Blowing up solutions for supercritical Yamabe problems on manifolds with umbilic boundary
aa r X i v : . [ m a t h . DG ] S e p BLOWING UP SOLUTIONS FOR SUPERCRITICAL YAMABEPROBLEMS ON MANIFOLDS WITH UMBILIC BOUNDARY
MARCO G. GHIMENTI AND ANNA MARIA MICHELETTI
Abstract.
We build blowing-up solutions for a supercritical perturbation ofthe Yamabe problem on manifolds with umbilic boundary provided the dimen-sion of the manifold is n ≥ and that the Weyl tensor W g is not vanishing on ∂M . Introduction
Let ( M, g ) be a smooth compact Riemannian manifold of dimension n ≥ witha smooth boundary ∂M . A well known problem in differential geometry is whether ( M, g ) can be conformally deformed in a constant scalar manifold with boundary ofconstant mean curvature. When the boundary is empty this is called the Yamabeproblem (see [4, 27]) which has been completely solved by Aubin [4], Schoen [24],and Trundinger [26]. Escobar [14] studied the problem in the context of manifoldswith boundary and gave an affirmative solution to the question in some cases. Theremaining cases were studied by Marques [21], Almaraz [1], Brendle and Chen [6],Mayer and Ndiaye [22].Once it is known that the problem admits solution, a natural question about thecompactness of the full set of solutions arises. Concerning the Yamabe problem,a necessary condition is that the manifold is not conformally equivalent to thestandard sphere S n , since the set of conformal transformation of the round sphereis not compact. The problem of compactness has been studied by Schoen in 1988[25] and by Brendle [5], Brendle and Marques [7], Khuri Marques and Schoen [13]in the last years.When the boundary of the manifold is not empty, a necessary condition is that M is not conformally equivalent to the standard ball B n . Compactness for boundaryYamabe problem has been studied firstly by Felli and Ould Ahmedou [10], Han andLi [20], Almaraz [3].In this context the case of scalar flat metrics is particularly interesting since itleads to study a linear equation in the interior with a critical nonlinear Neumann-type boundary condition(1) (cid:26) − ∆ g u + n − n − R g u = 0 on M ∂∂ν u + n − h g u = ( n − u nn − on ∂M where R g is the scalar curvature of M , h g is the mean curvature on ∂M and ν isthe outward normal to the boundary. The geometric meaning of (1) is that if u is asolution of (1) the scalar curvature of the conformal metric ˜ g = u n − g is zero andthe mean curvature of ˜ g on the boundary of M is n − . The Yamabe boundary Mathematics Subject Classification.
Key words and phrases.
Umbilic boundary, Yamabe problem, Compactness, Stability.The first authors was supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM) and byproject PRA from Univeristy of Pisa. problem in the case of scalar flat metrics can be also seen as the multidimensionalversion of the Riemann Mapping Theorem.Concerning problem (1), Felli and Ould Ahmedou in [10] have proved compact-ness when M is locally conformally flat and the boundary is umbilic and Almarazin [3] has proved compactness when n ≥ and the trace free second fundamentalform is non zero everywhere on ∂M , that is any point of the boundary is non um-bilic. In [12] Kim Musso and Wei showed that compactness continues to hold when n = 4 and when n = 6 , and the trace free second fundamental form is non zeroeverywhere on the boundary.Very recently, compactness has been proved by the authors in [15] for manifoldwith umbilic boundary when n > and the Weyl tensor of M is everywhere nonzero on the boundary ∂M . In [17] the authors extend the compactness result tomanifold of dimension n = 6 , , , when the boundary is umbilic and the Weyltensor of M is everywhere non zero on ∂M .An example of non compactness is given for n ≥ and manifold with umbilicboundary in [2]. We recall that the boundary of M is called umbilic if the tracefree second fundamental form of ∂M is zero everywhere on ∂M .Another interesting question is the stability problem. One can ask whether or notthe compactness property is preserved under perturbation of the equation. This isequivalent to having or not uniform a priori estimates for solutions of the perturbedproblem.In the following we consider the problem(2) (cid:26) L g v = 0 on MB g v + ( n − v nn − + ε = 0 on ∂M where ε is a positive real parameter, L g := ∆ g − n − n − R g is the conformal Laplacianand B g = − ∂∂ν v − n − h g ( x ) v is the conformal boundary operator. In the next wewill use a ( x ) := n − n − R g to simplify the notation.We study the question of stability of the problem (1). It is clear that the problemis not stable with respect to supercritical perturbation of the nonlinearity if we areable to build solutions v ε of the perturbed problem (2) which blow up at one pointof the manifold as the parameter ε goes to zero.Our main result is the following Theorem 1.
Let M be a manifold of positive type with umbilic boundary ∂M .Suppose n ≥ and that the Weyl tensor W g is not vanishing on ∂M .Then there exists a solution v ε of (2) such that v ε blows up when ε → + . Here M of positive type means that there exists C > such that Q ( u ) := R M (cid:16) |∇ u | + n − n − R g u (cid:17) dv g + R ∂M n − h g u dσ g (cid:18) R ∂M | u | n − n − dσ g (cid:19) n − n − ≥ C for any u ∈ H ( M ) r { } . We remark that this assumption on the positivity of Q in natural when we addressto compactness questions in Yamabe problems since if inf u ∈ H ( M ) r { } Q ( u ) ≤ , thenthe solution of Yamabe problem is unique.The stability of problem (1) with respect to the principal quantity of the bound-ary term has been studied in a series of paper by the authors and by Pistoia, bothin the case of non umbilic boundary and in the case of umbilic boundary with Weyltensor non vanishing on the boundary. Firstly, they studied what happens linearlyperturbing the mean curvature term. This problem present a strong analogy tothe Yamabe problem when perturbing the scalar curvature term (see, on this topic LOWING UP SOLUTIONS FOR SUPERCRITICAL YAMABE BOUNDARY PROBLEMS 3 [8, 9] and the references therein). In fact, we have that the set of solutions is