On perturbations of the fractional Yamabe problem

0.39 MB·English

by  Woocheol Choi

On perturbations of the fractional Yamabe problem
Woocheol Choi,SeunghyeokKim
February 9, 2015
Abstract
5 ThefractionalYamabeproblem,proposedbyGonza´lez-Qing(2013)[12],isageometric
1 question which concerns the existence of metrics with constant fractional scalar curvature.
0
It extends the phenomena which were discovered in the classical Yamabe problem and the
2
boundary Yamabe problem to the realm of nonlocal conformally invariant operators. We

b investigate a non-compactness property of the fractional Yamabe problem by constructing
e
bubblingsolutionstoitssmallperturbations.
F

6 2010MathematicsSubjectClassification.Primary:35R11,Secondary: 58J05,35B33,35B44.

KeywordsandPhrases. fractional Yamabe problem, blow-up solutions, nonlocal equations with
]
P criticalexponents
A
.
h
Contents
t
a
m
1 Introduction 2
[

2 2 Preliminaries 5
v 2.1 Reviewonconformalfractional Laplacians . . . . . . . . . . . . . . . . . . . . 5
1
2.2 Caffarelli-Silvestre’s result[10]andChang-Gonza´lez’s extension [12] . . . . . . 6
4
6 2.3 Sharptraceinequality anditsrelatedequations . . . . . . . . . . . . . . . . . . 8
0 2.4 Expansion ofthemetricneartheboundary . . . . . . . . . . . . . . . . . . . . . 8
0
.
1 3 Settingfortheproblem 9
0
3.1 Thefunction spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5
1 3.2 Theapproximate solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
:
v
i 4 Solvabilityoftheintermediateproblem 14
X
4.1 Estimatesfortheerror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
r
a 4.2 Lineartheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Derivationofasolutiontotheintermediate problem . . . . . . . . . . . . . . . . 20
5 Finitedimensionalreduction 21
6 Energyexpansion 22
6.1 TheC0-estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 TheC1-estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7 ConclusionoftheproofofTheorems1.1and1.2andsomeremarks 34
A ProofofLemma3.5 35
1
1 Introduction
Suppose that (XN+1,g+) is an asymptotically hyperbolic (A.H.) manifold with the conformal in-
finity (MN,[hˆ]) and Ps = Ps[g+,hˆ] is the fractional Paneitz operator with the principal symbol
hˆ
( ∆ )s. Weareconcerned withtwoloworderperturbations ofthefractional Yamabeequation
− hˆ
Phsˆu+ fu = uNN−+22ss±ǫ on(M,hˆ), u > 0 on(M,hˆ), (1±)
and
Psu+ǫfu = uNN+22ss on(M,hˆ), u > 0 on(M,hˆ) (2)
hˆ −
where f isaC1-functionon M,ǫ > 0isasmallparameterand s (0,1). (Equations(1+)and(1 )
∈ −
correspond to the supercritical and subcritical problem, respectively.) As one can observe, (1 )
±
is a manifold analogue of the fractional Lane-Emden-Fowler equation with a slightly subcritical
orsupercritical exponent, while(2)canbeviewedasaversion ofthefractional Brezis-Nirenberg
problemonA.H.manifolds.
For s (0,1), Gonza´lez-Qing [29] and Gonza´lez-Wang [30] studied the fractional Yamabe
∈
problem whichisageometric problem tofindametrichˆ intheconformal class[hˆ]ofhˆ withthe
0
constant fractional scalar curvature Qs = Ps (1). The existence of such a metric follows from a
hˆ0 hˆ0
solutionofthenon-local equation
Psu = cuNN+22ss on(M,hˆ), u > 0 on(M,hˆ) (3)
hˆ −
with some c R. As in the classical Yamabe problem, the sign of c depends on that of the
∈
fractional Yamabeinvariant
µs(M)= inf MQhsdvh = inf MuPhsˆudvhˆ , (4)
hˆ h∈[hˆ] RMdvh N−N2s u∈Cu∞>(0M), MRuN2−N2sdvhˆ N−N2s
(cid:16)R (cid:17) (cid:16)R (cid:17)
andthefractional Yamabeproblem issolvable iftheinequality
< µs(M)< µs SN
−∞ hˆ hc
(cid:16) (cid:17)
holds where the manifold (SN,h ) is the N-dimensional unit sphere with the canonical metric as
c
the boundary of the Poincare´ ball. In [29, 30], it is shown that the above inequality is valid for
A.H. manifolds with non-umbilic boundary or the non-locally conformally flat A.H. manifolds
withumbilicboundary undersomeadditional dimensional andtechnicalassumptions.
