proposition 5.3 and Theorem 0.2]). Let_{h ∈ C}¹(S^N)such thatµ1(h) > − ^{N −2}₂
2

and

(9)







maxS^N−1h > 0, if_{N ≥ 4,} R

S^N−1h ≥ 0, ifN = 3.

LetS(h)be defined in (8). ThenS(h) < SandS(h)is achieved.

The main difficulty in the minimization of the Rayleigh quotient in (3) is due to the lack of compactness of the embeddings_D^1,2(R^N) ,→ L^2∗(R^N)and_D^1,2(R^N) ,→

L² R^N, |x|⁻²h x/|x|
dx
, where, for _{h ∈ L}^∞(S^{N −1}), L² R^N, |x|⁻²h x/|x|
dx
is the the weighted Lebesgue space endowed with the norm

 Z

R^N|x|⁻²h x/|x|
u²(x) dx

1/2

.

Such a lack of compactness could produce non-convergence of minimizing se- quences and non-attainability of the infimum of the Rayleigh quotient in some cases. In [12], several configurations for which the infimum of the Rayleigh quo- tient is not attained are produced in the isotropic case, i.e., for allhiconstant; e.g.

the infimum in (3) is not attained if the coefficientshi are positive constants or if k = 2andh1andh2are constant.

A careful analysis of the behavior of minimizing sequences performed through the P. L. Lions Concentration-Compactness Principle [17, 18] clarifies the possible reasons for lack of compactness: concentration of mass at some non-singular point, at one of the singularities or at infinity; see Theorem 4.1. Extending analogous results of [12] for the isotropic case, Theorem 1.4 below provides sufficient conditions for minimizing sequences to stay at an energy level which

is strictly below all the energy thresholds at which the compactness can be lost.

The proof is based on a comparison between levels and is carried out by testing the energy functional associated to (1) with solutions to (8). On the other hand, while in the isotropic case the solutions to (8) are completely classified and can be explicitly written; in the anisotropic case such an explicit form is not available. We overcome this difficulty by exploiting the asymptotic analysis of the behavior near the singularities of solutions performed in [10], which allows us to estimate the behavior of minimizing sequences and to force their level to stay in the recovered compactness range.

From now on, for every_{h ∈ C}¹(S^{N −1}), we denote byµ1(h)the first eigenvalue of the operator _−∆S^N−1 − h(θ) on ^{SN −1} and by ψ₁^h the associated positive L²- normalized eigenfunction, and set

(10) σh:= −N − 2

2 +

s

 N − 2 2

2

+ µ1(h).

Theorem 1.4. Fori = 1, . . . , k, letai ∈ R^N,ai6= ajfor_{i 6= j}, andhi ∈ C¹(S^N) satisfy (7). If

(11) S(hk) = min{S(h^j) : j = 1, . . . , k},

(12) hk satisfies (9),

(13)



































k−1

X

i=1

hi a_k−a_i

|ak−ai|

|ak− ai|² > 0,

ifµ1(hk) ≥ − ^{N −2}2

2

+ 1,

k−1

X

i=1

Z

R^N

hi x

|x|

h

ψ^h₁^k _|x+a^x+ai_i^−a_−a^k

k|

i2

|x|²|x + ai− ak|^{2(σhk+N −2)} > 0,

if ₋ ^{N −2}₂
²< µ1(hk) < − ^{N −2}₂
2

+ 1,

(14) S(hk) ≤ S

 ^k X

i=1

hi

 ,

then the infimum in (3) is achieved and problem (1) admits a solution in_D^1,2(R^N). We note thatS(h) = S(h ◦ A)for any_{h ∈ C}¹(S^N)and any orthogonal matrix A ∈ O(N). Hence condition (14) is satisfied, for example, if there exists an

orthogonal matrix_{A ∈ O(N)}such that

k

X

i=1

hi(θ) ≤ h^k(A(θ)), for all_{θ ∈ S}^{N −1}.

