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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary

Mónica Clapp
  • Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 México City, Mexico
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  • De Gruyter OnlineGoogle Scholar
/ Marco GhimentiORCID iD: https://orcid.org/0000-0001-8777-1008 / Anna Maria Micheletti
Published Online: 2017-07-21 | DOI: https://doi.org/10.1515/anona-2017-0039

Abstract

We study the semiclassical limit to a singularly perturbed nonlinear Klein–Gordon–Maxwell–Proca system, with Neumann boundary conditions, on a Riemannian manifold 𝔐 with boundary. We exhibit examples of manifolds, of arbitrary dimension, on which these systems have a solution which concentrates at a closed submanifold of the boundary of 𝔐, forming a positive layer, as the singular perturbation parameter goes to zero. Our results allow supercritical nonlinearities and apply, in particular, to bounded domains in N. Similar results are obtained for the more classical electrostatic Klein–Gordon–Maxwell system with appropriate boundary conditions.

Keywords: Electrostatic Klein–Gordon–Maxwell–Proca system; semiclassical limit; boundary layer; Riemannian manifold with boundary; supercritical nonlinearity; Lyapunov–Schmidt reduction

MSC 2010: 35J60; 35J20; 35B40; 53C80; 58J32; 81V10

1 Introduction

On a compact smooth Riemannian manifold (𝔐,𝔤) with boundary, we consider the system

{-ε2Δ𝔤𝔲+α(x)𝔲=𝔲p-1+ω2(q𝔳-1)2𝔲on 𝔐,-Δ𝔤𝔳+Λ(𝔲)𝔳=q𝔲2on 𝔐,𝔲ν=0,𝔳ν=0 or 𝔳=0on 𝔐,(1.1)

where Δ𝔤=div𝔤𝔤 is the Laplace–Beltrami operator (without a sign), ε>0, q>0, ω, α𝒞2(𝔐) is a real-valued function which satisfies α(x)>ω2 on 𝔐, p(2,), and Λ is given by

Λ(𝔲)={1+q𝔲2if 𝔲ν=𝔳ν=0 on 𝔐,q𝔲2if 𝔲ν=𝔳=0 on 𝔐.

We are interested in studying the semiclassical limit to this system, i.e., the existence of positive solutions and their asymptotic profile, as ε0.

Solutions to system (1.1) correspond to standing waves of an electrostatic Klein–Gordon–Maxwell (KGM) system if Λ(𝔲)=q𝔲2, and of a Klein–Gordon–Maxwell–Proca (KGMP) system with Proca mass 1 if Λ(𝔲)=1+q𝔲2. For the physical meaning of these systems, we refer to [3, 4, 25].

The seminal paper [3] by Benci and Fortunato attracted the attention of the mathematical community, and motivated much of the recent activity towards the study of this type of systems. For ε=1, existence and nonexistence results for subcritical nonlinear terms have been obtained, e.g., in [1, 3, 6, 10, 13, 14, 15, 27] for systems in the entire space 3, or in a bounded domain in 3 with Dirichlet or Neumann boundary conditions. KGMP-systems on a closed (i.e., compact and without boundary) Riemannian manifold of dimension 3 or 4 have been recently investigated in [17, 24, 25] for subcritical or critical nonlinearities.

The existence and asymptotic behavior of semiclassical states in flat domains have been investigated, e.g., in [11, 12, 31]. In [11], D’Aprile and Wei constructed a family of positive radial solutions (𝔲ε,𝔳e) to a KGM-system in a 3-dimensional ball, with Dirichlet boundary conditions, such that 𝔲ε concentrates around a sphere which lies in the interior of the ball. For compact manifolds of dimensions 2 and 3, with or without boundary, the existence and multiplicity of positive semiclassical states, such that 𝔲ε concentrates at a point, have been exhibited, e.g., in [20, 21, 23], for subcritical nonlinearities. The concentration at a positive-dimensional submanifold for a KGMP-system on closed manifolds of arbitrary dimension, and for nonlinearities which include supercritical ones, was recently exhibited in [7].

Our aim is to extend the results in [7, 8] to manifolds with boundary, i.e., we will establish the existence of positive semiclassical states (𝔲ε,𝔳e) to system (1.1), on some compact Riemannian manifolds 𝔐 with boundary, such that 𝔲ε concentrates at a positive-dimensional submanifold as ε0. Our results apply, in particular, to systems with supercritical nonlinearities in bounded smooth domains Ω of N of any dimension.

The Neumann boundary condition 𝔳ν=0 on 𝔳 seems to be more meaningful from a physical point of view, as it gives a condition on the electric field on 𝔐. However, if the Proca mass is 0, i.e., if Λ(𝔲)=q𝔲2, and we set 𝔳ν=0, then the second equation in system (1.1) admits the trivial solution 𝔳=1q and the first equation reduces to a Schrödinger equation, making the coupling effect unnoticeable. This is why we impose a Dirichlet boundary condition on 𝔳 when Λ(𝔲)=q𝔲2.

The Neumann boundary condition 𝔲ν=0 on 𝔲 produces an effect of the boundary of 𝔐 on the existence and concentration of solutions to system (1.1). In fact, the solutions that we obtain form a positive layer which concentrates around a submanifold of 𝔐 as ε0.

As in [7], our approach consists in reducing system (1.1) to a similar system, with the same power nonlinearity, on a manifold of lower dimension. Solutions to the new system which concentrate at a point will give rise to solutions to the original system concentrating at a positive-dimensional submanifold. This approach was introduced by Ruf and Srikanth in [29] and has been used, for instance, in [9, 28, 30]. We begin by describing some of the reductions that we will use.

1.1 Reducing the dimension of the system

Let (M,g) be a compact smooth n-dimensional Riemannian manifold with boundary, let f:M(0,) be a 𝒞1-function, and let (N,h) be a compact smooth Riemannian manifold without boundary of dimension k1. The warped product M×f2N is the cartesian product M×N endowed with the Riemannian metric 𝔤:=g+f2h. It is a smooth Riemannian manifold of dimension n+k with boundary M×f2N.

For example, if Θ is a bounded smooth domain in n whose closure is contained in n-1×(0,), f(x1,,xn)=xn and 𝕊k is the standard k-sphere, then, up to isometry, the warped product Θ×f2𝕊k is

Θ×f2𝕊k{(y,z)n-1×k+1:(y,|z|)Θ},

which is a bounded smooth domain in n+k.

Let πM:M×f2NM be the projection, α^𝒞2(M) and α:=α^πM. A straightforward computation gives the following result; see, e.g., [16].

Proposition 1.1.

The functions uε,vε:MR solve the system

{-ε2divg(fkgu)+fkα^u=fkup-1+ω2fk(qv-1)2uon M,-divg(fkgv)+fkΛ(u)v=qfku2on M,uν=0,vν=0 or v=0on M(1.2)

if and only if the functions uε:=uεπM, vε:=vεπM:M×f2NR solve the system

{-ε2Δ𝔤𝔲+α𝔲=𝔲p-1+ω2(q𝔳-1)2𝔲on M×f2N,-Δ𝔤𝔳+Λ(𝔲)𝔳=q𝔲2on M×f2N,𝔲ν=0,𝔳ν=0 or 𝔳=0on (M×f2N).(1.3)

We stress that the exponent p is the same in both systems. Since k1, we have that 2n+k<2n, where 2d is the critical Sobolev exponent in dimension d, i.e., 2d:= if d=2 and 2d:=2dd-2 for d>2. So, if 2n+kp<2n, system (1.2) on M is subcritical, whereas system (1.3) on M×f2N is critical or supercritical. Moreover, if the solution uε of (1.2) concentrates at a point x0M as ε0, then the function 𝔲ε:=uεπM concentrates at the submanifold πM-1(x0)(N,f2(x0)h). Note also that 𝔲ε and 𝔳ε are positive if uε and vε are positive.

Another type of reduction is obtained from the Hopf maps. For N=2,4,8,16, we write N𝕂×𝕂, where 𝕂 is either the real numbers , the complex numbers , the quaternions , or the Cayley numbers 𝕆. The Hopf map 𝔥𝕂 is defined by

𝔥𝕂:2dim𝕂𝕂×𝕂𝕂×dim𝕂+1,𝔥𝕂(z):=(2z1¯z2,|z1|2-|z2|2)for z=(z1,z2)𝕂×𝕂.

This map is horizontally conformal with dilation λ(z)=2|z|. It is also invariant under the action of the units S𝕂:={ζ𝕂:|ζ|=1}, i.e., 𝔥𝕂(ζz)=𝔥𝕂(z) for all ζS𝕂, z𝕂×𝕂.

