Lénaïc Bagnères
email: lenaic.bagneres@inria.fr

Cédric Bastoul
email: cedric.bastoul@unistra.fr

Switchable Scheduling for Runtime Adaptation of Optimization

Parallel applications used to be executed alone until their termination on partitions of supercomputers: a very static environment for very static applications. The recent shift to multicore architectures for desktop and embedded systems as well as the emergence of cloud computing is raising the problem of the impact of the execution context on performance. The number of criteria to take into account for that purpose is significant: architecture, system, workload, dynamic parameters, etc. Finding the best optimization for every context at compile time is clearly out of reach. Dynamic optimization is the natural solution, but it is often costly in execution time and may offset the optimization it is enabling. In this paper, we present a static-dynamic compiler optimization technique that generates loop-based programs with dynamic auto-tuning capabilities with very low overhead. Our strategy introduces switchable scheduling, a family of program transformations that allows to switch between optimized versions while always processing useful computation. We present both the technique to generate self-adaptive programs based on switchable scheduling and experimental evidence of their ability to sustain high-performance in a dynamic environment.

Introduction

Static compilers are facing the challenge of generating efficient codes for increasingly dynamic execution environments. Two decades ago, optimizing compilation was referred as building "supercompilers for supercomputers" [START_REF] Wolfe | High performance compilers for parallel computing[END_REF]. Compiler techniques had to optimize aggressively for complex parallel machines but in a very static context: usually one program with few dynamic parameters, one well defined architecture/system and one user. Iterative compilation and autotuning approaches have been developed on top of static compilation as efficient solutions to find the best optimization parameters and to adapt to various (but fixed) architectures and problem sizes [START_REF] Bodin | Iterative compilation in a non-linear optimisation space[END_REF][START_REF] Whaley | Automated empirical optimization of software and the ATLAS project[END_REF][START_REF] Pouchet | Combined iterative and model-driven optimization in an automatic parallelization framework[END_REF]. The large adoption of multicore systems and the emergence of cloud computing brings new dynamic factors that are not captured by iterative compilation or auto-tuning, such as the existence of competing workloads or the possible migration of the process to another architecture. This situation raises the need for more dynamic optimization schemes.

Just in time compilation is a convenient solution to address dynamic execution environments. However, it requires very low algorithmic complexity of the underlying techniques to avoid to offset the optimization it is enabling. Current state-of-the-art static automatic optimization and parallelization techniques rely on an algebraic representation of programs that allows precise analyses as well as very aggressive program transformations to optimize codes, known as the polyhedral model [START_REF] Feautrier | Some efficient solutions to the affine scheduling problem, part II: multidimensional time[END_REF][START_REF] Bondhugula | A practical automatic polyhedral parallelizer and locality optimizer[END_REF][START_REF] Pouchet | Iterative optimization in the polyhedral model: Part II, multidimensional time[END_REF]. Unfortunately, most polyhedral-based techniques show exponential complexity [START_REF] Upadrasta | Sub-polyhedral scheduling using (unit-)two-variableper-inequality polyhedra[END_REF]. Hence, they are challenging to include in a dynamic compilation framework, except when a runtime analysis allows to use this model while it was not possible at static compile time [START_REF] Jimborean | VMAD: an Advanced Dynamic Program Analysis & Instrumentation Framework[END_REF]. Our proposal is a mixed static-dynamic technique, which benefits from the power of polyhedral frameworks at static compile time, while being able to change the optimization decision at runtime during the computation itself.

The potential benefit of such a technique is significant because the dynamic nature of the execution environment comes from several factors that directly impact performance. First of all, a compiled program may be run on different architectures with different features such as various cache memories or number of cores, which have dramatic impact on the best optimization choice. A decision at the early stage of the execution is not enough: virtual machines and cloud computing technology allow the architecture to change during execution. Next, the application may depend on dynamic parameters such as problem size (e.g., array size). Hence the best optimization is likely to be different depending on those parameters that will be known only at runtime. Finally, the operating system and the system workload are also paramount because processes may affect each other, e.g., through cache pollution or by stressing the system scheduler.

Our approach is to design at compile-time programs that can adapt at runtime to the execution context. The originality of our solution is to rely on switchable scheduling, a selected set of program restructuring which allows to swap between program versions at some meeting points without any rollback. A first step selects pertinent switchable versions according to their performance behavior on some execution contexts. The second step builds a self-adaptive program including selected versions. Then at runtime the program keeps choosing the best version thanks to a low overhead sampling and profiling of the versions, ensuring during the process that every computation contributes to the final result. We performed an experimental study on dozens of execution contexts and demonstrate superior adaptability of our generated codes with respect to stateof-the-art static optimization technique.

Background

The application domain of our technique is loop-based kernels with affine control and memory accesses, i.e., such that loop bounds, conditions and array subscripts are affine forms of outer loop counters and constant parameters. This class of computational kernels is known as SCoPs for Static Control Parts. SCoPs can be modeled using an algebraic representation called the polyhedral model. Because of the restriction on the input program form, each dynamic instance of a given SCoP statement can be modeled as an integer point in a union of polyhedra called the iteration domain of that statement. For example, let us consider the input code in Figure 1(a). Figure 1(b) shows the iteration domain of the statement S(i,j). Each loop enclosing the statement in the code corresponds to a dimension of the domain. Several compilers have the ability to raise SCoPs to a polyhedral form such as GNU GCC3 and LLVM 4 .

Once a SCoP is raised to the polyhedral model, an optimizer can compute a scheduling by means of scheduling relations that express logical execution dates for all statement instances, e.g., to achieve data locality or to expose parallelism while satisfying data dependences. In the following, we will only consider scheduling that does not alter the original program semantics. Figures 1(c1) and 1(c2) show two different possible scheduling relations. They map original input dimensions, which express original statement instances, to target output dimensions, which express their new order. Scheduling relations are expressive enough to encode a complex composition of program transformations (including, e.g., loop interchange, fusion, fission, skewing, tiling etc.) [START_REF] Girbal | Semi-automatic composition of loop transformations for deep parallelism and memory hierarchies[END_REF]. Those in Figures 1(c1) and 1(c2) correspond respectively to the identity transformation and to the reversal of the inner loop. Many efficient scheduling algorithms have been designed, notably the Pluto algorithm for automatic optimization and parallelization [START_REF] Bondhugula | A practical automatic polyhedral parallelizer and locality optimizer[END_REF] and the Letsee technique based on iterative optimization [START_REF] Pouchet | Iterative optimization in the polyhedral model: Part II, multidimensional time[END_REF].

