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Advances in Nonlinear Analysis

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


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Clustered solutions for supercritical elliptic equations on Riemannian manifolds

Wenjing Chen
Published Online: 2018-06-13 | DOI: https://doi.org/10.1515/anona-2017-0277

Abstract

Let (M,g) be a smooth compact Riemannian manifold of dimension n5. We are concerned with the following elliptic problem:

-Δgu+a(x)u=un+2n-2+ε,u>0 in M,

where Δg=divg() is the Laplace–Beltrami operator on M, a(x) is a C2 function on M such that the operator -Δg+a is coercive, and ε>0 is a small real parameter. Using the Lyapunov–Schmidt reduction procedure, we obtain that the problem under consideration has a k-peaks solution for positive integer k2, which blow up and concentrate at one point in M.

Keywords: Clustered solutions; supercritical elliptic equation on manifolds,Lyapunov–Schmidt reduction procedure

MSC 2010: 58G03; 58E30

1 Introduction

Let (M,g) be a smooth compact Riemannian manifold of dimension n5, where g denotes the metric tensor. We are interested in the following supercritical elliptic problem:

-Δgu+a(x)u=un+2n-2+ε,u>0 in M,(1.1)

where Δg=divg() is the Laplace–Beltrami operator on M, a(x) is a C2 function on M, and ε>0 is a real parameter with ε0.

There are many results about the existence and properties of solutions for nonlinear elliptic equations on compact Riemannian manifolds. Let us mention the following problem:

-ε2Δgu+u=|u|p-2uin M,(1.2)

where (M,g) is a compact, connected, Riemannian manifold of class C with Riemannian metric g, with dimension n3, 2<p<2nn-2 and ε is a positive parameter. The existence and multiplicity of solutions to problem (1.2) was considered in [4, 2, 25]. Moreover, the existence of peak solutions for (1.2) was obtained by Dancer, Micheletti and Pistoia [6, 15, 14].

The asymptotically critical case on a Riemannian manifold was studied by Micheletti, Pistoia and Vétois in [16]. They proved that problem (1.1) has blowing-up families of positive solutions, provided the graph of a(x) is distinct at some point from the graph of n-24(n-1)Scalg. Moreover, the existence of multi-peak solutions that are separate from each other for (1.1) was considered by Deng in [7]. Pistoia and Vétois [18] discovered the existence of sign-changing bubble towers for (1.1).

In the case an-24(n-1)Scalg, equation (1.1) is intensively studied as the Yamabe equation whose positive solutions u are such that the scalar curvature of the conformal metric u2-2g is constant (see [1, 21, 24]).

It is important to recall some results for the following linear perturbation of the Yamabe problem:

-Δgu+(n-24(n-1)Scalg+ε)u=un+2n-2in (M,g),(1.3)

where (M,g) is a non-locally conformally flat compact Riemannian manifold. Druet in [8] proved that problem (1.3) does not have any blowing-up solution when ε<0 and the dimension of the manifold is n=3,4,5 (except when the manifold is conformally equivalent to the round sphere). In case ε>0, if n=3, there is no blowing-up solutions to problem (1.3) as proved by Li and Zhu [13]. Esposito, Pistoia and Vétois in [9] showed that there exist blowing-up solutions for n6, and they built solutions which blow up at non-vanishing stable critical points ξ0 of the Weyl tensor, i.e., |Weylg(ξ0)|g0. Recently, Robert and Vétois [20], Pistoia and Vaira [17], Thizy and Vétois [23] provided several geometric and analytic settings in which linear perturbations to (1.3) have positive clustered bubbles that are non-isolated blowing-up solutions. In particular, Pistoia and Vaira in [17] investigated the existence of cluster solutions for problem (1.3). More precisely, they proved that for any point ξ0M, which is non-degenerate and a non-vanishing minimum point of the Weyl tensor, and for any integer k, there exists a family of solutions developing k peaks collapsing to ξ0 as ε goes to zero. Moreover, Thizy and Vétois [23] constructed clustering positive solutions for a perturbed critical elliptic equation on a closed manifold of dimension four and five.

Motivated by the previous consideration, in the present paper, we construct a family of cluster solutions for equation (1.1) with ε small enough.

Let Lq be the Banach space Lq(M) with the norm

|u|q=(M|u|qdμg)1q.

