Constant T-curvature conformal metrics on 4-manifolds with boundary
aa r X i v : . [ m a t h . A P ] A ug Constant T -curvature conformal metrics on -manifolds with boundary Cheikh Birahim NDIAYE
SISSA, via Beirut 2-4, 34014 Trieste, Italy.
Abstract
In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (
M, g ) withsmooth boundary there exists a metric conformal to g with constant T -curvature, zero Q -curvature andzero mean curvature under generic and conformally invariant assumptions. The problem amounts tosolving a fourth order nonlinear elliptic boundary value problem (BVP) with boundary conditions given bya third-order pseudodifferential operator, and homogeneous Neumann one. It has a variational structure,but since the corresponding Euler-Lagrange functional is in general unbounded from below, we look forsaddle points. In order to do this, we use topological arguments and min-max methods combined with acompactness result for the corresponding BVP. Key Words:
Geometric BVPs, Blow-up analysis, Variational methods, Min-max schemes Q -curvature, T -curvature, Conformal geometry, Topological methods. AMS subject classification: 35B33, 35J35, 53A30, 53C21
In recent years, there has been an intensive study of conformally covariant differential (or even pseudod-ifferential) operators on compact smooth Riemannian manifolds, their associated curvature invariants inorder to understand the relationships between analytic and geometric properties of such objects.A model example is the Laplace-Beltrami operator on compact closed surfaces (Σ , g ), which governsthe transformation laws of the Gauss curvature. In fact under the conformal change of metric g u = e u g ,we have ∆ g u = e − u ∆ g ; − ∆ g u + K g = K g u e u , (1)where ∆ g and K g (resp. ∆ g u and K g u ) are the Laplace-Beltrami operator and the Gauss curvature of(Σ , g ) (resp. of (Σ , g u )).Moreover we have the Gauss-Bonnet formula which relates R Σ K g dV g and the topology of Σ : Z Σ K g dV g = 2 πχ (Σ);where χ (Σ) is the Euler-Poincar´e characteristic of Σ. From this we deduce that R Σ K g dV g is a topologicalinvariant (hence also a conformal one). Of particular interest is the classical Uniformization Theorem
Ric g and scalar curvature R g ofthe Riemannian manifold ( M, g ) as follows P g ϕ = ∆ g ϕ + div g ( 23 R g g − Ric g ) dϕ ;(2)(3) Q g = −
112 (∆ g R g − R g + 3 | Ric g | ) , where ϕ is any smooth function on M .As the Laplace-Beltrami operator governs the transformation laws of the Gauss curvature, we alsohave that the Paneitz operator does the same for the Q-curvature. Indeed under a conformal change ofmetric g u = e u g we have P g u = e − u P g ; P g u + 2 Q g = 2 Q g u e u . Apart from this analogy, we also have an extension of the Gauss-Bonnet formula which is the Gauss-Bonnet-Chern formula Z M ( Q g + | W g | dV g = 4 π χ ( M ) , where W g denotes the Weyl tensor of ( M, g ), see [17]. Hence, from the pointwise conformal invarianceof | W g | dV g , it follows that the integral of Q g over M is also a conformal invariant one.As for the Uniformization Theorem for compact closed Riemannian surfaces, one can also ask if ev-ery closed compact four dimensional Riemannian manifolds carries a metric conformally related to thebackground one with constant Q -curvature.A first positive answer to this question was given by Chang-Yang[11] under the assumptions that P g non-negative and R M Q g dV g < π . Later Djadli-Malchiodi[17] extend Chang-Yang result to a large class ofcompact closed four dimensional Riemannian manifold assuming that P g has no kernel and R M Q g dV g isnot an integer multiple of 8 π .On the other hand, there are high-order analogues to the Laplace-Beltrami operator and to the Paneitzoperator for high-dimensional compact closed Riemannian manifolds and also to the associated curva-tures (called again Q -curvatures), see[20],[21] and [23].As for the question of the existence of constant Q -curvature conformal metrics on a given compact closedfour dimensional Riemannian manifold, regarding high-dimensional Q -curvature, one can still ask thesame question for a compact closed Riemannian manifolds of arbitrary dimensions.A first affirmative answer has been given by Brendle in the even dimensional case under the assumptionthat the high-dimensional analogue of the Paneitz operator is non-negative and the total integral of the Q -curvature is less than ( n − ω n ( where ω n is the area of the unit sphere S n of R n +1 ) using ageometric flow, see[6]. The result of Djadli-Malchiodi[17] (and the one in[6]) has been extended to alldimensions in [27].As for the case of compact closed Riemannian manifolds, many works have also been done in the studyof conformally covariant differential operators on compact smooth Riemannian manifolds with smoothboundary, their associated curvature invariants, the corresponding boundary operators and curvatures inorder also to understand the relationship between analytic and geometric properties of such objects.2 model example is the Laplace-Beltrami operator on compact smooth surfaces with smooth bound-ary (Σ , g ), and the Neumann operator on the boundary. Under a conformal change of metric the coupleconstituted by the Laplace-Beltrami operator and the Neumann operator govern the transformationlaws of the Gauss curvature and the geodesic curvature. In fact, under the conformal change of metric g u = e u g , we have ∆ g u = e − u ∆ g ; ∂∂n g u = e − u ∂∂n g ; and − ∆ g u + K g = K g u e u in Σ; ∂u∂n g + k g = k g u e u on ∂ Σ . where ∆ g (resp. ∆ g u ) is the Laplace-Beltrami operator of ( Σ , g ) (resp. (Σ , g u )) and K g (resp. K g u ) isthe Gauss curvature of (Σ , g ) (resp. of (Σ , g u )), ∂∂n g (resp ∂∂n gu ) is the Neumann operator of ( Σ , g ) (resp.of (Σ , g u )) and k g (resp. k g u ) is the geodesic curvature of ( ∂ Σ , g ) (resp of ( ∂ Σ , g u )) .Moreover we have the Gauss-Bonnet formula which relates R Σ K g dV g + R ∂ Σ k g dS g and the topology of Σ(4) Z Σ K g dV g + Z ∂ Σ k g dS g = 2 πχ (Σ) , where χ (Σ) is the Euler-Poincar´e characteristic of Σ, dV g is the element area of Σ and dS g is the lineelement of ∂ Σ. Thus R Σ K g dV g + R ∂ Σ k g dS g is a topological invariant, hence a conformal one.In this context, of particular interest is also an analogue of the classical Uniformization Theorem , namelygiven a compact Riemannian surface (Σ , g ) with boundary, does there exists metrics conformally relatedto g with constant Gauss curvature and constant geodesic curvature. This problem has been solvedthrough the following theorem (for a proof see [5]) Theorem 1.1
