On perturbations of the fractional Yamabe problem
aa r X i v : . [ m a t h . A P ] F e b On perturbations of the fractional Yamabe problem
Woocheol Choi, Seunghyeok KimJuly 15, 2018
Abstract
The fractional Yamabe problem, proposed by Gonz´alez-Qing (2013) [12], is a geometricquestion which concerns the existence of metrics with constant fractional scalar curvature.It extends the phenomena which were discovered in the classical Yamabe problem and theboundary Yamabe problem to the realm of nonlocal conformally invariant operators. Weinvestigate a non-compactness property of the fractional Yamabe problem by constructingbubbling solutions to its small perturbations.
Primary: 35R11, Secondary: 58J05, 35B33, 35B44.
Key words and Phrases. fractional Yamabe problem, blow-up solutions, nonlocal equations withcritical exponents
Contents ff arelli-Silvestre’s result [10] and Chang-Gonz´alez’s extension [12] . . . . . . 62.3 Sharp trace inequality and its related equations . . . . . . . . . . . . . . . . . . 82.4 Expansion of the metric near the boundary . . . . . . . . . . . . . . . . . . . . . 8 C -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 The C -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Introduction
Suppose that ( X N + , g + ) is an asymptotically hyperbolic (A.H.) manifold with the conformal in-finity ( M N , [ˆ h ]) and P s ˆ h = P s [ g + , ˆ h ] is the fractional Paneitz operator with the principal symbol( − ∆ ˆ h ) s . We are concerned with two low order perturbations of the fractional Yamabe equation P s ˆ h u + f u = u N + sN − s ± ǫ on ( M , ˆ h ) , u > M , ˆ h ) , (1 ± )and P s ˆ h u + ǫ f u = u N + sN − s on ( M , ˆ h ) , u > M , ˆ h ) (2)where f is a C -function on M , ǫ > s ∈ (0 , + ) 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 subcriticalor supercritical exponent, while (2) can be viewed as a version of the fractional Brezis-Nirenbergproblem on A.H. manifolds.For s ∈ (0 , h in the conformal class [ˆ h ] of ˆ h with theconstant fractional scalar curvature Q s ˆ h = P s ˆ h (1). The existence of such a metric follows from asolution of the non-local equation P s ˆ h u = cu N + sN − s on ( M , ˆ h ) , u > M , ˆ h ) (3)with some c ∈ R . As in the classical Yamabe problem, the sign of c depends on that of thefractional Yamabe invariant µ s ˆ h ( M ) = inf h ∈ [ˆ h ] R M Q sh d v h (cid:16)R M d v h (cid:17) N − sN = inf u ∈ C ∞ ( M ) , u > R M uP s ˆ h ud v ˆ h (cid:16)R M u NN − s d v ˆ h (cid:17) N − sN , (4)and the fractional Yamabe problem is solvable if the inequality −∞ < µ s ˆ h ( M ) < µ sh c (cid:16) S N (cid:17) holds where the manifold ( S N , h c ) is the N -dimensional unit sphere with the canonical metric asthe boundary of the Poincar´e ball. In [29, 30], it is shown that the above inequality is valid forA.H. manifolds with non-umbilic boundary or the non-locally conformally flat A.H. manifoldswith umbilic boundary under some additional dimensional and technical assumptions.Since the operator P s ˆ h = P s [ g + , ˆ h ] reduces to the conformal Laplacian if s = X , g + )is Poincar´e-Einstein (refer to (10)), the fractional Yamabe problem can be understood as a directgeneralization of the classical one towards the non-local conformally invariant operators. Thisfact being one of the reasons, recently intensive studies on the fractional conformal operators havebeen conducted by lots of researchers. We refer [28, 36, 1, 13, 33, 34, 35, 48] and referencestherein where closely related problems to ours, e.g., the singular fractional Yamabe problem, thefractional Yamabe flow and the fractional Nirenberg problem are investigated.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 compactnessof its solution set. Remarkably, it turned out that the solution set is indeed compact in the C -topology provided that the dimension of the background manifold is at most 24 [37], but it maybe false for some manifolds whose dimension is greater than or equal to 25 [6, 7].Furthermore, as a low order perturbation, equations (1 ± ) and (2) in the local case s = P h + f
2s coercive, then the solution set of (1 − ) should be compact when N ≥ f < M ([20]),but non-compact in the case that N ≥ M where f > s ∈ (0 ,
1) by considering (1 ± )and (2). As a result, a perturbation of the boundary Yamabe problem (corresponding to the case s = /
2) is partly covered here as a byproduct of our main results in the case of (1 ± ). For theexistence results of the boundary Yamabe problem in the Euclidean case and in the setting ofcompact Riemannian manifolds, see Adimurthi-Yadava [2], Escobar [23] and Marques [42]. Wealso should mention that equations with s = ff erential operators can be derived by exploiting the exten-sion theorem of Chang and Gonz´alez [12]. The authors of [29, 30] utilized this observation todeduce the existence result, instead finding a minimizer that attains the Yamabe invariant µ s ˆ h ( M )in a direct manner. After the fundamental extension result of Ca ff arelli and Silvestre [10] forthe fractional Laplacians on R N , such a standpoint, introducing and studying equivalent extendedlocal problems rather than considering nonlocal problems itself, has been highlighted by manyresearchers. See for example [9, 5, 50, 8, 16, 14] and references therein. In this paper, we keep onuse this strategy.According to [12] (see Proposition 2.1 below), it is natural to consider the following degenerateequation with the weighted Neumann boundary condition − div (cid:16) ρ − s ∇ U (cid:17) + E ( ρ ) U = X , ¯ g ) and ∂ s ν U = M , ˆ h ) (5)where ∂ s ν U : = − κ s · lim ρ → + ρ − s ∂ U ∂ρ with κ s : = Γ ( s )2 − s Γ (1 − s ) (6)( ν is the outward normal vector to M = ∂ X ) in order to understand equations with the fractionalPaneitz operator P s ˆ h . Let H be the trace of the second fundamental form π of ( M , ˆ h ) as the boundaryof ( X , ¯ g ) and H ( X ; ρ − s ) the weighted Sobolev space whose precise definition is given in Section3. Our paper deals with the situation when the first eigenvalue of (5) is positive (modulo the e ff ectof the function ˜ f to be introduced below), that is, there exists a constant C > Z X (cid:16) ρ − s |∇ U | g + E ( ρ ) U (cid:17) d v ¯ g + Z M ˜ f U d v ˆ h ≥ C Z X ρ − s U d v ¯ g (7)holds for arbitrary functions U ∈ H ( X ; ρ − s ), where the function ˜ f on M is defined to be˜ f = f if (1 ± ) is considered , . Under the coercivity assumption (7), we have the following non-compactness result for (1 ± ).Recall that for any C function ψ on M , a critical point x ∈ M is called to be C -stable if thereis a small neighborhood Λ of x in M such that ∇ ψ ( x ) = x ∈ Λ implies x = x anddeg( ∇ ψ, Λ , , C -stable critical point. Moreover, so is anondegenerate critical point if ψ is a C -function. Theorem 1.1.
Suppose that s ∈ (0 , , N > max { s , } and H = if s ∈ [1 / , . Assume also that (7) is true.1. If the function f possesses a C -stable critical point σ ∈ M such that f ( σ ) > , then forsu ffi ciently small ǫ > equation (1 + ) admits a positive solution u ǫ ∈ C ,β ( M ) which blowsup at σ as ǫ → . . If the function f possesses a C -stable critical point σ ∈ M such that f ( σ ) < , then forsu ffi ciently small ǫ > equation (1 − ) admits a positive solution u ǫ ∈ C ,β ( M ) which blowsup at σ as ǫ → .Here the H ¨older exponent β ∈ (0 , is determined by N and s. Furthermore, we can obtain an existence theorem for (2) where the geometric object H plays animportant role. Theorem 1.2.
Suppose that s ∈ (0 , / , N ≥ , as well as (7) hold. Also, let λ : M → [0 , ∞ ] bea function defined as λ ( σ ) = − N (1 − s ) sd f ( σ ) (cid:0) N ( N − + (cid:0) − s (cid:1)(cid:1) d s H ( σ ) ! − s if H ( σ ) , and f ( σ ) H ( σ ) ∈ ( −∞ , , ∞ otherwisewhere the positive numbers d and d s are given in Subsection 6.1. If ( λ , σ ) : = ( λ ( σ ) , σ ) is aC -stable critical point of the function e J ( λ, σ ) = d f ( σ ) λ s + N ( N − + (cid:16) − s (cid:17) N (1 − s ) d s H ( σ ) λ for ( λ, σ ) ∈ (0 , ∞ ) × Msuch that λ ( σ ) > , then for ǫ > small enough equation (2) has a positive solution u ǫ ∈ C ,β ( M ) which blows up at σ as ǫ → . Furthermore σ is necessarily a critical point of the function | f | / | H | s on M. The exponent β ∈ (0 , again depends on N and s. The analogous existence results to ours in the Euclidean setting, that is, a proof for the exis-tence of solutions for the fractional Lane-Emden-Fowler equation and the Brezis-Nirenberg prob-lem in smooth bounded domains of R N can be found in [14, 17]. While we are studying here a small perturbation of equation (3) defined on general manifolds to understand its non-compactnesscharacteristic, one may address a dual problem: to construct a particular metric for which original equation (3) has the solution set that is not L ∞ -bounded. It is investigated in [38], which extends[6, 7, 3, 52] to a nonlocal setting.To deduce our existence result, we shall employ the finite dimensional Lyapunov-Schmidt re-duction method. As far as we know, this paper is the first attempt to apply the reduction proceduretowards equations with the fractional Paneitz operators defined in general manifolds. For applica-tions of the reduction method to the fractional Laplacians in the Euclidean setting or the fractionalPaneitz operators under a particular choice of the metric, we refer to [14, 16, 38] and so on.Our problems require more delicate computations compared to problems on Euclidean spaces.The main reason making them harder is that the fractional Paneitz operator P s ˆ h = P s [ g + , ˆ h ] dependsnot only on the metric ˆ h on the boundary M , but also on the metric g + in the interior X . In otherwords, 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 theinterior X plays a role in our problem. This is achieved by inspecting the extended problem givenin Proposition 2.1. To overcome the other di ffi culties we face, we have to also establish a certainregularity 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) andothers. Notations. - An element of the upper half space R N + + is denoted by ( x , t ) where x ∈ R N and t > ff erentiable function U on R N + + , we denote ∇ x U = ( ∂ x U , · · · , ∂ x N U ) and ∇ U = ( ∇ x U , ∂ t U ). Also ∂ x i is often written as ∂ i .- B + r = B N + (0 , r ) ∩ R N + + is the ( N + r centered at theorigin.- u + = max { u , } and u − = max {− u , } .- Γ denotes the Gamma function.- For any N ∈ N and s ∈ (0 , min { , N / } ), we denote p = N + sN − s .- C > In this section, we present some geometric and analytical backgrounds to understand our problem.Most of materials are taken from [12, 29, 23, 10, 8].
Let ( X N + , g + ) be an ( N + M N .We call a function ρ on the closure X of X a defining function of the boundary M if ρ > X , ρ = M and d ρ , M . The manifold ( X , g + ) is said to be conformally compact (C.C.) ifthere is a defining function ρ making ( X , ¯ g ) be compact where ¯ g : = ρ g + . Also, given the metricˆ h = ¯ g | M , the boundary ( M , [ˆ h ]) with the conformal class [ˆ h ] of ˆ h is called the conformal infinity.A C.C. metric g + is asymptotically hyperbolic (A.H.) if the sectional curvature approaches to -1at the infinity M , whose model case is the hyperbolic space:( X , g + ) = ( H N + , g H ) = R N + + , | dx | + dt t ! or B N + , | dx | + dt )(1 − | x | − t ) ! . According to Graham-Lee [31], for an A.H. manifold X and a representative ˆ h for the confor-mal class on ( M , [ˆ h ]), there is a unique special defining function such that g + = ρ − (cid:16) d ρ + h ρ (cid:17) , h ρ = ˆ h + O ( ρ )near M . It is called the geodesic boundary defining function.Suppose that z ∈ C , Re( z ) > N / f ∈ C ∞ ( M ). Then, by [43, 32], unless z ( N − z ) is an L -eigenvalue of − ∆ g + , the following eigenvalue problem h − ∆ g + − z ( N − z ) i V = X (8)has a solution of the form V = F ρ N − z + G ρ z , F , G ∈ C ∞ ( X ) and F | ρ = = f . (9)Throughout the paper the existence of such a solution is always assumed. The scattering operatoron M is then defined to be S ( z ) f = G | M , which is a meromorphic family of pseudo-di ff erential operators in { z ∈ C : Re( z ) > N / } . Inaddition, we introduce its normalization so called the fractional Paneitz operator P s ˆ h , namely P s ˆ h = P s [ g + , ˆ h ] = − s s Γ ( s ) Γ (1 − s ) S (cid:18) N + s (cid:19) for s < N , ( − s s s !( s − · Res z = N / + s S ( z ) for s ∈ N , − ∆ ˆ h ) s . In the special case that ( X , g + ) is Poincar´e-Einstein(both C.C. and Einstein) and s = P h u = − ∆ ˆ h u + N − N − R ˆ h u (10)the usual conformal Laplacian, and P h u = ( − ∆ ˆ h ) u − div ˆ h (cid:16)(cid:16) ˜ c R ˆ h ˆ h − ˜ c Ric ˆ h (cid:17) du (cid:17) + N − Q ˆ h u (11)the Paneitz operator. Here Q stands for the Branson’s Q -curvature and ˜ c , ˜ c > P s ˆ h is that it is conformally covariant in the sense that P s ˆ hu N − s φ = u − N + sN − s P s ˆ h ( u φ ) for any function u > M . Finally, we set the fractional scalar curvature Q s ˆ h by P s ˆ h (1). ff arelli-Silvestre’s result [10] and Chang-Gonz´alez’s extension [12] In this subsection, we recall the observation of Chang and Gonz´alez [12] which identifies twofractional Laplacians arising in di ff erent contexts: one given as normalized scattering operators[32] described above and one originated from the Dirichlet-Neumann operators due to Ca ff arelliand Silvestre [10].For s ∈ (0 , D ( R N + + ; t − s ) be the completion of C ∞ c ( R N + + ) with respect to the weightedSobolev norms k U k D ( R N + + ; t − s ) : = Z R N + + t − s |∇ U ( x , t ) | dxdt ! / with the weight t − s . Furthermore, we designate by H s ( R N ) the standard fractional Sobolev spacegiven as H s (cid:16) R N (cid:17) = u ∈ L (cid:16) R N (cid:17) : k u k H s ( R N ) : = Z R N (cid:16) + | ξ | s (cid:17) | ˆ u ( ξ ) | d ξ ! / < ∞ where ˆ u denotes the Fourier transform of u , and define the fractional Laplacian ( − ∆ ) s : H s ( R N ) → H − s ( R N ) to be [ (( − ∆ ) s u )( ξ ) = (2 π | ξ | ) s ˆ u ( ξ ) for any ξ ∈ R N given u ∈ H s (cid:16) R N (cid:17) . In the celebrated work of Ca ff arelli and Silvestre [10], the authors found that if U ∈ D ( R N + + ; t − s )is a unique solution of the equation div (cid:16) t − s ∇ U (cid:17) = R N + + , U ( x , = u ( x ) for x ∈ R N , (12)provided a fixed function u ∈ H s ( R N ), then ( − ∆ ) s u = ∂ s ν U | R N where the definition of the weightednormal derivative ∂ s ν is given in (6). Let us call this U the s -harmonic extension of u and denote itby Ext s ( u ).It turned out that this extension result is a special case of the following proposition obtainedby Chang and Gonz´alez [12]. We also refer to Section 2 of [29].6 roposition 2.1. ([12, Theorem 5.1 and Theorem 4.3]) Let ( X N + , g + ) be an asymptotically hy-perbolic manifold with the conformal infinity ( M N , [ˆ h ]) and ρ the geodesic defining function of ˆ h.Assume also that H = if s ∈ (1 / , . For a smooth function u on M, if V is a solution of (8) andsatisfies (9) in which f is substituted with u, the function U : = ρ z − N V solves − div (cid:16) ρ − s ∇ U (cid:17) + E ( ρ ) U = in ( X , ¯ g ) and U = u on ( M , ˆ h ) given that E ( ρ ) : = ρ − − z ( − ∆ g + − z ( N − z )) ρ N − z , z : = N + s and ¯ g : = ρ g + . Moreover,P s ˆ h u = ( ∂ s ν U for s ∈ (0 , \ { / } ,∂ s ν U + N − N Hu for s = / . Here H denotes the trace of the second fundamental form ( π i j ) = (cid:16) −h∇ ∂ ρ ∂ i , ∂ j i ˆ h (cid:17) on M = ∂ X andthe operator ∂ s ν is the weighted normal derivative defined in (6) with t replaced by ρ .For su ffi ciently small r > , it also holds thatE ( ρ ) = N − s N h R ¯ g ρ − s − (cid:16) R g + + N ( N + (cid:17) ρ − − s i on M × (0 , r ) . (13) Remark 2.2.