Let Ω be a bounded smooth domain in 2dim𝕂{0} such that ζzΩ for all ζS𝕂, zΩ. Then Θ:=𝔥𝕂(Ω) is a bounded smooth domain in dim𝕂+1{0}. The main property of Hopf maps, for our purposes, is that they locally preserve the Laplace operator up to a factor, i.e.,

Δ(u𝔥𝕂)=λ2[(Δu)𝔥𝕂]in Ω for every u𝒞2(Θ).

Such maps are called harmonic morphisms; see [2]. This property allows us to reduce system (1.1) on 𝔐:=Ω to a system in Θ. Assume that α𝒞2(Ω) satisfies α(ζz)=α(z) for all ζS𝕂, zΩ. Then the map α^:Θ given by α^(x):=α(𝔥𝕂-1(x)) is well defined and of class 𝒞2. Note that λ2(𝔥𝕂-1(x))=4|x| for every xdim𝕂+1. The following proposition is an immediate consequence of these facts.

Proposition 1.2.

The functions uε,vε:ΘR solve the system

{-ε2Δu+α^(x)4|x|u=14|x|up-1+ω24|x|(qv-1)2uon Θ,-Δv+14|x|Λ(u)v=q4|x|u2on Θ,uν=0,vν=0 or v=0on Θ(1.4)

if and only if the functions uε:=uεhK, vε:=vεhK:ΩR solve the system

{-ε2Δ𝔲+α(x)𝔲=𝔲p-1+ω2(q𝔳-1)2𝔲on Ω,-Δ𝔳+Λ(𝔲)𝔳=q𝔲2on Ω,𝔲ν=0,𝔳ν=0 or 𝔳=0on Ω.(1.5)

Note again that, if p[22dim𝕂,2dim𝕂+1), system (1.4) is subcritical, whereas system (1.5) is critical or supercritical. And if the functions uε concentrate at a point ξ0Θ as ε0, then the functions 𝔲ε concentrate at the (dim𝕂-1)-dimensional sphere 𝔥𝕂-1(ξ0) in Ω.

Propositions 1.1 and 1.2 lead us to study the following problem.

1.2 The main results

Let (M,g) be a smooth compact Riemannian manifold with boundary of dimension n=2,3,4. We consider the subcritical system

{-ε2divg(c(x)gu)+a(x)u=b(x)up-1+b(x)ω2(qv-1)2uon M,-divg(c(x)gv)+b(x)Λ(u)v=b(x)qu2on M,uν=0,vν=0 or v=0on M,(1.6)

where ε,q>0, ω, a,b,c𝒞1(M) are strictly positive functions such that a(x)>ω2b(x) on M, and p(2,2n). As before, 2n:= if n=2 and 2n:=2nn-2 if n=3,4.

Theorem 1.3.

Let KM be a nonempty C1-stable critical set for the function Γ:MR, which is given by

Γ(ξ):=c(ξ)n2[a(ξ)-ω2b(ξ)]pp-2-n2b(ξ)2p-2.

Then, for ε small enough, system (1.6) has a positive solution (uε,vε) such that uε concentrates at a point ξ0K as ε goes to zero.

A 𝒞1-stable critical set is defined as follows.

Definition 1.4.

Let f𝒞1(M,). A subset 𝒦 of M is called a 𝒞1-stable critical set of f if 𝒦{xM:gf(x)=0} and if, for any μ>0, there exists δ>0 such that every function h𝒞1(M,) which satisfies

maxdistg(x,𝒦)μ(|f(x)-h(x)|+|gf(x)-gh(x)|g)δ

has a critical point x0 with distg(x0,𝒦)μ. Here distg denotes the geodesic distance associated to the Riemannian metric g.

Theorem 1.3, together with Propositions 1.1 and 1.2, yields the existence of solutions to the KGMP (or the KGM) system (1.1), which concentrate at a submanifold for subcritical, critical and supercritical exponents. The following two results illustrate this fact.

We write the points in n-1×(0,) as (y¯,yn) with y¯n-1 and yn(0,).

Theorem 1.5.

Let Θ be a bounded smooth domain in Rn whose closure is contained in Rn-1×(0,) for n=2,3,4, and let ωR and α^C2(Θ) be such that α^>ω2. Let

𝔐:={(y¯,z)n-1×k+1:(y¯,|z|)Θ}

and α(y¯,z):=α^(y¯,|z|). If K is a nonempty C1-stable critical set for the function Γ:ΘR defined by

Γ(y¯,yn):=ynk[α^(y¯,yn)-ω2]pp-2-n2,

then, for any q>0, p(2,2n) and ε small enough, system (1.1) has a positive solution (uε,vε) in M such that, for some point (ξ¯,ξn)K, uε concentrates at the k-dimensional sphere {(ξ¯,z)Rn-1×Rk+1:|z|=ξn}M as ε0.

Proof.

Set M:=Θ, a:=fkα^ and b:=fk=:c with f(y¯,yn):=yn. Theorem 1.3 yields a positive solution (uε,vε) to system (1.2) such that uε concentrates at a point (ξ¯,ξn)𝒦 as ε0. The result follows from Proposition 1.1. ∎

Theorem 1.6.

Let

𝔐:={z2:0<r<|z|<R}

and assume that αC2(M) satisfies α(ζz)=α(z)>ω2 for all ζC with |ζ|=1, zM. If K is a nonempty C1-stable critical set for the function Γ:(hC(M))R defined by

Γ(x):=2|x|[α(𝔥-1(x))-ω2]pp-2-32,

then, for any q>0, p(2,6) and ε small enough, system (1.1) has a positive solution (uε,vε) in M such that uε concentrates at the circle {ζz0:ζC,|ζ|=1}M, for some z0hC-1(K), as ε0.

Proof.

Set M:=𝔥(𝔐), a(x):=α^(x)2|x|, b(x):=12|x|, and c(x):=1 with α^(x):=α(𝔥-1(x)). Theorem 1.3 yields a positive solution (uε,vε) to system (1.4) such that uε concentrates at a point ξ0𝒦 as ε0. The result follows from Proposition 1.2. ∎

The rest of the paper is devoted to the proof of Theorem 1.3.

2 Preliminaries

2.1 Reducing system (1.6) to a single equation

In order to overcome the problems given by the competition between u and v, using an idea of Benci and Fortunato [3], we introduce the map Φ:Hg1(M)Hg1(M) which associates to each uHg1(M) the solution Φ(u) to the problem

{-divg(c(x)g[Φ(u)])+b(x)q2u2[Φ(u)]=b(x)qu2in M,Φ(u)=0on M(2.1)

for system (1.6) with Dirichlet boundary conditions, or to the problem

{-divg(c(x)g[Φ(u)])+b(x)(1+q2u2)[Φ(u)]=b(x)qu2in M,[Φ(u)]ν=0on M(2.2)

for system (1.6) with Neumann boundary conditions. It follows from standard variational arguments that Φ is well defined in Hg1(M). The proofs of the following two lemmas are contained in [17].

Lemma 2.1.

The map Φ:Hg1(M)Hg1(M) is of class C1 and its differential Φ(u)[h]=Vu[h] at uHg1(M) is the map defined by

-divg(c(x)g[Vu[h]])+b(x)q2u2[Vu[h]]=2b(x)qu(1-qΦ(u))h

for all hHg1(M), in case of Dirichlet boundary conditions, or by

-divg(c(x)g[Vu[h]])+b(x)(1+q2u2)[Vu[h]]=2b(x)qu(1-qΦ(u))h,

for all hHg1(M), in case of Neumann boundary conditions. Moreover,

0Φ(u)1q𝑎𝑛𝑑0Φ(u)[u]2q.

Lemma 2.2.

The function Θ:Hg1(M)R given by

Θ(u)=12Mb(x)(1-qΦ(u))u2𝑑μg

is of class C1, and its differential is given by

Θ(u)[h]=Mb(x)(1-qΦ(u))2uh𝑑μg

for any u,hHg1(M).

Now, we introduce the functionals Iε,Jε,Gε:Hg1(M) given by

Iε(u):=Jε(u)+ω22Gε(u),(2.3)

where

Jε(u):=12εnM[ε2c(x)|gu|2+d(x)u2]𝑑μg-1pεnMb(x)(u+)p𝑑μg

with d(x):=a(x)-ω2b(x), and

Gε(u):=qεnMb(x)Φ(u)u2𝑑μg.

From Lemma 2.2 we deduce that

12Gε(u)[φ]=1εnMb(x)[2qΦ(u)-q2Φ2(u)]uφ𝑑μg,

so

Iε(u)φ=1εnM[ε2c(x)gugφ+a(x)uφ-b(x)(u+)p-1φ-b(x)ω2(1-qΦ(u))2uφdμg].

