Veronica FELLI † and Angela PISTOIA ‡

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Existence of blowing-up solutions for a nonlinear elliptic equation with Hardy potential and critical

growth ^∗

Veronica FELLI ^† and Angela PISTOIA ^‡

January 8, 2004

Abstract We prove the existence of a positive solution to

−∆u − λ

|x| ² u = k(x)u ²

^∗

⁻¹ , u ∈ D ^1,2 (R ^N ),

which blows-up at a suitable critical point of k as the positive parameter λ goes to zero.

Keywords: Hardy potential, Sobolev critical exponent, perturbation methods.

AMS subject classification: 35J70, 35J20, 35B33.

0 Introduction

Let us consider the following nonlinear elliptic equation







−∆u − _|x| ^λ

₂

u = k(x)u ²

^∗

⁻¹ in R ^N , u ∈ D ^1,2 (R ^N ), u > 0 in R ^N ,

(0.1)

where N ≥ 3, 2 ^∗ = _{N −2} ^2N and the space D ^1,2 (R ^N ) is the closure of C ₀ ^∞ (R ^N ) with respect to the norm

kuk :=

Z

R

^N

|∇u| ² dx

1/2

. (0.2)

∗

The first author is supported by M.U.R.S.T. project “Variational Methods and Nonlinear Differential Equa- tions”. The second author is supported by M.U.R.S.T. project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

†

Dipartimento di Matematica e Applicazioni, Universit` a di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy. E-mail: [email protected].

‡

Dipartimento di Metodi e Modelli Matematici, Universit` a di Roma “La Sapienza”, 00100 Roma, Italy. E-mail:

[email protected].

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We point out that one of the main features of problem (0.1) is a lack of compactness due to the critical growth of the nonlinearity. Non-existence results for (0.1) can be obtained using the Pohozaev type identity proved in [13] ; more precisely it turns that any positive solution to (0.1) in the space D ^1,2 (R ^N ) satisfies the following identity

Z

R

^N

h∇k(x), xiu ²

^∗

dx = 0

provided k is continuously differentiable. This implies that ther are no such solution if h∇k(x), xi does not change sign and k is not constant .

When λ = 0 and k ≡ 1, it is well known (see [5], [7] and [14]) that all the solutions to (0.1) are given by

U _δ,ξ (x) := α N

δ

^{N −2}²

(δ ² + |x − ξ| ² )

^{N −2}²

, x ∈ R ^N , ξ ∈ R ^N , δ > 0, (0.3) where α N = [N (N − 2)] ^{(N −2)/4} .

The case λ 6= 0 and k ≡ 1 has been studied by Terracini in [15], where a solution to (0.1) is found explicitly provided 0 < λ < (N − 2) ² /4 , namely:

V _λ (x) := c _λ (N )

[|x| ^1−a (1 + |x| ^2a )]

^{N −2}²

,

where a := p1 − 4λ/(N − 2) ² and c _λ (N ) is a suitable positive constant. This solution is also unique, up to a conformal transformation of the form V _λ ^ε (x) := ε ^{(2−N )/2} V λ (x/ε) for some ε > 0.

In [13] Smets considers the case N = 4 and proves that if k satisfies the “global” condition k(0) = lim _|x|→+∞ k(x), then for any λ ∈ (0, 1) problem (0.1) has at least one solution.

In [8] Schneider and the first author consider the case N ≥ 3 (even for a more general class of differential operators related to Caffarelli-Kohn-Nirenberg inequalities) and assume that k is close to a constant, i.e. k(x) = 1 + εh(x), where h satisfies a suitable “global” condition and ε is a small real perturbation parameter. By using the perturbative method by Ambrosetti and Badiale [3], they find a solution to (0.1) which is close to a radial solution to the unperturbed problem, provided 0 < λ < (N − 2) ² /4 and ε small enough.

In [1] Abdellaoui, Peral, and the first author prove the existence of solutions to problem (0.1) blowing-up at global maximum points of k as the parameter λ goes to zero, under some suitable assumption about the local behaviour of k close to such maximum points.

Let us also mention that some related singular equations with Hardy type potential were also studied in [2, 9, 10, 12].

In this paper we are interested in finding solutions to problem (0.1) blowing-up at a suitable critical point (not necessarily a maximum point) ξ 0 of the function k, as the parameter λ goes to zero (see Definition 0.1).

Before stating our main result it is useful to introduce some notation. Let ξ ₀ ∈ R ^N be such that k(ξ 0 ) > 0 and set (see (0.3))

W _δ,ξ (x) := k(ξ ₀ ) −

_2∗−2¹

U _δ,ξ (x). (0.4)

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Then for δ > 0 and ξ ∈ R ^N the functions W _δ,ξ ∈ D ^1,2 (R ^N ) solve the limit equation







−∆W = k(ξ ₀ )W ²

^∗

⁻¹ in R ^N ,

W > 0 in R ^N .

We can introduce the following definition.

Definition 0.1 Let u λ be a solution to problem (0.1). We say that u λ blows-up at ξ 0 as λ goes to zero if there exist δ _λ > 0 and ξ _λ ∈ R ^N such that lim _λ→0 δ _λ = 0, lim _λ→0 ξ _λ = ξ 0 and u _λ − W _δ

_λ

_,ξ

_λ

→ 0 in D ^1,2 (R ^N ).

Let us assume k ∈ L ^∞ (R ^N ) ∩ C ⁰ (R ^N ). We state below our main assumption on k and ξ 0 .

(∗)



 

 

 

 

Let ξ 0 ∈ R ^N \ {0} be a critical point of k with k(ξ ₀ ) > 0 such that for some r > 0

k(x + ξ ₀ ) = k(ξ ₀ ) + Q _ξ

₀

(x) + R _ξ

₀

(x) ∀ x ∈ B(0, r) := {x ∈ R ^N : |x| < r}, (0.5) where Q _ξ

₀

: R ^N −→ R is a continuous function such that for some θ ∈ (2, N)

Q _ξ

₀

(tx) = t ^θ Q _ξ

₀

(x) ∀ x ∈ R ^N , t > 0, (0.6) and R ξ

0

: B(0, r) −→ R is a continuous function such that for some ∈ (0, N − θ)

|R _ξ

₀

(x)| ≤ c|x| ^θ+ ∀ x ∈ B(0, r). (0.7) Condition (∗) should be compared to the so-called flatness assumptions in the problems of prescribing scalar curvature arising in differential geometry (see for example [4] and [11]). Our main result is the following.

Theorem 0.2 Let us assume that (∗) holds with Q _ξ

₀

(x) = P _N

i=1 a _i x ^θ _i , x ∈ R ^N , where θ is an even integer. Assume either

θ ∈ (2, N ) and a _i < 0 for any i = 1, . . . , N (0.8) or

θ ∈

max

2, N − 2 4

, N

,

N

X

i=1

a _i < 0, and a _i 6= 0 for any i = 1, . . . , N. (0.9)

Then for λ small enough there exists a solution to problem (0.1) which blows-up at ξ 0 as λ goes to zero (see Definition 0.1).

