BUBBLING ALONG BOUNDARY GEODESICS FOR LANE …yzhou173/BC67.pdfcondition. Hence we extend the result...

BUBBLING ALONG BOUNDARY GEODESICS FOR

LANE-EMDEN-FOWLER PROBLEM NEAR THE SECOND

CRITICAL EXPONENT IN DIMENSIONS 6 AND 7

GUOYUAN CHEN, JUNCHENG WEI, AND YIFU ZHOU

Abstract. We construct geodesics bubbling solutions along a nondegenerateclosed geodesic Γ ⊂ ∂Ω with negative inner normal curvature for the Lane-

Emden-Fowler problem∆u+ u

n+1n−3

−ε= 0 in Ω,

u > 0 in Ω,u = 0 on ∂Ω,

where Ω ⊂ Rn with n = 6, 7, ∂Ω is smooth and bounded, and ε > 0 is a smallparameter. We prove that there exists a solution uε such that |∇uε|2 converges

to the Dirac measure on Γ as ε→ 0+ with ε satisfying a certain non-resonance

condition. Hence we extend the result in [del Pino, Musso and Pacard, J. Eur.Math. Soc. 12 (2010), 1553-1605] to lower dimension case n = 6, 7. The new

ingredient in our approach is a new inner-outer gluing method which works

for all dimensions n ≥ 6.

1. Introduction

1.1. The problem. In this paper, we consider the classical Lane-Emden-Fowlerproblem (see [23]) ∆u+ up = 0 in Ω,

u > 0 in Ω,u = 0 on ∂Ω,

(1.1)

where p > 1 and Ω ⊂ Rn with smooth bounded boundary.For 1 < p < n+2

n−2 , compactness of Sobolev embedding yields the existence ofsolution by minimizing the following functional

S(p) = infu∈H1

0 (Ω)\0

∫Ω|∇u|2

(∫

Ω|u|p+1)

2p+1

.

For subcritical case, an interesting phenomenon is the point bubbling if p = n+2n−2−ε

as ε → 0+. To be more precise, for ε > 0 small, there exists a solution to (1.1) inthe form

uε(x) = µ−n−2

2ε wn

(x− xεµε

)+ o(1), µε ∼ ε

1n−2 ,

2010 Mathematics Subject Classification. 35B40; 35J60; 58J26.Key words and phrases. Supercritical problem; Concentration along geodesic; nondegenerate

geodesic.

1

2 G. CHEN, J. WEI, AND Y. ZHOU

as ε→ 0+. Here wn is the Aubin-Talenti bubble

wn(x) =

(cn

1 + |x|2

)n−22

(1.2)

which is the bounded radial solution of

∆w + wn+2n−2 = 0, in Rn, (1.3)

where cn =√n(n− 2) (see [1, 35]). The blow-up point xε concentrates on a non-

degenerate critical point x0 of Robin’s function of Ω. For more related results seefor example [3, 7, 13,17,22,24,25,33] and the references therein.

For p ≥ n+2n−2 , Pohozaev identity [32] implies that there is no solution of (1.1) if the

domain Ω is star-shaped. For Ω with other geometry structures, solutions may exist.For example, in [26], Kazdan and Warner proved that if Ω is a symmetric annulus,then the compactness of Sobolev embedding can be recovered for all p > 1 withinthe radial function space, which verifies the existence of solutions to (1.1). In [2],Bahri and Coron obtained the existence of solutions to (1.1) for p = n+2

n−2 if Ω has

nontrivial topology. On the other hand, in [6], Brezis and Nirenberg recovered thecompactness by suitable linear perturbations for the critical exponent p = n+2

n−2 . For

p > n+2n−2 , variational method seems difficult to show the existence. A question raised

by Rabinowitz, stated by Brezis in [5], is whether the nontrivial topology of thedomain is sufficient for the solvability of (1.1) for p > n+2

n−2 . However, Passaseo [31]constructed a counterexample for this question by choosing the domain Ω to be athin tubular neighborhood of a copy of the unit sphere Sn−2 in Rn (n ≥ 4) withp ≥ n+1

n−3 . Here n+1n−3 is called the second critical exponent, which is strictly larger

than the critical exponent n+2n−2 .

In an interesting paper [18], del Pino, Musso and Pacard first constructed so-lutions to (1.1) when p = n+1

n−3 − ε with ε > 0 sufficiently small and n ≥ 8. Moreprecisely, they proved that if ∂Ω contains a nondegenerate closed geodesic Γ withstrictly negative inner normal curvature, then there exists a solution of (1.1) with aconcentration behavior as p→ (n+1

n−3 )− in the form of bubbling line which collapsesto Γ. A typical example of such domain is that Ω has a convex hole. This phenom-enon is called line bubbling. Note that the argument in [18] relies crucially on thedimension restriction n ≥ 8. The line bubbling phenomenon has also been discov-ered in supercritical problems with p = n+1

n−3 ± ε on compact Riemannian manifold

without boundary [12]. The concentration at higher dimensional boundary sub-manifolds was investigated in [21]. Also, the constructions in [12] and [21] rely on asimilar technical restriction that the codimension of the concentration submanifoldis no less than 7.

We remark that point bubbling is determined by the Green’s function whichrelies on the global information of the domain Ω, whereas from the constructionof [18], line bubbling only depends on the local structure of the domain near theconcentration curve Γ.

In this paper, we shall investigate the line bubbling for lower dimension casen = 6, 7. More precisely, we study the following problem

∆u+ un+1n−3−ε = 0 in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.

(1.4)

SUPERCRITICAL PROBLEM IN DIMENSIONS 6 AND 7 3

Let Γ ⊂ ∂Ω ⊂ Rn be a closed nondegenerate (see Definition 1.1 below) geodesicwith global negative curvature. We assume that a non-resonance condition

|k2ε2(n−2)n−3 − κ2| > δε

n−2n−3 , ∀k ∈ Z+ (1.5)

holds, where δ > 0 is a constant, and κ > 0 is a constant depending only on Γ (see(7.25)). Note that the resonance phenomenon has been found in higher dimensionalconcentration for many elliptic problems, see for example [29], [30], [28], [27], [14],[18], [12] and the references therein.

For simplicity, in the rest of this paper, we denote

N = n− 1 and p =n+ 1

n− 3=N + 2

N − 2.

1.2. Main result. To describe our main result precisely, we introduce some geom-etry notations. Let ∂Ω endow with the metric induced by the Euclidean metric and∇ be the associated connection. Near the geodesic Γ, we introduce the followingFermi coordinates. Let q ∈ ∂Ω and we split

Tq∂Ω = TqΓ⊕NqΓ

into the tangent and normal bundles over Γ. Assume that Γ is parameterized byarclength x0 with x0 → γ(x0), and the length of Γ is 2l. Let E0 be a unit tangentvector to Γ, Ei, i = 1, · · · , N−1, be an orthonormal basis of NqΓ which are parallelalong Γ, namely

∇E0Ei = 0, i = 1, · · · , N − 1.

Since Γ is geodesic, it holds that

∇E0E0 = 0.

In a neighborhood of Γ in ∂Ω, using exponential map exp∂Ω on ∂Ω, we introducea local coordinates

F (x0, x) := exp∂Ωγ(x0)(xiEi), x := (x1, x2, · · · , xN−1),

where we use Einstein summation over i = 1, · · · , N − 1 for simplicity. In theneighborhood of Γ in Ω, we give a local coordinates

G(x0, x) := F (x0, x)− xNn(F (x0, x)), x = (x, xN ) ∈ RN ,

where x is in a small neighborhood of 0 and n is the outward unit normal. Assumethat the curvature of Γ is given by

∂2x0γ = h00n,

where h00 is a strictly positive function on Γ.

Definition 1.1. In local coordinates, we say that the geodesic Γ is nondegenerateif

−d′′k +

N−1∑j=1

〈R(E0, Ej)E0, Ek〉dj = 0, x0 ∈ [−l, l], k = 1, · · · , N − 1, (1.6)

has only the trivial 2l-periodic solution d ≡ 0, where R denotes the Ricci tensor on∂Ω.

Our main result is as follows.


Theorem 1.1. Let n = 6, 7 and Ω ⊂ Rn be a domain with smooth bounded bound-ary ∂Ω. Assume Γ ⊂ ∂Ω is a closed nondegenerate geodesic with negative innernormal curvature. Then for all ε > 0 sufficiently small satisfying the non-resonancecondition (1.5) with δ > 0 fixed, there exists a solution uε of (1.4) such that

|∇uε|2 Sn−12

n−1 δΓ, as ε→ 0+,

in the measure sense, where δΓ denotes the Dirac measure supported on Γ. Fur-thermore, uε is in the following form

uε(x0, x) = µ−N−2

2ε w

(x− dεµε

)+ o(1),

where µε and dε are defined in (3.2)-(3.4), w := wN is the standard bubble given in(1.2) and N = n− 1.