Finally a code generator for scanning polyhedra such as CLooG [START_REF] Bastoul | Code generation in the polyhedral model is easier than you think[END_REF] can produce a syntactic program that implements the new scheduling from the iteration domains and the scheduling relations. Figures 1(d1) and 1(d2) present the programs generated back from the corresponding polyhedral representations after the code generation step. The complete Figure 1 summarizes the usual workflow of a polyhedral framework with two different scheduling relations that result in two versions of the input program. Most previous works aim at finding only one good version. Our work improves this scheme with dynamic capablities, to be able to chose the right version for the right execution context.

Switchable Scheduling

In a polyhedral compilation framework, a program version is generated from the input program information and a scheduling. The scheduling is in turn expressed as a list of scheduling relations, one for each statement. In this work, we focus on particular sets of scheduling called switchable scheduling. Two scheduling are switchable if and only if there exist meeting points in the corresponding generated versions such that it is possible to continue the execution from any of these versions at those meeting points without affecting the program result. Translated to the polyhedral model terminology, it means that there must exist a couple of logical dates called switching dates, one for each scheduling, such that the sets of instances that have been scheduled prior to these dates in each version is the same, regardless of their respective order. To simplify their computation, for (i = 0; i <= N; i++) for (j = 0; j <= N; j++) S(i, j);

(a) Input Code Raising DS(N ) = i j 0 ≤ i ≤ N 0 ≤ j ≤ N (b) Iteration Domain Scheduling 1 Scheduling 2 θS(N ) = i j → t1 t2 t1 = i t2 = j θS(N ) = i j → t1 t2 t1 = i t2 = -j (c1) Scheduling Relation 1 (c2) Scheduling Relation 2
Fig. 1: Polyhedral Transformation Workflow For Two Example Versions and without loss of generality, we require that switching dates correspond to existing instance schedules. The set of switching dates for a scheduling θ to a scheduling θ ′ is called its switching domain to θ ′ .

Property 1. To a given switching date in a scheduling there may exist only a unique corresponding switching date in another scheduling.

Explanation. Each instance of the original program has a unique image in the target program. Hence, given a set of already executed instances before a meeting point in a version, the corresponding meeting point in another version, if it exists, is the unique instance that will be executed directly after that set. ⊓ ⊔ Property 2. If the outermost dimensions of two scheduling are mapping input dimensions in the same order, then the first instance scheduled at any value of these outermost dimension belongs to the switching domain of the corresponding scheduling to the other scheduling.

Explanation. Logical dates are multidimensional like clocks: the first dimension may correspond to days (most significant) then the next one to hours (less significant), then the next one to minutes and so on. To each value of the outermost scheduling dimensions corresponds a set of scheduled instances. If the execution order of such sets is the same in any version, then at the beginning of each set it is possible to switch between versions, regardless of the scheduling order inside the set, i.e., of less significant scheduling dimensions.

⊓ ⊔

From these two properties we derive a practical technique to build a multiversion code. First for each version we compute a switching domain, as detailed in Section 3.1. Next we generate the code itself, inserting switching statements for each integer point of the switching domains, as explained in Section 3.2. Switching statements themselves rely on a low overhead runtime system described in Section 3.3.

Switching Domain Computation

We derive from Property 2 that a (subset of) the switching domain is the set of output vectors such that:

1. The outermost "common" output dimensions are expressed in the same way for every scheduling (this ensures that all versions are executing equivalent subsets of instances in the same order regardless of the order inside those subsets). This condition may be relaxed when information about the scheduling semantics is available. The most important case we are supporting is stripmining and, by extension, tiling, with a restriction on possible tile sizes. Tile sizes are chosen to be a multiple of the smallest tile size. Hence, we know statically that, e.g., an iteration at a given dimension in one version corresponds to n iterations of the same dimension in another version. We derive from this a simple affine constraint on the existence of meeting points. 2. The remaining output dimensions are set to the lexicographic minimum of the possible values (to ensure the logical date of the switching statement is at most the same as the first instance scheduled inside the subset). Moreover, we add another output dimension set to 0 to ensure the switching statement is executed before the first instance of the subset.

Switching domains are easy to compute from the scheduling using the PIP tool [START_REF] Feautrier | Parametric integer programming[END_REF] to compute the lexicographic minimum of the innermost output dimensions. Figures 2(d1) and 2(d2) show the switching domains corresponding to the scheduling in Figures 1(c1) and 1(c2): the first dimension has the same expression in both scheduling and has the same range, the second one is set to the minimum value for each version, and a new one has been added and set to 0.

The code generation step detailed in Section 3.2 uses switching domains to insert "switching statements" in the final code: to each integer point in this domain will correspond an execution of the switching statement. It is not desirable to execute the switching statement at each meeting point because of the overhead it may introduce. Switching domains can be easily restricted to fit the need. A first solution is to intersect it with a convenient lattice. In this way, switching statements will be executed at constant intervals along scheduling dimensions. A second solution with the same effect is to apply a special strip-mine onto some scheduling dimensions. In this case, selected scheduling dimensions are decoupled into three dimensions in the switching domains and the scheduling relations. The outer dimension iterates over strips, the middle one is set to 0 for the switching domain and to 1 for all the scheduling relations, and the inner one is set to 0 for the switching domain and iterates over integer points inside strips for the scheduling relations. This does not affect the order of the instances, but it inserts a switching date before each strip. While the first solution is simpler, the second one allows to consider switching along parallel dimensions: the dimension over strips has to be sequential, but the one over points inside strips may be parallel.

Multi-Version Code generation

Generating a code that includes multiple versions of the original program with the ability of switching between them is a three step process. First we extend the original scheduling with one innermost output dimension set to 1. It ensures that the switching statement will be executed before any existing instance if they are scheduled at the same logical date, since that output dimension has been set to 0 for the switching domain 5 . Figures 2(e1) and 2(e2) show the extended schedulings of Figures 1(c1) and 1(c2). Next, we generate the code from the original domains and scheduling as in a classical polyhedral framework, with the CLooG tool [START_REF] Bastoul | Code generation in the polyhedral model is easier than you think[END_REF]. The only difference is that we generate a code for each version and that we add the corresponding switching domain to each code generation problem. Each integer point of the switching domain corresponds to an execution of the switching statement. Finally some glue code is added to support switching: additional variables are created to communicate current common output coordinates while switching and labels/gotos are inserted to jump to the end of the code once one version terminates.