We quote the fact that we prove the existence of solutions to (0.1) blowing-up at a point ξ 0

which is a critical point of k with k(ξ 0 ) > 0, by only requiring a suitable condition on k in a

neighborhood of ξ ₀ .

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The proof of Theorem 0.2 is based on a Ljapunov-Schmidt reduction method (see, for ex- ample, [6]). We would like to point out that our method could also allow to construct solutions blowing-up at m different critical points ξ 1 , . . . , ξ m of the function k, provided assumptions (0.5–0.7) hold and either (0.8) or (0.9) are satisfied at each point ξ _i .

The paper is organized as follows. In Section 1 we reduce the problem to a finite dimensional one and in Section 2 we prove Theorem 0.2, by studying the reduced problem. In Appendix A and Appendix B we prove some technical results used in Section 1 and Section 2, respectively.

In Appendix C some useful estimates are given.

1 The finite dimensional reduction

Equation (0.1) is related to the Hardy inequality. For the reader’s convenience we recall it below referring for instance to [9] for a proof.

Lemma 1.1 (Hardy inequality) If u ∈ D ^1,2 (R ^N ), then _|x| ^u ∈ L ² (R ^N ) and C _N

Z

R

^N

u ²

|x| ² dx ≤ Z

R

^N

|∇u| ² dx, (1.1)

where C N = ^{N −2} ₂ 2

is optimal and not attained.

For any λ ∈ (0, C _N ) let us introduce the Hilbert space D _λ := D ^1,2 (R ^N ) equipped with the inner product

(u, v) λ :=

Z

R

^N

∇u∇v − λ

|x| ² uv

dx, which induces the norm

kuk _λ :=

Z

R

^N

|∇u| ² − λ

|x| ² u ²

dx

1/2

. By the Hardy inequality (1.1) we get that

1 − λ

C _N

1/2

kuk ≤ kuk _λ , (1.2)

hence k · k _λ and k · k are equivalent norms. By (1.2) we deduce that kuk _L

2∗

(R

^N

) ≤

S

1 − λ

C _N

−1/2

kuk _λ , (1.3)

where S is the best Sobolev costant of the embedding D ^1,2 (R ^N ) ,→ L ²

^∗

(R ^N ), i.e. the best constant in the Sobolev inequality

Skuk ² _L

2∗

(R

^N

) ≤ kuk ² _D

1,2

(R

^N

) . (1.4)

Let i _λ : D _λ ,→ L ²

^∗

(R ^N ) denote the embedding.

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Definition 1.2 Let i ^∗ _λ : L

^{N +2}^2N

(R ^N ) → D λ be the adjoint of i λ , namely (i ^∗ _λ u, v) _λ =

Z

R

^N

uv dx ∀ v ∈ D _λ , or equivalently w := i ^∗ _λ u is the unique solution to

−∆w − λ

|x| ² w = u in R ^N , w ∈ D ^1,2 (R ^N ).

From H¨ older inequality and (1.3) we obtain easily the following lemma.

Lemma 1.3 i ^∗ _λ : L

^{N +2}^2N

(R ^N ) → D λ is a continuous operator and

ki ^∗ _λ (u)k _λ ≤

S

1 − λ

C N

−1/2

kuk

L

2N

N +2

for any u ∈ L

^{N +2}^2N

(R ^N ).

Problem (0.1) is equivalent to the following one

u = i ^∗ _λ k(x)f (u)), u ∈ D _λ , (1.5)

where f (s) := (s ⁺ ) ²

^∗

⁻¹ . We will look for solutions to (1.5) of the following form u(x) = W _δ,ξ (x) + φ(x)

where the rest term φ is a lower order term belonging to a suitable space. Set for j = 1, . . . , N ψ _δ,ξ ^j (x) := ∂W _δ,ξ

∂ξ ^j (x) = α _N (N − 2)k(ξ ₀ ) −

¹

2∗−2

δ

^{N −2}²

x ^j − ξ ^j (δ ² + |x − ξ| ² )

^N²

(1.6) and

ψ _δ,ξ ⁰ (x) := ∂W _δ,ξ

∂δ (x) = α _N (N − 2)

2 k(ξ ₀ ) −

_2∗−2¹

δ

^{N −4}²

|x − ξ| ² − δ ²

(δ ² + |x − ξ| ² )

^N²

. (1.7) The N + 1 functions ψ _δ,ξ ^j solve the linearized problem







−∆ψ = (2 ^∗ − 1)k(ξ ₀ )W _δ,ξ ²

^∗

⁻² ψ in R ^N , ψ ∈ D ^1,2 (R ^N ).

(1.8)

Remark 1.4 It is known that the space of the solutions to problem (1.8) in D ^1,2 (R ^N ) is the linear space spanned by the hψ ^j _δ,ξ i j=0,1,...,N (see [4, Lemma 3.1]).

Let

K _δ,ξ ^λ = n

φ ∈ D _λ

ψ _δ,ξ ^j , φ

λ = 0 ∀ j = 0, 1, . . . , N o

(1.9)

and let Π ^λ _δ,ξ : D _λ −→ K _δ,ξ ^λ be the orthogonal projection.

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Proposition 1.5 There exist Z ⊂⊂ R ^N such that ξ ₀ ∈ Z, λ ₀ > 0 and δ ₀ > 0 such that for any λ ∈ (0, λ 0 ), δ ∈ (0, δ 0 ) and ξ ∈ Z there exists a unique φ ^λ _δ,ξ ∈ K _δ,ξ ^λ such that

Π ^λ _δ,ξ n

W δ,ξ + φ ^λ _δ,ξ − i ^∗ _λ h k(x)f

W δ,ξ + φ ^λ _δ,ξ

io

= 0. (1.10)

Moreover

kφ ^λ _δ,ξ k _λ =



 

 

 

 

O |ξ − ξ ₀ | ^θ + δ ^θ + λδ ²

if N ≥ 5, O |ξ − ξ 0 | ^θ + δ ^θ + λδ ² log δ

if N = 4, O |ξ − ξ 0 | ^θ + δ ^θ + λδ

if N = 3.

(1.11)

Proof. See Appendix B.

Let us introduce the functional J _λ : D ^1,2 (R ^N ) −→ R defined by J _λ (u) = 1

2 Z

R

^N

|∇u| ² dx − λ 2

Z

R

^N

u ²

|x| ² dx − 1 2 ^∗

Z

R

^N

k(x)(u ⁺ ) ²

^∗

dx.

It is well known that u is a critical point of J λ if and only if u is a solution to problem (0.1).

The following result allows to reduce the problem of existence of solutions to (0.1) to the existence of critical points of a real-valued function of N + 1 variables.