Since the operator Ps = Ps[g+,hˆ] reduces to the conformal Laplacian if s = 1 and (X,g+)
hˆ
is Poincare´-Einstein (refer to (10)), the fractional Yamabe problem can be understood as a direct
generalization of the classical one towards the non-local conformally invariant operators. This
factbeingoneofthereasons,recentlyintensivestudiesonthefractionalconformaloperatorshave
been conducted by lots of researchers. We refer [28, 36, 1, 13, 33, 34, 35, 48] and references
therein where closely related problems to ours, e.g., the singular fractional Yamabe problem, the
fractional Yamabeflowandthefractional Nirenberg problem areinvestigated.
After the classical Yamabe problem is completely solved by the contribution of Yamabe,
Trudinger, Aubin and Schoen [53, 51, 4, 49], Schoen proposed a question on the compactness
of its solution set. Remarkably, it turned out that the solution set is indeed compact in the C2-
topology provided that the dimension of the background manifold is at most 24 [37], but it may
befalseforsomemanifoldswhosedimension isgreaterthanorequalto25[6,7].
Furthermore, as a low order perturbation, equations (1 ) and (2) in the local case s = 1 have
±
beeninthelimelight(see[40,19,20,21,22,45,47,26]amongotherpossible references). Itwas
revealed that these equations also have aninteresting feature. Inparticular, iftheoperator P1 + f
hˆ
2
is coercive, then the solution set of (1 ) should be compact when N 3 and f < 0 in M ([20]),
− ≥
butnon-compact inthecasethat N 4andthereisaregionof M where f > 0([45,47]).
≥
The main objective of this paper is to extend previous results regarding the compactness or
stability property of conformal operators to the nonlocal setting s (0,1) by considering (1 )
∈ ±
and (2). As a result, a perturbation of the boundary Yamabe problem (corresponding to the case
s = 1/2) is partly covered here as a byproduct of our main results in the case of (1 ). For the
±
existence results of the boundary Yamabe problem in the Euclidean case and in the setting of
compact Riemannian manifolds, see Adimurthi-Yadava [2], Escobar [23] and Marques [42]. We
also should mention that equations with s = 2 (see (11)) were investigated in Deng-Pistoia [18]
andPistoia-Vaira[46].
FortheexistenceofsolutionstothefractionalYamabeproblem,equivalentminimizationprob-
lemsto(4)whichonlycontain local differential operators canbederived byexploiting theexten-
sion theorem of Chang and Gonza´lez [12]. The authors of [29, 30] utilized this observation to
deduce the existence result, instead finding a minimizer that attains the Yamabe invariant µs(M)
hˆ
in a direct manner. After the fundamental extension result of Caffarelli and Silvestre [10] for
thefractional Laplacians onRN,suchastandpoint, introducing andstudying equivalent extended
local problems rather than considering nonlocal problems itself, has been highlighted by many
researchers. Seeforexample[9,5,50,8,16,14]andreferences therein. Inthispaper,wekeepon
usethisstrategy.
Accordingto[12](seeProposition2.1below),itisnaturaltoconsiderthefollowingdegenerate
equationwiththeweightedNeumannboundary condition
div ρ1 2s U +E(ρ)U = 0 in(X,g¯) and ∂sU = 0 on(M,hˆ) (5)
− − ∇ ν
(cid:16) (cid:17)
where
∂U Γ(s)
∂sU := κ lim ρ1 2s withκ := (6)
ν − s·ρ 0+ − ∂ρ s 21 2sΓ(1 s)
→ − −
(ν is the outward normal vector to M = ∂X) in order to understand equations with the fractional
PaneitzoperatorPs. LetHbethetraceofthesecondfundamentalformπof(M,hˆ)astheboundary
hˆ
of(X,g¯)andH1(X;ρ1 2s)theweightedSobolevspacewhoseprecisedefinitionisgiveninSection
−
3. Ourpaperdealswiththesituationwhenthefirsteigenvalueof (5)ispositive(modulotheeffect
of the function f˜ to be introduced below), that is, there exists a constant C > 0 such that the
inequality
ρ1 2s U2+E(ρ)U2 dv + f˜U2dv C ρ1 2sU2dv (7)
Z − |∇ |g¯ g¯ Z hˆ ≥ Z − g¯
X(cid:16) (cid:17) M X
holdsforarbitrary functions U H1(X;ρ1 2s),wherethefunction f˜on M isdefinedtobe
−
∈
f if (1 )isconsidered,
f˜= ±

0 if (2)isconsidered.

