Solutions to a nonlinear Neumann problem in three-dimensional exterior domains

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Solutions to a nonlinear Neumann problem in three-dimensional exterior domains

Olivier Rey, Adélaïde Olivier

To cite this version:

Olivier Rey, Adélaïde Olivier. Solutions to a nonlinear Neumann problem in three-dimensional exterior

domains. Comptes Rendus. Mathématique, Académie des sciences (Paris), 2018, 356 (9), pp.933 -

956. �10.1016/j.crma.2018.07.005�. �hal-01869921�

THREE-DIMENSIONAL EXTERIOR DOMAINS SOLUTIONS POUR UN PROBL `EME DE NEUMANN NON-LIN ´EAIRE DANS DES DOMAINES EXT ´ERIEURS EN

DIMENSION 3

AD ´ELA¨IDE OLIVIER AND OLIVIER REY

Abstract. We prove the existence of multipeak solutions for an nonlinear elliptic Neumann problem involving nearly critical Sobolev exponent, in three- dimensional exterior domains.

R´esum´e. Nous d´emontrons, pour un probl`eme elliptique de Neunmann avec non-lin´earit´e presque critique, dans un domaine ext´erieur en dimension trois, l’existence de solutions qui se concentrent en plusieurs points de la fronti`ere lorsque la non-lin´earit´e devient critique.

To Prof. Norman Dancer

1. Introduction and Results

There have been innumerable articles devoted over the last three decades to the study of elliptic partial differential equations of second order with critical nonlin- earity. One thing, however, is to be noticed: virtually all articles consider problems in bounded domains. A work like Yan’s one [15] is an exception. In that paper, Yan considers the following Neumann problem:

(1.1)







−∆u=u^{2∗−1−ε} , u >0 in R^N \Ω

∂u

∂ν = 0 on ∂Ω

where Ω is a smooth and bounded domain in R^N, N ≥ 3, such that R^N \Ω is connected, 2^∗ = 2N/(N−2) is the limiting Sobolev exponent for the embedding of the Sobolev space W^1,2(Ω) into the L^p(Ω)-spaces, and ε is a strictly positive number assumed to be small. Although formally, the problem is subcritical, it is asymptotically critical, and the techniques to be used to study it asεgoes to zero are the same as those to be used in the case of critical nonlinearities.

Considering exterior domains is all but arbitrary. Indeed, various models lead to study the equation−∆u=u^p in domains with small holes. As the size of these

Key words and phrases. Nonlinear elliptic Neumann problems, Critical Sobolev exponent, Ex- terior domains.

Mots-cl´es. Probl`emes elliptiques de Neumann, Exposant critique de Sobolev, Domaines ext´erieurs.

1

holes goes to zero, the limiting problem which is obtained through a rescaling cen- tered on one of the holes looks precisely as (1.1).

In order to state the results proved by Yan, as well as those that we propose to establish, some notations have first to be introduced. Forλ∈R^∗+ andx∈R^N, we denote byU_λ,x the function defined inR^N by

(1.2) U_λ,x(y) = [N(N−2)]^N−24 λ^N−22 (1 +λ²|y−x|²)^N−22

TheUλ,x’s are the only nontrivial solutions to the equation−∆U =U^2∗−1, U ≥0 in R^N (see for example [1], [13] or [7]) and induce, asλgoes to infinity, a lack of compactness of the embedding of W^1,2 into L^2∗. In the following, D^1,2(R^N \Ω) refers to the completion of the set of smooth functions with compact support in R^N\Ω for the norm

kuk=< u, u >^1/2 with

< u, v >=

Z

R^N\Ω

∇u.∇v.

Lastly, we denote byH(y) the mean curvature of∂Ω at a pointyof this boundary.

Yan proves:

Theorem 1.1. [15]. Assume that N ≥4.

(1) [Case of a positive local maximum ofH.] Suppose thatS is a connected subset of ∂Ω satisfying: H(y) = Hm > 0 for any y ∈ S; and there exists δ > 0 such that H(y)< Hm for any y ∈ Sδ\S, andH has no critical point in Sδ \S, with Sδ = {y ∈ ∂Ω s.t. d(x, y) ≤δ}. Then, for any positive integer k, there exists ε0>0such that for anyε0∈(0, ε0), (1.1) has a solution

(1.3) uε=

k

X

i=1

αε,iUλ_ε,i,x_ε,i+vε

where, asεgoes to zero αε,i→1

ελε,i→c^∗Hm c^∗a positive constant depending onN only x_ε,i∈S_δ and x_ε,i →x_i∈S

for any i,i≤i≤k, and

v_ε→0 in D^1,2(R^N \Ω).

(2) [Case of a positive local minimum ofH.] Suppose thatS is a connected subset of ∂Ω satisfying: H(y) = H_m > 0 for any y ∈ S; and there exists δ > 0 such that H(y)> H_m for any y ∈ S_δ\S, andH has no critical point in S_δ \S, with S_δ = {y ∈ ∂Ω s.t. d(x, y) ≤δ}. Then, there exists ε₀ > 0 such that for any ε₀∈(0, ε₀), (1.1) has a solution

(1.4) u_ε=α_εU_λε,xε+v_ε

whereαε,i→1,ελε→c^∗Hm,xε→x0 ∈S and vε→0 inD^1,2(R^N \Ω)asεgoes to zero.