Every compact smooth Riemannian surface with smooth boundary (Σ , g ) carries a metricconformally related to g with constant Gauss curvature and constant geodesic curvature. As for compact closed four dimensional Riemannian manifolds, on four-manifolds with boundary wealso have the Paneitz operator P g and the Q -curvature. They are defined with the same formulas (see(2) and (3)) and enjoy the same invariance properties as in the case without boundary, see (1).Likewise, Chang and Qing[8] have discovered a boundary operator P g defined on the boundary of com-pact four dimensional smooth Riemannian manifolds and a natural third-order curvature T g associatedto P g as follows P g ϕ = 12 ∂ ∆ g ϕ∂n g + ∆ ˆ g ∂ϕ∂n g − H g ∆ ˆ g ϕ + ( L g ) ab ( ∇ ˆ g ) a ( ∇ ˆ g ) b + ∇ ˆ g H g . ∇ ˆ g ϕ + ( F − R g ∂ϕ∂n g .T g = − ∂R g ∂n g + 12 R g H g − < G g , L g > +3 H g − T r ( L ) + ∆ ˆ g H g , where ϕ is any smooth function on M , ˆ g is the metric induced by g on ∂M , L g = ( L g ) ab = − ∂g ab ∂n g is the second fundamental form of ∂M , H g = tr ( L g ) = g ab L ab ( g a,b are the entries of the in-verse g − of the metric g ) is the mean curvature of ∂M , R kbcd is the Riemann curvature tensor F = R anan , R abcd = g ak R kbcd ( g a,k are the entries of the metric g ) and < G g , L g > = R anbn ( L g ) ab .On the other hand, as the Laplace-Beltrami operator and the Neumann operator govern the transfor-mation laws of the Gauss curvature and the geodesic curvature on compact surfaces with boundary underconformal change of metrics, we have that the couple ( P g , P g ) does the same for ( Q g , T g ) on compactfour dimensional smooth Riemannian manifolds with smooth boundary. In fact, after a conformal changeof metric g u = e u g we have that ( P g u = e − u P g ; P g u = e − u P g ; and ( P g + 2 Q g = 2 Q g u e u in MP g + T g = T g u e u on ∂M. Z M ( Q g + | W g | dV g + Z ∂M ( T + Z ) dS g = 4 π χ ( M )where W g denote the Weyl tensor of ( M, g ) and
ZdS g (for the definition of Z see [8]) are pointwiseconformally invariant. Moreover, it turns out that Z vanishes when the boundary is totally geodesic (bytotally geodesic we mean that the boundary ∂M is umbilic and minimal).Setting κ P g = Z M Q g dV g , κ P g = Z ∂M T g dS g ;we have that thanks to (5), and to the fact that W g dV g and ZdS g are pointwise conformally invari-ant, κ P g + κ P g is conformally invariant, and will be denoted by(6) κ ( P ,P ) = κ P g + κ P g . The Riemann mapping Theorem is one of the most celebrated theorems in mathematics. It says thatan open, simply connected, proper subset of the plane is conformally diffeomorphic to the disk. So onecan ask if such a theorem remains true in dimension 4. Unfortunately in dimension 4 few regions areconformally diffeomorphic to the ball.However, in the spirit of the Uniformization Theorem (Theorem 1 . Q -curvature, constant T -curvature and zero mean curvature.In the context of the Yamabe problem, related questions were raised by Escobar [19].In this paper, we are interested to give an analogue of the Riemann mapping Theorem (in the spiritof Theorem 1 .
1) to compact four dimensional smooth Riemannian manifold with smooth boundary undergeneric and conformally invariant assumptions. Writting g u = e u g , the problem is equivalent to solvingthe following BVP: P g u + 2 Q g = 0 in M ; P g u + T g = ¯ T e u on ∂M ; ∂u∂n g − H g u = 0 on ∂M. where ¯ Q is a fixed real number and ∂∂n g is the inward normal derivative with respect to g .Due to a result by Escobar, [19], and to the fact that we are interested to solve the problem underconformally invariant assumptions, it is not restrictive to assume H g = 0, since this can be alwaysobtained through a conformal transformation of the background metric. Thus we are lead to solve thefollowing BVP with Neumann homogeneous boundary condition:(7) P g u + 2 Q g = 0 in M ; P g u + T g = ¯ T e u on ∂M ; ∂u∂n g = 0 on ∂M. Defining H ∂∂n as H ∂∂n = n u ∈ H ( M ) : ∂u∂n g = 0 o ;4nd P , g as follows, for every u, v ∈ H ∂∂n (cid:10) P , g u, v (cid:11) L ( M ) = Z M (cid:18) ∆ g u ∆ g v + 23 R g ∇ g u ∇ g v (cid:19) dV g − Z M Ric g ( ∇ g u, ∇ g v ) dV g − Z ∂M L g ( ∇ ˆ g u, ∇ ˆ g v ) dS g , we have that by the regularity result in Proposition 2 . II ( u ) = (cid:10) P , u, u (cid:11) L ( M ) + 4 Z M Q g udV g + 4 Z ∂M T g udS g − κ ( P ,P ) log Z ∂M e u dS g ; u ∈ H ∂∂n , which are weak solutions of (7) are also smooth and hence strong solutions.A similar problem has been adressed in [28], where constant Q -curvature metrics with zero T -curvature and zero mean curvature are found under generic and conformally invariant assumptions.In [29], using heat flow methods, it is proven that if the operator P , g is non-negative, KerP , g ≃ R , and κ ( P ,P ) < π the problem (7) is solvable.Here we are interested to extend the above result under generic and conformally invariant assumptions.Our main theorem is: Theorem 1.2
Suppose
KerP , g ≃ R . Then assuming κ ( P ,P ) = k π for k = 1 , , · · · , we have that ( M, g ) admits a conformal metric with constant T -curvature, zero Q -curvature and zero mean curvature. Remark 1.3 a) Our assumptions are conformally invariant and generic, so the result applies to a largeclass of compact -dimensional manifolds with boundary.b) From the Gauss-Bonnet-Chern formula, see (5) we have that Theorem . does NOT cover the caseof locally conformally flat manifolds with totally geodesic boundary and positive integer Euler-Poincar´echaracteristic. Our assumptions include the two following situations:(8) κ ( P ,P ) < π and (or) P , g possesses ¯ k negative eigenvalues (counted with multiplicity) κ ( P ,P ) ∈ (cid:0) kπ , k + 1) π (cid:1) , for some k ∈ N ∗ and (or) P , g possesses ¯ k negative eigenvalues(counted with multiplicity)(9) Remark 1.4
