Construction of Solutions for a Nonlinear Elliptic Problem on Riemannian Manifolds with Boundary

Marco Ghimenti; Anna Maria Micheletti

doi:10.1515/ans-2015-5036

Article Open Access

Construction of Solutions for a Nonlinear Elliptic Problem on Riemannian Manifolds with Boundary

Marco Ghimenti and Anna Maria Micheletti

Published/Copyright: April 7, 2016

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From the journal Advanced Nonlinear Studies Volume 16 Issue 3

Abstract

Let (M,g) be a smooth compact n-dimensional Riemannian manifold (n≥2) with smooth (n-1)-dimensional boundary ∂⁡M. We prove that the stable critical points of the mean curvature of the boundary generates H1⁢(M) solutions for the following singularly perturbed elliptic problem with Neumann boundary conditions:

- ε 2 ⁢ Δ g ⁢ u + u = u p - 1 ⁢ in ⁢ M , u > 0 ⁢ in ⁢ M , ∂ ⁡ u ∂ ⁡ ν = 0 ⁢ on ⁢ ∂ ⁡ M ,

when ε is small enough. Here p is subcritical.

Keywords: Riemannian Manifold with Boundary; Nonlinear Elliptic Equations; Neumann BoundaryCondition; Mean Curvature; Lyapunov–Schmidt

MSC 2010: 58J05; 35J60; 58E05

1 Introduction

Let (M,g) be a smooth compact n-dimensional Riemannian manifold (n≥2) with boundary ∂⁡M, which is the union of a finite number of smooth, connected and boundaryless n-1 submanifolds embedded in M. Here g denotes the Riemannian metric tensor. By the Nash theorem [17], we can consider (M,g) as a regular submanifold embedded in ℝN.

We consider the following Neumann problem:

(1.1) { - ε 2 ⁢ Δ g ⁢ u + u = u p - 1 in ⁢ M , u > 0 in ⁢ M , ∂ ⁡ u ∂ ⁡ ν = 0 on ⁢ ∂ ⁡ M ,

where p>2 if n=2 and 2<p<2*=2⁢nn-2 if n≥3, ν is the external normal to ∂⁡M and ε is a positive parameter.

We are interested in finding solutions u∈H1⁢(M) to problem (1.1), where

H 1 ( M ) = { u : M → ℝ : ∫ M | ∇ u | g 2 + u 2 d μ g < ∞ }

and μg denotes the volume form on M associated to g. More precisely, we want to show that, for ε sufficiently small, we can construct a solution which has a peak near a stable critical point of the scalar curvature of the boundary, as stated in the following.

Definition 1.1

Let f∈C1⁢(N,ℝ), where (N,g) is a Riemannian manifold. We say that K⊂N is a C1-stable critical set of f if K⊂{x∈N:∇⁡f⁢(x)=0} and for any μ>0 there exists δ>0 such that if h∈C1⁢(N,ℝ) with

max d g ⁢ ( x , K ) ≤ μ | f ( x ) - h ( x ) | + | ∇ f ( x ) - ∇ h ( x ) | ≤ δ ,

then h has a critical point x0 with dg⁢(x0,K)≤μ. Here dg denotes the geodesic distance associated to the Riemannian metric g.

Now we can state the main theorem.

Theorem 1.2

Assume K⊂∂⁡M is a C1-stable critical set of the mean curvature of the boundary. Then, there exists ε0>0 such that for any ε∈(0,ε0), problem (1.1) has a solution uε∈H1⁢(M) which concentrates at a point ξ0∈K as ε goes to zero.

Problem (1.1) in a flat domain has a long history. Starting from a problem of pattern formation in biology, Lin, Ni and Takagi [14, 18] showed the existence of a mountain pass solution for problem (1.1) and proved that this solution has exactly one maximum point which lies on the boundary of the domain. Moreover, in [19] Ni and Takagi proved that the maximum point of the solution approaches the maximum point of the mean curvature of the boundary when the perturbation parameter ε goes to zero.

Thenceforth, many papers were devoted to the study of problem (1.1) on flat domains. In particular, in [3, 20] it was proved that any stable critical point of the mean curvature of the boundary generated a single peaked solution whose peak approaches the critical point as ε vanishes. Moreover, in [10, 13, 22, 12] the existence of multipeak solutions whose peaks lies on the boundary was studied. We also mention the series of works [21, 11, 8, 9], in which the existence of solutions which have internal peaks was proved.

For the case of a manifold M, problem (1.1) was first studied in [2], where Byeon and Park proved the existence of a mountain pass solution when the manifold M is closed and when the manifold M has a boundary. They showed that for ε small, such a solution has a spike which approaches – as ε goes to zero – a maximum point of the scalar curvature when M is closed, and a maximum point of the mean curvature of the boundary when M has a boundary.

For the case of a closed manifold M, Benci, Bonanno and Micheletti [1] showed that problem (1.1) has at least cat⁡M+1 nontrivial positive solutions when ε goes to zero. Here cat⁡M denotes the Lusternik–Schnirelmann category of M. Moreover, in [15] the effect of the geometry of the manifold (M,g) was examined. In fact, it was shown that positive solution of the problem are generated by stable critical points of the scalar curvature of M.

More recently, for the case of a manifold M with boundary ∂⁡M, we proved in [6] that problem (1.1) has at least cat⁡∂⁡M non trivial positive solutions when ε goes to zero. We can compare the result of [6] with Theorem 1.2. In fact, in [7] the authors proved that, generically with respect to the metric g, the mean curvature of the boundary has nondegenerate critical points. More precisely, the set of metrics for which the mean curvature has only nondegenerate critical points is an open dense set among all the Ck metrics on M, k≥3. Thus, generically with respect to the metric, the mean curvature has P1⁢(∂⁡M) nondegenerate (hence stable) critical points, where P1⁢(∂⁡M) is the Poincaré polynomial of ∂⁡M, namely Pt⁢(∂⁡M) evaluated in t=1. So, generically with respect to the metric, problem (1.1) has P1⁢(∂⁡M) solution, and we have P1⁢(∂⁡M)≥cat⁡∂⁡M, where for many cases the strict inequality holds.

The paper is organized as follows. In Section 2 some preliminary notions are introduced, which are necessary for the comprehension of the paper. In Section 3 we study the variational structure of the problem, and we perform the finite dimensional reduction. In Section 4 the proof of Theorem 1.2 is sketched, while the expansion of the reduced functional is carried out in Section 5. Finally, the Appendix collects some technical lemmas.

2 Preliminary Results

In this section we give some general facts preliminary to our work. These results are widely present in the literature. We refer mainly to [2, 4, 5, 15] and the reference therein.

First of all we need to define a suitable coordinate chart on the boundary.

We know that on the tangent bundle of any compact Riemannian manifold ℳ it is defined the exponential map exp:T⁢ℳ→ℳ which is of class C∞. Moreover, there exist a constant RM>0, called radius of injectivity, and a finite number of xi∈ℳ such that ℳ=⋃i=1lBg⁢(xi,RM) and expxi:B⁢(0,RM)→Bg⁢(xi,RM) is a diffeormophism for all i. By choosing an orthogonal coordinate system (y1,…,yn) of ℝn and by identifying Tx0⁢ℳ with ℝn for x0∈ℳ, we can define using the exponential map the so-called normal coordinates. For x0∈ℳ, gx0 denotes the metric read through the normal coordinates. In particular, we have gx0⁢(0)=Id. We set |gx0⁢(y)|=det⁡(gx0⁢(y))i⁢j and gx0i⁢j⁢(y)=((gx0⁢(y))i⁢j)-1.

Definition 2.1

Assume that q belongs to the boundary ∂⁡M, and let y¯=(y1,…,yn-1) be the Riemannian normal coordinates on the (n-1)-manifold ∂⁡M at the point q. For a point ξ∈M close to q, there exists a unique ξ¯∈∂⁡M such that dg⁢(ξ,∂⁡M)=dg⁢(ξ,ξ¯). We set y¯⁢(ξ)∈ℝn-1 the normal coordinates for ξ¯ and yn⁢(ξ)=dg⁢(ξ,∂⁡M). Then, we define a chart ψq∂:ℝ+n→M such that (y¯⁢(ξ),yn⁢(ξ))=(ψq∂)-1⁢(ξ). These coordinates are called the Fermi coordinates at q∈∂⁡M. The Riemannian metric gq⁢(y¯,yn) read through the Fermi coordinates satisfies gq⁢(0)=Id.

We denote by dg∂ and exp∂, respectively, the geodesic distance and the exponential map on by ∂⁡M. By the compactness of ∂⁡M, there exist R∂ and a finite number of points qi∈∂⁡M, i=1,…,k, such that

I q i ( R ∂ , R M ) := { x ∈ M : d g ( x , ∂ M ) = d g ( x , ξ ¯ ) < R M , d g ∂ ( q i , ξ ¯ ) < R ∂ }

form a covering of (∂M)RM:={x∈M:dg(x,∂M)<RM}, and on every Iqi the Fermi coordinates are well defined. In the following, we choose R=min⁡{R∂,RM} in order to have the finite covering

M ⊂ { ⋃ i = 1 k B ⁢ ( q i , R ) } ∪ { ⋃ i = k + 1 l I ξ i ⁢ ( R , R ) } ,

where k,l∈ℕ, qi∈M∖∂⁡M and ξi∈∂⁡M.

For p∈∂⁡M, consider the projection πp:Tp⁢M→Tp⁢∂⁡M on the tangent space Tp⁢∂⁡M. For a pair of tangent vectors X,Y∈Tp⁢∂⁡M, we define the second fundamental form IIp⁢(X,Y):=∇X⁡Y-πp⁢(∇X⁡Y). The mean curvature at the boundary Hp, where p∈∂⁡M, is the trace of the second fundamental form.

If we consider the Fermi coordinates in a neighborhood of p, and denote by (hi⁢j)i,j=1,…,n-1 the matrix of the second fundamental form, then we have the well-known formulas

(2.1) g i ⁢ j ⁢ ( y ) = δ i ⁢ j + 2 ⁢ h i ⁢ j ⁢ ( 0 ) ⁢ y n + O ⁢ ( | y | 2 ) for ⁢ i , j = 1 , … , n - 1 ,

(2.2) g i ⁢ n ⁢ ( y ) = δ i ⁢ n ,

(2.3) g ⁢ ( y ) = 1 - ( n - 1 ) ⁢ H ⁢ ( 0 ) ⁢ y n + O ⁢ ( | y | 2 ) ,

where (y1,…,yn) are the Fermi coordinates and, by definition of hi⁢j,

(2.4) H = 1 n - 1 ⁢ ∑ i n - 1 h i ⁢ i .

Also, by [4, equation (3.2)], we have that

(2.5) ∂ 2 ∂ ⁡ y n ⁢ ∂ ⁡ y i ⁢ g ⁢ ( y ) | y = 0 = - ( n - 1 ) ⁢ ∂ ⁡ H ∂ ⁡ y i ⁢ ( 0 ) for ⁢ i = 1 , … , n - 1 .

It is well known that in ℝn there exists a unique positive radially symmetric function V⁢(y)∈H1⁢(ℝn) which satisfies

(2.6) - Δ ⁢ V + V = V p - 1 on ⁢ ℝ n .

Moreover, the function V as well as its derivative decay exponentially at infinity, that is,

lim | y | → ∞ ⁡ V ⁢ ( | y | ) ⁢ | y | n - 1 2 ⁢ e | y | = c and lim | y | → ∞ ⁡ V ′ ⁢ ( | y | ) ⁢ | y | n - 1 2 ⁢ e | y | = - c

for some c>0. On the half space ℝ+n={(y1,…,yn)∈ℝn:yn≥0}, we can define the function

U ⁢ ( y ) = V | y n ≥ 0 .

The function U satisfies in ℝ+n the following Neumann problem:

(2.7) { - Δ ⁢ U + U = U p - 1 in ⁢ ℝ + n , ∂ ⁡ U ∂ ⁡ y n = 0 on { y n = 0 } .