1.3. Main idea of proof. Our proof is based on the so-called inner-outer gluingprocedure which is a very useful tool in constructing higher dimensional concen-trating solutions for various elliptic problems, see, for example, [14–16,18] and thereferences therein. Recently, this method has been successfully applied to variouscritical heat equations, see, for example, [8–11,19,20,34] and the references therein.One of the key ingredients in this method is to prove proper a priori estimates forassociated linearized operators. Inspired by the linear theory in [18] and [11, Sec-tion 4], we develop a linear theory in the elliptic setting which shares the similarflavor of parabolic problems. Formally, in our problem, the tangential direction y0

plays a similar role as the time variable. More precisely, we consider the projectedequationa0∂

20φ+ ∆yφ+ Aφ = h+

N+1∑j=0

cj(ρy0)Zj(y), in Sρ ×DR,

φ(y0, y) = 0, ∀(y0, y) ∈ ∂(Sρ ×DR),

(1.7)

where a0 is a positive smooth 2l-periodic function of ρy0, A is a small coefficient

operator given by Lemma 3.1 below, ρ = εN−1N−2 , Sρ is the circle parameterized by

y0 ∈ [− lρ ,

lρ ],

DR = y = (y, yN ) ∈ RN : |y| < 2R,−dε,Nµε

(ρy0) < yN < 2R

with R = R(ε) = ε−θ∗ and θ∗ as in (7.13), Z0 is the first eigenfunction definedin (3.18), and Zi (i = 1, · · · , N,N + 1) are the bounded kernel functions of thelinearized operator of equation (1.3) around the standard bubble w = wN definedin (1.2), namely,

Zj = ∂jw for j = 1, · · · , N, and ZN+1 =N − 2

2w + x · ∇w,

(see for example [4]). Define the L∞-weighted norms

‖φ‖σ := supSρ×DR

〈y〉σ|φ(y0, y)|+ supSρ×DR

〈y〉1+σ|∇yφ(y0, y)|,

‖h‖2+σ := supSρ×DR

〈y〉2+σ|h(y0, y)|,(1.8)


where 0 < σ ≤ N − 4 with N = n − 1 and 〈y〉 :=√

1 + |y|2. It will be shownin Section 5 (see Proposition 5.2 below) that there exists a solution φ to equation(1.7) satisfying the following estimate

|φ| . Rτ−σ〈y〉−τ‖h‖2+σ (1.9)

where 2 < τ < N − 2. The main difference between Proposition 5.2 and the lineartheory in [18] is as follows: to obtain similar a prior estimates as in [18], the authorsassume that the solution φ is orthogonal to Zi (i = 0, 1, · · · , N,N + 1) in which adimension restriction n ≥ 8 is needed to guarantee the integrability of orthogonalityconditions in RN , whereas Proposition 5.2 is established by first proving a fastdecaying version (see Proposition 5.1) of the linear equation (1.7), then we applyit to the slow decaying version (see Proposition 5.2) and get the desired estimate(1.9). As a consequence, in the intermediate region |y| ∼ R, estimate (1.9) implies

‖φ‖σ . ‖h‖2+σ,

while in the interior, the estimate we get for the solution φ is deteriorated for lowdimension case n = 6, 7, namely that Rτ−σ appears in front. However, by choosingR appropriately and some further efforts, the linear theory is sufficient for us tocarry out the inner-outer gluing scheme.

We note that our new gluing method can also be used to reduce the co-dimensionsin [12] and [21]. We remark that, for dimension n = 4, 5 (i.e. N = 3, 4), similarresult as Proposition 5.2 does not hold and we need to make essential changes inthe argument. Furthermore in these low dimensions the equation for the scalingparameter may become nonlocal. We will return to this topic in a future work.

This paper is organized as follows. In Section 2, we recall some basic geometrynotations for local coordinates near the geodesic Γ. Section 3 is devoted to con-structing the approximate solution and computing the size of the error. In Section4, we set up the inner-outer gluing scheme and solve the outer problem. Before wesolve the reduced projected inner problem in Section 6, we shall develop a lineartheory of the associated linear problem in Section 5. Finally in Section 7, we ad-just the parameter functions such that the reduced system cj(ρy0) = 0 in (1.7) issatisfied for all y0 and j = 0, 1, · · · , N + 1 and prove Theorem 1.1.

Throughout this paper, the notation “.” always denotes “≤ c” where the con-stant c > 0 may differ from line to line but it is independent of ε.

2. Geometric Settings

In this section, we recall some geometric notations and results for the problemas in [18]. We refer the readers to [18] for detailed computations.

Consider the metric g on ∂Ω induced by the Euclidean metric in Rn. Denotethe associated connection by ∇. Then we introduce the Fermi coordinates in aneighborhood of Γ on ∂Ω. Given q ∈ Γ, we have the natural splitting

Tq∂Ω = TqΓ⊕NqΓ

into the tangent and normal bundles over Γ. Assume that Γ is parameterized bythe arclength x0 with x0 7→ γ(x0). Denote E0 by the unit tangent vector to Γ. In aneighborhood of q ∈ Γ, we denote E1, · · · , EN−1 by an orthonormal basis of NqΓ.We can assume that for each i = 1, · · · , N − 1

∇E0Ei = 0.


Since Γ is a geodesic,∇E0E0 = 0.

DefineF (x0, x) = exp∂Ω

γ(z0)(xiEi), x = (x1, · · · , xN−1),

where exp∂Ω is the exponential map on ∂Ω and the summation is from 1 to N − 1.The above Fermi coordinates are defined such that gab = δab along Γ, where gab

is the coefficient of g. Then the higher order terms in the Taylor expansion of themetric coefficients are estimated as follows, whose proof can be found in [18].

Proposition 2.1. At q = F (x0, x), the following estimates hold

g00 = 1 + 〈R(E0, Ek)E0, El〉xkxl +O(|x|3),

g0i = O(|x|2),

gij = δij +1

3〈R(Ei, Ek)Ej , El〉xkxl +O(|x|3),

where i, j, k, l = 1, · · · , N − 1, R is the curvature tensor and O(|x|s) is a smoothfunction not involving any term up to order s in xi, i = 1, · · · , N − 1.

For simplicity, we denote

Rijkl = 〈R(Ei, Ej)Ek, El〉. (2.1)

In order to parameterize a neighborhood near Γ, we define the following coordinates(x0, x) ∈ RN+1

G(x0, x) = F (x0, x)− xNn(F (x0, x)), x = (x, xN ) ∈ RN ,with x close to 0 and n denotes the unit outward normal.

The coefficients of the Euclidean metric in these coordinates

gaN = gNa = 0, gNN = 1 for a = 0, · · · , N − 1. (2.2)

Moreover, it holds that

gab = gab + 2habxN + kabx2N +O(x3

N ) for a, b = 0, · · · , N − 1,

where g is the metric on ∂Ω, h is the second fundamental form of ∂Ω with

hab = −Eb · ∇Ean = −Ea · ∇Ebn, (2.3)

andkab = (h⊗ h)ab =

∑c,d

hacgcdhdb. (2.4)

We remark that the normal curvature along the geodesic Γ in this setting is

∂2x0γ = ∇E0E0 = h00n.

Based on the above settings, we now compute the expansion of the Laplace-Beltrami operator

∆ =1√|g|∂xα(

√|g|gαβ∂xβ ) = gpq∂xα∂xβ + ∂pg

αβ∂xβ +1

2Trg(∂xαg)gαβ∂xβ

with the summation taken over α, β = 0, · · · , N . By (2.2), we have

∆ = ∂2xN +

1

2Trg(∂xN g)∂xN + gab∂xa∂xb + ∂xag

ab∂xb +1

2Trg(∂xag)gab∂xb ,

where the summation is taken over a, b = 0, · · · , N − 1.


Direct computations give the decomposition as follows

∆ = ∂2x0

+∑j

∂2xj + ∂2

xN +A00∂2x0

+∑j

A0j∂x0∂xj

+∑i,j

−1

3

∑k,l

〈R(Ei, Ek)Ej , El〉xkxl − 2hijxN +Aij

∂xi∂xj+B0∂x0 +

∑j

[∑k

(2

3〈R(Ei, Ej)Ei, Ek〉+ 〈R(E0, Ej)E0, Ek〉

)xk +Bj

]∂xj

+ (Trgh− TrgkxN +BN )∂xN ,

where R, g, h and k only depend on x0. The metric g, tensors h and k only dependon x0. All the rest functions Aij and Bj depend on x0, x1, · · · , xN and have furtherdecompositions as in [18, (4.13)]. For the reader’s convenience, we list below

A00 = A00N xN +

∑k,l

A00klxkxl,

Aij = AijNx2N +

(∑k

AijNkxk

)xN +

∑k,l,m

Aijklxkxlxm,

A0j = A0jN xN +

∑k,l

A0jklxkxl,

B0 = B0NxN +

∑k

B0kxk,

Bj = BjNxN +∑k,l

Bjklxkxl,

BN = BNN x2N +

(∑k

BNk xk

)xN +

∑j

BNj xj ,

(2.5)

where all the functions in (2.5) are smooth and depend on x0, · · · , xN .

3. Construction of the approximate solution

In this section, we shall construct an approximate solution to the following prob-lem

∆u+ uN+2N−2−ε = 0, in Ω

u = 0, on ∂Ω.

Our first approximate solution is based on the Aubin-Talenti bubble w satisfying

∆w + wN+2N−2 = 0 in RN ,

namely,

w(x) =

(cN

1 + |x|2

)N−22

with cN =√N(N − 2).


3.1. Scaling coordinates near the geodesic. We shall rescale and translate thebubble along a curve close to the geodesic Γ. Let (x0, x) ∈ RN+1 be the localcoordinates near the geodesic Γ. We perform the following change of variables

(y0, y) :=

(x0

ρ,x− dεµε

)and

u(G(x0, x)) = µ−N−2

2ε v

(x0

ρ,x− dεµε

), (3.1)

where v = v(y0, y), ρ = εN−1N−2 ,

µε(x0) = ρµε(x0), dε(x0) = εdε(x0) (3.2)

with

µε(x0) = µ0ε(x0) + εµ(x0), dε,j(x0) = εdj(x0), for j = 1, · · · , N − 1, (3.3)

and dε,N (x0) = d0ε,N (x0) + εdN (x0). In the above definitions, µ0

ε(x0) and d0ε,N (x0)

are given by

µ0ε(x0) = µ0(x0) + ε

1N−2µ1(x0), d0

ε,N (x0) = d0,N (x0) + ε1

N−2 d1,N (x0) (3.4)

with µ0(x0) = αh00(x0)

, d0,N (x0) = βh00(x0)

, where positive constants α and β only

depend on N , and h00 is the normal curvature along Γ. We assume Γ has globallynegative curvature, namely,

∂2x0γ = h00n,

where h00 is a smooth and strictly positive function along Γ, and n is the outwardunit normal.

The norms of the parameter functions µ(x0) and d(x0) = (d1, · · · , dN ) in (−l, l)are defined as follows

‖µ‖a = ‖εNN−2µ′′‖∞ + ‖ε

N2(N−2)µ′‖∞ + ‖µ‖∞, (3.5)

and

‖d‖d = ‖dN‖b +

N−1∑j=1

‖dj‖c, (3.6)

with

‖dN‖b = ‖εd′′N‖∞ + ‖ε 12 d′N‖∞ + ‖dN‖∞ (3.7)

and

‖dj‖c = ‖d′′j ‖∞ + ‖d′j‖∞ + ‖dj‖∞, for j = 1, · · · , N − 1. (3.8)

In the above definitions, the prime denotes ddx0

.