Therefore, if u is a critical point of the functional Iε, we have that

-ε2divg(c(x)gu)+d(x)u+ω2qb(x)Φ(u)(2-qΦ(u))u=b(x)(u+)p-1,(2.4)

with d(x):=a(x)-ω2b(x). In particular, if u0, by the maximum principle and regularity arguments we have that u>0. Thus, the pair (u,Φ(u)) is a positive solution to system (1.6).

This reduces solving system (1.6) to finding a solution uεHg1(M) to the single equation (2.4).

Some useful estimates involving the function Φ are contained in the appendix.

2.2 The approximate solution

We shall obtain a solution uε to equation (2.4) using the Lyapunov–Schmidt reduction method. It will be an approximation to a function Wε,ξ, which we introduce next.

If (M,g) is an n-dimensional compact smooth Riemannian manifold with boundary, its boundary M is a closed smooth Riemannian manifold of dimension n-1, possibly not connected. We fix R>0, smaller than the injectivity radius of M, such that for each point xM with distg(x,M)<R there exists a unique x¯M for which distg(x,x¯)=distg(x,M), where distg denotes the geodesic distance in (M,g). For ξM, we set

Qξ:={xM:distg(x,M)=distg(x,x¯)<R,x¯M,distg(ξ,x¯)<R}.

We write each point xQξ in Fermi coordinates (y1,,yn) at ξ, i.e., (y1,,yn-1) are normal coordinates for x¯ on M at the point ξ, and yn=distg(x,x¯) is the geodesic distance from x to M. We write ψξ:D+Qξ for the chart whose inverse is given by (ψξ)-1(x):=(y1,,yn), defined on

D+:=BRn-1(0)×[0,R),where BRn-1(0):={y¯n-1:|y¯|<R}.

The second fundamental form II(X,Y) of two vector fields X and Y on M is the component of XY which is normal to M, where is the covariant derivative operator in the ambient manifold M. In Fermi coordinates at q it is given by a matrix (hij)i,j=1,,n-1. One has the well-known formulas

gij(y)=δij+2hijyn+O(|y|2)for i,j=1,,n-1,(2.5)gin(y)=δin,(2.6)|g|(y)=1-(n-1)Hyn+O(|y|2),(2.7)

where y=(y1,,yn) are the Fermi coordinates, |g| is the determinant of g=(gij), gij are the coefficients of the inverse of (gij), and H=1n-1i=1n-1hii; see [5, 18, 19]. Abusing notation, we shall write (hij)i,j=1,,n for the matrix which coincides with the second fundamental form for i,j=1,,n-1 and has hi,n=hn,j=0 for i,j=1,,n.

Set d(x):=a(x)-ω2b(x). By assumption, this function is positive on M. Given ξM, we consider the unique positive radial solution V¯=V¯ξ to the equation

-c(ξ)ΔV¯+d(ξ)V¯=b(ξ)V¯p-1in n.(2.8)

By direct computation, one sees that

V¯ξ(y)=(d(ξ)b(ξ))1p-2U(d(ξ)c(ξ)y),

where U is the unique positive radial solution of

-ΔU+U=Up-1 in n.

In the following, we set

γ(ξ):=(d(ξ)b(ξ))1p-2andA(ξ):=d(ξ)c(ξ),

so

V¯ξ(y)=γ(ξ)U(A(ξ)y).

The restriction Vξ(y):=V¯ξ|+n of V¯ξ to the half-space +n:={yn0} solves the Neumann problem

{-c(ξ)ΔV+d(ξ)V=b(ξ)Vp-1in +n,Vyn=0on {yn=0}.

For ξM and ε>0, set Vεξ(y):=Vξ(yε). We define the functions Wε,ξ𝒞(M) by

Wε,ξ(x):={Vεξ((ψξ)-1(x))χ((ψξ)-1(x)),xQξ,0elsewhere.(2.9)

Here the function χ is a fixed cut-off function of the form χ(y¯,yn):=χ~(|y¯|)χ~(yn) for (y¯,yn)D+, where χ~:+[0,1] is a smooth function such that χ(s)1 for 0sR2, χ(s)0 for sR and |χ~(s)|1R.

Remark 2.3.

The following limits hold uniformly with respect to ξM,

limε01εn|Wε,ξ|p,gpC|U|pp,p2,limε01εn|ε2gWε,ξ|2,g2C|U|22,

where the constant C does not depend on ξ.

It is well known that the space of solutions to the linearized problem

{-Δφ+φ=(p-1)(Vξ)p-2φin +n,φyn=0on {yn=0},

is generated by the functions φi:=Vξyi for i=1,,n-1. The corresponding local functions on the manifold M are given by

Zε,ξi(x):={φεi((ψξ)-1(x))χ((ψξ)-1(x)),xQξ,0elsewhere,(2.10)

where φεi(y):=φi(yε) and χ is as above.

2.3 Proof of Theorem 1.3

As before, we set d(x):=a(x)-ω2b(x)>0. We denote by Hε the space Hg1(M) equipped with the scalar product

u,vε:=1εnMε2c(x)gugv+d(x)uvdμg

and the norm uε=u,uε1/2. Similarly, we write Lεp for the space Lgp(M) endowed with the norm

|u|ε,p=1εn(M|u|p𝑑μg)1p.

For any p[2,2n), the embedding iε:HεLε,p is compact and there is a positive constant C, independent of ε, such that |u|ε,pCuε. The adjoint operator iε:Lε,pHε, p:=pp-1, is defined by

u=iε(v)u,φε=1εnMvφ𝑑μgfor all φHg1(M)-ε2divg(c(x)gu)+d(x)u=v.

Note that, for some positive constant C independent of ε,

iε(v)εC|v|p,εfor all vLε,p.(2.11)

Using the adjoint operator, we can rewrite equation (2.4) as

u=iε(b(x)f(u)+ω2b(x)g(u)),

where

f(u):=(u+)p-1;g(u):=[q2Φ2(u)-2qΦ(u)]u.

For ξM and ε>0, let

Kε,ξ:=Span{Zε,ξ1,,Zε,ξn-1},

where the Zε,ξi are the functions defined in (2.10). This is an (n-1)-dimensional subspace of Hε. We denote its orthogonal complement with respect to ,ε by

Kε,ξ:={uHε:u,Zε,ξiε=0}.

We look for a solution to equation (2.4) of the form Wε,ξ+ϕ with ϕKε,ξ. Thus, Wε,ξ+ϕ solves the equations

Πε,ξ(Wε,ξ+ϕ-iε[b(x)f(Wε,ξ+ϕ)+ω2b(x)g(Wε,ξ+ϕ)])=0,(2.12)Πε,ξ(Wε,ξ+ϕ-iε[b(x)f(Wε,ξ+ϕ)+ω2b(x)g(Wε,ξ+ϕ)])=0,

where Πε,ξ:HεKε,ξ and Πε,ξ:HεKε,ξ are the orthogonal projections onto Kε,ξ and Kε,ξ, respectively.

The first step in the proof of Theorem 1.3 is to solve equation (2.12). To this end, we define the linear operator Lε,ξ:Kε,ξKε,ξ by

Lε,ξ(ϕ):=Πε,ξ(ϕ-iε[b(x)f(Wε,ξ)ϕ]).(2.13)

Lemma 3.1 yields the invertibility of Lε,ξ. Then we will use a contraction mapping argument to solve equation (2.12). In Section 3, we will prove the following result.

Proposition 2.4.

There exist ε0>0 and C>0 such that, for any ξM and any ε(0,ε0), there is a unique ϕ=ϕε,ξ which solves equation (2.12). This function satisfies

ϕε,ξεCε.

Moreover, ξϕε,ξ is a C1-map.

Now, for each ε(0,ε0), we introduce the reduced energy I~ε:M, defined by

I~ε(ξ):=Iε(Wε,ξ+ϕε,ξ),

where Iε is the functional defined in (2.3), whose critical points are the solutions to equation (2.4). It is easy to verify that ξε is a critical point of I~ε if and only if the function uε=Wε,ξε+ϕε,ξε is a weak solution to problem (2.4).

In Section 4, we will compute the asymptotic expansion of the reduced functional I~ε with respect to the parameter ε. We will show that

I~ε(ξ)=κc(ξ)n2d(ξ)pp-2-n2b(ξ)2p-2+o(1)

𝒞1-uniformly with respect to ξM as ε0, where

κ:=(12-1p)+nUp𝑑z.

If 𝒦 is a nonempty 𝒞1-stable critical set for the function Γ, then, by Definition 1.4, there exists a critical point ξεM of I~ε such that distg(ξε,𝒦)0 as ε0. Consequently, uε=Wε,ξε+ϕε,ξε is a solution of (2.4), and Theorem 1.3 is proved.