We set Uε⁢(y)=U⁢(yε).

Lemma 2.2

The space solution of the linearized problem

(2.8) { - Δ ⁢ φ + φ = ( p - 1 ) ⁢ U p - 2 ⁢ φ in ⁢ ℝ + n , ∂ ⁡ φ ∂ ⁡ y n = 0 on { y n = 0 }

is generated by a linear combination of

φ i = ∂ ⁡ U ∂ ⁡ y i ⁢ ( y ) for ⁢ i = 1 , … , n - 1 .

Proof.

It is trivial that every linear combination of φi is a solution of (2.8). We notice that ∂⁡U∂⁡yn is not a solution of (2.8) because the derivative on {yn=0} is not zero.

For the converse, suppose that φ¯⁢(y) is a solution of (2.8). Then, by even reflection around yn, we can construct a solution φ~ of

(2.9) - Δ ⁢ φ ~ + φ ~ = ( p - 1 ) ⁢ U p - 2 ⁢ φ ~ in ⁢ ℝ n

with ∂⁡φ~∂⁡yn=0 on yn=0. But all solution of (2.9) with zero derivative on yn=0 are linear combinations of ∂⁡V∂⁡yj with j=1,…,n-1. ∎

We endow H1⁢(M) with the scalar product

〈 u , v 〉 ε := 1 ε n ⁢ ∫ M ε 2 ⁢ g ⁢ ( ∇ ⁡ u , ∇ ⁡ v ) + u ⁢ v ⁢ d ⁢ μ g

and the norm ∥u∥ε=〈u,u〉ε1/2. We call Hε the space H1 equipped with the norm ∥⋅∥ε. We also define Lεp as the space Lp⁢(M) endowed with the norm |u|ε,p=(1εn⁢∫Mup⁢𝑑μg)1/p.

For any p∈[2,2*) if n≥3 or for all p≥2 if n=2, the embedding iε:Hε↪Lε,p is a compact, continuous map, and |u|ε,p≤c⁢∥u∥ε holds for some constant c not depending on ε. We define the adjoint operator iε*:Lε,p′↪Hε as

u = i ε * ⁢ ( v ) ⇔ 〈 u , φ 〉 ε = 1 ε n ⁢ ∫ M v ⁢ φ ⁢ 𝑑 μ g ,

so we can rewrite problem (1.1) in an equivalent formulation

u = i ε * ⁢ ( ( u + ) p - 1 ) .

Remark 2.3

We have that ∥iε*⁢(v)∥ε≤c⁢|v|p′,ε.

From now on we set, for the sake of simplicity,

f ⁢ ( u ) = ( u + ) p - 1 and f ′ ⁢ ( u ) = ( p - 1 ) ⁢ ( u + ) p - 2 .

We want to split the space Hε into a finite dimensional space generated by the solution of (2.8) and its orthogonal complement. For ξ∈∂⁡M and R>0 fixed, we consider on the manifold the functions

(2.10) Z ε , ξ i = { φ ε i ⁢ ( ( ψ ξ ∂ ) - 1 ⁢ ( x ) ) ⁢ χ R ⁢ ( ( ψ ξ ∂ ) - 1 ⁢ ( x ) ) , x ∈ I ξ ⁢ ( R ) := I ξ ⁢ ( R , R ) , 0 , elsewhere ,

where φεi⁢(y)=φi⁢(yε) and χR:Bn-1⁢(0,R)×[0,R)→ℝ+ is a smooth cut off function such that χR≡1 on Bn-1⁢(0,R/2)×[0,R/2) and |∇⁡χ|≤2.

In the following, for the sake of simplicity, we denote

(2.11) D + ⁢ ( R ) = B n - 1 ⁢ ( 0 , R ) × [ 0 , R ) .

Let

K ε , ξ := span ⁢ { Z ε , ξ 1 , … , Z ε , ξ n - 1 } .

We can split Hε into the sum of the (n-1)-dimensional space and its orthogonal complement with respect of 〈⋅,⋅〉ε, i.e.,

K ε , ξ ⊥ := { u ∈ H ε : 〈 u , Z ε , ξ i 〉 ε = 0 } .

We solve problem (1.1) by a Lyapunov–Schmidt reduction, defined by

W ε , ξ ⁢ ( x ) = { U ε ⁢ ( ( ψ ξ ∂ ) - 1 ⁢ ( x ) ) ⁢ χ R ⁢ ( ( ψ ξ ∂ ) - 1 ⁢ ( x ) ) , x ∈ I ξ ⁢ ( R ) := I ξ ⁢ ( R , R ) , 0 , elsewhere .

We look for a function of the form Wε,ξ+ϕ with ϕ∈Kε,ξ⊥ such that

(2.12) Π ε , ξ ⊥ ⁢ { W ε , ξ + ϕ - i ε * ⁢ [ f ⁢ ( W ε , ξ + ϕ ) ] } = 0 ,

(2.13) Π ε , ξ ⁢ { W ε , ξ + ϕ - i ε * ⁢ [ f ⁢ ( W ε , ξ + ϕ ) ] } = 0 ,

where Πε,ξ:Hε→Kε,ξ and Πε,ξ⊥:Hε→Kε,ξ⊥ are, respectively, the projection on Kε,ξ and Kε,ξ⊥. We see that Wε,ξ+ϕ is a solution of (1.1) if and only if Wε,ξ+ϕ solves (2.12) and (2.13).

Hereafter, we collect a series of results which will be useful in the paper.

Definition 2.4

Given ξ0∈∂⁡M, using the normal coordinates on the sub manifold ∂⁡M, we define

ℰ ⁢ ( y , x ) = ( exp ξ ⁢ ( y ) ∂ ) - 1 ⁢ ( x ) = ( exp exp ξ 0 ∂ ⁡ y ∂ ) - 1 ⁢ ( exp ξ 0 ∂ ⁡ η ¯ ) = ℰ ~ ⁢ ( y , η ¯ ) ,

where x,ξ⁢(y)∈∂⁡M, y,η¯∈B⁢(0,R)⊂ℝn-1 and ξ⁢(y)=expξ0∂⁡y, x=expξ0∂⁡η¯. Using the Fermi coordinates around ξ0, in a similar way, we define

ℋ ⁢ ( y , x ) = ( ψ ξ ⁢ ( y ) ∂ ) - 1 ⁢ ( x ) = ( ψ exp ξ 0 ∂ ⁡ y ∂ ) - 1 ⁢ ( ψ ξ 0 ∂ ⁢ ( η ¯ , η n ) ) = ℋ ~ ⁢ ( y , η ¯ , η n ) = ( ℰ ~ ⁢ ( y , η ¯ ) , η n ) ,

where x∈M, η=(η¯,ηn) with η¯∈B⁢(0,R)⊂ℝn-1 and 0≤ηn<R, ξ⁢(y)=expξ0∂⁡y∈∂⁡M and x=ψξ0∂⁢(η).

Lemma 2.5

Let x=ψξ0∂⁢(ε⁢z), where z=(z¯,zn) and ξ⁢(y)=expξ0∂⁡(y). Then, for j=1,…,n-1, we have

∂ ∂ ⁡ y j ⁢ W ε , ξ ⁢ ( y ) ⁢ ( x ) | y = 0 = ∑ k = 1 n - 1 [ 1 ε ⁢ χ R ⁢ ( ε ⁢ z ) ⁢ ∂ ∂ ⁡ z k ⁢ U ⁢ ( z ) + U ⁢ ( z ) ⁢ ∂ ∂ ⁡ z k ⁢ χ R ⁢ ( ε ⁢ z ) ] ⁢ ∂ ∂ ⁡ y j ⁢ ℰ k ~ ⁢ ( y , ε ⁢ z ¯ ) | y = 0 .

In order to prove Lemma 2.5, we need some preliminaries.

Lemma 2.6

We have that

ℰ ⁢ ( 0 , η ¯ ) = η ¯ for ⁢ η ¯ ∈ ℝ n - 1 ,

∂ ⁡ ℰ ~ k ∂ ⁡ η j ⁢ ( 0 , η ¯ ) = δ j ⁢ k for ⁢ y ∈ ℝ n - 1 , j , k = 1 , … , n - 1 ,

∂ ⁡ ℰ ~ k ∂ ⁡ y j ⁢ ( 0 , 0 ) = - δ j ⁢ k for ⁢ j , k = 1 , … , n - 1 ,

∂ 2 ⁡ ℰ ~ k ∂ ⁡ y j ⁢ ∂ ⁡ η h ⁢ ( 0 , 0 ) = 0 for ⁢ j , h , k = 1 , … , n .

Proof.

We recall that ℰ~⁢(y,η¯)=(expξ⁢(y)∂)-1⁢(expξ0∂⁡η¯), so the first claim is obvious. For y,η¯∈B⁢(0,R)⊂ℝn-1, let us introduce

F ⁢ ( y , η ¯ ) = ( exp ξ 0 ∂ ) - 1 ⁢ ( exp ξ ⁢ ( y ) ∂ ⁡ ( η ¯ ) ) and Γ ⁢ ( y , η ¯ ) = ( y , F ⁢ ( y , η ¯ ) ) .

We notice that Γ-1(y,β),=(y,ℰ~(y,β)). We can easily compute the derivative of Γ. Given y^,η^∈ℝn-1, we have

Γ ′ ⁢ ( y ^ , η ^ ) ⁢ [ y , β ] = ( Id ℝ n - 1 0 F y ′ ⁢ ( y ^ , η ^ ) F η ′ ⁢ ( y ^ , η ^ ) ) ⁢ ( y β ) ,

and thus

( Γ - 1 ) ′ ⁢ ( y ^ , η ^ ) ⁢ [ y , β ] = ( Id ℝ n - 1 0 - ( F η ′ ⁢ ( y ^ , η ^ ) ) - 1 ⁢ F y ′ ⁢ ( y ^ , η ^ ) ( F η ′ ⁢ ( y ^ , η ^ ) ) - 1 ) ⁢ ( y β ) .

Here y,β∈ℝn-1. Now by direct computation we have that

F η ′ ⁢ ( 0 , η ^ ) = Id ℝ n - 1 and F y ′ ⁢ ( y ^ , 0 ) = Id ℝ n - 1 ,

and so ∂⁡ℰ~k∂⁡ηj⁢(0,η^)=((Fη′⁢(0,η^))-1)j⁢k=δj⁢k and ∂⁡ℰ~k∂⁡yj⁢(0,0)=(-(Fη′⁢(0,0))-1⁢Fy′⁢(0,0))j⁢k=-δj⁢k. For the last claim, we refer to [15, Lemma 6.4]. ∎

Lemma 2.7

We have that

ℋ ~ ⁢ ( 0 , η ¯ , η n ) = ( η ¯ , η n ) for ⁢ η ¯ ∈ ℝ n - 1 , η n ∈ ℝ + ,

∂ ⁡ ℋ ~ k ∂ ⁡ y j ⁢ ( 0 , 0 , η n ) = - δ j ⁢ k for ⁢ j , k = 1 , … , n - 1 , η n ∈ ℝ + ,

∂ ⁡ ℋ ~ n ∂ ⁡ y j ⁢ ( y , η ¯ , η n ) = 0 for ⁢ j = 1 , … , n - 1 , y , η ¯ ∈ ℝ n - 1 , η n ∈ ℝ + ,

∂ ⁡ ℋ ~ k ∂ ⁡ η n ⁢ ( y , η ¯ , η n ) = 0 for ⁢ j , k = 1 , … , n - 1 , η ¯ ∈ ℝ n - 1 , η n ∈ ℝ + ,

∂ 2 ⁡ ℋ ~ k ∂ ⁡ η n ⁢ ∂ ⁡ y j ⁢ ( y , η ¯ , η n ) = 0 for ⁢ j , k = 1 , … , n - 1 , η ¯ ∈ ℝ n - 1 , η n ∈ ℝ + .