The arguments developed by Yan to prove Theorem 1.1allow him to consider also the problem

(1.5)







−∆u+µu=u^2∗−1 , u >0 in Ω

∂u

∂ν = 0 on ∂Ω

where µ is a positive number assumed to be large. Yan proves, for N ≥ 5 and a positive local minimum of H, an equivalent of Theorem 1.1 (1) as µ goes to infinity (1/µ plays a role similar to that of εpreviously). Such a result has been extended to the cases N = 3,4 by Wei and Yan in [14]. On the other hand, it has never been demonstrated until now that the statement of Theorem1.1itself is valid in the case N = 3. As Brezis and Nirenberg [6] have shown, there are problems involving elliptic equations with critical nonlinearity where casesN = 3 andN ≥4 are qualitatively different. This is not the case here, and we are able to prove:

Theorem 1.2. Let N = 3.

(1) [Case of a positive local maximum of H.] Suppose that S ⊂ ∂Ω is such that 0 <sup_y∈∂SH(y)< sup_y∈SH(y) = Hm. Then, for any positive integer k, there existsε₀>0such that for anyε₀∈(0, ε₀), (1.1) has a solution

(1.6) uε=

k

X

i=1

αε,iUλε,i,xε,i+vε

where, asεgoes to zero αε,i→1

lnλε,i

ελ_ε,i → π

16H_m i.e. λ_ε,i∼ 16Hm

πε ln1 ε

xε,i→xi∈S such that H(xi) = max

y∈S H(y) for any i,1≤i≤k, and

vε→0 in D^1,2(R^N \Ω).

(2) [Case of a positive local minimum of H.] Suppose that S ⊂ ∂Ω is such that 0 < inf_y∈SH(y) <inf_y∈∂SH(y) = H_m. Then, there exists ε₀ >0 such that for any ε0∈(0, ε0), (1.1) has a solution

(1.7) u_ε=α_εU_λε,xε+v_ε where α_ε,i →1, ^ln_ελ^λε,i

ε,i → _16H^π

m,x_ε→x₀ ∈S such that H(x_i) = min_y∈SH(y)and v_ε→0 inD^1,2(R^N\Ω)asεgoes to zero.

Remark 1. As ∂S is closed and bounded, sup_y∈∂SH(y) and infy∈∂SH(y) are achieved. The same holds for sup_y∈SH(y) in case (1) and inf_y∈SH(y) in case (2). Indeed, let us consider in case (1) a maximizing sequence (y_n) inS for H. A

subsequence of (yn) converges to some limit ¯y which satisfies H(¯y) =Hm. As, by assumption, sup_y∈∂SH(y)< Hm, ¯y has to be in the interior of S. Consequently, there exist one or several points y∈S such thatH(y) =Hm, and which are local maxima of H. We conclude in the same way in case (2).

Remark 2. As Ω is assumed to be bounded in R^N, H has a strictly positive maximum on ∂Ω, and case (1) always occurs (in the particular case of a ball, we can takeS=∂Ω,∂S=∅).

Remark 3. The method we use to prove Theorem1.2allows us to get rid of the unessential assumption in the statement of Theorem 1.1, that H has no critical point inSδ\S. Our argument to eliminate such an assumption applies as well to the caseN ≥4.

Problem (1.1) can be formulated in a variational way: formally,u∈D^1,2(R^N\Ω) solves (1.1) if and only ifuis a nontrivial critical point of the functional

(1.8) I_ε= 1

2 Z

R^N\Ω

|∇u|²− 1 2^∗−ε

Z

R^N\Ω

(u⁺)^2∗−ε withu⁺= max(u,0). Indeed, a critical point ofI_εsatisfies (1.9) −∆u= (u⁺)^2∗−εinR^N \Ω, ; ∂u

∂ν = 0 on∂Ω.

Multiplying the equation by u⁻ = max(−u,0) and integrating onR^N \Ω, we see that u⁻ ≡0, whence u≥ 0 inR^N \Ω. If u6≡ 0, the strong maximum principle implies thatu >0 inR^N \Ω; as a consequence, uis a solution of (1.1).

However, a difficulty arises: Iε is not defined in whole D^1,2(R^N \Ω) for, as R^N \Ω is not bounded, this space does not embed into L^2∗−ε(R^N \Ω). Our first task will therefore be, in the next section, to build a functional ˜Iε well defined in D^1,2(R^N \Ω), and whose critical points which write as in Theorem1.1 or1.2 are solutions of (1.1).

In Section3 we shall perform a parametrization of the problem in a neighbour- hood of the solutions of type (1.3) (1.6) we look for, in order to obtain a functional depending on theα_i’s,λ_i’s,x_i’s and v.

In Section4, an optimization of that functional with respect to theαi’s andvwill provide us with a function now dependent only on theλi’s and the xi’s. Then we will be able, in the last section (Section5), to deduce from the assumptions of The- orem1.2, that the reduced function has a critical point – whence, by construction, the existence of a solution of (1.1) with the properties specified in Theorem 1.2.

The proof of a number of technical results necessary for the exposition of the main argument is given in Appendix.

2. The variational formulation

Although this article is intended to establish Theorem1.2, and is therefore con- cerned only with space dimensionN= 3, we consider in this section any dimension N ≥3 – insofar the argument is identical in all dimensions. Moreover, we shall make explicit some points which are sketched in [15]. To obtain, from I_ε defined

by (1.8), a functional Ieε well defined in D^1,2(R^N \Ω), the nonlinearity has to be truncated at infinity. By adapting Yan’s strategy in [15], we proceed as follows.

We choose R > 0 such that Ω ⊂ B_R/2(0), and τ >0. Let ϕ : R+×R → R, 0≤ϕ≤1,ϕsmooth in R+×R+, such that

(2.1)







ϕ(s, t) = 0 if t <0

ϕ(s, t) = 1 if 0≤s≤Randt≥0, ors≥Rand 0≤s^N−2t≤τ ϕ(s, t) = 0 if s≥R+ 1 ands^N−2t≥τ+ 1.