3 The finite-dimensional reduction

In this section, we prove Proposition 2.4. Using the linear operator Lε,ξ:Kε,ξKε,ξ introduced in (2.13), equation (2.12) can be rewritten as

Lε,ξ(ϕ)=Nε,ξ(ϕ)+Rε,ξ+Sε,ξ(ϕ),

where

Nε,ξ(ϕ):=Πε,ξ(iε[b(x)(f(Wε,ξ+ϕ)-f(Wε,ξ)-f(Wε,ξ)ϕ)]),Rε,ξ:=Πε,ξ(iε[b(x)f(Wε,ξ)]-Wε,ξ),Sε,ξ(ϕ):=ω2Πε,ξ(iε[b(x)(q2Φ2(Wε,ξ+ϕ)-2qΦ(Wε,ξ+ϕ))(Wε,ξ+ϕ)]).

We refer to [26, Proposition 3.1], [7, Lemma 4.1] or [22, Lemma 10] for the proof of the following lemma.

Lemma 3.1.

There exist ε0 and C>0 such that, for any ξM and ε(0,ε0),

Lε,ξεCϕεfor every ϕKε,ξ.

We now estimate the remainder term Rε,ξ.

Lemma 3.2.

There exists ε0>0 such that, for any ξM and ε(0,ε0), one has

Rε,ξε=o(ε).

Proof.

Let Gε,ξ be the function such that Wε,ξ=iε(b(x)Gε,ξ), i.e.,

-ε2divg(c(x)gWε,ξ)+d(x)Wε,ξ=b(x)Gε,ξ.

Then, for xQξ and its Fermi coordinates y:=(ψξ)-1(x), setting c~(y):=c(x), d~(y):=d(x) and b~(y):=b(x), we have

b(x)Gε,ξ(x)=d~(y)Vεξ(y)χ(y)-ε2|g(y)|yj[|g(y)|gij(y)c~(y)yi(Vεξ(y)χ(y))]=d~(y)Vεξ(y)χ(y)-ε2gij(y)yj[c~(y)yi(Vεξ(y)χ(y))]-ε2|g(y)|yj[|g(y)|gij(y)]c~(y)yi(Vεξ(y)χ(y))=d~(y)Vεξ(y)χ(y)-ε2yi[c~(y)yiVεξ(y)]χ(y)-ε2c~(y)yiVεξ(y)yiχ(y)-ε2yi[c~(y)Vεξ(y)yiχ(y)]-ε2(gij(y)-δij)yj[c~(y)yi(Vεξ(y)χ(y))]-ε2|g(y)|yj[|g(y)|12gij(y)]c~(y)yi(Vεξ(y)χ(y)).

Moreover, by (2.8), we have

d~(y)Vεξ(y)χ(y)-ε2yi[c~(y)yi(Vεξ(y))]χ(y)=d(ξ)Vεξ(y)χ(y)-ε2c(ξ)ΔVεξ(y)χ(y)+[d~(y)-d(ξ)]Vεξ(y)χ(y)-ε2yi[(c~(y)-c(ξ))yi(Vεξ(y))]χ(y)=(b(ξ)(Vεξ(y))p-1+[d~(y)-d(ξ)]Vεξ(y)-ε2yi[(c~(y)-c(ξ))yi(Vεξ(y))])χ(y).

From the definition of Rε,ξ we obtain

Rε,ξεiε(b(x)f(Wε,ξ))-Wε,ξε=iε(b(x)[f(Wε,ξ)-Gε,ξ])ε.

Using (2.11), we estimate the right-hand side by

M|b(x)Wε,ξp-1-b(x)Gε,ξ|p𝑑μgCB(0,r)b~(y)p|(Vεξ(y))p-1-Gε,ξ(ψξ(y))|p𝑑yCD+|b~(y)-b(ξ)|p|Vεξ(y)|p𝑑y+CB(0,r)|d~(y)-d(ξ)|p|Vεξ(y)|p𝑑y+Cε2D+|yi[(c~(y)-c(ξ))yi(Vεξ(y))]|p𝑑y+Cε2D+|gij(y)-δij|p|vj[c~(y)yi(Vεξ(y))]|p𝑑y+Cε2D+1|g(y)|p2yj|[|g(y)|12gij(y)]|p|c~(y)yi(Vεξ(y))|p𝑑y.

By the usual change of variables y=εz, we can easily estimate almost all terms in the previous equation. The only term needing more attention is

I1=Cε2D+|gij(y)-δij|p|yj[c~(y)yi(Vεξ(y))]|p𝑑y.

We have

I1Cε2+n|gij(εz)-δij|p|1ε22zi2(Vξ(z))|pεn𝑑z+o(εp),

and by (2.5) we get

Rε,ξε=O(ε2+p+n-2pp)=O(ε2+np-1)=o(ε)

since p>2 and n2, so 2+np>2. ∎

Lemma 3.3.

There exist ε0>0 and C>0 such that, for any ξM, ε(0,ε0) and r>0, we have that

Sε,ξ(ϕ)εCε(3.1)

and

Sε,ξ(ϕ1)-Sε,ξ(ϕ2)εεϕ1-ϕ2ε(3.2)

for ϕ,ϕ1,ϕ2{vHε:vεrε}, with ε0 as ε0.

Proof.

Let us prove (3.1). From the definition of i and (2.11) it follows that

Sε,ξ(ϕ)εC|Φ2(Wε,ξ+ϕ)(Wε,ξ+ϕ)|ε,p+C|Φ(Wε,ξ+ϕ)(Wε,ξ+ϕ)|ε,pC|Φ(Wε,ξ+ϕ)(Wε,ξ+ϕ)|ε,p

since 0<Φ(u)<1q. Hence, for some t>2 if n=2 or for t=2n if n=3,4, we have that

Sε,ξ(ϕ)εC1εn/p(M|Φ(Wε,ξ+ϕ)|t)1t(|Wε,ξ+ϕ|tpt-p)t-ptpC1εn/pΦ(Wε,ξ+ϕ)Hg1|Wε,ξ+ϕ|tpt-p,gC1εn/pΦ(Wε,ξ+ϕ)Hg1εnt-ptp|Wε,ξ+ϕ|ε,tpt-pCε-ntΦ(Wε,ξ+ϕ)Hg1(1+ϕε)

by Remark 2.3. Now, for n=2, by (5.2) we have that

Sε,ξ(ϕ)εCεβ-2t,

and we can chose t>2 sufficiently large and β<2 sufficiently close to 2 to prove the claim. On the other hand, for n=3,4, recalling that t=2n and using (5.4), we have

Sε,ξ(ϕ)εCε-n-22εn+22=Cε2.

In every case, Sε,ξ(ϕ)εCε, and we have proved (3.1).

Let us prove (3.2). From (2.11), since 0<Φ(u)<1q, it follows that

Sε,ξ(ϕ1)-Sε,ξ(ϕ2)εC|Φ2(Wε,ξ+ϕ1)(Wε,ξ+ϕ1)-Φ2(Wε,ξ+ϕ2)(Wε,ξ+ϕ1)|ε,p   +C|Φ(Wε,ξ+ϕ1)(Wε,ξ+ϕ1)-Φ(Wε,ξ+ϕ2)(Wε,ξ+ϕ1)|ε,pC|Φ(Wε,ξ+ϕ1)(Wε,ξ+ϕ1)-Φ(Wε,ξ+ϕ2)(Wε,ξ+ϕ1)|ε,p=C|[Φ(Wε,ξ+ϕ1)-Φ(Wε,ξ+ϕ2)]Wε,ξ|ε,p+C|Φ(Wε,ξ+ϕ2)(ϕ1-ϕ2)|ε,p=:I1+I2.

In the light of Remark 2.3, for some θ(0,1) we have that

I1p=CεnM|Φ(Wε,ξ+θϕ1+(1-θ)ϕ2)(ϕ1-ϕ2)|p|Wε,ξ+ϕ1|pCεn(M|Φ(Wε,ξ+θϕ1+(1-θ)ϕ2)(ϕ1-ϕ2)|t)pt(M|Wε,ξ+ϕ1|ptt-p)t-pt=CεnΦ(Wε,ξ+θϕ1+(1-θ)ϕ2)(ϕ1-ϕ2)Hg1pεnt-pt|Wε,ξ+ϕ1|ε,ptt-ppCε-nptΦ(Wε,ξ+θϕ1+(1-θ)ϕ2)(ϕ1-ϕ2)Hg1p.

Here, as before, t>2 for n=2 and t=2n for n=3,4. Notice that, since p<2, we have

p22-p<2.