Proof.

The first three claims follow immediately by Definition 2.4 and Lemma 2.6. For the last two claims, observe that ℋ~k⁢(y,η¯,ηn)=ℰ~k⁢(y,η¯), which does not depends on ηn nor its derivatives. ∎

We now prove the claimed result.

Proof of Lemma 2.5.

By Definition 2.4, let x=ψξ0∂⁢(η)=ψξ0∂⁢(η¯,ηn) with η=(η¯,ηn)∈ℝn, and ξ⁢(y)=expξ0∂⁡(y) with y∈ℝn-1. Then, we have that

W ε , ξ ⁢ ( y ) ⁢ ( x ) = U ⁢ ( ℋ ~ ⁢ ( y , η ) ε ) ⁢ χ R ⁢ ( ℋ ~ ⁢ ( y , η ) ) .

For fixed j, by Lemma 2.7, we have

∂ ∂ ⁡ y j ⁢ W ε , ξ ⁢ ( y ) ⁢ ( x ) | y = 0 = ∑ k = 1 n ∂ ∂ ⁡ v k ⁢ [ χ R ⁢ ( ℋ ~ ⁢ ( y , η ) ) ⁢ U ε ⁢ ( ℋ ~ ⁢ ( y , η ) ) ] | ℋ ~ ⁢ ( 0 , η ) ⁢ ∂ ∂ ⁡ y i ⁢ ℋ ~ k ⁢ ( y , η ) | y = 0

= ∑ k = 1 n - 1 ∂ ∂ ⁡ η k ⁢ [ χ R ⁢ ( η ) ⁢ U ε ⁢ ( η ) ] ⁢ ∂ ∂ ⁡ y j ⁢ ℰ ~ k ⁢ ( y , η ¯ ) | y = 0

= ∑ k = 1 n - 1 ∂ ∂ ⁡ z k ⁢ [ χ R ⁢ ( ε ⁢ z ) ⁢ U ⁢ ( z ) ] ⁢ ∂ ∂ ⁡ y j ⁢ ℰ ~ k ⁢ ( y , ε ⁢ z ¯ ) | y = 0

Because ℋ~k⁢(y,η¯,ηn)=ℰk⁢(y,η¯) for k=1,…,n-1, using the change of variables η=ε⁢z=(ε⁢z¯,ε⁢zn), we get the claim. ∎

3 Reduction to Finite Dimensional Space

In this section we find a solution for equation (2.12). In particular, we prove that for all ε>0 and for all ξ∈∂⁡M there exists ϕε,ξ∈Kε,ξ⊥ solving (2.12). Here and hereafter, all the proofs are similar to [15]. So, for the sake of simplicity, we will underline the parts where differences appear, and sketch the rest of the proofs (we will provide precise references for each proof).

We introduce the linear operator Lε,ξ:Kε,ξ⊥→Kε,ξ⊥ with

L ε , ξ ⁢ ( ϕ ) := Π ε , ξ ⊥ ⁢ { ϕ - i ε * ⁢ [ f ′ ⁢ ( W ε , ξ ) ⁢ ϕ ] } .

Thus, we can rewrite equation (2.12) as follows:

L ε , ξ ⁢ ( ϕ ) = N ε , ξ ⁢ ( ϕ ) + R ε , ξ ,

where

N ε , ξ := Π ε , ξ ⊥ ⁢ { i ε * ⁢ [ f ⁢ ( W ε , ξ + ϕ ) - f ⁢ ( W ε , ξ ) - f ′ ⁢ ( W ε , ξ ) ⁢ ϕ ] }

is the nonlinear term and

R ε , ξ := Π ε , ξ ⊥ ⁢ { i ε * ⁢ [ f ⁢ ( W ε , ξ ) ] - W ε , ξ }

is the remainder term. The first step is to prove that the linear term is invertible.

Lemma 3.1

There exist ε0 and c>0 such that for any ξ∈∂⁡M and ε∈(0,ε0), we have that

∥ L ε , ξ ∥ ε ≥ c ⁢ ∥ ϕ ∥ ε for any ⁢ ϕ ∈ K ε , ξ ⊥ .

The proof of this lemma is given in the Appendix. We estimate now the remainder term Rε,ξ.

Lemma 3.2

There exists ε0>0 and c>0 such that for any ξ∈∂⁡M and for all ε∈(0,ε0),

∥ R ε , ξ ∥ ε ≤ c ⁢ ε 1 + n / p ′ .

Proof.

We proceed as in [15, Lemma 3.3]. We define on M the function Vε,ξ such that Wε,ξ=iε*⁢(Vε,ξ), and thus -ε2⁢Δg⁢Wε,ξ+Wε,ξ=Vε,ξ.

It is well known (see [16, p. 134]), by the definition of the Laplace–Beltrami operator, that in a local chart, we have that

- Δ ⁢ v = - Δ g ⁢ v + ( g ξ i ⁢ j - δ i ⁢ j ) ⁢ ∂ 2 ∂ ⁡ x i ⁢ ∂ ⁡ x j ⁢ v - g ξ i ⁢ j ⁢ Γ i ⁢ j k ⁢ ∂ ∂ ⁡ x k ⁢ v ,

where Δ is the Euclidean Laplace operator. Thus, if we define

V ~ ε , ξ ⁢ ( y ) = V ε , ξ ⁢ ( ψ ξ ∂ ⁢ ( y ) ) , y ∈ D + ⁢ ( R ) ,

then we have

V ~ ε , ξ ⁢ ( y ) = - ε 2 ⁢ Δ g ⁢ ( U ε ⁢ χ R ) + U ε ⁢ χ R

(3.1) = U ε p - 1 ⁢ χ R - ε 2 ⁢ U ε ⁢ Δ ⁢ χ R - 2 ⁢ ε 2 ⁢ ∇ ⁡ U ε ⁢ ∇ ⁡ χ R - ε 2 ⁢ ( g ξ i ⁢ j - δ i ⁢ j ) ⁢ ∂ 2 ∂ ⁡ y i ⁢ ∂ ⁡ y j ⁢ ( U ε ⁢ χ R ) + ε 2 ⁢ g ξ i ⁢ j ⁢ Γ i ⁢ j k ⁢ ∂ ∂ ⁡ y k ⁢ ( U ε ⁢ χ R ) .

Also, by Remark 2.3 and by the definition of Rε,ξ,

∥ R ε , ξ ∥ ε ≤ ∥ i ε * ⁢ f ⁢ ( W ε , ξ ) - W ε , ξ ∥ ε ≤ c ⁢ | W ε , ξ p - 1 - V ε , ξ | p ′ , ε .

Finally, by the definition of Wε,ξ and by (3.1), we get

| W ε , ξ p - 1 - V ε , ξ | p ′ , ε p ′ = ∫ D + ⁢ ( R ) | U ε p - 1 ⁢ ( y ) ⁢ χ R p - 1 ⁢ ( y ) - V ~ ε , ξ ⁢ ( y ) | p ′ ⁢ | g ξ ⁢ ( y ) | 1 / 2 ⁢ 𝑑 y

≤ c ⁢ ∫ D + ⁢ ( R ) | U ε p - 1 ⁢ ( y ) ⁢ ( χ R p - 1 ⁢ ( y ) - χ R ⁢ ( y ) ) | p ′ ⁢ 𝑑 y

+ c ε 2 ⁢ p ′ ∫ D + ⁢ ( R ) U ε p ′ | Δ χ R | p ′ d y + c ε 2 ⁢ p ′ ∫ D + ⁢ ( R ) | ∇ U ε ⋅ ∇ χ R | p ′ d y

+ c ε 2 ⁢ p ′ ∫ D + ⁢ ( R ) | g ξ i ⁢ j ( y ) - δ i ⁢ j | p ′ | ∂ 2 ∂ ⁡ y i ⁢ ∂ ⁡ y j ( U ε χ R ) ( y ) | p ′ d y

+ c ε 2 ⁢ p ′ ∫ D + ⁢ ( R ) | g ξ i ⁢ j ( y ) Γ i ⁢ j k ( y ) | p ′ | ∂ ∂ ⁡ y k ( U ε χ R ) ( y ) | p ′ d y .

By the definition of χr and the exponential decay, using (2.1) and (2.2), we have

≤ ε n ∫ ℝ + n | g ξ i ⁢ j ( ε z ) - δ i ⁢ j | p ′ | ∂ 2 ∂ ⁡ z i ⁢ ∂ ⁡ z j U ( z ) | p ′ d z + O ( ε n + p ′ )

≤ ε n + p ′ ⁢ ∫ ℝ + n | ∂ 2 ∂ ⁡ z i ⁢ ∂ ⁡ z j ⁢ U ⁢ ( z ) | p ′ ⁢ 𝑑 z + O ⁢ ( ε n + p ′ ) = O ⁢ ( ε n + p ′ ) .

The other terms can be estimated in a similar way. ∎

Using the fixed point theorem and the implicit function theorem, we can solve equation (2.12).

Proposition 3.3

There exist ε0>0 and c>0 such that for any ξ∈∂⁡M and for all ε∈(0,ε0), there exists a unique ϕε,ξ=ϕ⁢(ε,ξ)∈Kε,ξ⊥ which solves (2.12). Moreover,

∥ ϕ ε , ξ ∥ ε < c ⁢ ε 1 + n / p ′ .

Finally, ξ↦ϕε,ξ is a C1 map.

Proof.

The proof is similar to that of [15, Proposition 3.5], which we refer to for all details. We want to solve (2.12) by a fixed point argument. We define the operator

T ε , ξ : K ε , ξ ⊥ → K ε , ξ ⊥ , T ε , ξ ⁢ ( ϕ ) = L ε , ξ - 1 ⁢ ( N ε , ξ ⁢ ( ϕ ) + R ε , ξ ) .

By Lemma 3.1, Tε,ξ is well defined and we have

∥ T ε , ξ ⁢ ( ϕ ) ∥ ε ≤ c ⁢ ( ∥ N ε , ξ ⁢ ( ϕ ) ∥ ε + ∥ R ε , ξ ∥ ε ) ,

∥ T ε , ξ ⁢ ( ϕ 1 ) - T ε , ξ ⁢ ( ϕ 2 ) ∥ ε ≤ c ⁢ ( ∥ N ε , ξ ⁢ ( ϕ 1 ) - N ε , ξ ⁢ ( ϕ 2 ) ∥ ε )

for some suitable constant c>0. By the mean value theorem (and the properties of i*), we get

∥ N ε , ξ ( ϕ 1 ) - N ε , ξ ( ϕ 2 ) ∥ ε ≤ c | f ′ ( W ε , ξ + ϕ 2 + t ( ϕ 1 - ϕ 2 ) ) - f ′ ( W ε , ξ ) | p / ( p - 2 ) , ε ∥ ϕ 1 - ϕ 2 ∥ ε .

By [15, Remark 3.4], we have that |f′⁢(Wε,ξ+ϕ2+t⁢(ϕ1-ϕ2))-f′⁢(Wε,ξ)|p/(p-2),ε≪1, provided ∥ϕ1∥ε and ∥ϕ2∥ε are small enough. Thus, there exists 0<C<1 such that ∥Tε,ξ⁢(ϕ1)-Tε,ξ⁢(ϕ2)∥ε≤C⁢∥ϕ1-ϕ2∥ε. Also, with the same estimates we get

∥ N ε , ξ ⁢ ( ϕ ) ∥ ε ≤ c ⁢ ( ∥ ϕ ∥ ε 2 + ∥ ϕ ∥ ε p - 1 ) .

This, combined with Lemma 3.2, gives

∥ T ε , ξ ⁢ ( ϕ ) ∥ ε ≤ c ⁢ ( ∥ N ε , ξ ⁢ ( ϕ ) ∥ ε + ∥ R ε , ξ ∥ ε ) ≤ c ⁢ ( ∥ ϕ ∥ ε 2 + ∥ ϕ ∥ ε p - 1 + ε 1 + n / p ′ ) .