com-pact -and hence (1) is stable- perturbing the mean curvature from below while weconstruct a blowing up sequence when the perturbation is everywhere positive onthe boundary, and for a class of perturbation which are positive in at least onepoint on the boundary. The result of compactness is dealt in [16] both in umbilicand non umbilic case while for the construction of blowing up sequences for umbilicboundary manifold we refer to [19].Concerning the exponent of the nonlinearity, all the compactness results hold for p ≤ nn − , so the Yamabe boundary problem is stable from below with respect tothe critical exponent, while, in the present paper we have that small perturbationsabove the critical exponent imply blowing up solution when n ≥ , the boundary ∂M is umbilic and the Weyl tensor is non vanishing on ∂M .As a final remark, we notice that in [19] we ask that the manifold is umbilic, theWeyl tensor is never vanishing and that n ≥ . The assumption on the dimensionin this paper is technical, since we ask some integrability condition when performingthe Ljapounov Schmidt procedure. In deed, in the present paper we perform moreeffective computations in Lemma 6. This method could be applied verbatim inpaper [19], so we can reformulate the main result in dimension n ≥ .2. Preliminaries
We recall here a series of preliminary result which are useful for our result.Since the manifold is of positive type, then hh u, v ii g = Z M ( ∇ g u ∇ g v + auv ) dµ g + n − Z ∂M h g uvdσ g is an equivalent scalar product in H g , which induces to the equivalent norm k · k g .We define the exponent s ε = 2( n − n − nε and the Banach space H g := H ( M ) ∩ L s ε ( ∂M ) endowed with norm k u k H g = k u k g + | u | L sεg ( ∂M ) . By trace theorems, we have the following inclusion W ,τ ( M ) ⊂ L t ( ∂M ) for t ≤ τ n − n − τ .We recall the following result, by Nittka [23, Th. 3.14] Remark . Let nn +2 ≤ q < n , r > . Then there exists a constant c such that if f ∈ L q + r (Ω) , β bounded and measurable and g ∈ L ( n − qn − q + r ( ∂ Ω) and u ∈ H (Ω) is the unique weak solution of (cid:26) Lu = f on Ω ∂∂ν u + βu = g on ∂ Ω where L is a strictly elliptic second order operator, then u ∈ L nqn − q (Ω) , u | ∂ Ω L ( n − qn − q ( ∂ Ω) and | u | L nqn − q (Ω) + | u | L ( n − qn − q ( ∂ Ω) ≤ | f | L q + r (Ω) + | g | L ( n − qn − q + r ( ∂ Ω) We consider i : H ( M ) → L n − n − ( ∂M ) and its adjoint with respect to hh· , ·ii g i ∗ g : L n − n ( ∂M ) → H ( M ) defined by (cid:10)(cid:10) ϕ, i ∗ g ( f ) (cid:11)(cid:11) g = Z ∂M ϕf dσ g for all ϕ ∈ H MARCO G. GHIMENTI AND ANNA MARIA MICHELETTI so that v = i ∗ g ( g ) is the weak solution of the problem(3) (cid:26) − ∆ g v + a ( x ) v = 0 on M ∂∂ν v + n − h g ( x ) v = f on ∂M . By [23, Th. 3.14] (see Remark 2) we have that, if v ∈ H is a solution of (3),then for nn +2 ≤ q < n and r > it holds(4) | v | L ( n − qn − q ( ∂M ) = | i ∗ g ( f ) | L ( n − qn − q ( ∂M ) ≤ | f | L ( n − qn − q + r ( ∂M ) . By this result, we can choose q, r such that(5) ( n − qn − q = 2( n − n − nε and ( n − qn − q + r = 2( n −
1) + n ( n − εn + ( n − ε that is q = 2 n + n (cid:16) n − n − (cid:17) εn + 2 + 2 n (cid:16) n − n − (cid:17) ε and r = 2( n −
1) + n ( n − εn + ( n − ε − n −
1) + n ( n − εn + ( n − (cid:16) nn − (cid:17) ε . Set f ε ( v ) = ( n −
2) ( v + ) nn − + ε , we have that, if v ∈ L n − n − + nεg ( ∂M ) , then f ε ( v ) ∈ L n − n ( n − εn + ε ( n − g ( ∂M ) and, in light of (4), also i ∗ g ( f ε ( v )) ∈ L n − n − + nεg ( ∂M ) .Thus we can recast then Problem (2) as(6) v = i ∗ g ( f ε ( v )) , v ∈ H g . The problem has also a variational structure: we can associate to Problem (2)the following functional, which is well defined on H g . J ε,g ( v ) := 12 Z M |∇ g v | + av dµ g + n − Z ∂M h g v dσ g (7) − ( n − n −
1) + ε ( n − Z ∂M (cid:0) v + (cid:1) n − n − + ε dσ g . Remark . Since ∂M is umbilic for any q ∈ ∂M , there exists a metric ˜ g q = ˜ g ,conformal to g , ˜ g q = Λ n − q g q such that(8) | det ˜ g q ( y ) | = 1 + O ( | y | N ) (9) | ˜ h ij ( y ) | = o ( | y | )˜ g ij ( y ) = δ ij + 13 ¯ R ikjl y k y l + R ninj y n (10) + 16 ¯ R ikjl,m y k y l y m + R ninj,k y n y k + 13 R ninj,n y n + (cid:18)
120 ¯ R ikjl,mp + 115 ¯ R iksl ¯ R jmsp (cid:19) y k y l y m y p + (cid:18) R ninj,kl + 13 Sym ij ( ¯ R iksl R nsnj ) (cid:19) y n y k y l + 13 R ninj,nk y n y k + 112 ( R ninj,nn + 8 R nins R nsnj ) y n + O ( | y | ) (11) ¯ R ˜ g q ( y ) = O ( | y | ) and ∂ ii ¯ R ˜ g q ( q ) = − | ¯ W ( q ) | (12) ¯ R kl ( q ) = R nn ( q ) = R nk ( q ) = 0 LOWING UP SOLUTIONS FOR SUPERCRITICAL YAMABE BOUNDARY PROBLEMS 5 uniformly with respect to q ∈ ∂M and y ∈ T q ( M ) . Also,we have Λ q ( q ) = 1 and ∇ Λ q ( q ) = 0 . This results are contained in [21, 11]. Here ˜ h ij is the tensor of thesecond fundamental form referred to the metric ˜ g .The conformal Laplacian and the conformal boundary operator transform underthe change of metric ˜ g q = Λ n − q g q as follows: L ˜ g q ϕ = Λ − n +2 n − q L g (Λ q ϕ ) B ˜ g q ϕ = Λ − nn − q B g (Λ q ϕ ) . By these transformations we have that v := Λ q u is a positive solution of (2), if andonly if u is a positive solution of(13) (cid:26) L ˜ g q u = 0 in MB ˜ g q u + ( n − εq u nn − + ε = 0 on ∂M From now on we set ˜ f ε ( u ) = ( n − εq ( u + ) nn − + ε .Furthermore we have hh Λ q u, Λ q v ii g = hh u, v ii ˜ g and, consequently, k Λ q u k g = k u k ˜ g . In addition, we have that Λ q u ∈ L s ε g if and only if u ∈ L s ε ˜ g , so Λ q u ∈ H g if and onlyif u ∈ H ˜ g . Finally, we can define the functional J ε, ˜ g associated to (13), as J ε, ˜ g ( u ) := 12 Z M |∇ ˜ g u | + ˜ au dµ ˜ g + n − Z ∂M h ˜ g v dσ ˜ g − ( n − n −
1) + ε ( n −
2) Λ q Z ∂M (cid:0) u + (cid:1) n − n − + ε dσ ˜ g , where ˜ a = n − n − R ˜ g , and we get J ε,g (Λ q u ) = J ε, ˜ g ( u ) . so we can always switch between metrics g and ˜ g , and this will be useful in thenext. In the Section 3 we emphasize other equivalences of the same kind. As a lastremark, we notice also that a solution of (13) can be expressed by means of i ∗ ˜ g , infact u solves (13) if and only if u = i ∗ ˜ g ( ˜ f ε ( u )) . The finite dimensional reduction.