Proposition 1.6 There exist Z ⊂⊂ R ^N such that ξ ₀ ∈ Z, λ ₀ > 0 and δ ₀ > 0 such that for any λ ∈ (0, λ ₀ ), δ ∈ (0, δ ₀ ) and ξ ∈ Z the function W _δ,ξ + φ ^λ _δ,ξ is a solution to problem (0.1) if and only if (δ, ξ) is a critical point of the function

J e _λ (δ, ξ) := J _λ

W _δ,ξ + φ ^λ _δ,ξ

. (1.12)

Proof. See Appendix B.

2 The reduced problem

Let us introduce the function I _λ : R ^N × R ⁺ −→ R defined by (see (1.12))

I _λ (ζ, d) := e J _λ δ(d, λ), ξ(ζ, d, λ), (2.1) where

δ(d, λ) :=



 

 

 

 

dλ

^θ−1¹

if N = 3, dλ

^θ−2¹

| log λ|

^θ−2¹

if N = 4, dλ

^θ−2¹

if N ≥ 5,

(2.2)

and

ξ(ζ, d, λ) := ξ ₀ + δ(d, λ)ζ. (2.3)

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Proposition 2.1 Let K ⊂ R ^N be a compact set and b > a > 0 be fixed. If (∗) is satisfied and θ ∈ (2, N ), there holds

I _λ (d, ζ) =



 

 

 

 

a ₃ (ξ ₀ ) − Γ _ξ

₀

(d, ζ)λ

^θ−1^θ

+ o λ

^θ−1^θ

, if N = 3, a 4 (ξ 0 ) − Γ ξ

0

(d, ζ)λ

^θ−2^θ

| log λ|

^θ−2^θ

+ o λ

^θ−2^θ

| log λ|

^θ−2^θ

, if N = 4, a _N (ξ ₀ ) − Γ _ξ

₀

(d, ζ)λ

^θ−2^θ

+ o λ

^θ−2^θ

, if N ≥ 5,

(2.4)

C ⁰ -uniformly with respect to ζ in K and to d in [a, b]. If, in addition, θ ∈ max 2, ^{N −2} ₄ , N , estimate (2.4) holds C ¹ -uniformly with respect to ζ in K and to d in [a, b].

The function Γ _ξ

₀

: R ⁺ × R ^N → R in (2.4) is defined by

Γ ξ

0

(d, ζ) = b N (ξ 0 )d ^θ Z

R

^N

Q _ξ

₀

(y + ζ) (1 + |y| ² ) ^N dy +







c ₃ (ξ ₀ )d if N = 3, c _N (ξ ₀ )d ² if N ≥ 4,

(2.5)

and

a _N (ξ ₀ ) : = 1 2 − 1

2 ^∗

α ² _N

^∗

[k(ξ ₀ )] ⁻

^2∗−2²

Z

R

^N

dx

(1 + |x| ² ) ^N , (2.6)

b N (ξ 0 ) : = 1

2 ^∗ α ² _N

^∗

[k(ξ 0 )] ⁻

^2∗−2^2∗

, (2.7)

c _N (ξ ₀ ) : =



 

 

 

  1

2 α ² ₃ [k(ξ 0 )] ^−1/2 Z

R

^N

1 |x − ξ ₀ | ² 1

|x| ² dx, if N = 3, α ² ₄ k(ξ 0 ) ⁻¹

2(θ − 2)|ξ ₀ | ² , if N = 4,

1 2 α ² _N [k(ξ ₀ )] ⁻

^2∗−2²

1 |ξ ₀ | ² Z

R

^N

dx

(1 + |x| ² ) ^{N −2} , if N ≥ 5.

(2.8)

Proof. See Appendix C.

Definition 2.2 Let g : D → R be a C ¹ -function, where D ⊂ R ^m is an open set, m ∈ IN. Let x ₀ be a critical point of g.

(i) We say that x ₀ is a C ⁰ -stable critical point of g if for any sequence g _n : D → R with g _n → g C ⁰ -uniformly in D there exists x _n ∈ D such that ∇g _n (x _n ) = 0 and x _n → x ₀ . (ii) We say that x ₀ is a C ¹ -stable critical point of g if for any sequence g _n : D → R with

g _n → g C ¹ -uniformly in D there exists x _n ∈ D such that ∇g _n (x _n ) = 0 and x _n → x ₀ .

We point out that any strict local maximum point or strict local minimum point of g is a C ⁰ -

stable critical point of g. Moreover any non degenerate critical point of g is a C ¹ -stable critical

point of g. Let us remark that a non degenerate critical point is not necessarily a C ⁰ -stable

critical point as the following example shows.

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Example 2.3 Let g : R ² → R, g(x, y) = xy. (0, 0) is a C ¹ -stable critical point, but it is not a C ⁰ -stable critical point.

Proof. It is easy to verify that (0, 0) is a non degenerate critical point. Let us consider the sequence of functions g _n (x, y) := xy + ¹ _n sin(nx), n ∈ IN. The sequence {g _n } _n∈IN converges to g uniformly in R ² . If (x n , y n ) is a critical point of g n , then a direct computation shows that necessarily x n = 0 and y n = −1 for any n ∈ IN. Therefore there can not exist any sequence of critical points of g _n converging to (0, 0). Hence (0, 0) is not a C ⁰ -stable critical point.

Theorem 2.4 Assume either θ ∈ (2, N ) and (d ₀ , ζ ₀ ) is a C ⁰ -stable critical point of Γ _ξ

₀

(d, ζ) or θ ∈ max 2, ^{N −2} ₄ , N and (d ₀ , ζ 0 ) is a C ¹ -stable critical point of Γ ξ

0

(d, ζ). Then for λ small enough there exists a critical point (δ _λ , ξ _λ ) of the function e J defined in (1.12).

Proof. By Proposition 2.1 and Definition 2.2 it follows that there exists a critical point (d _λ , ζ _λ ) of I λ such that lim λ→0 d λ = d 0 and lim λ→0 ζ λ = ζ 0 . If δ λ is defined as in (2.2) and ξ λ is defined as in (2.3), it is easy to check that (δ _λ , ξ _λ ) is a critical point of e J .

Proposition 2.5 Assume Q _ξ

₀

(x) = P N

i=1 a _i x ^θ _i , x ∈ R ^N , where θ ∈ (2, N ) is an even integer. If a _i < 0 for any i = 1, . . . , N, there exists a strict maximum point of the function Γ _ξ

₀

, which is a C ⁰ -stable critical point according to Definition 2.2.