Case (8) includes the condition ( ¯ k = 0 ) under which in [29] it is proven existence of solu-tions to (7) , hence will not be considered here. However due to a trace Moser-Trudinger type inequality(see Proposition . below) it can be achieved using Direct Method of Calculus of Variations.In order to simplify the exposition, we will give the proof of Theorem . in the case where we are insituation (9) and ¯ k = 0 (namely P , g is non-negative). At the end of Section 4 a discussion to settlethe general case (9) and also case (8) is made. To prove Theorem 1 . II . Unless κ ( P ,P ) < π and ¯ k = 0, this Euler-Lagrange functional is unbounded from above and below (see Section 4), so it is necessary to find extremalswhich are possibly saddle points. To do this we will use a min-max method: by classical arguments incritical point theory, the scheme yields a Palais-Smale sequence , namely a sequence ( u l ) l ∈ H ∂∂n satisfyingthe following properties II ( u l ) → c ∈ R ; II ′ ( u l ) → l → + ∞ . Palais-Smale condition holds, namely that every Palais-Smale sequence has a converging subsequense ora similar compactness criterion. Since we do not know if the Palais-Smale condition holds, we will employStruwe’s monotonicity method, see [33], also used in [17] and [27]. The latter yields existence of solutionsfor arbitrary small perturbations of the given equation, so to consider the original problem one is lead tostudy compactness of solutions to perturbations of (7). Precisely we consider(10) P g u l + 2 Q l = 0 in M ; P g u l + T l = ¯ T l e u l on ∂M ; ∂u l ∂n g = 0 on ∂M. where(11) ¯ T l −→ ¯ T > in C ( ∂M ) T l −→ T in C ( ∂M ) Q l −→ Q in C ( M ); Remark 1.5
From the Green representation formula given in Lemma . below, we have that if u l is asequence of solutions to (10) , then u l satisfies u l ( x ) = − Z M G ( x, y ) Q l ( y ) dV g − Z ∂M G ( x, y ) T l ( y ) dS g ( y ) + 2 Z ∂M G ( x, y ) ¯ T l ( y ) e u l ( y ) dS g ( y ) . Therefore, under the assumption (11) , if sup ∂M u l ≤ C , then we have u l is bounded in C α for every α ∈ (0 , . In this context, due to Remark 1 . u l ) of solutions to (10) blows up if thefollowing holds:(12) there exist x l ∈ ∂M such that u l ( x l ) → + ∞ as l → + ∞ , and we prove the following compactness result. Theorem 1.6
Suppose
KerP , g ≃ R and that ( u l ) is a sequence of solutions to (10) with ¯ T l , T l and Q l satisfying (11) . Assuming that ( u l ) l blows up (in the sense of (12) ) and Z M Q dV g + Z ∂M T dS g + o l (1) = Z ∂M ¯ T l e u l dS g ;(13) then there exists N ∈ N \ { } such that Z M Q dV g + Z ∂M T dS g = 4 N π . From this we derive a corollary which will be used to ensure compactness of some solutions to a sequence ofapproximate BVP’s produced by the topological argument combined with Struwe’s monotonicity method.Its proof is a trivial application of Theorem 1 . . Corollary 1.7
Suppose
KerP , g ≃ R .a) Let ( u l ) be a sequence of solutions to (10) with ¯ T l , T l and Q l satisfying (11) . Assume also that Z M Q dV g + Z ∂M T dS g + o l (1) = Z ∂M ¯ T l e u l dV g ;6 nd k = Z M Q dV g + Z ∂M T dS g = 4 kπ k = 1 , , , . . . . then ( u l ) l is bounded in C α ( M ) for any α ∈ (0 , .b) Let ( u l ) be a sequence of solutions to (7) for a fixed value of the constant ¯ T . Assume also that κ ( P ,P ) =4 kπ , then ( u l ) l is bounded in C m ( M ) for every positive integer m .c) Let ( u ρ k ) ρ k → be a family of solutions to (7) with T g replaced by ρ k T g , Q g by ρ k Q g and ¯ T by ρ k ¯ T for a fixed value of the constant ¯ T . Assume also that κ ( P ,P ) = 4 kπ , then ( u ρ k ) k is bounded in C m ( M ) for every positive integer m .d) If κ ( P ,P ) = 4 kπ k = 1 , , , . . . , then the set of metrics conformal to g with constant T -curvaturethe constant being the same for all of them, and with zero Q -curvature and zero mean curvature is com-pact in C m ( M ) for positive integer m .f ) If κ ( P ,P ) = 4 kπ k = 1 , , , . . . , then the set of metrics conformal to g with constant T -curvature,zero Q -curvature, zero mean curvature and of unit boundary volume is compact in C m ( M ) for everypositive integer m . We are going to describe the main ideas to prove the above results. Since the proof of Theo-rem 1 . . . . u l on ∂M , see Remark 1 . . P , g is non-negative. First of all from κ ( P ,P ) ∈ ( k π , ( k + 1)4 π ) and considerations comingfrom an improvement of Moser-Trudinger inequality, it follows that if II ( u ) attains large negative valuesthen e u has to concentrate near at most k points of ∂M . This means that, if we normalize u so that R ∂M e u ds g = 1, then naively e u ≃ P ki =1 t i δ x i , x i ∈ ∂M, t i ≥ , P ki =1 t i = 1. Such a family of convexcombination of Dirac deltas are called formal barycenters of ∂M of order k , see Section 2 , and willbe denoted by ∂M k . With a further analysis (see Proposition (4.10) ), it is possible to show that thesublevel { II < − L } for large L has the same homology as ∂M k . Using the non contractibility of ∂M k ,we define a min-max scheme for a perturbed functional II ρ , ρ close to 1, finding a P-S sequence tosome levels c ρ . Applying the monotonicity procedure of Struwe, we can show existence of critical pointsof II ρ for a.e ρ , and we reduce ourselves to the assumptions of Theorem 1 . Acknowledgements:
I would like to thank Professor Andrea Malchiodi for several stimulating discus-sions.The author have been supported by M.U.R.S.T within the PRIN 2006 Variational methods and nonlineardifferential equations.
In this brief section we collect some useful notations, state a lemma giving the existence of the Greenfunction of the operator ( P g , P g ) with its asymptotics near the singularity and a trace analogue of thewell-known Moser-Trudinger inequality for the operator P , g when it is non-negative.In the following B p ( r ) stands for the metric ball of radius r and center p , B + p ( r ) = B p ( r ) ∩ M if p ∈ ∂M . Sometimes we use B + p ( r ) to denote B p ( r ) ∩ M even if p / ∈ ∂M .In the sequel, B x ( r ) will stand for the Euclidean ball of center x and radius r , B x + ( r ) = B x ( r ) ∩ R if x ∈ ∂ R . We use also B x + ( r ) to denote B x ( r ) ∩ R even if x / ∈ ∂ R . We denote by d g ( x, y ) the metric7istance between two points x and y of M and d ˆ g ( x, y ) the intrinsic distance of two points x and y of ∂M . Given a point x ∈ ∂M , and r > B ∂Mx ( r ) stands for the metric ball in ∂M with respectto the (intrinsic) distance d ˆ g ( · , · ) of center x and radius r . H ( M ) stands for the usual Sobolev spaceof functions on M which are of class H in each coordinate system. Large positive constants are alwaysdenoted by C , and the value of C is allowed to vary from formula to formula and also within the sameline. M stands for the Cartesian product M × M , while Diag ( M ) is the diagonal of M . Given afunction u ∈ L ( ∂M ), ¯ u ∂M denotes its average on ∂M , that is ¯ u ∂M = ( V ol ˆ g ( ∂M )) − R ∂M u ( x ) dS g ( x )where V ol ˆ g ( ∂M ) = R ∂M dS g . N denotes the set of non-negative integers. N ∗ stands for the set of positive integers. A l = o l (1) means that A l −→ l −→ + ∞ . A ǫ = o ǫ (1) means that A ǫ −→ ǫ −→ A δ = o δ (1) means that A δ −→ δ −→ A l = O ( B l ) means that A l ≤ CB l for some fixed constant C .. dV g denotes the Riemannian measure associated to the metric g . dS g stands for the Riemannian measure associated to the metric ˆ g induced by g on ∂M . dσ ˆ g stands for the surface measure on boundary of balls of ∂M . | · | ˆ g stands for the norm associated to g . f = f ( a, b, c, ... ) means that f is a quantity which depends only on a, b, c, ... .Next we let ∂M k denotes the family of formal sums(14) ∂M k = { k X i =1 t i δ x i , t i ≥ , k X i =1 t i = 1; x i ∈ ∂M } , It is known in the literature as the formal set of barycenters relative to ∂M of order k . We recall that ∂M k is a stratified set namely a union of sets of different dimension with maximum one equal to 4 k − Lemma 2.1 (well-known) For any k ≥ one has H k − ( ∂M k ; Z ) = 0 . As a consequence ∂M k isnon-contractible. If ϕ ∈ C ( ∂M ) and if σ ∈ ∂M k , we denote the action of σ on ϕ as h σ, ϕ i = k X i =1 t i ϕ ( x i ) , σ = k X i =1 t i δ x i . Moreover, if f is a non-negative L function on ∂M with R ∂M f ds g = 1, we can define a distance of f from ∂M k in the following way(15) d ( f, ∂M k ) = inf σ ∈ ∂M k sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂M f ϕdS g − h σ, ϕ i (cid:12)(cid:12)(cid:12)(cid:12) | k ϕ k C ( ∂M ) = 1 (cid:27) . We also let(16) D ε,k = (cid:8) f ∈ L ( ∂M ) : f ≥ , k f k L ( ∂M ) = 1 , d ( f, ∂M k ) < ε (cid:9) . Now we state a Lemma which asserts the existence of the Green function of ( P g , P g ) with homoge-neous Neumann condition. Its proof can be found in [27]. Lemma 2.2
Assume that
KerP , g ≃ R , then the Green function G ( x, y ) of ( P g , P g ) exists in thefollowing sense :a) For all functions u ∈ C ( M ) , ∂u∂n g = 0 , we have u ( x ) − ¯ u = Z M G ( x, y ) P g u ( y ) dV g ( y ) + 2 Z ∂M G ( x, y ′ ) P g u ( t ) dS g ( y ′ ) x ∈ M ) G ( x, y ) = H ( x, y ) + K ( x, y ) is smooth on M \ Diag ( M ) , K extends to a C α function on M and H ( x, y ) = π f ( r ) log r if B δ ( x ) ∩ ∂M = ∅ ; π f ( r )(log r + log r ) otherwise . where f ( · ) = 1 in [ − δ , δ ] and f ( · ) ∈ C ∞ ( − δ, δ ) , δ ≤ min { δ , δ } , δ is the injectivity radius of M in ˜ M , and δ = δ , r = d g ( x, y ) and ¯ r = d g ( x, ¯ y ) . Next we give a regularity result corresponding to boundary value problems of the type of BVP (7)and high order a priori estimates for sequences of solutions to BVP like (10) when they are bounded fromabove. Its proof is a trivial adaptation of the arguments of Proposition 2.3 in [28]
Lemma 2.3
Let u ∈ H ∂∂n be a weak solution to ( P g u = h in M ; P g u + f = ¯ f e u on ∂M. with f ∈ C ∞ ( ∂M ) , h ∈ C ∞ ( M ) and ¯ f a real constant. Then we have that u ∈ C ∞ ( M ) .Let u l ∈ H ∂∂n be a sequence of weak solutions to ( P g u l = h l in M ; P g u l + f l = ¯ f l e u l on ∂M. with f l → f in C k ( ∂M ) , ¯ f l → ¯ f in C k ( ∂M ) and h l → h in C k ( M ) for some fixed k ∈ N ∗ . Assuming sup ∂M u l ≤ C we have that || u l || C k +3+ α ( M ) ≤ C for any α ∈ (0 , . Now we give a Proposition which is a trace Moser-Trudinger type inequality when the operator P , g isnon-negative with trivial kernel. Its proof can be found in [29], but for the reader convenience we willrepeat it here. Proposition 2.4
Assume P , g is a non-negative operator with KerP , g ≃ R . Then we have that forall α < π there exists a constant C = C ( M, g, α ) such that (17) Z ∂M e α ( u − ¯ u∂M )2 h P , g u,u i L M,g ) dS g ≤ C, for all u ∈ H ∂∂n , and hence (18) log Z ∂M e u − ¯ u ) dS g ≤ C + 94 α (cid:10) P , g u, u (cid:11) L ( M,g ) ∀ u ∈ H ∂∂n . Proof . First of all, without loss of generality we can assume ¯ u ∂M = 0. Following the same argumentas in Lemma 2.2 in [9]. we get ∀ β < π there exists C = C ( β, M ) Z M e βv R M | ∆ gv | dVg dV g ≤ C, ∀ v ∈ H ∂∂n with ¯ v ∂M = 0 . Z M e βv h P , g v,v i L M ) dV g ≤ C, ∀ v ∈ H ∂∂n with ¯ v ∂M = 0 . Now let X be a vector field extending the the outward normal at the boundary ∂M . Using the divergencetheorem we obtain Z ∂M e αu dS g = Z M div g (cid:16) Xe αu (cid:17) dV g . Using the formula for the divergence of the product of a vector fied and a function we get(20) Z ∂M e αu dS g = Z M ( div g X + 2 uα ∇ g u ∇ g X ) e αu dV g . Now we suppose < P , g u, u > L ( M ) ≤
1, then since the vector field X is smooth we have(21) (cid:12)(cid:12)(cid:12)(cid:12)Z M div g Xe αu dV g (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ;thansk to (19). Next let us show that (cid:12)(cid:12)(cid:12)(cid:12)Z M αu ∇ g u ∇ g Xe αu dV g . (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Let ǫ > p = 43 − ǫ , p = 4 , p = 4 ǫ . It is easy to check that 1 p + 1 p + 1 p = 1 . Using Young’s inequality we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z M αu ∇ g u ∇ g Xe αu dV g (cid:12)(cid:12)(cid:12)(cid:12) ≤ C || u || L ǫ ||∇ g u || L (cid:18)Z M e α − ǫ u dV g (cid:19) − ǫ . On the other hand, Lemma 2.8 in [28] and Sobolev embedding theorem imply || u || L ǫ ≤ C ;and ||∇ g u || L ≤ C. Furthermore from the fact that α < π , by taking ǫ sufficiently small and using (19), we obtain (cid:18)Z M e α − ǫ u dV g (cid:19) − ǫ . Thus we arrive to(22) (cid:12)(cid:12)(cid:12)(cid:12)Z M αu ∇ g u ∇ g Xe αu dV g (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. Hence (20), (21) and (22) imply Z ∂M e αu dS g ≤ C, as desired. So the first point of the Lemma is proved.Now using the algebraic inequality 3 ab ≤ γ a + 3 b γ , we have that the second point follows directly from the first one. Hence the Lemma is proved.10 Proof of Theorem . This section is concerned about the proof of Theorem 1 .
6. We use the same strategy as in [27] and [28].Hence in many steps we will be sketchy and referring to the corresponding arguments in [27]. However,in contrast to the situation in [28], due remark 1 .
5, we have only to take care of the behaviour of therestriction of the sequence u l to the boundary M . Proof of Theorem 1 . Theorem 3.1 ( [ ] ) There exists a dimensional constant σ > such that, if u ∈ C ( R ) is solution ofthe integral equation u ( x ) = Z R σ log (cid:18) | y || x − y | (cid:19) e u ( y ) dy + c , where c is a real number, then e u ∈ L ( R ) implies, there exists λ > and x ∈ R such that u ( x ) = log (cid:18) λλ + | x − x | (cid:19) . Now, if σ in Theorem 3 . k = 2 π σ and γ = 2( k ) We divide the proof in 5-steps as in [27].