After the change of variables (3.1), the original cylinder close to Γ is transformedinto the following region

(y0, y) ∈ D :=

(y0, y, yN ) : −dε,N

µε< yN <

δ

ρ, |y| < δ

ρ

(3.9)

with some fixed δ > 0.


3.2. Equation in the local scaling coordinates. After performing the changeof variable (3.1), the Laplacian becomes

µN+2

2ε ∆u = A(v),

where Av = a0∂20v + ∆yv + Av with a0 = µ2

ε and µε is defined in (3.2). The

differential operator A can be expressed as

Av =∑α,β

aα,β∂α,βv +∑α

bα∂αv + cv

with

aα,β = O(ε+ ρ2|y|2), if α 6= 0, β 6= 0, a0,β = O(ε), a0,0 = 0

and

bα = ρO(ε+ ρ|y|), c = ρ2O(1).

The more specific expression of A is given by the following Lemma, whose proofcan be found in [18, Lemma 5.1].

Lemma 3.1. After the change of variables (3.1), it holds that

µN+2

2ε ∆u = A(v) := a0∂

20v + ∆yv +

5∑k=0

Akv +B(v),

whereA0(v) = (µ′ε)

2[Dyyv[y]2 + 2(1 + γ)Dyv[y] + γ(1 + γ)v

]+ µ′ε [Dyyv[y] + γDyv] [d′ε] +Dyyv[d′ε]

2

− 2µε

[ε−

N−1N−2Dy(∂0v)[µ′εy + d′ε] + γµ′εε

−N−1N−2 ∂0v

]− µεDyv[d′′ε ]− µεµ′′ε (γv +Dyv[y]),

(3.10)

A1v =∑i,j

[− 1

3Rikjl(µεyk + dε,k)(µεyl + dε,l)− 2hij(µεyN + dε,N )

+∑k

aijNk(µεyk + dε,k)(µεyN + dε,N )]∂ijv,

(3.11)

A2v =∑j

[− 4h0j(µεyN + dε,N )

× (−Dy(∂jv)[d] + µεε−N−1N−2 ∂0jv − (γ∂jv +Dy(∂jv)[y])µ′ε)

],

(3.12)

A3v =

(∑k

b0k[µεyk + dε,k] + b0N (µεyN + dε,N )

)×µε

[−Dyv[d′ε] + µεε

−N−1N−2 ∂0v − µ′ε(γv +Dyv[y])

],

(3.13)

A4v =∑j

[∑k

(2

3Rijik +R0j0k)(µεyk + dε,k) + bjN (µεyN + dε,N )

]µε∂jv, (3.14)

A5v =(Trgh− Trgk(µεyN + dε,N )

)µε∂Nv, (3.15)


B(v) = O(|µεy + dε|2 + (µεyN + dε,N ) + (µεyN + dε,N )(µεy + dε)

)A0(v)

+O(|µεy + dε|3 + (µεyN + dε,N )|µεy + dε|2 + (µεyN + dε,N )2

)∂ijv

+O(|µεy + dε|2 + (µεyN + dε,N )|µεy + dε|+ (µεyN + dε,N )2

)×[µεε−N−1N−2 ∂0jv + µεε

−N−1N−2 ∂0v −Dy(∂jv)[dε]

− (γ∂jv +Dy(∂jv)[y])µ′ε −Dyvd′ε − µ′ε(γv +Dyv[y]) + µε∂jv

]+O

(|µεy + dε|2 + (µεyN + dε,N )(µεy + dε) + (µεyN + dε,N )2

)µε∂Nv.

(3.16)In the above expressions (3.10)-(3.16), Rikjl is defined in (2.1), hij is defined in

(2.3), k is defined in (2.4), and aijNk is a smooth function of ρy0 with

AijNk = aijNkxN +O(x2N ),

where AijNk is defined in (2.5). b0k is a smooth function of ρy0 with

B0k = b0kxN +O(x2

N ),

where B0k is defined in (2.5). bjN is a smooth function of ρy0 with

BjN = bjNxN +O(x2N ),

where B0k is defined in (2.5).

After performing the change of variable (3.1) under the local coordinates alongΓ, the original equation becomes

Av + µN−2

2 εε v

N+2N−2−ε = 0.

Define the error of w by

Sε(w) = Aw + µN−2

2 εε w

N+2N−2−ε.

3.3. The first approximate solution. We first define a smooth cut-off function

χ(s) =

1, if s < δ

0, if s > 2δ

and

χε(y) = χ(ε1

N−2 |y|),where δ in (3.9) is chosen such that

χε(y,−dε,Nµε

) = 0. (3.17)

Denote Z0 by the eigenfunction corresponding to the only negative eigenvalue λ0

of the following eigenvalue problem

∆yφ+ pw(y)p−1φ+ λφ = 0, φ ∈ L∞(RN ),

namely,

∆yZ0 + pw(y)p−1Z0 + λ0Z0 = 0 (3.18)

with λ0 < 0.Our first approximate solution close to the geodesic Γ is

w = w + eε(ρy0)χε(y)Z0, (3.19)


where w is defined by

w(y) = (1 + αε)[w(y)− w(y)],

with the Aubin-Talenti bubble w, αε := µ− (N−2)2ε

8ε − 1 and

w(y) = w(y, yN +2dε,Nµε

).

Observe that (1 + αε)w satisfies

∆[(1 + αε)w] + µN−2

2 εε [(1 + αε)w]

N+2N−2 = 0 in RN ,

and w = 0 on yN = −dε,Nµε . In (3.19), eε(ρy0) is defined as

eε = εeε (3.20)

with

eε = e0ε + εe and e0

ε = e0 + ε1

N−2 e1, (3.21)

where e1 is a smooth function uniformly bounded in ε, and

e0 =2∫RN ∂iiwZ0

|λ0|(Trgh− h00)d0,N . (3.22)

The purpose of the parameter function e is to eliminate the resonance which canproduce large noise in the tangential direction. We shall choose e in the final sectionand the norm of e is defined as

‖e‖e = ‖ε2+ 2N−2 e′′‖∞ + ‖ε1+ 1

N−2 e′‖∞ + ‖e‖∞. (3.23)

3.4. Error of the first approximate solution w. Suppose that our parameterfunctions µ, d and e satisfy

‖(µ, d, e)‖ := ‖µ‖a + ‖d‖d + ‖e‖e ≤ c, (3.24)

where the definitions of the above norms are given in (3.5), (3.6) and (3.23), c > 0is a constant independent of ε.


By a similar computation as in [18, (5.33)], the expansion of the error Sε(w) forsmall ε is given by

Sε(w) =− pwp−1w − εwp logw + ε(−2hijd0ε,N∂ijw + |λ0|e0

εZ0)

+ εN−1N−2µ0

ε(−2hijyN∂ijw + Trgh∂Nw)

+ ε2[(ρ2a0e

′′ + |λ0|e)Z0 − 2hijdN∂ijw

+∑i,j

(d′id′j −

1

3Rijkldkdl + aijNkdkd

0ε,N + 4h0jdid

0ε,N )∂ijw + Υε

]+ ε

2N−3N−2 µ0

ε

[−∑j

∂jw · d′′j + (−∑i,j

1

3Rijklykdl∂ijw + 2aijNkykd

0ε,N∂ijw)

+ (2

3Rijik +R0j0k)dk∂jw + 4h0jd

′iyN∂ijw

]+ ε

3N−5N−2

[− µ0

ε∂Nw · d′′N −1

3µ0εRijklykdl∂ijw + µ

(2

3Rijik +R0j0k

)dk∂jw

+ (µ0εdN + µd0

ε,N )(2aijNkyk∂ijw + bjN∂jw − Trgh∂Nw)

+ (µ0εe+ µe0

ε)(−2hijyN∂ijZ0 + Trgh∂NZ0)]

+ ε3N−4N−2

[− µ′′µZN+1 + 2µµ0

ε

(− 1

3Rikjlykyl∂ijw

+

(2

3Rijik +R0j0k

)yk∂jw + bjNyN∂jw − TrgkyN∂Nw

)]+ ε4(log ε)r,

(3.25)where

Υε = Υ0 + ε1

N−2 Υ1ε

withΥ0 = −2hijd0,Ne0∂ijZ0 + p(p− 1)e2

0wp−2Z2

0 + pe0wp−1 logwZ0,

and Υ1ε is a smooth function of the form

f1(ρy0)f2(µ, d, e)f3(y).

In the above expression, f1 is smooth and uniformly bounded in ε, f2 is smoothand uniformly bounded in ε and f3 is smooth with

supD〈y〉N−2|f3(y)| < +∞.

Note that f2 depends linearly on µ′′, d′′ and e′′. We refer the reader to [18, Appen-dix] for the detailed computations of (3.25).

From (3.25), we can write

Sε(w) = εS0 + ε2(ρ2a0e′′ + |λ0|e)χεZ0 + ε2S1, (3.26)

where

εS0(ρy0) :=− pwp−1w − εwp logw + ε(−2hijd0ε,N∂ijw + |λ0|e0

εZ0)

+ εN−1N−2µ0

ε(−2hijyN∂ijw + Trgh∂Nw),

S0 is smooth and uniformly bounded in ε, and S1 depends on µ, d and e.


3.5. Correction of the approximate solution. Now we add an extra correctionΠ to get rid of εS0 in (3.26), namely, Π solves the following linear problema0∂

20Π + ∆yΠ + AΠ + pwp−1Π = −εS0 +

N+1∑i=0

αiZi in D

Π(y0, y) = 0 on ∂Dy0 for all y0,

where

Dy0 := y ∈ RN : (y0, y) ∈ Dwith D defined in (3.9). By choosing suitable parameters µ, d and e at mainorder, namely µ0

ε , d0ε,N and e0

ε (see (3.3), (3.21) and (3.22) for their definitions), theorthogonality conditions∫

Dy0S0Zidy = 0, for all y0 and i = 0, 1, · · · , N + 1,

are achieved. The argument is the same as that of [18, Appendix]. We omit thedetails.