Let us describe in more detail the case in which the singularities are generated by electric dipoles, i.e.,hi(θ) = λiθ · di, for someλi > 0and^di ∈ R^N with_|di| = 1. For anyλ > 0and^d_{∈ R}^N with_{|d| = 1}, let

µ^λ₁ = min

ψ∈H¹(S^N−1)\{0}

R

S^N−1|∇^SN−1ψ(θ)|²dV (θ) − λR

S^N−1(θ · d)ψ²(θ) dV (θ) R

S^N−1ψ²(θ) dV (θ)

be the first eigenvalue of the operator_−∆S^N−1− λ (θ · d) on^{SN −1}. By rotation invariance, it is easy to verify that the above minimum does not depend on ^d. Moreover, condition (6) can be explicitly expressed as a bound on the dipole magnitudes; indeed,

µ^λ₁ > − N − 2 2

2

if and only if λ < 1 ΛN

whereΛN is the best constant in the dipole Hardy-type inequality, i.e.,

ΛN := sup

u∈D^1,2(R^N)\{0}

R

RN

x·d

|x|³u²(x) dx R

RN|∇u(x)|²dx;

see [10]. By rotation invariance,ΛN does not depend on the unit vector^dand, by the classical Hardy’s inequality,ΛN < 4/(N − 2)². For everyλ > 0, let us denote

σ^λ:= −N − 2

2 +

r

N − 2 2

2

+ µ^λ₁.

Corollary 1.5. Fori = 1, . . . , k, letai∈ R^N,ai 6= a^j for_{i 6= j},^di∈ R^N with

|dⁱ| = 1, and

0 < λ1≤ λ2≤ · · · ≤ λk < Λ⁻¹_N .

Assume that the quadratic form

u 7→

Z

R^N|∇u(x)|²dx −

k

X

i=1

λi(x − ai) · di

|x − aⁱ|³ u²(x) dx

is positive definite and that

(15)



































k−1

X

i=1

λid_i·_|a^ak_k^−a_−aⁱ_i|

|ak− ai|² > 0,

ifµ^λ₁^k≥ − ^{N −2}2

2

+ 1,

k−1

X

i=1

Z

R^N

λi x

|x|· dⁱh

ψ₁^λkθ·dk _|x+a^x+ai_i^−a_−a^k

k|

i2

|x|²|x + aⁱ− a^k|^{2(σλk+N −2)} > 0,

if ₋ ^{N −2}₂
²< µ^λ₁^k< − ^{N −2}₂
2

+ 1,

(16)

k

X

i=1

λid_i    ≤ λk. Then the infimum

u∈D^1,2inf(R^N)\{0}

R

R^N|∇u(x)|²dx −Pk i=1

λi(x−ai)·di

|x−ai|³ u²(x) dx R

RN|u|^2∗dx
2/2^∗

is achieved and the problem

(17)







−∆u −Pk i=1

λi(x−ai)·di

|x−ai|³ u = u^2∗−1 u > 0 in^RN_{\ {a}1, . . . , ak} admits a solution in_D^1,2(R^N).

With respect to the isotropic case, the possibility of orientating the dipoles helps in finding the balance between the strength and the locations of the singularities required in assumptions (15) and (16). Consider, for example, the case of two dipoles k = 2. Assume that 0 < λ1 ≤ λ², λ2 is small andN is large in such a way that the associated quadratic form is positive definite andµ^λ₁² ≥ − ^{N −2}₂
2

+ 1. Then condition (15) becomes

(a2− a¹) · d¹> 0, while (16) becomes

d₁· d2< − λ1

2λ2

.

In this case, if the first dipoleλ1d₁is fixed at pointa1, (15) gives a constraint on the location of the second dipole while (16) gives a condition on its orientation.

In particular, it is possible to construct many configurations ensuring the existence

of ground state solutions to (17), unlike the isotropic case where problem (1) with k = 2andh1andh2constants has no ground state solutions, as observed in [12, Theorem 1.3].

In bounded domains, concentration of mass at infinity is no longer possible and an existence result similar to Theorem 1.4 can be obtained without assumption (14).

Theorem 1.6. Assume that Ω is a bounded smooth domain, _{ai}^ki=1 ⊂ Ω, hi∈ C¹(S^N),i = 1, . . . , k, such that the quadratic form

(18) _QΩ(u) =:=

Z

Ω|∇u(x)|²dx −

k

X

i=1

Z

R^N

hi x−ai

|x−ai|

|x − aⁱ|² u²(x) dx

is positive definite inH₀¹(Ω),hksatisfies (9),S(hk) = min{S(hj) : j = 1, . . . , k},

µ1(hk) ≥ − N − 2 2

2

+ 1, and

k−1

X

i=1

hi a_k−a_i

|ak−ai|

|ak− ai|² > 0.