Given q ∈ ∂M and ψ ∂q : R n + → M the Fermi coordinates in a neighborhood of q ;we define W δ,q ( ξ ) = U δ (cid:16)(cid:0) ψ ∂q (cid:1) − ( ξ ) (cid:17) χ (cid:16)(cid:0) ψ ∂q (cid:1) − ( ξ ) (cid:17) == 1 δ n − U (cid:16) yδ (cid:17) χ ( y ) = 1 δ n − U ( x ) χ ( δx ) where y = ( z, t ) , with z ∈ R n − and t ≥ , δx = y = (cid:0) ψ ∂q (cid:1) − ( ξ ) and χ is a radialcut off function, with support in ball of radius R .Here U δ ( y ) = δ n − U (cid:0) yδ (cid:1) is the one parameter family of solution of the problem(14) ( − ∆ U δ = 0 on R n + ; ∂U δ ∂t = − ( n − U nn − δ on ∂ R n + . MARCO G. GHIMENTI AND ANNA MARIA MICHELETTI and U ( z, t ) := 1[(1 + t ) + | z | ] n − is the standard bubble in R n + .Now, let us consider the linearized problem(15) − ∆ φ = 0 on R n + , ∂φ∂t + nU n − φ = 0 on ∂ R n + ,φ ∈ H ( R n + ) . and it is well know that every solution of (15) is a linear combination of the functions j , . . . , j n defined by . j i = ∂U∂x i , i = 1 , . . . n − j n = n − U + n X i =1 y i ∂U∂y i . (16)Given q ∈ ∂M we define, for b = 1 , . . . , nZ bδ,q ( ξ ) = 1 δ n − j b (cid:18) δ (cid:0) ψ ∂q (cid:1) − ( ξ ) (cid:19) χ (cid:16)(cid:0) ψ ∂q (cid:1) − ( ξ ) (cid:17) and we decompose H ( M ) in the direct sum of the following two subspaces ˜ K δ,q = Span (cid:10) Λ q Z δ,q , . . . , Λ q Z nδ,q (cid:11) ˜ K ⊥ δ,q = n ϕ ∈ H ( M ) : (cid:10)(cid:10) ϕ, Λ q Z bδ,q (cid:11)(cid:11) g = 0 , b = 1 , . . . , n o and we define the projections ˜Π = H ( M ) → ˜ K δ,q and ˜Π ⊥ = H ( M ) → ˜ K ⊥ δ,q . In order to give a good ansatz on the shape of the solution we need to introducethe function v q : R n + → R which is a solution of the linear problem(17) ( − ∆ v q = (cid:2) ¯ R ijkl ( q ) y k y l + R ninj ( q ) y n (cid:3) ∂ ij U on R n + ∂v∂y n = − nU n − v q on ∂ R n + This function is a key tool for several estimates in what follows. In fact, a goodchoice of v q we allow us to get the correct size of the remainder term in the finitedimensional reduction (Lemma 6). Remark . There exists a unique v q : R n + → R solution of the problem (17) L ( R n + ) -ortogonal to j b for all b = 1 , . . . , n . Moreover it holds(18) |∇ τ v q ( y ) | ≤ C (1 + | y | ) − τ − n for τ = 0 , , , (19) Z ∂ R n + U nn − ( t, z ) v q ( t, z ) dz = 0 and(20) Z ∂ R n + v q ( t, z )∆ v q ( t, z ) dz ≤ , where y ∈ R n + , y = ( t, z ) with t ≥ and z ∈ R n − . In addition, the map q v q isin C ( ∂M ) .The proof of this remark can be found in [19, Lemma 3] and will be omitted.At this point, given q ∈ ∂M we define, similarly to W δ,q , the function V δ,q ( ξ ) = 1 δ n − v q (cid:18) δ (cid:0) ψ ∂q (cid:1) − ( ξ ) (cid:19) χ (cid:16)(cid:0) ψ ∂q (cid:1) − ( ξ ) (cid:17) and ( v q ) δ ( y ) = 1 δ n − v q (cid:16) yδ (cid:17) , LOWING UP SOLUTIONS FOR SUPERCRITICAL YAMABE BOUNDARY PROBLEMS 7 where v q is chosen as in Remark 4.We look for solution of (6) having the form v = Λ q u = ˜ W δ,q + δ ˜ V δ,q + ˜ φ with ˜ φ ∈ ˜ K ⊥ δ,q ∩ H . where we used the intuitive notation ˜ W δ,q = Λ q W δ,q ˜ V δ,q = Λ q V δ,q and ˜ φ = Λ q φ We can rewrite, in light of the previous orthogonal decomposition, Problem (6)(and so Problem (2)) as ˜Π n ˜ W δ,q + δ ˜ V δ,q + ˜ φ − i ∗ g h f ε ( ˜ W δ,q + δ ˜ V δ,q + ˜ φ ) io = 0 (21) ˜Π ⊥ n ˜ W δ,q + δ ˜ V δ,q + ˜ φ − i ∗ g h f ε ( ˜ W δ,q + δ ˜ V δ,q + ˜ φ ) io = 0 . (22)We stress out than we can proceed in analogous way in the manifold ( M, ˜ g ) . Inthis case we should define K δ,q = Span (cid:10) Z δ,q , . . . , Z nδ,q (cid:11) K ⊥ δ,q = n ϕ ∈ H ( M ) : (cid:10)(cid:10) ϕ, Z bδ,q (cid:11)(cid:11) ˜ g = 0 , b = 1 , . . . , n o , and we should ask that u = W δ,q + δ V δ,q + φ , recasting (13) as the couple ofequations Π n W δ,q + δ V δ,q + φ − i ∗ ˜ g h ˜ f ε ( W δ,q + δ V δ,q + φ ) io = 0 (23) Π ⊥ n W δ,q + δ V δ,q + φ − i ∗ ˜ g h ˜ f ε ( W δ,q + δ V δ,q + φ ) io = 0 . (24)Roughly speaking, we are allowed to move the tilde symbol from solutions to prob-lems and vice versa, so in any moment we can choose in which metric and withwhich functional it is more convenient to work.Coming back to problem (22), we define the linear operator L : ˜ K ⊥ δ,q ∩ H g → ˜ K ⊥ δ,q ∩ H g as(25) L ( ˜ φ ) = ˜Π ⊥ n ˜ φ − i ∗ g (cid:16) f ′ ε ( ˜ W δ,q + δ ˜ V δ,q )[ ˜ φ ] (cid:17)o , a nonlinear term N ( ˜Φ) and a remainder term R as N ( ˜ φ ) = ˜Π ⊥ n i ∗ g (cid:16) f ε ( ˜ W δ,q + δ ˜ V δ,q + ˜ φ ) − f ε ( ˜ W δ,q + δ ˜ V δ,q ) − f ′ ε ( ˜ W δ,q + δ ˜ V δ,q )[ ˜ φ ] (cid:17)o (26) R = ˜Π ⊥ n i ∗ g (cid:16) f ε ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17) − ˜ W δ,q − δ ˜ V δ,q o , (27)so equation (22) becomes L ( ˜ φ ) = N ( ˜ φ ) + R. The rest of this section is devoted to show that for any choice of δ, q a solution ˜ φ of (22) exists. Lemma 5.
Let δ = ε λ . For a, b ∈ R , < a < b there exists a positive constant C = C ( a, b ) such that, for ε small, for any q ∈ ∂M , for any λ ∈ [ a, b ] and for any φ ∈ K ⊥ δ,q ∩ H there holds k L δ,q ( φ ) k H ≥ C k φ k H . Proof.
The proof of this Lemma is very similar to the proof of [18, Lemma 2] andwill be omitted. (cid:3)
MARCO G. GHIMENTI AND ANNA MARIA MICHELETTI
Lemma 6.
It holds k R k H g = ( δ − O + ( ε ) (cid:8) O (cid:0) δ log δ (cid:1) + O ( ε log δ ) + O ( ε ) (cid:9) if n = 8 O (cid:0) δ (cid:1) + δ − O + ( ε ) { O ( ε log δ ) + O ( ε ) } if n > where < O + ( ε ) < Cε for some positive constant C . In addition, with the choice δ = ε λ we have that k R k H g = O (cid:16) ε log ε (cid:17) if n = 8 O (cid:16) ε (cid:17) if n > . Proof.
Step 1.