The switching statement itself is made of two parts. First, the switching source includes calling the runtime to decide about switching or not, communicating of current common output coordinates and actual switching (through goto statements). Second, the switching sink includes a label to be used as the target of a switch, receiving the common output coordinates and setting back the remaining output coordinates to the lexicographic minimum. Figure 2(f) shows the final code (spanning two columns) for our running example started in Figure 1. The switching source corresponds to the if part of the switching statement while the sink corresponds to the else part.

Runtime

The runtime switching decision system is as simple as possible to minimize the overhead. It is based only on the execution time and has two modes called

Switching Domain

Switching Domain

Dsw(N ) = t1 t2 t3 0 ≤ t1 ≤ N t2 = 0 t3 = 0 D ′ sw (N ) = t1 t2 t3 0 ≤ t1 ≤ N t2 = -N t3 = 0 (d1) Switching Domain 1 (d2) Switching Domain 2
Extended Scheduling Extended Scheduling watching and sampling. In watching mode, the runtime simply checks that the performance is stable by measuring the time spent between two calls. Since switching statements are inserted at constant strides along output dimensions and SCoP execution time is typically not affected by data values, this measure is precise enough for our purpose. If it is the first call to the runtime or if the watching mode detected a performance variation, due to, e.g., changes on the execution context or on the workload executed between two calls to the runtime, the sampling mode is enabled. This mode switches quickly between versions to detect the best performing one. Then a switch is performed to that version while the runtime is set back to the watching mode. A very important property of this strategy is that every computation contributes to the final result: no rollback is necessary if a bad optimization decision has been made.

θ × S (N ) = i j → t1 t2 t3 t1 = i t2 = j t3 = 1 θ ′× S (N ) = i j → t1 t2 t3 t1 = i t2 = -j t3 = 1 (

Selecting Pertinent Versions

A key aspect of our optimization strategy is the selection and the ordering of the switchable versions to be part of the multi-version code. For this purpose we rely on a dedicated version generation phase and on an extensive empirical study of the version behavior.

To generate versions, we rely on the polyhedral compiler PoCC6 which uses both the Pluto algorithm [START_REF] Bondhugula | A practical automatic polyhedral parallelizer and locality optimizer[END_REF] and the Letsee iterative optimization engine [START_REF] Pouchet | Iterative optimization in the polyhedral model: Part II, multidimensional time[END_REF] to compute efficient scheduling. Generating switchable versions is done by enforcing additional constraints discussed in Section 3.1: from a base version, other versions are generated by calling Letsee or Pluto with different strategies and/or tile sizes, such that they share common output dimensions. Different scheduling may often end up to the same executable code (a shifting on an output dimension may be removed by a loop normalization by the compiler). Such versions are discarded.

Once a set of versions has been generated for a given input code, they are evaluated separately by running them on pre-defined contexts. Contexts include various architectures, data sizes and system workloads. One context is a combination of these factors. Only the versions that are the best in at least one context are considered to be selected. Our results show that they are still too many. Some of them are performing the same way in several contexts: those duplicates are detected and discarded (in our study, we accept a performance loss of 10%). Finally to select a pre-defined maximum number of versions (in our study, 8), we associate an "efficiency" coefficient to each version on each context (depending on how far it is from the best version) and we model and solve the choice as a linear optimization problem to maximize the overall efficiency.

The order in which the selected versions are used during sampling by the runtime described in Section 3.3 is critical: small loops are likely to be entirely executed before the sampling is done. For this reason, best performing versions in most contexts including small problem sizes are used for sampling first.

Experimental Results

We evaluate the switchable scheduling approach on a selection of realistic execution contexts. Experimental results demonstrate the ability of this technique to generate programs that can adapt themselves to their environment. Overall, its geomean speedup over a fixed optimization of a state-of-the-art automatic optimization and parallelization is 1.49 for our test cases.

Our experimental setup is three-dimensional. First, target architectures includes one ARM and several flavours of Intel x86 architectures: Olimex A20 ARM Cortex-A7 dual-core, Intel Core2 Quad CPU Q9550 2.83GHz, Intel Core2 Quad CPU Q6600 2.40GHz and Intel Core2 Quad CPU Q8200 2.33GHz. This selection notably spans different number of cores and cache sizes. Next, problem size ranges are small and medium as they are defined in the target benchmarks. Lastly, 5 workloads have been investigated: the target process may be running alone, with low (one process) or high (one process per core) computation intensive workload and with low or high memory access intensive workload.

We consider 12 benchmarks, typical compute-intensive kernels extracted from the PolyBench suite7 . Our selection focuses on kernels including one main loop since it is the main target of our technique. We report below for all benchmarks a short description. Column #versions gives the number of different versions that have been generated using PoCC (duplicates have been removed); #best reports the number of best versions reported in the 40 contexts; and #nodup removes from the previous column the versions that behave in the same way as another one if we accept up to a 10% performance loss. It illustrates that the best version is indeed dependant on the execution context, but also that a limited number of versions is enough most of the time, hence with a reasonable impact on the generated code size. Figure 3 reports normalized mean performance for all execution contexts for each benchmark, worst corresponds to the worse (context-wise) version, baseline is the mean of all versions, roughly corresponding to the average performance a random strategy is likely to provide, best corresponds to the best (context-wise) solution, pluto is the default static Pluto (version 0.10) solution and switchable is the switchable scheduling solution. Overall, the difference between baseline and best with geomean 4.98 is the maximum speedup of the solution, it corresponds to an iterative compilation strategy, a high potential already demonstrated by previous work [START_REF] Pouchet | Combined iterative and model-driven optimization in an automatic parallelization framework[END_REF]. switchable corresponds to our solution with an overall geomean speedup of 4.36 against a random strategy, including a sensible yet acceptable overhead of the switching strategy, and of 1.49 over the default Pluto solution. size growth shows the compiled switchable scheduling kernel size growth with respect to Pluto's solution, a limited increase. Sampling on bad versions may degrade performance significantly (e.g., gemm case). Also in jacobi-1d case, our strategy has lower performance than Pluto. This corresponds to situations where Pluto's solution is good enough while the overhead of switchable scheduling overcomes its benefits. We may complement our technique with a dynamic test as Pradelle et al. suggested [START_REF] Pradelle | Adaptive Runtime Selection of Parallel Schedules in the Polytope Model[END_REF] to prevent using switchable scheduling in such situation. 6 Related work