Since it holds that R g + = − N ( N + + N ρ∂ ρ log(det h ( ρ )) + ρ R ¯ g on M × (0 , r )and ∂ ρ log(det h ( ρ )) (cid:12)(cid:12)(cid:12) ρ = = Tr (cid:16) h ( ρ ) − ∂ ρ h ( ρ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ρ = = − H , the remainder term E ( ρ ) in (13) is reduced to E ( ρ )( z ) = − N − s ! ∂ ρ log(det h ( ρ ))( σ ) ρ − s = − N − s ! ∂ ρ log(det h ( ρ )) (cid:12)(cid:12)(cid:12) ρ = ( σ ) ρ − s + O (cid:16) ρ − s (cid:17) = N − s ! H ( σ ) ρ − s + O (cid:16) ρ − s (cid:17) (14)for z = ( σ, ρ ) ∈ M × (0 , r ).In particular, our main equation (1 ± ) is equivalent to the problem − div (cid:16) ρ − s ∇ U (cid:17) + E ( ρ ) U = X , ¯ g ) ,∂ s ν U = u p ± ǫ − f u on ( M , ˆ h ) , U = u > M , ˆ h ) (15 ± )and it remains the same as well except the second equation in (15 ± ) is replaced by ∂ s ν U = u p − ǫ f u for s ∈ (0 , /
2) on ( M , ˆ h ) (16)if we deal with (2).In [12], it is also proved that given a geodesic defining function ρ , there is another specialdefining function ρ ∗ such that E ( ρ ∗ ) = Proposition 2.3. ([12, Theorem 4.7], [29, Proposition 2.2]) Assume that H = if s ∈ (1 / , .For a smooth function u on M, if V satisfies (8) as well as (9) in which f is substituted with u, thefunction U : = ( ρ ∗ ) z − N V is a solution of − div (cid:16)(cid:0) ρ ∗ (cid:1) − s ∇ U (cid:17) = in ( X , g ∗ ) and U = u on ( M , ˆ h ) (17) where g ∗ : = ( ρ ∗ ) g + . Moreover g ∗ | M = ˆ h, ( ρ ∗ /ρ ) | M = andP s ˆ h u = ∂ s ν U + Q s ˆ h u (18) where Q s ˆ h is the fractional scalar curvature and the operator ∂ s ν is defined in (6) with t substitutedwith ρ ∗ . This observation is useful in showing a priori L ∞ -estimate or the strong maximum principle of theoperator P s ˆ h . Refer to [29, Section 3]. (cf. Lemma 3.3 and Proposition 5.1 below)7 .3 Sharp trace inequality and its related equations Given any number δ > σ = ( σ , · · · , σ N ) ∈ R N , let w δ,σ ( x ) = ˜ κ N , s δδ + | x − σ | ! N − s for x ∈ R N with ˜ κ N , s = N − s Γ (cid:16) N + s (cid:17) Γ (cid:16) N − s (cid:17) N − s s . (19)Its constant multiples attain the equality for the sharp Sobolev inequality Z R N | u | NN − s dx ! N − s N ≤ S N , s Z R N (cid:12)(cid:12)(cid:12) ( − ∆ ) s / u (cid:12)(cid:12)(cid:12) dx ! where S N , s > − ∆ ) s u = u p , u > R N and lim | x |→∞ u ( x ) = W δ,σ = Ext s ( w δ,σ ), the s -harmonic extension of w δ,σ . Then we observe thatextremal functions of Sobolev trace inequality Z R N | U ( x , | NN − s dx ! N − s N ≤ S N , s √ κ s Z ∞ Z R N t − s |∇ U ( x , t ) | dxdt ! , (21)have the form U ( x , t ) = cW δ,σ ( x , t ) for any c > , δ > σ ∈ R N , where κ s > W δ,σ solves div (cid:16) t − s ∇ U (cid:17) = R N + + ,∂ s ν U = U p on R N × { } , U = w δ,σ on R N × { } (22)and as an immediate consequence we have κ s Z R N + + t − s |∇ W δ,σ | dxdt = Z R N w NN − s δ,σ dx . (23)On the other hand, in the work of D´avila, del Pino and Sire [15], it was revealed that the set ofsolutions bounded on Ω × { } to the equation div (cid:16) t − s ∇ Φ (cid:17) = R N + + ,∂ s ν Φ = p w p − δ,σ Φ on R N × { } , (24)consists of the linear combinations of Z δ,σ : = ∂ W δ,σ ∂σ , · · · , Z N δ,σ : = ∂ W δ,σ ∂σ N and Z δ,σ : = ∂ W δ,σ ∂δ . (25)This fact is crucial in applying the reduction method to our problem. Hereafter, we will denote w δ = w δ, , W δ = W δ, , z i δ = z i δ, and Z i δ = Z i δ, for i = , · · · , N . Suppose that ( X , ¯ g ) is a compact Riemannian manifold and 0 ∈ M = ∂ X . Let x = ( x , · · · , x N )be normal coordinates on M at the point 0 and ( x , · · · , x N , t ) be the Fermi coordinates on X at 0where x , · · · , x N ∈ R and t >
0. Also, we denote¯ g = dt + h i j ( x , t ) dx i dx j so that h = ¯ g | T M . Then the following asymptotic expansion of the metric near 0 is valid.8 emma 2.4. [23, Lemma 3.1, 3.2] For x , · · · , x N and t : = x N + small, it holds that p | ¯ g | = p | h | = − Ht + (cid:16) H − k π k h − Ric ( ∂ t ) (cid:17) t − H i x i t − R i j x i x j + O (cid:16) | ( x , t ) | (cid:17) and h i j = δ i j + π i j t − R i jkl x k x l + h i j , ( N + k x k t + (cid:16) π ik π jm + R i jn n (cid:17) t + O (cid:16) | ( x , t ) | (cid:17) where π is the second fundamental form of M = ∂ X, H is its trace, i.e., N times of the meancurvature, R i j denotes a component of the Ricci tensor, R i jkl is a component of the Riemanniantensor and Ric ( ∂ t ) = g i j R i ( N + j ( N + . Also, the indices i , j and k run from 1 to N. As before, let ( X N + , g + ) be an A.H. manifold with the boundary ( M n , ˆ h ) and ρ the geodesic defin-ing function, so that ( X , ¯ g ) where ¯ g = ρ g + is a compact manifold. Denote by H ( X ; ρ − s ) theweighted Sobolev space endowed with the inner product h U , V i H ( X ; ρ − s ) : = Z X ρ − s h ( ∇ U , ∇ V ) ¯ g + UV i d v ¯ g and the norm k U k H ( X ; ρ − s ) : = Z X ρ − s (cid:16) |∇ U | g + U (cid:17) d v ¯ g ! / . (26)By applying (21) and the standard partition of unity argument, we obtain a manifold version of theweighted Sobolev trace inequality k U k L NN − s ( M ) ≤ C k U k H ( X ; ρ − s ) (27)where C > s , N and X . In addition, the embedding H ( X ; ρ − s ) ֒ → L q ( M ) is compact for any 1 ≤ q < NN − s . The next two lemmas provide equivalent norms to the H ( X ; ρ − s )-norm. Lemma 3.1.
The norm (cid:16)R X ρ − s |∇ U | g d v g + R M U d v ˆ h (cid:17) / is equivalent to the norm k U k H ( X ; ρ − s ) defined in (26) .Proof. We first consider a function U defined on B + R for some R > B + R = { ( x , t ) ∈ R N + + : | ( x , t ) | < R , t > } . For each 0 ≤ t ≤ R , using the elementary calculus and H ¨older’s inequality wehave | U ( x , t ) | ≤ | U ( x , | + Z t | ∂ r U ( x , r ) | dr ≤ | U ( x , | + Z t r s − dr ! / Z t r − s | ∂ r U ( x , r ) | dr ! / = | U ( x , | + t s √ s Z R r − s | ∂ r U ( x , r ) | dr ! / . For any given number a ∈ ( − , Z R Z | x |≤ R t a | U ( x , t ) | dx dt ≤ Z R t a dt ! Z | x |≤ R | U ( x , | dx + s Z R t a + s dt ! Z | x |≤ R Z R r − s | ∂ r U ( x , r ) | dr dx ≤ C Z | x |≤ R | U ( x , | dx + Z R Z | x |≤ R t − s |∇ U ( x , t ) | dx dt ! . (28)9mploying this inequality with a = − s in each local chart, we can obtain that Z X ρ − s | U | d v ¯ g ! / ≤ C Z X ρ − s |∇ U | g d v ¯ g + Z M U d v ˆ h ! / . On the other hand, the weighted trace inequality (27) and H ¨older’s inequality yield Z M | U | d v ˆ h ! / ≤ C Z X ρ − s (cid:16) |∇ U | g + U (cid:17) d v ¯ g ! / . These two estimates enable us to get the equivalence of the two norms, concluding the proof. (cid:3)
Lemma 3.2.
Suppose that the trace of the second fundamental form H of M = ∂ X vanishes ifs ∈ [1 / , . Under the assumption that (7) holds, k U k ˜ f : = κ s Z X (cid:16) ρ − s |∇ U | g + E ( ρ ) U (cid:17) d v ¯ g + Z M ˜ f U d v ˆ h ! / (29) gives an equivalent norm to (26) . Hence one can define the inner product h· , ·i ˜ f from the norm k · k ˜ f through the polarization identity.Proof. Suppose first that s ∈ [1 / , H = | E ( ρ ) | ≤ C ρ − s by (14). Using this fact and (27) also, we immediately obtain that k U k H ( X ; ρ − s ) ≥ C k U k ˜ f .If s ∈ (0 , / U near the boundary by taking a = − s in(28) and applying (27). Additionally, by realizing that ρ is bounded away from 0 in any compactsubset of X , it is possible to manage the integral of U in the interior of X . Combining the bothestimates, we deduce the same inequality k U k H ( X ; ρ − s ) ≥ C k U k ˜ f .Suppose that the opposite inequality does not hold. Then there is a sequence { U n } ∞ n = such that k U n k ˜ f → n → ∞ but k U n k H ( X ; ρ − s ) = n ∈ N . Let us first claim that R X E ( ρ ) U n → R X ρ − s U n →
0, so the claim is verified at once if H =
0. If s ∈ (0 , /
2) and H ,
0, then the main order of E ( ρ ) is ρ − s as (14) indicates. In this situation, we take a < n →∞ Z X ρ − s U n ≤ lim n →∞ Z X ρ − s U n ! η Z X ρ − a U n ! − η = η = a − sa − s + ∈ (0 , k U n k H ( X ; ρ − s ) =
1, (27) and (28) guarantee boundedness of the value nR X ρ − a U n o ∞ n = . Now if we let U ∞ be the H ( X ; ρ − s )-weak limit of U n , then U ∞ ≡
0. Thus compactness of the trace embedding gives usthat R M ˜ f U n → R M ˜ f U ∞ =
0. However, it is a contradiction because previous computations showthat R X ρ − s |∇ U n | should converge to both 0 and 1. This proves that k U k ˜ f ≥ C k U k H ( X ; ρ − s ) . (cid:3) By (27), we know that the trace operator i : H ( X ; ρ − s ) → L p + ( M ) given as i ( U ) = U | M : = u iswell-defined and continuous. Thus the adjoint operator i ∗ ˜ f : L p + p ( M ) → H ( X ; ρ − s ) defined bythe equation − div (cid:16) ρ − s ∇ U (cid:17) + E ( ρ ) U = X , ¯ g ) ,∂ s ν U = v − ˜ f u on ( M , ˆ h ) , U = u on ( M , ˆ h ) , (30)with U = i ∗ ˜ f ( v ) is bounded in light of Lemma 3.2. Furthermore, i : H ( X ; ρ − s ) → L q ( M ) ⊃ L p + ( M ) for 1 ≤ q < p + + ) or (15 + ), wemust restrict the space H ( X ; ρ − s ) so that the trace of the each element belongs to L p + + ǫ ( M ) for ǫ > q ǫ = ( p + + N s ǫ, which implies q ǫ p + ǫ = Nq ǫ N + sq ǫ . (31)Then let us introduce a Banach space H ǫ = n U ∈ H ( X ; ρ − s ) : i ( U ) ∈ L q ǫ ( M ) o (32)equipped with the norm k · k ˜ f ,ǫ defined by k U k ˜ f ,ǫ = k U k ˜ f + k i ( U ) k L q ǫ ( M ) for any U ∈ H ǫ . (33)The following estimate explains why it is plausible to work with the space H ǫ . Lemma 3.3.