By direct computation, one sees that uHg1ε(n-2)/2uε for n=2,3,4. Thus, in case n=2, from Lemma 5.2 we obtain that

I1Cε-2t(εβ+ϕ1ε+ϕ2ε)ϕ1-ϕ2εCεβ-2tϕ1-ϕ2ε

and, choosing t sufficiently large, we conclude that I1εϕ1-ϕ2ε with ε0. For n=3,4, again by Lemma 5.2, we have

I1Cε-n2(ε2+εn-22(ϕ1ε+ϕ2ε))εn-22ϕ1-ϕ2εC(ε2+εn-22(ϕ1ε+ϕ2ε))ϕ1-ϕ2ε,

and since ϕ1ε+ϕ2εCε, we have again that I1εϕ1-ϕ2ε with ε0. To estimate I2 we proceed in a similar way, obtaining

I2p=CεnM|Φ(Wε,ξ+ϕ2)|p|ϕ1-ϕ2|pCεn(M|Φ(Wε,ξ+ϕ2)|t)pt(M|ϕ1-ϕ2|ptt-p)t-ptCε-nptΦ(Wε,ξ+ϕ2)Hg1pϕ1-ϕ2εp.

For n=2, we have by (5.2) that

I2Cε-2tεβ(1+ϕ2ε)ϕ1-ϕ2εCεβ-2tϕ1-ϕ2ε

and, since t may be chosen arbitrarily large, we have εβ-2/t0. For n=3,4, again by (5.2) we conclude that

I2Cε-n2ε2(1+ϕ2ε)ϕ1-ϕ2εCε2ϕ1-ϕ2ε,

so I2εϕ1-ϕ2ε with ε0. Collecting the estimates for I1 and I2, we get (3.2). ∎

Sketch of the proof of Proposition 2.4.

Since, by Lemma 3.1, Lε,ξ is invertible, the map

Tε,ξ:Kε,ξKε,ξ,Tε,ξ(ϕ):=Lε,ξ-1(Nε,ξ(ϕ)+Rε,ξ+Sε,ξ(ϕ))

is well defined. As

Tε,ξ(ϕ)εC(Nε,ξ(ϕ)ε+Sε,ξ(ϕ)ε+Rε,ξε)

and

Tε,ξ(ϕ1)-Tε,ξ(ϕ2)εCNε,ξ(ϕ1)-Nε,ξ(ϕ2)ε+CSε,ξ(ϕ1)-Sε,ξ(ϕ2)ε,

we deduce from Lemmas 3.2 and 3.3 that Tε,ξ is a contraction in the ball centered at 0 with radius Cε in Kε,ξ for a suitable constant C. Then Tε,ξ has a unique fixed point. The proof that the map ξϕε,ξ is a 𝒞1-map uses the implicit function theorem. This part of the proof is standard. ∎

4 The reduced energy

In this section, we obtain the expansion of the functional I~ε(ξ) with respect to ε. Recall the notation introduced in Section 2.2.

Lemma 4.1.

The expression

I~ε(ξ)=Iε(Wε,ξ+ϕε,ξ)=Iε(Wε,ξ)+o(1)=Jε(Wε,ξ)+ω22Gε(Wε,ξ)+o(1)

holds true C0-uniformly with respect to ξ as ε goes to zero. Moreover, setting ξ(z¯):=expξ(z¯) for z¯BRn-1(0), we have that

(z¯hI~ε(ξ(z¯)))|z¯=0=(z¯hIε(Wε,ξ(z¯)+ϕε,ξ(z¯)))|z¯=0=(z¯hIε(Wε,ξ(z¯)))|z¯=0+o(1)=(z¯hJε(Wε,ξ(z¯)))|z¯=0+ω22(z¯hGε(Wε,ξ(z¯)))|z¯=0+o(1)

𝒞0 -uniformly with respect to ξ as ε goes to zero, for every h=1,,n-1.

Proof.

As in [7, Lemma 5.1], we obtain the estimates

Jε(Wε,ξ(z¯)+ϕε,ξ(z¯))-Jε(Wε,ξ(z¯))=o(1),(Jε(Wε,ξ(z¯)+ϕε,ξ(z¯))-Jε(Wε,ξ(z¯)))[(z¯hWε,ξ(z¯))|z¯=0]=o(1).

To complete the proof we need the following estimates:

Gε(Wε,ξ+ϕε,ξ)-Gε(Wε,ξ)=o(1),(4.1)[Gε(Wε,ξ+ϕε,ξ)-Gε(Wε,ξ)][(z¯hWε,ξ(z¯))|z¯=0]=o(1),(4.2)(Jε(Wε,ξ(z¯)+ϕε,ξ(z¯))+ω22Gε(Wε,ξ(z¯)+ϕε,ξ(z¯)))[z¯hϕε,ξ(z¯)]=o(1).(4.3)

The proof of (4.1), (4.2) and (4.3) is technical and it is postponed to the appendix. With these estimates, one can prove the claim following the argument of [7, Lemma 5.1]. ∎

Lemma 4.2.

The estimate

Jε(Wε,ξ)=(12-1p)c(ξ)n2d(ξ)pp-2-n2b(ξ)2p-2+nUp𝑑z+O(ε)

holds true C1-uniformly with respect to ξM.

Proof.

For yD+, setting c~(y):=c(x), d~(y):=d(x) and b~(y):=b(x) with x:=ψξ(y)Qξ, we have

Jε(Wε,ξ)=ε22εnD+c~(y)i,j=1ngij(y)(Vεξ(y)χ(y))yi(Vεξ(y)χ(y))yj|g(y)|12dy+12εnD+d~(y)(Vεξ(y)χ(y))2|g(y)|12𝑑y-1pεnD+b~(y)(Vεξ(y)χ(y))p|g(y)|12𝑑y.

Using the change of variables y=εζ, from the expansions (2.5), (2.6) and (2.7) we immediately obtain

Jε(Wε,ξ)=12+nc(ξ)|Vξ(ζ)|2+d(ξ)(Vξ(ζ))2dζ-1p+nb(ξ)(Vξ(ζ))p𝑑ζ+O(ε).

From the definitions of Vξ and U we get

Jε(Wε,ξ)=c(ξ)n2d(ξ)pp-2-n2b(ξ)2p-2[12+n(|U|2+U2)-1p+nUp𝑑ζ]+O(ε)=(12-1p)c(ξ)n2d(ξ)pp-2-n2b(ξ)2p-2+nUp𝑑ζ+O(ε)

𝒞0-uniformly with respect to ξM. For the sake of readability, the 𝒞1-convergence is postponed to the appendix, where a proof is given in full detail. ∎

Lemma 4.3.

The expression

Iε(Wε,ξ)=(12-1p)c(ξ)n2d(ξ)pp-2-n2b(ξ)2p-2+nUp𝑑z+o(1)

holds true C1-uniformly with respect to ξM.

Proof.

In Lemma 4.2 we proved that

Jε(Wε,ξ)=(12-1p)c(ξ)n2d(ξ)pp-2-n2b(ξ)2p-2nUp𝑑z+O(ε).

It is enough to show now that Gε(Wε,ξ)=o(1) holds true 𝒞1-uniformly with respect to ξM. For the 𝒞0-convergence, by Remark 2.3 and since Φ(Wε,ξ)εCε, we have that

|Gε(Wε,ξ)|Cεn|MΦ(Wε,ξ)Wε,ξ2𝑑μg|Cεn|Φ(Wε,ξ)|2,g|Wε,ξ|4,g2C|Φ(Wε,ξ)|ε,2|Wε,ξ|ε,42CΦ(Wε,ξ)εCε.

For the 𝒞1-convergence, we estimate

|z¯hGε(Wε,ξ(z¯))|z¯=0||Cεnz¯hMΦ(Wε,ξ(z¯))Wε,ξ(z¯)2|z¯=0dμg||CεnMΦ(Wε,ξ(z¯))2Wε,ξ(h)(z¯hWε,ξ(z¯))|z¯=0dμg|+|CεnMWε,ξ(h)2Φ(Wε,ξ(z¯))[z¯hWε,ξ(z¯)|z¯=0]𝑑μg|=:I1+I2.

Now, by Remark 2.3 and since

z¯hWε,ξ(z¯)|z¯=0ε=O(1ε),

we have

I1Cεn|Φ(Wε,ξ(z¯))|3,g|Wε,ξ(z¯)|3,g|z¯hWε,ξ(z¯)|z¯=0|3,gCε23nεnΦ(Wε,ξ(z¯))Hg1|Wε,ξ(z¯)|ε,3z¯hWε,ξ(z¯)|z¯=0εCε-n3-1Φ(Wε,ξ(z¯))Hg1.