We assume that the operator -Δg+a is coercive, and the Sobolev space Hg1(M) is endowed with the scalar product ,a, defined by

u,va=M(u,vg+auv)𝑑μg

for all u,vHg1(M). We let a be the norm induced by ,a. This norm is equivalent to the standard norm on Hg1(M). Let uε be a family of solutions of (1.1). We say that uε blows up at k points which collapse to ξ0 as ε0 if there exists ξ1,ε,ξ2,ε,,ξk,εM and λ1,ε,λ2,ε,,λk,ε+ such that ξj,εξ0, λj,ε0, j=1,2,,k, and

uε(x)-j=1kλj,ε-n-22U(expξj,ε-1(x)λj,ε)a0as ε0,

where the function

U(z)=U(|z|)=αn(1+|z|2)-n-22,αn=(n(n-2))n-24,

is the solution of the limit equation

-ΔU=U2-1in n,(1.4)

It is known that [1, 22] the functions λ2-n2U(λ-1z) satisfy equation (1.4).

Our main result can be stated as follows.

Theorem 1.1.

Let (M,g) be a smooth compact Riemannian manifold of dimension n5, let a(x) be a C2 function on M such that the operator -Δg+a is coercive, and let ξ0 be a nondegenerate maximum point of the function

φ(ξ):=a(ξ)-n-14(n-2)Scalg(ξ),(1.5)

with φ(ξ0)<0. For any given integer k2, if ε>0 is small enough, then problem (1.1) has a solution uε blowing up at k points which collapse to ξ0 as ε0.

Remark 1.2.

We can get the same result for the subcritical case. Precisely, if (M,g) is a smooth compact Riemannian manifold of dimension n5, a(x) is a C2 function on M such that the operator -Δg+a is coercive, and ξ0 is a nondegenerate minimum point of the function φ(ξ) with φ(ξ0)>0, then for any ε<0 small enough, problem (1.1) has a clustered solution uε, which blows up at k points which collapse to ξ0 as ε0. We will not give the details of the proof in this case.

The proof of our result relies on a very well known finite dimensional Lyapunov–Schmidt reduction procedure, introduced in [10, 19] and used in many of the quoted papers. In particular, we refer to [6, 15, 14] for nonlinear elliptic problems on Riemannian manifolds, [7, 16] for asymptotically critical elliptic equations on Riemannian manifolds, [9, 17] for linear perturbations on the Yamabe problems, and recently this method has been used to study the fractional Yamabe problem by Choi and Kim in [5], and Kim, Musso and Wei in [12].

This paper is organized as follows. In Section 2, we introduce the framework and present some preliminary results. The proof of the main result is given in Section 3. Section 4 contains the asymptotic expansion of the energy functional.

2 The framework and preliminary results

Let r be a positive real number less than iM, where iM is the injectivity radius of M, and χr be a smooth cut-off function such that 0χr1 in n, χr(z)=1 if zB(0,r2), χr(z)=0 if znB(0,r), and |χr(z)|2r, |2χr(z)|2r2, where B(0,r) denotes the ball in TξM centered at 0 with radius r. For any point ξ in M and any positive real number λ, we define the function Wλ,ξ on M by

Wλ,ξ(x):={χr(expξ-1(x))λ2-n2U(λ-1expξ-1(x))if xBg(ξ,r),0otherwise.(2.1)

It is clear that the embedding i:Hg1(M)L2nn-2(M) is a continuous map. Let i:L2nn+2(M)Hg1(M) be the adjoint operator of the embedding i. Then the embedding i is a continuous map such that for any wL2nn+2(M), the function u=i(w) in Hg1(M) is the unique solution of the equation -Δgu+au=w in M. By the continuity of the embedding Hg1(M) into L2nn-2(M), we have

i(w)aC|w|2nn+2

for some positive constant C independent of w.

By standard elliptic estimates [11], given a real number s>2nn-2, that is, nsn+2s>2nn+2, for any wLnsn+2s(M), the function i(w) belongs to Ls(M) and satisfies

|i(w)|sC|w|nsn+2s(2.2)

for some positive constant C independent of w. For ε small, we set

sε:=2nn-2+n2ε,

and let ε=Hg1(M)Lsε(M) be the Banach space equipped with the norm

ua,sε=ua+|u|sε.

Taking into account that

nsεn+2sε=sε2-1+ε,

and by (2.2), we can write problem (1.1) as

u=i(fε(u)),uε,(2.3)

where fε(u)=u+n+2n-2+ε and u+=max{u,0}.

It is known, see [3, 19], that every solution of the linear equation

-Δv=n+2n-2U4n-2v,v𝒟01,2(n),

is a linear combination of the functions

V0(z)=d(λ2-n2U(λ-1z))dλ|λ=1=12αn(n-2)|z|2-1(1+|z|2)n2

and

Vi(z)=-Uzi(z)=αn(n-2)zi(1+|z|2)n2for i=1,2,,n.

Let us define on M the functions

Zλ,ξi(x):={χr(expξ-1(x))λ2-n2Vi(λ-1expξ-1(x))if xBg(ξ,r),0otherwise,

for i=0,1,2,,n.