So, there exists c>0 such that Tε,ξ maps a ball of center 0 and radius c⁢ε1+n/p′ in Kε,ξ⊥ into itself and it is a contraction. So there exists a fixed point ϕε,ξ with norm ∥ϕε,ξ∥ε≤ε1+n/p′.

The regularity of ϕε,ξ with respect to ξ is proved via the implicit function theorem. Let us define the functional

G : ∂ ⁡ M × H ε → ℝ , G ⁢ ( ξ , u ) = Π ε , ξ ⊥ ⁢ { W ε , ξ + Π ε , ξ ⊥ ⁢ u + i ε * ⁢ [ f ⁢ ( W ε , ξ + Π ε , ξ ⊥ ⁢ u ) ] } + Π ε , ξ ⁢ u .

We have that G⁢(ξ,ϕε,ξ)=0, and that the operator ∂∂⁡u⁢G⁢(ξ,ϕε,ξ):Hε→Hε is invertible. This concludes the proof. ∎

4 Sketch of the Proof of Theorem 1.2

In Proposition 3.3 we found a function ϕε,ξ solving (2.12). In order to solve (2.13), we define the functional

J ε : H 1 ⁢ ( M ) → ℝ , J ε ⁢ ( u ) = 1 ε n ⁢ ∫ M 1 2 ⁢ ε 2 ⁢ | ∇ ⁡ u | g 2 + 1 2 ⁢ u 2 - 1 p ⁢ ( u + ) p ⁢ d ⁢ μ g .

In what follows, we will often use the notation F⁢(u)=1p⁢(u+)p.

By Jε, we define the reduced functional J~ε on ∂⁡M as

J ~ ε ⁢ ( ξ ) = J ε ⁢ ( W ε , ξ + ϕ ε , ξ ) ,

where ϕε,ξ is uniquely determined by Proposition 3.3.

Remark 4.1

Our goal is to find the critical points for J~ε, since any critical point ξ for J~ε corresponds to a function ϕε,ξ+Wε,ξ which solves equation (2.13).

At this point we give the expansion for the functional J~ε with respect to ε. By Lemma 5.1 and Lemma 5.2,

(4.1) J ~ ε ⁢ ( ξ ) = C - ε ⁢ H ⁢ ( ξ ) + o ⁢ ( ε ) ,

C1 uniformly with respect to ξ∈∂⁡M as ε goes to zero. Here H⁢(ξ) is the mean curvature of the boundary ∂⁡M at ξ. If ξ0 is a C1-stable critical point for H, then in light of (4.1) and by the definition of C1-stability, we have that for ε small enough, there exists a critical point ξε for J~ε close to ξ0 , and we can prove Theorem 1.2.

5 Asymptotic Expansion of the Reduced Functional

In this section we study the asymptotic expansion of J~ε⁢(ξ) with respect to ε.

Lemma 5.1

We have that

(5.1) J ~ ε ⁢ ( ξ ) = J ε ⁢ ( W ε , ξ + ϕ ε , ξ ) = J ε ⁢ ( W ε , ξ ) + o ⁢ ( ε ) ,

uniformly with respect to ξ∈∂⁡M as ε goes to zero. Moreover, by setting ξ⁢(y)=expξ∂⁡(y), y∈Bn-1⁢(0,r), we have

∂ ∂ ⁡ y h ⁢ J ~ ε ⁢ ( ξ ⁢ ( y ) ) | y = 0 = ∂ ∂ ⁡ y h ⁢ J ε ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) | y = 0

(5.2) = ∂ ∂ ⁡ y h ⁢ J ε ⁢ ( W ε , ξ ⁢ ( y ) ) | y = 0 + o ⁢ ( ε )

Proof.

We split the proof into several steps. Step 1: We prove (5.1). Using (2.12), we get

J ~ ε ⁢ ( ξ ) - J ε ⁢ ( W ε , ξ ) = 1 2 ⁢ ∥ ϕ ε , ξ ∥ ε 2 + 1 ε n ⁢ ∫ M ε 2 ⁢ g ⁢ ( ∇ ⁡ W ε , ξ , ∇ ⁡ ϕ ε , ξ ) + W ε , ξ ⁢ ϕ ε , ξ - f ⁢ ( W ε , ξ ) ⁢ ϕ ε , ξ ⁢ d ⁢ μ g

- 1 ε n ⁢ ∫ M F ⁢ ( W ε , ξ + ϕ ε , ξ ) - F ⁢ ( W ε , ξ ) - f ⁢ ( W ε , ξ ) ⁢ ϕ ε , ξ

= - 1 2 ⁢ ∥ ϕ ε , ξ ∥ ε 2 + 1 ε n ⁢ ∫ M [ f ⁢ ( W ε , ξ + ϕ ε , ξ ) - f ⁢ ( W ε , ξ ) ] ⁢ ϕ ε , ξ ⁢ 𝑑 μ g

- 1 ε n ⁢ ∫ M F ⁢ ( W ε , ξ + ϕ ε , ξ ) - F ⁢ ( W ε , ξ ) - f ⁢ ( W ε , ξ ) ⁢ ϕ ε , ξ .

By the mean value theorem, we obtain that

for some t1,t2∈(0,1). Now, by the properties of f′, we can conclude that

| J ~ ε ⁢ ( ξ ) - J ε ⁢ ( W ε , ξ ) | ≤ c ⁢ ( ∥ ϕ ε , ξ ∥ ε 2 + ∥ ϕ ε , ξ ∥ ε p ) ,

and in light of Proposition 3.3, we obtain (5.1). Step 2: In order to prove (5.2), consider that

∂ ∂ ⁡ y h ⁢ J ε ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) - ∂ ∂ ⁡ y h ⁢ J ε ⁢ ( W ε , ξ ⁢ ( y ) )

= J ε ′ ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) ⁢ [ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) + ∂ ∂ ⁡ y h ⁢ ϕ ε , ξ ⁢ ( y ) ] - J ε ′ ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ [ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ]

= [ J ε ′ ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) - J ε ′ ⁢ ( W ε , ξ ⁢ ( y ) ) ] ⁢ [ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ] + J ε ′ ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) ⁢ [ ∂ ∂ ⁡ y h ⁢ ϕ ε , ξ ⁢ ( y ) ]

= : L 1 + L 2 .

Step 3: We estimate L2. By (2.12), we have that

J ε ′ ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) ⁢ [ ∂ ∂ ⁡ y h ⁢ ϕ ε , ξ ⁢ ( y ) ] = ∑ l = 1 n - 1 c ε l ⁢ 〈 Z ε , ξ ⁢ ( y ) l , ∂ ∂ ⁡ y h ⁢ ϕ ε , ξ ⁢ ( y ) 〉 ε .

We prove that

(5.3) ∑ l = 1 n - 1 | c ε l | = O ( ε ) .

Indeed, by (2.12) and (A.10), for some positive constant C, we have

(5.4) J ε ′ ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) ⁢ [ Z ε , ξ ⁢ ( y ) s ] = ∑ l = 1 n - 1 c ε l ⁢ 〈 Z ε , ξ ⁢ ( y ) l , Z ε , ξ ⁢ ( y ) s 〉 ε = C ⁢ ∑ l = 1 n - 1 c ε l ⁢ ( δ l ⁢ s + o ⁢ ( 1 ) ) .

Also, since ϕε,ξ⁢(y)∈Kε,ξ⁢(y)⊥, we have

J ε ′ ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) ⁢ [ Z ε , ξ ⁢ ( y ) s ] = 1 ε n ⁢ ∫ M ε 2 ⁢ g ⁢ ( ∇ ⁡ W ε , ξ ⁢ ( y ) , ∇ ⁡ Z ε , ξ ⁢ ( y ) s ) + W ε , ξ ⁢ ( y ) ⁢ Z ε , ξ ⁢ ( y ) s - f ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ Z ε , ξ ⁢ ( y ) s ⁢ d ⁢ μ g

- 1 ε n ⁢ ∫ M [ f ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) - f ⁢ ( W ε , ξ ⁢ ( y ) ) ] ⁢ Z ε , ξ ⁢ ( y ) s ⁢ 𝑑 μ g .

By (2.1), (2.2) and (2.3), after a change of variables, we have

1 ε n ⁢ ∫ M ε 2 ⁢ g ⁢ ( ∇ ⁡ W ε , ξ ⁢ ( y ) , ∇ ⁡ Z ε , ξ ⁢ ( y ) s ) + W ε , ξ ⁢ ( y ) ⁢ Z ε , ξ ⁢ ( y ) s - f ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ Z ε , ξ ⁢ ( y ) s ⁢ d ⁢ μ g

= ∫ ℝ + n ∇ ⁡ U ⁢ ∇ ⁡ φ l + U ⁢ φ l - f ⁢ ( U ) ⁢ φ l ⁢ d ⁢ z + O ⁢ ( ε ) = O ⁢ ( ε ) .

Besides, by the mean value theorem, for some t∈(0,1),

| 1 ε n ⁢ ∫ M [ f ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) - f ⁢ ( W ε , ξ ⁢ ( y ) ) ] ⁢ Z ε , ξ ⁢ ( y ) s ⁢ 𝑑 μ g | = | 1 ε n ⁢ ∫ M [ f ′ ⁢ ( W ε , ξ ⁢ ( y ) + t ⁢ ϕ ε , ξ ⁢ ( y ) ) ] ⁢ Z ε , ξ ⁢ ( y ) s ⁢ ϕ ε , ξ ⁢ ( y ) ⁢ 𝑑 μ g |

≤ c ⁢ 1 ε n ⁢ ∫ M ( | W ε , ξ ⁢ ( y ) | p - 2 + | ϕ ε , ξ ⁢ ( y ) | p - 2 ) ⁢ | Z ε , ξ ⁢ ( y ) s | ⁢ | ϕ ε , ξ ⁢ ( y ) | ⁢ 𝑑 μ g

≤ c ⁢ ( ∥ W ε , ξ ⁢ ( y ) ∥ ε p - 2 + ∥ ϕ ε , ξ ⁢ ( y ) ∥ ε p - 2 ) ⁢ ∥ Z ε , ξ ⁢ ( y ) s ∥ ε ⁢ ∥ ϕ ε , ξ ⁢ ( y ) ∥ ε

= O ⁢ ( ε 1 + n / p ′ ) = o ⁢ ( ε ) .

Hence, Jε′(Wε,ξ⁢(y)+ϕε,ξ⁢(y))[Zε,ξ⁢(y)s]=Oε) and, comparing with (5.4), we get (5.3). At this point, by (A.5), (5.3) and Proposition (3.3), we have that

| L 2 | ≤ | ∑ l = 1 n - 1 c ε l 〈 Z ε , ξ ⁢ ( y ) l , ∂ ∂ ⁡ y h ϕ ε , ξ ⁢ ( y ) 〉 ε |

= | ∑ l = 1 n - 1 c ε l 〈 ∂ ∂ ⁡ y h Z ε , ξ ⁢ ( y ) l , ϕ ε , ξ ⁢ ( y ) 〉 ε |

≤ ( ∑ l = 1 n - 1 | c ε l | ) ⁢ ∥ ∂ ∂ ⁡ y h ⁢ Z ε , ξ ⁢ ( y ) l ∥ ε ⁢ ∥ ϕ ε , ξ ⁢ ( y ) ∥ ε

≤ O ⁢ ( ε 1 + n / p ′ ) = o ⁢ ( ε ) .