Proof. Since necessarily N ≥ 5, we can write function Γ _ξ

₀

as Γ(d, ζ) = b _N d ^θ f (ζ)+c _N d ² , where

f (ζ) :=

Z

R

^N

Q _ξ

₀

(y + ζ) (1 + |y| ² ) ^N dy =

N

X

i=1

a _i Z

R

^N

(y _i + ζ _i ) ^θ

(1 + |y| ² ) ^N dy. (2.9) We claim that

f (ζ) < f (0) < 0 ∀ ζ ∈ R ^N \ {0}. (2.10) Indeed

Z

R

^N

(y _i + ζ _i ) ^θ (1 + |y| ² ) ^N dy =

Z

R

^N

y _i ^θ

(1 + |y| ² ) ^N dy +

θ

X

k=2 keven

θ k

Z

R

^N

y ^θ−k _i ζ _i ^k

(1 + |y| ² ) ^N dy ≥ Z

R

^N

y _i ^θ (1 + |y| ² ) ^N dy with equality only for ζ _i = 0 and claim (2.10) follows, since a _i < 0 for any i. Let us set d 0 :=

h −2c

_N

b

_N

θf (0)

i

_θ−2¹

. Note that the definition of d 0 makes sense because of (2.10). We claim that (d ₀ , 0) is a strict maximum point of Γ _ξ

₀

. (2.11) Indeed by (2.10) we deduce that for any ζ it holds

Γ(d, ζ) ≤ Γ(d, 0) ≤ Γ(d ₀ , 0) = max

d∈R

⁺

Γ(d, 0).

If (d, ζ) 6= (d 0 , 0) then either d 6= d 0 and hence Γ(d, 0) < Γ(d 0 , 0) or ζ 6= 0 and hence from (2.10)

Γ(d, ζ) < Γ(d, 0). Therefore for any (d, ζ) 6= (d 0 , 0) we have Γ(d, ζ) < Γ(d 0 , 0).

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Proposition 2.6 Assume Q _ξ

₀

(x) = P N

i=1 a _i x ^θ _i , x ∈ R ^N , where θ ∈ (2, N ) is an even integer. If P N

i=1 a _i < 0 and a _i 6= 0 for any i = 1, . . . , N, there exists a non degenerate critical point of the function Γ ξ

0

, which is a C ¹ -stable critical point according to Definition 2.2.

Proof. As above, since N ≥ 5, we can write function Γ _ξ

₀

as Γ(d, ζ) = b _N d ^θ f (ζ) + c _N d ² , where f is defined in (2.9). We point out that

f (0) = d N N

X

i=1

a i < 0,

where d N := R

R

^N

y ₁ ^θ (1 + |y| ² ) ^−N dy. Therefore it makes sense to set d 0 :=

h −2c

_N

b

_N

θf (0)

i

_θ−2¹

. We claim that

(d ₀ , 0) is a nondegenerate critical point of Γ _ξ

₀

. (2.12) Let us compute

∂f

∂ζ i

(0) = a i θ Z

R

^N

y _i ^θ−1

(1 + |y| ² ) ^N dy = 0 (2.13)

and

∂ ² f

∂ζ i ∂ζ j

(0) =







a _i θ(θ − 1)e _N if i = j,

0 if i 6= j,

(2.14)

where e _N := R

R

^N

y ₁ ^θ−2 (1 + |y| ² ) ^−N dy. By (2.13) we deduce that (d ₀ , 0) is a critical point of Γ, since

∂Γ

∂d (d ₀ , 0) = 0 and ∇ _ζ Γ(d ₀ , 0) = 0.

Moreover

∂ ² Γ

∂d ² (d 0 , 0) = b N θ(θ − 2)d ^θ−2 ₀ f (0) < 0 ,

∂ ² Γ

∂d∂ζ i

(d 0 , 0) = b N θd ^θ−1 ₀ ∂f

∂ζ i

(0) = 0,

∂ ² Γ

∂ζ _i ∂ζ _j (d 0 , 0) = b N d ^θ ₀ ∂ ² f

∂ζ _i ∂ζ _j (0). (2.15)

Finally (2.12) follows by (2.15) and (2.14).

Proof of Theorem 0.2. It follows from Proposition 2.6 and Proposition 2.5, taking into account Proposition 1.6 and Theorem 2.4.

3 Appendix A

In this appendix we prove Proposition 1.5 and Proposition 1.6. Let K _δ,ξ ^λ be defined in (1.9) and

let Π ^λ _δ,ξ : D λ −→ K _δ,ξ ^λ denote the orthogonal projection.

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Lemma 3.1 It holds

Π ^λ _δ,ξ (u)

λ ≤ kuk _λ ∀ u ∈ D _λ . Let L ^λ _δ,ξ : K _δ,ξ ^λ −→ K _δ,ξ ^λ be defined by

L ^λ _δ,ξ (φ) = Π ^λ _δ,ξ n

φ − i ^∗ _λ h

(2 ^∗ − 1)k(ξ ₀ )W _δ,ξ ²

^∗

⁻² φ io

.

Lemma 3.2 For any Z ⊂⊂ R ^N \ {0} there exist C > 0, λ ₀ > 0 and δ 0 > 0 such that for any λ ∈ (0, λ ₀ ), δ ∈ (0, δ ₀ ) and ξ ∈ Z there holds

L ^λ _δ,ξ (φ)

_λ ≥ Ckφk ∀ φ ∈ K _δ,ξ ^λ .

Proof. We argue by contradiction. Assume there exist sequences λ _n → 0, δ _n → 0, ξ _n → ξ 6= 0, φ n , ψ n ∈ K _δ ^λ

ⁿ

n

,ξ

n

such that

kφ _n k _λ

_n

= 1, ψ _n = L ^λ _δ

ⁿ

n

,ξ

n

(φ _n ), kψ _n k _λ

_n

→ 0.

More precisely φ _n and ψ _n solve φ _n − i ^∗ _λ

n

(2 ^∗ − 1)k(ξ ₀ )W _δ ²

^∗

⁻²

n

,ξ

n

φ _n = ψ _n + w _n , (3.1) where w _n = P N

i=0 c ⁱ _n ψ _δ ⁱ

n

,ξ

n

for some real numbers c ⁱ _n . First of all we claim that

w _n −→ 0 in D ^1,2 (R ^N ). (3.2)

If we multiply (3.1) by w _n we get for large n kw _n k ² _λ

n

= −(2 ^∗ − 1) Z

R

^N

U _δ ²

^∗

⁻²

n

,ξ

n

φ _n w _n ≤ (2 ^∗ − 1)

Z

R

^N

U _δ ²

_n^∗

_,ξ

_n

2/N

kφ _n k _L

2∗

(R

^N

) kw _n k _L

2∗

(R

^N

)

≤ const

S

1 − λ _n

C N

−1

kw _n k _λ

_n

and hence kw _n k _λ

_n

≤ const . Therefore for large n

kw _n k ² = Z

R

^N

|∇w _n | ² =

N

X

i=0

c ⁱ _n ² kψ ⁱ _δ

_n

_,ξ

_n

k ² ≤ const .