It holds(28) k R k g = (cid:26) O (cid:0) δ log δ (cid:1) + O ( ε log δ ) + O ( ε ) if n = 8 O (cid:0) δ (cid:1) + O ( ε log δ ) + O ( ε ) if n > We have k R k g ≤ (cid:13)(cid:13)(cid:13) i ∗ g (cid:16) f ε ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17) − i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17)(cid:13)(cid:13)(cid:13) g + (cid:13)(cid:13)(cid:13) i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17) − ˜ W δ,q − δ ˜ V δ,q (cid:13)(cid:13)(cid:13) g , and we start by estimating the second term. By definition of i ∗ g there exists Γ = i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17) , that is a function Γ solving(29) (cid:26) − ∆ g Γ + a ( x )Γ = 0 on M ∂∂ν Γ + n − h g ( x )Γ = f ( ˜ W δ,q + δ ˜ V δ,q ) on ∂M . So we have (cid:13)(cid:13)(cid:13) i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q (cid:17) − ˜ W δ,q − δ ˜ V δ,q (cid:13)(cid:13)(cid:13) g = k Γ − ˜ W δ,q − δ ˜ V δ,q k g = Z M h − ∆ g (Γ − ˜ W δ,q − δ ˜ V δ,q ) + a (Γ − ˜ W δ,q − δ ˜ V δ,q ) i (Γ − ˜ W δ,q − δ ˜ V δ,q ) dµ g + Z ∂M h g (Γ − ˜ W δ,q − δ ˜ V δ,q ) dσ g + Z ∂M (cid:20) ∂∂ν (Γ − ˜ W δ,q − δ ˜ V δ,q ) (cid:21) (Γ − ˜ W δ,q − δ ˜ V δ,q ) dσ g = Z M h ∆ g ( ˜ W δ,q + δ ˜ V δ,q ) − a ( ˜ W δ,q + δ ˜ V δ,q ) i (Γ − ˜ W δ,q − δ ˜ V δ,q ) dµ g − Z ∂M h g ( ˜ W δ,q + δ ˜ V δ,q )(Γ − ˜ W δ,q − δ ˜ V δ,q ) dσ g + Z ∂M (cid:20) f ( ˜ W δ,q + δ ˜ V δ,q ) − ∂∂ν ( ˜ W δ,q + δ ˜ V δ,q ) (cid:21) (Γ − ˜ W δ,q − δ ˜ V δ,q ) dσ g =: I + I + I . We have I = Z ∂M h ˜ g ( W δ,q + δ V δ,q )(Λ − q R ) dσ ˜ g ≤ C | h ˜ g ( W δ,q + δ V δ,q ) | L n − n ˜ g ( ∂M ) k Λ − q R k ˜ g Set B n − /δ = (cid:8) z ∈ R n − , | z | ≤ /δ (cid:9) , we have | h ˜ g ( W δ,q + δ V δ,q ) | L n − n ˜ g ( ∂M ) = O ( δ ) (cid:12)(cid:12) h ˜ g ( δz )( U ( z ) − δ v q ( z )) (cid:12)(cid:12) L n − n ( B n − /δ ) . LOWING UP SOLUTIONS FOR SUPERCRITICAL YAMABE BOUNDARY PROBLEMS 9
Since z ≤ /δ we have that δ (1 + | z | ) = O (1) . We recall that |∇ τ v q ( y ) | ≤ C (1 + | y | ) − τ − n by (18) and that |∇ τ U ( y ) | ≤ C (1 + | y | ) − τ − n for τ = 0 , , . By 9 wehave also that h ˜ g q ( q ) = h ˜ g q ,i ( q ) = h ˜ g q ,ik ( q ) = 0 , so, | h ˜ g ( W δ,q + δ V δ,q ) | L n − n ˜ g ( ∂M ) = O ( δ ) (cid:12)(cid:12) h ˜ g ( δz )(1 + | z | ) − n (cid:12)(cid:12) L n − n ( B n − /δ ) = O ( δ ) (cid:12)(cid:12) | z | (1 + | z | ) − n (cid:12)(cid:12) L n − n ( B n − /δ ) = O ( δ ) , (30)since | z | (1+ | z | ) − n ≤ (1+ | z | ) − n and (cid:12)(cid:12) (1 + | z | ) − n (cid:12)(cid:12) L n − n ( B n − /δ ) is bounded when n > or (cid:12)(cid:12) (1 + | z | ) − n (cid:12)(cid:12) L n − n ( B n − /δ ) = O (log δ ) when n = 8 . Thus I = O ( δ ) k Λ − q R k ˜ g = (cid:26) O ( δ log δ ) k R k g if n = 8 O ( δ ) k R k g if n > . For I we proceed in a similar way, having I = Z M (cid:2) ∆ ˜ g ( W δ,q + δ V δ,q ) − ˜ a ( W δ,q + δ V δ,q ) (cid:3) (Λ − q R ) dµ ˜ g ≤ (cid:12)(cid:12) ∆ ˜ g ( W δ,q + δ V δ,q ) − ˜ a ( W δ,q + δ V δ,q ) (cid:12)(cid:12) L nn +2˜ g ( M ) k Λ − q R k ˜ g . Set B n /δ = { z ∈ R n , | z | ≤ /δ } . By [21, page 1609], we have R ˜ g (0) = 0 , so we get (cid:12)(cid:12) ˜ a ( W δ,q + δ V δ,q ) (cid:12)(cid:12) L nn +2˜ g ( M ) = O ( δ ) (cid:12)(cid:12) R ˜ g ( δx )( U ( x ) + δ v q ( x )) (cid:12)(cid:12) L nn +2˜ g ( B n /δ ) O ( δ ) (cid:12)(cid:12) | x | (1 + | x | ) − n (cid:12)(cid:12) L nn +2˜ g ( B n /δ ) = (cid:26) O ( δ log δ ) if n = 8 O ( δ ) if n > since (cid:12)(cid:12) (1 + | x | ) − n (cid:12)(cid:12) L nn +2˜ g ( B n /δ ) is bounded when n > and (cid:12)(cid:12) (1 + | x | ) − n (cid:12)(cid:12) L nn +2˜ g ( B n /δ ) = O (log δ ) when n = 8 .For the Laplacian term, in local charts we have ∆ ˜ g q = ∆ euc + [˜ g ijq ( y ) − δ ij ] ∂ ij + " ∂ i ˜ g ijq ( y ) + ˜ g ijq ( y ) ∂ i | ˜ g q | ( y ) | ˜ g q | ( y ) ∂ j + ∂ n | ˜ g q | ( y ) | ˜ g q | ( y ) ∂ n . Thus by the expansion of the metric given in (8), (10) and since v q solves (17) and ∆ euc U = 0 , we have that(31) (cid:12)(cid:12) ∆ ˜ g ( W δ,q + δ V δ,q ) (cid:12)(cid:12) L nn +2˜ g ( M ) = O (1) (cid:12)(cid:12) ∆ ˜ g ( U + δ v q ) (cid:12)(cid:12) L nn +2 ( B n /δ ) = O (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ U + [˜ g ijq ( δx ) − δ ij ] ∂ ij U + δ ∆ euc v q + δ [˜ g ijq ( δx ) − δ ij ] ∂ ij v q + " ∂ i ˜ g ijq ( δx ) + ˜ g ijq ( δx ) ∂ i | ˜ g q | ( δx ) | ˜ g q | ( δx ) ∂ j ( U + δ v q )+ ∂ n | ˜ g q | ( δx ) | ˜ g q | ( δx ) ∂ n ( U + δ v q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L nn +2 ( B n /δ ) = O (1) (cid:12)(cid:12) δ | x | ∂ ij U + δ | x | ∂ ij v q + δ | x | ∂ j ( U + δ v q ) + δ | x | ∂ n ( U + δ v q ) (cid:12)(cid:12) L nn +2 ( B n /δ ) = O (1) (cid:12)(cid:12) δ (1 + | x | ) − n + δ (1 + | x | ) − n (cid:12)(cid:12) L nn +2 ( B n /δ ) = O ( δ ) (cid:12)(cid:12) (1 + | x | ) − n (cid:12)(cid:12) L nn +2 ( B n /δ ) = (cid:26) O ( δ log δ ) if n = 8 O ( δ ) if n > , and we conclude that I = (cid:26) O ( δ log δ ) k R k g if n = 8 O ( δ ) k R k g if n > . For the last integral I we have I ≤ C (cid:12)(cid:12)(cid:12)(cid:12) ( n − (cid:0) ( W δ,q + δ V δ,q ) + (cid:1) nn − − ∂∂ν ( W δ,q + δ V δ,q ) (cid:12)(cid:12)(cid:12)(cid:12) L n − n ˜ g ( ∂M ) k R k g ≤ C ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ( W δ,q + δ V δ,q ) + (cid:1) nn − − ( W δ,q ) nn − − δ ∂∂ν V δ,q (cid:12)(cid:12)(cid:12)(cid:12) L n − n ˜ g ( ∂M ) k R k g + C (cid:12)(cid:12)(cid:12)(cid:12) ( n −
2) ( W δ,q ) nn − − ∂∂ν W δ,q (cid:12)(cid:12)(cid:12)(cid:12) L n − n ˜ g ( ∂M ) k R k g . (32)Since U is a solution of (14) one can easily obtain(33) (cid:12)(cid:12)(cid:12)(cid:12) ( n −
2) ( W δ,q ) nn − − ∂∂ν W δ,q (cid:12)(cid:12)(cid:12)(cid:12) L n − n ˜ g ( ∂M ) = O ( δ ) . Finally we have, using (17), and expanding (cid:0) ( U + δ v δ,q ) + (cid:1) nn − near U ,(34) (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ( W δ,q + δ V δ,q ) + (cid:1) nn − − ( W δ,q ) nn − − δ ∂∂ν V δ,q (cid:12)(cid:12)(cid:12)(cid:12) L n − n ˜ g ( ∂M ) = O (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ( U + δ v δ,q ) + (cid:1) nn − − U nn − − δ ∂∂ν v q (cid:12)(cid:12)(cid:12)(cid:12) L n − n ( B n − /δ ) = O ( δ ) (cid:12)(cid:12)(cid:12)(cid:0) ( U + θδ v δ,q ) + (cid:1) n − v q − U nn − v q (cid:12)(cid:12)(cid:12) L n − n ( B n − /δ ) . By the decay estimates (18) we have that U + θδv q > in B n − /δ provided δ smallenough. So, expanding again we have(35) O ( δ ) (cid:12)(cid:12)(cid:12)(cid:0) ( U + θδ v δ,q ) + (cid:1) n − v q − U nn − v q (cid:12)(cid:12)(cid:12) L n − n ( B n − /δ ) = O ( δ ) (cid:12)(cid:12)(cid:12)(cid:12) δ (cid:0) ( U + θ δ v δ,q ) + (cid:1) − nn − v q (cid:12)(cid:12)(cid:12)(cid:12) L n − n ( B n − /δ ) = O ( δ ) (cid:12)(cid:12)(cid:12) δ (1 + | y | ) − n (cid:12)(cid:12)(cid:12) L n − n ( B n − /δ ) = O ( δ ) (cid:12)(cid:12)(cid:12) (1 + | y | ) − n (cid:12)(cid:12)(cid:12) L n − n ( B n − /δ ) = O ( δ ) since n ≥ and we get I = O ( δ ) k R k g and, consequently, (cid:13)(cid:13)(cid:13) i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q (cid:17) − ˜ W δ,q − δ ˜ V δ,q (cid:13)(cid:13)(cid:13) g = O ( δ ) k R k g . To conclude the first part of the proof we estimate the term (cid:13)(cid:13)(cid:13) i ∗ g (cid:16) f ε ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17) − i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17)(cid:13)(cid:13)(cid:13) g It is useful to recall the following Taylor expansions with respect to εU ε = 1 + ε ln U + 12 ε ln U + o ( ε ) (36) δ − ε n − = 1 − ε n −
22 ln δ + ε ( n − δ + o ( ε ln δ ) (37) LOWING UP SOLUTIONS FOR SUPERCRITICAL YAMABE BOUNDARY PROBLEMS 11
We have that, recalling that Λ q (0) = 0 ,(38) (cid:13)(cid:13)(cid:13) i ∗ g (cid:16) f ε ( ˜ W δ,q + δ ˜ V δ,q (cid:17) − i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q (cid:17)(cid:13)(cid:13)(cid:13) g = (cid:13)(cid:13)(cid:13) i ∗ ˜ g (cid:16) ˜ f ε ( W δ,q + δ V δ,q (cid:17) − i ∗ ˜ g (cid:0) f ( W δ,q + δ V δ,q (cid:1)(cid:13)(cid:13)(cid:13) ˜ g ≤ C (cid:12)(cid:12)(cid:12) Λ εq (cid:0) W δ,q + δ V δ,q (cid:1) nn − + ε − (cid:0) W δ,q + δ V δ,q (cid:1) nn − (cid:12)(cid:12)(cid:12) L n − n ˜ g ( ∂M ) = O (1) (cid:12)(cid:12)(cid:12) Λ εq ( δy ) (cid:0) U + δ v q (cid:1) nn − + ε − (cid:0) U + δ v q (cid:1) nn − (cid:12)(cid:12)(cid:12) L n − n ( B n − /δ ) = O (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Λ εq ( δy ) δ ε n − ( U + δ v q ) ε − (cid:19) ( U + δ v q ) nn − (cid:12)(cid:12)(cid:12)(cid:12) L n − n ( B n − /δ ) ≤ O (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − n − ε ln δ + ε ln( U + δ v q ) + O ( ε ln δ ) (cid:19) U nn − (cid:12)(cid:12)(cid:12)(cid:12) L n − n ( B n − /δ ) = O ( ε log δ ) + O ( ε ) , and we have proved (28). Step 2.