Let ξ0 be a nondegenerate local maximum point of

φ(ξ)=a(ξ)-n-14(n-2)Scalg(ξ),

and ξ¯=(ξ1,ξ2,,ξk)Mk. Given η>0,R>0, we define the open set

𝒪η,Rk={(d¯,τ¯)=(d1,,dk,τ1,,τk)[η-1,η]k×(Bn(0,R)¯)k:|τi-τj|R-1 for all ij}(2.4)

and the parameters

λj=εαdj>0andξj=expξ0(εβτj)M(2.5)

for each j=1,,k, where expξ0:Bn(0,iM)M is the exponential map whose base point is ξ0 and

α=12andβ=n-42n.

Let

Kλ¯,ξ¯=Span{Zλj,ξji:i=0,1,2,,n,j=1,2,,k}

and

Kλ¯,ξ¯={ϕε:ϕ,Zλj,ξjia=0,i=0,1,2,,n,j=1,2,,k}.

3 Scheme of the proof of Theorem 1.1

We will look for a solution to (2.3), or equivalently to (1.1), of the form

uε=𝒲d¯,τ¯+ϕε,d¯,τ¯,with 𝒲d¯,τ¯=j=1kWλj,ξj,(3.1)

for (d¯,τ¯)𝒪η,Rk, where the rest term ϕε,d¯,τ¯ belongs to the space Kd¯,τ¯ and the functions Wλj,ξj are defined in (2.1).

Let Πd¯,τ¯:εKd¯,τ¯ and Πd¯,τ¯:εKd¯,τ¯ be the orthogonal projections. In order to solve problem (2.3), we will solve the system

Πd¯,τ¯{𝒲d¯,τ¯+ϕε,d¯,τ¯-i[fε(𝒲d¯,τ¯+ϕε,d¯,τ¯)]}=0,(3.2)Πd¯,τ¯{𝒲d¯,τ¯+ϕε,d¯,τ¯-i[fε(𝒲d¯,τ¯+ϕε,d¯,τ¯)]}=0.(3.3)

The first step in the proof consists in solving equation (3.2). This requires Proposition 3.1 below. We skip the proof of this result which is rather standard in the literature on Lyapunov–Schmidt reduction; we refer to [16] and [7].

Proposition 3.1.

For any (d¯,τ¯)Oη,Rk, there exists a positive constant C such that for ε small enough, equation (3.2) admits a unique solution ϕε,d¯,τ¯ in Kλ¯,ξ¯ satisfying

ϕε,d¯,τ¯a,sεCi(fε(𝒲d¯,τ¯))-𝒲d¯,τ¯a,sεCε|lnε|.(3.4)

Moreover, ϕε,d¯,τ¯ is continuously differentiable with respect to τ¯ and d¯.

We introduce the functional Jε:Hg1(M), defined by

Jε(u)=12M|gu|2dμg+12Ma(x)u2dμg-12+εMu+2+εdμg,

where 2*=2nn-2 denotes the Sobolev critical exponent. It is well known that any critical point of Jε is a solution to problem (1.1). We define the functional J~ε:(+)k×(n)k by

J~ε(d¯,τ¯)=Jε(𝒲d¯,τ¯+ϕε,d¯,τ¯),

where 𝒲d¯,τ¯ is as (3.1) and ϕε,d¯,τ¯ is given by Proposition 3.1.

The next result, whose proof is postponed to Section 4, allows us to solve equation (3.3), by reducing the problem to a finite dimensional one.

Proposition 3.2.

  • (i)

    For ε small, if (d¯,τ¯) is a critical point of the functional J~ε , then 𝒲d¯,τ¯+ϕε,d¯,τ¯ is a solution of ( 2.3 ), or equivalently of problem ( 1.1 ).

  • (ii)

    If n5 and (λ¯,τ¯)𝒪η,Rk satisfies ( 2.4 ), then

    J~ε(d¯,τ¯)=kc0+εk(c1+c2lnε)+j=1k[c3ln(dj)+c4dj2φ(ξ0)]ε+ε2(n-2)n[c42j=1kdj2D2φ(ξ0)[τj,τj]-c5j<ldln-22djn-22|τl-τj|n-2]+o(ε2(n-2)n)(3.5)

    as ε0, C0 -uniformly with respect to τ¯ in (n)k and to d¯ in compact subsets of (+)k , where ci , for i=0,1,,5 , are positive constants and φ is defined in ( 1.5 ).

Proof of Theorem 1.1.

From Proposition 3.2 (i), it follows that 𝒲d¯,τ¯+ϕε,d¯,τ¯, where 𝒲d¯,τ¯ is defined in (3.1) and ϕε,d¯,τ¯, whose existence is guaranteed by Proposition 3.1, is a solution of (1.1) if (d¯,τ¯) is a critical point of the functional J~ε, which is a consequence of finding a maximum point of

ε(d¯,τ¯)=j=1k[c3ln(dj)+c4dj2φ(ξ0)]+εn-4n[c42j=1kdj2D2φ(ξ0)[τj,τj]-c5j<ldln-22djn-22|τl-τj|n-2]+o(ε2(n-2)n)

in the interior of 𝒪η,Rk.