Step 1
There exists N ∈ N ∗ , N converging points ( x i,l ) ⊂ ∂M i = 1 , ..., N , N with limit points x i ∈ ∂M ,sequences ( µ i,l ) i = 1; ... ; N ; of positive real numbers converging to 0 such that the following hold: a ) d g ( x i,l , x j,l ) µ i,l −→ + ∞ i = j i, j = 1 , .., N and ¯ T l ( x i,l ) µ i.l e u l ( x i,l ) = 1; b )For every iv i,l ( x ) = u l ( exp x i,l ( µ i,l x )) − u l ( x i,l ) −
13 log( k ) −→ V ( x ) in C loc ( R ) , V | ∂ R ( x ) := log( 4 γ γ + | x | );and lim R → + ∞ lim l → + ∞ Z B + xi,l ( Rµ i,l ) ∩ ∂M ¯ T l ( y ) e u l ( y ) ds g ( y ) = 4 π ; c ) T here exists C > such that inf i =1 ,...,N d g ( x i,l , x ) e u l ( x ) ≤ C ∀ x ∈ ∂M, ∀ l ∈ N . Proof of Step 1
First of all let x l ∈ ∂M be such that u l ( x l ) = max x ∈ ∂M u l ( x ), then using the fact that u l blows up weinfer u l ( x l ) −→ + ∞ . Now since ∂M is compact, without loss of generality we can assume that x l → ¯ x ∈ ∂M .Next let µ l > T l ( x l ) µ l e u l ( x l ) = 1. Since ¯ T l −→ ¯ T C ( ∂M ), ¯ T > u l ( x l ) −→ + ∞ , we have that µ l −→ . B ( δµ − l ) be the half Euclidean ball of center 0 and radius δµ − l , with δ > x ∈ B ( δµ − l ), we set v l ( x ) = u l ( exp x l ( µ l x )) − u l ( x l ) −
13 log( k );(23) ˜ Q l ( x ) = Q l ( exp x l ( µ l x ));(24) ˜¯ Q l ( x ) = ¯ Q l ( exp x l ( µ l x ));(25) g l ( x ) = (cid:0) exp ∗ x l g (cid:1) ( µ l x ) . (26)Now from the Green representation formula we have, u l ( x ) − ¯ u l = Z M G ( x, y ) P g u l ( y ) dV g ( y ) + 2 Z ∂M G ( x, y ′ ) P g u l ( y ′ ) dS g ( y ′ ); ∀ x ∈ M, (27)where G is the Green function of ( P g , P g ) (see Lemma 2 . x we obtain that for k = 1 , |∇ k u l | g ( x ) ≤ Z ∂M |∇ k G ( x, y ) | g ¯ T l ( y ) e u l ( y ) dV g + O (1) , since T l −→ T in C ( ∂M ) and Q l → Q in C ( M ).Now let y l ∈ B + x l ( Rµ l ) , R > Z ∂M |∇ k G ( y l , y ) | g e u l ( y ) dV g ( y ) = O ( µ − kl )(28)Hence we get(29) |∇ k v l | g ( x ) ≤ C. Furthermore from the definition of v l (see (23)), we get(30) v l ( x ) ≤ v l (0) = −
13 log( k ) ∀ x ∈ R Thus we infer that ( v l ) l is uniformly bounded in C ( K ) for all compact subsets K of R . Hence byArzel`a-Ascoli theorem we derive that v l −→ V in C loc ( R ) , (31)On the other hand (30) and (31) imply that(32) V ( x ) ≤ V (0) = −
13 log( k ) ∀ x ∈ R . Moreover from (29) and (31) we have that V is Lipschitz.On the other hand using the Green’s representation formula for ( P g , P g ) we obtain that for x ∈ R fixedand for R big enough such that x ∈ B ( R )(33) u l ( exp x l ( µ l x )) − ¯ u l = Z M G ( exp x l ( µ l x ) , y ) P g u l ( y ) dV g ( y )+ 2 Z ∂M G ( exp x l ( µ l x ) , y ′ ) P g u l ( y ′ ) dS g ( y ′ ) . I l ( x ) = 2 Z B + xl ( Rµ l ) ∩ ∂M ( G ( exp x l ( µ l x ) , y ′ ) − G ( exp x l (0) , y ′ )) ¯ T l ( y ) e u l ( y ) dS g ( y ′ );II l ( x ) = 2 Z ∂M \ ( B + xl ( Rµ l ) ( G ( exp x l ( µ l x ) , y ′ ) − G ( exp x l (0) , y ′ )) ¯ T l ( y ′ ) e u l ( y ) dS g ( y ′ );III l ( x ) = 2 Z ∂M ( G ( exp x l ( µ l x ) , y ′ ) − G ( exp x l (0) , y ′ )) T l ( y ) dS g ( y ′ );and IIII l ( x ) = 2 Z M ( G ( exp x l ( µ l x ) , y ) − G ( exp x l (0) , y )) Q l ( y ) dV g ( y ) . Using again the same argument as in [27] (see formula (45)- formula (51)) we get(34) v l ( x ) = I l ( x ) + II l ( x ) − III l ( x ) − IIII l ( x ) −
14 log(3) . Moreover following the same methods as in [27]( see formula (53)-formula (62)) we obtain(35) lim l I l ( x ) = Z B ( R ) ∩ ∂R σ log (cid:18) | z || x − z | (cid:19) e V ( z ) dz. (36) lim sup l II l ( x ) = o R (1) . (37) III l ( x ) = o l (1)and(38) IIII l ( x ) = o l (1) . Hence from (31), (34)-(38) by letting l tends to infinity and after R tends to infinity, we obtain V | R (that for simplicity we will always write by V ) satisfies the following conformally invariant integral equa-tion on R V ( x ) = Z R σ log (cid:18) | z || x − z | (cid:19) e V ( z ) dz −
13 log( k ) . (39)Now since V is Lipschitz then the theory of singular integral operator gives that V ∈ C ( R ).On the other hand by using the change of variable y = exp x l ( µ l x ), one can check that the following holdslim l −→ + ∞ Z B + xl ( Rµ l ) ∩ ∂M ¯ T l e u l dV g = k Z B +0 ( R ) ∩ ∂R e V dx ;(40)Hence (13) implies that e V ∈ L ( R ) . Furthermore by a classification result by X. Xu, see Theorem 3 . V ( x ) = log (cid:18) λλ + | x − x | (cid:19) (41)for some λ > x ∈ R .Moreover from V ( x ) ≤ V (0) = − log( k ) ∀ x ∈ R , we have that λ = 2 k and x = 0 namely, V ( x ) = log( 4 γ γ + | x | ) .
13n the other hand by letting R tends to infinity in (40) we obtainlim R → + ∞ lim l → + ∞ Z B + xl ( Rµ l ) ∩ ∂R ¯ T l ( y ) e u l ( y ) dS g ( y ) = k Z R e V dx. (42)Moreover from a generalized Pohozaev type identity by X.Xu [34] (see Theorem 1.1) we get σ Z R e V ( y ) dy = 2 , hence using (42) we derive thatlim R → + ∞ lim l → + ∞ Z B + xl ( Rµ l ) ∩ ∂M ¯ T l ( y ) e u l ( y ) dS g ( y ) = 4 π Now for k ≥ H k ) holds if there exists k converging points ( x i,l ) l ⊂ ∂M i = 1 , ..., k , k sequences ( µ i,l ) i = 1 , ..., k of positive real numbers converging to 0 such that the following hold (cid:0) A k (cid:1) d ˆ g ( x i,l , x j,l ) µ i,l −→ + ∞ i = j i, j = 1 , .., k and ¯ T l ( x i,l ) µ i.l e u l ( x i,l ) = 1; (cid:0) A k (cid:1) For every i = 1 , · , k x i,l → ¯ x i ∈ ∂M ; v i,l ( x ) = u l ( exp x i,l ( µ i,l x )) − u l ( x i,l ) −
13 log( k ) −→ V ( x ) in C loc ( R ) , V | ∂ R := log( 4 γ γ + | x | )and lim R → + ∞ lim l → + ∞ Z B + xi,l ( Rµ i,l ) ∩ ∂M ¯ T l ( y ) e u l ( y ) = 4 π Clearly, by the above arguments ( H ) holds. We let now k ≥ H k ) holds. We alsoassume that sup ∂M R k,l ( x ) e u l ( x ) −→ + ∞ as l −→ + ∞ , (43)where R k,l ( x ) = min i =1; .. ; k d g ( x i,l , x ) . Now using the same argument as in [18],[27] and the arguments which have rule out the possibility ofinterior blow up above that also apply for local maxima, one can see easily that ( H k +1 ). Hence since (cid:0) A k (cid:1) and (cid:0) A k (cid:1) of H k imply that Z ∂M ¯ T l ( y ) e u l ( y ) dS g ( y ) ≥ k π + o l (1) . Thus (13) imply that there exists a maximal k , 1 ≤ k ≤ π (cid:0)R M Q ( y ) dV g ( y ) + R ∂M T ( y ′ ) dS g ( y ′ ) (cid:1) ,such that ( H k ) holds. Arriving to this maximal k , we get that (43) cannot hold. Hence setting N = k the proof of Step 1 is done. 14 tep 2 There exists a constant