It holds(39) | R ε,δ,q | L sε ( ∂M ) = ( δ − O + ( ε ) (cid:8) O (cid:0) δ log δ (cid:1) + O ( ε log δ ) + O ( ε ) (cid:9) if n = 8 O (cid:0) δ (cid:1) + δ − O + ( ε ) { O ( ε log δ ) + O ( ε ) } if n > . As in the previous case we consider | R | L sεg ( ∂M ) ≤ (cid:12)(cid:12)(cid:12) i ∗ g (cid:16) f ε ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17) − i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17)(cid:12)(cid:12)(cid:12) L sεg ( ∂M ) + (cid:12)(cid:12)(cid:12) i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17) − ˜ W δ,q ( x ) − δ ˜ V δ,q (cid:12)(cid:12)(cid:12) L sεg ( ∂M ) and we start estimating the second term. Taking again Γ = i ∗ g ( f ( W δ,q + δV δ,q ) thesolution of (29), the function Γ − W δ,q − δV δ,q solves the problem − ∆ g (Γ − ˜ W δ,q − δ ˜ V δ,q ) + a ( x )(Γ − ˜ W δ,q − δ ˜ V δ,q )= − ∆ g ( ˜ W δ,q + δ ˜ V δ,q ) + a ( x )( ˜ W δ,q + δ ˜ V δ,q ) on M ∂∂ν (Γ − ˜ W δ,q − δ ˜ V δ,q ) = f ( ˜ W δ,q + δ ˜ V δ,q ) − ∂∂ν ( ˜ W δ,q + δ ˜ V δ,q ) on ∂M . We choose q = n + n ( n − n − ) εn +2+2 n ( n − n − ) ε and r = ε , so, by Remark 2, we get | Γ − ˜ W δ,q − δ ˜ V δ,q | L sεg ( ∂M ) ≤| − ∆ g ( ˜ W δ,q + δ ˜ V δ,q ) + a ( x )( ˜ W δ,q + δ ˜ V δ,q ) | L q + εg ( M ) + (cid:12)(cid:12)(cid:12)(cid:12) f ( ˜ W δ,q + δ ˜ V δ,q ) − ∂∂ν ( ˜ W δ,q + δ ˜ V δ,q ) (cid:12)(cid:12)(cid:12)(cid:12) L ( n − qn − q + εg ( ∂M ) . We remark that with our choice we can write q = nn +2 + O + ( ε ) , q + ε = n +22 n − O + ( ε ) and ( n − qn − q + ε = n − n + O + ( ε ) where < O + ( ε ) < Cε for some positive constant C . We proceed as in the first part, obtaining | a ( x )( ˜ W δ,q + δ ˜ V δ,q ) | L q + εg ( M ) = O (cid:16) δ − O + ( ε ) (cid:17) (cid:12)(cid:12) (1 + | x | ) − n (cid:12)(cid:12) L nn +2 + O +( ε )˜ g ( B n /δ ) . At this point we have that, if n > , (cid:12)(cid:12) (1 + | x | ) − n (cid:12)(cid:12) L nn +2 + O +( ε )˜ g ( B n /δ ) = O (1) , while,if n = 8 , (cid:12)(cid:12) (1 + | x | ) − n (cid:12)(cid:12) L nn +2 + O +( ε )˜ g ( B n /δ ) = O (cid:16) δ − O + ( ε ) log δ (cid:17) . So we get | a ( x )( ˜ W δ,q + δ ˜ V δ,q ) | L q + εg ( M ) = ( O (cid:16) δ − O + ( ε ) log δ (cid:17) if n = 8 O (cid:0) δ (cid:1) if n > . In the same spirit one can check that | − ∆ g ( ˜ W δ,q + δ ˜ V δ,q ) | L q + εg ( M ) = ( O (cid:16) δ − O + ( ε ) log δ (cid:17) if n = 8 O (cid:0) δ (cid:1) if n > (cid:12)(cid:12)(cid:12)(cid:12) f ( ˜ W δ,q + δ ˜ V δ,q ) − ∂∂ν ( ˜ W δ,q + δ ˜ V δ,q ) (cid:12)(cid:12)(cid:12)(cid:12) L ( n − qn − q + εg ( ∂M ) = ( O (cid:16) δ − O + ( ε ) log δ (cid:17) if n = 8 O (cid:0) δ (cid:1) if n > . To finish the proof we estimate (cid:12)(cid:12)(cid:12) i ∗ g (cid:16) f ε ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17) − i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17)(cid:12)(cid:12)(cid:12) L sεg ( ∂M ) Again, by Remark 2, we have (cid:12)(cid:12)(cid:12) i ∗ g (cid:16) f ε ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17) − i ∗ g (cid:16) f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:17)(cid:12)(cid:12)(cid:12) L sεg ( ∂M ) ≤ (cid:12)(cid:12)(cid:12) f ε ( ˜ W δ,q + δ ˜ V δ,q ) − f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:12)(cid:12)(cid:12) L n − n + O +( ε ) g ( ∂M ) and, proceeding as in (38) we obtain (cid:12)(cid:12)(cid:12) f ε ( ˜ W δ,q + δ ˜ V δ,q ) − f ( ˜ W δ,q + δ ˜ V δ,q ) (cid:12)(cid:12)(cid:12) L n − n + O +( ε ) g ( ∂M ) = δ − O + ( ε ) { O ( ε | ln δ | ) + O ( ε ) } , and we have proved (39).The last claim follows by direct computation. (cid:3) Proposition 7.
Let δ = λε For a, b ∈ R , < a < b there exists a positive constant C = C ( a, b ) such that, for ε small, for any q ∈ ∂M and for any λ ∈ [ a, b ] thereexists a unique φ δ,q which solves (21) with k φ δ,q k H g = O (cid:16) ε log ε (cid:17) if n = 8 O (cid:16) ε (cid:17) if n > . Moreover the map q φ δ,q is a C ( ∂M, H g ) map.Proof. First we prove that the nonlinear operator N defined (26) is a contractionon a suitable ball of H . Recalling that k N ( φ ) − N ( φ ) k H = k N ( φ ) − N ( φ ) k H + | N ( φ ) − N ( φ ) | L sε ( ∂M ) we estimate the two right hand side terms separately.By the continuity of i ∗ : L n − n ( ∂M ) → H , and by Lagrange theorem we have k N ( φ ) − N ( φ ) k H ≤ k ( f ′ ε ( W δ,q + θφ + (1 − θ ) φ + δV δ,q ) − f ′ ε ( W δ,q + δV δ,q )) [ φ − φ ] k L n − n ( ∂M ) LOWING UP SOLUTIONS FOR SUPERCRITICAL YAMABE BOUNDARY PROBLEMS 13 and, since | φ − φ | n − n ∈ L nn − ( ∂M ) and | f ′ ε ( · ) | n − n ∈ L n ( ∂M ) , we have k N ( φ ) − N ( φ ) k H ≤ k ( f ′ ε ( W δ,q + θφ + (1 − θ ) φ + δV δ,q ) − f ′ ε ( W δ,q ) + δV δ,q ) k L n − ( ∂M ) k φ − φ k H = γ k φ − φ k H where we can choose γ := k ( f ′ ε ( W δ,q + θφ + (1 − θ ) φ + δV δ,q ) − f ′ ε ( W δ,q + δV δ,q )) k L n − n − ( ∂M ) < , provided k φ k H and k φ k H sufficiently small.For the second term we argue in a similar way and, recalling that, by (4), | i ∗ ( g ) | L sε ( ∂M ) ≤ | g | L n − n ( n − εn +( n − ε ( ∂M ) , we have | N ( φ ) − N ( φ ) | L sε ( ∂M ) ≤ | ( f ′ ε ( W δ,q + θφ + (1 − θ ) φ + δV δ,q ) − f ′ ε ( W δ,q + δV δ,q )) [ φ − φ ] | L n − n ( n − εn +( n − ε ( ∂M ) Since φ , φ , W δ,q V δ,q ∈ L s ε we have that | φ − φ | n − n ( n − εn +( n − ε ∈ L n +( n − εn − ( ∂M ) and | f ′ ( · ) | n − n ( n − εn +( n − ε ∈ L n +( n − ε n − ε ( ∂M ) . So we conclude as above that we canchoose | φ | L sε ( ∂M ) , | φ | L sε ( ∂M ) sufficiently small in order to get | N ( φ ) − N ( φ ) | L sε ( ∂M ) ≤ γ | φ − φ | L sε ( ∂M ) . So k N ( φ ) − N ( φ ) k H ≤ γ k φ − φ k H with γ < , provided k φ k H , k φ k H small enough.With the same strategy it is possible to prove that if k φ k H is sufficiently smallthere exists ¯ γ < such that k N ( φ ) k H ≤ ¯ γ k φ k H .At this point, recalling Lemma 5 and Lemma 6, it is not difficult to prove thatthere exists a constant C > such that, if n > and k φ k H ≤ Cε then the map T ( φ ) := L − ( N ( φ ) + R ) is a contraction from the ball k φ k H ≤ Cε in itself. We proceed analogously for n = 8 and we get the first claim by the Contraction Mapping Theorem. Theregularity claim can be proven via the Implicit Function Theorem. (cid:3) The reduced problem
For any choice of ( δ, q ) Proposition 7 states the we can solve the infinite dimen-sional problem (22). Now, set δ = λε , we look for a critical point for the functional J ε,g having the form ˜ W λε ,q + λ ε ˜ V λε ,q + ˜ φ λε ,q .We define the function I ε ( λ, q ) : = J ε,g (cid:16) ˜ W λε ,q + λ ε ˜ V λε ,q + ˜ φ λε ,q (cid:17) I ε :[ a, b ] × ∂M → R Which will useful in the next
Lemma 8.