We only need to consider the leading term in εS0

h0 = −pwp−1w − εwp logw + ε(−2hijd0ε,N∂ijw + |λ0|e0

εZ0),

since other terms are of sufficiently fast decay. Note that

|wp(y)| . 1

1 + |y|N+2,

∣∣−2hijd0ε,N∂ijw + |λ0|e0

εZ0

∣∣ . 1

1 + |y|N,

and

|wp−1w|(y0, y) .1

1 + |y|41

1 + |y|N−2 + (yN + ε−1

N−2 )N−2.

In order to apply the linear theory as in [18, Section 3], we can gain enough dacayin y by losing a little bit ε. To be more precise, for fixed ϑ > 0 close to 0, it holdsthat

|wp−1w| = |(wp−1wϑ)w1−ϑ| . ε1−ϑ

1 + |y|4+(N−2)ϑ,

in D. We consider the problema0∂

20Π + ∆yΠ + AΠ + pwp−1Π = −εS0 +

N+1∑i=0

αiZi in D

Π(y0, y) = 0 on ∂Dy0 for all y0∫Dy0

Π(y0, y)Zidy = 0, for all y0 and i = 0, 1, · · · , N + 1.

(3.27)

Since

‖∂0(dε,Nµε

)‖∞ . ρε−1

N−2 (ε‖µ′‖∞ + ε‖d′N‖∞) = o(1)

and

ρ−1‖∂00(dε,Nµε

)‖∞ . ρε−1

N−2 (ε‖µ′′‖∞ + ε‖d′′N‖∞) = o(1)


as ε→ 0+, problem (3.27) satisfies assumptions (5.3) and (5.4) in Proposition 5.1.Therefore, from Proposition 5.1, there exists a unique tuple (αi,Π) solving theelliptic problem (3.27). Moreover, the solution Π satisfies

|Π(y0, y)| . ε1−ϑ

1 + |y|2+(N−2)ϑ, ∀(y0, y) ∈ D, (3.28)

namely

‖Π‖σ . ε1−ϑ,

where we have used the facts that 0 < σ ≤ N − 4 and N = 5, 6. Since Π onlydepends on µ and d, by Lemma 3.1, we can estimate

‖Πµ1,d1 −Πµ2,d2‖σ ≤ cε2−ϑ‖(µ1 − µ2, d1 − d2)‖.

Moreover, using similar computations as in [18, (5.47)], we have

sup |αi| ≤ o(1)ε3. (3.29)

Let ψ = ∂0Π. Then by differentiating the equation (3.27) with respect to y0, wehave

a0∂20ψ + ∆yψ + Aψ + pwp−1ψ + ρa′0∂0ψ = h in D∫

Dy0ψ(y0, y)Zidy = 0 for all y0 and i = 0, 1, · · · , N + 1

ψ(y0, y, yN )− ∂0

(dεNµε

)∂NΠ(y0, y, yN ) = 0 on ∂Dy0 for all y0,

where h = −ερ∂0S0− (∂0A)Π +N+1∑i=0

∂0αiZi. Then by a similar argument as above,

we find that

‖ψ‖σ . ρε1−ϑ.Therefore, with the correction Π which eliminates the term εS0 in (3.26), the newerror for our new approximate solution

W = w + Π

is the following

Sε(W) = ε2S1 + ε2(ρ2a0e′′ + |λ0|e)χεZ0 +N1(Π) +

N+1∑i=0

αiZi, (3.30)

where

N1(Π) = µ−N−2

2 εε [(w + Π)p−ε − wp−ε]− pwp−1Π. (3.31)

The dependence of S1 on the parameter function µ, d and e is given by

‖S1(µ1, d1, e1)− S1(µ2, d2, e2)‖2+σ ≤ c‖(µ1 − µ2, d1 − d2, e1 − e2)‖.

Further, by (3.28), we know that w & |Π| gives

|y| . εϑ−1

N−4−(N−2)ϑ

and vice versa. By the definition (1.8) and R = R(ε) = ε−θ∗ with θ∗ given in (7.13),we see that

R εϑ−1

N−4−(N−2)ϑ ,


where we have used ϑ ≈ 0. Thus by a direct Taylor expansion, ‖N1(Π)‖2+σ can beestimated as follows

‖N1(Π)‖2+σ = supSρ×DR

〈y〉2+σ|N1(Π)| . sup

|y|.εϑ−1

N−4−(N−2)ϑ

〈y〉2+σwp−2Π2

= sup

|y|.εϑ−1

N−4−(N−2)ϑ

〈y〉2+σ〈y〉N−6〈y〉−4−2(N−2)ϑε2(1−ϑ)

. ε2(1−ϑ).

(3.32)

This finishes the construction of the approximate solution W and the estimate ofthe new error Sε(W).

4. The inner-outer gluing procedure

In this section, we shall apply the inner-outer gluing scheme to find a true solutionbased on the approximate solution W we built in Section 3.

Letting u(z) = ρ−N−2

2 v( zρ ), equation (1.4) becomes∆v + ρ

(N−2)ε2 vp−ε = 0, in Ωρ

v > 0, in Ωρ

v = 0, on ∂Ωρ

where p = N+2N−2 and Ωρ = 1

ρΩ. With a slight abuse of notation, we replace v by u,

namely, ∆u+ ρ

(N−2)ε2 up−ε = 0, in Ωρ

u > 0, in Ωρ

u = 0, on ∂Ωρ

In this problem, we have local coordinates near the geodesic Γ and the correspond-ing Laplace-Beltrami operator encodes the geometric information of the geodesic Γ,while in the region far away from the geodesic Γ, we use the usual Euclidean coordi-nates. Therefore, after introducing some suitable cut-off functions, it is natural todecompose the full problem into the inner and outer parts in which the inner-outergluing procedure can be carried out.

4.1. The global approximate solution. In a small neighborhood of the geodesicΓ, we denote

f(z) = µ−N−2

2ε (ρy0)f(y0, y), where z =

1

ρG(ρy0, ρµε(ρy0)y + εdε(ρy0)),

or equivalently,

f(y0, y) = µN−2

2ε (ρy0)f(z),

where z ∈ RN+1 is the original variable in Ωρ. Near the geodesic, the approximatesolution we construct in previous section W now becomes w in this setting. Indeed,recall that after the change of variables (3.1), we have

Av + µ(N−2)ε

2ε vp−ε = 0.

Observe that w is only defined locally on a small neighborhood near the ge-odesic. In order to get a global approximate solution, we first introduce some

cut-off functions. Let δ > 0 be a fixed number with 4δ < δ, where δ is chosen in


(3.17). Consider a standard cut-off function ζδ(s) with ζδ(s) = 1 for 0 < s < δ andζδ(s) = 0 for s > 2δ. We define

ζεδ(y0, y) = ζδ(|G(ρy0, µε(ρy0)ρy + εdε(ρy0))|),and

ηεδ,2R(z) = ζεδ(y0, y/R(ε)),

whereR(ε) = ε−θ∗ with 0 < θ∗ < 1 (4.1)

which we will specify later. For simplicity, we write R(ε) as R in the sequel. Bya zero extension of W far away from 1

ρΓ, it is natural to choose our first global

approximate solution asw(z) = ηεδ,2R(z)w(z).

We look for a solution u = w + Φ where Φ = η2δ,2Rφ+ψ and φ is such that φ is inprinciple defined only in D. Here D is defined in (3.9). Then Φ satisfies

∆Φ + pwp−1Φ +N(Φ) + E = 0, in Ωρ

Φ = 0, on ∂Ωρ(4.2)

where

N(Φ) = ρ(N−2)ε

2 (w + Φ)p−ε −wp−ε − pwp−1Φ and E = ∆w + wp−ε. (4.3)

Recall that near the stretched geodesic Γρ = 1ρΓ, the error is

Sε(w) = Aw + µ(N−2)ε

2ε wp−ε.

Then near Γρ, Sε(u) = 0 implies that the equation for Φ is

AΦ + pwp−1Φ + N(Φ) + Sε(w) = 0,

where

N(Φ) = µ(N−2)ε

2ε (w + Φ)p−ε − µ

(N−2)ε2

ε wp−ε − pwp−1Φ.

4.2. Inner and outer problems. It is direct to see that Φ = η2δ,2Rφ + ψ solves

(4.2) if the tuple (φ, ψ) solves the following two coupled equations

• φ solves the so-called inner problemAφ+ pwp−1φ = −N(ζε2δφ+ ψ)− Sε(w)− pwp−1ψ, in Sρ ×DR

φ = 0, on ∂(Sρ ×DR)(4.4)

• ψ solves the so-called outer problem∆ψ = −(1− ηεδ,2R)pwp−1ψ −2∇φ · ∇ηεδ,2R − φ∆ηεδ,2R

−(1− ηεδ,2R)N(ηεδ,2Rφ+ ψ), in Ωρ

ψ = 0, on ∂Ωρ

(4.5)

Define the following norm

‖φ‖∗ := supSρ×DR

Rσ−τ 〈y〉τ |φ(y0, y)| (4.6)

with 0 < σ ≤ N − 4 and 2 < τ < N − 2.We will first solve the outer problem (4.5) for given φ with sufficiently small ‖·‖∗-

norm. After we get the outer solution ψ(φ), the inner problem (4.4) is reduced to anonlinear nonlocal problem. In order to solve the reduced inner problem, we shall


develop a linear theory for the solvability of the associated model problem of theprojected inner problem. Then by applying the linear theory and the ContractionMapping Theorem, we solve the projected inner problem. Finally, we shall adjustour parameter functions µ, d and e such that the reduced system cj(ρy0) = 0 issatisfied for all y0 and j = 0, 1, · · · , N + 1.