The root of our work belongs to compiler optimization in the polyhedral model [START_REF] Feautrier | Some efficient solutions to the affine scheduling problem, part II: multidimensional time[END_REF] and loop versioning [START_REF] Byler | Multiple version loops[END_REF]. The Pluto algorithm is a state-of-the-art compiler technique relying on the polyhedral model to build complex loop transformations with excellent parallelism-locality trade-offs using a target independent cost model [START_REF] Bondhugula | A practical automatic polyhedral parallelizer and locality optimizer[END_REF]. It has been coupled with iterative frameworks to optimize for specific targets [START_REF] Pouchet | Combined iterative and model-driven optimization in an automatic parallelization framework[END_REF]. Those techniques create unspecialized or overspecialized optimization which may not be adequate for various execution contexts. Static compiler techniques have been used to help runtime systems to optimize dynamically. The ADAPT framework provides runtime generation and specialization of code sections [START_REF] Voss | ADAPT: Automated de-coupled adaptive program transformation[END_REF]. Because of the runtime overhead it fits well to programs with large execution time while we are using static techniques as much as possible to minimize runtime costs. Qilin provides adaptive mapping for parallel programs [START_REF] Luk | Qilin: Exploiting parallelism on heterogeneous multiprocessors with adaptive mapping[END_REF]. Unlike our method, it is not addressing the dynamic workload dimension of the execution context. Emani et al. proposed an adaptive mapping technique which primarily targets dynamic workload variations [START_REF] Emani | Smart, adaptive mapping of parallelism in the presence of external workload[END_REF]. It impacts the OpenMP runtime behaviour whereas we target code restructuring.

Aggressive dynamic optimization techniques include thread-level speculation [START_REF] Rauchwerger | The LRPD test: speculative run-time parallelization of loops with privatization and reduction parallelization[END_REF][START_REF] Steffan | The stampede approach to thread-level speculation[END_REF]. They generate an optimistically optimized version and in case of mistake, they rollback to a conservative version. In comparison, we target a different program class that can be analyzed precisely at compile time, and in case of a bad choice, no rolling back is necessary since every computation is useful by construction. Dynamic optimization involving polyhedral compilation is emerging. EvolveTile is a framework to perform a dynamic tile size selection [START_REF] Tavarageri | Dynamic selection of tile sizes[END_REF]. Our approach also supports such optimization but with more restrictions on tile sizes and shapes because of the switchable scheduling class constraints. However, our technique supports a wider range of optimizations. Pradelle et al. target the same program class as our technique and involve versioning as well [START_REF] Pradelle | Adaptive Runtime Selection of Parallel Schedules in the Polytope Model[END_REF]. Their approach is to use profiling to build predictive tests according to dynamic factors to choose the best version of a kernel before executing it. Our approach is acting at a finer grain as we focus on switching from kernel versions during computation. VMAD is an infrastructure for dynamic profiling with the unique ability to discover static behavior, which is not visible at static compilation time [START_REF] Jimborean | VMAD: an Advanced Dynamic Program Analysis & Instrumentation Framework[END_REF]. VMAD supports dynamic version selection. Some forms of switchable scheduling are possible within this framework and are under investigation.

Conclusion

This paper addresses the problem of taking advantage of the best optimization while computing in an ever more dynamic environment, focusing on static control loop nests. Our proposal differs from just-in-time compilation approaches which have to rely on low-overhead techniques as well as static compilation approaches that generate a code which can be either too generic or too specialized. Instead, we propose a mixed static-dynamic scheme which builds on state-ofthe-art static polyhedral compilation techniques with empirical study to select pertinent optimizations and a low-overhead runtime mechanism to switch to the best optimization during computation, depending on the current execution context. Our technique introduces a special class of optimization called switchable scheduling and a code generation method to build a program that takes advantage of multiple such optimizations. Experimental evidence demonstrate both the potential of this approach and its effectiveness at generating codes that perform well on various environments.

Ongoing work includes a code generation technique to allow versions to lie inside their own functions, to benefit from per-version low-level compiler optimization options. More aggressive versioning and switchable-scheduling generation under time constraint are also under investigation.

Fig. 2 :

 2 Fig. 2: (Our Alternative End of Fig. 1) Generation of a Multi-Version Code

Fig. 3 :

 3 Fig.3: Potential and Operational Performance Results (mean of all contexts, the baseline is the mean performance of all versions in all contexts)

http://gcc.gnu.org/wiki/Graphite

http://polly.llvm.org

If the last output dimension is not a common dimension, another solution without scheduling extension is to subtract 1 to its expression in the switching domain.

http://pocc.sf.net

http://polybench.sf.net

Code Generation

Code Generation for (t1 = 0; t1 <= N; t1++) for (t2 = 0; t2 <= N; t2++) i = t1; j = t2; S(i, j);

for (t1 = 0; t1 <= N; t1++) for (t2 = -N; t2 <= 0; t2++) i = t1; j = -t2; S(i, j);

(d1) Output Code 1 (d2) Output Code 2


F R Éd

Éric Robert

J Ér Ôme

V Étois

Jérôme Vétois
email: vetois@unice.fr

SIGN-CHANGING SOLUTIONS TO ELLIPTIC SECOND ORDER EQUATIONS: GLUEING A PEAK TO A DEGENERATE CRITICAL MANIFOLD

We construct blowing-up sign-changing solutions to some nonlinear critical equations by glueing a standard bubble to a degenerate function. We develop a new method based on analyticity to perform the glueing when the critical manifold of solutions is degenerate and no Bianchi-Egnell type condition holds.