Step 4: We estimate L1. We have

L 1 = 〈 ϕ ε , ξ ⁢ ( y ) , ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) 〉 ε - 1 ε n ⁢ ∫ M [ f ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) - f ⁢ ( W ε , ξ ⁢ ( y ) ) ] ⁢ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ⁢ 𝑑 μ g

= 〈 ϕ ε , ξ ⁢ ( y ) - i ε * ⁢ [ f ′ ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ ϕ ε , ξ ⁢ ( y ) ] , ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) 〉 ε

- 1 ε n ⁢ ∫ M [ f ⁢ ( W ε , ξ ⁢ ( y ) + ϕ ε , ξ ⁢ ( y ) ) - f ⁢ ( W ε , ξ ⁢ ( y ) ) - f ′ ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ ϕ ε , ξ ⁢ ( y ) ] ⁢ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ⁢ 𝑑 μ g

= 〈 ϕ ε , ξ ⁢ ( y ) - i ε * ⁢ [ f ′ ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ ϕ ε , ξ ⁢ ( y ) ] , ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) + 1 ε ⁢ Z ε , ξ ⁢ ( y ) h 〉 ε

- 1 ε ⁢ 〈 ϕ ε , ξ ⁢ ( y ) , Z ε , ξ ⁢ ( y ) h - i ε * ⁢ [ f ′ ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ ϕ ε , ξ ⁢ ( y ) ] 〉 ε

= : A 1 + A 2 + A 3 .

For the first term, by (A.7), we have

A 1 ≤ ∥ ϕ ε , ξ ⁢ ( y ) - i ε * ⁢ [ f ′ ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ ϕ ε , ξ ⁢ ( y ) ] ∥ ε ⁢ ∥ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) + 1 ε ⁢ Z ε , ξ ⁢ ( y ) h ∥ ε

≤ c ⁢ ε ⁢ ∥ ϕ ε , ξ ⁢ ( y ) - i ε * ⁢ [ f ′ ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ ϕ ε , ξ ⁢ ( y ) ] ∥ ε

≤ c ε ( ∥ ϕ ε , ξ ⁢ ( y ) ∥ ε + - | f ′ ( W ε , ξ ⁢ ( y ) ) ϕ ε , ξ ⁢ ( y ) | p ′ , ε )

≤ ε ⁢ ∥ ϕ ε , ξ ⁢ ( y ) ∥ ε = o ⁢ ( ε ) .

For the second term, in light of Proposition 3.3 and equation (A.4), we have

A 2 ≤ 1 ε ⁢ ∥ ϕ ε , ξ ⁢ ( y ) ∥ ε ⁢ ∥ Z ε , ξ ⁢ ( y ) h - i ε * ⁢ [ f ′ ⁢ ( W ε , ξ ⁢ ( y ) ) ⁢ Z ε , ξ ⁢ ( y ) h ] ∥ ε = O ⁢ ( ε 1 + 2 ⁢ n / p ′ ) = o ⁢ ( ε ) .

In order to estimate the last term, we have to consider separately the cases where 2≤p<3 and p≥3.

We recall from [15, Remark 3.4] that

| f ′ ⁢ ( W ε , ξ + v ) - f ′ ⁢ ( W ε , ξ ) | ≤ { c ⁢ ( p ) ⁢ | v | p - 2 , 2 < p < 3 , c ⁢ ( p ) ⁢ [ W ε , ξ p - 3 ⁢ | v | + | v | p - 2 ] , p ≥ 3 .

For p≥3, by the growth properties of f and using (A.5), we get

≤ ∥ ϕ ε , ξ ⁢ ( y ) ∥ ε 2 ⁢ ∥ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ∥ ε + ∥ ϕ ε , ξ ⁢ ( y ) ∥ ε p - 1 ⁢ ∥ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ∥ ε

≤ O ⁢ ( ε 1 + 2 ⁢ n / p ′ ) + O ⁢ ( ε p - 2 + ( p - 1 ) ⁢ n / p ′ ) = o ⁢ ( ε ) ,

since p≥3.

For 2<p<3, in a similar way, we get

A 3 ≤ c ε n ⁢ ∫ M ϕ ε , ξ ⁢ ( y ) p - 1 ⁢ | ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) | ⁢ 𝑑 μ g ≤ ∥ ϕ ε , ξ ⁢ ( y ) ∥ ε p - 1 ⁢ ∥ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ∥ ε = o ⁢ ( ε ) ,

which concludes the proof.∎

Lemma 5.2

We have that

J ε ⁢ ( W ε , ξ ) = C - ε ⁢ α ⁢ H ⁢ ( ξ ) + o ⁢ ( ε ) ,

C 0 -uniformly with respect to ξ as ε goes to zero, where

C := ∫ ℝ + n 1 2 ⁢ | ∇ ⁡ U ⁢ ( z ) | 2 + 1 2 ⁢ U 2 ⁢ ( z ) - 1 p ⁢ U p ⁢ ( z ) ⁢ d ⁢ z 𝑎𝑛𝑑 α := ( n - 1 ) 2 ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z n 3 ⁢ 𝑑 z

Proof.

By the definition of Jε, we have

J ε ⁢ ( W ε , ξ ) = 1 2 ⁢ ∫ D + ⁢ ( R / ε ) ∑ i , j = 1 n g i ⁢ j ⁢ ( ε ⁢ z ) ⁢ ∂ ⁡ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) ∂ ⁡ z i ⁢ ∂ ⁡ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) ∂ ⁡ z j ⁢ | g ⁢ ( ε ⁢ z ) | 1 / 2 ⁢ d ⁢ z

+ ∫ D + ⁢ ( R / ε ) [ 1 2 ⁢ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) 2 - 1 p ⁢ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) p ] ⁢ | g ⁢ ( ε ⁢ z ) | 1 / 2 ⁢ 𝑑 z .

By (2.1), (2.2) and (2.3), we easily get

J ε ⁢ ( W ε , ξ ) = ∫ D + ⁢ ( R / ε ) 1 2 ⁢ ∑ i = 1 n ∂ ⁡ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) ∂ ⁡ z i ⁢ ∂ ⁡ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) ∂ ⁡ z i ⁢ d ⁢ z + ∫ D + ⁢ ( R / ε ) 1 2 ⁢ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) 2 - 1 p ⁢ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) p ⁢ d ⁢ z

+ ε ⁢ ∫ D + ⁢ ( R / ε ) ∑ i , j = 1 n - 1 h i ⁢ j ⁢ ( 0 ) ⁢ z n ⁢ ∂ ⁡ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) ∂ ⁡ z i ⁢ ∂ ⁡ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) ∂ ⁡ z j ⁢ d ⁢ z

- n - 1 2 ⁢ ε ⁢ ∫ D + ⁢ ( R / ε ) H ⁢ ( ξ ) ⁢ z n ⁢ ∑ i = 1 n ∂ ⁡ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) ∂ ⁡ z i ⁢ ∂ ⁡ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) ∂ ⁡ z i ⁢ d ⁢ z

- n - 1 2 ⁢ ε ⁢ ∫ D + ⁢ ( R / ε ) H ⁢ ( ξ ) ⁢ z n ⁢ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) 2 ⁢ 𝑑 z + n - 1 p ⁢ ε ⁢ ∫ D + ⁢ ( R / ε ) H ⁢ ( ξ ) ⁢ z n ⁢ ( U ⁢ ( z ) ⁢ χ R / ε ⁢ ( z ) ) p ⁢ 𝑑 z + o ⁢ ( ε )

= ∫ ℝ + n 1 2 ⁢ | ∇ ⁡ U ⁢ ( z ) | 2 + 1 2 ⁢ U 2 ⁢ ( z ) - 1 p ⁢ U p ⁢ ( z ) ⁢ d ⁢ z - ε ⁢ ( n - 1 ) ⁢ H ⁢ ( ξ ) ⁢ ∫ ℝ + n z n ⁢ ( 1 2 ⁢ | ∇ ⁡ U ⁢ ( z ) | 2 + 1 2 ⁢ U 2 ⁢ ( z ) - 1 p ⁢ U p ⁢ ( z ) ) ⁢ 𝑑 z

+ ε ⁢ ∑ i , j = 1 n - 1 h i ⁢ j ⁢ ( 0 ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z i ⁢ z j ⁢ z n ⁢ 𝑑 z + o ⁢ ( ε ) .

Using Lemma A.1, finally we have

J ε ⁢ ( W ε , ξ ) = ∫ ℝ + n 1 2 ⁢ | ∇ ⁡ U ⁢ ( z ) | 2 + 1 2 ⁢ U 2 ⁢ ( z ) - 1 p ⁢ U p ⁢ ( z ) ⁢ d ⁢ z

- ε ⁢ ( n - 1 ) ⁢ H ⁢ ( ξ ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z n 3 ⁢ 𝑑 z + ε ⁢ ∑ i , j = 1 n - 1 h i ⁢ j ⁢ ( 0 ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z i ⁢ z j ⁢ z n ⁢ 𝑑 z + o ⁢ ( ε ) .

Now, by symmetry arguments and by (2.4) we have that

∑ i , j = 1 n - 1 h i ⁢ j ⁢ ( 0 ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z i ⁢ z j ⁢ z n ⁢ 𝑑 z = ∑ i , j = 1 n - 1 h i ⁢ j ⁢ ( 0 ) ⁢ δ i ⁢ j ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z i ⁢ z j ⁢ z n ⁢ 𝑑 z

= ∑ i = 1 n - 1 h i ⁢ i ⁢ ( 0 ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z i 2 ⁢ z n ⁢ 𝑑 z

= ( n - 1 ) ⁢ H ⁢ ( ξ ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z 1 2 ⁢ z n ⁢ 𝑑 z ,

and, by simple computations in polar coordinates,

(5.5) ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z 1 2 ⁢ z n ⁢ 𝑑 z = 1 2 ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z n 3 ⁢ 𝑑 z .

Concluding, we get

J ε ⁢ ( W ε , ξ ) = ∫ ℝ + n 1 2 ⁢ | ∇ ⁡ U ⁢ ( z ) | 2 + 1 2 ⁢ U 2 ⁢ ( z ) - 1 p ⁢ U p ⁢ ( z ) ⁢ d ⁢ z - ε ⁢ H ⁢ ( ξ ) ⁢ [ ( n - 1 ) 2 ⁢ ∫ ℝ + n ( U ′ ⁢ ( | z | ) | z | ) 2 ⁢ z n 3 ⁢ 𝑑 z ] + o ⁢ ( ε ) . ∎

Lemma 5.3

Let ξ⁢(y)=expξ∂⁡(y) with y∈Bn-1⁢(0,r). Then, we have

∂ ∂ ⁡ y h ⁢ J ε ⁢ ( W ε , ξ ⁢ ( y ) ) | y = 0 = - ε ⁢ α ⁢ ( ∂ ∂ ⁡ y h ⁢ H ⁢ ( ξ ⁢ ( y ) ) ) | y = 0 + o ⁢ ( ε ) ,

uniformly with respect to ξ as ε goes to zero.

Proof.

For simplicity, we prove the claim for h=1. The cases where h=2,…,n-1 are straightforward. Let us consider first

∂ ∂ ⁡ y 1 ⁢ ∫ I ξ ⁢ ( R ) 1 2 ⁢ ε n ⁢ W ε , ξ ⁢ ( y ) 2 ⁢ 𝑑 μ g | y = 0 = ∫ I ξ ⁢ ( R ) 1 ε n ⁢ W ε , ξ ⁢ ∂ ∂ ⁡ y 1 ⁢ W ε , ξ ⁢ ( y ) | y = 0 ⁢ d ⁢ μ g .