4.3. Solving the outer problem. Recall the definition of ‖ · ‖∗ in (4.6). Given φ

such that ‖φ‖∗ . ε2(1−ϑ) in D, we solve the outer problem (4.5) first.Case 1. Assume that Ω is bounded. Then the following simple problem

−∆ψ = h1, in Ωρ

ψ = 0, on ∂Ωρ

has a unique solution ψ = (−∆)−1h1 for h1 ∈ L∞(Ωρ). Further, we have

‖ψ‖∞ . ‖h1‖∞.

Now we consider each term on the right hand side of the first equation of (4.5).Due to the effect of cut-off functions, we obtain from (4.6) that

‖φ‖L∞(R<|y|<2R) ∼ R−σ‖φ‖∗

and

‖φ∆ηεδ,2R‖∞ .1

R2‖φ‖L∞(R<|y|<2R) .

1

R2+σ‖φ‖∗. (4.7)

Similarly, we have

‖∇φ · ∇ηεδ,2R‖∞ .1

R2+σ‖φ‖∗. (4.8)

According to decay of φ, we assume ‖ψ‖∞ ≤ ΛR−σ‖φ‖∗ with Λ fixed sufficientlylarge and set

M(ψ) := (1− ηεδ,2R)N(ηεδ,2Rφ+ ψ),

where N(ηεδ,2Rφ+ ψ) is defined in (4.3). Then for ε small, we have

‖M(ψ)‖∞ . (1 + Λp)1

R(N+2)σN−2

‖φ‖p∗. (4.9)

Moreover,

‖(1− ηεδ,2R)pwp−1ψ‖∞ .1

R4‖ψ‖∞. (4.10)

By the Contraction Mapping Theorem, for R sufficiently large (by (4.1) this ispossible as ε is small enough), the fixed point problem

ψ = T (ψ)

: = (−∆)−1(M(ψ) + (1− ηεδ,2R)pwp−1ψ + φ∆ηεδ,2R + 2∇φ · ∇ηεδ,2R

) (4.11)

has a unique solution ψ = ψ(φ) in the function spaceN = ψ : ‖ψ‖∞ ≤ ΛR−σ‖φ‖∗provided ‖φ‖∗ . ε2(1−ϑ). Indeed, from (4.7)-(4.11) we have that

‖T (ψ)‖∞ . R−σ‖φ‖∗ for N = 5, 6, (4.12)


where we have used the assumptions ‖φ‖∗ . ε2(1−ϑ), R is sufficiently large and εis small. Therefore, the mapping T (ψ) maps N to itself. On the other hand, from(4.9), we see that for ψ(1), ψ(2) ∈ N

‖M(ψ(1))−M(ψ(2))‖∞ .(‖φ‖L∞(R<|y|<2R) + Λ

1

Rσ‖φ‖∗

)p−1

‖ψ(1) − ψ(2)‖∞

. (1 + Λ)4

N−21

R4σN−2

‖φ‖4

N−2∗ ‖ψ(1) − ψ(2)‖∞.

(4.13)Therefore, from (4.10) and (4.13) we have

‖T (ψ(1))− T (ψ(2))‖∞ .[(1 + Λ)

4N−2

1

R4σN−2

‖φ‖4

N−2∗ +

1

R4

]‖ψ(1) − ψ(2)‖∞,

which implies T is a contraction mapping in the space N for ε sufficiently small andR sufficiently large. Thus, the Contraction Mapping Theorem implies the existenceof such ψ ∈ N .

Moreover, from (4.11) and (4.12), the Lipschitz dependence of ψ on φ is givenby

‖ψ(φ1)− ψ(φ2)‖∞ . R−σ‖φ1 − φ2‖∗ for N = 5, 6.

Case 2. Next we consider the unbounded case and let Ω = RN+1\Υ with Υbounded.

Observe that the coupling term −2∇φ · ∇ηεδ,2R − φ∆ηεδ,2R in the outer problem

(4.5) is supported in Sρ × (DR\DR/2), where

DR\DR/2 = y ∈ RN : R < |y| < 2R.

So we decompose the outer problem into the following two equations∆ψ1 = −2∇φ · ∇ηεδ,2R − φ∆ηεδ,2R, in Sρ × (DR\DR/2)

ψ1 = 0, on ∂Sρ × (DR\DR/2)(4.14)

and∆ψ2 = −(1− ηεδ,2R)pwp−1ψ2 − (1− ηεδ,2R)N(ηεδ,2Rφ+ ψ), in Ωρ

ψ2 = 0, on ∂Ωρ.(4.15)

For equation (4.14), by a same argument as in the Case 1, we obtain that thereexists solution ψ1 ∈ N , namely,

‖ψ1‖∞ . R−σ‖φ‖∗ (4.16)

.We pull back the equation (4.15) for ψ2 from Ωρ to Ω. Define f(z) = f( zρ ). Then

the equation (4.15) becomes∆ψ2 = −ρ−2(1− ηεδ,2R)pwp−1ψ2 − ρ−2(1− ηεδ,2R)(ηεδ,2Rφ+ ψ)p in Ω

ψ2 = 0 on ∂Ω

namely,∆ψ2 = −O(ρ2)(1−χ)ψ2

ρ4+|z|4 − ρ−2(1− χ)(O(R−σ)‖φ‖∗χ+ ψ

)pin Ω,

ψ2 = 0 on ∂Ω(4.17)


where χ is a smooth function with compact support. In the exterior domain,after a perform of Kelvin transform with respect to a given point q in the interiorΥ ⊂ RN+1 = Rn

Kψ2(x) = |x− q|2−(N+1)ψ2(z) with z =x− q|x− q|2

∈ Ω and q ∈ Υ,

we see that the equation (4.17) becomes the following equation in the boundeddomain K(Ω)

∆(Kψ2(x)) =− O(ρ2)(1− χ1)

1 + |x− q|4Kψ2(x)− 1

ρ2|x− q|N+3

[O(R−σ)‖φ‖∗χ3

+ |x− q|N−1Kψ(1− χ4)]p, in K(Ω)

Kψ2(x) = 0, on ∂K(Ω)

(4.18)

which has a solution Kψ2 = (−∆)−1h2 with

‖Kψ2(x)‖∞ . ‖h2(x)‖∞ < +∞, x ∈ K(Ω),

namely ∥∥∥|z|N−1ψ2 (z)∥∥∥∞. ‖h2(x)‖∞ < +∞, x ∈ K(Ω), (4.19)

where h2 is the nonhomogeneous term in (4.18). Here χ1 and χ4 are cut-off functionssupported in neighborhood of the geodesic Γ, χ2 and χ3 are two cut-off functions

supported in bounded annulus. If ‖ψ2‖∞ ≤ ΛR−σ‖φ‖∗, we see that

‖h2(x)‖∞ ∼ maxρ2R−σ‖φ‖∗, ρ−2R−σp‖φ‖p∗ (4.20)

for ε small. Since K(Ω) is bounded and ‖φ‖∗ . ε2(1−ϑ) with ϑ > 0 close to 0, from(4.20) we get

‖h2‖∞ . R−σ‖φ‖∗, for N = 5, 6,

where θ∗ cannot be chosen too small. For example, in the case N = 6, directcomputation shows that θ∗ > 1/4. This is valid since we will choose θ∗ in (7.13)at last. By a similar fixed point argument, we can solve the problem (4.15) in the

function space N = ψ : ‖ψ‖∞ ≤ ΛR−σ‖φ‖∗ whenever ‖φ‖∗ . ε2(1−ϑ) for ε small,and the solution ψ2 satisfies

‖ψ2‖∞ . R−σ‖φ‖∗, for N = 5, 6. (4.21)

Combining (4.12) and (4.21), we get that the solution ψ = ψ1 + ψ2 of the outerproblem (4.5) satisfies

‖ψ‖∞ . R−σ‖φ‖∗, for N = 5, 6. (4.22)

This finishes the argument of the outer problem.

4.4. The reduced inner problem. As a conclusion, substituting ψ = ψ(φ) in theinner problem (4.4), the full problem reduces to the following nonlinear nonlocalequation

Aφ+ pwp−1φ = −N(ζε2δφ+ ψ(φ))− Sε(w)− pwp−1ψ(φ), in Sρ ×DR

φ = 0, on ∂(Sρ ×DR)

(4.23)


Instead of directly solving (4.23), we shall solve the following projected problemAφ+ pwp−1φ = H(φ, ψ, µ, d, e) +N+1∑j=0

cj(ρy0)Zj(y), in Sρ ×DR

φ = 0, on ∂(Sρ ×DR),

(4.24)

where

H(φ, ψ, µ, d, e) := −N(ζε2δφ+ ψ(φ))− Sε(w)− pwp−1ψ(φ).

We shall develop a linear theory concerning the solvability for the associated modelproblem of (4.24) in Section 5. In Section 6, we will solve the projected innerproblem (4.24) by the linear theory and the Contraction Mapping Theorem. InSection 7, we will derive and solve the reduced system of µ, d and e such thatcj(ρy0) = 0 for j = 0, 1, · · · , N + 1.

5. The linear theory for N = 5 and 6

In this section, we shall develop the linear theory concerning the existence anda priori estimates of the following linear problemAφ+ pwp−1φ = h+

N+1∑j=0

cj(ρy0)Zj(y), in D1

φ = 0, on ∂D1

(5.1)

in the following domain

D1 = (y0, y, yN ) ∈ RN+1 : −dε,Nµε

(ρy0) < yN < M(ε), |y| < M(ε),

where M(ε) > 0 depends on ε. Here h satisfies

‖h‖2+σ < +∞ with 0 < σ ≤ N − 4,

where ‖ · ‖2+σ norm is defined in (1.8) in the domain D1. Recall from Section 3.2that for (y0, y) ∈ D1,

Av = a0∂20v + ∆yv + Av,

where

a0 = µ2ε = (µ0 + ε

1N−2µ1 + εµ)2,

and

Av =∑α,β

aα,β∂α,βv +∑α

bα∂αv + cv

with

aα,β = O(ε+ ρ2|y|2) = O(ε) for α 6= 0, β 6= 0,

a0,β = O(ε) and a0,0 = 0,

bα = ρO(ε+ ρ|y|) = ρO(ε) and c = ρ2O(1).