Then, we defineg_ε:R^N ×R→Ras

(2.2) gε(y, t) =|t|^{2∗−1−ε}ϕ(|y|, t)

which, forεsmall enough, isC¹, and we consider the problem (2.3)







−∆u=gε(y, u) in R^N \Ω

∂u

∂ν = 0 on ∂Ω.

By construction, g_ε(y, u) = (u⁺)^{2∗−1−ε} for|y| ≤R, or|y| ≥Rand |y|^N−2u⁺≤ τ;gε(y, u) = 0 for|y| ≥R+ 1 and|y|^N−2u⁺≥τ+ 1. We notice that, according to definition (1.2) of theUλ,x’s, whenx∈∂Ω (whence|x|< R), |y|^N−2Uλ,x(y) goes to zero asλ goes to infinity, uniformly with respect to y, |y| ≥R. Consequently, when the xi’s belong to ∂Ω, the αi’s are close to 1 and the λi’s are large enough, we have

g_ε y,

k

X

i=1

α_iU_λi,xi(y)

= ^k

X

i=1

α_iU_λi,xi(y)

^2∗−1−ε

in allR^N\Ω : the truncation does not affect the nonlinearity whenu=Pk

i=1α_iU_λi,xi, in the neighborhood of which we look for a solution of (1.1). We set now

(2.4) Ie_ε(u) =1 2

Z

R^N\Ω

|∇u|²− Z

R^N\Ω

G_ε(y, u) with

G_ε(y, u) = Z u

0

g_ε(y, t)dt.

According to the definition of gε, Gε(y, u) = 0 if u≤0, Gε(y, u) = ₂∗¹−εu^2∗−ε if u≥0 and|y| ≤R, and|Gε(y, u)| ≤ ₂∗¹−ε|u|^2∗−ε everywhere. Moreover, (2.1) and (2.2) imply that if|y| ≥R+ 1 and|y|^N−2|u| ≥τ+ 1,g_ε(y, u) = 0, and if|y| ≥R+ 1 and|y|^N−2|u| ≤τ+ 1

(2.5)

g_ε(y, u) u

≤ u

2^∗−2−ε

≤(τ+ 1)^{N−24 −ε}

|y|^{4−(N−2)ε} . Then, using Young’s inequality, we can write, for|y| ≥R+ 1,

0≤Gε(y, u)≤ u²(τ+ 1)^{N−24 −ε}

2|y|^{4−(N−2)ε} ≤(τ+ 1)^{N−24 −ε} |u|^2∗

2^∗ + 1

N|y|^{2N−N(N−2)2 ε}

. As D^1,2(R^N \Ω) embeds intoL^2∗(R^N \Ω), we see that Ieε is well defined in this functional space.

Arguing as we did before for the solutions of (1.8), we know that a nontrivial solution of (2.3) is strictly positive in R^N \Ω. Let us show that a solutionuε of (2.3), writing as (1.3) (1.6), is actually a solution of (1.1). To this end, we shall show that|y|^N−2u_ε(y) goes uniformly to 0 inR^N\B_R(0) asεgoes to zero – this entails, by definition ofgε, thatgε(y, uε) =u²_ε^∗−1−ε inR^N \Ω, whence the result.

Setting

wε(y) = 1

|y|^N−2uε

y

|y|²

this amounts to showing w_ε goes uniformly to 0 in B_1/R(0). Asu_ε is assumed to solve (2.3),w_ε satisfies, inB_2/R(0)\ {0}

−∆wε= 1

|y|^N+2∆uε

y

|y|²

= 1

|y|^N+2gε

y

|y|²,|y|^N−2wε(y)

. We may also write

(2.6) −∆w_ε=a_ε(y)w_ε in B_2/R(0)\ {0}

where, according to the definition (2.1) (2.2) ofg_ε,a_εsatisfies (2.7) 0< a_ε(y)≤ 1

|y|^(N−2)εw²_ε^∗−2−ε in B_2/R(0) and also, because of (2.5)

(2.8) 0< a_ε(y)≤ C

|y|^(N−2)ε in B_1/(R+1)(0) whereC is a constant depending only onτ andN.

Actually, we can check that (2.6) holds in whole B_2/R(0). To this end, we consider ψ∈C^∞(R^N,R) such thatψ(y) = 0 if|y| ≤1/2 andψ(y) = 1 if |y| ≥1, and we set, for n ∈ N^∗, ψn(y) = ψ(ny). For any ϕ ∈ C₀^∞(B2/R(0)), ψnϕ ∈ C₀^∞(B2/R(0)\ {0}), and (2.6) yields

h−∆wε, ψnϕi= Z

B_2/R(0)

aε(y)wεψnϕ.

Through dominated convergence, it is easily checked that the right hand side goes toR

B_2/R(0)aε(y)wεϕasngoes to infinity. On the left hand side we have h−∆wε, ψnϕi=−hwε, ψn∆ϕ+ 2∇ψn.∇ϕ+ϕ∆ψni and

−hwε, ϕ∆ψni= Z

1/2n≤|y|≤1/n

wεϕ∆ψn

= O n² Z

1/2n≤|y|≤1/n

w_ε^2∗ ₂¹∗ Z

1/2n≤|y|≤1/n

dy ^N+2_2N !

. As

Z

1/2n≤|y|≤1/n

w²_ε^∗ = Z

n≤|y|≤2n

1

|y|^2Nw²_ε^∗ y

|y|²

= Z

n≤|y|≤2n

u²_ε^∗ we see that

−hwε, ϕ∆ψ_ni= o(n^−N−22 ).