By Lemma 2.5 and by the exponential decay of U, we have

∂ ∂ ⁡ y 1 ⁢ ∫ 1 2 ⁢ ε n ⁢ W ε , ξ ⁢ ( y ) 2 ⁢ 𝑑 μ g | y = 0 = ∫ ℝ + n U ⁢ ( z ) ⁢ χ R ⁢ ( ε ⁢ z ) ⁢ [ U ⁢ ( z ) ⁢ ∂ ⁡ χ R ∂ ⁡ z k ⁢ ( ε ⁢ z ) + 1 ε ⁢ ∂ ⁡ U ∂ ⁡ z k ⁢ ( z ) ⁢ χ R ⁢ ( ε ⁢ z ) ] ⁢ ∂ ∂ ⁡ y 1 ⁢ ℰ ~ ⁢ ( y , ε ⁢ z ) | y = 0 ⁢ | g ⁢ ( ε ⁢ z ) | 1 / 2 ⁢ d ⁢ z

= 1 ε ⁢ ∫ ℝ + n U ⁢ ( z ) ⁢ ∂ ⁡ U ∂ ⁡ z k ⁢ ( z ) ⁢ ∂ ∂ ⁡ y 1 ⁢ ℰ ~ ⁢ ( y , ε ⁢ z ) | y = 0 ⁢ | g ⁢ ( ε ⁢ z ) | 1 / 2 ⁢ d ⁢ z + o ⁢ ( ε )

= 1 ε ⁢ ∫ ℝ + n U ⁢ ( z ) ⁢ U ′ ⁢ ( z ) | z | ⁢ z k ⁢ ∂ ∂ ⁡ y 1 ⁢ ℰ k ~ ⁢ ( y , ε ⁢ z ) | y = 0 ⁢ | g ⁢ ( ε ⁢ z ) | 1 / 2 ⁢ d ⁢ z + o ⁢ ( ε ) ,

where ℰ~ is defined in Definition 2.4. Expanding in ε, by Lemma 2.6 and by (2.3), we obtain

∂ ∂ ⁡ y 1 ⁢ ∫ 1 2 ⁢ ε n ⁢ W ε , ξ ⁢ ( y ) 2 ⁢ 𝑑 μ g | y = 0 = 1 ε ⁢ ∫ ℝ + n U ⁢ ( z ) ⁢ U ′ ⁢ ( z ) | z | ⁢ z k ⁢ ( - δ 1 ⁢ k + 1 2 ⁢ ε 2 ⁢ E i ⁢ j k ⁢ z i ⁢ z j ) ⁢ ( 1 - ε ⁢ ( n - 1 ) ⁢ H ⁢ z n + 1 2 ⁢ ε 2 ⁢ G l ⁢ s ⁢ z l ⁢ z s ) ⁢ 𝑑 z + o ⁢ ( ε )

= 1 ε ⁢ ∫ ℝ + n U ⁢ ( z ) ⁢ U ′ ⁢ ( z ) | z | ⁢ [ - z 1 ⁢ ( 1 - ε ⁢ ( n - 1 ) ⁢ H ⁢ z n + 1 2 ⁢ ε 2 ⁢ G l ⁢ s ⁢ z l ⁢ z s ) + 1 2 ⁢ z k ⁢ ε 2 ⁢ E i ⁢ j k ⁢ z i ⁢ z j ] ⁢ 𝑑 z + o ⁢ ( ε ) ,

where

E i ⁢ j k = ∂ 2 ∂ ⁡ z i ⁢ ∂ ⁡ z j ⁢ ∂ ∂ ⁡ y 1 ⁢ ℰ ~ k ⁢ ( y , z ) | y = 0 , z = 0 and G l ⁢ s = ∂ 2 ∂ ⁡ z i ⁢ ∂ ⁡ z j ⁢ | g ⁢ ( z ) | 1 / 2 | z = 0 .

By symmetry, the only terms remaining are the ones containing zr2⁢zn, thus

∂ ∂ ⁡ y 1 ⁢ ∫ 1 2 ⁢ ε n ⁢ W ε , ξ ⁢ ( y ) 2 ⁢ 𝑑 μ g | y = 0 = ε ⁢ ∫ ℝ + n U ⁢ ( z ) ⁢ U ′ ⁢ ( z ) | z | ⁢ [ - z 1 ⁢ G n ⁢ 1 ⁢ z n ⁢ z 1 + z k ⁢ E k ⁢ n k ⁢ z k ⁢ z n ] ⁢ 𝑑 z + o ⁢ ( ε ) .

By (2.5) we have that Gn⁢1⁢(0)=-(n-1)⁢∂⁡H∂⁡z1⁢(0), and in light of Lemma 2.7 we get Ek⁢nk=0. We conclude that

∂ ∂ ⁡ y 1 ⁢ ∫ 1 2 ⁢ ε n ⁢ W ε , ξ ⁢ ( y ) 2 ⁢ 𝑑 μ g | y = 0 = ε ⁢ ∫ ℝ + n U ⁢ ( z ) ⁢ U ′ ⁢ ( z ) | z | ⁢ ( n - 1 ) ⁢ ( ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ) ⁢ z n ⁢ z 1 2 ⁢ 𝑑 z + o ⁢ ( ε )

= ε ⁢ ∫ ℝ + n ∂ ∂ ⁡ z 1 ⁢ ( 1 2 ⁢ U 2 ⁢ ( z ) ) ⁢ ( n - 1 ) ⁢ ( ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ) ⁢ z n ⁢ z 1 ⁢ 𝑑 z + o ⁢ ( ε )

= - ε ⁢ ( n - 1 ) ⁢ ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ⁢ ∫ ℝ + n 1 2 ⁢ U 2 ⁢ ( z ) ⁢ z n ⁢ 𝑑 z + o ⁢ ( ε ) .

In the same way, we get that

∂ ∂ ⁡ y 1 ⁢ ∫ 1 p ⁢ ε n ⁢ W ε , ξ ⁢ ( y ) p ⁢ 𝑑 μ g | y = 0 = ε ⁢ ∫ ℝ + n U p - 1 ⁢ ( z ) ⁢ U ′ ⁢ ( z ) | z | ⁢ ( n - 1 ) ⁢ ( ∂ ∂ ⁡ z 1 ⁢ H ⁢ ( 0 ) ) ⁢ z n ⁢ z 1 2 ⁢ 𝑑 z + o ⁢ ( ε )

= - ε ⁢ ( n - 1 ) ⁢ ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ⁢ ∫ ℝ + n 1 p ⁢ U p ⁢ ( z ) ⁢ z n ⁢ 𝑑 z + o ⁢ ( ε ) .

Now we look at the last term

I := ∂ ∂ ⁡ y 1 ⁢ ∫ I ξ ⁢ ( R ) ε 2 2 ⁢ ε n ⁢ | ∇ ⁡ W ε , ξ ⁢ ( y ) | g 2 ⁢ 𝑑 μ g | y = 0 = ∫ I ξ ⁢ ( R ) ε 2 ε n ⁢ g ⁢ ( ∇ ⁡ W ε , ξ ⁢ ∇ ⁡ ∂ ∂ ⁡ y 1 ⁢ W ε , ξ ⁢ ( y ) ) | y = 0 ⁢ d ⁢ μ g ,

and, again using Lemma 2.5 and the decay of U, we have

I = 1 ε ⁢ ∫ ℝ + n g i ⁢ j ⁢ ( ε ⁢ z ) ⁢ ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z j ⁢ ( ∂ ⁡ U ∂ ⁡ z k ⁢ ∂ ∂ ⁡ y 1 ⁢ ℰ k ~ ⁢ ( y , ε ⁢ z ) ) | y = 0 ⁢ | g ⁢ ( ε ⁢ z ) | 1 / 2 ⁢ d ⁢ z + o ⁢ ( ε ) .

Recall (2.1) (2.2) and (2.3), and set, with abuse of notation, hi⁢n=hn⁢j=0 for all i,j=1,…,n. Then, we have

I = 1 ε ⁢ ∫ ℝ + n ( δ i ⁢ j + 2 ⁢ ε ⁢ h i ⁢ j ⁢ z n + 1 2 ⁢ ε 2 ⁢ γ r ⁢ t i ⁢ j ⁢ z r ⁢ z t ) ⁢ ( 1 - ε ⁢ ( n - 1 ) ⁢ H ⁢ z n + 1 2 ⁢ ε 2 ⁢ G l ⁢ s ⁢ z l ⁢ z s )

× ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z j ⁢ ( ∂ ⁡ U ∂ ⁡ z k ⁢ ( - δ 1 ⁢ k + 1 2 ⁢ ε 2 ⁢ E v ⁢ w k ⁢ z v ⁢ z w ) ) ⁢ d ⁢ z + o ⁢ ( ε ) ,

where

E v ⁢ w k = ∂ 2 ∂ ⁡ z v ⁢ ∂ ⁡ z w ⁢ ∂ ∂ ⁡ y 1 ⁢ ℰ ~ k ⁢ ( y , z ) | y = 0 , z = 0 , G l ⁢ s = ∂ 2 ∂ ⁡ z l ⁢ ∂ ⁡ z s ⁢ | g ⁢ ( z ) | 1 / 2 | z = 0 and γ r ⁢ t i ⁢ j = ∂ 2 ∂ ⁡ z r ⁢ ∂ ⁡ z t ⁢ g i ⁢ j ⁢ ( z ) | z = 0 .

More explicitly,

I = - 1 ε ⁢ ∫ ℝ + n ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z i ⁢ ∂ ⁡ U ∂ ⁡ z 1 ⁢ 𝑑 z + ∫ ℝ + n ( n - 1 ) ⁢ H ⁢ z n ⁢ ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z i ⁢ ∂ ⁡ U ∂ ⁡ z 1 ⁢ 𝑑 z - 1 2 ⁢ ε ⁢ ∫ ℝ + n G l ⁢ s ⁢ z l ⁢ z s ⁢ ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z i ⁢ ∂ ⁡ U ∂ ⁡ z 1 ⁢ 𝑑 z

- 2 ⁢ ∫ ℝ + n h i ⁢ j ⁢ z n ⁢ ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z j ⁢ ∂ ⁡ U ∂ ⁡ z 1 ⁢ 𝑑 z + 2 ⁢ ε ⁢ ∫ ℝ + n ( n - 1 ) ⁢ H ⁢ h i ⁢ j ⁢ z n 2 ⁢ ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z j ⁢ ∂ ⁡ U ∂ ⁡ z 1 ⁢ 𝑑 z - 1 2 ⁢ ε ⁢ ∫ ℝ + n γ r ⁢ t i ⁢ j ⁢ z r ⁢ z t ⁢ ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z j ⁢ ∂ ⁡ U ∂ ⁡ z 1 ⁢ 𝑑 z

+ 1 2 ⁢ ε ⁢ ∫ ℝ + n ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z i ⁢ ( ∂ ⁡ U ∂ ⁡ z k ⁢ E v ⁢ w k ⁢ z v ⁢ z w ) ⁢ 𝑑 z + o ⁢ ( ε )

= : I 1 + I 2 + I 3 + I 4 + I 5 + I 6 + I 7 .

We easily have

I 1 = - 1 ε ⁢ ∫ ℝ + n ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ∂ ⁡ z i ⁢ ∂ ⁡ U ∂ ⁡ z 1 ⁢ 𝑑 z = - 1 2 ⁢ ε ⁢ ∫ ℝ + n ∂ ∂ ⁡ z 1 ⁢ | ∇ ⁡ U | 2 ⁢ 𝑑 z = 0 ,

and, in a similar way and by integration by parts,

I 2 = ( n - 1 ) ⁢ H ⁢ ( 0 ) ⁢ 1 2 ⁢ ∫ ℝ + n z n ⁢ ∂ ∂ ⁡ z 1 ⁢ | ∇ ⁡ U | 2 ⁢ 𝑑 z = 0 ,

I 3 = - 1 4 ⁢ ε ⁢ ∫ ℝ + n G l ⁢ s ⁢ ( 0 ) ⁢ z l ⁢ z s ⁢ ∂ ∂ ⁡ z 1 ⁢ | ∇ ⁡ U | 2 ⁢ 𝑑 z = ε ⁢ 1 2 ⁢ ∫ ℝ + n G 1 ⁢ s ⁢ ( 0 ) ⁢ z s ⁢ | ∇ ⁡ U | 2 ⁢ 𝑑 z .