Multiplying (3.1) by ψ _δ ^j

_n

_,ξ

_n

, in view of (1.8) we get

w n , ψ _δ ^j

n

,ξ

n

λ

n

= −(2 ^∗ − 1) Z

R

^N

U _δ ²

^∗

⁻²

n

,ξ

n

φ n ψ _δ ^j

n

,ξ

n

= −λ n

Z

R

^N

φ n ψ _δ ^j

n

,ξ

n

|x| ² = O(λ n )kψ _δ ^j

n

,ξ

n

k. (3.3)

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On the other hand

w _n , ψ ^j _δ

n

,ξ

n

λ

n

=

N

X

i=0

c ⁱ _n

ψ ⁱ _δ

_n

_,ξ

_n

, ψ ^j _δ

n

,ξ

n

λ

n

=

N

X

i=0

c ⁱ _n

"

Z

R

^N

∇ψ ⁱ _δ

n

,ξ

n

· ∇ψ ^j _δ

n

,ξ

n

− λ _n Z

R

^N

ψ _δ ⁱ

n

,ξ

n

ψ ^j _δ

n

,ξ

n

|x| ²

#

= kψ ^j _δ

n

,ξ

n

k ² c ^j _n − λ _n

N

X

i=0

c ⁱ _n Z

R

^N

ψ ⁱ _δ

n

,ξ

n

ψ _δ ^j

n

,ξ

n

|x| ² and hence from Hardy’s inequality (1.1)

kψ ^j _δ

n

,ξ

n

k ² c ^j _n + C _N ⁻¹ λ _n kψ ^j _δ

n

,ξ

n

k

N

X

i=0

|c ⁱ _n |kψ _δ ⁱ

n

,ξ

n

k

≥ w n , ψ ^j _δ

n

,ξ

n

λ

n

≥ kψ ^j _δ

n

,ξ

n

k ² c ^j _n − C _N ⁻¹ λ n kψ _δ ^j

n

,ξ

n

k

N

X

i=0

|c ⁱ _n |kψ ⁱ _δ

_n

_,ξ

_n

k. (3.4)

From (3.3) and (3.4) we deduce that

|c ^j _n |kψ _δ ^j

n

,ξ

n

k ≤ cλ _n 1 +

N

X

i=0

|c ⁱ _n |kψ ⁱ _δ

_n

_,ξ

_n

k

! ,

which implies that c ^j n kψ ^j _δ

n

,ξ

n

k → 0 for any j. Therefore (3.2) follows. Now let φ ˜ n (y) := δ

N −2

n

2

φ n (δ n y + ξ n ), k ˜ φ n k = kφ _n k ∈

1, 1 − λ n /C N

−1/2

(3.5) because of (1.3), and

ψ ˜ n (y) := δ

N −2

n

2

ψ n (δ n y + ξ n ), k ˜ ψ n k = kψ _n k → 0, (3.6)

˜

w _n (y) := δ

N −2

n

2

w _n (δ _n y + ξ _n ), k ˜ w _n k = kw _n k → 0. (3.7) Up to a subsequence, we can assume that ˜ φ _n → φ weakly in D ^1,2 (R ^N ). Given ψ ∈ C _c ^∞ (R ^N ), let v n (x) = ψ ^x−ξ _δ

ⁿ

n

. Note that

kv _n k = δ

N −2

n

2

kψk.

Testing (3.1) with v n , we find Z

R

^N

∇ ˜ φ _n ∇ψ − λ _n δ ¹⁻

N

n

2

Z

R

^N

φ _n v _n

|x| ² − (2 ^∗ − 1) Z

R

^N

U _1,0 ²

^∗

⁻² φ ˜ _n ψ

= Z

R

^N

∇( ˜ ψ _n + ˜ w _n ) · ∇ψ − λ _n δ ¹⁻

N

n

2

Z

R

^N

(ψ n + w n )v n

|x| ² . (3.8)

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From Hardy inequality we have that

δ ¹⁻

N

n

2

Z

R

^N

|φ _n v _n |

|x| ² ≤ δ ¹⁻

N

n

2

Z

R

^N

|φ _n | ²

|x| ²

¹₂

Z

R

^N

|v _n | ²

|x| ²

¹₂

≤ C _N ⁻¹

1 − λ _n

C N

−

¹₂

kψk ≤ const (3.9)

and

δ ¹⁻

N

n

2

Z

R

^N

|ψ _n + w n | |v _n |

|x| ² ≤ C _N ⁻¹ kψ _n + w n kkψk −→

n→+∞ 0. (3.10)

In view of (3.9) and (3.10), passing to the limit in (3.8) we get that φ is a (weak) solution to

−∆φ = (2 ^∗ − 1)U _1,0 ²

^∗

⁻² φ in R ^N . (3.11) Moreover, since φ n ∈ K _δ ^λ

ⁿ

n

,ξ

n

, it is easy to see that φ, ∂U _δ,ξ

∂δ

(δ,ξ)=(1,0)

!

= φ, ∂U _δ,ξ

∂ξ i

(δ,ξ)=(1,0)

!

= 0, i = 1, . . . , N. (3.12)

By (3.11), (3.12), and Remark 1.4 we deduce that φ = 0. On the other hand, if we multiply (3.1) by φ n we get

1 =kφ n k ² _λ

n

= (2 ^∗ − 1) Z

R

^N

U _δ ²

^∗

⁻²

n

,ξ

n

φ ² _n + (ψ n + w n , φ n ) λ

n

= (2 ^∗ − 1) Z

R

^N

U _1,0 ²

^∗

⁻² φ ˜ ² _n + o(1) hence by letting n → ∞ we find 1 = (2 ^∗ − 1) R

R

^N

U _1,0 ²

^∗

⁻² φ ² and a contradiction arises.

Proof of Proposition 1.5. First of all we point out that φ solves equation (1.10) if and only if φ is a fixed point of the operator T _δ,ξ ^λ : K _δ,ξ ^λ −→ K _δ,ξ ^λ defined by

T _δ,ξ ^λ (φ) =

L ^λ _δ,ξ −1

◦ Π ^λ _δ,ξ ◦ i ^∗ _λ

k(x) f (W _δ,ξ + φ) − f (W _δ,ξ ) − f ⁰ (W _δ,ξ ) φ + [k(x) − k(ξ ₀ )] f ⁰ (W _δ,ξ ) φ

+

L ^λ _δ,ξ −1

◦ Π ^λ _δ,ξ

{i ^∗ _λ [k(x)f (W _δ,ξ )] − W _δ,ξ } . (3.13) By Lemma 1.3, Lemma 3.1 and Lemma 3.2 we deduce that for some positive constant ¯ c and for any φ, φ 1 , φ 2 ∈ D _λ it holds

kT _δ,ξ ^λ (φ)k _λ ≤ ¯ c kk(x) f (W _δ,ξ + φ) − f (W _δ,ξ ) − f ⁰ (W _δ,ξ ) φ k

L

^{N +2}^2N

+ ¯ c k [k(x) − k(ξ ₀ )] f ⁰ (W _δ,ξ ) φk

L

2N

N +2

+ ¯ cki ^∗ _λ [k(x)f (W _δ,ξ )] − W _δ,ξ k _λ (3.14) and

kT _δ,ξ ^λ (φ 1 − φ ₂ )k λ ≤ ¯ c kk(x) f (W _δ,ξ + φ 1 ) − f (W δ,ξ + φ 2 ) − f ⁰ (W δ,ξ ) (φ 1 − φ ₂ ) k

L

2N N +2

+ ¯ c k [k(x) − k(ξ 0 )] f ⁰ (W _δ,ξ ) (φ 1 − φ ₂ )k

L

2N

N +2

. (3.15)

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Let us note that the following inequalities

(a + b) ⁺ s−1

−a ^s−1 − (s − 1)a ^s−2 b ≤







C(a ^s−3 |b| ² + |b| ^s−1 ) if s ≥ 3 C |b| ^s−1 if 2 < s < 3,

(3.16) _s

and

(a + b) ⁺ s−1

− a ^s−1

≤ C|b| ^s−1 if 1 < s ≤ 2 (3.17) s

where C = C(s) > 0, hold for any b ∈ R, a ∈ R ⁺ . From (3.16) s with s = 2 ^∗ it follows that

f (W _δ,ξ + φ) −f (W _δ,ξ ) − f ⁰ (W _δ,ξ ) φ ≤





 C

W

6−N N −2

δ,ξ |φ| ² + |φ|

^{N +2}^{N −2}

if N ≤ 6 C |φ|

^{N +2}^{N −2}

if N > 6.