Introduction and statement of the results

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3, and let h ∈ C 0,θ (M ) (θ ∈ (0, 1)) be such that ∆ g +h is coercive where ∆ g = -div g (∇) is the Laplace-Beltrami operator. In [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF], we addressed the question of the existence of a family (u ε ) ε>0 ∈ C 2,θ (M ) of blowing-up solutions of type (u 0 -B) to [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] ∆

g u ε + hu ε = |u ε | 2 ⋆ -2-ε u ε in M,
where 2 ⋆ := 2n n-2 . Concerning terminology, we say that (u ε ) ε is of type (u 0 -B) when there exists a function u 0 ∈ C 2,θ (M ) positive that is a solution to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF] ∆ g u 0 + hu 0 = u 2 ⋆ -1 0 in M and such that

u ε = u 0 -B ε + o(1),
where (B ε ) ε is a bubble as defined in [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R N with critical growth[END_REF] below and lim ε→0 o(1) = 0 in H 2 1 (M ), the completion of C ∞ (M ) for the norm u → u H 2 1 := u 2 + ∇u 2 . Solutions of type (u 0 -B) are sign-changing. When h ≡ c n R g , where c n := n-2 4(n-1) and R g is the scalar curvature, equation ( 2) is the Yamabe equation, and ∆ g + h is coercive if and only if (M, g) has positive Yamabe invariant. There is an extensive literature on the existence of positive blowing-up solutions to equations of type [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF]: see for instance Rey [START_REF] Rey | The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF] for a historical reference, Brendle-Marques [START_REF] Brendle | Blow-up phenomena for the Yamabe equation. II[END_REF] for the Yamabe equation, Druet-Hebey [START_REF] Druet | Blow-up examples for second order elliptic PDEs of critical Sobolev growth[END_REF] and Esposito-Pistoia-Vétois [START_REF] Esposito | The effect of linear perturbations on the Yamabe problem[END_REF] for perturbations of the Yamabe equation, Chen-Wei-Yan [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R N with critical growth[END_REF] and Hebey-Wei [START_REF] Hebey | Resonant states for the static Klein-Gordon-Maxwell-Proca system[END_REF] for equations on the sphere, and the references therein. Sign-changing blowing-up solutions to [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] on the canonical sphere have been constructed by del Pino-Musso-Pacard-Pistoia [START_REF] Del Pino | Large energy entire solutions for the Yamabe equation[END_REF][START_REF]Torus action on S n and sign-changing solutions for conformally invariant equations[END_REF] and Pistoia-Vétois [START_REF] Pistoia | Sign-changing bubble towers for asymptotically critical elliptic equations on Riemannian manifolds[END_REF]. We refer to Robert-Vétois [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF] for a discussion and references on the compactness of solutions to [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF].

Date: September 8th, 2014. The research of the first author is supported by INSMI (CNRS).

In [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF], we gave sufficient conditions to get blowing-up solutions of type (u 0 -B) to [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] provided that u 0 is a nondegenerate solution to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF], that is K 0 = {0} where (3)

K 0 := {ϕ ∈ C 2,θ (M )/ ∆ g ϕ + hϕ = (2 ⋆ -1)u 2 ⋆ -2 0 ϕ in M }.
When u 0 is degenerate, the situation can be different. In [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF], we showed that there is no blowing-up solutions of type (u 0 -B) to the constant scalar curvature equation on the canonical sphere: in this case, u 0 is necessarily degenerate.

The present article is devoted to the analysis of the degenerate case, that is when K 0 = {0}. We say that u 0 ∈ C 2,θ (M ) \ {0} is a strict local minimizer of I 0 if there exists ν > 0 such that I 0 (u) > I 0 (u 0 ) for all u ∈ B ν (u 0 ) \ Ru 0 , where

I 0 (u) := M |∇u| 2 g + hu 2 dv g M |u| 2 ⋆ dv g 2 2 ⋆
for all u ∈ H 2 1 (M ) \ {0}. Our main result is the following:

Theorem 1.1. We let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 with positive Yamabe invariant and we fix h ≡ n-2 4(n-1) R g . We assume that there exists u 0 ∈ C 2,θ (M ) that is a positive solution to (2) and a strict local minimizer of I 0 . We assume either that {3 ≤ n ≤ 9} or that {(M, g) is locally conformally flat}. Then there exists a solution of type (u 0 -B) to [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF].

It follows from the compactness results of Schoen [START_REF] Schoen | On the number of constant scalar curvature metrics in a conformal class, Differential geometry[END_REF] and Khuri-Marques-Schoen [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF] (see also Druet [START_REF] Druet | Compactness for Yamabe metrics in low dimensions[END_REF]) that blowing-up solutions to (1) must change sign under the assumptions of Theorem 1.1.

As a remark, any nondegenerate local minimizer of I 0 is a strict local minimizer, so we recover the main theorem of [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF]. Moreover no solution of the Yamabe equation on the sphere is a strict local minimizer. However, as soon as one takes the product of a sphere with another manifold, one gets examples of degenerate strict local minimizers. We refer to Section 7 for such examples, in particular to Corollary 7.1.

We prove Theorem 1.1 by performing a finite-dimensional reduction modeled on (u -B) where B is a bubble and u ∈ M, and where M is a suitable finitedimensional analytic manifold containing u 0 . We construct the manifold M such that its elements are as close as possible to solutions of (2): this is done using a first finite-dimensional reduction. The manifold M is locally parametrized by K 0 , and the tangent space of M at u 0 is K 0 . The general construction in Robert-Vétois [START_REF]A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations via Lyapunov-Schmidt's reduction[END_REF] reduces the proof of Theorem 1.1 to finding stable critical points to a functional that is the sum of two terms: the first is an explicit local well involving essentially the bubble, the second is the restriction to M of a nontrivial global functional J 0 . Solutions to (2) around u 0 are all in M. However, in general, the elements of M are not all solutions to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF], that is M is not a critical manifold of the problem. Following the terminology of Chapter 2 of the monograph Ambrosetti-Malchiodi [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF], a critical manifold around u 0 is a finite-dimensional manifold Z ∋ u 0 of solutions to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF]. A critical manifold Z is nondegenerate if its tangent space at any u ∈ Z is exactly Ker(I ′′ 0 (u)), the kernel of the Hessian of I 0 at u. The existence of a nondegenerate critical manifold around u 0 is equivalent to the existence of ũ ∈

C 1 (B 1 (0) ⊂ K 0 , H 2 1 (M )) such that    ũ(z) is a solution to (2) for all z ∈ B 1 (0) ⊂ K 0 , ũ(0) = u 0 , K 0 = Span{∂ zi ũ(0)/ i = 1, ..., d}, where d := dim(K 0 ).    (BE)
Condition (BE) (for Bianchi-Egnell type condition) is a standard and natural assumption in the finite-dimensional reduction. It is satisfied when M = R n and h ≡ 0 (see the classical references Rey [START_REF] Rey | The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF] and Bianchi-Egnell [START_REF] Bianchi | A note on Sobolev inequality[END_REF]), also for some sign-changing solutions (see the recent example of Musso-Wei [START_REF] Musso | Nondegeneracy of nonradial nodal solutions to Yamabe problem[END_REF]). We refer to Ambrosetti-Malchiodi [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] for an abstract general setting for the use of nondegenerate critical manifolds.