Suppose that N > s and v ∈ L q ( M ) for some q ∈ (1 , N s ) . If U = i ∗ ˜ f ( v ) and u = i ( U ) ,then there exists C = C ( q ) > such that k u k L q ( M ) ≤ C k v k L q ( M ) with q > NN − s satisfying q = q − sN . In other words, we have k u k L q ( M ) ≤ C k v k L NqN + sq ( M ) for any q ∈ ( NN − s , ∞ ) .Proof. Instead of giving consideration to (30) directly, we shall use the observation coming fromPropositions 2.1 and 2.3 that e U = ( ρ ∗ /ρ ) z − N U is a solution of (17) and e U = U = u on M . Forany number L >
0, let us denote e U L = min (cid:8) | e U | , L (cid:9) . Due to (18), if we multiply (17) by e U β − L e U forsome β >
1, we get κ s Z X ( ρ ∗ ) − s (cid:16) ∇ e U , ∇ (cid:16) e U β − L e U (cid:17)(cid:17) g ∗ d v g ∗ = Z M v u β − L ud v ˆ h − Z M (cid:16) ˜ f + Q s ˆ h (cid:17) u β − L u d v ˆ h where u L = min {| u | , L } . Therefore we have Z X ( ρ ∗ ) − s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) e U β − L e U (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ∗ d v g ∗ ≤ C (cid:13)(cid:13)(cid:13)(cid:13) u β − L u (cid:13)(cid:13)(cid:13)(cid:13) L ( β + p + β ( M ) k v k L q ′ ( M ) + C k u k β + L β + ( M ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u β − L u (cid:19) ββ + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( β + p + β ( M ) k v k L q ′ ( M ) + C k u k β + L β + ( M ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u β − L u (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p + ( M ) + C k v k β + L q ′ ( M ) + C k u k β + L β + ( M ) , where q ′ satisfies q ′ + ( N − s ) β N ( β + = C > N and s . Also, weused Young’s inequality to derive the third inequality. Using this, Lemma 3.1 and the weightedtrace inequality, we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u β − L u (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p + ( M ) ≤ Z X ( ρ ∗ ) − s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) e U β − L e U (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ∗ d v g ∗ + Z M (cid:18) u β − L u (cid:19) d v ˆ h ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u β − L u (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p + ( M ) + C k v k β + L q ′ ( M ) + C k u k β + L β + ( M ) . (34)11aking L → ∞ in this estimate, we may deduce k u k L N ( β + N − s ( M ) ≤ C (cid:16) k v k L q ′ ( M ) + k u k L β + ( M ) (cid:17) . Letting q = N ( β + N − s we have k u k L q ( M ) ≤ C (cid:18) k v k L q ′ ( M ) + k u k L ( N − s ) qN ( M ) (cid:19) . (35)One may check that q = q ′ − sN . Besides, since we took β >
1, it holds that q ′ > NN + s and q > p + e U L ) β − e U for 0 < β ≤ e U L : = max (cid:8) | e U | , L (cid:9) andfollow the above argument except taking L → L → ∞ , then we obtain (35) for1 < q ′ ≤ NN + s and NN − s < q ≤ p + k u k L q ( M ) ≤ C k v k L q ′ ( M ) holds for some C >
0. To show this inequality,we assume that it does not hold for any C . Then, we can find a sequence of functions v n ∈ L q ′ ( M ), U n = i ∗ ¯ f ( v n ) and u n = i ( U n ) such that k u n k L q ( M ) = n →∞ k v n k L q ′ ( M ) =
0. By the compactnessproperty whose proof is postponed to below, u n converges strongly in L ( N − s ) qN ( M ). We let u be itslimit. Applying (35) with u n and v n , and then taking the limit n → ∞ , we obtain1 ≤ C (cid:18) lim n →∞ k v n k L q ′ ( M ) + k u n k L ( N − s ) qN ( M ) (cid:19) = C k u k L ( N − s ) qN ( M ) . (36)On the other hand, by employing Lemma 3.2, the weighted trace inequality and H ¨older’s inequal-ity, we find k u n k L NN − s ( M ) ≤ C k U n k ˜ f ≤ C k v n k L NN + s ( M ) ≤ C k v n k L q ′ ( M ) . From this estimate and lim n →∞ k v n k L q ′ ( M ) =
0, we have k u k L NN − s ( M ) = lim n →∞ k u n k L NN − s ( M ) = u ≡
0. However it contradicts to (36). Hence the assertion that k u k L q ( M ) ≤ C k v k L q ′ ( M ) should hold for some C > { u n } ∞ n = in L ( N − s ) qN ( M ). By (34), we get Z X ρ − s (cid:12)(cid:12)(cid:12)(cid:12) ∇| U n | β + (cid:12)(cid:12)(cid:12)(cid:12) g d v ¯ g + Z M | U n | β + d v ˆ h ≤ C (cid:16) k v n k L q ′ ( M ) + k u n k L β + ( M ) (cid:17) β + . Owing to Lemma 3.1, it follows that (cid:26) | U n | β + (cid:27) ∞ n = is a bounded subset of H ( X ; ρ − s ). Thus (cid:26) | U n | β + (cid:27) ∞ n = is a compact set in L NN − s − ζ ( M ) for any small ζ >
0, which in turn implies that { U n } ∞ n = is a compact set in L N ( β + N − s − ζ ( M ) = L q − ζ ( M ) for every small ζ >
0, hence in L ( N − s ) qN ( M ). The proofis finished. (cid:3) Corollary 3.4.
Fix any q > NN + s . Then the adjoint map i ∗ f : L q ( M ) → H ǫ is compact forsu ffi ciently small ǫ > .Proof. It easily follows from the previous lemma and its proof. We leave the details to the reader. (cid:3)
By Lemma 3.3, if u ∈ L q ǫ p + ǫ ( M ), then i ( i ∗ ˜ f ( u )) ∈ L q ǫ ( M ). Hence one may attempt to solve equation(1 + ) by writing U = i ∗ ˜ f (cid:0) u p + ǫ (cid:1) and U = u > M for U ∈ H ǫ .To unify the notation, we will use ( H ǫ , k · k ˜ f ,ǫ ) to denote ( H ( X ; ρ − s ) , k · k ˜ f ) from now even ifwe study the subcritical problem (1 − ) and the critical one (2). Notice that if equations (1 − ) and (2)are considered, then q ǫ in (31) should be read as NN − s − N s ǫ and NN − s , respectively. Hence in thiscase the Banach spaces ( H ǫ , k · k ˜ f ,ǫ ) (defined according to (32) and (33)) and ( H ( X ; ρ − s ) , k · k ˜ f )are equivalent to each other, justifying our expression.12 .2 The approximate solutions Recalling the number r selected in (13), we choose r < r a positive number less than the quarterof the injectivity radius of ( M , ˆ h ). Let χ : (0 , ∞ ) → [0 ,
1] be a smooth function such that χ = , r ) and 0 in (2 r , ∞ ). Noting that any element z ∈ X near the boundary can be denoted as z = ( ˆ σ, ρ ) for some ˆ σ ∈ M and ρ ∈ (0 , ∞ ), we define the function W δ,σ on X (provided δ > σ ∈ M ) by W δ,σ ( z ) = W δ,σ ( ˆ σ, ρ ) = χ ( d ( z , σ )) W δ (cid:16) exp − σ ( ˆ σ ) , ρ (cid:17) if d ( z , σ ) < r for some σ ∈ M , , (37)where W δ = Ext s ( w δ ) is the function defined in Subsection 2.3, d M ( · , σ ) denotes the geodesicdistance from σ on ( M , ˆ h ), d ( · , σ ) is a positive function defined near the boundary of ( X , ¯ g ) by therelation d ( z , σ ) = d (( ˆ σ, ρ ) , σ ) = d M ( ˆ σ, σ ) + ρ and exp is the exponential map on ( M , ˆ h ). Thusthe parameter δ can be regarded as a concentration rate, while σ expresses a blow-up point. Weset δ = ǫ α λ where λ > ǫ -independent number. The number α is chosen to be α = / (2 s ) for problems (15 ± ) , / (1 − s ) for problem (16) . (38)In this paper, we search for solutions of (15 ± ) and (16) of the form W ǫ α λ,σ + Φ where Φ isa function defined on X whose H ǫ -norm is su ffi ciently small. Because we regard the equationsas perturbations of the limit equation (22), it is important to understand their linearized equations.Hence it is natural to introduce Z i δ,σ ( z ) = Z i δ,σ ( ˆ σ, ρ ) = χ ( d ( z , σ )) Z i δ (cid:16) exp − σ ( ˆ σ ) , ρ (cid:17) if d ( z , σ ) < r for some σ ∈ M , , for i = , · · · , N , where Z i δ,σ is the function whose definition is presented in (25). For each ǫ > H ǫ K ǫλ,σ = Span n Z i ǫ α λ,σ : i = , · · · , N o and its orthogonal complement with respect to the inner product h· , ·i ˜ f (cid:16) K ǫλ,σ (cid:17) ⊥ = (cid:26) U ∈ H ǫ : D U , Z i ǫ α λ,σ E ˜ f = i = , · · · , N (cid:27) . Furthermore, denote by Π ǫλ,σ : H ǫ → K ǫλ,σ and (cid:16) Π ǫλ,σ (cid:17) ⊥ : H ǫ → (cid:16) K ǫλ,σ (cid:17) ⊥ the orthogonal projections onto K ǫλ,σ and ( K ǫλ,σ ) ⊥ , respectively.As mentioned before, we will apply the finite dimensional reduction method. Namely, for asmall fixed ǫ >
0, we first solve an intermediate problem (in Section 4) (cid:16) Π ǫλ,σ (cid:17) ⊥ (cid:20) ( W ǫ α λ,σ + Φ ǫ α λ,σ ) − i ∗ ˜ f (cid:0) i (cid:0) g ǫ ( W ǫ α λ,σ + Φ ǫ α λ,σ ) (cid:1)(cid:1)(cid:21) = λ, σ ) ∈ (0 , ∞ ) × M by employing the contraction mapping theorem, where ( g ǫ ( u ) = u p ± ǫ + and ˜ f = f if we consider (15 ± ) ,g ǫ ( u ) = u p + − ǫ f u and ˜ f = . . (40)Then we choose an appropriate ( λ ǫ , σ ǫ ) which makes Π ǫλ ǫ ,σ ǫ (cid:20) ( W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ ) − i ∗ ˜ f (cid:0) i (cid:0) g ǫ ( W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ ) (cid:1)(cid:1)(cid:21) = , ∞ ) × M correspondingto the above problem (41). This is conducted in Section 6. Observe that we modified the nonlinearterm in (39) and (41) because we want to find a positive solution.Before concluding this section, we provide a lemma regarding the decay property of W δ and Z i δ , which will be used throughout the paper. We defer its proof to Appendix A. Lemma 3.5.
Assume that N > s, fix any < R < R and set A + ( R , R ) = B + R \ B + R . Then as δ → we have the following estimates. Z R N + + \ B + R t − s |∇ W δ | dxdt = O (cid:16) δ N − s (cid:17) . Z B + R t − s |∇ W δ | dxdt = O ( δ ) for N > s + , O (cid:0) δ | log δ | (cid:1) for N = s + , O (cid:16) δ N − s (cid:17) for N < s + . Z A + ( R , R t − s W δ dxdt = O (cid:16) δ N − s (cid:17) for N , s + , O (cid:16) δ | log δ | (cid:17) for N = s + . (42) Besides, the followings are also true. Z R N + + \ B + R t − s (cid:12)(cid:12)(cid:12) ∇ Z i δ (cid:12)(cid:12)(cid:12) dxdt = O (cid:16) δ N − s (cid:17) for i = , · · · , N , O (cid:16) δ N − s − (cid:17) for i = . Z A + ( R , R t − s (cid:16) Z i δ (cid:17) dxdt = O (cid:16) δ N − s (cid:17) for i = , · · · , N , O (cid:16) δ N − s − (cid:17) for i = and N , s + , O (cid:0) | log δ | (cid:1) for i = and N = s + . (43) We also know Z B + R t − s O (cid:16) | ( x , t ) | (cid:17) |∇ W δ | dxdt = O (cid:16) δ (cid:17) for N > s + , O (cid:16) δ | log δ | (cid:17) for N = s + , O (cid:16) δ N − s (cid:17) for N < s + . (44) This section is devoted to solvability of the intermediate problem (39).
In this subsection, we shall obtain a uniform bound of the H ǫ -norm of the error term W ǫ α λ,σ − i ∗ ˜ f ( i ( g ǫ ( W ǫ α λ,σ ))) where ( λ, σ ) ∈ ( λ − , λ ) × M and ǫ > λ > α was set in (38). Lemma 4.1.
Assume that N > max { s , } for (15 ± ) and N ≥ for (16) . Given a fixed λ > , itholds that (cid:13)(cid:13)(cid:13)(cid:13) W ǫ α λ,σ − i ∗ ˜ f (cid:0) i ( g ǫ ( W ǫ α λ,σ )) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) ˜ f ,ǫ = O (cid:0) ǫ γ (cid:1) (45) where γ = − ζ for problems (15 ± ) if < s < , − s s − ζ for problems (15 ± ) if ≤ s < , N − s s − ζ for problems (15 ± ) with s < N ≤ s + if ≤ s < , s − ζ for problems (15 ± ) with N > s + if ≤ s < , − s − s − ζ for problem (16) (46)14 niformly ( λ, σ ) ∈ ( λ − , λ ) × M. Here ζ > can be taken to be arbitrarily small. Before starting the proof, we remark that γ > / ± ), while γ > / (2(1 − s )) forproblem (16). Proof.
Let us take into account the subcritical problem (15 − ), recalling the notation δ = ǫ s λ ∈ (cid:16) ǫ s λ − , ǫ s λ (cid:17) . Here we will use the dual characterization of the norm k U k f = sup n h U , Φ i f : k Φ k f ≤ o which holds for any U ∈ H ( X ; ρ − s ). For a fixed Φ ∈ H ( X ; ρ − s ) such that k Φ k f ≤
1, we have (cid:10) W δ,σ , Φ (cid:11) f − D i ( W p ± ǫδ,σ ) , φ E L pp + ( M ) = κ s Z B + ¯ g ( σ, r ) (cid:16) ρ − s ( ∇W δ,σ , ∇ Φ ) ¯ g + E ( ρ ) W δ,σ Φ (cid:17) d v ¯ g + Z B ˆ h ( σ, r ) (cid:16) f W δ,σ − W p ± ǫδ,σ (cid:17) φ d v ˆ h (47)where B + ¯ g ( σ, r ) : = { z ∈ X : d ( z , σ ) < r } , B ˆ h ( σ, r ) : = { ˆ σ ∈ M : d M ( ˆ σ, σ ) < r } (48)and φ = i ( Φ ). Note that in Section 3 the distance functions d ( · , σ ) and d M ( · , σ ) were introducedin setting the first approximation W δ,σ for a solution, for each fixed σ ∈ M (see (37)). Since thedomains of the above integrations are small neighborhoods of the point σ in X and M , respectively,we may replace Φ by χ ( d ( · , σ ) / Φ for instance without a ff ecting on the value of the integrations,where χ is a cut-o ff function introduced for (37). Moreover, by the equivalence of two norms k·k f and k · k H ( X ; ρ − s ) , it can be easily seen that k χ ( d ( · , σ ) / Φ k f ≤ C k Φ k f ≤ C where C > Φ . Therefore, to obtain (45), we may without any loss ofgenerality regard Φ (or φ ) as a function on R N + + (or R N ) and assume that its support is containedin B + ¯ g : = B + ¯ g ( σ, r ) ⊂ R N + + (cid:16) or B ˆ h : = B ˆ h ( σ, r ) ⊂ R N (cid:17) .Now we shall estimate each of the right-hand side of (47). For this objective, we denote Φ δ − ( z ) = δ N − s Φ ( δ z ) for all z ∈ R N + + and φ δ − = i ( Φ δ − ), for which it holds that (cid:13)(cid:13)(cid:13) Φ δ − (cid:13)(cid:13)(cid:13) D ( R N + + ; t − s ) = Z R N + + t − s |∇ Φ δ − ( z ) | dxdt ≤ C (49)by the scaling invariance. Firstly, from (42) and the estimate that Z B + ¯ g ρ − s | z ||∇W δ,σ ||∇ Φ | dz ≤ C Z B + r \ B + r t − s W δ dz + Z B + r t − s | z | |∇ W δ | dz = O ( δ ) = O (cid:16) ǫ s (cid:17) if N > s + , O (cid:16) δ | log δ | (cid:17) = O (cid:16) ǫ s | log ǫ | (cid:17) if N = s + , O (cid:16) δ N − s (cid:17) = O (cid:16) ǫ N − s s (cid:17) if N < s + ,