Under the coercivity assumption (7), we have the following non-compactness result for (1 ).
Recall that for any C1 function ψ on M, a critical point x M is called to be C1-stable if the±re
0
∈
is a small neighborhood Λ of x in M such that ψ(x) = 0 for some x Λ implies x = x and
0 0
∇ ∈
deg( ψ,Λ,0) , 0 (see [39]). Here deg denotes the Brouwer degree. It is well-known that any
∇
isolated local minimum point and maximum point is aC1-stable critical point. Moreover, so is a
nondegenerate criticalpointifψisaC2-function.
Theorem1.1. Supposethat s (0,1),N > max 4s,1 andH = 0if s [1/2,1). Assumealsothat
∈ { } ∈
(7)istrue.
1. If the function f possesses aC1-stable critical point σ M such that f(σ ) > 0, then for
0 0
∈
sufficiently small ǫ > 0 equation (1+)admits a positive solution uǫ C1,β(M)which blows
∈
upatσ asǫ 0.
0
→
3
2. If the function f possesses aC1-stable critical point σ M such that f(σ ) < 0, then for
0 0
∈
sufficiently small ǫ > 0 equation (1 )admits a positive solution u C1,β(M)which blows
ǫ
− ∈
upatσ asǫ 0.
0
→
HeretheHo¨lderexponent β (0,1)isdeterminedby N and s.
∈
Furthermore, we can obtain an existence theorem for (2) where the geometric object H plays an
importantrole.
Theorem 1.2. Suppose that s (0,1/2), N 2, aswellas(7)hold. Also, letλ : M [0, ]be
∈ ≥ → ∞
afunction definedas
1
−4N(1−2s)sd1f(σ) 1−2s if H(σ), 0and f(σ) ( ,0],
λ(σ)=  (cid:0)2N(N −1)+(cid:0)1−4s2(cid:1)(cid:1)dsH(σ)! otherwise H(σ) ∈ −∞
where the positiv∞e numbers d and d are given in Subsection 6.1. If (λ ,σ ) := (λ(σ ),σ ) is a
1 s 0 0 0 0
C1-stablecriticalpointofthefunction
d 2N(N 1)+ 1 4s2
J(λ,σ)= 1 f(σ)λ2s + − − d H(σ)λ for(λ,σ) (0, ) M
2  4N(1 (cid:16)2s) (cid:17) s ∈ ∞ ×
e  − 
suchthatλ(σ )> 0,thenforǫ > 0smallenoughequation(2)hasapositivesolutionu C1,β(M)
0 ǫ
∈
which blows up at σ as ǫ 0. Furthermore σ is necessarily a critical point of the function
0 0
→
f /H2s on M. Theexponent β (0,1)againdepends onN and s.
| | | | ∈
The analogous existence results to ours in the Euclidean setting, that is, a proof for the exis-
tenceofsolutions forthefractionalLane-Emden-FowlerequationandtheBrezis-Nirenberg prob-
lem in smooth bounded domains of RN can be found in [14, 17]. While we are studying here a
smallperturbationofequation(3)definedongeneralmanifoldstounderstanditsnon-compactness
characteristic,onemayaddressadualproblem: toconstructaparticularmetricforwhichoriginal
equation (3)has the solution set that isnot L -bounded. Itis investigated in [38], which extends
∞
[6,7,3,52]toanonlocal setting.
Todeduce ourexistence result, weshallemploythefinitedimensional Lyapunov-Schmidt re-
ductionmethod. Asfarasweknow,thispaperisthefirstattempttoapplythereductionprocedure
towardsequations withthefractional Paneitzoperatorsdefinedingeneralmanifolds. Forapplica-
tionsofthereductionmethodtothefractionalLaplaciansintheEuclideansettingorthefractional
Paneitzoperators underaparticular choiceofthemetric,wereferto[14,16,38]andsoon.
OurproblemsrequiremoredelicatecomputationscomparedtoproblemsonEuclideanspaces.
ThemainreasonmakingthemharderisthatthefractionalPaneitzoperatorPs = Ps[g+,hˆ]depends
hˆ
not only on the metric hˆ on the boundary M, but also on the metric g+ in the interior X. In other
words, the boundary M does not contain whole information in contrast with problems with frac-
tional Laplacians ( ∆)s on the Euclidean spaces, and so it is inevitable to look carefully how the
−
interior X plays aroleinourproblem. Thisisachievedbyinspecting theextended problem given
inProposition 2.1. Toovercome the other difficulties weface, wehave to also establish a certain
regularity result (Lemma 3.3), compute decay of the s-harmonic extensions of the bubbles (19)
(Lemma 3.5), use the weighted Sobolev trace inequality (27) for compact manifolds elaborately,
employ the dual characterization of the norm (29) in estimating the error term (Lemma 4.1) and
others.