In case condition (BE) holds, the manifold M is the nondegenerate critical manifold, and minimizing J 0|M exactly amounts to minimizing I 0|M , which is a considerable simplification for our problem. However, in general, the Bianchi-Egnell condition (BE) does not hold. It is even exceptional: in Section 7, we exhibit examples of degenerate minimizers u 0 that are isolated among solutions to (2), and therefore, the only possible critical manifold is {u 0 } and is degenerate (see Propositions 7.1 and 7.3). Therefore, the classical methods using nondegenerate critical manifolds (see again the monograph Ambrosetti-Malchiodi [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF]) are ineffective here. We refer to Del Pino-Felmer [START_REF] Del Pino | Local mountain passes for semilinear elliptic problems in unbounded domains[END_REF], Jeanjean-Tanaka [START_REF] Jeanjean | Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities[END_REF], Byeon-Jeanjean [START_REF] Byeon | Standing waves for nonlinear Schrodinger equations with a general nonlinearity[END_REF], and Dancer [START_REF]Peak solutions without non-degeneracy conditions[END_REF] for an analysis on R n without condition (BE) based on topological arguments.

Our aim in the present article is to develop a new method to deal with the absence of nondegenerate critical manifold (that is when the Bianchi-Egnell condition (BE) does not hold) by using analyticity. Indeed, due to our choice of the manifold M, we are able to compare precisely all the terms in the analytic expansions of I 0 and J 0 on M. As a consequence, we prove that the restriction of J 0 to M has a strict local minimum at u 0 if and only if u 0 is a strict local minimizer of I 0 (Theorem 6.1). This allows us to get a stable critical point for our problem. This article is organized as follows. In Section 2, we state byproducts of our analysis. In Section 3, we define bubbles, we state the general construction theorem via finite-dimensional reduction and we recall existing results. In Section 4, we perform a first Lyapunov-Schmidt reduction to construct the analytic manifold M of approximations of u 0 . In Section 5, we reduce the proof of Theorem 1.1 to obtaining a stable well for J 0 restricted to M. In Section 6, we use the analyticity to prove the equivalence of strict local minimization for I 0 and J 0 on M. In Section 7, we construct examples of degenerate strict local minimizers.

Acknowledgement: the authors thank the referee for careful reading of this manuscript and useful remarks.

Miscellaneous further results

Theorem 1.1 is a particular case of Theorem 2.1 below: Theorem 2.1. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Let h ∈ C 0,θ (M ) be such that ∆ g + h is coercive. Assume that there exists u 0 ∈ C 2,θ (M ) that is a solution to (2) and a strict local minimizer of I 0 . Assume that one of the following situations holds:

(4)            3 ≤ n ≤ 5, n = 6 and c n R g -h < 2u 0 , 3 ≤ n ≤ 9 and h ≡ c n R g , n = 10 , h ≡ c n R g and u 0 > 5 567 |Weyl g | 2 g , n ≥ 3, (M, g) is locally conformally flat and h ≡ c n R g .           
Then there exist a solution of type (u 0 -B) to (1).

We are also in position to construct positive solutions in dimension n = 6.

Theorem 2.2. Let (M, g) be a smooth compact Riemannian manifold of dimension n = 6 and let h ∈ C 0,θ (M ) be such that ∆ g + h is coercive. Assume that there exists u 0 ∈ C 2,θ (M ) that is both a solution to (2) and an strict local minimizer of I 0 . Assume that

(5) h -c 6 R g > 2u 0 > 0 in M.
Then for ε > 0 small, equation (1) admits a solution

u ε > 0 such that u ε = u 0 + B ε + o(1)
, where (B ε ) ε is a bubble and lim ε→0 o(1) = 0 in H 2 1 (M ).

Bubbles, general existence theorem and preliminary computations

This section essentially collects existing results from Robert-Vétois [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF][START_REF]A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations via Lyapunov-Schmidt's reduction[END_REF].

3.1. Bubbles. We follow the terminology in [START_REF]A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations via Lyapunov-Schmidt's reduction[END_REF]. We say that (B ε ) ε is a bubble if there exists (x ε ) ε ∈ M and (µ ε ) ε ∈ (0, +∞) such that lim ε→0 µ ε = 0 and (6)

B ε (x) := n(n -2)µ ε µ 2 ε + d g (x, x ε ) 2 n-2 2
for all x ∈ M.

There exists r 0 ∈ (0, i g (M )) and Λ ∈ C ∞ (M × M ) such that (ξ, x) → Λ ξ (x) > 0, Λ ξ (ξ) = 1 and :

(i) If (M, g) is locally conformally flat (lcf), then g ξ = Λ 4/(n-2) ξ g is flat in B ξ (r 0 ). (ii) If (M, g) is not locally conformally flat (non lcf) then g ξ := Λ 4 n-2 ξ g satisfies dv g ξ = (1+O(d g ξ (ξ, •) n )) dx in a geodesic normal chart. An immediate con- sequence is that R g ξ (ξ) = |∇R g ξ (ξ)| g ξ = 0 and ∆ g ξ R g ξ (ξ) = 1 6 |Weyl g (ξ)| 2 g . Moreover, ∇Λ ξ (ξ) = 0.
This change of metric is due to Lee-Parker [START_REF] Lee | The Yamabe problem[END_REF]. We let χ be a smooth cutoff function such that 0 ≤ χ ≤ 1 in R, χ = 1 in [-r 0 /2, r 0 /2], and χ = 0 in R\(-r 0 , r 0 ). For any κ ∈ {-1, 1}, any positive real number δ and any point ξ in M , we define the function W κ,δ,ξ on M by

W κ,δ,ξ (x) := κχ(d g ξ (x, ξ))Λ ξ (x) n(n -2)δ δ 2 + d g ξ (x, ξ) 2 n-2 2
, where d g ξ is the geodesic distance on M associated with the metric g ξ , the exponential map is taken with respect to the same metric g ξ . As one checks, for any family (δ ε ) ε ∈ (0, +∞) going to 0 as ε → 0, there exists a bubble (B ε ) ε such that

(7) W κ,δε,ξε = κB ε + o(1)
in H 2 1 (M ) when ε → 0. Bubbles like W κ,δ,ξ with a modification of the metric were introduced by Lee-Parker for an alternate resolution of the Yamabe problem. Using these bubbles smoothly depending on ξ for finite dimensional reduction was first used in the article [START_REF] Esposito | The effect of linear perturbations on the Yamabe problem[END_REF] by the second author and his collaborators.