15e find κ s Z B + ¯ g ρ − s ( ∇W δ,σ , ∇ Φ ) ¯ g d v ¯ g = κ s Z R N + + t − s ∇ W δ · ∇ Φ dz + O Z B + ¯ g ρ − s | z ||∇W δ,σ ||∇ Φ | dz + O (cid:16) δ N − s (cid:17) if N , s + , O (cid:16) δ | log δ | (cid:17) if N = s + , = κ s Z R N + + t − s ∇ W · ∇ Φ δ − dz + O ( δ ) if N > s + , O (cid:16) δ | log δ | (cid:17) if N = s + , O (cid:16) δ N − s (cid:17) if N < s + , = Z R N w p ( x ) φ δ − ( x ) dx + O (cid:16) ǫ s (cid:17) if N > s + , O (cid:16) ǫ s | log ǫ | (cid:17) if N = s + , O (cid:16) ǫ N − s s (cid:17) if N < s + . (50)Also, if 1 / ≤ s < H =
0, then (14) and the Cauchy-Schwarz inequality imply (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ s Z B + ¯ g E ( ρ ) W δ,σ Φ d v ¯ g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z B + ¯ g ρ − s W δ,σ d v ¯ g · Z B + ¯ g ρ − s Φ d v ¯ g ≤ C Z B + r t − s W δ ( z ) dz = O ( δ ) = O (cid:16) ǫ s (cid:17) if N > s + , O (cid:16) δ | log δ | (cid:17) = O (cid:16) ǫ s | log ǫ | (cid:17) if N = s + , O (cid:16) δ N − s (cid:17) = O (cid:16) ǫ N − s s (cid:17) if N < s + . (51)In the case that 0 < s < /
2, we take ζ > − ( s + ζ ) > /
2. It follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ s Z B + ¯ g E ( ρ ) W δ,σ Φ d v ¯ g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z B + ¯ g ρ − s |W δ,σ || Φ | d v ¯ g ≤ C Z B + ¯ g ρ − s − s + ζ ) W δ,σ d v ¯ g · Z B + ¯ g ρ − + ζ Φ d v ¯ g ≤ C Z B + r t − s − s + ζ ) W δ ( z ) dz = O (cid:16) δ − ( s + ζ ) (cid:17) = O (cid:16) ǫ (1 − ( s + ζ )) / s (cid:17) for N ≥ . (52)On the other hand, if ζ is a number chosen to be ζ = NN − s + ζ ′ for 4 s < N ≤ s where ζ ′ > , NN + s for N > s , then thanks to the Sobolev trace inequality (27), it can be computed that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ˆ h f W δ,σ φ d v ˆ h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k f k L ∞ ( M ) k w δ k L ζ ( R N ) k Φ k H ( X ; ρ − s ) = O (cid:16) δ N − s − ζ ′′ (cid:17) = O ǫ N − s s − ζ ′′ s ! for 4 s < N ≤ s , O (cid:16) δ s (cid:17) = O ( ǫ ) for N > s . (53)Here ζ ′′ > ζ ′ . Moreover one has − Z B ˆ h W p ± ǫδ,σ φ d v ˆ h = − Z B ˆ h W p δ,σ φ d v ˆ h + O ( ǫ | log ǫ | ) = − Z R N w p ( x ) φ δ − ( x ) dx + O ( ǫ | log ǫ | ) . (54)16onsequently, combining all computations (47) and (50)-(54), we obtain the validity of the firstestimate of (45).The error estimate (45) for problem (16) can be handled in a similar way and we omit it.Now we are left to handle the supercritical problems (15 + ). To obtain the conclusion, it su ffi cesto show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w δ,σ − i (cid:18) i ∗ ˜ f ( g ǫ ( w δ,σ )) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ǫ ( M ) = O (cid:0) ǫ γ (cid:1) . (55)By the trace inequality (27) and the computations made above, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w δ,σ − i (cid:18) i ∗ ˜ f ( g ǫ ( w δ,σ )) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ǫ ( M ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w δ,σ − i (cid:18) i ∗ ˜ f ( g ǫ ( w δ,σ )) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − r ǫ L p + ( M ) · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w δ,σ − i (cid:18) i ∗ ˜ f ( g ǫ ( w δ,σ )) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r ǫ L p + ( M ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) W δ,σ − i ∗ ˜ f (cid:0) g ǫ ( w δ,σ ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) − r ǫ f · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w δ,σ − i (cid:18) i ∗ ˜ f ( g ǫ ( w δ,σ )) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r ǫ L p + ( M ) ≤ C ǫ γ (1 − r ǫ ) · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w δ,σ − i (cid:18) i ∗ ˜ f ( g ǫ ( w δ,σ )) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r ǫ L p + ( M ) (56)where r ǫ ∈ (0 ,
1) satisfies 1 − r ǫ p + + r ǫ p + = q ǫ , which leads to r ǫ = Ns [ ( p + + N s ǫ ] ǫ . Applying Lemma 3.3 we see that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w δ,σ − i (cid:18) i ∗ ˜ f ( g ǫ ( w δ,σ )) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p + ( M ) ≤ (cid:13)(cid:13)(cid:13) w δ,σ (cid:13)(cid:13)(cid:13) L p + ( M ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) i (cid:18) i ∗ ˜ f ( g ǫ ( w δ,σ )) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p + ( M ) ≤ C (cid:18) ǫ − N − s s + (cid:13)(cid:13)(cid:13) g ǫ ( w δ,σ ) (cid:13)(cid:13)(cid:13) L NN + s ( M ) (cid:19) ≤ C (cid:18) ǫ − N − s s + ǫ N − s s (cid:19) . Using this and the fact that ǫ − ǫ = O (1), we deduce the desired estimate (55) from (56). (cid:3) To solve (39), it is important to understand the linear operator L ǫλ,σ ( Φ ) : = Φ − ( Π ǫλ,σ ) ⊥ i ∗ ˜ f ( i ( g ′ ǫ ( W ǫ α λ,σ ) Φ )) for Φ ∈ (cid:16) K ǫλ,σ (cid:17) ⊥ (57)where the function g ǫ and ˜ f are defined in (40). Letting Ψ = L ǫλ,σ ( Φ ), we see that the expression Φ − i ∗ f ( i ( g ′ ǫ ( W ǫ α λ,σ ) Φ )) = Ψ + P Ni = c i Z i ǫ α λ,σ in X , D Φ , Z i ǫ α λ,σ E ˜ f = i = , · · · , N (58)with certain pair of constants ( c , · · · , c N ) ∈ R N + , is equivalent to (57).This subsection is devoted to deduce that for a fixed Ψ ∈ ( K ǫλ,σ ) ⊥ , there are a unique function Φ ∈ ( K ǫλ,σ ) ⊥ and an ( N + c , · · · , c N ) ∈ R N + satisfying (58). This is the content ofProposition 4.4. It comes from the fact that the operators L ǫλ,σ : ( K ǫλ,σ ) ⊥ → ( K ǫλ,σ ) ⊥ have theinverses whose norms are uniformly bounded for ( λ, σ ) ∈ ( λ − , λ ) × M and su ffi ciently small ǫ > almost orthogonality of Z i δ,σ ’s with respect to the innerproduct h· , ·i f . As before, we use δ = ǫ α λ . 17 emma 4.2. For each i , j ∈ { , · · · , N } , we have D Z i δ,σ , Z j δ,σ E ˜ f = δ (cid:16) β i δ i j + o (1) (cid:17) as ǫ → where β i > .Proof. Recalling that Z i ’s are solutions of (24), we compute with estimates (44) and (43) that δ D Z i δ,σ , Z j δ,σ E ˜ f = κ s δ Z X (cid:18) ρ − s (cid:16) ∇Z i δ,σ , ∇Z j δ,σ (cid:17) ¯ g + E ( ρ ) Z i δ,σ Z j δ,σ (cid:19) d v ¯ g + δ Z M ˜ f Z i δ,σ Z j δ,σ d v ˆ h = Z R N + + t − s ∇ Z i · ∇ Z j dxdt + o (1) ! + O (cid:16) δ (cid:17) + O (cid:16) δ s (cid:17) = p Z R N w p − z i z j dx + o (1) , which implies (59). (cid:3) From the above lemma and the nondegeneracy result of [15] described in Subsection 2.3, thefollowing invertibility result of the linear operator L ǫλ,σ can be deduced. Lemma 4.3.
Suppose that N > s, ( λ, σ ) ∈ ( λ − , λ ) × M and ǫ > is small enough. Then thereexists a constant C > independent of the choice of ( λ, σ ) and ǫ such that k L ǫλ,σ ( Φ ) k ˜ f ,ǫ ≥ C k Φ k ˜ f ,ǫ (60) for all Φ ∈ ( K ǫλ,σ ) ⊥ .Proof. We only inspect the case when g ǫ ( u ) = u p ± ǫ + (and ˜ f = f ). The other case, namely, when g ǫ ( u ) = u p + − ǫ f u (and ˜ f =
0) is covered in a parallel way.Assume that (60) does not hold so that there are sequences ǫ n → λ n → λ ∞ ∈ [ λ − , λ ], δ n = ǫ α n λ n , σ n → σ ∞ ∈ M , Φ n ∈ ( K ǫ n λ n ,σ n ) ⊥ and Ψ n = L ǫ n λ n ,σ n ( Φ n ) with k Ψ n k f ,ǫ → k Φ n k f ,ǫ = n → ∞ . (61)We may further assume that σ ∞ = σ ∞ in M and that of theorigin in R N . According to (58) and Lemma 4.2, it is true that − δ n ( p ± ǫ ) Z M W p − ± ǫδ n ,σ n Z j δ n ,σ n Φ n d v ˆ h = δ n D Ψ n , Z j δ n ,σ n E f + N X i = ( c i ) n (cid:16) β i δ i j + o (1) (cid:17) for each j = , · · · , N . Following the assertion in the proof of Lemma 4.1, it is possible to regard Φ n as a function in R N + + whose support is included in the small half ball B + ¯ g ( σ n , r ) ⊂ B + ¯ g : = B + ¯ g (0 , r ) satisfying k Φ n k f ≤ C for a fixed constant C >
0. We define e Φ n ( z ) = δ N − s n Φ n ( δ n x + σ n , δ n t ) for all z ∈ R N + + . Then as in (49), one can check that k e Φ n k D ( R N + + ; t − s ) is bounded in n ∈ N and in particular e Φ n ⇀ e Φ ∞ weakly in D ( R N + + ; t − s ). Hence the compactness property of thetrace operator tells us that e Φ n → e Φ ∞ strongly in L q loc ( R N ) for any q < NN − s and so − δ n ( p ± ǫ ) Z M W p − ± ǫδ n ,σ n Z j δ n ,σ n Φ n d v ˆ h = − δ n Z R N p w p − z j e Φ ∞ dx + o (1) ! = o ( δ n ) . Here the second equality holds, for the assumption Φ n ∈ ( K ǫ n λ n ,σ n ) ⊥ gives0 = δ n D Φ n , Z j δ n ,σ n E f = δ n κ s Z X ρ − s ( ∇Z j δ n ,σ n , ∇ Φ n ) ¯ g d v ¯ g + O (cid:16) δ sn (cid:17) = Z R N + + t − s ∇ Z j · ∇ e Φ ∞ dxdt + o (1) = Z R N p w p − z j e Φ ∞ dx + o (1) . (62)18ince (cid:12)(cid:12)(cid:12)(cid:12) δ n D Ψ n , Z j δ n ,σ n E f (cid:12)(cid:12)(cid:12)(cid:12) = o ( δ n ) by (61), it follows that | ( c i ) n | = o ( δ n ) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X i = ( c i ) n Z i δ n ,σ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f = o (1) . (63)Therefore, if we define Ξ n ( z ) = δ − N − s n Ξ ( δ − n ( x − σ n ) , δ − n t ) for any function Ξ ∈ C ∞ c ( R N + ) andregard it as a function in the open half ball B + ¯ g ⊂ X , which is possible for n ∈ N large enough, wesee κ s Z B + ¯ g h t − s ( ∇ Φ n , ∇ Ξ n ) ¯ g + E ( t ) Φ n Ξ n i p | ¯ g | dxdt + Z B ˆ h h f − ( p ± ǫ ) W p − ± ǫδ n ,σ n i Φ n Ξ n q | ˆ h | dx = * Ψ n + N X i = ( c i ) n Z i δ n ,σ n , Ξ n + f = o (1)where B ˆ h : = B ˆ h (0 , r ) ⊂ R N . Note that {k Ξ n k f } ∞ n = is bounded and that (28) implies Z B + ¯ g | E ( t ) || Φ n || Ξ n | dxdt ≤ C Z B + ¯ g t − s | Φ n || Ξ n | dxdt ≤ C Z B + ¯ g t − s Φ n dxdt Z B + ¯ g t − s Ξ n dxdt ≤ C k Φ n k f · δ Z R N + + t − s Ξ dxdt ! = o (1)for s ∈ (0 , / R B + ¯ g | E ( t ) || Φ n || Ξ n | dxdt = o (1) when s ∈ [1 / ,
1) and H = n → ∞ , we obtain from Lemma 2.4 that κ s Z R N + + t − s ∇ e Φ ∞ · ∇ Ξ dxdt = p Z R N w p − e Φ ∞ Ξ dx , which means that e Φ ∞ is a weak solution of (24). On the other hand, the D ( R N + + ; t − s )-norm of e Φ ∞ is finite, so the Moser iteration argument works and it reveals that e Φ ∞ is L ∞ ( R N )-bounded (seethe proof of Lemma 5.1 in [14]). Thus with (61) the linear nondegeneracy result in [15], touchedin Subsection 2.3, implies e Φ ∞ = R N . Now we have that Z B ˆ h W p − ± ǫδ n ,σ n Φ n q | ˆ h | dx = δ ∓ ( N − s ) ǫ n Z R N χ p − ± ǫ ( δ n x ) w p − ± ǫ ( x ) e Φ n ( x ) q | ˆ h | ( δ n x + σ n ) dx = o (1) . Putting
Φ = Φ n into (58) shows then k Φ n k f = ( p ± ǫ ) Z B ˆ h W p − ± ǫδ n ,σ n Φ n q | ˆ h | dx + * Ψ n + N X i = ( c i ) n Z i δ n ,σ n , Φ n + f = o (1) , and particularly k Φ n k L p + ( M ) = o (1). At this point, we claim that k Φ n k L q ǫ ( M ) = o (1). Once we verifyit, together the previous estimate, it will yield that k Φ n k f ,ǫ → n → ∞ . Therefore we willreach a contradiction and our desired inequality (60) should have the validity. Since the assertionclearly holds in the subcritical or critical cases, it su ffi ces to consider the supercritical case only.In this situation, by applying Lemma 3.3 and using (58), (61) and (63), we get k Φ n k L q ǫ ( M ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) i ∗ f (cid:0) i (cid:0) g ′ ǫ (cid:0) W δ n ,σ n (cid:1) Φ n (cid:1)(cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L q ǫ ( M ) + (cid:13)(cid:13)(cid:13)(cid:13) Φ n − i ∗ f (cid:0) i (cid:0) g ′ ǫ (cid:0) W δ n ,σ n (cid:1) Φ n (cid:1)(cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L q ǫ ( M ) ≤ (cid:13)(cid:13)(cid:13) i (cid:0) g ′ ǫ (cid:0) W δ n ,σ n (cid:1) Φ n (cid:1)(cid:13)(cid:13)(cid:13) L Nq ǫ N + sq ǫ ( M ) + o (1) . (64)19ccording to H ¨older’s inequality, (cid:13)(cid:13)(cid:13) i (cid:0) g ′ ǫ (cid:0) W δ n ,σ n (cid:1) Φ n (cid:1)(cid:13)(cid:13)(cid:13) L Nq ǫ N + sq ǫ ( M ) ≤ (cid:13)(cid:13)(cid:13) g ′ ǫ ( w δ n ,σ n ) (cid:13)(cid:13)(cid:13) L ˜ r ǫ ( M ) k Φ n k L p + ( M ) , (65)where r ǫ + p + = N + sq ǫ Nq ǫ . Since ˜ r ǫ = N s + O ( ǫ ), we have (cid:13)(cid:13)(cid:13) g ′ ǫ ( w δ n ,σ n ) (cid:13)(cid:13)(cid:13) L ˜ r ǫ ( M ) = O (1). Thus we getfrom (65) that (cid:13)(cid:13)(cid:13) i (cid:0) g ′ ǫ (cid:0) W δ n ,σ n (cid:1) Φ n (cid:1)(cid:13)(cid:13)(cid:13) L Nq ǫ N + sq ǫ ( M ) = o (1), which gives k Φ n k L q ǫ ( M ) = o (1) with (64). (cid:3) As a result, we can construct a solution of (58).
Proposition 4.4.