Notations.
-AnelementoftheupperhalfspaceRN+1 isdenoted by(x,t)where x RN andt > 0.
+
∈
4
- For any weakly differentiable function U on RN+1, we denote U = (∂ U, ,∂ U) and
+ ∇x x1 ··· xN
U = ( U,∂U). Also∂ isoftenwrittenas∂.
-∇B+ =∇BxN+1(t0,r) RN+1xiis the (N + 1)-dimeinsional half open ball of radius r centered at the
r ∩ +
origin.
-u+ = max u,0 andu = max u,0 .
-Γdenotes{theG} amma−function{−. }
-Forany N Nand s (0,min 1,N/2 ),wedenote p= N+2s.
∈ ∈ { } N 2s
-C > 0isageneric constant, whichmaychangelinebylin−e.
2 Preliminaries
Inthissection,wepresentsomegeometricandanalyticalbackgroundstounderstandourproblem.
Mostofmaterialsaretakenfrom[12,29,23,10,8].
2.1 Reviewonconformal fractional Laplacians
Let (XN+1,g+) be an (N +1)-dimensional smooth Riemannian manifold with the boundary MN.
We call a function ρ on the closure X of X a defining function of the boundary M if ρ > 0 in X,
ρ = 0 on M and dρ , 0 on M. The manifold (X,g+) is said to be conformally compact (C.C.)if
there is a defining function ρ making (X,g¯)be compact where g¯ := ρ2g+. Also, given the metric
hˆ = g¯ , the boundary (M,[hˆ]) with the conformal class [hˆ] of hˆ is called the conformal infinity.
M
| +
A C.C. metric g is asymptotically hyperbolic (A.H.) if the sectional curvature approaches to -1
attheinfinity M,whosemodelcaseisthehyperbolic space:
dx2 +dt2 4(dx2 +dt2)
(X,g+)= (HN+1,gH) = R+N+1, | | or BN+1, | | .
t2 ! (1 x2 t2)2!
−| | −
According toGraham-Lee[31],foranA.H.manifold X andarepresentative hˆ fortheconfor-
malclasson(M,[hˆ]),thereisauniquespecialdefiningfunctionsuchthat
g+ = ρ 2 dρ2+h , h = hˆ +O(ρ)
− ρ ρ
(cid:16) (cid:17)
near M. Itiscalledthegeodesic boundary definingfunction.
Suppose that z C, Re(z) > N/2 and f C (M). Then, by [43, 32], unless z(N z) is an
∞
∈ ∈ −
L2-eigenvalue of ∆g+,thefollowingeigenvalue problem
−
∆g+ z(N z) V = 0inX (8)
− − −
h i
hasasolutionoftheform
V = FρN−z +Gρz, F,G C∞(X)and Fρ=0 = f. (9)
∈ |
Throughout thepapertheexistence ofsuchasolution isalwaysassumed. Thescattering operator
on M isthendefinedtobe
S(z)f =G ,
M
|
which is a meromorphic family of pseudo-differential operators in z C : Re(z) > N/2 . In
{ ∈ }
addition, weintroduce itsnormalization socalledthefractional Paneitzoperator Ps,namely
hˆ
sΓ(s) N
22s S + s for s< N,
Phsˆ = Ps[g+,hˆ]= −( 1)sΓ2(21ss−!(ss) 1(cid:18))2! Res(cid:19)z=N/2+sS(z) for s N,
− − · ∈

5
whose principal symbol is exactly ( ∆ )s. In the special case that (X,g+) is Poincare´-Einstein
− hˆ
(bothC.C.andEinstein)and s= 1or2,wehave
N 2
P1u = ∆ u+ − R u (10)
hˆ − hˆ 4(N 1) hˆ
−
theusualconformalLaplacian, and
N 4
P2u = ( ∆ )2u div c˜ R hˆ c˜ Ric du + − Q u (11)
hˆ − hˆ − hˆ 1 hˆ − 2 hˆ 2 hˆ
(cid:16)(cid:16) (cid:17) (cid:17)
the Paneitz operator. Here Q stands for the Branson’s Q-curvature and c˜ ,c˜ > 0 are constants.