The term involving∇ψn.∇ϕmay be treated in the same way, so that

n→∞limh−∆wε, ψnϕi=− lim

n→∞hwε, ψn∆ϕi=−hwε,∆ϕi=h−∆wε, ϕi and (2.6) holds in wholeB_2/R(0), as announced.

Now, in view (2.7)and(2.8), for anyy0∈B_3/(2R)(0) and anyδ, 0< δ < _4R¹ , we can write

Z

Bδ(y0)

a^N/2_ε = Z

Bδ(y0)∩B_1/(R+1)(0)

a^N/2_ε + Z

Bδ(y0)\B_1/(R+1)(0)

a^N/2_ε

≤C1

Z

B_δ(y₀)

|y|^{−N(N−2)2 ε}dy+C2(R+ 1)^{N(N−2)2 ε} Z

B_7/(4R)(0)

w

N

2(2^∗−2−ε) ε

≤C3δ^{N−N(N−2)2 ε}+C4

Z

B_7/(4R)(0)

w_ε^2∗

^1−N−2₄ ^ε

where C1, . . . , C4 are constants depending only on τ, R and N. Furthermore, we have

(2.9)

Z

B_7/(4R)(0)

w²_ε^∗= Z

R^N\B_4R/7(0)

u²_ε^∗

and, when uε write as (1.3) (1.6), the last integral goes to zero as ε goes to zero (remember that theUλ_ε,i,x_ε,i’s concentrate at points located on the boundary of Ω, with Ω⊂B_R/2(0), andvεgoes to zero inD^1,2(R^N\Ω) whence also inL^2∗(R^N\Ω)).

Consequently, we see that for every η > 0 we can chooseδ > 0 such that, for ε small enough,

(2.10)

Z

B_δ(y₀)

a^N/2_ε < η

uniformly with respect toy0∈B3/(2R)(0). This will imply thatwε goes to zero in L^∞(B1/R(0)).

Indeed, let us consider δ⁰, 0< δ⁰ < δ, and ψ ∈C^∞(R^N) such that 0≤ψ≤ 1 andψ≡1 inBδ⁰(0),ψ≡0 inR^N\Bδ(0). Fory0∈B_3/(2R)(0), letψy₀ ∈C^∞(R³) be the function defined byψy₀(y) =ψ(y−y0). Multiplying (2.6) by ψ_y²₀w^γ_ε,γ >1, and integrating inB_δ(y₀), we obtain

(2.11) −

Z

Bδ(y0)

∆wε.ψ²_y0w_ε^γ= Z

Bδ(y0)

aεψ²_y0w_ε^γ+1

provided that the integrals are well defined. On the one hand, integrating by parts, we have

− Z

B_δ(y0)

∆wε.ψ_y²₀w^γ_ε = 4γ (γ+ 1)²

Z

B_δ(y0)

∇(ψy₀w

γ+1

ε2 )

2

−4(γ−1) (γ+ 1)²

Z

B_δ(y₀)

w

γ+1

ε2 ∇ψy₀.∇(ψy₀w

γ+1

ε2 )

− 4

(γ+ 1)² Z

B_δ(y0)

∇ψ_y0

2w^γ+1_ε .

On the other hand, H¨older’s inequality yields Z

B_δ(y₀)

aεψ_y²₀w^γ+1_ε ≤ Z

B_δ(y₀)

a^N/2_{ε N}² Z

B_δ(y₀)

ψ_y^2∗₀w

N N−2(γ+1) ε

^N−2_N . Coming back to (2.11), we deduce from (2.10) and Cauchy-Schwarz inequality

4γ (γ+ 1)²

Z

B_δ(y₀)

∇(ψy₀w

γ+1

ε2 )

2

−4(γ−1) (γ+ 1)²

Z

B_δ(y₀)

∇ψy₀

2w^γ+1_ε ¹₂ Z

B_δ(y₀)

∇(ψy₀w

γ+1

ε2 )

2¹₂

≤ 4

(γ+ 1)² Z

B_δ(y₀)

∇ψy0

2w^γ+1_ε +η^N2 Z

B_δ(y₀)

ψ²_y^∗₀w

N N−2(γ+1) ε

^N−2₂ . Therefore, there exists a constantC such that

Z

B_δ(y₀)

∇(ψy₀w

γ+1

ε2 )

2≤C Z

B_δ(y₀)

w_ε^γ+1+η^N2 Z

B_δ(y₀)

ψ²_y^∗₀w

N N−2(γ+1) ε

^N−2₂ . Still assuming that the integrals are well defined,ψy₀w

γ+1

ε2 ∈W₀^1,2(Bδ(y0)), and as W₀^1,2(Bδ(y0)) embeds continuously intoL^2∗(Bδ(y0)), we have

Z

Bδ(y0)

ψ_y²₀^∗w

N N−2(γ+1) ε

^N−2₂

≤ 1 S

Z

Bδ(y0)

∇(ψy₀w

γ+1

ε2 )

2

where S is a strictly positive constant (which depends only on N). We see that, choosingη small enough, there exists a constantC⁰, depending only on δ andN, such that

(2.12)

Z

B_δ(y₀)

ψ²_y^∗₀w

N N−2(γ+1) ε

^N−2₂

≤C⁰ Z

B_δ(y₀)

w^γ+1_ε . Withγ= 2^∗−1, we obtain

(2.13)

Z

B_δ0(y0)

w

N N−22^∗ ε

^N−2₂

≤C⁰ Z

Bδ(y0)

w²_ε^∗.