Given N > s, fix a point ( λ, σ ) ∈ ( λ − , λ ) × M and a small parameter ǫ > suchthat Lemma 4.3 holds. For each Ψ ∈ ( K ǫλ,σ ) ⊥ , there exists a unique solution ( Φ , ( c , · · · , c N )) ∈ ( K ǫλ,σ ) ⊥ × R N + to equation (58) such that estimate (60) is satisfied.Proof. Firstly let us show that the linear map L ǫλ,σ on H ǫ is the sum of the identity and a compactoperator, that is to say, the map Φ ( Π ǫλ,σ ) ⊥ i ∗ ˜ f ( i ( g ′ ǫ ( W ǫ α λ,σ ) Φ )) for Φ ∈ H ǫ is compact. Denote ζ = N N + s . Then, by Corollary 3.4, we observe that i ∗ ˜ f : L ζ ( M ) → H ǫ is a compact operatorgiven ǫ > i ( W ǫ α λ,σ ) is in L ∞ ( M ), it holds that i ( g ′ ǫ ( W ǫ α λ,σ ) Φ ) ∈ L ζ ( M ) for any Φ ∈ H ǫ . Consequently, our assertion is true and the proposition follows from astandard argument utilizing the previous lemma and the Fredholm alternative. (cid:3) From the unique existence result for the linear problem (58) stated in Proposition 4.4, we are nowable to derive that (39) is solvable for any given ( λ, σ ) ∈ ( λ − , λ ) × M provided ǫ > ffi cientlysmall. Let us rewrite problem (39) as L ǫλ,σ ( Φ ) = − E ǫλ,σ + N ǫλ,σ ( Φ ) : = − (cid:16) Π ǫλ,σ (cid:17) ⊥ (cid:18) W ǫ α λ,σ − i ∗ ˜ f (cid:0) g ǫ ( W ǫ α λ,σ ) (cid:1)(cid:19) + (cid:16) Π ǫλ,σ (cid:17) ⊥ (cid:18) i ∗ ˜ f (cid:0) g ǫ ( W ǫ α λ,σ + Φ ) − g ǫ ( W ǫ α λ,σ ) − g ′ ǫ ( W ǫ α λ,σ ) Φ (cid:1)(cid:19) . (66) Proposition 4.5.
Under the assumption of Proposition 4.4 equation (66) possesses a unique solu-tion
Φ = Φ ǫ α λ,σ ∈ ( K ǫλ,σ ) ⊥ such that k Φ ǫ α λ,σ k ˜ f ,ǫ = O (cid:0) ǫ γ (cid:1) (67) where the exponent γ is defined in (46) .Proof. We define an operator T ǫλ,σ : ( K ǫλ,σ ) ⊥ → ( K ǫλ,σ ) ⊥ by T ǫλ,σ ( Φ ) = (cid:16) L ǫλ,σ (cid:17) − (cid:16) − E ǫλ,σ + N ǫλ,σ ( Φ ) (cid:17) . A direct computation using Lemmas 3.3 and 4.1 shows that it is a contraction map on the set B = n Φ ∈ ( K ǫλ,σ ) ⊥ : k Φ k ˜ f ,ǫ ≤ M ǫ γ o for some large M > . Consequently, it admits a unique fixed point Φ ǫ α λ,σ ∈ B , which becomes a solution to (66). Thiscompletes the proof. (cid:3) Finite dimensional reduction
We keep using notations g ǫ ( u ), ˜ f in (40). Define also G ǫ ( u ) = R t g ǫ ( t ) dt .It is notable that equations (15 ± )-(16) have the variational structure. In other words, U ∈ H ǫ is a weak solution of (15 ± )-(16) if it is a critical point of the energy functional I ǫ ( U ) : = κ s Z X (cid:16) ρ − s |∇ U | g + E ( ρ ) U (cid:17) d v ¯ g + Z M ˜ f U d v ˆ h − Z M G ǫ ( U ) d v ˆ h where d v ¯ g and d v ˆ h denote the volume forms on ( X , ¯ g ) and its boundary ( M , ˆ h ), respectively. Basedon the previous observations, we define a reduced energy functional by J ǫ ( λ, σ ) = I ǫ (cid:0) W ǫ α λ,σ + Φ ǫ α λ,σ (cid:1) (68)for any ( λ, σ ) ∈ (0 , ∞ ) × M where the exponent α > Φ ǫ α λ,σ denotes thefunction determined in Proposition 4.5.The next proposition claims that the well-known finite dimension reduction procedure is stillapplicable in our setting. Proposition 5.1.
Assume that ǫ > is small enough. Then the reduced energy J ǫ : (0 , ∞ ) × M → R is continuously di ff erentiable. Moreover, if J ′ ǫ ( λ ǫ , σ ǫ ) = for some element ( λ ǫ , σ ǫ ) ∈ (0 , ∞ ) × M,then the function W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ solves problems (15 ± ) - (16) (according to the choice of thenonlinearity g ǫ ). Its trace on M is in C ,β ( M ) for some β ∈ (0 , determined by N and s.Proof. Fix ǫ > L ǫ (( λ, σ ) , U ) = U + (cid:16) Π ǫλ,σ (cid:17) ⊥ (cid:20) W ǫ α λ,σ − i ∗ ˜ f ( i ( g ǫ ( W ǫ α λ,σ + U ))) (cid:21) for (( λ, σ ) , U ) ∈ (0 , ∞ ) × M × H ǫ . Then L ǫ (( λ, σ ) , Φ ǫ α λ,σ ) = ∂ L ǫ ∂ U (( λ, σ ) , U ) = U − (cid:16) Π ǫλ,σ (cid:17) ⊥ (cid:20) i ∗ ˜ f (cid:0) i (cid:0) g ′ ǫ (cid:0) W ǫ α λ,σ (cid:1) U (cid:1)(cid:1)(cid:21) . By elliptic regularity, i ( g ′ ǫ ( W ǫ α λ,σ ) Φ ǫ α λ,σ ) ∈ L q ( M ) for some q > NN + s . (Refer to the latter part ofthis proof.) Hence we know from Corollary 3.4 that ∂ L ǫ ∂ U (( λ, σ ) , Φ ǫ α λ,σ ) : H ǫ → H ǫ is a Fredholmoperator of index 0. Moreover, using (67), one can check that it is also injective. Therefore ∂ L ǫ ∂ U (( λ, σ ) , Φ ǫ α λ,σ ) is invertible and the implicit function theorem shows that the mapping ( λ, σ ) ∈ (0 , ∞ ) × M Φ ǫ α λ,σ ∈ H ǫ is C . This leads that J ǫ is a C map. Furthermore it is a standard stepto show that J ′ ǫ ( λ ǫ , σ ǫ ) = I ′ ǫ (cid:0) W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ (cid:1) = ± ). The other equation (16) can bedealt with similarly. One has then κ s Z X h ρ − s (cid:0) ∇ (cid:0) W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ (cid:1) , ∇ Ξ (cid:1) ¯ g + E ( ρ ) (cid:0) W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ (cid:1) Ξ i d v ¯ g + Z M f (cid:0) W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ (cid:1) Ξ d v ˆ h = ( p ± ǫ ) Z M (cid:0) W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ (cid:1) p − ± ǫ + Ξ d v ˆ h for any Ξ ∈ H ( X ; ρ − s ). Putting Ξ = ( W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ ) − into the above identity and thenapplying (7) verifies that W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ ≥ X . By Propositions 2.1 and 2.3, equation (17)is solved by the nonnegative function U = ( ρ ∗ /ρ ) z − N ( W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ ) defined in X and itstrace u ≥ M . Also, U is not identically zero since kW ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ k f ≥kW ǫ α λ ǫ ,σ ǫ k f − k Φ ǫ α λ ǫ ,σ ǫ k f ≥ C + O ( ǫ γ ) >
0, and it is strictly positive in X , for (17) is a uniformlyelliptic equation in divergence form away from the boundary. Suppose now that u ( z ) = z ∈ M . Then by the Hopf lemma [29, Theorem 3.5], we have ( ρ ∗ ) − s ∂ ρ ∗ U > z , while(18) gives κ s ( ρ ∗ ) − s ∂ ρ ∗ U = − ∂ s ν U = Q s ˆ h u − P s ˆ h u = (cid:16) Q s ˆ h + f − u p ± ǫ (cid:17) u = z . Therefore a contradiction arises and the functions U and W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ should be strictlypositive in X .Finally, if the nonlinearity of the problem is subcritical, then [29, Theorem 3.4] implies that U is a locally bounded function in X . Then the regularity property of W ǫ α λ ǫ ,σ ǫ + Φ ǫ α λ ǫ ,σ ǫ (cid:12)(cid:12)(cid:12) M followsdirectly by the result of [29, Proposition 3.2]. If our problem is critical or supercritical one, thenthe nonlinear term is given by g ǫ ( u ) = u N + sN − s + ǫ + = u sN − s + ǫ · u for ǫ ≥
0. Note that N s · sN − s + ǫ ! = NN − s + N s ǫ = q ǫ (see (31)). Therefore u ∈ L q ǫ ( M ) means that u sN − s + ǫ ∈ L N s ( M ), and so one can modify the proofof Lemmas 3.5 and 3.8 in [14] slightly to show that U is L ∞ -bounded. The regularity of U againfollows from [29, Proposition 3.2] now. (cid:3) C -estimates We set d = R R N w p + dx , d = R R N w dx , d = R R N w p + log w dx and d s = κ s R R N + + t − s |∇ W | dxdt (whose finiteness for N > s + Proposition 6.1.
Suppose that ǫ > is su ffi ciently small, and H = if s ∈ [1 / , . In addition,we remind the reduced energy functional J ǫ defined in (68) .(i) Assume further that N > max { s , } . If problems (15 ± ) is concerned for s ∈ (0 , , then it holdsJ ǫ ( λ, σ ) = sd N ± ǫ p + "( N − s s ! log ǫ − p + ) d − d + ǫ " d f ( σ ) λ s ± ( N − s ) d N log λ + o (1) . (69) (ii) Let us consider equation (16) under the assumption that s ∈ (0 , / and N ≥ . Then itfollows thatJ ǫ ( λ, σ ) = sd N + ǫ − s d f ( σ ) λ s + N ( N − + (cid:16) − s (cid:17) N (1 − s ) d s H ( σ ) λ + o (1) . (70) In the above estimates, o (1) tends to 0 uniformly for ( λ, σ ) ∈ ( λ − , λ ) × M. To prove this, we need the following lemmas.
Lemma 6.2.
Fix any small λ ∈ (0 , . Given δ = ǫ α λ , we haveJ ǫ ( λ, σ ) = I ǫ (cid:0) W δ,σ (cid:1) + o ( ǫ ) for problems (15 ± ) , o (cid:16) ǫ − s (cid:17) for problem (16) , (71) uniformly for ( λ, σ ) ∈ ( λ − , λ ) × M. roof. Putting Φ δ,σ into (39) and then applying Φ δ,σ ∈ ( K ǫλ,σ ) ⊥ and Taylor’s theorem, we obtain J ǫ ( λ, σ ) − I ǫ (cid:0) W δ,σ (cid:1) = (cid:10) W δ,σ + Φ δ,σ , Φ δ,σ (cid:11) ˜ f − Z M (cid:0) G ǫ (cid:0) W δ,σ + Φ δ,σ (cid:1) − G ǫ (cid:0) W δ,σ (cid:1)(cid:1) = Z M (cid:0) g ǫ (cid:0) W δ,σ + Φ δ,σ (cid:1) − g ǫ (cid:0) W δ,σ (cid:1)(cid:1) Φ δ,σ − Z M (cid:0) G ǫ (cid:0) W δ,σ + Φ δ,σ (cid:1) − G ǫ (cid:0) W δ,σ (cid:1) − g ǫ (cid:0) W δ,σ (cid:1) Φ δ,σ (cid:1) = O (cid:18) k Φ δ,σ k f (cid:19) . Therefore the conclusion follows by Proposition 4.5. (cid:3)
Lemma 6.3.
Suppose that s ∈ (0 , / and N > s + . Then Z R N + + t − s |∇ W | dxdt = + s Z R N + + t − s |∇ x W | dxdt = − s Z R N + + t − s W dxdt < ∞ . (72) Proof.
The argument we will use here is based on the proof of Lemma 7.2 in [29]. We will onlyprove the first identity, because the second identity can be justified in a similar manner.If we denote the Fourier transform of W with respect to the x -variable by b W , then we have b W ( ξ, t ) = ˆ w ( ξ ) φ (2 π | ξ | t ) where φ ( t ) is a solution of the equation φ ′′ ( t ) + − st φ ′ ( t ) − φ ( t ) = R + , φ (0) = , lim t →∞ φ ( t ) = . (73)Thus we have Z R N + + t − s |∇ x W | dxdt = Z R N + + (2 π | ξ | ) t − s (cid:12)(cid:12)(cid:12) b W ( ξ, t ) (cid:12)(cid:12)(cid:12) d ξ dt = Z R N (2 π | ξ | ) s − | ˆ w ( ξ ) | d ξ · Z ∞ t − s | φ ( t ) | dt (74)and Z R N + + t − s ( ∂ t W ) dxdt = Z R N + + (2 π | ξ | ) t − s | ˆ w ( ξ ) | | φ (2 π | ξ | t ) | d ξ dt = Z R N (2 π | ξ | ) s − | ˆ w ( ξ ) | d ξ · Z ∞ t − s | φ ′ ( t ) | dt . (75)Since φ ( t ) = − s t s K s ( t ) / Γ ( s ) where K s is the modified Bessel function of the second kind (see[27, Lemma 14] for its derivation), φ decays exponentially as t goes to ∞ and φ ′ ( t ) ∼ t − near 0.Hence after multiplying (73) by t − s φ ′ ( t ), which converges to 0 as t →
0, and applying integrationby parts, we discover that3 − s Z ∞ t − s φ = − (1 − s ) Z ∞ t − s ( φ ′ ) − Z ∞ t − s φ ′ φ ′′ = − (1 − s ) Z ∞ t − s ( φ ′ ) + − s Z ∞ t − s ( φ ′ ) − t − s ( φ ′ ) i ∞ t = = + s Z ∞ t − s ( φ ′ ) . Putting this with (74) and (75) gives the first estimate of (72). (cid:3) roof of Proposition 6.1. We will accomplish the proof in 3 steps. We use the notation δ = ǫ α λ , ∂ i = ∂ x i for i = , · · · , N , and t to denote ρ near the boundary.S tep
1. We initiate the proof by computing κ s R X ρ − s |∇W δ,σ | g d v ¯ g . By (42), we have κ s Z X ρ − s |∇W δ,σ | g d v ¯ g = κ s Z B + r t − s h ¯ g i j ∂ i W δ ( x , t ) ∂ j W δ ( x , t ) + ( ∂ t W δ ( x , t )) i p | ¯ g | dxdt + o ( ǫ ) for (15 ± ) , o (cid:16) ǫ − s (cid:17) for (16) , where r is the small positive number chosen in Section 3. Also Lemma 2.4 implies that ¯ g i j = δ i j + π i j t + O ( | ( x , t ) | ) and p | ¯ g | = − Ht + O ( | ( x , t ) | ). Hence we can compute κ s Z B + r t − s h ¯ g i j ∂ i W δ ( x , t ) ∂ j W δ ( x , t ) + ( ∂ t W δ ( x , t )) i p | ¯ g | dxdt = κ s Z B + r t − s |∇ W δ | dxdt + κ s π i j ( σ ) Z B + r t − s ∂ i W δ ∂ j W δ dxdt − H ( σ ) Z B + r t − s |∇ W δ | dxdt + Z B + r t − s O (cid:16) | ( x , t ) | (cid:17) |∇ W δ | dxdt (76)where the last term of the right-hand side is negligible by (44).On the other hand, since ∂ i W is odd in x i and π i j = ˆ h ik π kl ˆ h l j = δ ik π kl δ l j = π i j so that π i j δ i j = π i j ˆ h i j = H at the point σ (for we are using now the normal coordinate of ˆ h at σ ), it holds2 π i j ( σ ) Z R N + + t − s ∂ i W ∂ j W dxdt = N H ( σ ) Z R N + + t − s |∇ x W | dxdt (77)(which is finite provided that N > s + ± ) first. It would be convenient to divide thecases according to the magnitude of s .- If s ∈ (0 , / R B + r t − s |∇ W δ | dxdt = o ( δ s ) = o ( ǫ ).- If s ∈ [1 / , B + r . Thus by the hypothesis that H = M if s ∈ [1 / , N ≥
2, then N > s + s ∈ (0 , / κ s Z X ρ − s |∇W δ,σ | g d v ¯ g = R R N w p + dx + o (cid:16) δ s (cid:17) for (15 ± ) , R R N w p + dx + δκ s H ( σ ) · (cid:16) + s − N N (cid:17) R R N + + t − s |∇ W | dxdt + o ( δ ) for (16) . (78)S tep
2. Next, we calculate κ s R X E ( ρ ) W δ,σ d v ¯ g . Assume that s ∈ [1 / ,
1) and H =
0. Then | E ( ρ ) | ≤ C ρ − s , so we get κ s Z X | E ( ρ ) |W δ,σ d v ¯ g ≤ C Z B + r t − s W δ dxdt = O (cid:16) δ min { , N − s } (cid:17) if N , s + , O (cid:16) δ | log δ | (cid:17) if N = s + . (79)24n the other hand, if s ∈ (0 , / κ s R X E ( ρ ) W δ,σ d v ¯ g = O ( δ ) = o ( δ s ) so thatwe can neglect this term if problems (15 ± ) is considered. If N > s +
1, we have more accurateestimate κ s Z X E ( ρ ) W δ,σ d v ¯ g = κ s N − s ! Z X H ρ − s W δ,σ d v ¯ g + O (cid:16) δ min { , N − s } (cid:17) if N , s + , O (cid:16) δ | log δ | (cid:17) if N = s + , = κ s N − s ! H ( σ ) δ Z R N + + t − s W dxdt + o ( δ ) , (80)which is needed for problem (16).S tep
3. Finally, we turn to estimate R M G ǫ ( W δ,σ ) d v ˆ h . To deal with whole cases, it su ffi ces tocompute that I : = Z M W p + δ,σ d v ˆ h , I : = Z M W p + ± ǫδ,σ p + ± ǫ − W p + δ,σ p + ! d v ˆ h and I : = Z M f W δ,σ d v ˆ h . Since d v ˆ h = √| h | dx = − R i j x i x j + O ( | x | ), under the assumption that N > s ∈ (0 ,
1) that I = Z R N w p + dx + o (cid:16) δ max { s , } (cid:17) and I = δ s f ( σ ) Z R N w dx + o (1) ! . (81)Besides one can calculate the integral I by applying Taylor’s theorem and the expansion ( a ǫ ) b ǫ = + b ǫ log( a ǫ ) + O ( ǫ | log ǫ | ) which holds for a > b ∈ R and small ǫ >
0, yielding I = Z R N ( λǫ α ) ∓ ( N − s ) ǫ w p + ± ǫ p + ± ǫ − w p + p + ! dx + O (cid:16) δ | log δ | (cid:17) = ± ǫ " p + Z R N w p + log w dx − ( α log ǫ + log λ ) · ( N − s ) N · Z R N w p + dx ∓ ǫ ( p + Z R N w p + dx + O (cid:16) ǫ | log ǫ | (cid:17) + O (cid:16) δ | log δ | (cid:17) . (82)From (78)-(82) and (71), estimations (69) and (70) can be deduced at once. This concludesthe proof. (cid:3) C -estimates The aim of this subsection is to improve Proposition 6.1 by showing that the o (1)-terms go to0 in C -sense. Unfortunately there is some technical di ffi culty in obtaining the C -estimates,because the estimate k Φ ǫ α λ,σ k ¯ f = O ( ǫ γ ) in (67) (and k Φ ǫ α λ,σ k L q ( M ) = O ( ǫ γ ) for 1 ≤ q ≤ q ǫ )of the remainder term Φ ǫ α λ,σ is not so small compared with the blow up rate ǫ − α of the bubbles W ǫ α λ,σ , especially when s is close to zero. In fact, the standard argument for the C -estimates of J ǫ (see e.g. [44]) provides only the bound O ( ǫ − α + γ ) for the error term, which is not tolerated in(83) and (85) below. Nevertheless, we can achieve the C -estimates by modifying some ideas inEsposito-Musso-Pistoia [24]. Proposition 6.4.