1 2
Theimportantproperty ofPs isthatitisconformally covariant inthesensethat
hˆ
PhsˆuN42sφ = u−NN−+22ssPhsˆ(uφ) foranyfunctionu > 0on M.
−
Finally,wesetthefractional scalarcurvature Qs byPs(1).
hˆ hˆ
2.2 Caffarelli-Silvestre’s result[10]and Chang-Gonza´lez’s extension [12]
In this subsection, we recall the observation of Chang and Gonza´lez [12] which identifies two
fractional Laplacians arising in different contexts: one given as normalized scattering operators
[32] described above and one originated from the Dirichlet-Neumann operators due to Caffarelli
andSilvestre[10].
For s (0,1),letD1(RN+1;t1 2s)bethecompletionofC (RN+1)withrespecttotheweighted
∈ + − c∞ +
Sobolevnorms
1/2
kUkD1(R+N+1;t1−2s) := ZRN+1t1−2s|∇U(x,t)|2dxdt!
+
withtheweightt1 2s. Furthermore,wedesignatebyHs(RN)thestandardfractionalSobolevspace
−
givenas
1/2
Hs RN = u L2 RN : u := 1+ ξ2s uˆ(ξ)2dξ <
(cid:16) (cid:17)  ∈ (cid:16) (cid:17) k kHs(RN) ZRN (cid:16) | | (cid:17)| | ! ∞
whereuˆ denotestheFouriertransformofu,anddefinethefractionalLaplacian( ∆)s: Hs(RN)
− →
H s(RN)tobe
−
(([∆)su)(ξ)= (2πξ)2suˆ(ξ) foranyξ RN givenu Hs RN .
− | | ∈ ∈
(cid:16) (cid:17)
InthecelebratedworkofCaffarelliandSilvestre[10],theauthorsfoundthatifU D1(RN+1;t1 2s)
+ −
∈
isauniquesolution oftheequation
div t1 2s U = 0 inRN+1,
− ∇ + (12)
 U(x(cid:16),0) = u(x(cid:17)) for x RN,
 ∈

providedafixedfunctionu Hs(RN),then( ∆)su = ∂sU wherethedefinitionoftheweighted
ν RN
∈ − |
normalderivative∂s isgivenin(6). LetuscallthisU the s-harmonicextension ofuanddenoteit
ν
byExts(u).
It turned out that this extension result is a special case of the following proposition obtained
byChangandGonza´lez[12]. WealsorefertoSection2of[29].
6
Proposition 2.1. ([12, Theorem 5.1 and Theorem 4.3]) Let (XN+1,g+) be an asymptotically hy-
perbolic manifold withthe conformal infinity (MN,[hˆ])and ρthe geodesic defining function ofhˆ.
AssumealsothatH = 0if s (1/2,1). Forasmoothfunctionuon M,ifV isasolutionof (8)and
satisfies(9)inwhich f issub∈stituted withu,thefunction U := ρz NV solves
−
div ρ1 2s U +E(ρ)U = 0 in(X,g¯) and U = u on(M,hˆ)
−
− ∇
giventhat E(ρ) := ρ−(cid:16)1−z( ∆g+(cid:17) z(N z))ρN−z,2z := N +2sandg¯ := ρ2g+. Moreover,
− − −
∂sU for s (0,1) 1/2 ,
Psu = ν ∈ \{ }
hˆ ( ∂νsU + N2−N1Hu for s= 1/2.
Here H denotes thetrace ofthe second fundamental form(π ) = ∂,∂ on M = ∂X and
ij −h∇∂ρ i jihˆ
theoperator ∂s istheweightednormalderivative definedin(6)wi(cid:16)thtreplaced b(cid:17)yρ.
ν
Forsufficiently smallr > 0,italsoholdsthat
1
N 2s
E(ρ) = − Rg¯ρ1−2s Rg+ +N(N +1) ρ−1−2s on M (0,r1). (13)
4N − ×
h (cid:16) (cid:17) i
Remark2.2. Sinceitholdsthat
Rg+ = N(N +1)+Nρ∂ρlog(deth(ρ))+ρ2Rg¯ on M (0,r1)
− ×
and
∂ log(deth(ρ)) = Tr h(ρ) 1∂ h(ρ) = 2H,
theremainder term E(ρ)ρin(13)isredu(cid:12)(cid:12)(cid:12)ρc=e0dto (cid:16) − ρ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)ρ=0 −
N 2s
E(ρ)(z) = − ∂ log(deth(ρ))(σ)ρ 2s
ρ −
− 4 !