Then the infimum in

(19) SΩ(h1, h2, . . . , hk) = inf

u∈H₀¹(Ω)\{0}

Q^Ω(u) kuk²_L2∗(Ω)

is achieved and equation

(20)







−∆u −Pk i=1

hi x−ai

|x−ai|

|x−ai|² u = u^2∗−1

u > 0 in_{Ω \ {a}1, . . . , ak}, u = 0 on∂Ω admits a solution inH₀¹(Ω).

The further assumptionµ1(hk) ≥ − ^{N −2}₂
2

+ 1of Theorem 1.6 is not technical but quite natural when working in bounded domains. Indeed, it plays the role of a critical dimension for Brezis–Nirenberg type problems in bounded domains; see [4, 15].

The paper is organized as follows. Section 2 contains the proofs of Propositions 1.1 and 1.2. In section 3 some interaction estimates are first deduced and then applied to a comparison of energy levels of minimizing sequences. Section 4 provides a local Palais–Smale condition which is used to prove Theorem 1.4 and Corollary 1.5. Finally, in section 5 we analyze the problem in bounded domains.

Notation. We list some notation used throughout the paper.

- B(a, r)denotes the ball_{{x ∈ R}^N : |x − a| < r}in^RN with center ataand radiusr.

- For any_{A ⊂ R}^N,^χAdenotes the characteristic function ofA.

- Sis the best constant in the Sobolev inequality_Skuk²_L2∗(R^N)≤ kuk²D^1,2(R^N). - ωN denotes the volume of the unit ball in^RN.

- O(N )denotes the group of orthogonal_{N × N}matrices.

2 Necessity of the positivity of the quadratic form

In the present section we discuss the necessity of the positivity of the quadratic form for the solvability of (1), by proving Propositions 1.1 and 1.2.

Proof of Proposition 1.1. Let ube a positive classical solution to (1) in R^N\ {a1, . . . , ak}. For any_{φ ∈ Cc}^∞(R^N\ {a1, . . . , ak}), by testing equation (9) with φ²/uwe obtain

2 Z

RN

φ

u∇φ · ∇u dx − Z

RN

φ²

u²|∇u|²dx

−

k

X

i=1

Z

R^N

hi x−a_i

|x−ai|

|x − ai|² u²(x) dx − Z

R^N

φ²u^2∗−2dx = 0.

From the elementary inequality2^φ_u∇φ · ∇u −^φ_u²²|∇u|²≤ |∇φ|², we deduce Q(φ) ≥

Z

R^N

φ²u^2∗−2dx ≥ 0 for all_{φ ∈ Cc}^∞(R^N \ {a¹, . . . , ak}).

From the density ofC_c^∞(R^N \ {a1, . . . , ak})in_D^1,2(R^N)(see [7, Lemma 2.1]), we obtain that_Qis positive semidefinite in_D^1,2(R^N). _ Proof of Proposition 1.2. Assume that (6) holds and let_{u ∈ D}^1,2(R^N)be a positive_D^1,2(R^N)-solution to (1). By Proposition 1.1, it follows that

µ(h1, . . . , hk, a1, . . . , ak) ≥ 0,

where µ(h1, . . . , hk, a1, . . . , ak) has been defined in (7). Let us assume by contradiction that µ(h1, . . . , hk, a1, . . . , ak) = 0. From [11, Proposition 4.1], µ(h1, . . . , hk, a1, . . . , ak) = 0 is attained by some _{v ∈ D}^1,2(R^N), _{v ≥ 0} a.e. in R^N,_{v 6≡ 0}, which then satisfies

−∆v −

k

X

i=1

hi x−ai

|x−ai|

|x − ai|² v = 0 weakly in_D^1,2(R^N).

Testing the above equation withu, we obtain that Z

R^N

u^2∗−1(x)v(x) dx = 0,

which contradictes the positivity ofu. 