Estimates (69) and (70) are valid C -uniformly for ( λ, σ ) ∈ ( λ − , λ ) × M. Pre-cisely, the following holds for each fixed point σ ∈ M. Suppose that y ∈ R N is a point near theorigin.(i) Under the assumption of (i) in Proposition 6.1, we have ∂∂y k J ǫ ( λ, exp σ ( y )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = = ǫ ∂∂y k " d f (exp σ ( y )) λ s y = + o ( ǫ ) (83)25 or each ≤ k ≤ N and ∂∂λ J ǫ ( λ, σ ) = ǫ " d s f ( σ ) λ s − ± ( N − s ) d N λ + o ( ǫ ) . (84) (ii) Under the assumption of (ii) in Proposition 6.1, we have ∂∂y k J ǫ ( λ, exp σ ( y )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = = ǫ − s ∂∂y k d f (exp σ ( y )) λ s + N ( N − + (cid:16) − s (cid:17) N (1 − s ) d s H (exp σ ( y )) λ y = + o (cid:18) ǫ − s (cid:19) (85) for each ≤ k ≤ N and ∂∂λ J ǫ ( λ, σ ) = ǫ − s d s f ( σ ) λ s − + N ( N − + (cid:16) − s (cid:17) N (1 − s ) d s H ( σ ) y = + o (cid:18) ǫ − s (cid:19) . (86)Let us note that ∂∂λ W ǫ α λ is a even function in x ∈ R N like the bubble W ǫ α λ and has the samedecaying property as W ǫ α λ . From this fact we can see that all the error estimates in the proof ofProposition 6.1 hold exactly in the same manner even if they are di ff erentiated in the λ -variable.This tells us that (69) and (70) hold in C -sense with respect to λ , i.e., (84) and (86) are true. Thusit only remains to show that (69) and (70) also hold in C -sense with respect to σ , or equivalently,(83) and (85) are valid.We fix σ ∈ M and set σ ( y ) = exp σ ( y ) for y ∈ B N (0 , r ) (recall that 4 r > M ) for conciseness. For the proof, we first need to establishseveral preliminary lemmas. Lemma 6.5.
Recall the definition of the truncation function χ which was introduced in (37) , andthe fact that any point z ∈ X located su ffi ciently close to σ ∈ M can be described as z = ( σ ( x ) , t ) for some x ∈ B N (0 , r ) and t ∈ (0 , r ) . Also fix any ≤ k ≤ N.1. For any z = ( σ ( x ) , t ) near the point σ , it holds that ∂∂y k W δ,σ ( y ) ! y = ( z ) = − χ ( | ( x , t ) | ) ∂ k W δ ( x , t ) + ̺ ( σ ( x ) , t ) , (87) where ̺ is a function on X supported on the half ball B + ¯ g ( σ , r ) (defined in (48) ) satisfying k ̺ k ˜ f ,ǫ = O ( δ ) .2. For any z near the point σ and ≤ i ≤ N, we have ∂∂y k Z i δ,σ ( y ) ! y = ( z ) = − χ ( | ( x , t ) | ) ∂ k Z i δ ( x , t ) + ̺ i ( σ ( x ) , t ) (88) where ̺ i is a function on X supported on B + ¯ g ( σ , r ) such that k ̺ i k ˜ f ,ǫ = O (1) .Proof. Using the chain rule and Lemma A.2, we compute ∂∂y k W δ,σ ( y ) ( z ) = χ ( d ( z , σ ( y ))) ∂ W δ ∂y k ( E ( y, x ) , t ) + ∂χ ∂y k ( d ( z , σ ( y ))) W δ ( E ( y, x ) , t ) = χ ( d ( z , σ ( y ))) N X j = " ∂ j W δ ( E ( y, x ) , t ) ∂ E j ( y, x ) ∂y k + O δ N − s · |∇ χ | ( | ( x − y, t ) | ) | ( x − y, t ) | N − s ! E ( y, x ) = exp − σ ( y ) ( σ ( x )) = ( E ( y, x ) , · · · , E N ( y, x )) ∈ R N . Therefore replacing( σ ( x ) , t ) with (exp σ ( y ) ( x ) , t ) in the previous inequalities, we obtain ∂∂y k W δ,σ ( y ) (cid:16) exp σ ( y ) ( x ) , t (cid:17) = χ (cid:16) d ((exp σ ( y ) ( x ) , t ) , σ ( y )) (cid:17) N X j = " ∂ j W δ ( x , t ) ∂ E j ∂y k (cid:16) y, σ − (cid:16) exp σ ( y ) (cid:17) ( x ) (cid:17) + O δ N − s |∇ χ | ( | ( x , t ) | ) | ( x , t ) | N − s ! . (89)By (6.12) of [44], it holds that ∂ E j ∂y k (cid:16) y, σ − (cid:16) exp σ ( y ) (cid:17) ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = = ∂ E j ∂y k (0 , x ) = − δ k j + O (cid:16) | x | (cid:17) . Taking y = ∂∂y k W δ,σ ( y ) ! y = (cid:16) exp σ ( x ) , t (cid:17) = − χ ( | ( x , t ) | ) ∂ k W δ ( x , t ) + χ ( | ( x , t ) | ) N X j = h ∂ j W δ ( x , t ) i O (cid:16) | x | (cid:17) + O δ N − s |∇ χ | ( | ( x , t ) | ) | ( x , t ) | N − s !| {z } : = ̺ ( x , t ) . We readily find that k ̺ k ˜ f ,ǫ = O ( δ ), and thus arrive at the first equality (87).The same argument can be applied to prove the second equality (88). The proof is completed. (cid:3) We remind from Proposition 4.5 that Φ ǫ α λ,σ solves equation (39). Hence for some constants c i ∈ R , 0 ≤ i ≤ N , we have Φ ǫ α λ,σ = −W ǫ α λ,σ + i ∗ ˜ f (cid:0) i (cid:0) g ǫ ( W ǫ α λ,σ + Φ ǫ α λ,σ ) (cid:1)(cid:1) + N X i = c i Z i ǫ α λ,σ . (90) Lemma 6.6. In (90) , we have that c i = O ( ǫ γ + α ) for each ≤ i ≤ N.Proof.
Fixing any i ∈ { , · · · , N } and taking the inner product D · , Z i δ,σ E ˜ f on (90), we get that c i D Z i δ,σ , Z i δ,σ E ˜ f + X j , i c j D Z j δ,σ , Z i δ,σ E ˜ f = D W δ,σ , Z i δ,σ E ˜ f − Z M g ǫ ( W δ,σ + Φ δ,σ ) Z i δ,σ = D W δ,σ , Z i δ,σ E ˜ f − Z M g ǫ ( W δ,σ ) Z i δ,σ ! + Z M ( g ǫ ( W δ,σ ) − g ǫ ( W δ,σ + Φ δ,σ )) Z i δ,σ ! (91)where δ = ǫ α λ . Replacing Φ by Z i δ,σ in the proof of Lemma 4.1 and using the estimate (cid:13)(cid:13)(cid:13)(cid:13) Z i δ,σ (cid:13)(cid:13)(cid:13)(cid:13) ˜ f ,ǫ = O ( ǫ − α ) instead of k Φ k ˜ f = O (1), we may deduce that D W δ,σ , Z i δ,σ E ˜ f − Z M g ǫ ( W δ,σ ) Z i δ,σ = O (cid:0) ǫ γ − α (cid:1) . Next we apply H ¨older’s inequality to ascertain Z M ( g ǫ ( W δ,σ ) − g ǫ ( W δ,σ + Φ δ,σ )) Z i δ,σ = O (cid:18)(cid:13)(cid:13)(cid:13) g ′ ǫ ( W δ,σ ) (cid:13)(cid:13)(cid:13) L N s ( M ) · (cid:13)(cid:13)(cid:13) Φ δ,σ (cid:13)(cid:13)(cid:13) L NN − s ( M ) · (cid:13)(cid:13)(cid:13) Z i δ,σ (cid:13)(cid:13)(cid:13) L NN − s ( M ) (cid:19) = O (cid:0) ǫ γ − α (cid:1) . c i ǫ − α + X j , i c j o (cid:16) ǫ − α (cid:17) = O (cid:0) ǫ γ − α (cid:1) , which yields the desired estimate c i = O ( ǫ γ + α ). (cid:3) Recall a fixed point σ ∈ M and the map σ ( y ) = exp σ ( y ) defined for y ∈ B N (0 , r ). In thenext lemma, we shall replace the derivatives ∂ y k W δ,σ ( y ) and ∂ y k Φ δ,σ ( y ) with respect to the parame-ters by the derivatives ∂ k W δ,σ ( y ) and ∂ k Φ δ,σ ( y ) with respect to the spatial variables in the expressionof ∂ y k J ǫ ( λ, σ ( y )) | y = . This will permit us to take integration by parts to evaluate ∂ y k J ǫ ( λ, σ ( y )) | y = .This idea was introduced in [24] where existence of the bubbling solutions for the two dimensionalLane-Emden-Fowler equation was examined.Take a cut-o ff function χ : (0 , ∞ ) → [0 ,
1] such that χ = , r ) and 0 on (4 r , ∞ ).Then we see that χ = χ ). We also set a function b Φ δ,σ : R n + + → R by b Φ δ,σ ( x , t ) = χ ( d (( σ ( x ) , t ) , σ )) Φ δ,σ ( σ ( x ) , t ) , which satisfies supp( b Φ δ,σ ) ⊂ B + r , and a function e Φ k δ,σ : X → R ( k = , · · · , N ) by e Φ k δ,σ ( z ) = (cid:16) ∂ k b Φ δ,σ (cid:17) ( x , t ) if z ∈ X is near M so that it can be written as z = (exp σ ( x ) , t ) , . Then we have the following result.
Lemma 6.7.