The right hand side integral is well defined, which proves that the left hand side is also well defined. (Actually, to make the previous argument perfectly rigorous, we should have multiplied (2.11) byψ²_y0w_ε,n^γ , where, forn∈N∗,wε,n(y) =wε(y) if w_ε(y)≤n, w_ε,n(y) = notherwise. Then, all the integrals in the previous compu- tations are well defined and, lettingngo to infinity, we obtain (2.13).)

The right hand side of (2.13) goes to zero asεgoes to zero, as (2.9) proves. As a consequence, wε goes to zero inL^{N−2N 2∗} Bδ⁰(y0)

, uniformly with respect toy0, and thus inL^{N−2N 2∗} B_3/(2R)(0)

. Iterating the process, we find through (2.12) that wεgoes to zero in everyL^p B_3/(2R)(0)

,p <∞. This implies, taking into account (2.7), that a_ε goes to zero in L^q B_3/(2R)(0)

for someq > N/2 (actually, for any q <∞). Then, standard theory for elliptic equations (seee.g. Theorem 8.24 in [9]) ensures that wε goes to zero in L^∞ B_1/R(0)

. Therefore, gε(y, uε) = u²_ε^∗−1−ε in R^N \Ω andu_ε is a solution of (1.1).

3. Parametrization of the variational problem

In this section, we shall still consider the general case N ≥3. Letk ∈N^∗. We set

(3.1) Dε=n

(Λ, X)∈(R^∗+)^k×(∂Ω)^k s.t. f₁(ε)< λ_i< f₂(ε)

and |x_i−x_j|> h(ε),1≤i, j≤k, i6=jo where Λ = (λ₁, . . . , λ_n), X = (x₁, . . . , x_n) and f₁, f₂, h are positive functions such that f₁(ε) and f₂(ε) go to infinity, h(ε) goes to zero as ε goes to zero, and whose precise expression will be determined later. We define also, for (Λ, X) ∈ (R^∗+)^k×(∂Ω)^k

(3.2) E_Λ,X =n

v∈D^1,2(R^N \Ω) s.t. hv, Uii= v,∂U_i

∂λi

= v, ∂U_i

∂τi,j

= 0, 1≤i≤k,1≤j≤N−1o whereU_i=U_λi,xi, and (τ_i,1, . . . , τ_i,j) is an orthogonal system of coordinates of the tangent space to∂Ω atx_i. Lastly, forδ >0, we define

(3.3) Mε,δ=n

(A,Λ, X, v)∈R^k×(R^∗+)^k×(∂Ω)^k×D^1,2(R^N\Ω)

s.t. |αi−1|< δ,1≤i≤k; (Λ, X)∈ Dε ;v∈EΛ,X,kvk< δo whereA= (α1, . . . , αk), and we consider onMε,δ the functional

(3.4) Jε(A,Λ, X, v) =Ieε

^k X

i=1

αiUi+v

.

According to [3, 4], we know that for ε and δ small enough, (A,Λ, X, v) is a critical point of Jε in Mε,δ if and only if u=Pk

i=1αiUi+v is a critical point of Ie_εin D^1,2(R^N \Ω). Let us remark that, in consideration of (3.2), (A,Λ, X, v) is a critical point ofJ_εin M_ε,δ if and only if there exists Lagrange multipliersA_i, B_i, Cij, 1≤i≤k, 1≤j≤N−1 such that

∂Jε

∂α_i = 0 (3.5)

∂Jε

∂λ_i =Bi∂²Ui

∂λ²_i , v +

N−1

X

`=1

Ci` ∂²Ui

∂λ_i∂τ_ij, v (3.6)

∂Jε

∂τij

=Bi

∂²Ui

∂λi∂τij

, v +

N−1

X

`=1

Ci`

∂²Ui

∂τij∂τi`

, v (3.7)

∂J_ε

∂v =

k

X

i=1

AiUi+

k

X

i=1

Bi

∂U_i

∂λi

+

k

X

i=1 N−1

X

`=1

Ci`

∂U_i

∂τi`

. (3.8)

In order to find critical points ofJεinMε,δwe shall proceed in two steps. Firstly we shall eliminate the non-significant parameters : theαi’s andv. More precisely, we shall prove the existence of a C¹-map which to every (Λ, X) ∈ Dε associates A_ε(Λ, X)∈R^k, each α_ε,i close to 1, and v_ε ∈E_Λ,X, kvεk close to zero, such that (3.5) and (3.8) are satisfied. It will be left to us, in a second time, to show, using

a topological argument, that the function (Λ, X)7→ Jε Aε(Λ, X),Λ, X, vε(Λ, X) has a critical point inDε.

4. The reduced functional

We first perform a change of variables concerning the parametersαi’s, setting

(4.1) A⁰=A−1

where1= (1, . . . ,1), that isA⁰ = (α⁰₁, . . . , α⁰_k), with

(4.2) α⁰_i=α_i−1, 1≤i≤k.

We define also, onR^k×D^1,2(R^N \Ω), the scalar product

(4.3) (A, f),(B, g)=

k

X

i=1

a_ib_i+ Z

R^N\Ω

∇f.∇g

and|||·|||will denote the associated norm. Let us expandJε(A,Λ, X, v) with respect tow= (A⁰, v) in a neighborhood of (1,0) in R^k×E_Λ,X. We can write, using Riesz representation theorem

(4.4) J_ε(A,Λ, X, v) =J_ε(1,Λ, X,0)+f_ε,Λ,X, w+1

2 Q_ε,Λ,Xw, w +Rε,Λ,X(w) wheref_ε,Λ,X ∈R^k×E_Λ,X is such that

(4.5) f_ε,Λ,X, w=

k

X

i=1

k

X

j=1

U_j, U_i

− Z

R^N\Ω

U_ig_ε y,

k

X

j=1

U_j

α⁰_i

− Z

R^N\Ω

gε y,

k

X

j=1

Uj v, Qε,Λ,X is an endomorphism ofR^k×EΛ,X such that

(4.6) Qε,Λ,Xw, w=

k

X

i,j=1

hUi, Uji − Z

R^N\Ω

UiUj

∂g_ε

∂t y,

k

X

`=1

U`

α⁰_iα⁰_j

−

k

X

i=1

Z

R³\Ω

∂gε

∂t y,

k

X

`=1

U` Uiv

α⁰_i

+kvk²− Z

R^N\Ω

∂gε

∂t y,

k

X

`=1

U` v², andRε,Λ,X satisfies

(4.7) R^(m)_ε,Λ,X(w) = O kwk^{min(2∗−ε,3)−m}

, m= 0,1,2.