3 Interaction estimates and comparison of energy levels

By Theorem 1.3, for every function_{h ∈ C}¹(S^N)verifyingµ1(h) > − ^{N −2}2

2

and (9), there exists someφh ∈ D^1,2(R^N), φh ≥ 0, φh 6≡ 0, such thatφh attainsS(h), i.e.,

(21) S(h) =

R

RN

h|∇φh(x)|²−^h(x/|x|)_|x|² φ²_h(x)i dx R

R^N|φh|^2∗
2/2∗ , and solves

(22) _−∆φh−h(x/|x|)

|x|² φh= φ²_h^∗−1, in^RN. Moreover, Kelvin’s transformwh(x) := |x|^{−(N −2)}φh(x/|x|²)solves

−∆wh−h(x/|x|)

|x|² wh= w_h^2∗−1 in^RN. From [10], it follows that, withσhdefined in (10), the functions

x 7→ φh(x)

|x|^σhψ₁^h(x/|x|), x 7→ wh(x)

|x|^σhψ^h₁(x/|x|) = φh(x/|x|²)

|x|^{σh+N −2}ψ₁^h(x/|x|) are continuous in^RN and admit positive limits as_{|x| → 0}, i.e.,

c^h₀ := lim

|x|→0

φh(x)

|x|^σhψ₁^h(x/|x|) ∈ (0, +∞) and c^h_∞:= lim

|x|→+∞

φh(x)

|x|^{−σh−N +2}ψ^h₁(x/|x|) ∈ (0, +∞).

(23)

Hence there exists a positive constantC(h) > 0such that

(24) ¹

C(h)

|x|^σh

1 + |x|^{2σh+N −2} ≤ φ^h(x) ≤ C(h) |x|^σh

1 + |x|^{2σh+N −2}, for all_{x ∈ R}^N_{\ {0}.}

For anyµ > 0, we denoteφ^h_µ(x) := µ^{−(N −2)/2}φh(x/µ).

Lemma 3.1. Let_{h, k ∈ C}¹(S^N)such thathsatisfies (9) and µ1(h) > −N − 2

2

2

+ 1.

Then, for every_{a ∈ R}^N _{\ {0}}, Z

R^N

φ²_h(x) dx ∈ (0, +∞) and

Z

R^N

k(_|x−a|^x−a)

|x − a|²|φ^hµ(x)|²dx = µ²" k ^−a_|a|

|a|² Z

R^N

φ²_h(x) dx + o(1)

#

as_{µ → 0}⁺.

Proof. From (24) and the assumptionµ1(h) > − ^{N −2}₂
²

+ 1, it follows that φh∈ L²(R^N). We have

Z

R^N

k(_|x−a|^x−a)

|x − a|²|φ^hµ(x)|²dx = µ² Z

|x|<^|a|_2µ

k(_|µx−a|^µx−a)

|µx − a|²φ²_h(x) dx (25)

+ µ^{−N +2} Z

|x+a|≥^|a|₂

k(_|x|^x)

|x|² φ²_hx + a µ

dx.

Since

χB 0,^|a|_2µ
(x)k _|µx−a|^µx−a

|µx − a|²     

≤ 4

|a|²kkkL^∞(S^N−1)

andφh∈ L²(R^N), we deduce from the Dominated Convergence Theorem that

µ→0lim⁺ Z

|x|<^|a|_2µ

k(_|µx−a|^µx−a )

|µx − a|²φ²_h(x) dx = lim

µ→0⁺

Z

R^N

χB 0,^|a|_2µ
(x)k(_|µx−a|^µx−a)

|µx − a|²φ²_h(x) dx (26)

= k ^−a_|a|

|a|² Z

R^N

φ²_h(x) dx.

Moreover, from (24) andµ1(h) > − ^{N −2}₂
2

+ 1, it follows that

(27)    

µ^−N Z

|x+a|≥^|a|₂

k(_|x|^x)

|x|² φ²_hx + a µ

dx    

≤ µ^{2σh+N −4}kkkL^∞(S^N−1)(C(h))² Z

|x+a|≥^|a|₂

1

|x|²|x + a|^{2(σh+N −2)}dx = o(1) as_{µ → 0}⁺. The conclusion then follows from (25), (26), and (27). _

Lemma 3.2. Let_{h, k ∈ C}¹(S^N)such thathsatisfies (9), and µ1(h) = −N − 2

2

2

+ 1.