Assume n ≥ and δ = λε . It holds (cid:12)(cid:12)(cid:12) I ε ( λ, q ) − J ε,g (cid:16) ˜ W λε ,q + λ ε ˜ V λε ,q (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) ˜ φ λε ,q (cid:13)(cid:13)(cid:13) H g + C (cid:16) ε | log ε | + ε (cid:17) (cid:13)(cid:13)(cid:13) ˜ φ λε ,q (cid:13)(cid:13)(cid:13) H g = o ( ε ) C -uniformly for q ∈ ∂M and λ in a compact set of (0 , + ∞ ) . The proof of this result is similar to prove of [19, Lemma 6] and will be postponedin the AppendixAt this point we can prove the main result of this section.
Proposition 9.
Assume n ≥ and δ = λε . It holds J ε ( ˜ W λε ,q + λ ε ˜ V λε ,q ) = A + B ( ε ) + ελ ϕ ( q ) + Cε ln λ + o ( ε ) ,C -uniformly for q ∈ ∂M and λ in a compact set of (0 , + ∞ ) , where A = 12 Z R n + |∇ U ( t, z ) | dtdz − ( n − n − Z R n − U (0 , z ) n − n − dzB ( ε ) = ε (cid:20) ( n − n − Z R n − U n − n − ( z, dz − ( n − n − Z R n − U n − n − ( z,
0) ln U ( z, dz (cid:21) − ε | ln ε | ( n − n − Z R n − U n − n − ( z, dzϕ ( q ) = 12 Z R n + v q ∆ v q dtdz − n − n − | ¯ W ( q ) | Z R n + | z | U ( t, z ) dtdz − ( n − n − n − R ninj ( q ) Z R n + t | z | ((1 + t ) + | z | ) n dtdz.C = ( n − n − Z R n − U n − n − dz > . Proof.
We write J ε,g ( ˜ W δ,q + δ ˜ V δ,q ) = J ,g ( ˜ W δ,q + δ ˜ V δ,q ) − ( n − n −
1) + ε ( n − Z ∂M (cid:16) ( ˜ W δ,q + δ ˜ V δ,q ) + (cid:17) n − n − + ε dσ g + ( n − n − Z ∂M (cid:16) ( ˜ W δ,q + δ ˜ V δ,q ) + (cid:17) n − n − dσ g , where J ,g ( v ) := 12 Z M |∇ g v | + av dµ g + n − Z ∂M h g v dσ g − ( n − n − Z ∂M (cid:0) v + (cid:1) n − n − dσ g . Since ( n − n − ε ( n − = ( n − n − − ε ( n − n − + o ( ε ) we have J ε,g ( ˜ W δ,q + δ ˜ V δ,q ) = J ,g ( ˜ W δ,q + δ ˜ V δ,q ) − ( n − n − Z ∂M (cid:16) ( ˜ W δ,q + δ ˜ V δ,q ) + (cid:17) n − n − + ε − (cid:16) ( ˜ W δ,q + δ ˜ V δ,q ) + (cid:17) n − n − dσ g + (cid:20) ε ( n − n −
1) + o ( ε ) (cid:21) Z ∂M (cid:16) ( ˜ W δ,q + δ ˜ V δ,q ) + (cid:17) n − n − + ε dσ g . For J ,g we proceed as in [19, Lemma 8] (see also [15]) obtaining that J ,g ( ˜ W δ,q + δ ˜ V δ,q ) = A + δ ϕ ( q ) + o ( δ ) LOWING UP SOLUTIONS FOR SUPERCRITICAL YAMABE BOUNDARY PROBLEMS 15 where A = 12 Z R n + |∇ U ( t, z ) | dtdz − ( n − n − Z R n − U (0 , z ) n − n − dzϕ ( q ) = 12 Z R n + v q ∆ v q dtdz − n − n − | ¯ W ( q ) | Z R n + | z | U ( t, z ) dtdz − ( n − n − n − R ninj ( q ) Z R n + t | z | ((1 + t ) + | z | ) n dtdz. Using again (36) and (37), proceeding similarly to (38), and recalling that δ = λε we have Z ∂M (cid:16) ( ˜ W δ,q + δ ˜ V δ,q ) + (cid:17) n − n − + ε − (cid:16) ( ˜ W δ,q + δ ˜ V δ,q ) + (cid:17) n − n − dσ g = Z ∂M Λ εq (cid:0) ( W δ,q + δ V δ,q ) + (cid:1) n − n − + ε − (cid:0) ( W δ,q + δ V δ,q ) + (cid:1) n − n − dσ ˜ g = (1 + o (1)) Z | z | < δ Λ εq ( δy ) δ ε n − (cid:0) ( U + δ v q ) ε − (cid:1) ( U + δ v q ) n − n − dz = Z R n − (cid:18) − n − ε ln ε − n − ε ln λ + ε ln( U ) + o ( ε ) (cid:19) U n − n − dz = n − ε | ln ε | Z R n − U n − n − dz + ε Z R n − U n − n − ln( U ) dz − n − ε ln λ Z R n − U n − n − dz + o ( ε ) . Finally, with the same technique, (cid:20) ε ( n − n −
1) + o ( ε ) (cid:21) Z ∂M (cid:16) ( ˜ W δ,q + δ ˜ V δ,q ) + (cid:17) n − n − + ε dσ = (cid:20) ε ( n − n −
1) + o ( ε ) (cid:21) Z | z | < δ Λ q ( δy ) δ ε n − ( U + δv q ) n − n − ( U + δv q ) ε dz + o ( δ )= ε ( n − n − Z R n − U n − n − + o ( ε ) . (cid:3) Proof of Theorem 1.
By the following result we prove that once we have a critical point of the reducedfunctional I ε ( λ, q ) , we solve Problem (2). The proof of this result is very similar tothe proof of [18, Claim (i) of Prop. 5], and we will omit it for the sake of brevity. Lemma 10. If (¯ λ, ¯ q ) ∈ (0 , + ∞ ) × ∂M is a critical point for the reduced functional I ε ( λ, q ) , then the function ˜ W ¯ λε , ¯ q + ¯ λ ε ˜ V λε , ¯ q + ˜ φ λε ,q is a solution of (2). Here ˜ φ λε ,q is defined in Proposition 7. Lemma 11.
Assume n ≥ and that the Weyl tensor W g is not vanishing on ∂M .Then the function ϕ ( q ) defined in Proposition 9 is strictly negative on ∂M .Proof. By Proposition 9 we have that ϕ ( q ) = 12 Z R n + v q ∆ v q dtdz − C | ¯ W ( q ) | − ( n − C R ninj ( q ) , where C , C are positive constants. We recall (see [21, page 1618]) that when theboundary is umbilic W ( q ) = 0 if and only if ¯ W ( q ) and R ninj ( q ) are both zero, so,by our assumption, we have that at least one among ¯ | W ( q ) | and R nlnj ( q ) which isstrictly positive. This, combined with (20), implies that, ϕ ( q ) is strictly negativefor n > .When n = 8 the same strategy leads only to the weak inequality ϕ ( q ) ≤ . Toovercome this difficulty, a delicate analysis of the term R R n + v q ∆ v q dtdz is needed.An improvement of the estimate (20) is performed in [17], where the authors givea more precise description of the function v q as a sum of an harmonic functionwith explicit rational functions. This description, for n = 8 , leads to the inequality,proved in [17, Lemma 19], Z R v q ∆ v q dy ≤ − C R i j ( q ) , where C > . Thus for n = 8 we have ϕ ( q ) ≤ − C | ¯ W ( q ) | − C R i j ( q ) < , which completes the proof. (cid:3) Proof of Theorem 1.