From Lemma 5.1, choosing 53<β<2 if n=2, we get I1Cεβ-2/3-1=o(1), and for n=3,4 we get

I1Cεn+22-n3-1=εn6=o(1).

Using Remark 2.3 and choosing 2nn+2<t<2, we obtain

I2Cεn|Wε,ξ(z¯)|2t,g2|Φ(Wε,ξ(z¯))[z¯hWε,ξ(z¯)|z¯=0]|t,gCεntεn|Wε,ξ(z¯)|ε,2t2Φ(Wε,ξ(z¯))[z¯hWε,ξ(z¯)|z¯=0]Hg1Cεnt-nΦ(Wε,ξ(z¯))[z¯hWε,ξ(z¯)|z¯=0]Hg1.

Finally, using Lemma 5.2 and noting that uHg1Cε(n-2)/2uε, for n=2 and 3-2t<β<2 we have

I2Cε2t-2εβz¯hWε,ξ(z¯)|z¯=0εCεβ+2t-3=o(1),

while for n=3,4 we get

I2Cεnt-nε2εn-22z¯hWε,ξ(z¯)|z¯=0εCεnt-n2=o(1)

since t<2. ∎

5 Appendix

We collect a series of technical results that were used previously.

5.1 Key estimates for the function Φ

Lemma 5.1.

For ε>0, ξM and φHg1(M), we have the following estimates:

For n=2 and 1<β<2, we have

Φ(Wε,ξ+φ)Hg1C1(εβ+φHg12),(5.1)Φ(Wε,ξ+φ)Hg1C1εβ(1+φε2),(5.2)

and for n=3,4 we have

Φ(Wε,ξ+φ)Hg1C1(εn+22+φHg12),(5.3)Φ(Wε,ξ+φ)Hg1C1εn+22(1+φε2),(5.4)

where the constant C1 does not depend on ε, ξ and φ.

Proof.

To simplify the notation we set v:=Φ(Wε,ξ+φ). By (2.1) or (2.2) we have

vHg12CMc(x)|gv|2+b(x)q2(Wε,ξ+φ)2v2=CqMb(x)(Wε,ξ+φ)2vC(Mvt)1t(M(Wε,ξ+φ)2t)1tCvHg1Wε,ξ+φLg2t2CvHg1(Wε,ξLg2t2+φLg2t2),

where t=2n for n=3,4 and t2 for n=2. We recall (see Remark 2.3) that

limε01εn|Wε,ξ|qqC|U|qquniformly with respect to ξM.

Thus, we have

vHg1C1(εnt+|φ|2t,g2)C1(εnt+φHg12).(5.5)

Notice that for n=2, since t2, we have that 12t<2, while for n=3,4 we have t=2nn+2, which proves (5.1) and (5.3). In the light of (5.5), we also obtain that

vHg1C1(εnt+|φ|2t,g2)C1εnt(1+|φ|2t,ε2)C1εnt(1+φε2),

which proves the other two inequalities (5.2) and (5.4). ∎

Lemma 5.2.

For ε>0, ξM and h,kHg1(M), we have the following estimates:

For n=2 and β(0,2), we have

Φ(Wε,ξ+k)[h]Hg1ChHg1(εβ+kHg1),

and for n=3,4 we have

Φ(Wε,ξ+k)[h]Hg1ChHg1(ε2+kHg1),

where the constant C does not depend on ε, ξ, h and k.

Proof.

From Lemma 2.1 we obtain

Φ(Wε,ξ+k)[h]Hg12=2qMb(x)(Wε,ξ+k)(1-qΦ(Wε,ξ+k))hΦ(Wε,ξ+k)[h]-q2Mb(x)(Wε,ξ+k)2(Φ(Wε,ξ+k)[h])2CMWε,ξ|h||Φ(Wε,ξ+k)[h]|+M|k||h||Φ(Wε,ξ+k)[h]|.

We call the last two integrals I1 and I2, respectively, and we estimate each of them separately. We have, by Remark 2.3, that

I2kLg3hLg3Φ(Wε,ξ+k)[h]Lg3kHg1hHg1ΦHg1

and

I1Φ(Wε,ξ+k)[h]LgthLgtWε,ξLgtt-2εnt-2tΦHg1hHg1,

where t=2n for n=3,4 and t>2 for n=2. ∎

5.2 Change of coordinates along M

For ξM, we consider the chart ψξ:D+Qξ, introduced in Section 2.2, whose inverse (ψξ)-1(x)=y expresses a point xQξM in Fermi coordinates y=(y1,,yn) around ξ.

For z¯BRn-1(0) and xQξQexpξ(z¯), we consider the change of coordinates map

(z¯,x)=(ψexpξ(z¯))-1(x)=(ψexpξ(z¯))-1ψξ(y)=~(z¯,y).(5.6)

Since yn=distg(x,M), writing y=(y¯,yn) with y¯n-1 and yn[0,), we have that

~(z¯,y¯,yn)=(expexpξ(z¯)-1expξ(y¯),yn).(5.7)

Lemma 5.3.

The derivatives of E at (0,ξ) are given by

kyh(0,ξ)=~kyh(0,0)=-δhkfor h=1,,n-1,k=1,,n,2kηjyh(0,ξ)=2~kηjyh(0,0)=0for h=1,,n-1,j,k=1,,n.

Proof.

This follows from [26, Lemma 6.4] by using the expression (5.7). ∎

For z¯BRn-1(0), we set ξ(z¯):=expξ(z¯)M. The function Wε,ξ(z¯), defined in (2.9), can now be written as

Wε,ξ(z¯)(x)=γ(ξ(z¯))Uε(A(ξ(z¯))(ψξ(z¯))-1(x))χ((ψξ(z¯))-1(x))=γ~(z¯)Uε(A~(z¯)(z¯,x))χ((z¯,x)),

where A~(z¯):=A(expξ(z¯)) and γ~(z¯):=γ(expξ(z¯)). Thus, we have

z¯sWε,ξ(z¯)|z¯=0=(z¯sγ~(z¯)|z¯=0)U(1εA~(0)(0,x))χ((0,x))+γ~(0)U(1εA~(0)(0,x))z¯sχ((z¯,x))|z¯=0+γ~(0)χ((0,x))z¯sU(1εA~(z¯)(z¯,x))|z¯=0.

If x:=ψξ(εy), ξ:=ξ(0), then (0,x)=εy, and we have

z¯sWε,ξ(z¯)|z¯=0=(z¯sγ~(z¯)|z¯=0)U(A~(0)y)χ(εy)+γ~(0)U(A~(0)y)χηk(εy)z¯sk(z¯,ψξ0(εy))|z¯=0+γ~(0)χ(εy)A~(0)εUηk(A~(0)y)z¯sk(z¯,ψξ0(εy))|z¯=0+γ~(0)χ(εy)Uηk(A~(0)y)z¯sA~(z¯)|z¯=0yk,(5.8)

where fηk() denotes the derivative of the function f with respect to its k-th variable.

5.3 The pending proofs in Section 4

Conclusion of the proof of Lemma 4.1.

To finish the proof of this lemma we need to prove (4.1), (4.2) and (4.3).

Proof of (4.1). For some θ[0,1], we have

Gε(Wε,ξ+ϕε,ξ)-Gε(Wε,ξ)=1εnMb(x)[Φ(Wε,ξ+ϕε,ξ)(Wε,ξ+ϕε,ξ)2-Φ(Wε,ξ)(Wε,ξ)2]=1εnMb(x)Φ(Wε,ξ+θϕε,ξ)[ϕε,ξ](Wε,ξ)2+1εnMb(x)Φ(Wε,ξ+ϕε,ξ)[2ϕε,ξWε,ξ+ϕε,ξ2].

Since ϕε,ξεCε and 0<Φ(u)<1q, from Remark 2.3 we obtain

|Gε(Wε,ξ+ϕε,ξ)-Gε(Wε,ξ)|Cεn|Φ(Wε,ξ+θϕε,ξ)[ϕε,ξ]|2,g|Φε,ξ|4,g2+Cεn|ϕε,ξ|2,g(|Wε,ξ|2,g+|ϕε,ξ|2,g)Cεn/2εnΦ(Wε,ξ+θϕε,ξ)[ϕε,ξ]Hg1Φε,ξε2+CΦε,ξε(Wε,ξε+Φε,ξε)ε2εn/2Φ(Wε,ξ+θϕε,ξ)[ϕε,ξ]Hg1+Cε(1+ε).

Using Lemma 5.2, we conclude that

|Gε(Wε,ξ+ϕε,ξ)-Gε(Wε,ξ)|C(ε3-n2+ε)Cε.