Then, for every_{a ∈ R}^N _{\ {0}}, N ωN | log µ|

|C(h)|²(1 + o(1)) ≤ Z

|x|<^|a|_2µ

φ²_h(x) dx (28)

≤ Nω^N|C(h)|²| log µ| (1 + o(1)), as_{µ → 0}⁺, whereωN is the volume of the standard unitN-ball, and

(29) Z

R^N

k(_|x−a|^x−a)

|x − a|²|φ^hµ(x)|²dx = µ²

k ^−a_|a|

|a|² + o(1)

 Z

|x|<¹_µ

φ²_h(x) dx



as_{µ → 0}⁺.

Proof. Estimate (28) follows from (24) and direct calculations. We have

Z

R^N

k(_|x−a|^x−a)

|x − a|²|φ^hµ(x)|²dx (30)

= µ²

k ^−a_|a|

|a|² Z

|x|<^|a|_2µ

φ²_h(x) dx + Z

|x|<^|a|_2µ

k µx − a

|µx − a|

 1

|µx − a|² − 1

|a|²



φ²_h(x) dx

+ 1

|a|² Z

|x|<^|a|_2µ



k µx − a

|µx − a|

− k−a

|a|



φ²_h(x) dx

+ µ^−N Z

|x+a|≥^|a|₂

k(_|x|^x )

|x|² φ²_hx + a µ

dx

 .

Since 

1

|µx − a|² − 1

|a|²    ≤ 4

|a|⁴ µ²|x|²+ 2µ|a||x|

for_{|x| <} ^|a|

2µ, it follows from (24) that

(31) Z

|x|<^|a|_2µ

k µx − a

|µx − a|

 1

|µx − a|²− 1

|a|²



φ²_h(x) dx = O(1) as_{µ → 0}⁺. Since_{k ∈ C}¹(S^N), for some positive constantCdepending onk

k µx − a

|µx − a|

− k−a

|a|

   ≤ C

µx − a

|µx − a|−−a

|a|

= C√

2 p|µx − a|

r

|µx − a| − |a| + µa · x

|a|

≤ C√

2

p|µx − a|p2µ|x| ≤ 2 C√

2 õp|x|

p|a| for_{|x| <} ^|a|

2µ; hence, from (24), it follows that

(32) Z

|x|<^|a|_2µ



k µx − a

|µx − a|

− k−a

|a|



φ²_h(x) dx = O(1) as_{µ → 0}⁺. From (24), we deduce that

µ^−N Z

|x+a|≥^|a|₂

k(_|x|^x)

|x|² φ²_hx + a µ

dx    

≤ kkkL^∞(S^N−1)(C(h))² Z

|x+a|≥^|a|₂

1

|x|²|x + a|^N, hence

(33) µ^−N

Z

|x+a|≥^|a|₂

k(_|x|^x )

|x|² φ²_hx + a µ

dx = O(1) as_{µ → 0}⁺.

From (28), (30), (31), (32), and (33) it follows that (34)

Z

R^N

k(_|x−a|^x−a)

|x − a|²|φ^hµ(x)|²dx = µ²

k ^−a_|a|

|a|² + o(1)

 Z

|x|<^|a|_2µ

φ²_h(x) dx



as_{µ → 0}⁺. From (24) and the assumptionµ1(h) = − ^{N −2}₂
2

+ 1, we obtain 

    Z

|x|<^|a|_2µ

φ²_h(x) dx − Z

|x|<_µ¹

φ²_h(x) dx     

≤ NωN(C(h))²     

Z ^|a|_2µ

1 µ

r⁻¹dr     

= N ωN(C(h))²    

log|a|

2    

; hence, taking into account that, under the assumption µ1(h) = − ^{N −2}2

2

+ 1, φh6∈ L²(R^N),

(35) Z

|x|<^|a|_2µ

φ²_h(x) dx = Z

|x|<¹_µ

φ²_h(x) dx + O(1) = (1 + o(1)) Z

|x|<_µ¹

φ²_h(x) dx

as_{µ → 0}⁺. The conclusion (29) follows from (34) and (35). _ Lemma 3.3. Let_{h, k ∈ C}¹(S^N)such thathsatisfies (9) and

−N − 2 2

2

< µ1(h) < −N − 2 2

2

+ 1.