By Lemma 11 we have that the function ϕ ( q ) defined inProposition 9 is strictly negative on ∂M . We recall as well, that the number C defined in the same proposition is positive. Then, defined I : [ a, b ] × ∂M → R I ( λ, q ) = λ ϕ ( q ) + C log λ we have that for any M < there exist a, b such that I ( λ, q ) < M for any q ∈ ∂M, λ [ a, b ] and ∂I∂λ ( a, q ) = 0 , ∂I∂λ ( a, q ) = 0 ∀ q ∈ ∂M. Then the function I admits a absolute maximum on [ a, b ] × ∂M . This maximumis also C -stable. in other words, if ( λ , q ) is the maximum point for I , for anyfunction f ∈ C ([ a, b ] × ∂M ) with k f k C sufficiently small, then the function I + f on [ a, b ] × ∂M admits a maximum point (¯ λ, ¯ q ) close to ( λ , q ) .Then, taken an ε sufficiently small, in light of Proposition 8 and Proposition9, there exists a pair ( λ ε , q ε ) maximum point for I ε ( λ, q ) . Thus, by Lemma 10, v ε := ˜ W ¯ λε , ¯ q + ¯ λ ε ˜ V λε , ¯ q + ˜ φ λε ,q ∈ H g is a solution of (2). By construction v ε blows up at q ε → q when ε → . (cid:3) LOWING UP SOLUTIONS FOR SUPERCRITICAL YAMABE BOUNDARY PROBLEMS 17 Appendix
Proof of Lemma 8.
We have, for some θ ∈ (0 , J ε, ˜ g q ( W δ,q + δ V δ,q + φ δ,q ) − ˜ J ε, ˜ g q ( W δ,q + δ V δ,q ) = ˜ J ′ ε, ˜ g q ( W δ,q + δ V δ,q )[ φ δ,q ]+ 12 ˜ J ′′ ε, ˜ g q ( W δ,q + δ V δ,q + θφ δ,q )[ φ δ,q , φ δ,q ]= Z M (cid:0) ∇ ˜ g q W δ,q + δ ∇ ˜ g q V δ,q (cid:1) ∇ ˜ g q φ δ,q + ˜ a ( x ) (cid:0) W δ,q + δ V δ,q (cid:1) φ δ,q dµ ˜ g q − ( n − Z ∂M Λ εq (cid:16)(cid:0) W δ,q + δ V δ,q (cid:1) + (cid:17) nn − + ε φ δ,q dσ ˜ g q + n − Z ∂M h ˜ g q (cid:0) W δ,q + δ V δ,q (cid:1) φ δ,q dσ ˜ g q + 12 Z M |∇ ˜ g q φ δ,q | + ˜ a ( x ) φ δ,q dµ ˜ g q + n − Z ∂M h ˜ g q φ δ,q dσ ˜ g q − n + ε ( n − Z ∂M Λ εq (cid:16)(cid:0) W δ,q + δ V δ,q + θφ δ,q (cid:1) + (cid:17) n − + ε φ δ,q dσ ˜ g q . Immediately we have, by definition of k · k ˜ g , Z M |∇ ˜ g q φ δ,q | + ˜ aφ δ,q dµ ˜ g q + Z ∂M n − h ˜ g q φ δ,q dσ = C k φ δ,q k g q = o ( ε ) . By Holder inequality one can easily obtain Z M ˜ aW δ,q φ δ,q dµ ˜ g q ≤ C | W δ,q | L nn +2˜ g | φ δ,q | L nn − g ≤ Cδ k φ δ,q k ˜ g = o ( ε ) and δ Z M ˜ aV δ,q φ δ,q dµ ˜ g q ≤ Cδ | V δ,q | L g | φ δ,q | L g ≤ Cδ k φ δ,q k ˜ g = o ( ε ) . In addition, notice that (cid:16)(cid:0) W δ,q + δ V δ,q + θφ δ,q (cid:1) + (cid:17) n − + ε belongs to L n − n ( n − ε n − ε ˜ g andthat (cid:16) n − n ( n − ε n − ε (cid:17) ′ = n − n ( n − ε n − n − n − ε < s ε , so, again by Holder inequalitywe have Z ∂M Λ εq (cid:16)(cid:0) W δ,q + δ V δ,q + θφ δ,q (cid:1) + (cid:17) n − + ε φ δ,q dσ ˜ g q ≤ C (cid:18)(cid:12)(cid:12) W δ,q + δ V δ,q + θφ δ,q (cid:12)(cid:12) n − L sε ˜ g (cid:19) k φ δ,q k g = o ( ε ) . and by(30) Z ∂M h ˜ g q ( W δ,q + δ V δ,q ) φ δ,q dσ ˜ g q ≤ | h ˜ g ( W δ,q + δ V δ,q ) | L n − n ˜ g ( ∂M ) k φ δ,q k ˜ g = O ( δ ) k φ δ,q k ˜ g = o ( ε ) . By integration by parts we have(40) Z M (cid:0) ∇ ˜ g q W δ,q + δ ∇ ˜ g q V δ,q (cid:1) ∇ ˜ g q φ δ,q dµ ˜ g q = − Z M ∆ ˜ g q (cid:0) W δ,q + δ V δ,q (cid:1) φ δ,q dµ ˜ g q + Z ∂M (cid:18) ∂∂ν W δ,q + δ ∂∂ν V δ,q (cid:19) φ δ,q dσ ˜ g q . and, as in (31), we get Z M ∆ ˜ g q (cid:0) W δ,q + δ V δ,q (cid:1) φ δ,q dµ ˜ g q ≤ | ∆ ˜ g q ( W δ,q + δ V δ,q ) | L nn +2˜ g k φ δ,q k ˜ g = O ( δ ) k φ δ,q k ˜ g = o ( ε ) , and for the boundary term in (40), we have, in light of (32), (33), (34) and (35),that Z ∂M (cid:20)(cid:18) ∂∂ν W δ,q + δ ∂∂ν V δ,q (cid:19) − ( n − (cid:16)(cid:0) W δ,q + δ V δ,q (cid:1) + (cid:17) nn − (cid:21) φ δ,q dσ ˜ g q = (cid:12)(cid:12)(cid:12)(cid:12) ( n − (cid:16)(cid:0) W δ,q + δ V δ,q (cid:1) + (cid:17) nn − − ∂∂ν W δ,q (cid:12)(cid:12)(cid:12)(cid:12) L n − n ˜ g | φ δ,q | L n − n − g = O ( δ ) k φ δ,q k ˜ g = o ( ε ) . At this point it remains to estimate Z ∂M (cid:20) Λ εq (cid:16)(cid:0) W δ,q + δ V δ,q (cid:1) + (cid:17) nn − + ε − (cid:16)(cid:0) W δ,q + δ V δ,q (cid:1) + (cid:17) nn − (cid:21) φ δ,q dσ ˜ g q and we proceed as in (38) to get Z ∂M (cid:20) Λ εq (cid:16)(cid:0) W δ,q + δ V δ,q (cid:1) + (cid:17) nn − + ε − (cid:16)(cid:0) W δ,q + δ V δ,q (cid:1) + (cid:17) nn − (cid:21) φ δ,q dσ ˜ g q ≤ (cid:12)(cid:12)(cid:12)(cid:12) Λ εq (cid:16)(cid:0) W δ,q + δ V δ,q (cid:1) + (cid:17) nn − + ε − (cid:16)(cid:0) W δ,q + δ V δ,q (cid:1) + (cid:17) nn − (cid:12)(cid:12)(cid:12)(cid:12) L n − n ˜ g k φ δ,q k ˜ g = o ( ε ) , ending the proof. (cid:3) References [1] S. Almaraz,
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E-mail address : [email protected] A. M. Micheletti,Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5, 56126 Pisa,Italy
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