(3.18)

From (3.18) and from the inequality

(α + β) ^γ ≤ 2 ^γ−1 (α ^γ + β ^γ ) for any α, β ≥ 0, γ ≥ 1, we find

f (W δ,ξ + φ) −f (W δ,ξ ) − f ⁰ (W δ,ξ ) φ

2N N +2

≤



 

  C(W

2(6−N )N (N −2)(N +2)

δ,ξ |φ|

^{N +2}^4N

+ |φ|

^{N −2}^2N

) if N ≤ 6 C |φ|

^{N −2}^2N

if N > 6.

(3.19)

From (3.19) and H¨ older’s inequality we have

k(x)f W _δ,ξ + φ − f W _δ,ξ − f ⁰ W δ,ξ φ

L

2N N +2

≤



 

  C

kφk

4N N +2

λ + kφk

2N N −2

λ

^{N +2}_2N

if N ≤ 6 Ckφk

N +2 N −2

λ if N > 6.

(3.20)

Moreover by using H¨ older’s inequality and Lemma 5.7 we get for some constant c ^∗ > 0 k [k(x) − k(ξ ₀ )] f ⁰ (W _δ,ξ ) φk

L

2N

N +2

≤ c ^∗ kφk _λ |ξ − ξ ₀ | ^θ + δ ^θ . (3.21) From (3.20), (3.21), and (5.18) we deduce that there exists a positive constant ˜ c such that for any φ ∈ ¯ B ^λ _δ,ξ (0, ρ) := {v ∈ K _δ,ξ ^λ : kvk _λ ≤ ρ}, ρ ≤ 1,

kT _δ,ξ ^λ (φ)k _λ ≤ ˜ c

ρ ^min{2,

^{N +2}^{N −2}

^} + ρ |ξ − ξ 0 | ^θ + δ ^θ + |ξ − ξ ₀ | ^θ + δ ^θ + h(λ, δ)

, (3.22) where

h(λ, δ) =



 

 

 

 

λδ ² if N ≥ 5,

λδ ² log δ if N = 4,

λδ if N = 3.

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Since R

R

^N

(W _δ,ξ +z) ⁺ 2

^∗

−2

−W _δ,ξ ²

^∗

⁻²

N/2 goes to 0 as z → 0 in D _λ by the Dominated Convergence Theorem, we can choose K, δ ₀ , and λ ₀ such that for any λ ∈ (0, λ ₀ ), δ ∈ (0, δ ₀ ), and ξ ∈ Z



 



 



1 4˜ c − |ξ − ξ ₀ | ^θ − δ ^θ > 0 and

|ξ − ξ ₀ | ^θ + δ ^θ + h(λ, δ) ≤ min n

1 4˜ c

1 4˜ c − |ξ − ξ ₀ | ^θ − δ ^θ max { 1,

^{N −2}₄

} , _4c ¹

∗

¯ c

o sup

kzk

_λ

≤3ρ

^λ_δ,ξ

R

^N

(W _δ,ξ + z) ⁺ 2

^∗

−2

− W _δ,ξ ²

^∗

⁻²

N/2 2/N

< ^[S(1−λ/C _4¯ _c(2

∗

−1)kkk

^N

^)]

_L∞^1/2

(3.23)

where

¯

ρ ^λ _δ,ξ = 4˜ c |ξ − ξ 0 | ^θ + δ ^θ + h(λ, δ). (3.24) From (3.22) and (3.23) we find

kT _δ,ξ ^λ (φ)k λ ≤ ¯ ρ ^λ _δ,ξ for any φ ∈ ¯ B _δ,ξ ^λ (0, ¯ ρ ^λ _δ,ξ ).

In particular T _δ,ξ ^λ maps B _δ,ξ ^λ (0, ¯ ρ ^λ _δ,ξ ) into itself; in order to prove that it is a contraction there we have also to estimate T _δ,ξ ^λ (φ ₁ − φ ₂ ), i.e. the right hand side in (3.15). By H¨ older inequality and (3.23) we have

kk(x)f (W _δ,ξ + φ ₁ ) − f (W _δ,ξ + φ ₂ ) − f ⁰ (W _δ,ξ ) (φ ₁ − φ ₂ )k

L

^{N +2}^2N

≤ kkk _L

^∞

Z 1 0

d

dt f (W δ,ξ + φ 2 + t(φ 1 − φ ₂ )

− f ⁰ (W δ,ξ )(φ 1 − φ ₂ ) L

2N N +2

≤ kkk _L

^∞

Z 1 0

f ⁰ (W _δ,ξ + φ ₂ + t(φ ₁ − φ ₂ )) − f ⁰ (W _δ,ξ )(φ ₁ − φ ₂ ) _L

_{N +2}^2N

≤ kkk _L

^∞

kφ ₁ − φ ₂ k _L

2∗

sup

kzk

_λ

≤3 ¯ ρ

^λ_δ,ξ

kf ⁰ (W _δ,ξ + z) − f ⁰ (W _δ,ξ )k _L

N/2

≤ 1

4¯ c kφ ₁ − φ ₂ k _λ . (3.25) Moreover from (3.21) and (3.23) we have

k [k(x) − k(ξ ₀ )] f ⁰ (W δ,ξ ) (φ 1 − φ ₂ )k

L

2N

N +2

≤ c ^∗ kφ ₁ − φ ₂ k _λ |ξ − ξ ₀ | ^θ + δ ^θ ≤ 1

4¯ c kφ ₁ − φ ₂ k _λ . (3.26) From (3.15), (3.25), and (3.26) we obtain

kT _δ,ξ ^λ (φ 1 − φ ₂ )k _λ ≤ 1

2 kφ ₁ − φ ₂ k _λ for any φ 1 , φ 2 ∈ B ^λ _δ,ξ (0, ¯ ρ ^λ _δ,ξ )

namely T _δ,ξ ^λ is a contraction in B _δ,ξ ^λ (0, ¯ ρ ^λ _δ,ξ ) and hence T _δ,ξ ^λ has a unique fixed point φ ^λ _δ,ξ in B _δ,ξ ^λ (0, ¯ ρ ^λ _δ,ξ ). In particular, in view of (3.24), φ ^λ _δ,ξ satisfies estimate (1.11).