The dimension restriction in the linear theory of [18, Section 3] is made suchthat the orthogonality conditions∫

RNφ(y0, y)Zj(y)dy = 0, j = 0, 1, · · · , N + 1


are well-defined. We have the following (see [18, Proposition 3.2])

Proposition 5.1. Assume that N = 5, 6 and ‖h‖2+τ < +∞ with 2 < τ < N − 2.For the linear projected problem

Aφ+ pwp−1φ = h+N+1∑j=0


φ = 0, on ∂D1∫D1φ(y0, y)Zj(y)dy = 0, for all y0 ∈ R, j = 0, 1, · · · , N + 1,

(5.2)

if for all indices α, β = 0, 1, · · · , N + 1,

‖∂0(dε,Nµε

)‖∞ +M(ε)‖∂00(dε,Nµε

)‖∞ +M(ε)‖∂0a0‖∞ + ‖aα,β‖∞ + ‖Daα,β‖∞

+‖〈y〉bα‖∞ + ‖〈y〉2c‖∞ < δ,

(5.3)and

δ−1 <dε,Nµε

(ρy0) < M(ε)δ for all y0 ∈ R (5.4)

for some positive constant δ, then for any ‖h‖2+τ < +∞ there exists a uniquesolution φ = T (h) which defines a linear operator of h with ‖φ‖τ < +∞. Moreover,it holds that

‖φ‖τ . ‖h‖2+τ .

Denote

h = Rτ−σh and φ = Rτ−σφ,

where 2 < τ < N − 2. Since τ > σ for N = 5, 6, we see that

‖h‖2+τ ≤ 〈y〉2+τRσ−τ 〈y〉−2−σ‖h‖2+σ ≤ ‖h‖2+σ. (5.5)

Since problem (5.2) is linear, multiplying equation (5.2) with Rτ−σ yields that φsolves

Aφ+ pwp−1φ = h+N+1∑j=0


φ = 0, on ∂D1∫D1φ(y0, y)Zj(y)dy = 0, for all y0 ∈ R, j = 0, 1, · · · , N + 1,

with cj(ρy0) = Rτ−σ cj(ρy0). From Proposition 5.1 and (5.5), we obtain

|φ(y0, y)| . Rτ−σ〈y〉−τ‖h‖2+σ.

The above argument concludes the following proposition.

Proposition 5.2. Assume that 0 < σ ≤ N − 4, 2 < τ < N − 2, N = 5, 6 and‖h‖2+σ < +∞. If for all indices α, β = 0, 1, · · · , N + 1,

‖∂0(dε,Nµε

)‖∞ +M(ε)‖∂00(dε,Nµε

)‖∞ +M(ε)‖∂0a0‖∞ + ‖aα,β‖∞ + ‖Daα,β‖∞

+‖〈y〉bα‖∞ + ‖〈y〉2c‖∞ < δ,

and

δ−1 <dε,Nµε

(ρy0) < M(ε)δ for all y0 ∈ R


for some positive constant δ, then there exists a unique solution φ to equation (5.1)satisfying ∫

D1

φ(y0, y)Zj(y)dy = 0, ∀y0 ∈ R, j = 0, 1, · · · , N + 1.

Furthermore, one has

|φ(y0, y)| . Rτ−σ〈y〉−τ‖h‖2+σ. (5.6)

Remark 5.1. (1) In the intermediate region |y| ∼ R, by Proposition 5.2, it followsthat

‖φ‖σ . ‖h‖2+σ.

(2) The estimate (5.6) is deteriorated in the interior. However, when we apply thelinear theory to solve the reduced inner problem (4.24), taking τ close to 2 will besufficient for our purpose.

6. The inner problem

In this section we drop all the tildes in the projected inner problem (4.24) forsimplicity. We solve problem (4.24) in a 2l/ρ-period (in y0) manner as follows

L(φ) = Sε(w) + N(φ) +N+1∑j=0


φ(y0, y) = φ(y0 + 2lρ , y), for all (y0, y) ∈ Sρ ×DR

φ = 0, on ∂(Sρ ×DR)

where L(φ) = Aφ+ pwp−1φ and

N(φ) = p(wp−1 −wp−1)φ− N(ζε2δφ+ ψ(φ)) + (ζε2δ)p−1pwp−1ψ(φ) (6.1)

with

N(φ) = µ− (N−2)ε

2ε (w + φ)p−ε − µ−

(N−2)ε2

ε wp−ε − pwp−1φ.

Recall from (3.30) that

Sε(w) = Sε(W) = ε2(ρ2a0e

′′(ρy0) + |λ0|e(ρy0))χεZ0 + E,

where E := ε2S1 +N1(Π) +N+1∑i=0

αiZi with N1(Π) defined in (3.31).

We consider the inner problema0∂

20φ+ ∆yφ+ Aφ+ pwp−1φ = H(φ, ψ, µ, d, e) +

N+1∑j=0


φ(y0, y) = φ(y0 + 2lρ , y), for all (y0, y) ∈ Sρ ×DR

φ(y0, y) = 0, on ∂(Sρ ×DR)∫DR

φ(y0, y)Zi(y)dy = 0, for all i = 0, 1, · · · , N + 1 and all y0 ∈ Sρ,(6.2)

where

H(φ, ψ, µ, d, e) := Sε(W) + N(φ).

Our aim is to find φ by using the linear theory developed in Section 5. Here µ, dand e satisfy (3.24). Note that the Lipschitz dependence of E on µ, d and e is

‖E(µ1, d1, e1)− E(µ2, d2, e2)‖∞ . ε2‖(µ1 − µ2, d1 − d2, e1 − e2)‖.


Consider the fixed point problem

φ = T (Sε(W) + N(φ)) := A(φ). (6.3)

We solve the above problem in the function space

M = φ : ‖φ‖∗ ≤ Λ1ε2(1−ϑ),

where the norm ‖ · ‖∗ is defined as in (4.6) and ϑ > 0 is close to 0. The first factwe show is that the mapping A maps M to itself. We estimate term by term asfollows.

From (3.30) and (3.32), we observe that Sε(W) is independent of φ and it satisfies

‖Sε(W)‖2+σ . ε2(1−ϑ). (6.4)

The nonlinear term satisfies

‖N(φ)‖2+σ . ‖(wp−1 −wp−1)φ‖2+σ + ‖ηεδ,2RN(ηεδ,2Rφ+ ψ(φ))‖2+σ

+ ‖(ηεδ,2R)p−1wp−1ψ(φ)‖2+σ.

By (4.6), we estimate

‖(wp−1 −wp−1)φ‖2+σ . ‖((w + εeZ0 + Π)p−1 − wp−1

)φ‖2+σ

. ‖wp−2(εeZ0 + Π)φ‖2+σ

. supSρ×DR

〈y〉2+σ〈y〉N−6(εeZ0 + ε〈y〉−σ)|φ|

. εRτ−σ‖φ‖∗.

(6.5)

Since ‖ψ‖∞ . R−σ‖φ‖∗, we get that for (y0, y) ∈ Sρ ×DR

|ηεδ,2Rφ+ ψ| .(R−σ +Rτ−σ〈y〉−τ

)‖φ‖∗ . Rτ−σ〈y〉−τ‖φ‖∗. (6.6)

When N = 5, 6 we know that w & |ηεδ,2Rφ+ ψ| if

|y| . Rσ−τ

N−2−τ ‖φ‖−1

N−τ−2∗ ,

and vice versa. Recall that R = R(ε) = ε−θ∗ with 0 < θ∗ < 1. We denote

R1 := Rσ−τ

N−2−τ ‖φ‖−1

N−τ−2∗ ∼ ε

θ∗(τ−σ)N−2−τ −

2(1−ϑ)N−τ−2 . (6.7)

Therefore, we can perform the Taylor expansion as follows. Using (6.6), we obtainthat

‖ηεδ,2RN(ηεδ,2Rφ+ ψ(φ))‖2+σ . sup|y|.R1

|〈y〉2+σwp−2(φ+ ψ)2|

+ supR1.|y|.R

〈y〉2+σ(Rτ−σ〈y〉−τ‖φ‖∗

)p. R2(τ−σ)‖φ‖2∗ +R2+σ−pτ

1 Rp(τ−σ)‖φ‖p∗.

(6.8)

For the last term in the nonlinear term, since |ψ| . R−σ‖φ‖∗, we get

‖(ηεδ,2R)p−1wp−1ψ(φ)‖2+σ . R−σ sup|y|.R

〈y〉σ−2‖φ‖∗ = R−σ‖φ‖∗ (6.9)

Collecting all the terms above from (6.5), (6.8) and (6.9), we obtain that forR(ε) = ε−θ∗ with θ∗ small

‖N(φ)‖2+σ . εRτ−σ‖φ‖∗+R2(τ−σ)‖φ‖2∗+R

2+σ−pτ1 Rp(τ−σ)‖φ‖p∗+R−σ‖φ‖∗. (6.10)


Therefore, by (6.4) , (6.10), (6.3) and Proposition 5.2, we have

‖A(φ)‖∗ . ‖H‖2+σ . ε2 + εRτ−σ‖φ‖∗ +R2(τ−σ)‖φ‖2∗ +R2+σ−pτ1 Rp(τ−σ)‖φ‖p∗

+R−σ‖φ‖∗.(6.11)

We observe from (6.11) that, in order to make A map M to itself, the followingrelations should be satisfied

εRτ−σ ≤ CR2(τ−σ)‖φ‖∗ ≤ CR2+σ−pτ

1 Rp(τ−σ)‖φ‖p−1∗ ≤ C

for some constant C when ε is small. Recall that R = R(ε) = ε−θ∗ . By (6.7) and‖φ‖∗ ∼ ε2(1−ϑ), we obtain that

1− θ∗(τ − σ) > 0

2(1− ϑ)− 2θ∗(τ − σ) > 0θ∗(τ−σ)−2(1−ϑ)

N−2−τ (2 + σ − pτ)− θ∗p(τ − σ) + 2(1− ϑ)(p− 1) > 0.