N 2s N 2s
= − ∂ log(deth(ρ)) (σ)ρ 2s +O ρ1 2s = − H(σ)ρ 2s +O ρ1 2s
− 4 ! ρ ρ=0 − − 2 ! − −
(cid:12) (cid:16) (cid:17) (cid:16) (cid:17)
(cid:12) (14)
(cid:12)
forz = (σ,ρ) M (0,r ).
1
∈ ×
Inparticular, ourmainequation (1 )isequivalent totheproblem
±
div ρ1 2s U +E(ρ)U = 0 in(X,g¯),
−
− ∇
 ∂sU =(cid:16) up ǫ f(cid:17)u on(M,hˆ), (15 )
 Uν= u> 0± − on(M,hˆ) ±
anditremainsthesameaswellexceptthesecondequation in(15 )isreplaced by
±
∂sU = up ǫfu for s (0,1/2) on(M,hˆ) (16)
ν
− ∈
ifwedealwith(2).
In [12], it is also proved that given a geodesic defining function ρ, there is another special
definingfunction ρ suchthat E(ρ ) = 0.
∗ ∗
Proposition 2.3. ([12, Theorem 4.7], [29, Proposition 2.2]) Assume that H = 0 if s (1/2,1).
∈
Forasmoothfunction uon M,ifV satisfies (8)aswellas(9)inwhich f issubstituted withu,the
functionU := (ρ )z NV isasolution of
∗ −
div ρ 1 2s U = 0 in(X,g ) and U = u on(M,hˆ) (17)
∗ − ∗
− ∇
whereg := (ρ )2g+. Mo(cid:16)(cid:0)reo(cid:1)verg (cid:17)= hˆ,(ρ /ρ) = 1and
∗ ∗ ∗ M ∗ M
| |
Psu = ∂sU +Qsu (18)
hˆ ν hˆ
whereQs isthefractional scalar curvature andtheoperator ∂s isdefinedin(6)withtsubstituted
hˆ ν
withρ .
∗
Thisobservationisusefulinshowingapriori L -estimateorthestrongmaximumprincipleofthe
∞
operator Ps. Referto[29,Section3]. (cf. Lemma3.3andProposition 5.1below)
hˆ
7
2.3 Sharp trace inequality andits related equations
Givenanynumberδ > 0andpointσ = (σ , ,σ ) RN,let
1 N
··· ∈
N 2s
N 2s Γ N+2s 4−s
wδ,σ(x)= κ˜N,s δ2+ xδ σ2! −2 for x ∈ RN withκ˜N,s = 2N−22s Γ(cid:16)N22s(cid:17) . (19)
| − |  (cid:16) −2 (cid:17)
Itsconstant multiplesattaintheequalityforthesharpSobolevinequality
N 2s 1
ZRN |u|N2−N2sdx! 2−N ≤ SN,s ZRN (cid:12)(−∆)s/2u(cid:12)2dx!2
(cid:12) (cid:12)
(cid:12) (cid:12)
where > 0istheoptimalSobolevconstant, andinparticular solve
N,s
S
( ∆)su= up, u > 0 inRN and lim u(x) = 0 (20)
− x
| |→∞
(see [41]). Set also W = Exts(w ), the s-harmonic extension of w . Then we observe that
δ,σ δ,σ δ,σ
extremalfunctions ofSobolevtraceinequality
N 2s 1
ZRN |U(x,0)|N2−N2sdx! 2−N ≤ S√Nκ,ss Z0∞ZRN t1−2s|∇U(x,t)|2dxdt!2 , (21)
havetheformU(x,t) = cW (x,t)foranyc > 0, δ > 0andσ RN,whereκ > 0istheconstant
δ,σ s
∈
definedin(6). Moreover, byitsdefinition, W solves
δ,σ
div t1 2s U = 0 inRN+1,
− +
∇
 ∂sU(cid:16)= Up (cid:17) onRN 0 , (22)
 Uν= wδ,σ onRN ×{0}
andasanimmediateconsequence wehave ×{ }
2N
κ t1 2s W 2dxdt = wN 2sdx. (23)
sZRN+1 − |∇ δ,σ| ZRN δ,σ−
+
Ontheotherhand,intheworkofDa´vila,delPinoandSire[15],itwasrevealedthatthesetof
solutions bounded onΩ 0 totheequation
×{ }
div t1 2s Φ = 0 inRN+1,
− +
∇ (24)
 ∂νsΦ(cid:16)= pwδp,−σ1(cid:17)Φ onRN ×{0},
consistsofthelinearcombinations of
∂W ∂W ∂W
Z1 := δ,σ, , ZN := δ,σ and Z0 := δ,σ. (25)
δ,σ ∂σ ··· δ,σ ∂σ δ,σ ∂δ
1 N
This fact is crucial in applying the reduction method to our problem. Hereafter, we will denote
w = w ,W = W ,zi = zi andZi = Zi fori = 0, ,N.