Lemma 3.3 There exist Z ⊂⊂ R ^N such that ξ ₀ ∈ Z, λ ₀ > 0 and δ ₀ > 0 such that for any

λ ∈ (0, λ ₀ ), the function (δ, ξ) 7→ φ ^λ _δ,ξ given by Proposition 1.5 is of class C ¹ on (0, δ ₀ ) × Z.

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Proof. From (1.10) it follows that for any v ∈ K _δ,ξ ^λ

W δ,ξ + φ ^λ _δ,ξ − i ^∗ _λ (k(x)f (W δ,ξ + φ ^λ _δ,ξ )), v) λ = 0 i.e.

(W _δ,ξ + φ ^λ _δ,ξ , v) _λ − Z

R

^N

k(x)f (W _δ,ξ + φ ^λ _δ,ξ )v dx = 0, which can be written in term of the functional J _λ as

(J _λ ⁰ (W _δ,ξ + φ ^λ _δ,ξ ), v) _λ = 0 for any v ∈ K _δ,ξ ^λ

where J _λ ⁰ (u) denotes the Fr´ echet derivative of J _λ at u (identified with an element of D _λ through the canonical identification of the Hilbert space D _λ with its dual). Hence φ ^λ _δ,ξ satisfies

Π ^λ _δ,ξ J _λ ⁰ (W δ,ξ + φ ^λ _δ,ξ ) = 0. (3.27) For λ fixed consider the map

R ⁺ × R ^N × D _λ −→ D _λ

(δ, ξ, φ) 7−→ Π ^λ _δ,ξ J _λ ⁰ (W _δ,ξ + φ). (3.28) Let us remark that J _λ ∈ C ² (D _λ , R) and the map (δ, ξ) 7→ W δ,ξ which parametrizes the manifold of the solutions to the limit problem is of class C ² . Morover the projection map Π ^λ _δ,ξ can be written in the form

Π ^λ _δ,ξ (u) = u −

N

X

j=0

u, Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ

λ

Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ where

Υ ⁰ _λ,δ,ξ = ψ _δ,ξ ⁰ kψ _δ,ξ ⁰ k _λ , Υ ^j _λ,δ,ξ = ψ _δ,ξ ^j

kψ _δ,ξ ^j k _λ −

j−1

X

i=0

ψ _δ,ξ ^j

kψ ^j _δ,ξ k _λ , Υ ⁱ _λ,δ,ξ kΥ ⁱ _λ,δ,ξ k _λ

λ

Υ ⁱ _λ,δ,ξ

kΥ ⁱ _λ,δ,ξ k _λ , j = 1, . . . , N. (3.29) Note that Υ ^j _λ,δ,ξ , j = 0, 1, . . . , N generate the linear space span{ψ _δ,ξ ^j : j = 0, 1, . . . , N }, solve equation (1.8), and satisfy

Υ ^j _λ,δ,ξ , Υ ⁱ _λ,δ,ξ

λ = 0, i 6= j. It is easy to verify that (δ, ξ) 7→ Υ ^j _λ,δ,ξ is a C ¹ −map, and consequently the projection map Π ^λ _δ,ξ is of class C ¹ . Hence we deduce that the map defined in (3.28) is of class C ¹ , as it is shown in the diagram below

(δ, ξ, φ) ^C

2

7−→ W _δ,ξ 7−→ W ^C

^∞

_δ,ξ + φ ^C

1

7−→ J _λ ⁰ (W δ,ξ + φ) ^C

1

7−→ Π ^λ _δ,ξ J _λ ⁰ (W δ,ξ + φ).

From (3.27), the regularity of the map defined in (3.28), and the Implicit Function Theorem the

statement follows.

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Lemma 3.4 J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )u = L ^λ _δ,ξ (Π ^λ _δ,ξ u) + kuk _λ o(1) as δ, λ, |ξ − ξ ₀ | → 0.

Proof. We have that for any v ∈ D _λ

(J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )u, v) _λ − (L ^λ _δ,ξ (Π ^λ _δ,ξ u), v) _λ = (J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )Π ^λ _δ,ξ u, Π ^λ _δ,ξ v) _λ

− (L ^λ _δ,ξ (Π ^λ _δ,ξ u), Π ^λ _δ,ξ v) _λ + (J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )u, v − Π ^λ _δ,ξ v) _λ

+ (J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )(u − Π ^λ _δ,ξ u, Π ^λ _δ,ξ v) _λ . (3.30) For simplicity of notation, let us set f _δ,ξ ^j := ^Υ

j λ,δ,ξ

kΥ

^j_λ,δ,ξ

k

_λ

. Note that by H¨ older inequality (J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )u, v − Π ^λ _δ,ξ v) _λ

=

N

X

j=0

v, f _δ,ξ ^j

λ

u, f _δ,ξ ^j

λ − (2 ^∗ − 1) Z

R

^N

k(x)(W _δ,ξ + φ ^λ _δ,ξ ) ²

^∗

⁻² f _δ,ξ ^j u

=

N

X

j=0

v, f _δ,ξ ^j

λ

− (2 ^∗ − 1) Z

R

^N

k(x)[(W _δ,ξ + φ ^λ _δ,ξ ) ²

^∗

⁻² − W _δ,ξ ²

^∗

⁻² ]f _δ,ξ ^j u

− (2 ^∗ − 1) Z

R

^N

[k(x) − k(ξ ₀ )]W _δ,ξ ²

^∗

⁻² f _δ,ξ ^j u − λ Z

R

^N

f _δ,ξ ^j u

|x| ²

≤ constkuk _λ kvk _λ λ + k(k(x) − k(ξ ₀ ))W _δ,ξ ²

^∗

⁻² k _L

N/2

+ k(W _δ,ξ + φ ^λ _δ,ξ ) ²

^∗

⁻² − W _δ,ξ ²

^∗

⁻² k _L

N/2

.

Hence from (3.16) s and (3.17) s with s = 2 ^∗ − 1 and Lemma 5.7, we get (J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )u,v − Π ^λ _δ,ξ v) _λ

≤ constkuk _λ kvk _λ λ + |ξ − ξ 0 | ^ϑ + δ ^ϑ + kφ ^λ _δ,ξ k _λ + kφ ^λ _δ,ξ k ^{4/(N −2)} _λ . (3.31) Arguing in a similar way we can show that

(J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )(u − Π ^λ _δ,ξ u), Π ^λ _δ,ξ v) _λ

≤ constkuk _λ kvk _λ λ + |ξ − ξ ₀ | ^ϑ + δ ^ϑ + kφ ^λ _δ,ξ k _λ + kφ ^λ _δ,ξ k ^{4/(N −2)} _λ

(3.32) and

(J _λ ⁰⁰ (W δ,ξ + φ ^λ _δ,ξ )Π ^λ _δ,ξ u,Π ^λ _δ,ξ v) λ − (L ^λ _δ,ξ (Π ^λ _δ,ξ u), Π ^λ _δ,ξ v) λ

= −(2 ^∗ − 1)

Z

R

^N

k(x)[(W _δ,ξ + φ ^λ _δ,ξ ) ²

^∗

⁻² − W _δ,ξ ²

^∗

⁻² ]Π ^λ _δ,ξ uΠ ^λ _δ,ξ v +

Z

R

^N

(k(x) − k(ξ ₀ ))W _δ,ξ ²

^∗

⁻² Π ^λ _δ,ξ uΠ ^λ _δ,ξ v

≤ constkuk _λ kvk _λ |ξ − ξ ₀ | ^ϑ + δ ^ϑ + kφ ^λ _δ,ξ k _λ + kφ ^λ _δ,ξ k ^{4/(N −2)} _λ . (3.33)

The lemma follows from (3.30), (3.31), (3.32), and (3.33).