(In the special caseN = 3, 2^∗= 6 andR^(m)_ε,Λ,X(w) = O kwk^3−m

,m= 0,1,2.) The fact that equations (3.5) and (3.8) are satisfied is equivalent to

(4.8) f_ε,Λ,X +Q_ε,Λ,Xw+R_ε,Λ,X⁰ = 0.

In the special case N = 3 we choose in the definition (3.1) ofDε (for reasons that will become clear later)

(4.9) f1(ε) =a₁ ε ln1

ε, f2(ε) =a₂

ε ln¹_ε, h(ε) =b ln1 ε

−3/4

where a1, a2, b, with a1 < a2, are strictly positive constants which will be de- termined later. (For N ≥ 4, following [15], we would choose f1(ε) = a1/ε, f2(ε) =a2/ε,h(ε) =ε^{1−N−21+θ} withθa small positive number.) Then it is proved in Appendix B1 that

(4.10) |||fε,Λ,X|||= O

ε^1/2 ln1 ε

−1/2

.

Moreover it is proved in Appendix B2 thatQε,Λ,X is invertible and there existsρ, independent ofεand (Λ, X)∈ Dε, such that

(4.11)

Q⁻¹_ε,Λ,X

≤ρ.

Let us consider now the mapFε,Λ,X, from a neighborhood of (0,0) in (R^k×EΛ,X)² toR^k×EΛ,X defined by

Fε,Λ,X(f, w) =f+Qε,Λ,Xw+R⁰_ε,Λ,X(w).

We have

F_ε,Λ,X(0,0) = 0 and ∂Fε,Λ,X

∂w =Q_ε,Λ,X+R⁰⁰_ε,Λ,X(w).

From (4.11) and (4.7), we deduce the existence ofη >0 such that forεsmall enough and|||w|||< η, ^∂F_∂w^ε,Λ,X is invertible. Then, taking into account (4.10), the implicit function theorem allows us to state:

Proposition 4.1. There exists ε0 >0 such that for every ε∈ (0, ε0), a C¹-map exists which to every(Λ, X)∈ Dεassociateswε,Λ,X = (A⁰_ε,Λ,X, vε,Λ,X)∈R^k×EΛ,X

such that Fε,Λ,X(fε,Λ,X, wε,Λ,X) = 0. This means that (3.5)and (3.8)are satisfied when theαⁱ’s andv are such thatA=1+A⁰_ε,Λ,X andv=vε,Λ,X. Moreover, when N = 3,

|α⁰_i;ε,Λ,X|=|αi;ε,Λ,X−1|= O ε^1/2 ln¹_ε^−1/2

, 1≤i≤k, (4.12)

kvε,Λ,Xk= O ε^1/2 ln¹_ε−1/2 (4.13)

asεgoes to zero.

Remark. Actually, estimate (4.12) could be improved: we could prove that

|α⁰_i|= O(εln¹_ε). However, (4.12) is sufficient for our purposes.

We consider now the reduced functional

(4.14) Je_ε(Λ, X) =J_ε α_ε(Λ, X),Λ, X, v_ε(Λ, X)

in Dε. ForJe_ε and its derivatives with respect to theλ_i’s, the following expansion holds:

Proposition 4.2. Let N = 3. Forε∈(0, ε0)and(Λ, X)∈ Dε, we have:

Jeε(Λ, X) =kK1,ε+

k

X

i=1

K2H(xi)lnλ_i λi

+K3εlnλi

(4.15)

−K4 k

X

i,j=1 i6=j

1

λ^1/2_i λ^1/2_j |xi−xj|+ O ε ln¹_ε−1 ,

∂Jeε

∂λ_i(Λ, X) =−K2H(xi)lnλi

λ²_i +K3

ε

λ_i + O ε² ln¹_ε^−5/4 (4.16)

whereKε,1,K2,K3,K4 are strictly positive constants.

This proposition is proved in Appendix A. We are now able to prove Theorem1.2.

5. Proof of Theorem1.2

For sake of simplicity, we always assume now thatN = 3. We shall concentrate our attention on part (1) of Theorem1.2(the proof of part (2) will then be straight- forward). We look for a critical point (Λ, X)∈ D_εofJe_ε. Thex_i’s will be supposed to belong to the subset S which, according to the assumptions of Theorem1.2, is such that

0<max

y∈∂SH(y)<max

y∈S H(y) =Hm.

More precisely, thex_i’s will be supposed to converge, asεgoes to zero, to points

¯

x_i which satisfy H(¯x_i) =H_m. Then according to (4.16), the λ_i’s should be close toλ(ε) defined by

(5.1) lnλ(ε)

λ(ε) = K3ε K₂H_m. A simple computation yields the expansions

λ(ε) =K2Hm

K₃ε ln1

ε+K2Hm

K₃ε ln ln1 ε+ O 1

ε , (5.2)

lnλ(ε) = ln1

ε+ ln ln1

ε+ O(1).