δ δ,0 δ δ,0 δ δ,0 δ δ,0 ···
2.4 Expansionofthe metricnear the boundary
Suppose that (X,g¯) is a compact Riemannian manifold and 0 M = ∂X. Let x = (x , ,x )
1 N
∈ ···
be normal coordinates on M at the point 0 and (x , ,x ,t)be the Fermicoordinates on X at 0
1 N
···
where x , ,x Randt > 0. Also,wedenote
1 N
··· ∈
g¯ = dt2+h (x,t)dxdx
ij i j
sothath = g¯ . Thenthefollowingasymptotic expansion ofthemetricnear0isvalid.
TM
|
8
Lemma2.4. [23,Lemma3.1,3.2]For x1, ,xN andt := xN+1 small,itholdsthat
···
1 1
g¯ = h = 1 Ht+ H2 π 2 Ric(∂) t2 H xt R x x +O (x,t)3
| | | | − 2 −k kh− t − i i − 6 ij i j | |
p p (cid:16) (cid:17) (cid:16) (cid:17)
and
1
hij = δij+2πijt Ri jx x +hij x t+ 3πikπ j+Ri j t2+O (x,t)3
− 3 kl k l ,(N+1)k k m n n | |
(cid:16) (cid:17) (cid:16) (cid:17)
where π is the second fundamental form of M = ∂X, H is its trace, i.e., N times of the mean
curvature, R denotes a component of the Ricci tensor, R is a component of the Riemannian
ij ijkl
tensorandRic(∂t)= gijRi(N+1)j(N+1). Also,theindices i, jandkrunfrom1toN.
3 Setting for the problem
3.1 The function spaces
Asbefore,let(XN+1,g+)beanA.H.manifoldwiththeboundary(Mn,hˆ)andρthegeodesicdefin-
ing function, so that (X,g¯) where g¯ = ρ2g+ is a compact manifold. Denote by H1(X;ρ1 2s) the
−
weightedSobolevspaceendowedwiththeinnerproduct
U,V := ρ1 2s ( U, V) +UV dv
H1(X;ρ1 2s) − g¯ g¯
h i − Z ∇ ∇
X h i
andthenorm
1/2
U := ρ1 2s U2+U2 dv . (26)
k kH1(X;ρ1−2s) Z − |∇ |g¯ g¯!
X (cid:16) (cid:17)
Byapplying(21)andthestandardpartitionofunityargument,weobtainamanifoldversionofthe
weightedSobolevtraceinequality
U C U (27)
2N H1(X;ρ1 2s)
k kLN 2s(M) ≤ k k −
−
whereC > 0isaconstant determined by s, N and X. Inaddition, theembedding H1(X;ρ1 2s) ֒
−
→
Lq(M) is compact for any 1 q < 2N . The next two lemmas provide equivalent norms to the
≤ N 2s
H1(X;ρ1 2s)-norm. −
−
1/2
Lemma3.1. Thenorm ρ1 2s U2dv + U2dv is equivalent tothe norm U
X − |∇ |g¯ g M hˆ k kH1(X;ρ1−2s)
definedin(26). (cid:16)R R (cid:17)
Proof. WefirstconsiderafunctionU definedon B+ forsomeR > 0where B+ = (x,t) RN+1 :
2R 2R { ∈ +
(x,t) < 2R, t > 0 . Foreach0 t R,usingtheelementarycalculusandHo¨lder’sinequality we
| | } ≤ ≤
have
t t 1/2 t 1/2
U(x,t) U(x,0) + ∂ U(x,r)dr U(x,0) + r2s 1dr r1 2s∂ U(x,r)2dr
r − − r
| | ≤ | | Z | | ≤ | | Z ! Z | | !
0 0 0
ts R 1/2
= U(x,0) + r1 2s∂ U(x,r)2dr .
− r
| | √2s Z0 | | !
