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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 infinity (MN,[hˆ]) and Ps = Ps[g+,hˆ] is the fractional Paneitz operator with the principal symbol hˆ ( ∆ )s. Weareconcerned withtwoloworderperturbations ofthefractional Yamabeequation − hˆ Phsû+ 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 MuPhsûdvhˆ , (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 fõn 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 problem 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 fractional 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 PhsûN42sφ = 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 hyperbolic 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)= κÑ,s δ2+ xδ σ2! −2 for x ∈ RN withκÑ,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