We haveI ′ ǫ ( W δ,σ ( y ) + Φ δ,σ ( y ) ) ∂ y k Φ δ,σ ( y ) (cid:12)(cid:12)(cid:12) y = = − I ′ ǫ ( W δ,σ + Φ δ,σ ) e Φ k δ,σ + O (cid:16) ǫ γ (cid:17) andI ′ ǫ ( W δ,σ ( y ) + Φ δ,σ ( y ) ) ∂ y k W δ,σ ( y ) (cid:12)(cid:12)(cid:12) y = = − I ′ ǫ ( W δ,σ + Φ δ,σ ) ∂ k ( χ ( | ( x , t ) | ) W δ ( x , t )) + o ( ǫ ) for (15 ± ) , O (cid:0) ǫ γ + α (cid:1) for (16) , where z = ( σ ( x ) , t ) ∈ X satisfies d ( z , σ ) = | ( x , t ) | ≤ r .Proof. From (90) and the fact that D Z i δ,σ , Φ δ,σ E ˜ f = σ ∈ M , we see that I ′ ǫ ( W δ,σ ( y ) + Φ δ,σ ( y ) )( ∂ y k Φ δ,σ ( y ) ) = N X i = c i D Z i δ,σ ( y ) , ∂ y k Φ δ,σ ( y ) E ˜ f = − N X i = c i D ∂ y k Z i δ,σ ( y ) , Φ δ,σ ( y ) E ˜ f = − N X i = c i " κ s Z R N + + t − s (cid:16) ∇ ∂ y k Z i δ,σ ( y ) ( σ ( x ) , t ) , ∇ b Φ δ,σ ( y ) ( x , t ) (cid:17) ¯ g p | ¯ g | dxdt + κ s Z R N + + E ( t ) ∂ y k Z i δ,σ ( y ) ( σ ( x ) , t ) b Φ δ,σ ( y ) ( x , t ) p | ¯ g | dxdt + Z R N ˜ f ( σ ( x )) ∂ y k Z i δ,σ ( y ) ( σ ( x ) , b Φ δ,σ ( y ) ( x , q | ˆ h | dx . (92)28n the other hand, I ′ ǫ ( W δ,σ + Φ δ,σ ) e Φ k δ,σ = N X i = c i D Z i δ,σ , e Φ k δ,σ E ˜ f = N X i = c i " κ s Z R N + + t − s (cid:16) ∇ (cid:16) χ ( | ( x , t ) | ) Z i δ ( x , t ) (cid:17) , ∇ (cid:16) ∂ k b Φ δ,σ ( x , t ) (cid:17)(cid:17) ¯ g p | ¯ g | dxdt + κ s Z R N + + E ( t ) χ ( | ( x , t ) | ) Z i δ ( x , t ) ∂ k b Φ δ,σ ( x , t ) p | ¯ g | dxdt + Z R N ˜ f ( σ ( x )) χ ( | x | ) z i δ ( x ) ∂ k b Φ δ,σ ( x , q | ˆ h | dx . (93)Let us compare (92), for which y = ∂ k p | ¯ g | = O ( | ( x , t ) | ) which stems from Lemma 2.4, and applying the integrationby parts, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N + + E ( t ) (cid:16) ∂ y k Z i δ,σ ( y ) ( σ ( x ) , t ) (cid:17) y = b Φ δ,σ ( x , t ) p | ¯ g | dxdt − Z R N + + E ( t ) χ ( | ( x , t ) | ) Z i δ ( x , t ) ∂ k b Φ δ,σ ( x , t ) p | ¯ g | dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R N + + O (cid:16) t − s (cid:17) (( | χ | + | ∂ k χ | )( | ( x , t ) | ) + O ( | ( x , t ) | )) (cid:12)(cid:12)(cid:12)(cid:12) Z i δ ( x , t ) b Φ δ,σ ( x , t ) (cid:12)(cid:12)(cid:12)(cid:12) dxdt + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N + + E ( t ) ̺ i ( x , t ) b Φ δ,σ ( x , t ) p | ¯ g | dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:13)(cid:13)(cid:13) t − s Z i δ (cid:13)(cid:13)(cid:13) L ( B + r ) (cid:13)(cid:13)(cid:13) Φ δ,σ (cid:13)(cid:13)(cid:13) ˜ f + C (cid:13)(cid:13)(cid:13) ̺ i (cid:13)(cid:13)(cid:13) ˜ f · (cid:13)(cid:13)(cid:13) Φ δ,σ (cid:13)(cid:13)(cid:13) ˜ f = O (cid:0) ǫ γ − α (cid:1) for s ∈ (0 , / s ∈ [1 / ,
1) and H =
0, the above term has a better bound O ( ǫ γ ). Similarly, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N ˜ f ( σ ( x )) (cid:16) ∂ y k Z i δ,σ ( y ) ( σ ( x ) , (cid:17) y = b Φ δ,σ ( x , q | ˆ h | dx − Z R N ˜ f ( σ ( x )) χ ( | x | ) z i δ ( x ) ∂ k b Φ δ,σ ( x , q | ˆ h | dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ k ˜ f ( σ ( x )) χ ( | x | ) q | ˆ h | !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) z i δ ( x ) b Φ δ,σ ( x , (cid:12)(cid:12)(cid:12)(cid:12) dx + Z R N (cid:12)(cid:12)(cid:12)(cid:12) ˜ f ( σ ( x )) ̺ i ( x , b Φ δ,σ ( x , (cid:12)(cid:12)(cid:12)(cid:12) q | ˆ h | dx ≤ C (cid:18)(cid:13)(cid:13)(cid:13) z i δ (cid:13)(cid:13)(cid:13) L NN + s ( B N (0 , r )) (cid:13)(cid:13)(cid:13) Φ δ,σ (cid:13)(cid:13)(cid:13) L NN − s ( M ) + (cid:13)(cid:13)(cid:13) ̺ i ( · , (cid:13)(cid:13)(cid:13) L NN − s ( M ) (cid:13)(cid:13)(cid:13) Φ δ,σ (cid:13)(cid:13)(cid:13) L NN − s ( M ) (cid:19) = O (cid:16) ǫ (2 s − α + γ + ǫ γ (cid:17) . Finally we use Lemmas 6.5, 2.4 and 3.5 to get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N + + t − s (cid:16) ∇ ∂ y k Z i δ,σ ( y ) ( σ ( x ) , t ) , ∇ b Φ δ,σ ( y ) ( x , t ) (cid:17) ¯ g p | ¯ g | dxdt − Z R N + + t − s (cid:16) ∇ (cid:16) χ ( | ( x , t ) | ) Z i δ ( x , t ) (cid:17) , ∇ (cid:16) ∂ k b Φ δ,σ ( x , t ) (cid:17)(cid:17) ¯ g p | ¯ g | dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R N + + t − s (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:16) χ ( | ( x , t ) | ) Z i δ ( x , t ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) O ( | ( x , t ) | ) + (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:16) ∂ k χ ( | ( x , t ) | ) Z i δ ( x , t ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) · (cid:12)(cid:12)(cid:12)(cid:12) ∇ b Φ δ,σ ( x , t ) (cid:12)(cid:12)(cid:12)(cid:12) dxdt + Z R N + + t − s (cid:12)(cid:12)(cid:12) ∇ ̺ i ( x , t ) (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) ∇ b Φ δ,σ ( x , t ) (cid:12)(cid:12)(cid:12)(cid:12) p | ¯ g | dxdt = O (cid:0) ǫ γ (cid:1) . I ′ ǫ ( W δ,σ + Φ δ,σ ) (cid:16) ∂ y k Φ δ,σ ( y ) (cid:17) + I ′ ǫ ( W δ,σ + Φ δ,σ ) e Φ k σ = O (cid:0) ǫ γ + α (cid:1) · O (cid:16) ǫ γ − α + ǫ (2 s − α + γ (cid:17) = O (cid:16) ǫ γ (cid:17) . It proves the first identity.We turn to prove the second identity. For this we apply Lemmas 3.5, 6.5 and 6.6 to certify I ′ ǫ ( W δ,σ ( y ) + Φ δ,σ ( y ) ) ∂ y k W δ,σ ( y ) + I ′ ǫ ( W δ,σ + Φ δ,σ ) ∂ k ( χ ( | ( x , t ) | ) W δ ( x , t )) = N X i = c i (cid:28) Z i δ,σ , ∂ y k W δ,σ ( y ) (cid:12)(cid:12)(cid:12) y = + χ ( | ( x , t ) | ) ∂ k W δ ( x , t ) + χ ′ ( | ( x , t ) | ) ∂ k | ( x , t ) | W δ ( x , t ) (cid:29) ˜ f = N X i = c i D Z i δ,σ , ̺ i + χ ′ ( | ( x , t ) | ) ∂ k | ( x , t ) | W δ ( x , t ) E ˜ f = O N X i = | c i | (cid:13)(cid:13)(cid:13) Z i δ,σ (cid:13)(cid:13)(cid:13) ˜ f · (cid:13)(cid:13)(cid:13) ̺ i (cid:13)(cid:13)(cid:13) ˜ f + O (cid:16) | c i | ǫ − α | log ǫ | (cid:17) for (15 ± ) , o ( | c i | ) for (16) = O (cid:0) ǫ γ + α ǫ − α ǫ α (cid:1) + O (cid:16) ǫ + γ | log ǫ | (cid:17) for (15 ± ) , o (cid:0) ǫ γ + α (cid:1) for (16) = o ( ǫ ) for (15 ± ) , O (cid:0) ǫ γ + α (cid:1) for (16) . Here we also used D Z i δ,σ , χ ′ ( | ( x , t ) | ) ∂ k | ( x , t ) | W δ ( x , t ) E ˜ f = O (cid:16) δ N − s − | log δ | (cid:17) = O (cid:16) δ s − | log δ | (cid:17) if N > s , o (1) if N > s + . Our assertion is proved. (cid:3)
Now we are ready to establish the desired C -estimates of the reduced energy functional J ǫ . Proof of Proposition 6.4.
For the sake of simplicity, we identify e Φ k δ,σ = ∂ k b Φ δ,σ and use an ab-breviation ( χ ∂ k W δ )( z ) = χ ( | ( x , t ) | ) W δ ( x , t ) defined for z = ( σ ( x ) , t ) ∈ X near σ ∈ M . We mayassume that the domain of these functions is the Euclidean space R N + + . By the previous lemma,we have I ′ ǫ ( W δ,σ ( y ) + Φ δ,σ ( y ) ) (cid:16) ∂ y k W δ,σ ( y ) + ∂ y k Φ δ,σ ( y ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) y = = − I ′ ǫ ( W δ,σ + Φ δ,σ ) ∂ k (cid:16) χ W δ + b Φ δ,σ (cid:17) + o ( ǫ ) for problems (15 ± ) , o ( ǫ α ) for problem (16) . . Let us decompose I ′ ǫ ( W δ,σ + Φ δ,σ ) ∂ k (cid:16) χ W δ + b Φ δ,σ (cid:17) = I ′ + I ′ + I ′ − I ′ , where I ′ = κ s Z R N + + t − s (cid:16) ∇ (cid:0) χ W δ + Φ δ,σ (cid:1) , ∇ ∂ k (cid:16) χ W δ + b Φ δ,σ (cid:17)(cid:17) ¯ g p | ¯ g | dxdt , I ′ = κ s Z R N + + E ( t ) (cid:0) χ W δ + Φ δ,σ (cid:1) ∂ k (cid:16) χ W δ + b Φ δ,σ (cid:17) p | ¯ g | dxdt , I ′ = Z R N ˜ f ( σ ( x )) (cid:0) χ w δ + Φ δ,σ (cid:1) ∂ k (cid:16) χ w δ + b Φ δ,σ (cid:17) q | ˆ h | dx and I ′ = Z R N g ǫ (cid:0) χ w δ + Φ δ,σ (cid:1) ∂ k (cid:16) χ w δ + b Φ δ,σ (cid:17) q | ˆ h | dx .
30e will calculate each term to conclude the proof of Proposition 6.4.1. E stimate of I ′ . In this step, we only consider problem (16) in order to ensure the finiteness ofthe value d s defined in the beginning of Subsection 6.1. To handle the other case (15 ± ) is an easiertask.Direct computation shows that Z R N + + t − s (cid:16) ∇ (cid:0) (1 − χ ) Φ δ,σ (cid:1) , ∇ ∂ k (cid:16) χ W δ + b Φ δ,σ (cid:17)(cid:17) ¯ g p | ¯ g | dxdt = O (cid:16) ǫ γ (cid:17) . Thus we have I ′ = I ′ + I ′ + O ( ǫ γ ) where I ′ = κ s Z R N + + t − s ¯ g i j ∂ i (cid:16) χ W δ + b Φ δ,σ (cid:17) ∂ j ∂ k (cid:16) χ W δ + b Φ δ,σ (cid:17) p | ¯ g | dxdt and I ′ = κ s Z R N + + t − s ∂ t (cid:16) χ W δ + b Φ δ,σ (cid:17) ∂ t ∂ k (cid:16) χ W δ + b Φ δ,σ (cid:17) p | ¯ g | dxdt . We shall compute the term I ′ first. By (42), (44) and (67), we discover I ′ = − κ s Z R N + + t − s ∂ k (cid:16) ¯ g i j p | ¯ g | (cid:17) ∂ i (cid:16) χ W δ + b Φ δ,σ (cid:17) ∂ j (cid:16) χ W δ + b Φ δ,σ (cid:17) dxdt = − κ s Z R N + + t − s ∂ k (cid:16) ¯ g i j p | ¯ g | (cid:17) ∂ i ( χ W δ ) ∂ j ( χ W δ ) dxdt + O Z R N + + t − s |∇ ( χ W δ ) | (cid:12)(cid:12)(cid:12)(cid:12) ∇ b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) | ( x , t ) | dxdt + Z R N + + t − s (cid:12)(cid:12)(cid:12)(cid:12) ∇ b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) dxdt ! = − κ s Z B + r t − s ∂ k (cid:16) ¯ g i j p | ¯ g | (cid:17) ∂ i W δ ∂ j W δ dxdt + O (cid:16) ǫ γ (cid:17) + o (cid:16) ǫ s α (cid:17) for N > s , o ( ǫ α ) for N > s + . (94)Also Lemma 2.4 implies that ∂ k p | ¯ g | = − H k t −
16 ( R kl + R lk ) x l + O (cid:16) | ( x , t ) | (cid:17) and ∂ k h i j = − (cid:16) R i jkl + R i jlk (cid:17) x l + h i j , ( N + k t + O (cid:16) | ( x , t ) | (cid:17) , from which we obtain ∂ k (cid:16) ¯ g i j p | ¯ g | (cid:17) = ∂ k ¯ g i j p | ¯ g | + ¯ g i j ∂ k p | ¯ g | = − " (cid:16) R i jkl + R i jlk (cid:17) + δ i j ( R kl + R lk ) x l + (cid:16) h i j , ( N + k − δ i j H k (cid:17) t + O (cid:16) | ( x , t ) | (cid:17) . Inserting this into (94) and then applying (44) as well as the relations h ii , ( N + k = π ii , k = H k and Z B + r t − s x l ∂ i W δ ∂ j W δ dxdt = W δ in the x , · · · , x N variables) ,
31e get I ′ = κ s (cid:16) δ i j H k − h i j , ( N + k (cid:17) δ i j N Z R N + + t − s |∇ x W δ | dxdt + κ s " (cid:16) R i jkl + R i jlk (cid:17) + δ i j ( R kl + R lk ) B + r t − s x l ∂ i W δ ∂ j W δ dxdt + O Z B + r t − s | ( x , t ) | |∇ W δ | dxdt + O (cid:16) ǫ γ (cid:17) + o (cid:16) ǫ s α (cid:17) for N > s , o ( ǫ α ) for N > s + , = κ s N − N ! H k δ Z R N + + t − s |∇ x W | dxdt + O (cid:16) ǫ γ (cid:17) + o (cid:16) ǫ s α (cid:17) for N > s , o ( ǫ α ) for N > s + . (95)Next the term I ′ is to be considered. In fact, one can observe that I ′ = κ s Z B + r t − s ∂ k (cid:16) ∂ t (cid:16) χ W δ + b Φ δ,σ (cid:17)(cid:17) p | ¯ g | dxdt = − κ s Z B + r t − s (cid:16) ∂ t (cid:16) χ W δ + b Φ δ,σ (cid:17)(cid:17) ∂ k p | ¯ g | dxdt = κ s Z B + r t − s ( ∂ t W δ ) " H k t +
16 ( R kl + R lk ) x l + O (cid:16) | ( x , t ) | (cid:17) dxdt + O (cid:16) ǫ γ (cid:17) + o (cid:16) ǫ s α (cid:17) for N > s , o ( ǫ α ) for N > s + , = κ s H k Z R N + + t − s ( ∂ t W δ ) dxdt + O (cid:16) ǫ γ (cid:17) + o (cid:16) ǫ s α (cid:17) for N > s , o ( ǫ α ) for N > s + . (96)Consequently, (95), (96) and (72) give us that I ′ = κ s H k λǫ α Z R N + + t − s " N − N ! |∇ x W | + ( ∂ t W ) dxdt + o (cid:0) ǫ α (cid:1) = κ s H k λǫ α N − s − N ! Z R N + + t − s |∇ W | dxdt + o (cid:0) ǫ α (cid:1) . (97)2. E stimate of I ′ . Performing the integration by parts, we have I ′ = − κ s Z R N + + ∂ k (cid:16) E ( t ) p | ¯ g | (cid:17) (cid:16) χ W δ + b Φ δ,σ (cid:17) dxdt + O (cid:16) ǫ γ (cid:17) . If s ∈ (1 / ,
1) and H =
0, then (cid:12)(cid:12)(cid:12) I ′ (cid:12)(cid:12)(cid:12) = O Z R N + + t − s | W δ | dxdt ! + O Z R N + + t − s (cid:12)(cid:12)(cid:12)(cid:12)b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) dxdt ! = O (cid:16) ǫ α (cid:17) + O (cid:16) ǫ γ (cid:17) = o ( ǫ ) for (15 ± ) , o ( ǫ α ) for (16) . (98)If s ∈ (0 , / | I ′ | = O ( δ ) + O ( ǫ γ ) = o ( ǫ ) for equations (15 ± ). Further-more, if N > s + I ′ = − κ s N − s ! Z R N + + ∂ k H ( σ ( x )) t − s W δ dxdt + O Z R N + + t − s (cid:16) W δ + b Φ δ,σ (cid:17) | ( x , t ) | dxdt ! + O Z R N + + t − s (cid:18)(cid:12)(cid:12)(cid:12) χ − (cid:12)(cid:12)(cid:12) W δ + χ W δ (cid:12)(cid:12)(cid:12)(cid:12)b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) + e Φ δ,σ (cid:19) dxdt ! = − κ s N − s ! ∂ k ( H ( σ ( x ))) | x = δ Z R N + + t − s W dxdt + o ( δ ) (99)32or equation (16), by utilizing (52).3. E stimate of I ′ . We have I ′ = − Z R N ∂ k ˜ f ( σ ( x )) q | ˆ h | ! (cid:16) χ w δ + b Φ δ,σ (cid:17) dx + O (cid:16) ǫ γ (cid:17) = − Z R N ∂ k (cid:16) ˜ f ( σ ( x )) (cid:17) w δ dx + Z R N O ( | x | ) (cid:12)(cid:12)(cid:12)(cid:12) χ w δ + b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) dx − Z R N ∂ k (cid:16) ˜ f ( σ ( x )) (cid:17) (cid:16) ( χ − w δ + χ w δ b Φ δ,σ + b Φ δ,σ (cid:17) q | ˆ h | dx + O (cid:16) ǫ γ (cid:17) = − Z R N ∂ k (cid:16) ˜ f ( σ ( x )) (cid:17) w δ dx + O Z B N (0 , r ) | W δ | (cid:12)(cid:12)(cid:12)(cid:12)b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) dx ! + o (cid:16) ǫ s α (cid:17) + O (cid:16) ǫ γ (cid:17) . Applying H ¨older’s inequality, we estimate Z B N (0 , r ) w δ (cid:12)(cid:12)(cid:12)(cid:12)b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) dx = O (cid:16) k w δ k L ( R N ) k Φ δ,σ k L ( M ) (cid:17) = O (cid:0) ǫ s α + γ (cid:1) . Hence it follows that I ′ = − λ s ǫ ∂ k (cid:16) ˜ f ( σ ( x )) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) x = R R N w dx + o ( ǫ ) for (15 ± ) , . (100)4. E stimate of I ′ . We will deal with the cases (15 ± ) only. The remaining case (16) is similar,and especially, the small linear term ǫ f u of g ǫ ( u ) for this problem (see (40)) can be taken intoconsideration as in the previous step. One has I ′ = Z R N ∂ k G ǫ (cid:16) χ W δ + b Φ δ,σ (cid:17) q | ˆ h | dx + O (cid:16) ǫ γ (cid:17) = − Z R N G ǫ (cid:16) χ W δ + b Φ δ,σ (cid:17) ∂ k q | ˆ h | dx + O (cid:16) ǫ γ (cid:17) = − Z R N G ǫ ( χ W δ ) ∂ k q | ˆ h | dx + Z R N h G ǫ ( χ W δ ) − G ǫ (cid:16) χ W δ + b Φ δ,σ (cid:17)i ∂ k q | ˆ h | dx + o ( ǫ ) . With the observation that ∂ k q | ˆ h | = − ( R kl + R lk ) x l + O ( | x | ), we estimate the second term as Z R N (cid:12)(cid:12)(cid:12)(cid:12) G ǫ ( χ W δ ) − G ǫ (cid:16) χ W δ + b Φ δ,σ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) O ( | x | ) dx ≤ Z R N (cid:18) ( χ W δ ) p ± ǫ (cid:12)(cid:12)(cid:12)(cid:12)b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) p + ± ǫ (cid:19) O ( | x | ) dx ≤ C Z B N (0 , r ) W p + ± ( NN + s ) ǫδ | x | NN + s dx ! N + s N Z R N (cid:12)(cid:12)(cid:12)(cid:12)b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) p + dx ! p + + C Z R N (cid:12)(cid:12)(cid:12)(cid:12)b Φ δ,σ (cid:12)(cid:12)(cid:12)(cid:12) p + ± ǫ dx ! = O (cid:0) ǫ α + γ (cid:1) + O (cid:16) ǫ ( p + γ (cid:17) = o ( ǫ ) . In addition, we find Z R N G ǫ ( χ W δ ) ∂ k q | ˆ h | dx = O Z R N W p + ± ǫδ | x | dx ! = O (cid:16) ǫ γ | log ǫ | (cid:17) = o ( ǫ )given that N ≥ C -estimates for J ǫ . (cid:3) Conclusion of the proof of Theorems 1.1 and 1.2 and some remarks
In this section, we complete the proof of our main results.