(5.3)

Now, we are able to fix a1 and a2 in the definition (4.9) of functions f1 and f2

which occur in (3.1): we choosea1 anda2 such that

(5.4) 0< a₁< K₂H_m

K3

< a₂.

(For example, in view of (A.23), we may takea1= 15Hm/π anda2= 17Hm/π.) In order to prove the theorem, we are going to argue by contradiction. The general strategy is the following: we define two levels,cε,1 andcε,2,cε,1< cε,2, and assume thatJe_ε has no critical value between them. Then we use the gradient flow to deform the level sets ofJe_εcorresponding toc_ε,2andc_ε,1one into the other. The difference of topology between the two level sets provides us with a contradiction.

Let us make the argument precise. Firstly, we define the two levelsc_ε,1 andc_ε,2 as follows. In view of (4.15), we set

(5.5) c_1,ε=kK_ε,1+

k

X

i=1

K₂H_mlnλ(ε)

λ(ε) +K₃εlnλ(ε)

−ε ln1 ε

−1/4

.

Taking into account (5.1), we can also write (5.6) c_1,ε =kK_ε,1+kK₃ε lnλ(ε) + 1

−ε ln1 ε

−1/4

.

Concerningc2,ε, we just setc2,ε =kKε,1+τwhereτis some small positive constant.

It is clear that forε small enough,c2,ε> c1,ε. Next, we have to define the subset ofDεon which the deformation argument will be implemented. We set

(5.7) S⁰=n

x∈S s.t. H(x)> Hm−a0 ln1 ε

−1/4o

wherea₀ is a strictly positive constant to be determined later, (5.8) M =n

X∈(∂Ω)^k s.t. xi∈S⁰ and |xi−xj|> b ln1 ε

−3/4

,

1≤i, j≤k, i6=jo (b, which already occurs in the definition (4.9), will be determined later) andT is the interval

(5.9) T =

1−D ln1 ε

−1/4 λ(ε),

1 +D ln1 ε

−1/4 λ(ε)

where D is a large constant which will be chosen later. We note that for ε small enough, every λ∈T satisfies ^a_ε¹ ln¹_ε < λ < ^a_ε² ln¹_ε, and T^k×M ⊂ Dε. We note also that for εsmall enough, T^k×M ⊂Jeε

cε,2

, whereJeε

ω,ω ∈R, is the level set {(Λ, X)∈ Dε s.t. Jeε(Λ, X)< ω}.

We consider now the gradient flow (5.10)





 d

dt Λ(t), X(t)

=−∇Jeε Λ(t), X(t) Λ(0), X(0)

∈(T^k×M) Arguing by contradiction, we assume :

(H) Jeε has no critical value between cε,1 and cε,2 in T^k×M.

The following holds:

Lemma 5.1. The flow line Λ(t), X(t)

defined by (5.8)and starting from a point of T^k×M does not leaveT^k×M before it reachesJeε

cε,1

.

Proof. We have to check that for any (Λ, X) ∈ ∂(T^k ×M) either −∇Jeε points inwards, orJeεis less than cε,1.

Case 1. X ∈∂M.

Case 1a. There is somej such thatH(xj) =Hm−a0 ln¹_ε−1/4

. According to (4.15), we have

(5.11) Je_ε(Λ, X)≤kK_1,ε+

k

X

i=1

K₂H(x_i)lnλ_i λi

+K₃εlnλ_i +K2 H(xj)−Hmlnλj

λ_j + O

ε ln¹_ε⁻¹ .

Whenλi=λ(ε)(1 +ζi),ζi small, a simple computations yields K2Hm

lnλi

λi

+K3εlnλi=K2Hm

lnλ(ε)

λ(ε) +K3εlnλ(ε) + O _λ(ε)^ζi +εζ_i² and, ifζi = O

ln¹_ε^−1/4

, as it is the case whenλi∈T

ζ_i

λ(ε)+εζ_i²= O

ε ln¹_ε−1/2 . Moreover, we have, using (5.9) and (5.1) (5.2)

lnλj

λj

=lnλ(ε) λ(ε) + O

ζj

lnλ(ε) λ(ε)

= K3ε K2Hm

+ O

ε ln¹_ε−1/4 . Therefore, taking into account (5.5),

Je_ε(Λ, X)≤c_ε,1+ε ln¹_ε−1/4

−a₀K₃ Hm

ε ln¹_ε−1/4

+ O

ε ln¹_ε−1/2 so that Jeε(Λ, X)< cε,1 for εsmall enough, provided that a0 > ^H_K^m

3. For example we set in (5.7)

(5.12) a0= 2Hm

K₃ (or, in view of (A.23),a0= ^√64

3π²Hm).

Case 1b. There is somei, j,i6=j, such that|xi−xj|=b ln¹_ε−3/4

.

According to (4.15) and using the previous computations, we have in that case Jeε(Λ, X)≤cε,1+ε ln¹_ε−1/4

− K4

bλ^1/2_i λ^1/2_j

ε ln¹_ε^3/4 + O

ε ln¹_ε−1/2 . As we have also, from (5.9)

1 λ^1/2_i λ^1/2_j

= 1

λ(ε)

1 + O

ln¹_ε−1/4 and from (5.2)

(5.13) 1

λ(ε) = K₃ε K2Hm

1

ln¹_ε + O _{ln ln}¹

ε ln1

ε

²

, we obtain

Je_ε(Λ, X)≤c_ε,1+ε ln¹_ε−1/4

− K3K4

bK₂H_mε ln¹_ε−1/4

+ o

ε ln¹_ε−1/4 . Therefore, Je_ε(Λ, X) < c_ε,1 for ε small enough, provided that b < _K^K3K4

2Hm. For example, we set in (4.9) and (5.8)

(5.14) b= K3K4

2K2Hm

(or, in view of (A.23),b=

√3π² 32H_m).