Moreover, by symmetry, the only nonzero contribution comes from the term containing zn, so by (2.5),

I 3 = ε ⁢ 1 2 ⁢ ∫ ℝ + n G 1 ⁢ n ⁢ ( 0 ) ⁢ z n ⁢ | ∇ ⁡ U | 2 ⁢ 𝑑 z = - ε ⁢ ( n - 1 ) ⁢ ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ⁢ ∫ ℝ + n 1 2 ⁢ | ∇ ⁡ U | 2 ⁢ z n ⁢ 𝑑 z .

Since hi⁢j is symmetric, we have

I 4 = - ∫ ℝ + n h i ⁢ j ⁢ ( 0 ) ⁢ z n ⁢ ∂ ∂ ⁡ z 1 ⁢ ( ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ⁡ U ∂ ⁡ z j ) ⁢ 𝑑 z = 0 ,

and, similarly, by integration by parts, we also obtain that I5=0.

For I7, we have

I 7 = 1 2 ⁢ ε ⁢ ∫ ℝ + n ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ⁡ z k ⁢ E v ⁢ w k ⁢ z v ⁢ z w ⁢ 𝑑 z + 1 2 ⁢ ε ⁢ ∫ ℝ + n ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ⁡ U ∂ ⁡ z k ⁢ ( E v ⁢ i k ⁢ z v + E i ⁢ w k ⁢ z w ) ⁢ 𝑑 z ,

and, since ∂⁡U∂⁡zi=U′⁢(z)|z|⁢zi,

(5.6) ∫ ℝ + n ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ⁡ U ∂ ⁡ z k ⁢ E v ⁢ i k ⁢ z v = ∫ ℝ + n ( U ′ ⁢ ( z ) | z | ) 2 ⁢ E v ⁢ i k ⁢ z i ⁢ z k ⁢ z v ⁢ 𝑑 z .

The only nonzero integrals are the ones of the form zr2⁢zn and, since Er⁢rn=0 (Lemma 2.7), all the terms in (5.6) are 0. With a similar argument, since

∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ⁡ z k = U ′ ⁢ ( z ) | z | ⁢ δ i ⁢ k + U ′′ ⁢ ( z ) ⁢ | z | - U ′ | z | 3 ⁢ z k ⁢ z i ,

we can conclude that I7=0.

Finally, let us consider I6. Since gi⁢j is symmetric, integrating by parts gives

I 6 = - 1 4 ⁢ ε ⁢ ∫ ℝ + n γ r ⁢ t i ⁢ j ⁢ z r ⁢ z t ⁢ ∂ ∂ ⁡ z 1 ⁢ ( ∂ ⁡ U ∂ ⁡ z i ⁢ ∂ ⁡ U ∂ ⁡ z j ) ⁢ 𝑑 z = 1 2 ⁢ ε ⁢ ∫ ℝ + n γ 1 ⁢ t i ⁢ j ⁢ ( U ′ ⁢ ( z ) | z | ) 2 ⁢ z i ⁢ z j ⁢ z t ⁢ 𝑑 z .

By (2.2), we have that γ1⁢tn⁢j=γ1⁢ti⁢n=0 for all i,j,t=1,…,n. In addition, by symmetry, only the term which contains zr2⁢zn gives a nonzero contribution. Thus, by (2.1) and (5.5),

I 6 = 1 2 ⁢ ε ⁢ ∑ i = 1 n - 1 ∫ ℝ + n γ 1 ⁢ n i ⁢ i ⁢ ( U ′ ⁢ ( z ) | z | ) 2 ⁢ z i 2 ⁢ z n ⁢ 𝑑 z

= ε ⁢ ∑ i = 1 n - 1 ∂ ⁡ h i ⁢ i ∂ ⁡ z 1 ⁢ ( 0 ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( z ) | z | ) 2 ⁢ z i 2 ⁢ z n ⁢ 𝑑 z

= ε ⁢ ( n - 1 ) ⁢ ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( z ) | z | ) 2 ⁢ z 1 2 ⁢ z n ⁢ 𝑑 z

= 1 2 ⁢ ε ⁢ ( n - 1 ) ⁢ ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( z ) | z | ) 2 ⁢ z n 3 ⁢ 𝑑 z .

Concluding, we have

∂ ∂ ⁡ y 1 ⁢ J ε ⁢ ( W ε , ξ ⁢ ( y ) ) | y = 0 = - ε ⁢ ( n - 1 ) ⁢ ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ⁢ ∫ ℝ + n [ 1 2 ⁢ | ∇ ⁡ U | 2 + 1 2 ⁢ | U | 2 - 1 p ⁢ | U | p ] ⁢ z n ⁢ 𝑑 z

+ 1 2 ⁢ ε ⁢ ( n - 1 ) ⁢ ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( z ) | z | ) 2 ⁢ z n 3 ⁢ 𝑑 z ,

and, by Lemma A.1,

∂ ∂ ⁡ y 1 ⁢ J ε ⁢ ( W ε , ξ ⁢ ( y ) ) | y = 0 = - 1 2 ⁢ ε ⁢ ( n - 1 ) ⁢ ∂ ⁡ H ∂ ⁡ z 1 ⁢ ( 0 ) ⁢ ∫ ℝ + n ( U ′ ⁢ ( z ) | z | ) 2 ⁢ z n 3 ⁢ 𝑑 z . ∎

Funding source: Istituto Nazionale di Alta Matematica ”Francesco Severi”

Award Identifier / Grant number: Gruppo Nazionale per l’Analisi Matematica

Award Identifier / Grant number: la Probabilità e le loro Applicazioni (GNAMPA)

Funding statement: The authors were supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM).

A Technical Lemmas

Here we collect a series of estimates that we used in the paper as well as the proofs of some lemmas which were previously claimed.

Lemma A.1

We have that

∫ ℝ + n ( ∂ z n ⁡ U ) 2 ⁢ z n ⁢ 𝑑 z = ∫ ℝ + n ( U ′ ⁢ ( z ) | z | ) 2 ⁢ z n 3 ⁢ 𝑑 z

= 1 2 ∫ ℝ + n | ∇ U | 2 z n d z + 1 2 ∫ ℝ + n U 2 z n d z - 1 p ∫ ℝ + n U p z n d z .

Proof.

We multiply -Δ⁢U by zn2⁢∂zn⁡U and integrate by parts over ℝ+n, obtaining

- ∫ ℝ + n Δ ⁢ U ⁢ z n 2 ⁢ ∂ z n ⁡ U ⁢ d ⁢ z = ∫ ℝ + n ( Δ ⁢ ∂ z n ⁡ U ) ⁢ z n 2 ⁢ U ⁢ 𝑑 z + 2 ⁢ ∫ ℝ + n Δ ⁢ U ⁢ z n ⁢ U ⁢ 𝑑 z

= - ∫ ℝ + n ∑ i = 1 n ( ∂ z i ⁡ ∂ z n ⁡ U ) ⁢ z n 2 ⁢ ∂ z i ⁡ U ⁢ d ⁢ z - 2 ⁢ ∫ ℝ + n ∑ i = 1 n ( ∂ z i ⁡ ∂ z n ⁡ U ) ⁢ z n ⁢ δ i ⁢ n ⁢ U ⁢ d ⁢ z

- 2 ⁢ ∫ ℝ + n ∑ i = 1 n ( ∂ z i ⁡ U ) ⁢ z n ⁢ ∂ z i ⁡ U ⁢ d ⁢ z - 2 ⁢ ∫ ℝ + n ∑ i = 1 n ( ∂ z i ⁡ U ) ⁢ δ i ⁢ n ⁢ U ⁢ d ⁢ z

= - ∫ ℝ + n ∑ i = 1 n ( ∂ z i ⁡ ∂ z n ⁡ U ) ⁢ z n 2 ⁢ ∂ z i ⁡ U ⁢ d ⁢ z - 2 ⁢ ∫ ℝ + n ( ∂ z n 2 ⁡ U ) ⁢ z n ⁢ U ⁢ 𝑑 z

- 2 ⁢ ∫ ℝ + n | ∇ ⁡ U | 2 ⁢ z n ⁢ 𝑑 z - 2 ⁢ ∫ ℝ + n ( ∂ z n ⁡ U ) ⁢ U ⁢ 𝑑 z .

Now, again by integration by parts,

- ∫ ℝ + n ∑ i = 1 n ( ∂ z i ⁡ ∂ z n ⁡ U ) ⁢ z n 2 ⁢ ∂ z i ⁡ U ⁢ d ⁢ z = ∫ ℝ + n ( ∂ z n ⁡ U ) ⁢ z n 2 ⁢ Δ ⁢ U ⁢ 𝑑 z + 2 ⁢ ∫ ℝ + n ∑ i = 1 n ( ∂ z n ⁡ U ) ⁢ δ i ⁢ n ⁢ z n ⁢ ∂ z i ⁡ U ⁢ d ⁢ z

= ∫ ℝ + n ( ∂ z n ⁡ U ) ⁢ z n 2 ⁢ Δ ⁢ U ⁢ 𝑑 z + 2 ⁢ ∫ ℝ + n ( ∂ z n ⁡ U ) 2 ⁢ z n ⁢ 𝑑 z .

Thus,

- ∫ ℝ + n Δ ⁢ U ⁢ z n 2 ⁢ ∂ z n ⁡ U ⁢ d ⁢ z = ∫ ℝ + n ( ∂ z n ⁡ U ) ⁢ z n 2 ⁢ Δ ⁢ U ⁢ 𝑑 z + 2 ⁢ ∫ ℝ + n ( ∂ z n ⁡ U ) 2 ⁢ z n ⁢ 𝑑 z

- 2 ∫ ℝ + n ( ∂ z n 2 U ) z n U d z - 2 ∫ ℝ + n | ∇ U | 2 z n d z - 2 ∫ ℝ + n ( ∂ z n U ) U d z ,

that is,

- ∫ ℝ + n Δ U z n 2 ∂ z n U d z = ∫ ℝ + n ( ∂ z n U ) 2 z n d z - ∫ ℝ + n ( ∂ z n 2 U ) z n U d z - ∫ ℝ + n | ∇ U | 2 z n d z - ∫ ℝ + n ( ∂ z n U ) U d z .

Now

0 = ∫ ℝ + n ∂ z n ⁡ ( z n ⁢ U ⁢ ∂ z n ⁡ U ) ⁡ d ⁢ z = ∫ ℝ + n U ⁢ ∂ z n ⁡ U + z n ⁢ ( ∂ z n ⁡ U ) 2 + z n ⁢ U ⁢ ∂ z n 2 ⁡ U ⁢ d ⁢ z ,

and we get

(A.1) - ∫ ℝ + n Δ U z n 2 ∂ z n U d z = 2 ∫ ℝ + n ( ∂ z n U ) 2 z n d z - ∫ ℝ + n | ∇ U | 2 z n d z .

In a similar way, we prove that

(A.2) ∫ ℝ + n U ⁢ z n 2 ⁢ ∂ z n ⁡ U ⁢ d ⁢ z = - ∫ ℝ + n U 2 ⁢ z n ⁢ 𝑑 z ,

(A.3) ∫ ℝ + n U p - 1 ⁢ z n 2 ⁢ ∂ z n ⁡ U ⁢ d ⁢ z = - 2 p ⁢ ∫ ℝ + n U p ⁢ z n ⁢ 𝑑 z .

Now, multiplying by zn2⁢∂zn⁡U both terms of (2.7), integrating over ℝ+n and using (A.1)–(A.3), we obtain the claim. ∎

The following lemma collects several estimates on Zε,ξj.

Lemma A.2

There exist ε0>0 and c>0 such that for any ξ0∈∂⁡M and for any ε∈(0,ε0), we have

(A.4) ∥ Z ε , ξ h - i * ⁢ [ f ′ ⁢ ( W ε , ξ ) ⁢ Z ε , ξ h ] ∥ ε ≤ c ⁢ ε 1 + N / p ′ ,

(A.5) ∥ ∂ ∂ ⁡ y h ⁢ Z ε , ξ ⁢ ( y ) l ∥ ε = O ⁢ ( 1 ε ) , ∥ ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ∥ ε = O ⁢ ( 1 ε ) ,

(A.6) 〈 Z ε , ξ 0 l , ( ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ) | y = 0 〉 = - 1 ε ⁢ c ⁢ δ l ⁢ h + o ⁢ ( 1 ε ) ,

(A.7) ∥ 1 ε ⁢ Z ε , ξ 0 h + ( ∂ ∂ ⁡ y h ⁢ W ε , ξ ⁢ ( y ) ) | y = 0 ∥ ε ≤ c ⁢ ε

for h=1,…,n-1, l=1,…,n.