To the contrary, assume that the maximum of ε is achieved only on the boundary 𝒪η,Rk. There are three possibilities, namely,

  • (1)

    one of d1,,dk is equal to either η-1 or η,

  • (2)

    one of τ1,,τk is located on Bn(0,R),

  • (3)

    there exist 1ijk such that |τi-τj|=R-1.

We will exclude each of them, thereby showing the existence of the maximum point in the interior.

Suppose first that (1) occurs. Define the function

h(d):=c3lnd+c4d2φ(ξ0)

By the assumption φ(ξ0)<0, we have that h attains the maximum M0 at the point dmax=(-c32c4φ(ξ0))1/2. Let us take η>0 so large that max{h(η),h(η-1)}min{-1,2M0} and d0(η-1,η). Then we obtain

min{-1,2M0}+(k-1)M0+o(1)max(d¯,τ¯)𝒪η,Rkε(d¯,τ¯)max{ε(d0,,d0,τ¯):τ¯(Bn(0,R)¯)k,|τi-τj|R-1 for 1ijk}=kM0+o(1),

which gives that min{-1,2M0}M0+o(1). This in turn implies that 0M0<-1. Hence, a contradiction arises and (1) cannot occur.

We next prove that cases (2) and (3) never take place, provided that R>0 is sufficiently large. Because Dg2φ(ξ0) is negative definite, for any M1>0, one can choose R1 so large that

Dg2φ(ξ0)(σ,σ)-M1on {σn:|σ|=R}(3.6)

and

-1|τi-τj|n-2-M1for τi,τjn, with |τi-τj|=R-1.(3.7)

Let τ¯0=(τ01,,τ0k) be any point in (n)k which constitutes of vertices of a k-regular polygon whose center and circumradius are 0 and 1/sin(π/k), respectively. Then it is easy to see that

|τ01|==|τ0k|=1/sin(π/k)and|τ0i-τ0j|=2 for each 1ijk.

For each small ε>0, let also d¯ε=(dε1,,dεk)(η-1,η)k be an element attaining the maximum value of the map j=1kh(dj). Observe that dεid0 as ε0 for all i=1,,k. From (3.6) and (3.7), we get an upper bound of the maximum of ε as follows:

max(d¯,τ¯)𝒪η,Rkε(d¯,τ¯)j=1kh(djε)+εn-4nmax(d¯,τ¯)𝒪η,Rk[c42j=1kdj2D2φ(ξ0)[τj,τj]-c5j<ldln-22djn-22|τl-τj|n-2]j=1kh(djε)-εn-4nkM1[c42η-2+k-12c5η-(n-2)].

However, given M1>0 large enough, the above estimate does not make sense, since we can derive

max(d¯,τ¯)𝒪η,Rkε(d¯,τ¯)ε(d¯ε,τ¯0)j=1kh(djε)-εn-4nk[|Dg2φ(ξ0)|c42sin(π/k)2d02+k-12n-2+1c5d0n-2+o(1)]

at the same time.

Thus, (d¯,τ¯)𝒪η,Rk. Then problem (1.1) has a solution of the form uε=𝒲d¯,τ¯+ϕε,d¯,τ¯ for ε>0 sufficiently small, which we call a solution blowing-up at k points that collapse to ξ0 as ε0. By taking into account the definition of the approximate solution ϕε,d¯,τ¯ and (3.4), the proof is completed. ∎

4 Proof of Proposition 3.2: Expansion of the energy

This section is devoted to the proof of Proposition 3.2. As a first step, we need the following lemma.

Lemma 4.1.

For ε small, if (d¯,τ¯) is a critical point of the functional J~ε, then Wd¯,τ¯+ϕε,d¯,τ¯ is a solution of (2.3), or equivalently of problem (1.1).

Proof.

The proof is the same as that of [7, Lemma 5.1]. ∎

Lemma 4.2.

If n5 and (λ¯,τ¯)Oη,Rk satisfies (2.4), then

J~ε(d¯,τ¯)=Jε(𝒲d¯,τ¯)+o(ε2(n-2)n)

as ε0, C0-uniformly with respect to τ¯ in (Rn)k and to d¯ in compact subsets of (R+)k.

Proof.