(6.12)

In our setting, τ is chosen slightly larger than 2 and ϑ > 0 is close to 0. WhenN = 5, σ ≈ 1. When N = 6, σ ≈ 2. Elementary computations show that (6.12) issatisfied if θ∗ < 1. Thus, by θ∗ < 1 and (6.11), it follows that for ε small

‖A(φ)‖∗ ≤ C‖φ‖∗

where C is some positive constant. Therefore, we conclude that for φ ∈ M withΛ1 fixed sufficiently large, A(φ) ∈M.

It remains to show that A is a contraction mapping in M. From (6.3), (6.10)and (6.11), we see that for φ(1), φ(2) ∈M

‖A(φ(1))− A(φ(2))‖∗ . εRτ−σ‖φ(1) − φ(2)‖∗ + ε2R2(τ−σ)‖φ(1) − φ(2)‖∗+ ε2(p−1)R2+σ−pτ

1 Rp(τ−σ)‖φ(1) − φ(2)‖∗ +R−σ‖φ(1) − φ(2)‖∗.

By our choices of θ∗, τ , ϑ and σ in the system (6.12), we already see that

‖A(φ(1))− A(φ(2))‖∗ ≤ o(1)‖φ(1) − φ(2)‖∗,

where o(1) → 0 as ε → 0+. It then follows that A is a contraction mapping inthe function space M for ε sufficiently small. Therefore, the Contraction MappingTheorem implies the existence of the solution φ.

Moreover, by a similar argument as in [18], we find that the Lipschitz dependenceof T on the parameters is given by

‖T(µ1,d1,e1) − T(µ2,d2,e2)‖ . R−σ‖(µ1 − µ2, d1 − d2, e1 − e2)‖,

and for ‖φ‖∗ . ε2(1−ϑ), we have

‖N(µ1,d1,e1)(φ)−N(µ2,d2,e2)(φ)‖2+σ . R−σε2(1−ϑ)‖(µ1 − µ2, d1 − d2, e1 − e2)‖.

By a fixed point argument, we see that

‖φ(µ1,d1,e1) − φ(µ2,d2,e2)‖∗ . R−σε2(1−ϑ)‖(µ1 − µ2, d1 − d2, e1 − e2)‖.

We have thus proved the existence of the following


Proposition 6.1. For sufficiently small ε and suitable parameters µ, d and e sat-isfying

‖(µ, d, e)‖ = ‖µ‖a + ‖d‖d + ‖e‖e . 1,

the problem (6.2) has a unique solution φ = φ(µ, d, e) satisfying

‖φ‖∗ . ε2(1−ϑ).

Furthermore, φ depends Lipschitz continuously on µ, d and e with

‖φ(µ1,d1,e1) − φ(µ2,d2,e2)‖∗ . R−σε2(1−ϑ)‖(µ1 − µ2, d1 − d2, e1 − e2)‖.

7. Choice of the parameter functions µ, d and e

In this section, we shall choose the parameter functions µ, d and e such that

cj(ρy0) = 0, j = 0, 1, · · · , N + 1. (7.1)

are satisfied. Multiplying equation (6.2) with Zj and integrating in y over DR implythat the reduced system (7.1) is equivalent to∫

DR

(a0∂

20φ+ ∆yφ+ Aφ+ pwp−1φ−H

)Zjdy = 0 (7.2)

for all y0 ∈ Sρ and j = 0, 1, · · · , N + 1. Recall from problem (6.2) that

H = Sε(W) + N(φ),

where Sε(W) and N(φ) are defined in (3.30) and (6.1), respectively. Since Sε(W) andN(φ) involve µ, d and e, our full problem is reduced to a system involving theseparameter functions. Recall that

N(φ) = p(wp−1 −wp−1)φ− N(ηεδ,2Rφ+ ψ(φ)) + (ηεδ,2R)p−1pwp−1ψ(φ)

where

N(φ) = µ(N−2)ε

2ε (w + φ)p−ε − µ

(N−2)ε2

ε wp−ε − pwp−1φ,

and

Sε(W) = ε2S1 + ε2(ρ2a0e

′′(ρy0) + |λ0|e(ρy0))χεZ0 +N1(Π) +

N+1∑i=0

αiZi,

where N1(Π) is the nonlinear term defined as

N1(Π) = µN−2

2 εε [(w + Π)p−ε − wp−ε]− pwp−1Π. (7.3)

Next we expand (7.2) in terms of µ, d, e.

7.1. Projections of Sε(W) on Zj, j = 0, 1, · · · , N + 1. For j = 0, 1, · · · , N + 1,one has∫

DR

Sε(W)Zj = ε2∫DR

S1Zj +

∫DR

ε2(ρ2a0e

′′(ρy0) + λ1e(ρy0))χεZ0Zj

+

∫DR

N1(Π)Zj + αj

∫DR

Z2j

=

∫DR

Sε(w)Zj +

∫DR

N1(Π)Zj + αj

∫DR

Z2j ,

(7.4)

where Sε(w) is defined in (3.26).


It turns out that the size of projections on Z0, ZN and ZN+1 are much larger thanthat of Zj directions with j = 1, · · · , N−1. We proceed to compute the projectionsof Sε(w) and N1(Π) along different directions as follows. Here we mainly follow theresults in [18, Section 5].

7.2. Projections of Sε(w) on Zj, j = 0, · · · , N+1. We use the expansion of Sε(w)(3.25) to compute the projection of Sε(w) onto Zi, i = 0, 1, · · · , N + 1. Suppose theparameter functions µ, d and e satisfy

‖(µ, d, e)‖ := ‖µ‖a + ‖d‖d + ‖e‖e ≤ c,where the norms are defined in (3.5), (3.6), (3.7), (3.8) and (3.23). With suitablechoices of µ0

ε and dε,N as in (3.4) and e0 as in (3.22), the main order terms in theprojections of Sε(w) on Zj are eliminated, and thus we obtain∫

DR

Sε(w)Zk = ε2+ 1N−2

( ∫RN

Z2k

)[µ0(−d′′k +R0j0kdj) + αk(ρy0)

+ εβk(ρy0;µ, d, e)]

+ ε3r

(7.5)

$

∫DR

Sε(w)ZN = ε2+ 1N−2 [Bh00µ+ Ch00dN + αN (ρy0) + εβN (ρy0;µ, d, e)]

− ε3+ 1N−2

(∫RN

Z2N

)µ0d′′N + ε4r

(7.6)

∫DR

Sε(w)ZN+1 = ε2[Ah00µ+Bh00dN + αN+1(ρy0) + εβN+1(ρy0;µ, d, e)]

− ε3+ 2N−2

(∫RN

Z2N+1

)µ0µ

′′ + ε4r

(7.7)

∫DR

Sε(w)Z0 = ε2(∫

RNZ2

0

)ρ2a0e

′′ + |λ0|e+ α0(ρy0)

− 2(Trgh− h00)

(∫RN

∂iiwZ0

)dN +

∑i

[(d′i)

2

− 1

3Rikildkdl + aiiNkdkd0,N + 4h0jdjd0,N

](∫RN

∂iiwZ0

)+ ε2β0(ρy0;µ, d, e)

+ ε4r.

(7.8)

In the above expressions (7.5)-(7.8), Rijkl is the component of the curvature tensordefined in (2.1), αk and βk are smooth and uniformly bounded in ε. Note that αkand βk does not depend on µ′, d′ and e′. Function r is of the following form

h0(ρy0) [h1(µ, d, e, µ′, d′, e′) + o(1)h2(µ, d, e, µ′, d′, e′, µ′′, d′′, e′′)] ,

where h0, h1 and h2 are smooth and uniformly bounded in ε. A,B and C areconstants depending only on the dimension with AC − B2 > 0. $ is a constantdepending on the dimension and the smooth functions

µ0, d0,N , e0, µ1, d1,N , e1 : (−l, l)→ Rare defined in (3.4), (3.20) and (3.22).


Since the projections (7.5)-(7.8) are exactly the same as that of [18], we omit theproof here. A proof can be found in [18, Appendix].

7.3. Projections of N1(Π) and∑αiZi. By the computations in Section 3.5

about the estimates of Π, we have the following. For ε sufficiently small andN = 5, 6, it holds that∫

DR

N1(Π)Zjdy = ε2(1−ϑ)h0(ρy0), for j = 0, 1, · · · , N + 1, (7.9)

where N1(Π) is defined in (7.3), h0(ρy0) is a smooth function of ρy0. Indeed, by(3.32) and the decay of Zj , j = 0, 1, · · · , N + 1, we can easily get the desiredestimate.

Note that in (7.9), h0(ρy0) is independent of µ, d and e. From a quite similarargument as in [18, Appendix], we can eliminate the largest term ε2(1−ϑ)h0(ρy0) bysolving a system like [18, Appendix (9.17)]. Therefore, the new projection becomes∫

DR

N1(Π)Zjdy = o(1)ε3, for j = 0, 1, · · · , N + 1. (7.10)

On the other hand, by (3.29) and the fact that ϑ > 0 is close to 0, we see that

αj

∫DR

Z2j = o(1)ε3−ϑ, (7.11)

which is of smaller order compared with the projections of Sε(w)’s for ε small.

7.4. Projections of N(φ). From (6.10), ‖φ‖∗ ∼ ε2(1−ϑ) and the definition of R =R(ε) as in (4.1), one has

‖N(φ)‖2+σ . ε3−2ϑ−θ∗(τ−σ) + ε4(1−ϑ)−2θ∗(τ−σ)

+ εθ∗(τ−σ)−2(1−ϑ)

N−2−τ (2+σ−pτ)−θ∗p(τ−σ)+2(1−ϑ)p + ε2(1−ϑ)+σθ∗ .