Notations: Here and in the sequel, (∆ g + h) -1 denotes the inverse of the natural isometric isomorphism

∆ g + h : H 2 1 (M ) → (H 2 1 (M )) ′ φ → τ → M ((∇φ, ∇τ ) g + hφτ ) dv g . Any function f ∈ L 2n n+2 (M ) = (L 2 ⋆ (M ))
′ is seen as a linear form on H 2 1 (M ). In the sequel C will denote a constant independent of ξ, δ, ϕ, ε. The value of C can change from one line to the other for simplicity.

3.2. General existence theorem. For any ν 0 > 0 and ε > 0, we define

D ε (ν 0 ) := {(δ, ξ) ∈ (0, ν 0 ) × M / |δ ε -1| < ν 0 }. We define for ǫ ∈ [0, 2 ⋆ -2) J ε (u) := 1 2 M |∇u| 2 g + hu 2 dv g - 1 2 ⋆ -ε M |u| 2 ⋆ -ε dv g = 1 2 u 2 h -F ε (u) for all u ∈ H 2 1 (M )
, where

u 2 h = (u, u) h = M |∇u| 2 g + hu 2 dv g and F ε (u) := 1 2 ⋆ -ε M H(u) 2 ⋆ -ε dv g .
Here,

H(u) := |u| if κ = -1 and H(u) := u + if κ = 1.
For any closed subspace L ⊂ H 2 1 (M ), Π L will denote the orthogonal projection onto L and L ⊥ the orthogonal complement of L with respect to the Hilbert structure (•,

•) h . We let u ∈ C 1 (B ν0 (0) ⊂ K 0 , H 2 1 (M )) be such that u(0) = u 0 and (8) |det(Π K0 ∂ 1 u(ϕ), • • • , Π K0 ∂ d u(ϕ))| ≥ c 0 d i=1 ∂ i u(ϕ) H 2 1
for some c 0 > 0 and all ϕ ∈ B ν0 (0) ⊂ K 0 . Here, d := dim R (K 0 ) and derivatives refer to a fixed basis of K 0 . The following existence theorem is a consequence of Theorem 1.1 in Robert-Vétois [START_REF]A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations via Lyapunov-Schmidt's reduction[END_REF]:

Theorem 3.1. There exists ν 0 > 0 and there exists

φ ε ∈ C 1 (B ν0 (0) × D ε (ν 0 ), K ⊥ 0 ) such that for all ϕ ∈ B ν0 (0) ⊂ K 0 , (δ, ξ) ∈ D ε (ν 0 ), the function u ε (ϕ, δ, ξ) := u(ϕ) + W κ,δ,ξ + φ ε (ϕ, δ, ξ) is a critical point for J ε if and only if (ϕ, δ, ξ) is a critical point of (ϕ, δ, ξ) → J ε (u ε (ϕ, δ, ξ)). Moreover, φ ε (ϕ, δ, ξ) h ≤ C • R ε (ϕ, δ, ξ) where (9) R ε (ϕ, δ, ξ) := Π K ⊥ δ,ξ u(ϕ) + W κ,δ,ξ -(∆ g + h) -1 (F ′ ε (u(ϕ) + W κ,δ,ξ )) h . The space K δ,ξ is defined below. The projection onto K ⊥ δ,ξ in the rest R ε (ϕ, δ, ξ) follows from Subsection 5.3 in [25]. The function φ ε is defined implicitely as follows: given (ϕ, δ, ξ) ∈ B ν0 (0) × D ε (ν 0 ), φ ε (ϕ, δ, ξ) is the sole element of K ⊥ δ,ξ such that Π K ⊥ δ,ξ u ε (ϕ, δ, ξ) -(∆ g + h) -1 (F ′ ε (u ε (ϕ, δ, ξ)) = 0 . The linear space K δ,ξ is defined as K δ,ξ := Span {ϕ , Z δ,ξ , Z δ,ξ,X , ϕ ∈ K 0 and X ∈ T ξ M } , where Z δ,ξ (x) := χ(d g ξ (x, ξ))Λ ξ (x)δ n-2 2 d g ξ (x, ξ) 2 -δ 2 (δ 2 + d g ξ (x, ξ) 2 ) n 2 , Z δ,ξ,X (x) := χ(d g ξ (x, ξ))Λ ξ (x)δ n 2 (exp g ξ ξ ) -1 (x), X g ξ (ξ) (δ 2 + d g ξ (x, ξ) 2 ) n 2
for all x ∈ M .

3.3.

Estimate of the error term. For simplicity, we will often write W := W κ,δ,ξ and φ := φ ε (ϕ, δ, ξ) in this section. It follows from [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF]Sections 5 and 7], that (10)

F ′ ε (u(ϕ) + W ) -F ′ ε (u(ϕ)) -F ′ ε (W ) H 2 1 (M ) ′ ≤ C • ε 1 (δ), ( 11
) F ε (u(ϕ) + W ) -F ε (u(ϕ)) -F ε (W ) -F ′ ε (u(ϕ))W -F ′ ε (W )u(ϕ) = O (ε 2 (δ)) , and (12)
W -

(∆ g + h) -1 (F ′ ε (W )) h ≤ C • ε ln 1 δ + ε 1 (δ) + 1 {n≥7} h -c n R g ∞ δ 2 + 1 {n≥15 and non lcf} δ 4 ,
where

(13) ε 1 (δ) :=      δ n-2 2 if n < 6 δ 2 ln 1 δ 2 3 if n = 6 δ n+2 4 if n > 6      and ε 2 (δ) :=    δ if n = 3 δ 2 ln 1 δ if n = 4 δ n 2 if n ≥ 5    .
Plugging [START_REF] Del Pino | Large energy entire solutions for the Yamabe equation[END_REF] and ( 12) in (9) yields

R ε (ϕ, δ, ξ) ≤ C • Π K ⊥ δ,ξ u(ϕ) -(∆ g + h) -1 (F ′ 0 (u(ϕ)) h (14) +O ε ln 1 δ + ε 1 (δ) + 1 {n≥7} h -c n R g ∞ δ 2 + 1 {n≥15
and non lcf} δ 4 .