It can be proved by the same argument used in the proof of [16, Lemma 4.2].∎

We next give the expansion of Jε(𝒲d¯,τ¯) for (d¯,τ¯)𝒪η,Rk and ε0. We denote Kn the sharp constant for the embedding of 𝒟1,2(n) into L2(n), that is,

Kn=4n(n-2)ωn1/n,

where ωn is the volume of the unit n-sphere. We have

Jε(𝒲d¯,τ¯)=12M|g(j=1kWλj,ξj)|2dμg+12Ma(x)(j=1kWλj,ξj)2dμg-12+εM(j=1kWλj,ξj)+2+εdμg=j=1kJε(Wλj,ξj)-j<lMWλl,ξl2-1+εWλj,ξj𝑑μg+l<jkM[gWλj,ξjgWλl,ξl+a(x)Wλj,ξjWλl,ξl-Wλl,ξl2-1+εWλj,ξj]𝑑μg-12+εM[(j=1kWλj,ξj)2+ε-j=1kWλj,ξj2+ε-(2+ε)jlWλl,ξl2-1+εWλj,ξj]𝑑μg.(4.1)

We will estimate each term in the following lemmas.

Lemma 4.3 ([16, Lemma 4.1]).

For any j=1,2,,k, we have

Jε(Wλj,ξj)=Kn-nn(1+(n-2)28εlnε+(n-2)24εln(dj)+Cnε)+Kn-nn2(n-1)(n-2)(n-4)dj2(a(ξj)-n-24(n-1)Scalg(ξj))ε+o(ε)

as ε0, C0-uniformly with respect to τ¯ in (Rn)k and to d¯ in compact subsets of (R+)k, where Cn is the positive constant

Cn=2n-3(n-2)2ωn-1ωn0+tn-22ln(1+t)(1+t)n𝑑t+(n-2)24(1-nlnn(n-2)).(4.2)

Lemma 4.4.

We have

-jlMWλl,ξl2-1+εWλj,ξj𝑑μg=-ε(n-2)(α-β)(jldln-22djn-22|τl-τj|n-2αnnU(z)2-1𝑑z+o(1))(4.3)

as ε0, C0-uniformly with respect to τ¯ in (Rn)k and to d¯ in compact subsets of (R+)k.

Proof.

By the definition (2.1) of the functions Wλl,ξl and Wλj,ξj, for lj, we have

MWλl,ξl2-1+ε(x)Wλj,ξj(x)𝑑μg=M[χr(expξl-1(x))λl2-n2U(expξl-1(x)λl)]2-1+εχr(expξj-1(x))λj2-n2U(expξj-1(x)λj)𝑑μg=B(0,r)[χr(y)λl2-n2U(yλl)]2-1+εχr(y)λj2-n2U(expξj-1(expξl(y))λj)|g(expξl(y))|𝑑y,

where we set expξl-1(x)=y. Let θ be a positive constant, satisfying β<θ<α, and we divide the domain B(0,r) into two parts, i.e., B(0,r)=B(0,εθ)(B(0,r)B(0,εθ)). Then we have

B(0,r)Wλl,ξl2-1+ε(x)Wλj,ξj(x)𝑑μg=B(0,εθ)[χr(y)λl2-n2U(yλl)]2-1+εχr(y)λj2-n2U(expξj-1(expξl(y))λj)|g(expξl(y))|𝑑y   +B(0,r)B(0,εθ)[χr(y)λl2-n2U(yλl)]2-1+εχr(y)λj2-n2U(expξj-1(expξl(y))λj)|g(expξl(y))|𝑑y=:L1+L2.(4.4)

We estimate each term as follows. Let us introduce the transition map mlj:Bn(0,iM)n, defined as mlj(z)=expξj-1(expξl(z)), so as to estimate the interaction between Wλl,ξl and Wλj,ξj with lj. We point out that

|mlj(0)|=|expξj-1(ξl)|=dg(ξl,ξj)=εβ(|τl-τj|+o(1)),(4.5)

where o(1)0 as ε0, and the last equality comes from (2.5). For L1, setting y=λlz, we have

L1=λln-22(1+ε)B(0,εθλl-1)[χr(λlz)U(z)]2-1+εχr(mlj(λlz))λj2-n2U(mlj(λlz)λj)|g(expξl(λlz))|𝑑z=λln-22(1+ε)λjn-22B(0,εθλl-1)[χr(λlz)U(z)]2-1+εχr(mlj(λlz))αn(λj2+|mlj(λlz)|2)n-22|g(expξl(λlz))|𝑑z=αnλln-22(1+ε)λjn-22(λj2+|mlj(0)|2)n-22B(0,εθλl-1)[χr(λlz)U(z)]2-1+εχr(mlj(λlz))(λj2+|mlj(0)|2λj2+|mlj(λlz)|2)n-22|g(expξl(λlz))|𝑑z,

where αn=(n(n-2))n-24. Using (2.5) and (4.5), it follows that

λln-22(1+ε)λjn-22(λj2+|mlj(0)|2)n-22=(λlλjλj2+|mlj(0)|2)n-22λln-22ε=(ε2αdldjε2αdj2+ε2β(|τl-τj|2+o(1)))n-22(1+O(ε|lnε|))=ε(n-2)(α-β)dln-22djn-22|τl-τj|n-2(1+o(1)).