Proof of (4.2). Recall that ξ(z¯):=expξ(z¯) for z¯BRn-1(0). Since 0<Φ(u)<1q, for some θ[0,1] we have

|[Gε(Wε,ξ+ϕε,ξ)-Gε(Wε,ξ)][(z¯hWε,ξ(z¯))|z¯=0]||CεnM[Φ(Wε,ξ+ϕε,ξ)-Φ(Wε,ξ)]Wε,ξ(z¯hWε,ξ(z¯))|z¯=0|   +|CεnM[qΦ2(Wε,ξ+ϕε,ξ)-qΦ2(Wε,ξ)]Wε,ξ(z¯hWε,ξ(z¯))|z¯=0|   +|CεnM[Φ(Wε,ξ+ϕε,ξ)-qΦ2(Wε,ξ+ϕε,ξ)]ϕε,ξ(z¯hWε,ξ(z¯))|z¯=0||CεnM[Φ(Wε,ξ+ϕε,ξ)-Φ(Wε,ξ)]Wε,ξ(z¯hWε,ξ(z¯))|z¯=0|   +|CεnMΦ(Wε,ξ+ϕε,ξ)ϕε,ξ(z¯hWε,ξ(z¯))|z¯=0||CεnMΦ(Wε,ξ+θϕε,ξ)(ϕε,ξ)Wε,ξ(z¯hWε,ξ(z¯))|z¯=0|   +|CεnMΦ(Wε,ξ+θϕε,ξ)(ϕε,ξ)ϕε,ξ(z¯hWε,ξ(z¯))|z¯=0|   +|CεnMΦ(Wε,ξ)ϕε,ξ(z¯hWε,ξ(z¯))|z¯=0|=:I1+I2+I3.

From (5.8) and a straightforward computation we derive that

|(z¯hWε,ξ(z¯))|z¯=0|ε,3(n[k=1n|1εUyk(y)|]3dy)13=O(1ε).

Now, recalling that ϕε,ξ(z¯)εCε and that uHg1Cε(n-2)/2uε, from Remark 2.3 and Lemma 5.2 we get that

I1Cε2n3εn(M|Φ(Wε,ξ+ϕε,ξ)(ϕε,ξ)|3)13(1εnMWε,ξ3)13(1εnM|(z¯hWε,ξ(z¯))|z¯=0|3)13Cε-n3-1Φ(Wε,ξ+ϕε,ξ)(ϕε,ξ)Hg1Cε-n3-1ϕε,ξHg12Cε-n3-1εn=o(1).

The term I2 can be estimated in the same way, while for I3 we have

I3Cε2n3εn(M|Φ(Wε,ξ)|3)13(1εnM|ϕ|ε,ξ3)13(1εnM|(z¯hWε,ξ(z¯))|z¯=0|3)13Cε-n3Φ(Wε,ξ)H1.

Now, if n=2, by (5.1) we have I3Cε=β-n/3o(1), choosing β wisely. If n=3,4, by (5.2) we get

I3Cεn+22-n3=Cεn6+1.

This proves (4.2).

Proof of (4.3). Following the proof of [7, Lemma 5.1, step 2], we just have to prove that

|Gε(Wε,ξ(z¯)+ϕε,ξ(z¯))[Zε,ξ(z¯)l]|=o(1),

Since 0<Φ(u)<1q,

|Gε(Wε,ξ(z¯)+ϕε,ξ(z¯))[Zε,ξ(z¯)l]||CεnMΦ(Wε,ξ(z¯)+ϕε,ξ(z¯))(Wε,ξ(z¯)+ϕε,ξ(z¯))Zε,ξ(z¯)l|+|1εnMΦ2(Wε,ξ(z¯)+ϕε,ξ(z¯))(Wε,ξ(z¯)+ϕε,ξ(z¯))Zε,ξ(z¯)l||CεnMΦ(Wε,ξ(z¯)+ϕε,ξ(z¯))(Wε,ξ(z¯)+ϕε,ξ(z¯))Zε,ξ(z¯)l|=:I4.

By (2.10) it can be proved easily that Zε,ξ(z¯)lε=O(1). So we have

I4ε-n3(M|Φ(Wε,ξ+ϕε,ξ)|3)13(1εnM|Wε,ξ+ϕε,ξ|3)13(1εnM|Zε,ξ(z¯)l|3)13ε-n3Φ(Wε,ξ+ϕε,ξ)Hg1(Wε,ξ3,ε+ϕε,ξε)Zε,ξ(z¯)lεε-n3Φ(Wε,ξ+ϕε,ξ)Hg1.

Now, if n=2, by (5.2) we have that

I3cεβ-n/3=o(1),

and if n=3,4 by (5.2) we have that

I3cεn+22-n3=cεn6+1.

Conclusion of the proof of Lemma 4.2.

To finish the proof of this lemma we need to prove the 𝒞1-convergence. We do this for the first partial derivative. We set ξ(z¯):=expξ(z¯) for z¯BRn-1(0). Then we have

z¯1Jε(Wε,ξ(z¯))|z¯=0=Jε(Wε,ξ(z¯))[z¯1Wε,ξ(z¯)]|z¯=0=ε2εnMc(x)gWε,ξ(z¯)gz¯1Wε,ξ(z¯)dμg|z¯=0+1εnMd(x)Wε,ξ(z¯)z¯1Wε,ξ(z¯)𝑑μg|z¯=0+1εnMb(x)Wε,ξ(z¯)p-1z¯1Wε,ξ(z¯)𝑑μg|z¯=0=:I1+I2+I3.

Next, we estimate each term. Set x:=ψξ(y) and c~(y):=c(ψξ(y))=c(x). By (5.8), we have

I1=ε2εnMc(x)gWε,ξ(z¯)g(z¯1Wε,ξ(z¯))dμg|z¯=0=ε2εn+nc~(y)|gξ(y)|12gξij(y)yi[γ~(0)Uε(A~(0)~(0,y))χ(~(0,y))]×yjz¯1[γ~(z¯)Uε(A~(z¯)~(z¯,y))χ(~(z¯,y))]|z¯=0dy=nc~(εζ)|gξ(εζ)|12gξij(εζ)ζi[γ~(0)Uε(A~(0)~(0,εζ))χ(~(0,εζ))]×ζjz¯1[γ~(z¯)Uε(A~(z¯)~(z¯,εζ))χ(~(z¯,εζ))]|z¯=0dζ.

Using the definition (5.6) of ~, we obtain

I1=+nc~(εζ)γ~(0)|gξ(εζ)|12gξij(εζ)[(ζiU(A~(0)ζ))χ(εζ)+U(A~(0)ζ)ζiχ(εζ)]×ζjz¯1[γ~(z¯)Uε(A~(z¯)~(z¯,εζ))χ(~(z¯,εζ))]|z¯=0dζ+O(ε)=+nc~(εζ)γ~(0)|gξ(εζ)|12gξij(εζ)[(ζiU(A~(0)ζ))χ(εζ)+U(A~(0)ζ)ζiχ(εζ)]×ζj[z¯1γ~(z¯)|z¯=0U(A~(0)ζ)χ(εζ)+γ~(0)U(A~(0)ζ)χζk(εζ)z¯1k(z¯,ψξ(εζ))|z¯=0   +γ~(0)χ(εζ)A~(0)εUζk(A~(0)ζ)z¯1k(z¯,ψξ(εζ))|z¯=0   +γ~(0)χ(εζ)Uζk(A~(0)ζ)z¯1A~(z¯)|z¯=0ζk]dζ+O(ε)=:D1+D2+D3+D4+O(ε),

where fζk() denotes the derivative of the function f with respect to its k-th variable. Expanding c~(εζ), by the exponential decay of U and its derivative, and by (2.5), (2.6) and (2.7), we get

D1=γ~(0)+n(c~(0)δij+O(ε|ζ|))[ζi(U(A~(0)ζ))χ(εζ)+O(ε|ζ|)]×z¯1γ~(z¯)|z¯=0[ζj(U(A~(0)ζ))χ(εζ)+O(ε2|ζ|2)]dζ=12z¯1(γ~(z¯))2|z¯=0c~(0)n|ζ(U(A~(0)ζ))|2𝑑ζ+O(ε).

Similarly, D2=O(ε). Also, we have

D4=γ~2(0)+n(c~(0)δij+O(ε|ζ|))[ζi(U(A~(0)ζ))χ(εζ)+O(ε2|ζ|2)]×ζj[χ(εζ)Uζk(A~(0)ζ)z¯1A~(z¯)|z¯=0ζk]dζ=c~(0)γ~2(0)z¯1A~(z¯)|z¯=0+nζi(U(A~(0)ζ))ζi[Uζk(A~(0)ζ)ζk]𝑑ζ+O(ε).