Foranygivennumbera ( 1,1),weapplytheaboveestimatetoget
∈ −
R
taU(x,t)2dxdt
Z Z | |
0 x R
| |≤
R 1 R R
2 tadt U(x,0)2dx+ ta+2sdt r1 2s∂ U(x,r)2drdx (28)
− r
≤ Z !Z | | s Z !Z Z | |
0 x R 0 x R 0
| |≤ | |≤
R
C U(x,0)2dx+ t1 2s U(x,t)2dxdt .
−
≤ Z | | Z Z |∇ | !
x R 0 x R
| |≤ | |≤
9
Employingthisinequality witha = 1 2sineachlocalchart,wecanobtainthat
−
1/2 1/2
ρ1 2sU2dv C ρ1 2s U2dv + U2dv .
Z − | | g¯! ≤ Z − |∇ |g¯ g¯ Z hˆ!
X X M
Ontheotherhand,theweightedtraceinequality (27)andHo¨lder’sinequality yield
1/2 1/2
U2dv C ρ1 2s U2+U2 dv .
Z | | hˆ! ≤ Z − |∇ |g¯ g¯!
M X (cid:16) (cid:17)
Thesetwoestimatesenableustogettheequivalence ofthetwonorms,concluding theproof. (cid:3)
Lemma 3.2. Suppose that the trace of the second fundamental form H of M = ∂X vanishes if
s [1/2,1). Undertheassumption that (7)holds,
∈
1/2
U := κ ρ1 2s U2+E(ρ)U2 dv + f˜U2dv (29)
k kf˜ sZ − |∇ |g¯ g¯ Z hˆ!
X(cid:16) (cid:17) M
gives an equivalent norm to (26). Hence one can define the inner product , from the norm
h· ·if˜
throughthepolarization identity.
k·kf˜
Proof. Suppose first that s [1/2,1). In this case, the condition H = 0 is assumed, so E(ρ)
∈ | | ≤
Cρ1 2s by(14). Usingthisfactand(27)also,weimmediately obtainthat U C U .
If s− (0,1/2),thenonecancontroltheintegralvalueofU neartheboundakrykbHy1t(Xak;ρi1n−2gs)a≥= k2skifn˜
∈ −
(28)and applying (27). Additionally, by realizing that ρisbounded awayfrom 0inany compact
subset of X, it is possible to manage the integral of U in the interior of X. Combining the both
estimates,wededucethesameinequality U C U .
k kH1(X;ρ1−2s) ≥ k kf˜
Supposethattheoppositeinequalitydoesnothold. Thenthereisasequence U suchthat
{ n}∞n=1
U 0asn but U = 1foralln N. Letusfirstclaimthat E(ρ)U2 0.
k nkf˜ → → ∞ k nkH1(X;ρ1−2s) ∈ X n →
By (7), we have ρ1 2sU2 0, so the claim is verified at once if H = 0. If sR (0,1/2) and
X − n → ∈
H , 0,thenthemRainorder of E(ρ)isρ 2s as(14)indicates. Inthissituation, wetakea < 1close
−
to1andusetheHo¨lder’sinequality toget
η 1 η
lim ρ 2sU2 lim ρ1 2sU2 ρ aU2 − = 0
n→∞ZX − n ≤ n→∞ ZX − n! ZX − n!
where η = a 2s (0,1), so we can justify our claim again. Observe that U =
a −2s+1 ∈ k nkH1(X;ρ1−2s)
1, (27) and (2−8) guarantee boundedness of the value ρ aU2 ∞ . Now if we let U be the
X − n n=1 ∞
H1(X;ρ1 2s)-weak limit ofU , then U 0. Thus comnRpactnessoof the trace embedding gives us
− n
that f˜U2 f˜U2 = 0. However∞, it≡isacontradiction because previous computations show
thatRMρ1 2ns→URM2 sho∞uldconverge toboth0and1. Thisprovesthat U C U . (cid:3)
X − |∇ n| k kf˜ ≥ k kH1(X;ρ1−2s)
R
By(27),weknowthatthetraceoperator i : H1(X;ρ1 2s) Lp+1(M)givenasi(U) = U := uis
− M
→ p+1 |
well-defined and continuous. Thus the adjoint operator i∗f˜ : L p (M) → H1(X;ρ1−2s) defined by
theequation
div ρ1 2s U +E(ρ)U = 0 in(X,g¯),
−
− ∇
 ∂sU =(cid:16) v f˜u (cid:17) on(M,hˆ), (30)
 Uν= u − on(M,hˆ),
with U = i (v) is bounded in light of Lemma 3.2. Furthermore, i : H1(X;ρ1 2s) Lq(M)
∗f˜ − → ⊃
Lp+1(M)for1 q < p+1iscompact.
≤
10