Note that

limr+|ξl-ξj|0supzB(0,r){0}|mlj(z)-mlj(0)||z|=1.(4.6)

Using this and (4.5), we then have that for zB(0,rεθλl-1),

(λj2+|mlj(0)|2λj2+|mlj(λlz)|2)n-22=(λj2+|mlj(0)|2λj2+(|mlj(0)|+O(λlz))2)n-22=(ε2αdj2+ε2β|τl-τj|2+o(ε2β)ε2αdj2+(εβ|τl-τj|+o(εβ)+O(εθ))2)n-221

as ε0. Moreover, since mlj converges to the identity map and |g(expξl(λlz))|1 pointwise as ε0, by the dominated convergence theorem, we get

B(0,εθλl-1)[χr(λlz)U(z)]2-1+εχr(mlj(λlz))(λj2+|mlj(0)|2λj2+|mlj(λlz)|2)n-22|g(expξl(λlz))|𝑑znU(z)2-1𝑑z

as ε0. Thus, we derive

L1=ε(n-2)(α-β)dln-22djn-22|τl-τj|n-2αnnU(z)2-1𝑑z+o(ε(n-2)(α-β)).(4.7)

On the other hand,

|L2|=B(0,r)B(0,εθ)[χr(y)λl2-n2U(yλl)]2-1+εχr(y)λj2-n2U(expξj-1(expξl(y))λj)|g(expξl(y))|𝑑yCB(0,r)B(0,εθ)[λl2-n2U(yλl)]2-1+ελj2-n2U(expξj-1(expξl(y))λj)𝑑y

CB(0,r)B(0,εθ)λln+22+n-22ε(λl2+|y|2)n+22+n-22ελjn-22(λj2+|mlj(y)|2)n-22𝑑yCλln+22+n-22ελjn-22B(0,r)B(0,εθ)1|y|n+2+(n-2)ε1|mlj(y)|n-2𝑑y.

Since β<θ<α, we have

|mlj(y)|=|expξj-1(expξl(y))|=|mlj(0)|+O(|y|)=εβ|τl-τj|+o(εβ)+O(|y|)εβ|τl-τj|+o(εβ)+O(εθ)(since yB(0,r)B(0,εθ))εβ|τl-τj|+o(εβ)(since β<θ)εβR-1+o(εβ)(since |τl-τj|>R-1).

Thus, we find

|L2|Cεnα-(n-2)βB(0,r)B(0,εθ)1|y|n+2+(n-2)ε𝑑yCεnα-(n-2)β-2θnB(0,1)1|z|n+2+(n-2)ε𝑑zCε(n-2)(α-β)+2(α-θ)=o(ε(n-2)(α-β)).(4.8)

By (4.4), (4.7) and (4.8), we get (4.3). ∎

Lemma 4.5.

We have

ljM[gWλj,ξjgWλl,ξl+a(x)Wλj,ξjWλl,ξl-Wλl,ξl2-1+εWλj,ξj]𝑑μg=O(ε(n-2)(α-β)+2β)=o(ε(n-2)(α-β))(4.9)

as ε0, C0-uniformly with respect to τ¯ in (Rn)k and to d¯ in compact subsets of (R+)k.

Proof.

Using the standard properties of the exponential map, for xB(0,r),

-gu=-u-(gij-δij)ij2u+gijΓijkku,

with

gij=δij-13Riabjxaxb+O(|x|3)andgijΓijk=lΓiikxl+O(|x|2).

Thus,

|ΔgWλl,ξl+a(x)Wλl,ξl-Wλl,ξl2-1+ε|(expξ0(x))Cλl2-n2U(expξl-1(x)λl).

For lj, we then have

M[gWλj,ξjgWλl,ξl+a(x)Wλj,ξjWλl,ξl-Wλl,ξl2-1+εWλj,ξj]𝑑μg=M[ΔgWλl,ξl+a(x)Wλl,ξl-Wλl,ξl2-1+ε]Wλj,ξj𝑑μgCBg(0,r)λl2-n2U(expξl-1(x)λl)λj2-n2U(expξj-1(x)λj)𝑑μgCBg(0,r)λln-22(λl2+|expξl-1(x)|2)n-22λjn-22(λj2+|expξj-1(x)|2)n-22𝑑μgCB(0,r)λln-22(λl2+|y|2)n-22λjn-22(λj2+|mlj(y)|2)n-22𝑑y.(4.10)

We divide the domain B(0,r) into three disjoint sets, i.e., we set

Ω1:={y:yB(0,r),|mlj(y)|εβ|τl-τj|2},Ω2:=B(0,10Rεβ)Ω1,Ω3:=B(0,r)B(0,10Rεβ),

where R is given in (2.4). From (4.6), for yΩ1, we have

|y|910(|mlj(0)|-|mlj(y)|)910(910-12)εβ|τl-τj|210Rεβ.