Then, for every_{a ∈ R}^N _{\ {0}} and_{A ∈ O(N)} such that Ae1 = a/|a|, with e1 = (1, 0, . . . , 0) ∈ R^N,

Z

R^N

k(_|x−a|^x−a)

|x − a|²|φ^hµ(x)|²dx

= µ^{2σh+N −2}





(c^h_∞)² Z

R^N

k _|x|^x
h

ψ^h_{1 |x+a|}^x+a
i²

|x|²|x + a|^{2(σh+N −2)}dx + o(1)







= µ^{2σh+N −2}

|a|^{2σh+N −2}





(c^h_∞)² Z

R^N

(k ◦ A) _|x|^x
h

(ψ₁^h◦ A) _|x+e^x+e1_1|
i2

|x|²|x + e1|^{2(σh+N −2)} dx + o(1)





 as_{µ → 0}⁺, wherec^h_∞is defined in (23).

Proof. A direct calculation yields (36)

Z

R^N

k(_|x−a|^x−a)

|x − a|²|φ^hµ(x)|²dx

= µ^{2σh+N −2} Z

RN

k _|x|^x
h

ψ^h_{1 |x+a|}^x+a
i2

|x|²|x + a|^{2(σh+N −2)} · φ²_h ^x+a_µ


^x+a_µ  

2(2−σh−N )h

ψ^h_{1 |x+a|}^x+a
i²dx.

From (24), it follows that the function

x 7→ φ²_h ^x+a_µ


^x+a_µ  

2(2−σ_h−N )h

ψ^h_{1 |x+a|}^x+a
i2

is bounded a.e. in^RN uniformly with respect toµ > 0, whereas (23) implies that, for a.e. _{x ∈ R}^N,

(37) lim

µ→0

φ²_h ^x+a_µ


^x+a_µ  

2(2−σ_h−N )h

ψ₁^h _|x+a|^x+a
i2 = (c^h_∞)². Since the assumptionµ1(h) < − ^{N −2}₂
2

+ 1ensures that

x 7→ k _|x|^x
h

ψ1^h x+a

|x+a|

i2

|x|²|x + a|^{2(σh+N −2)} ∈ L¹(R^N),

from (36), (37), and the Dominated Convergence Theorem we deduce that Z

R^N

k(_|x−a|^x−a)

|x − a|²|φ^hµ(x)|²dx = µ^{2σh+N −2}





(c^h_∞)² Z

R^N

k _|x|^x
h

ψ₁^h _|x+a|^x+a
i2

|x|²|x + a|^{2(σh+N −2)}dx + o(1)





 as_{µ → 0}. By the change of variable_{x = |a|Ay}, we obtain

Z

R^N

k _|x|^x
h

ψ₁^h _|x+a|^x+a
i2

|x|²|x + a|^{2(σh+N −2)}dx = |a|^{−N −2σh+2} Z

R^N

(k ◦ A) |y|^y

h

ψ^h₁ A _|y+e^y+e1_1|

i2

|y|²|y + e1|^{2(σh+N −2)} dx,

thus completing the proof. 

The interaction estimates provided by Lemmas 3.1, 3.2, and 3.3 allow us to compare the ground state level of the multisingular problem with the ground state level of the single dipole problem.

Proposition 3.4. Let hi ∈ C¹(S^N), i = 1, . . . , k and j ∈ {1, 2, . . . , k}. We assume thathj verifies (9), and one of the following assumptions is satisfied

µ1(hj) ≥ −N − 2 2

²

+ 1 and

k

X

i=1i6=j

hi aj−ai

|aj−ai|

|a^j− aⁱ|² > 0, (38)



















− N − 2 2

2

< µ1(hj) < − N − 2 2

2

+ 1 and

k

X

i=1i6=j

Z

R^N

hi x

|x|

h

ψ₁^hj _|x+a^x+ai_i^−a_−a^j_j|
i2

|x|²|x + ai− aj|^{2(σhj+N −2)} > 0.

(39)

ThenS(h1, . . . , hk) < S(hj).