C > R l ( x ) |∇ g u l | g ( x ) ≤ C ∀ x ∈ M and ∀ l ∈ N ; ∀ x ∈ ∂M (44)where R l ( x ) = min i =1 ,..,N d g ( x i,l , x );and the x i,l ’s are as in Step 1. Proof of Step 2
First of all using the Green representation formula for ( P g , P g ) see Lemma 2 . u l ( x ) − ¯ u l = Z M G ( x, y ) P g u l ( y ) dV g ( y ) + 2 Z ∂M G ( x, y ′ ) P g u l ( y ′ ) dS g ( y ′ ) . Now using the BVP (7) we get u l ( x ) − ¯ u l = − Z M G ( x, y ) Q l dV g ( y ) − Z ∂M G ( x, y ′ ) T l ( y ′ ) u l ( y ′ ) dS g ( y ′ )+2 Z ∂M G ( x, y ) ¯ T l ( y ′ ) e u l ( y ′ ) dS g ( y ′ ) . (45)Thus differentiating with respect to x (45) and using the fact that Q l → Q , ¯ Q l → ¯ Q and T l → T in C , we have that for x l ∈ ∂M |∇ g u l ( x l ) | g = O (cid:18)Z ∂M d g ( x l , y ) e u l ( y ) dS g ( y ) (cid:19) + O (1) . Hence at this stage following the same argument as in the proof of Theorem 1.3, Step 2 in [27], we obtain Z ∂M d g ( x l , y )) e u l ( y ) dV g ( y ) = O (cid:18) R l ( x l ) (cid:19) ;hence since x l is arbitrary, then the proof of Step 2 is complete. Step 3
Set R i,l = min i = j d g ( x i,l , x j,l );we have that1) There exists a constant C > ∀ r ∈ (0 , R i,l ] ∀ s ∈ ( r , r ] | u l (cid:0) exp x i,l ( rx ) (cid:1) − u l (cid:0) exp x i,l ( sy ) (cid:1) | ≤ C f or all x, y ∈ ∂ R such that | x | , | y | ≤ . (46)2) If d i,l is such that 0 < d i,l ≤ R i,l and d i,l µ i,l −→ + ∞ then we have thatif Z B + xi,l ( d i,l ) ∩ ∂M ¯ T l ( y ) e u l ( y ) dS g ( y ) = 4 π + o l (1);(47)then Z B + xi,l (2 d i,l ) ∩ ∂M ¯ T l ( y ) e u l ( y ) ds g ( y ) = 4 π + o l (1) .
15) Let R be large and fixed. If d i,l > d i,l −→ d i,l µ i,l −→ + ∞ , and d i,l < R i,l R then if Z B + xi,l ( di,l R ) ∩ ∂M ¯ Q l ( y ) e u l ( y ) dS g ( y ) = 4 π + o l (1);then by setting ˜ u l ( x ) = u l ( exp x i,l ( d i,l x )); x ∈ A +2 R ;where A +2 R = ( B (2 R ) \ B ( R )) ∩ ∂ R , we have that, || d i,l e u l || C α ( A + R ) → as l → + ∞ ;for some α ∈ (0 ,
1) where A + R = ( B ( R ) \ B ( R )) ∩ ∂ R . Proof of Step 3
We have that property 1 follows immediately from Step 2 and the definition of R i,l . In fact we can join rx to sy by a curve whose length is bounded by a constant proportional to r .Now let us show point 2. Thanks to d i,l µ i,l −→ + ∞ , point c) of Step 1 and (47) we have that Z B + xi,l ( d i,l ) ∩ ∂M \ B + xi,l ( di,l ) ∩ ∂M e u l ( y ) dS g ( y ) = o l (1) . (48)Thus using (46), with s = r and r = 2 d i,l we get Z B + xi,l (2 d i,l ) ∩ ∂M \ B + xi,l ( d i,l ) ∩ ∂M e u l ( y ) ds g ( y ) ≤ C Z B + xi,l ( d i,l ) ∩ ∂M \ B + xi,l ( di,l ) ∩ ∂M e u l ( y ) dS g ( y );Hence we arrive Z B + xi,l (2 d i,l ) ∩ ∂M \ B + xi,l ( d i,l ) ∩ ∂M e u l ( y ) dS g ( y ) = o l (1) . So the proof of point 2 is done. On the other hand by following in a straightforward way the proof ofpoint 3 in Step 3 of Theorem 1.3 in [27] one gets easily point 3. Hence the proof of Step 3 is complete.
Step 4
There exists a positive constant C independent of l and i such that Z B + xi,l ( Ri,lC ) ∩ ∂M ¯ T l ( y ) e u l ( y ) dS g ( y ) = 4 π + o l (1) . Proof of Step 4
The proof is an adaptation of the arguments in Step 4 ([27])
Step 5 :Proof of Theorem . Z ∂M \ ( ∪ i = Ni =1 B + xi,l ( Ri,lC ) ∩ ∂M ) e u l ( y ) dS g ( y ) = o l (1) . So since B + x i,l ( R i,l C ) ∩ ∂M are disjoint then the Step 4 implies that, Z ∂M ¯ T l ( y ) e u l ( y ) dS g ( y ) = 4 N π + o l (1) , Z M Q ( y ) dV g ( y ) + Z ∂M T ( y ′ ) dS g ( y ′ ) = 4 N π . ending the proof of Theorem 1 . . This section deals with the proof of Theorem 1 .
2. It is divided into four Subsections. The first oneis concerned with an improvement of the Moser-Trudinger type inequality (see Proposition 2 .
4) and itscorollaries. The second one is about the existence of a non-trivial global projection from some negativesublevels of II onto ∂M k (for the definition see Section 2 formula 14). The third one deals with theconstruction of a map from ∂M k into suitable negative sublevels of II . The last one describes themin-max scheme. In this Subsection we give an improvement of the Moser-Trudinger type inequality, see Proposition 2 . u ∈ H ∂∂n such that II ( u ) attains large negative values, e u can concentrate at most at k points of ∂M . (see Lemma 4 . e u (for some functions u suitably normalized) from ∂M k .As said in the introduction of the Subsection, we start by the following Lemma giving an improvement ofthe Moser-Trudinger type inequality (Proposition 2 . Lemma 4.1
For a fixed l ∈ N , let S · · · S l +1 , be subsets of ∂M satisfying, dist ( S i , S j ) ≥ δ for i = j ,let γ ∈ (0 , l + l ) .Then, for any ¯ ǫ > , there exists a constant C = C (¯ ǫ, δ , γ , l, M, ) such that the following hods1) log Z ∂M e u − ¯ u ∂M ) ≤ C + 316 π ( 1 l + 1 − ¯ ǫ ) (cid:10) P , g u, u (cid:11) L ( M ) ; for all the functions u ∈ H ∂∂n satisfying (49) R S i e u dSg R ∂M e u dSg ≥ γ , i ∈ { , .., l + 1 } . In the next Lemma we show a criterion which implies the situation described in the first condition in(49). The result is proven in [17] Lemma 2.3.
Lemma 4.2
Let l be a given positive integer, and suppose that ǫ and r are positive numbers. Supposethat for a non-negative function f ∈ L ( ∂M ) with k f k L ( ∂M ) = 1 there holds Z ∪ ℓi =1 B ∂Mr ( p i ) f dS g < − ε for every ℓ -tuples p , . . . , p ℓ ∈ ∂M Then there exist ε > and r > , depending only on ε, r, ℓ and ∂M (but not on f ), and ℓ + 1 points p , . . . , p ℓ +1 ∈ ∂M (which depend on f ) satisfying Z B ∂Mr ( p ) f dS g > ε, . . . , Z B ∂Mr ( p ℓ +1 ) f dS g > ε ; B ∂M r ( p i ) ∩ B ∂M r ( p j ) = ∅ for i = j.