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Lemma 3.5 There holds

∂φ ^λ _δ,ξ

∂δ _λ

=



 

 

 

 

δ ⁻¹ O λδ + |ξ − ξ ₀ | ^θ + δ ^θ + kφ ^λ _δ,ξ k min{1,4/(N −2)}

λ

if N ≥ 5, δ ⁻¹ O λδp| log δ| + |ξ − ξ ₀ | ^θ + δ ^θ + kφ ^λ _δ,ξ k _λ

if N = 4, δ ⁻¹ O λδ ^1/2 + |ξ − ξ ₀ | ^θ + δ ^θ + kφ ^λ _δ,ξ k _λ

if N = 3,

(3.34)

and for N ≥ 3 and i = 1, . . . , N

∂φ ^λ _δ,ξ

∂ξ i

λ

= δ ⁻¹ O

λδ + |ξ − ξ 0 | ^θ + δ ^θ + kφ ^λ _δ,ξ k min{1,4/(N −2)}

λ

. (3.35)

Proof. Let us first note that from standard calculations, it is easy to verify that for some positiver constants C 0 (N, k(ξ 0 )) and C 1 (N, k(ξ 0 ))

kψ _δ,ξ ⁰ k = C ₀ (N, k(ξ 0 ))δ ⁻¹ , kψ _δ,ξ ^j k = C ₁ (N, k(ξ 0 ))δ ⁻¹ , j = 1, . . . , N,

∂ψ ⁰

∂δ ,

∂ψ ⁰

∂ξ _i ,

∂ψ ^j

∂δ ,

∂ψ ^j

∂ξ _i

= O(δ ⁻² ). (3.36)

From (3.27) it follows that there exist N + 1 real valued C ¹ -function α _j (λ, δ, ξ) such that J _λ ⁰ (W _δ,ξ + φ ^λ _δ,ξ ) =

N

X

j=0

α j (λ, δ, ξ) Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ , where Υ ^j _λ,δ,ξ are defined in (3.29). Since

α _j (λ, δ, ξ) +

N

X

i=0 i6=j

α _i (λ, δ, ξ) Υ ⁱ _λ,δ,ξ

kΥ ⁱ _λ,δ,ξ k _λ , Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ

!

λ

=

J _λ ⁰ (W _δ,ξ + φ ^λ _δ,ξ ), Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ

λ

= − λ

kΥ ^j _λ,δ,ξ k _λ Z

R

^N

W _δ,ξ Υ ^j _λ,δ,ξ

|x| ² dx

− Z

R

^N

k(x)((W δ,ξ + φ ^λ _δ,ξ ) ²

^∗

⁻¹ − W _δ,ξ ²

^∗

⁻¹ ) Υ ^j _λ,δ,ξ

kΥ ^j _λ,δ,ξ k _λ dx − Z

R

^N

(k(x) − k(ξ 0 ))W _δ,ξ ²

^∗

⁻¹ Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ dx, using (3.36), Lemma 5.2, Lemma 5.5, (3.29), and (3.20) we deduce that for any j = 0, 1, . . . , N

|α _j (λ, δ, ξ)| =



 

 

O λδ ² + |ξ − ξ 0 | ^θ + δ ^θ + kφ ^λ _δ,ξ k _λ

if N ≥ 5, O λδ ² | log δ| + |ξ − ξ ₀ | ^θ + δ ^θ + kφ ^λ _δ,ξ k _λ

if N = 4, O λδ + |ξ − ξ ₀ | ^θ + δ ^θ + kφ ^λ _δ,ξ k _λ

if N = 3.

(3.37)

The function (λ, δ, ξ) 7→ (φ ^λ _δ,ξ , α ₀ (λ, δ, ξ), . . . , α _N (λ, δ, ξ)) is implicitly given by H(δ, ξ, φ, α, λ) = 0 where

H : (0, δ 0 ) × R ^N × D _λ × R ^{N +1} × (0, λ ₀ ) → D _λ × R ^{N +1} (δ, ξ, φ, α, λ) 7→

J _λ ⁰ (W _δ,ξ + φ) −

N

X

j=0

α _j Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ ,

φ, Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ

j=0,...,N λ

.

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Since

∂H 1

∂(φ, α) (φ,α)

u β

= J _λ ⁰⁰ (W δ,ξ + φ)u −

N

X

j=0

β j

Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ

∂H 2

∂(φ, α) (φ,α)

u β

=

u, Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ

j=0,...,N λ

,

in view of Lemma 3.4 and (3.31) we find for any (u, β) ∈ D _λ × R ^{N +1}

∂H

∂(φ, α)

(φ

^λ_δ,ξ

,α(λ,δ,ξ))

u β

2 =

J _λ ⁰⁰ (W δ,ξ + φ ^λ _δ,ξ )u −

N

X

j=0

β j

Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ

2 λ

+

N

X

j=0

u, Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ

2 λ

=

J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )u

2 λ +

N

X

j=0

β _j ² + ku − Π ^λ _δ,ξ uk ² − 2

N

X

j=0

β _j

J _λ ⁰⁰ (W _δ,ξ + φ ^λ _δ,ξ )u, Υ ^j _λ,δ,ξ kΥ ^j _λ,δ,ξ k _λ

λ

≥ (kL ^λ _δ,ξ (Π ^λ _δ,ξ u)k − kuk _λ o(1)) ² + |β| ² + ku − Π ^λ _δ,ξ uk ² − 2

N

X

j=0

|β _j |kuk _λ o(1).

Hence from Lemma 3.2 we obtain for λ, δ, |ξ − ξ ₀ | small

∂H

∂(φ, α)

(φ

^λ_δ,ξ

,α(λ,δ,ξ))

u β

2 ≥ const

u β

2 .

Therefore there exist δ 0 > 0, λ 0 > 0, a small neighbourhood Z of ξ 0 , and a positive constant C ^∗ , such that for any δ ∈ (0, δ ₀ ), λ ∈ (0, λ ₀ ), ξ ∈ Z,

Veronica FELLI † and Angela PISTOIA ‡

Existence of blowing-up solutions for a nonlinear elliptic equation with Hardy potential and critical

growth ∗