Then

Ω1λln-22(λl2+|y|2)n-22λjn-22(λj2+|mlj(y)|2)n-22𝑑yCε(n-2)αΩ11|y|n-21|mlj(y)|n-2𝑑yCε(n-2)(α-β){|mlj(y)|εβ|τl-τj|/2}1|mlj(y)|n-2𝑑y.

Take the change of variable mlj(y)z, and use the facts that |τl-τj||τl|+|τj|2R, the map mlj is injective in its domain B(0,iM) and minB(0,r)|Dmlj(z)|12 for ε>0 and r>0 small enough. Thus, we find

Ω1λln-22(λl2+|y|2)n-22λjn-22(λj2+|mlj(y)|2)n-22𝑑yCε(n-2)(α-β)minB(0,r)|Dmlj(z)|{|z|εβ|τl-τj|/2}1|z|n-2𝑑zCε(n-2)(α-β)+2β=o(ε(n-2)(α-β)).(4.11)

If yΩ2, we have that |y|10Rεβ and |mlj(y)|>εβ|τl-τj|2εβ2R. Then

Ω2λln-22(λl2+|y|2)n-22λjn-22(λj2+|mlj(y)|2)n-22𝑑yCε(n-2)αΩ21|y|n-21|mlj(y)|n-2𝑑yCε(n-2)(α-β){|y|10Rεβ}1|y|n-2𝑑yCε(n-2)(α-β)+2β=o(ε(n-2)(α-β)).(4.12)

If yΩ3, we have that |y|<r and |y|10Rεβ, From (4.5), (4.6) and |τl|R,|τj|R, for yΩ3, we get

|mlj(y)||mlj(y)-mlj(0)|-|mlj(0)|=(|y|+o(1))-εβ(|τl-τj|+o(1))910|y|-2Rεβ710|y|.

Thus,

Ω3λln-22(λl2+|y|2)n-22λjn-22(λj2+|mlj(y)|2)n-22𝑑yCε(n-2)αΩ31|y|n-21|mlj(y)|n-2𝑑yCε(n-2)α{10Rεβ|y|r}1|y|2(n-2)𝑑yCε(n-2)αε-(n-4)β=Cε(n-2)(α-β)+2β=o(ε(n-2)(α-β)).(4.13)

Therefore, (4.9) follows from (4.10)–(4.13). ∎

Lemma 4.6.

We have

M[(j=1kWλj,ξj)2+ε-j=1kWλj,ξj2+ε-(2+ε)jlWλl,ξl2-1+εWλj,ξj]𝑑μg=o(ε(n-2)(α-β))(4.14)

as ε0, C0-uniformly with respect to τ¯ in (Rn)k and to d¯ in compact subsets of (R+)k.

Proof.

Let Bh:=Bg(ξh,εβ4R), where Bg(ξh,εβ4R) denotes the geodesic ball in (M,g) with center at ξh and R>0 is given in (2.4). We write

M[(j=1kWλj,ξj)2+ε-j=1kWλj,ξj2+ε-(2+ε)jlWλl,ξl2-1+εWλj,ξj]𝑑μg=Mh=1kBh[(j=1kWλj,ξj)2+ε-j=1kWλj,ξj2+ε-(2+ε)jlWλl,ξl2-1+εWλj,ξj]𝑑μg   +h=1kBh[(j=1kWλj,ξj)2+ε-j=1kWλj,ξj2+ε-(2+ε)jlWλl,ξl2-1+εWλj,ξj]𝑑μg:=I+h=1kIh.(4.15)

We first have

|I|CjlMh=1kBhWλl,ξl2-2+εWλj,ξj2𝑑μgCjlM(BlBj)Wλl,ξl2-2+εWλj,ξj2𝑑μgCjlBg(0,r)(Bg(ξl,εβ4R)Bg(ξj,εβ4R))λl2+n-22ε(λl2+|expξl-1(x)|2)2+n-22ελjn-2(λj2+|expξj-1(x)|2)n-2𝑑μgCjlBg(0,r)(Bg(ξl,εβ4R)Bg(ξj,εβ4R))λl2+n-22ε|expξl-1(x)|4+n-2ελjn-2|expξj-1(x)|2(n-2)𝑑μgCjlB(0,r)(B(0,εβ4R)expξl-1(Bg(ξj,εβ4R)))λl2+n-22ε|y|4+(n-2)ελjn-2|mlj(y)|2(n-2)dy(y=expξl-1(x))Cεnαε-4βjlB(0,r)(B(0,εβ4R)expξl-1(Bg(ξj,εβ4R)))1|mlj(y)|2(n-2)𝑑yCεnαε-4βjl|mlj(y)|εβ4R1|mlj(y)|2(n-2)𝑑yCεnαε-4βjl|z|εβ4R1|z|2(n-2)𝑑yCεn(α-β).(4.16)