17n interesting consequence of Lemma 4 . H ∂∂N for which the value of II is large negative. Lemma 4.3
Under the assumptions of Theorem . , and for k ≥ given by (9) , the following propertyholds. For any ǫ > and any r > there exists large positive L = L ( ǫ, r ) such that for any u ∈ H ∂∂n with II ( u ) ≤ − L, R ∂M e u dS g = 1 there exists k points p ,u , . . . , p k,u ∈ ∂M such that (50) Z ∂M \∪ ki =1 B ∂Mpi,u ( r ) e u dS g < ǫ Proof . Suppose that by contradiction the statement is not true. Then there exists ǫ > r >
0, anda sequence ( u n ) ∈ H ∂n such that R ∂M e u n dS g = 1, II ( u n ) → −∞ as n → + ∞ and such thatfor any k tuples of points p , . . . , p k ∈ ∂M ,we have(51) Z ( ∪ ki =1 B ∂Mpi,u ( r )) e u dS g < − ǫ ;Now applying Lemma 4 . f = e u n , and after Lemma 4 . δ = 2¯ r , S i = B ∂M ¯ p i (¯ r ), and γ = ¯ ǫ where ¯ ǫ , ¯ r , ¯ p i are given as in Lemma 4 .
2, we have for every ˜ ǫ > C depending on ǫ , r ,and ˜ ǫ such that II ( u n ) ≥ (cid:10) P , g u n , u n (cid:11) + 4 Z M Q g u n dV g + 4 Z ∂M T g u n dS g − κ ( P ,P ) π ( k + 1 − ˜ ǫ ) (cid:10) P , g u n , u n (cid:11) − Cκ ( P ,P ) − κ ( P ,P ) u n∂M where C is independent of n . Using elementary simplifications, the above inequality becomes II ( u n ) ≥ (cid:10) P , g u n , u n (cid:11) + 4 Z M Q g u n dV g + 4 Z ∂M T g u n dS g − κ P ,P π ( k + 1 − ˜ ǫ ) (cid:10) P , g u n , u n (cid:11) − Cκ P ,P − κ P ,P u n∂M . So, since κ P ,P < ( k + 1)4 π , by choosing ˜ ǫ small we get II ( u n ) ≥ β (cid:10) P , g u n , u n (cid:11) − C (cid:10) P , g u n , u n (cid:11) − Cκ P ,P ;thanks to H¨older inequality, to Sobolev embedding, to trace Sobolev embedding and to the fact that KerP , g ≃ R (where β = 1 − κ P ,P π ( k +1 − ˜ ǫ ) > II ( u n ) ≥ − C. So we reach a contradiction. Hence the Lemma is proved.Next we give a Lemma which is a direct consequence of the previous one. It gives the distance of thefunctions e u , from ∂M k for u belonging to low energy levels of II such that R ∂M e u dS g = 1. Itsproof is the same as the one of corollary in [17]. Corollary 4.4
Let ε be a (small) arbitrary positive number and k be given as in (9) . Then there exists L > such that, if II ( u ) ≤ − L and R ∂M e u dS g = 1 , then we have that d ( e u , ∂M k ) ≤ ε . II into ( M ∂ ) k In this short Subsection we show that one can map in a non trivial way some appropriate low energysublevels of the Euler-Lagrange functional II into ∂M k .First of all arguing as in Proposition 3.1 in [17], we have the following Lemma.18 emma 4.5 Let m be a positive integer, and for ε > let D ε,m be as in (16) . Then there exists ε m > ,depending on m and ∂M such that, for ε ≤ ε k there exists a continuous map Π m : D ε,m → ∂M m . Using the above Lemma we have the following non-trivial continuous global projection form low energysublevels of II into ∂M k . Proposition 4.6
For k ≥ given as in (9) , there exists a large L > and a continuous map Ψ fromthe sublevel { u : II ( u ) < − L, R ∂M e u dS g = 1 } into ∂M k which is topologically non-trivial. By the non-contractibility of ∂M k , the non-triviality of the map is apparent from b) of Proposi-tion 4 .
10 below.
Proof . We fix ε k so small that Lemma 4 . m = k . Then we apply Corollary 4.4 with ε = ε k . We let L be the corresponding large number, so that if II ( u ) ≤ − L and R ∂M e u dS g = 1, then d ( e u , ∂M k ) < ε k . Hence for these ranges of u , since the map u e u is continuous from H ( M ) into L ( ∂M ), then the projections Π k from H (Σ) onto ∂M k is well defined and continuous. . ∂M k into sublevels of II In this Subsection we will define some test functions depending on a real parameter λ and give estimateof the quadratic part of the functional II on those functions as λ tends to infinity. And as a corollarywe define a continuous map from ∂M k into large negative sublevels of II .For δ > χ δ : R + → R satisfying the followingproperties (see [17]): χ δ ( t ) = t, f or t ∈ [0 , δ ]; χ δ ( t ) = 2 δ, f or t ≥ δ ; χ δ ( t ) ∈ [ δ, δ ] , f or t ∈ [ δ, δ ] . Then, given σ = ∈ ∂M k , σ = P ki =1 t i δ x i and λ >
0, we define the function ; ϕ λ,σ : M → R as follows(52) ϕ λ,σ ( y ) = 13 log " k X i =1 t i (cid:18) λ λ χ δ ( d i ( y )) (cid:19) ;where we have set d i ( y ) = d g ( y, x i ) , x i ∈ ∂M, y ∈ M, ;with d g ( · , · ) denoting the Riemannian distance on M .Now we state a Lemma giving an estimate (uniform in σ ∈ ∂M k ) of the quadratic part (cid:10) P , g ϕ λ,σ , ϕ λ,σ (cid:11) ofthe Euler functional II as λ → + ∞ . Its proof is a straightforward adaptation of the arguments in Lemma4.5 in [27]. Lemma 4.7
Suppose ϕ λ,σ as in (52) and let ǫ > small enough. Then as λ → + ∞ one has (53) (cid:10) P , g ϕ λ,σ , ϕ λ,σ (cid:11) ≤ (16 π k + ǫ + o δ (1)) log λ + C ǫ,δ Next we state a lemma giving estimates of the remainder part of the functional II along ϕ σ,λ . Theproof is the same as the one of formulas (40) and (41) in the proof of Lemma 4.3 in [17]. Lemma 4.8
Soppose ϕ σ,λ as in (52) . Then as λ → + ∞ one has Z M Q g ϕ σ,λ dV g = − κ P g log λ + O ( δ log λ ) + O (log δ ) + O (1); Z ∂M T g ϕ σ,λ dV g = − κ P g log λ + O ( δ log λ ) + O (log δ ) + O (1); and log Z ∂M e ϕ σ,λ = O (1) . λ > λ : ∂M k → H ∂∂n by the following formula ∀ σ ∈ ∂M k Φ λ ( σ ) = ϕ σ,λ . We have the following Lemma which is a trivial application of Lemmas 4 . . Lemma 4.9
For k ≥ (given as in (9) ), given any L > large enough, there exists a small δ and alarge ¯ λ such that II (Φ ¯ λ ( σ )) ≤ − L for every σ ∈ ∂M k . Next we state a proposition giving the existence of the projection from ∂M k into large negative sublevelsof II , and the non-triviality of the map Ψ of the proposition (4.6). Proposition 4.10
Let Ψ be the map defined in proposition . . Then assuming k ≥ (given as in (9) ), for every L > sufficiently large (such that proposition . applies), there exists a map Φ ¯ λ : ∂M k −→ H ∂∂n with the following propertiesa) II (Φ ¯ λ ( z )) ≤ − L for any z ∈ ∂M k ; b) Ψ ◦ Φ ¯ λ is homotopic to the identity on ∂M k . Proof . The statement (a) follows from Lemma 4 .