Now, an elementary computation yields

12z¯1|ζU(A~(z¯)ζ)|2=ζi(U(A~(z¯)ζ))ζiz¯1(U(A~(z¯)ζ))=ζi(U(A~(z¯)ζ))ζi[Uξk(A~(z¯)ζ)ζkz¯1A~(z¯)].

Hence,

D4=12c~(0)γ~2(0)+nz¯1(|ζU(A~(z¯)ζ)|2)|z¯=0dζ+O(ε).

We conclude that

D1+D4=12c~(0)+nz¯1(|γ~(z¯)ζU(A~(z¯)ζ)|2)|z¯=0dζ+O(ε)=12c(ξ)+nz¯1(|ζVξ(z¯)(ζ)|2)|z¯=0dζ+O(ε).

The term D3 is more delicate since the factor 1ε forces us to expand all factors up to the second order. In the light of (2.5), (2.6) and (2.7), with the convention that the matrix (hij)i,j=1,,n coincides with the second fundamental form when i,j=1,,n-1 and hi,n=hn,j=0 for i,j=1,,n, we obtain

D3=γ~2(0)A~(0)+n1εc~(εζ)|gξ(εζ)|12gξij(εζ)[ζi(U(A~(0)ζ))χ(εζ)+O(ε2|ζ|2)]×ζj[χ(εζ)Uξk(A~(0)ζ)z¯1k(z¯,ψξ(εζ))|z¯=0]dζ=γ~2(0)A~(0)+n[δijc~(0)ε+2c~(0)hijζn-c~(0)(n-1)δijHζn+δijc~ζl(0)ζl]×ζi(U(A~(0)ζ))ζj[Uξk(A~(0)ζ)z¯1k(z¯,ψξ(εζ))|z¯=0]dζ+O(ε).

By Lemma 5.3, we have

z¯1k(z¯,ψξ(εζ))|z¯=0=-δ1k+O(ε2|ζ|2),ζj(z¯1k(z¯,ψξ(εζ))|z¯=0)=O(ε2|ζ|2).

Moreover, since U is radial,

Uζi(A~(0)ζ)=U(A~(0)ζ)ζi|ζ|,(5.9)ζ1(U(A~(0)ζ)|ζ|)=(A~(0)U′′(ζ)|ζ|2-U(ζ)|ζ|3)ζ1,(5.10)

where U=Ur, U′′=2Ur2 and r=|ζ|. Thus, we get

D3=-γ~2(0)A~(0)+n[δijc~(0)ε+2c~(0)hijζn-c~(0)(n-1)δijHζn+δijc~ξl(0)ζl]×U(A~(0)ζ)|ζ|ζiζj(U(A~(0)ζ)|ζ|ζk)δ1kdζ+O(ε)=-γ~2(0)A~(0)+n[δijc~(0)ε+2c~(0)hijζn-c~(0)(n-1)δijHζn+δijc~ξl(0)ζl]×(U(A~(0)ζ)|ζ|)2ζiδj1+U(A~(0)ζ)|ζ|(A~(0)U′′(ζ)|ζ|2-U(ζ)|ζ|3)ζjζiζ1dζ+O(ε).

Now, by symmetry considerations, for i=1,,n-1, any term containing ζi to an odd power vanishes, and, since hin=hnj=0, we get that

D3=-γ~2(0)A~(0)c~ξl(0)+nδij(U(A~(0)ζ)|ζ|)2ζiζlδj1+δijU(A~(0)ζ)|ζ|(A~(0)U′′(ζ)|ζ|2-U(ζ)|ζ|3)ζjζiζ1ζldζ+O(ε)=-γ~2(0)A~(0)c~ξ1(0)+n(U(A~(0)ζ)|ζ|)2ζ12+U(A~(0)ζ)|ζ|(A~(0)U′′(ζ)|ζ|2-U(ζ)|ζ|3)ζiζiζ12dζ+O(ε)

Notice that, by (5.9) and (5.10), we have

12ζ1|ζU(A~(0)ζ)|2=A~(0)(U(A~(0)ζ)|ζ|)2ζ1+A~(0)U(A~(0)ζ)|ζ|(A~(0)U′′(ζ)|ζ|2-U(ζ)|ζ|3)ζiζiζ1,

so

D3=-γ~2(0)c~ξ1(0)12+nζ1|ζU(A~(0)ζ)|2ζ1𝑑ζ=γ~2(0)c~ξ1(0)12+nζ1|ζU(A~(0)ζ)|2𝑑ζ=c~ξ1(0)12+n|Vξ(ζ)|2𝑑ζ=z¯1c(ξ(z¯))|z¯=012+n|Vξ(ζ)|2𝑑ζ+O(ε).

Consequently, we obtain

I1=12+nz¯1[c(ξ(z¯))|Vξ(z¯)(ζ)|2]|z¯=0dζ+O(ε).

For the second term, setting d~(y)=d(ψξ(y))=d(x), we obtain in an analogous way

I2=+nd~(εζ)|gξ(εζ)|12γ~(0)U(A~(0)ζ)χ(εζ)z¯1[γ~(z¯)Uε(A~(z¯)~(z¯,εζ))χ(~(z¯,εζ))]|z¯=0dζ=nd~(εζ)|gξ(εζ)|12γ~(0)U(A~(0)ζ)χ(εζ)×[z¯1γ~(z¯)|z¯=0U(A~(0)ζ)χ(εζ)+γ~(0)χ(εζ)A~(0)εUξk(A~(0)ζ)z¯1k(z¯,ψξ(εζ))|z¯=0   +γ~(0)χ(εζ)Uξk(A~(0)ζ)z¯1A~(z¯)|z¯=0ζk+γ~(0)U(A~(0)ζ)z¯1[χ(~(z¯,εζ))]|z¯=0]dζ=:B1+B2+B3+B4.

Expanding d~(εζ), by the exponential decay of U and its derivative, and by (2.7) and the definition of ~, we get

B1=+nd~(0)γ~(0)z¯1γ~(z¯)|z¯=0U2(A~(0)ζ)dζ+O(ε)=12d~(0)z¯1γ~2(z¯)|z¯=0nU2(A~(0)ζ)𝑑ζ.

As before, we obtain that B4=O(ε) and

B3=+nd~(0)γ~2(0)U(A~(0)ζ)Uξk(A~(0)ζ)z¯1A~(z¯)|z¯=0ζkdζ+O(ε)=12+nd~(0)γ~2(0)z¯1(U(A~(z¯)ζ))2|z¯=0dζ+O(ε).

Thus,

B1+B3=12d~(0)+nz¯1(γ~2(z¯)U(A~(z¯)ζ))2|z¯=0dζ+O(ε)=12d(ξ)+nz¯1(Vξ(z¯)(ζ))2|z¯=0dζ+O(ε).

Again, we have to pay particular attention to the term containing 1ε as a factor. From (2.7) and Lemma 5.3 we get

B2=γ~2(0)A~(0)+nd~(εζ)ε|gξ(εζ)|12U(A~(0)ζ)Uξk(A~(0)ζ)(-δ1k+O(ε2|ζ|2))𝑑ζ+O(ε)=-γ~2(0)A~(0)+n(d~(0)ε+d~ξl(0)ζl)U(A~(0)ζ)Uξ1(A~(0)ζ)𝑑ζ+O(ε)

and, by (5.9),

B2=-γ~2(0)A~(0)+n(d~(0)ε+d~ξl(0)ζl)U(A~(0)ζ)U(A~(0)ζ)|ζ|ζ1𝑑ζ+O(ε)=-γ~2(0)A~(0)+nd~ξ1(0)U(A~(0)ζ)U(A~(0)ζ)|ζ|ζ12𝑑ζ+O(ε)

due to the symmetry. So,

B2=-γ~2(0)d~ξ1(0)+n12ζ1[U(A~(0)ζ)]2ζ1𝑑ζ+O(ε)=γ~2(0)d~ξ1(0)+nU2(A~(0)ζ)𝑑ζ+O(ε)=12z¯1d(ξ)|z¯=0+n(Vξ(z¯)(ζ))2𝑑ζ

and

I2=12+nz¯1{d(ξ(z¯))(Vξ(z¯)(ζ))2}|z¯=0dζ+O(ε).

In a similar way we proceed for I3, completing the proof. ∎

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About the article

Received: 2017-02-24

Accepted: 2017-04-25

Published Online: 2017-07-21


The first author is supported by CONACYT grant 237661 (México) and PAPIIT-DGAPA-UNAM grant IN104315 (Mexico). The second and third authors are partially supported by the GNAMPA project by INDAM. The second author is partially supported by the PRA project of the university of Pisa.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 559–582, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0039.

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