Proof.

The proof of (A.4) is similar to that of Lemma 3.2, and will be omitted. The other three estimates are similar to [15, Lemmas 6.1–6.3], which we refer to for the proof of the claim. ∎

Proof of Lemma 3.1.

We will prove it by contradiction. We assume that there exist sequences εk→0, ξk∈∂⁡M with ξk→ξ∈∂⁡M and ϕk∈Kεk,ξk⊥ with ∥ϕ∥εk=1 such that

L ε k , ξ k ⁢ ( ϕ k ) = ψ k with ⁢ ∥ ψ k ∥ ε k → 0 ⁢ for ⁢ k → + ∞ .

By the definition of Lεk,ξk, there exists ζk∈Kεk,ξk such that

(A.8) ϕ k - i ε k * ⁢ [ f ′ ⁢ ( W ε k , ξ k ) ⁢ ϕ k ] = ψ k + ζ k .

We prove that ∥ζk∥εk→0 for k→+∞. Let ζk=∑j=1n-1ajk⁢Zεk,ξkj with Zε,ξi being defined in (2.10). By using the fact that ϕk,ψk∈Kεk,ξk⊥ and (A.8), we have

∑ j = 1 n - 1 a j k ⁢ 〈 Z ε k , ξ k j , Z ε k , ξ k h 〉 ε k = - 〈 i ε k * ⁢ [ f ′ ⁢ ( W ε k , ξ k ) ⁢ ϕ k ] , Z ε k , ξ k h 〉 ε k

(A.9) = - 1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) ⁢ ϕ k ⁢ Z ε k , ξ k h ⁢ 𝑑 μ g .

By elementary properties of φj, we have that

(A.10) 〈 Z ε k , ξ k j , Z ε k , ξ k j 〉 ε k = C ⁢ δ j ⁢ h + o ⁢ ( 1 ) for all ⁢ j , h = 1 , … , n - 1 ,

where C is a positive constant.

We set

ϕ ~ k := { ϕ k ⁢ ( ψ ξ k ∂ ⁢ ( ε k ⁢ z ) ) ⁢ χ R ⁢ ( ε k ⁢ z ) , z ∈ D + ⁢ ( R ) , 0 , otherwise .

We get easily that ∥ϕ~k∥H1⁢(ℝn)≤c⁢∥ϕk∥εk≤c for some positive constant c. Thus, there exists ϕ~∈H1⁢(ℝn) such that ϕ~k→ϕ~ weakly in H1⁢(ℝn) and strongly in Llocp⁢(ℝn) for all 2≤p<2* if n≥3 or p≥2 if n=2.

We recall that ϕk∈Kεk,ξk⊥, so

- 1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) ⁢ ϕ k ⁢ Z ε k , ξ k h ⁢ 𝑑 μ g = 〈 ϕ k , Z ε k , ξ k h 〉 ε - 1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) ⁢ ϕ k ⁢ Z ε k , ξ k h ⁢ 𝑑 μ g

= 1 ε k n ⁢ ∫ M [ ε k 2 ⁢ ∇ ⁡ ϕ k ⁢ ∇ ⁡ Z ε k , ξ k h + Z ε k , ξ k h ⁢ ϕ k - f ′ ⁢ ( W ε k , ξ k ) ⁢ ϕ k ⁢ Z ε k , ξ k h ] ⁢ 𝑑 μ g

= ∫ D + ⁢ ( R / ε k ) [ ∑ l , m = 1 n g l ⁢ m ( ε k z ) ∂ ⁡ ϕ ~ k ∂ ⁡ z l ∂ ⁡ ( φ h ⁢ ( z ) ) ∂ ⁡ z m + ϕ ~ k φ h ( z ) ] | g ( ε k z ) | 1 / 2 d y

- ∫ D + ⁢ ( R / ε k ) f ′ ⁢ ( U ⁢ ( z ) ⁢ χ R ⁢ ( ε k ⁢ z ) ) ⁢ ϕ ~ k ⁢ φ h ⁢ ( z ) ⁢ | g ⁢ ( ε k ⁢ z ) | 1 / 2 ⁢ 𝑑 z + o ⁢ ( 1 )

= ∫ ℝ n ∇ ⁡ ϕ ~ ⁢ ∇ ⁡ φ h + ϕ ~ ⁢ φ h - f ′ ⁢ ( U ) ⁢ ϕ ~ ⁢ φ h ⁢ d ⁢ z + o ⁢ ( 1 ) = o ⁢ ( 1 ) ,

because φh is a weak solution of the linearized problem (2.8). So we can rewrite (A.9), obtaining

c ⁢ a h k + o ⁢ ( 1 ) = ∑ j = 1 n - 1 a j k ⁢ 〈 Z ε k , ξ k j , Z ε k , ξ k h 〉 ε k = - 1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) ⁢ ϕ k ⁢ Z ε k , ξ k h ⁢ 𝑑 μ g = o ⁢ ( 1 ) ,

so ahk→0 for all h while k→+∞, and thus ∥ζk∥εk→0 for k→+∞.

Setting uk:=ϕk-ψk-ζk, (A.8) can be read as

(A.11) { - ε k 2 ⁢ Δ g ⁢ u k + u k = f ′ ⁢ ( W ε k , ξ k ) ⁢ u k + f ′ ⁢ ( W ε k , ξ k ) ⁢ ( ψ k + ζ k ) in ⁢ M , ∂ ⁡ u k ∂ ⁡ ν = 0 on ⁢ ∂ ⁡ M .

Multiplying (A.11) by uk and integrating by parts, we get

(A.12) ∥ u k ∥ ε k = 1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) ⁢ u k 2 + f ′ ⁢ ( W ε k , ξ k ) ⁢ ( ψ k + ζ k ) ⁢ u k .

By Hölder’s inequality, and recalling that |u|ε,p≤c⁢∥u∥ε, we have

1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) ⁢ ( ψ k + ζ k ) ⁢ u k ≤ ( 1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) n / 2 ) 2 / n ⁢ | u k | ε k , 2 ⁢ n / ( n - 2 ) ( n - 2 ) / ( 2 ⁢ n ) ⁢ | ψ k + ζ k | ε k , 2 ⁢ n / ( n - 2 ) ( n - 2 ) / ( 2 ⁢ n )

(A.13) ≤ c ⁢ ( 1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) n / 2 ) 2 / n ⁢ ∥ u k ∥ ε k ( n - 2 ) / ( 2 ⁢ n ) ⁢ ∥ ψ k + ζ k ∥ ε k ( n - 2 ) / ( 2 ⁢ n ) .

Now,

1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) n / 2 ⁢ 𝑑 μ g ≤ 1 ε k n ⁢ ∫ I ξ k ⁢ ( R ) ( U ε ⁢ ( ( ψ ξ k ∂ ) - 1 ⁢ ( x ) ) ) n ⁢ ( p - 2 ) / 2 ⁢ 𝑑 μ g

(A.14) ≤ c ⁢ ∫ D + ⁢ ( R / ε ) ( U ⁢ ( z ) ) n ⁢ ( p - 2 ) / 2 ⁢ 𝑑 z ≤ c

for some positive constant c.

Combining (A.12), (A.13) and (A.14), and recalling that ∥uk∥εk→1 and ∥ψk+ζk∥εk→0 while k→+∞, we get

(A.15) 1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ) ⁢ u k 2 → 1 while ⁢ k → + ∞ .

We will see how this leads us to a contradiction.

We set

u ~ k ⁢ ( z ) := u k ⁢ ( ψ ξ k ∂ ⁢ ( ε k ⁢ z ) ) ⁢ χ R ⁢ ( ε k ⁢ z ) for ⁢ z ∈ ℝ + n ,

and have

∥ u ~ k ∥ H 1 ⁢ ( ℝ n ) ≤ c ⁢ ∥ u k ∥ ε k ≤ c .

So, up to subsequence, there exists u~∈H1⁢(ℝ+n) such that u~k→u~ weakly in H1⁢(ℝ+n)and strongly in Llocp⁢(ℝ+n), p∈(2,2*) if n≥3 or p>2 if n=2. By (A.11) we deduce that

(A.16) { - Δ ⁢ u ~ + u ~ = f ′ ⁢ ( U ) ⁢ u ~ in ⁢ ℝ + n , ∂ ⁡ u ~ ∂ ⁡ x n = 0 on { x n = 0 } .

We prove also that

(A.17) 〈 φ h , u ~ 〉 H 1 = 0 for all ⁢ h ∈ 1 , … , n - 1 .

In fact, since ϕk,ψk∈Kε,ξ⊥ and ∥ζk∥εk→0, we have

(A.18) | 〈 Z ε k , ξ k h , u k 〉 ε k | = | 〈 Z ε k , ξ k h , ζ k 〉 ε k | ≤ ∥ Z ε k , ξ k h ∥ ε k ⁢ ∥ ζ k ∥ ε k = o ⁢ ( 1 ) .

On the other hand, by direct computation, we get

〈 Z ε k , ξ k h , u k 〉 ε k = 1 ε k n ⁢ ∫ M ε k 2 ⁢ g ⁢ ( ∇ ⁡ Z ε k , ξ k h ⁢ ∇ ⁡ u k ) + Z ε k , ξ k h ⁢ u k

= ∫ D + ⁢ ( R / ε k ) ∑ l , m = 1 n g l ⁢ m ⁢ ( ε k ⁢ z ) ⁢ ∂ ⁡ ( φ h ⁢ ( z ) ⁢ χ R ⁢ ( ε k ⁢ z ) ) ∂ ⁡ z l ⁢ ∂ ⁡ u ~ k ∂ ⁡ z m ⁢ | g ⁢ ( ε k ⁢ z ) | 1 2 ⁢ d ⁢ z + ∫ D + ⁢ ( R / ε k ) φ h ⁢ ( z ) ⁢ χ R ⁢ ( ε k ⁢ z ) ⁢ u ~ k ⁢ | g ⁢ ( ε k ⁢ z ) | 1 2 ⁢ 𝑑 z

(A.19) = ∫ ℝ + n ( ∇ ⁡ φ h ⁢ ∇ ⁡ u ~ + φ h ⁢ u ~ ) ⁢ 𝑑 z + o ⁢ ( 1 ) .

So, by (A.18) and (A.19), we obtain (A.17).

Now (A.17) and (A.16) imply that u~=0. Thus,

1 ε k n ⁢ ∫ M f ′ ⁢ ( W ε k , ξ k ⁢ ( x ) ) ⁢ u k 2 ⁢ ( x ) ⁢ 𝑑 μ g ≤ 1 ε k n ⁢ ∫ I g ⁢ ( R ) f ′ ⁢ ( U ε ⁢ ( ( ψ ξ k ∂ ) - 1 ⁢ ( x ) ) ) ⁢ u k 2 ⁢ ( x ) ⁢ 𝑑 μ g = c ⁢ ∫ D + ⁢ ( R / ε k ) f ′ ⁢ ( U ⁢ ( z ) ) ⁢ u ~ k 2 ⁢ ( z ) = o ⁢ ( 1 ) ,

which contradicts (A.15). This concludes the proof. ∎

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Received: 2015-04-21

Accepted: 2015-12-04

Published Online: 2016-04-07

Published in Print: 2016-08-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2015-5036

Keywords for this article

Riemannian Manifold with Boundary; Nonlinear Elliptic Equations; Neumann BoundaryCondition; Mean Curvature; Lyapunov–Schmidt

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