Now we choose θ∗ such that the projections of N(φ) are of smaller order comparedwith the leading order of Sε(w)’s for R sufficiently large (namely ε sufficiently small).More precisely, by (7.5), the projection of Sε(w) along Zj (j = 1, · · · , N − 1, N) is

of order ε2+ 1N−2 . Thus, θ∗ satisfies the following inequalities

3− 2ϑ− θ∗(τ − σ) > 2 + 1N−2

4(1− ϑ)− 2θ∗(τ − σ) > 2 + 1N−2

θ∗(τ−σ)−2(1−ϑ)N−2−τ (2 + σ − pτ)− θ∗p(τ − σ) + 2(1− ϑ)p > 2 + 1

N−2

2(1− ϑ) + σθ∗ > 2 + 1N−2

(7.12)

We know that τ ≈ 2, σ ≈ N − 4 and ϑ ≈ 0. To make the projection of N(φ) alongZj comparatively smaller than Sε(w)’s, a sound choice of θ∗ satisfying system (7.12)is

θ∗ =1 + ν

(N − 2)σwith R(ε) = ε−θ∗ and ν > 0 small. (7.13)

We can easily check the other directions Z0 and ZN+1 in a similar way.In conclusion, with such θ∗ in (7.13), we obtain that for i = 0, 1, · · · , N,N + 1∫

DR

N(φ)Zidy = o(1)

∫DR

Sε(w)Zidy. (7.14)


7.5. Projections of L(φ). Recall that L(φ) = Aφ + pwp−1φ and Aφ = a0∂20φ +

∆yφ+ Aφ. Since the differential operator A is a small perturbation of ∆y of orderε, we have

|Aφ| . ε|φ| . εRτ−σ‖φ‖∗,where we have used the definition of the norm ‖ · ‖∗ in (4.6). By ‖φ‖∗ ∼ ε2(1−ϑ)

and the choice of θ∗ in (7.13), we obtain

|Aφ| . ε3−2ϑ−θ∗(τ−σ) = o(1)ε2+ 1N−2 (7.15)

for ε small. For the remaining terms, since the solution φ we get is orthogonal toZj in DR, we have∫

DR

(a0∂

20φ+ ∆yφ+ pwp−1φ

)Zjdy =

∫DR

(∆yφ+ pwp−1φ

)Zjdy

=

∫DR

(∆yZj + pwp−1Zj

)φdy +

∫∂DR

Zj∂νφdS −∫∂DR

φ∂νZjdS

. R−σ‖φ‖∗ = o(1)ε2+ 1N−2

(7.16)

for ε small, where we have used the integration-by-parts formula and (7.13). Inconclusion, combining (7.15) and (7.16), we have∫

DR

L(φ)Zjdy = o(1)ε2+ 1N−2 (7.17)

for j = 0, 1, · · · , N + 1.

7.6. Reduced equations for µ, d, e. By the computations above, the reducedsystem (7.2) are equivalent to a system of ODEs for µ, d, e. We assume that

‖(µ, d, e)‖ := ‖µ‖a + ‖d‖d + ‖e‖e ≤ c. (7.18)

By collecting (7.4), (7.5), (7.6), (7.7), (7.8), (7.10), (7.11), (7.14) and (7.17), weknow that the reduced system (7.2) are achieved if (e, d, µ) satisfies the followingsystem of ODEsL0(e) := ρ2a0e

′′ + |λ0|e+ γ0dN = −α0(ρy0)−Q0(d) + ε2M0(ρy0;µ, d, e)Lk(dk) := −d′′k +R0j0kdj = −αk(ρy0) + εMk(ρy0;µ, d, e), k = 1, · · · , N − 1,LN (dN ) := −εCN$µ0d

′′N +Bh00µ+ Ch00dN = −αN (ρy0) + εMN (ρy0;µ, d, e)

LN+1(µ) := −εNN−2CN+1µ0µ

′′ +Ah00µ+Bh00dN= −αN+1(ρy0) + εMN+1(ρy0;µ, d, e),

(7.19)

where

CN :=

∫RN

Z2N , CN+1 :=

∫RN

Z2N+1,

γ0 := −2(Trgh− h00)

(∫RN

∂iiwZ0

), (7.20)

and

Q0(d) =∑i

[(d′i)

2 − 1

3Rikildkdl + aiiNkdkd0,N + 4h0jdjd0,N

](∫RN

∂iiwZ0

).

For j = 0, 1, · · · , N,N + 1, the operator Mj(ρy0;µ, d, e) can be decomposed intothe following form

Mj(ρy0;µ, d, e) = Aj(ρy0;µ, d, e) +Kj(ρy0;µ, d, e)


where Kj is uniformly bounded in L∞(−l, l) for (µ, d, e) satisfying (7.18) and iscompact, Aj depends on (µ, d, e, µ′, d′, e′, µ′′, d′′, e′′) and satisfies

‖Aj(µ1, d1, e1)−Aj(µ2, d2, e2)‖∞ . o(1)‖(µ1, d1, e1)− (µ2, d2, e2)‖,in which the dependence of Aj on µ′′, d′′ and e′′ is linear.

7.7. Linear theory for the ODE system (7.19). For j = 0, 1, · · · , N,N + 1, wefirst develop a linear theory concerning the invertibility of Lj in a L∞ manner.

We seek 2l-periodic solutions of the following problem

LN+1(µ) = h1, LN (d) = h2, (7.21)

where‖h1‖∞+‖h2‖∞ < +∞. We have the following existence and a priori estimatesfor the problem (7.21).

Lemma 7.1. Assume that A > 0, C > 0 and AC −B2 > 0. If ‖h1‖∞ + ‖h2‖∞ <+∞, then there exists a 2l-periodic solution (µ, d) of (7.21) such that

‖µ‖∞ + ‖d‖∞ + εN

2(N−2) ‖µ′‖∞ + ε12 ‖d′‖∞ . ‖h1‖∞ + ‖h2‖∞.

Proof. The associated energy functional for the operators LN and LN+1 is givenby

F (µ, d) =

∫ l

−l

[ε

NN−2µ0(µ′)2 + εµ0(d′)2 + ε

NN−2µ′0µµ

′ + εµ′0dd′

+ (Aµ2 + 2Bdµ+ Cd2)h00 + h1µ+ h2d]dx0.

Since A > 0, C > 0 and AC −B2 > 0, for ε > 0 small, we have

F (µ, d) ≥ c,where c is a positive constant. Hence, the existence of solution to (7.21) follows.

The proof of the a priori estimate is the same as that in [18, Lemma 8.1].

Now we consider the invertibility of

L0(e) := ρ2a0e′′ + |λ0|e+ γ0d = f. (7.22)

We perform the Liouville transform as follows.

m =

∫ l

−l

1√a0(s)

ds, t =π∫ s−l

(√a0(θ)

)−1

dθ

m,

λ0 =m2

π2|λ0|, y(t) = a

− 14

0 (s)e(s), q(t) =m2

π2

(a

140

)′′a

340 .

After the Liouville transform, equation (7.22) for e gets reduced toρ2(y′′ + q(t)y) + λ0y = f , in (0, π)

y(0) = y(π), y′(0) = y′(π)(7.23)

By directly applying the Sturm-Liouville theory to (7.23) together with the non-resonance condition

|k2ε2N−1N−2 − κ2| > δε

N−1N−2 (7.24)

and

κ =

√|λ0|2π

∫ l

−l

1√a0(s)

ds, (7.25)


we obtain the following existence and a priori estimates for e.

Lemma 7.2. Assume that ε satisfies the non-resonance condition (7.24). If f ∈C(−l, l) ∩ L∞(−l, l), then there exists a unique 2l-periodic solution e of (7.22)satisfying

ρ2‖e′′‖∞ + ρ‖e′‖∞ + ‖e‖∞ . ρ−1‖f‖∞.

Furthermore, if f ∈ C2(−l, l), then

ρ2‖e′′‖∞ + ρ‖e′‖∞ + ‖e‖∞ . ‖f ′′‖∞ + ‖f ′‖∞ + ‖f‖∞.

Proof. See [18, Lemma 8.2].

7.8. Final argument.

Proof of Theorem 1.1. From the nondegenerate condition of the geodesic Γ (1.6),we have that for any f ∈ L∞(−l, l), k = 1, · · · , N − 1, there exists a 2l-periodicfunction dk such that Lk(dk) = f with

‖d′′k‖∞ + ‖d′k‖∞ + ‖dk‖∞ . ‖f‖∞. (7.26)

Let (µ0, d0,N , d0,k) be a solution to Lk(d0,k) = αk, k = 1, · · · , N − 1,

LN (d0,N ) = αNLN+1(µ0) = αN+1.

By Lemma 7.1 and (7.26), we obtain that

ε‖d′′0,N‖∞ + ε12 ‖d′0,N‖∞ + ‖d0,N‖∞ ≤ c, (7.27)

‖d′′0,k‖∞ + ‖d′0,k‖∞ + ‖d0,k‖∞ ≤ c, (7.28)

and

εNN−2 ‖µ′′0‖∞ + ε

N2(N−2) ‖µ′0‖∞ + ‖µ0‖∞ ≤ c.

Now we consider

L0(e0) = −γ0d0,N − α0 −Q0(d0)

where d0 = (d0,1, · · · , d0,N ) and γ0 is defined in (7.20). Since α0 and Q0(d0) areregular, by (7.27), (7.28) and Lemma 7.2, we have that

ε2N−2N−2 ‖e′′0‖∞ + ε

N−1N−2 ‖e′0‖∞ + ‖e0‖∞ ≤ c. (7.29)

Summarizing (7.27), (7.28) and (7.29), it holds that

‖(µ0, d0, e0)‖ ≤ c.

We assume that

µ = µ0 + µ1, d = d0 + d1, e = e0 + e1.

Then the original system (7.19) reduces toL0(e1) = −γ0d1,N + ε2M0(ρy0;µ, d, e)

Lk(d1,k) = εMk(ρy0;µ, d, e), k = 1, · · · , N − 1,

LN (d1,N ) = εMN (ρy0;µ, d, e)LN+1(µ1) = εMN+1(ρy0;µ, d, e).

(7.30)


A direct use of Schauder’s fixed point theorem establishes the existence of (µ1, d1, e1)solving system (7.30), whose proof can be found in [18]. We omit the details.

Acknowledgements

G. Chen is partially supported by Zhejiang Provincial Science Foundation ofChina (No. LY18A010023) and Zhejiang University of Finance and Economics. J.Wei is partially supported by NSERC of Canada.

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School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou310018, Zhejiang, P. R. China

E-mail address: [email protected]

Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada,

V6T 1Z2


Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada,V6T 1Z2


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