Moreover,

|Ih|=|Bh[(Wλh,ξh+jhkWλj,ξj)2+ε-Wλh,ξh2+ε-jhkWλj,ξj2+ε-(2+ε)jlWλl,ξl2-1+εWλj,ξj]𝑑μg|Cjh[BhWλh,ξh2-2+εWλj,ξj2dμg+BhWλj,ξj2+εdμg+jlBhWλj,ξj2-1+εWλl,ξldμg].(4.17)

Since

distg(Bh,ξj)dg(ξj,ξh)-εβ4Rεβ4Rfor jh,

we have

BhWλh,ξh2-2+εWλj,ξj2𝑑μgCBhλh2+n-22ε(λh2+|expξh-1(x)|2)2+n-22ελjn-2(λj2+|expξj-1(x)|2)n-2𝑑μgCBhλh2+n-22ε|expξh-1(x)|4+n-2ελjn-2|expξj-1(x)|2(n-2)𝑑μgCB(0,εβ4R)λh2+n-22ε|y|4+(n-2)ελjn-2|mlj(y)|2(n-2)dy(y=expξl-1(x))Cεnαε-(2(n-2)βB(0,εβ4R)1|y|4+(n-2)ε𝑑yCεn(α-β).

For jh,

BhWλj,ξj2+ε𝑑μgCBhλjn+n-22ε(λj2+|expξj-1(x)|2)n+n-22ε𝑑μgCεnαB(o,r)B(0,εβ4R)1|y|2n+(n-2)ε𝑑yCεn(α-β).

For the third term in (4.17), if lh, using Hölder’s inequality and the fact that the estimate for the second integral is bounded by Cεn(α-β), then we can get that the third term is bounded by Cεn(α-β). Moreover, for jh and l=h, we have

BhWλl,ξl2-1+εWλj,ξj𝑑μg=BhWλj,ξj2-1+εWλh,ξh𝑑μgCBhλjn+22+n-22ε(λj2+|expξj-1(x)|2)n+22+n-22ελhn-22(λh2+|expξh-1(x)|2)n-22𝑑μgCεnαBh1|expξj-1(x)|n+2+(n-2)ε1|expξh-1(x)|n-2𝑑μgCεnαε-(n+2)βBh1|expξj-1(x)|n-2𝑑μgCεnαε-(n+2)βB(0,εβ4R)1|y|(n-2)𝑑yCεn(α-β),

where we used the fact that

|expξj-1(x)|=distg(x,ξj)distg(Bh,ξj)dg(ξj,ξh)-εβ4Rεβ4R

for jh and xBh. Thus, we have |Ih|Cεn(α-β). Therefore, (4.14) follows from (4.15), (4.16) and the last inequality. ∎

Proof of Proposition 3.2 (ii).

From (4.1) and Lemmas 4.24.6, we get that if n5 and (d¯,τ¯)𝒪η,Rk satisfies (2.4), then

J~ε(d¯,τ¯)=kc0+εk(c1+c2lnε)+j=1k[c3ln(dj)+c4dj2φ(ξj)]ε-c5ε(n-2)(α-β)j<ldln-22djn-22|τl-τj|n-2+o(ε(n-2)(α-β))(4.18)

as ε0, C0-uniformly with respect to τ¯ in (n)k and to d¯ in compact subsets of (+)k, where

c0=Kn-nn,c1=c0Cn,c2=c0(n-2)28,c3=(n-2)24c0,c4=2(n-1)(n-2)(n-4)c0,c5=CnnU(z)2-1𝑑z,

with Cn being a positive constant, which is defined in (4.2), and φ given by (1.5). By Taylor’s theorem and the assumptions φ(ξ0)=0, we have

φ(ξj)=φ(ξ0)+ε2βD2φ(ξ0)[τj,τj]+o(ε2β).(4.19)

Therefore, (3.5) follows from (4.18) and (4.19). ∎

Acknowledgements

The author would like to thank Seunghyeok Kim for useful comments.

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About the article

Received: 2017-12-12

Revised: 2018-03-14

Accepted: 2018-03-22

Published Online: 2018-06-13


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11501468

The author has been supported by NSFC No. 11501468 and Chongqing Research Program of Basic Research and Frontier Technology cstc2016jcyjA0323 and Fundamental Research Funds for the Central Universities XDJK2017C049.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1213–1226, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0277.

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