Proof of Theorem 1.1.
Suppose that σ ∈ M is a C -stable critical point of f such that f ( σ ) > e J ( λ, σ ) = d f ( σ ) λ s − ( N − s ) d N log λ for ( λ, σ ) ∈ (0 , ∞ ) × M and λ = ( N − s ) d N f ( σ ) d ! s > , then it follows from the invariance of the Brouwer degree under a homotopy that ( λ , σ ) is a C -stable critical point of e J (refer to the proof of Theorem 1.1 in [45]). Therefore, by Propositions 6.1and 6.4, there exists a critical point ( λ ǫ , σ ǫ ) ∈ (0 , ∞ ) × M of J ǫ in (69) for su ffi ciently small ǫ > λ ǫ , σ ǫ ) → ( λ , σ ) as ǫ →
0. This fact and Proposition 5.1 imply that (15 − ) attains apositive solution. As a consequence, we see from Proposition 2.1 that its trace on M solves (1 − ),deducing the conclusion.If there is a C -stable critical point σ ∈ M of f such that f ( σ ) <
0, then the same argumentprovides solutions of equations (15 + ), and so those of (1 + ). This concludes the proof of Theorem1 . (cid:3) Proof of Theorem 1.2.
Under our assumptions existence of solutions to (2) follows from Proposi-tions 6.1, 6.4, 5.1 and 2.1. Observe that λ ( σ ) is the unique value such that ∂ e J ∂λ ( λ ( σ ) , σ ) = σ ∈ M fixed, and ∇ e J ( λ ( σ ) , σ ) = σ is a critical point of the function b J ( σ ) : = e J ( λ ( σ ) , σ ) = ± ( d s ) − s N (1 − s ) (cid:0) N ( N − + (cid:0) − s (cid:1)(cid:1) d s ! s − s − s s ! | f ( σ ) || H ( σ ) | s ! − s . Hence σ should be a critical point of | f | / | H | s . The proof is finished. (cid:3) We conclude this section, raising some additional questions regarding our main result.First of all, one may ask the compactness issue for equations (1 ± ) with f =
0. For the localcase ( s = N of a manifold M satisfies N ≤
24, the positive mass theoremholds for M and the nonlinearity is slightly subcritical or critical, then the solution set for (1 ± ) ispre-compact as shown by Khuri, Marques and Schoen [37]. On the other hand, if N ≥ ± ) which blow-up at a maximum point of the function x → k Weyl ˆ h ( x ) k ˆ h defined for x ∈ ( M , ˆ h ). We think that a similar phenomenon may happen for the nonlocal case too, but do nothave any definitive answer yet.Secondly, the behavior of equation (2) in the case H = H ,
0, is ǫ − s whoseexponent is well-defined (namely, positive) only if s ∈ (0 , / P s ˆ h in terms of extension problemsis valid for any H only if s ∈ (0 , / s = / H =
0, thecorrect choice of α in (38) and the main order of the energy expansion would be − s ) and ǫ − s ) ,respectively, hence it makes sense for any s ∈ (0 , ffi culty also arose in the local cases ( s = ,
2) in [26] and [46].In both problems, we suspect that the governing function for the blow-up location has a rela-tionship with the norm of the second fundamental form k π k ˆ h or that of the Weyl tensor k Weyl ˆ h k ˆ h .34n [30, 38], one can observe how the Weyl tensor carries out its role in the fractional Yamabeproblem.Currently a theory for the higher order fractional Paneitz operator ( γ ∈ (1 , ± ) should have bubble-tower type solutions as in [47]. Acknowledgement.
The authors would like to express their sincere gratitude to Professor A.Pistoia for her valuable comments. W. Choi was supported by the Global Ph.D Fellowship of theGovernment of South Korea 300-20130026. Also, S. Kim has been supported by FONDECYTGrant 3140530, Chile.
A Proof of Lemma 3.5
In this appendix, we justify Lemma 3.5 which describes the decay of the bubble W δ . The proofwill be achieved once we combine Lemmas A.1 and A.2. Lemma A.1.
Let < s < and a ∈ R . Also fix < R < R and denote A + δ − = B + R δ − \ B + R δ − .Then, as δ → , we have the estimates Z A + δ − t − s | ( x , t ) | N + − s + a dxdt = O ( δ a ) for a , , O (cid:0) | log δ | (cid:1) for a = , and Z A + δ − t s − | ( x , t ) | N + s + a dxdt = O ( δ a ) for a , , O (cid:0) | log δ | (cid:1) for a = . Proof.
The second inequality follows from the first inequality by substituting s with 1 − s . Toprove the first inequality, we decompose the domain of integration A + δ − = (cid:16) A + δ − ∪ {| t | ≥ | x |} (cid:17) ∪ (cid:16) A + δ − ∪ {| t | ≤ | x |} (cid:17) and estimate each part separately. If | t | ≥ | x | , then it holds that | t | ≤ | ( x , t ) | ≤ √ | t | . Hence we get Z A + δ − ∪{| t |≥| x |} t − s | ( x , t ) | N + − s + a dxdt ≤ max n , √ s − o Z A + δ − ∪{| t |≥| x |} | ( x , t ) | N + + a dxdt ≤ C Z A + δ − | ( x , t ) | N + + a dxdt = O ( δ a ) for a , , O (cid:0) | log δ | (cid:1) for a = . If | t | ≤ | x | , then we have that δ − √ ≤ √ | ( x , t ) | ≤ | x | ≤ | ( x , t ) | ≤ δ − for ( x , t ) ∈ A + δ − . Consequently, Z A + δ − ∪{| t |≤| x |} t − s | ( x , t ) | N + − s + a dxdt ≤ Z (cid:26) δ − √ ≤| x |≤ δ − (cid:27) Z {| t |≤| x |} t − s | x | N + − s + a dtdx = − s Z (cid:26) δ − √ ≤| x |≤ δ − (cid:27) | x | − s | x | N + − s + a dx = − s Z (cid:26) δ − √ ≤| x |≤ δ − (cid:27) | x | N + a dx = O ( δ a ) for a , , O (cid:0) | log δ | (cid:1) for a = . Combination of the above two estimates yields the desired inequality, concluding the proof. (cid:3)
Lemma A.2.
Assume that | ( x , t ) | ≥ R for some fixed R > large. Then we have the validity of i) W ( x , t ) ≤ C | ( x , t ) | N − s and |∇ W ( x , t ) | ≤ C | ( x , t ) | N − s + + Ct s − | ( x , t ) | N + s . (ii) | ∂ i W ( x , t ) | ≤ C | ( x , t ) | N − s + and |∇ ∂ i W ( x , t ) | ≤ C | ( x , t ) | N − s + + Ct s − | ( x , t ) | N + s + for i = , · · · , N.(iii) | ∂ δ W ( x , t ) | ≤ C | ( x , t ) | N − s and |∇ ∂ δ W ( x , t ) | ≤ C | ( x , t ) | N − s + + Ct s − | ( x , t ) | N + s for some C > determined by N, s and R .Proof. We initiate the proof with recalling Green’s representation formula W δ ( x , t ) = a N , s Z R N w δ ( y ) N + sN − s | ( x − y, t ) | N − s d y = b N , s Z R N δδ + | y | ! N + s | ( x − y, t ) | N − s d y (101)where a N , s and b N , s are positive constants depending only on N and s (see [14, Subsection 2.3]).The proof consists of 3 steps. Step 1: Estimates of W . We split the situation into two cases.Case 1. Assume that | x | ≤ | t | . Since | ( x , t ) | ≤ √ | t | , we obtain W ( x , t ) ≤ b N , s Z R N + | y | ) N + s | t | N − s d y = C | t | N − s ≤ C | ( x , t ) | N − s . Case 2. Assume next that | x | ≥ | t | . Then we observe from | ( x , t ) | ≤ √ | x | that W ( x , t ) ≤ b N , s Z R N + | y | ) N + s | x − y | N − s d y = w ( x ) ≤ C | x | N − s ≤ C | ( x , t ) | N − s . Putting these two estimates together, we get the first inequality of (i).
Step 2: Estimates of |∇ W | . Again we deal with the two mutually exclusive cases.Case 1. Suppose | x | ≤ | t | . Then, from we have | ( x , t ) | ≤ √ | t | , we see that |∇ ( x , t ) W ( x , t ) | ≤ b N , s Z R N + | y | ) N + s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ( x , t ) | ( x − y, t ) | N − s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d y ≤ C Z R N + | y | ) N + s | ( x − y, t ) | N − s + d y ≤ C Z R N + | y | ) N + s | t | N − s + d y = C | t | N − s + ≤ C | ( x , t ) | N − s + . (102)Case 2. Assume that | x | ≥ | t | so that we get | ( x , t ) | ≤ √ | x | . By integration by parts, we deduce ∇ x W ( x , t ) = − b N , s Z R N + | y | ) N + s ∇ y | ( x − y, t ) | N − s ! d y = − Z | y − x |≥ | x | + | y | ) N + s ∇ y | ( x − y, t ) | N − s ! d y + Z | y − x |≤ | x | ∇ y + | y | ) N + s d y | ( x − y, t ) | N − s − Z | y − x | = | x | + | y | ) N + s ν y dS y | ( x − y, t ) | N − s where ν y and dS y is the outward unit normal vector and the surface measure on the sphere | y − x | = | x | , respectively. Hence, realizing that | y | ≥ | x | if | y − x | ≤ | x | , we derive from the above that |∇ x W ( x , t ) |≤ C | x | N − s + Z | y − x |≥ | x | + | y | ) N + s d y + C | x | N + s + Z | y − x |≤ | x | | ( x − y, t ) | N − s d y + O | x | N − | x | N ! = O | x | N − s + ! + O | x | N + s + · | x | s ! + O | x | N − | x | N ! ≤ C | x | N − s + ≤ C | ( x , t ) | N − s + , (103)36hich with (102) implies the first inequality of (ii).On the other hand, for | x | ≥ | t | and | y − x | ≥ | x | , we have Z | y − x |≥ | x | + | x − y | ) N + s t | ( y, t ) | N − s + d y ≤ | x | N + s Z R N t | ( y, t ) | N − s + d y = | x | N + s Z R N t · t N t N − s + | ( y, | N − s + d y = Ct s − | x | N + s ≤ Ct s − | ( x , t ) | N + s . (104)Moreover for | x | ≥ | t | and | y − x | ≤ | x | , it holds that | y | ≥ | x | , from which we find Z | y − x |≤ | x | + | x − y | ) N + s t | ( y, t ) | N − s + d y ≤ t | x | N − s + Z R N + | x − y | ) N + s d y = Ct | x | N − s + ≤ Ct | ( x , t ) | N − s + ≤ C | ( x , t ) | N − s + . (105)As a result, thanks to (104) and (105), we obtain | ∂ t W ( x , t ) | ≤ C Z + | x − y | ) N + s t | ( y, t ) | N − s + d y ≤ t s − | ( x , t ) | N + s + | ( x , t ) | N − s + . (106)Now (102), (103) and (106) give us the second inequality of (i). Step 3: Estimates of |∇ ∂ i W | , | ∂ δ W | and |∇ ∂ δ W | . Following the same procedure which wasapplied to W and ∇ W in Steps 1 and 2, one can find an upper bound of |∇ ∂ i W | for each i = , · · · , N , and in particular the second inequality of (ii).Meanwhile, we discover from (101) that ∂ δ W ( x , t ) = b N , s N + s ! Z R N | y | − + | y | ) N + s + | ( x − y, t ) | N − s d y. Because (cid:12)(cid:12)(cid:12) | y | − (cid:12)(cid:12)(cid:12) (1 + | y | ) N + s + ≤ + | y | ) N + s for any y ∈ R N , we can get (iii) by adopting the argument in Steps 1 and 2 once more. This completes the proof. (cid:3) References [1] W. Abdelhedi, H. Chtioui and H. Hajaiej,
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