9. To prove (b) it is sufficient to consider the familyof maps T λ : ∂M k → ∂M k defined by T λ ( σ ) = Ψ(Φ λ ( σ )) , σ ∈ ∂M k We recall that when λ is sufficiently large, then this composition is well defined. Therefore , since e ϕσ,λ R ∂M e ϕσ,λ dS g ⇀ σ in the weak sens of distributions, letting λ → + ∞ we obtain an homotopy betweenΨ ◦ Φ and Id ∂M k . This concludes the proof. In this Subsection, we describe the min-max scheme based on the set ∂M k in order to prove Theorem 1 . II ρ for which we can prove existenceof solutions in a dense set of the values of ρ . Following a idea of Struwe ( see [33]), this is done byproving the a.e differentiability of the map ρ → II ρ ( where II ρ is the minimax value for the functional II ρ ).We now introduce the minimax scheme which provides existence of solutions for (8). Let [ ∂M k denote the(contractible) cone over ∂M k , which can be represented as [ ∂M k = ( ∂M k × [0 , ∂M k × L be so large that Proposition 4 . L , and then let ¯ λ be so largethat Proposition 4 .
10 applies for this value of L . Fixing ¯ λ , we define the following class.(54) II ¯ λ = { π : [ ∂M k → H ∂∂n : π is continuous and π ( · ×
1) = Φ ¯ λ ( · ) } . We then have the following properties.
Lemma 4.11
The set II ¯ λ is non-empty and moreover, letting II ¯ λ = inf π ∈ II ¯ λ sup m ∈ \ ∂M k, II ( π ( m )) , there holds II ¯ λ > − L . roof . The proof is the same as the one of Lemma 5.1 in [17]. But we will repeat it for the reader’sconvenience.To prove that II ¯ λ is non-empty, we just notice that the following map¯ π ( · , t ) = t Φ ¯ λ ( · )belongs to II ¯ λ . Now to prove that II ¯ λ > − L , let us argue by contradiction. Suppose that II ¯ λ ≤ − L :then there exists a map π ∈ II ¯ λ such that sup m ∈ [ ∂M k II ( π ( m )) ≤ − L . Hence since Proposition 4 . L , writing m = ( z, t ) with z ∈ ∂M k we have that the map t → Ψ ◦ π ( · , t )is an homotopy in ∂M k between Ψ ◦ Φ ¯ λ and a constant map. But this is impossible since ∂M k is non-contractible and Ψ ◦ Φ ¯ λ is homotopic to the identity by Proposition 4 . ρ in a smallneighborhood of 1, [1 − ρ , ρ ], we define the modified functional II ρ : H ∂∂n → R (54) II ρ ( u ) = (cid:10) P , g u, u (cid:11) + 4 ρ Z M Q g udV g + 4 ρ Z ∂M T g udS g − ρκ ( P ,P ) log Z ∂M e u dS g ; u ∈ H ∂∂n . Following the estimates of the previous section, one easily checks that the above minimax scheme appliesuniformly for ρ ∈ [1 − ρ , ρ ] and for ¯ λ sufficiently large. More precisely, given any large number L >
0, there exist ¯ λ sufficiently large and ρ sufficiently small such that(55) sup π ∈ II ¯ λ sup m ∈ ∂ [ ∂M k II ( π ( m )) < − L ; II ρ inf π ∈ II ¯ λ sup m ∈ [ ∂M k II ρ ( π ( m )) > − L ρ ∈ [1 − ρ , ρ ] , where II ¯ λ is defined as in (54). Moreover, using for example the test map, one shows that for ρ sufficiently small there exists a large constant ¯ L such that(56) II ρ ≤ ¯ L, for every ρ ∈ [1 − ρ , ρ ] . We have the following result regarding the dependence in ρ of the minimax value II ρ . Lemma 4.12
Let ¯ λ and ρ such that (55) holds. Then the function ρ → II ρ ρ is non-increasing in [1 − ρ , − ρ ] Proof . For ρ ≥ ρ ′ , there holds II ρ ( u ) ρ − II ρ ′ ( u ) ρ ′ = (cid:18) ρ − ρ ′ (cid:19) (cid:10) P , g u, u (cid:11) Therefore it follows easily that also II ρ ρ − II ρ ′ ρ ′ ≤ , hence the Lemma is proved.From this Lemma it follows that the function ρ → II ρ ρ is a.e. differentiable in [1 − ρ , ρ ], and weobtain the following corollary. 21 orollary 4.13 Let ¯ λ and ρ be as in Lemma . , and let Λ ⊂ [1 − ρ , ρ ] be the (dense) set of ρ for which the function II ρ ρ is differentiable. Then for ρ ∈ Λ the functional II ρ possesses a boundedPalais-Smale sequence ( u l ) l at level II ρ . Proof . The existence of Palais-Smale sequence ( u l ) l at level II ρ follows from (55) and the boundedis proved exactly as in [15], Lemma 3.2.Next we state a Proposition saying that bounded Palais-Smale sequence of II ρ converges weakly (up toa subsequence) to a solution of the perturbed problem. The proof is the same as the one of Proposition5.5 in [17]. Proposition 4.14
Suppose ( u l ) l ⊂ H ∂∂n is a sequence for which II ρ ( u l ) → c ∈ R ; II ′ ρ [ u l ] → Z ∂M e u l dS g = 1 k u l k H ( M ) ≤ C. Then ( u l ) has a weak limit u (up to a subsequence) which satisfies the following equation: P g u + 2 ρQ g = 0 in M ; P g u + ρT g = ρκ ( P ,P ) e u on ∂M ; ∂u∂n g = 0 on ∂M. Now we are ready to make the proof of Theorem 1 . Proofof Theorem . ρ l → u l such that the following holds : P g u l + 2 ρ l Q g = 0 in M ; P g u l + ρ l T g = ρκ ( P ,P ) e u l ; on ∂M ; ∂u l ∂n g = 0 on ∂M. Now since κ ( P ,P ) = R M Q g dV g + R ∂M T g dS g then applying corollary 1 . Q l = ρ l Q g , T l = ρ l T g and¯ T l = ρ l κ ( P ,P ) we have that u l is bounded in C α for every α ∈ (0 , C ( M ) to a solution of (7). Hence Theorem 1 . Remark 4.1
As said in the introduction, we now discuss how to settle the general case.First of all, to deal with the remaining cases of situation 1 , we proceed as in [17]. To obtain Moser-Trudinger type inequality and its improvement we impose the additional condition k ˆ u k ≤ C where ˆ u isthe component of u in the direct sum of the negative eigenspaces. Furthermore another aspect has to beconsidered, that is not only e u can concentrate but also k ˆ u k can also tend to infinity. And to deal withthis we have to substitute the set ∂M k with an other one, A k, ¯ k which is defined in terms of the integer k (given in (9) ) and the number ¯ k of negative eigenvalues of P , g , as is done in [ ] . This also requiressuitable adaptation of the min-max scheme and of the monotonicity formula in Lemma . , which ingeneral becomes ρ → II ρ ρ − Cρ is non-increasing in [1 − ρ , ρ ]; for a fixed constant C > .As already mentioned in the introduction, see Remark, to treat the situation 1 , we only need to considerthe case ¯ k = 0 . In this case the same arguments as in [17] apply without any modifications. E-mail addresses: [email protected] eferences [1] Aubin T., Nonlinear Analysis on manifold, Monge-Ampere equations , Springer-Verlag, 1982.[2] Branson T.P.,
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