Case 2. Λ∈∂T.

Case 2a. There is somej such thatλ_j=

1 +D ln¹_ε−1/4 λ(ε).

In view of (4.16), we compute, using the fact that H(xi)≤ Hm, (5.1)–(5.3) and (5.13)

−K2H(xi)lnλ_j λ²_j +K3

ε λj

≥ −K2Hm

lnλ_j λ²_j +K3

ε λj

≥ K₃² K2Hm

Dε² ln1 ε

−5/4

+ o

ε² ln¹_ε−5/4 . From (4.16), we know that there exists a constantC, independent ofD, such that

∂Jeε

∂λ_i(Λ, X)≥ −K2H(xi)lnλi

λ²_i +K3

ε

λ_i −C ε² ln¹_ε^−5/4 . Therefore, ifDis chosen large enough, so that _K^K32

2H_mD > C, the inequality implies

∂fJε

∂λ_j(Λ, X)>0, so that −∇Jeε(Λ, X) is directed toward the interior ofT^k×M. Case 2b. There is somej such thatλ_j=

1−D ln¹_ε−1/4 λ(ε).

According to (4.16), (5.7) and (5.12), we have

∂Je_ε

∂λj

(Λ, X)≤ −K2Hm

1− 2

K3

ln1 ε

−1/4lnλ_j λ²_j +K3

ε λj

+ O

ε² ln¹_ε−5/4 and the same kind of computations as in the previous subcase yield

∂Je_ε

∂λj

(Λ, X)≤ − K₃² K2Hm

D− 2 K3

ε² ln1 ε

−5/4

+ O

ε² ln¹_ε−5/4

where once again the last term denotes a quantity whose absolute value is less than Cε² ln¹_ε−5/4

for some constantCindependent ofD. Consequently, ifDis chosen large enough, ^∂f_∂λ^Jε

j(Λ, X)<0, so that−∇Jeε(Λ, X) is directed toward the interior of

T^k×M.

We can now complete the proof of Theorem1.2. We defineS⁰⁰⊂S⁰andM⁰⊂M as

(5.15) S⁰⁰=n

x∈S s.t. H(x)> H_m− H_m 4kK3

ln1 ε

−1/4o

and

(5.16) M⁰ =n

X ∈(∂Ω)^k s.t. xi∈S⁰⁰

and |xi−xj|>4k(k−1)K₃K₄ K2Hm

ln1 ε

−3/4

,1≤i, j≤k, i6=jo . We claim that forεsmall enough

(5.17) ∀(Λ, X)∈T^k×M⁰ Je_ε(Λ, X)> c_ε,1.

Let us assume that (5.17) is true. For X ∈ M⁰, we take (Λ(ε), X) as initial value in (5.10) – with Λ(ε) = λ(ε), . . . , λ(ε)

. Lemma 5.1 and (5.17) imply that the flow line has to meet∂M⁰ before Λ(t), X(t)

reachesJeε c_ε,1

. Consequently the flow, projected onto the X-variable, provides us with a deformation of M⁰ onto

∂M⁰. However,M⁰ is topologically different from∂M⁰ – see Proposition B1 in [8]

– hence a contradiction. Consequently, assumption (H) is not true – that is,Je_εhas

a critical point inT^k×M, which provides us with a solution of (1.1) satisfying the statements of Theorem1.2 (1).

It only remains to prove (5.17). Let (Λ, X)∈T^k×M⁰. According to (4.15), we have

Je_ε(Λ, X) =kK_1,ε+

k

X

i=1

K₂H_mlnλ_i λi

+K₃εlnλ_i

−

k

X

i=1

K₂ H_m−H(x_i)lnλ_i λi

−K4 k

X

i,j=1 i6=j

1

λ^1/2_i λ^1/2_j |xi−xj|+ O

ε ln¹_ε⁻¹ .

We remark that forλi∈T lnλ_i

λi

= K₃ε K2Hm

+ O

ε ln¹_ε−1/4 so that, using (5.15),

k

X

i=1

K2 Hm−H(xi)lnλi

λi

≤ ε 4 ln1

ε ^−1/4

+ O

ε ln¹_ε^−1/2 . We have also, using (5.16) and (5.2),

K4 k

X

i,j=1 i6=j

1

λ^1/2_i λ^1/2_j |xi−xj| ≤ ε 4 ln1

ε −1/4

+ O

ε ln¹_ε−5/4

ln ln¹_ε .

Lastly, we remark that in view of (5.1) and (5.9)

K2Hm

lnλi

λi

+K3εlnλi=K2Hm

lnλ(ε)

λ(ε) +K3εlnλ(ε) + O

ε ln¹_ε−1/4 . Then, taking into account definition (5.5) ofcε,1, we obtain

Je_ε(Λ, X)≥c_ε,1+ε 2 ln1

ε −1/4

+ O

ε ln¹_ε−1/2 and (5.17) follows. This ends the proof of the first part of Theorem1.2.

The proof of the second part is straightforward. Indeed, in that case, the only thing we have to do is to minimizeJe_ε(λ, x) inT×S⁰. One can easily deduce from the previous computations that such a minimum cannot lie on the boundary of T×S⁰ – whence the existence of a critical point ofJe_ε which provides us with the desired solution of (1.1).

Appendix A.

We begin this appendix by a number of integral estimates, which will be useful in establishing Proposition4.2.