Proof. Sincehj satisfies (9), by Theorem 1.3 there exists φhj ∈ D^1,2(R^N), φhj ≥ 0,φhj 6≡ 0, attainingS(hj), i.e., satisfying (21) and (22) with h = hj. Set zµ(x) = φ^hµ^j(x − aj). Then

S(h1, . . . , hk)

≤ Z

RN|∇z^µ(x)|²dx − Z

RN

hj x−aj

|x−a_j|

|x − aj|² z_µ²(x) dx kz^µk²_L^2∗_(R^N₎

− X

i6=j

Z

R^N

hi x−ai

|x−ai|

|x − ai|² z_µ²(x) dx kz^µk²_L^2∗_(RN)

= Z

R^N|∇φ^hj(x)|²dx − Z

R^N

hj x

|x|

|x|² φ²_hj(x) dx kφ^hjk²_L^2∗_(RN)

−X

i6=j

Z

R^N

hi x−(ai−aj)

|x−(ai−aj)|

|x − (aⁱ− a^j)|²(φ^h_µ^j(x))²dx kφhjk²_L2∗(R^N)

.

From the above and Lemmas 3.1, 3.2 and 3.3, we deduce the following estimate:

S(h1, . . . , hk) ≤ S(h^j) − kφ^hjk⁻²_L^2∗_(R^N₎ (40)

×



















































µ² R

R^Nφ²_hj(x)
X

i6=j

hi aj −ai

|aj −ai|

|aj−ai|² + o(1)

!

ifµ1(hj) > − ^{N −2}₂
2

+ 1

µ² R

|x|<_µ¹φ²_hj(x)
X

i6=j

hi aj −ai

|aj −ai|

|aj−ai|² + o(1)

!

ifµ1(hj) = − ^{N −2}₂
² + 1

µ^{2σhj+N −2}(c^h∞^j)² X

i6=j

Z

R^N hi(_|x|^x)

ψ^hj₁ ^x+ai−aj

|x+ai−aj |

²

|x|²|x+a_i−a_j|^{2(σhj+N −2)} + o(1)

!

ifµ1(hj) < − ^{N −2}2

2

+ 1 as_{µ → 0}⁺. Takingµ small enough in (40), we obtain from (38) and (39) that

S(h1, . . . , hk) < S(hj). _

Remark 3.5. For₋ ^{N −2}₂
² < µ1(hj) < − ^{N −2}₂
²

+ 1, assumption (39) can be rewritten as

Z

R^N

X

i6=j

(hi◦ A^ij) _|x|^x

|aⁱ− a^j|^{2(σhj+N −2)}

h(ψ^h₁^j◦Aij) x + e1

|x + e¹|

i2!

dx

|x|²|x + e¹|^{2(σhj+N −2)} > 0, whereAij ∈ O(N)are such thatAije1= _|a^ai−aj

i−aj|.

4 The Palais–Smale condition and proof of Theorem 1.4

If _{u ∈ D}^1,2(R^N), u > 0 a.e. in ^RN, is a critical point of the functional J : D^1,2(R^N) → R,

J(v) =1 2

Z

R^N|∇v|²dx −1 2

k

X

i=1

Z

R^N

hi x−ai

|x−ai|

|x − ai|² v²(x) dx (41)

−S(h1, h2, . . . , hk) 2^∗

Z

R^N

|v|^2∗dx,

then w = S(h1, h2, . . . , hk)^1/(2∗−2)u is a solution to equation (1) (weakly in D^1,2(R^N)and classically in^RN_\{a1, . . . , ak}). From now on, for any_{u ∈ D}^1,2(R^N), J⁰(u) ∈ (D^1,2(R^N))^?will denote the Fr´echet derivative ofJatuand_{h·, ·i}will remain as the duality product between_D^1,2(R^N)and its dual space_(D^1,2(R^N))^?.

The Concentration-Compactness analysis of the behavior of Palais–Smale se- quences provides the following local compactness result.

Theorem 4.1. Let (7) hold and _{un}n∈N ⊂ D^1,2(R^N) be a Palais–Smale sequence forJ, namely

n→∞lim J(un) = c < ∞in^R and lim

n→∞J⁰(un) = 0in the dual space_(D^1,2(R^N))^?. If

(42) c < 1

NS(h1, h2, . . . , hk)^1−N2

 min



S, S(h1), . . . , S(hk), S X^k

j=1hj

^N/2 ,

then_{un}n∈Nadmits a subsequence strongly converging in_D^1,2(R^N).

Proof. Let_{un}^n∈N be a Palais–Smale sequence forJ at levelc. Then from (7) there exists some positive constantc1such that

c1kuⁿk²D^1,2(R^N)≤ Q(uⁿ) = N J(un) −N − 2

2 hJ⁰(un), uni

= N c + o(kuⁿkD^1,2(R^N)) + o(1)