3.4.

First expansion of the energy J ε . The Taylor expansion of J ε , the control of φ ε in Theorem 3.1 and the definition (9) of R ε (ϕ, δ, ξ) yield

J ε (u(ϕ) + W + φ) =J ε (u(ϕ) + W ) + (u(ϕ) + W -(∆ g + h) -1 (F ′ ε (u(ϕ) + W )), φ) h + O( φ 2 h ) =J ε (u(ϕ) + W ) + (Π K ⊥ δ,ξ (u(ϕ) + W -(∆ g + h) -1 (F ′ ε (u(ϕ) + W ))), φ) h + O( φ 2 h ) =J ε (u(ϕ) + W ) + O(R ε (ϕ, δ, ξ) 2 ).
It then follows from [START_REF]Torus action on S n and sign-changing solutions for conformally invariant equations[END_REF] and ( 13) that

(15) J ε (u(ϕ) + W + φ) = J ε (u(ϕ)) + J ε (W κ,δ,ξ ) + u(ϕ) -(∆ g + h) -1 (F ′ ε (u(ϕ))), W h -F ′ ε (W )u(ϕ) + O R ε (ϕ, δ, ξ) 2 + ε 2 (δ) . Since ϕ → u(ϕ) > 0 is C 1 , u(0) = u 0 is a solution to (2), we get that (16) u(ϕ) -(∆ g + h) -1 (u(ϕ) 2 ⋆ -1-ε ), W h = f 1 (ϕ, ξ)δ n-2 2 + o(δ n-2 2 )
when δ, ε → 0 and

f 1 ∈ C 1 (B ν0 (0) × M, R) (B ν0 (0) ⊂ K 0 ) and f 1 (0, ξ) = 0 for all ξ ∈ M . It follows from [24] that (17) F ′ ε (W )u(ϕ) = κ2 n ω n-1 K -n n n(n(n -2)) n-2 4 ω n u(ϕ)[ξ]δ n-2 2 + O(δ n-2 2 (o(1) + |δ ε -1|))
when (δ, ε) → 0. Here, ω k is the volume of the canonical unit k-sphere in R k+1 and K n is the best constant of the Sobolev inequality u 2 ⋆ ≤ K ∇u 2 in R n . Finally, expanding J ε (u(ϕ)) with respect to ε and collecting ( 15), ( 16) and ( 17) yield

(18) J ε (u(ϕ) + W + φ) = J 0 (u(ϕ)) + εf 2 (ϕ) + J ε (W κ,δ,ξ ) + f 1 (ϕ, ξ) - κ2 n ω n-1 K -n n n(n(n -2)) n-2 4 ω n u(ϕ)[ξ] δ n-2 2 + O R ε (ϕ, δ, ξ) 2 + ε 2 (δ) + δ n-2 2 (o(1) + |δ ε -1|) + o(ε) when δ, ε → 0. Here, f 2 ∈ C 1 (B ν0 (0) ⊂ K 0 , R) 3.5. Expansion of J ε (W κ,δ,ξ
). The following result was obtained in [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF]: there exists

β n > 0 such that (19) J ε (W κ,δ,ξ ) = K -n n n 1 -β n ε - (n -2) 2 4 (δ ε -1) + O εδ 2 + ε 2 + (δ ε -1) 2 + O 1 {n≤5 or lcf} δ n-2 + K -n n n            O ( h -c 3 R g C 0,θ δ) if n = 3 3(h -c 4 R g )(ξ)δ 2 ln 1 δ + O h -c 4 R g C 0,θ δ 2 if n = 4 2(n-1) (n-2)(n-4) (h -c n R g )(ξ)δ 2 + O h -c n R g C 0,θ δ 2+θ if n ≥ 5            + K -n n n    -1 64 |Weyl g (ξ)| 2 g δ 4 ln 1 δ + O δ 4 if n = 6 and non lcf - 1 24(n-4)(n-6) |Weyl g (ξ)| 2 g δ 4 + O(δ 5 ) if n ≥ 7 and non lcf    .

Suitable approximation of u 0 and analyticity

In [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF], the blowing-up solutions of type (u 0 -B) are directly modeled on a nondegenerate function u 0 . When u 0 is degenerate, the kernel K 0 plays a role in the finite-dimensional reduction and we consider a manifold of functions around u 0 parametrized locally by K 0 . Proposition 4.1. There exist ν 0 > 0 small and φ ∈ C 1 (B ν0 (0) ⊂ K 0 , K ⊥ 0 ) such that for all ϕ ∈ K 0 and ψ ∈ K ⊥ 0 satisfying ϕ h , ψ h < ν 0 , we have that

Π K ⊥ 0 (u 0 + ϕ + ψ -(∆ g + h) -1 (F ′ 0 (u 0 + ϕ + ψ))) = 0 ⇔ ψ = φ(ϕ).
In particular, φ vanishes up to order 1 at 0. Moreover, taking ν 0 smaller if necessary, u 0 + ϕ + φ(ϕ) ∈ C 2,θ (M ) is positive for all ϕ ∈ B ν0 (0) and φ : B ν0 (0) → C 2,θ (M ) is analytic with respect to the associated topologies.

The analytic manifold of approximation is

M := {u 0 +ϕ+φ(ϕ)/ ϕ ∈ B ν0 (0) ⊂ K 0 }.
Proposition 4.1 is a particular case of a more general result. Some definitions and notations are required in order to state the general result. We fix f ∈ C 1 (R) and we assume that there exists

u 0 ∈ C 2,θ (M ) such that (20) ∆ g u 0 + hu 0 = f (u 0 ) in M.
We define ( 21)

K 0 := {ϕ ∈ C 2,θ (M )/ ∆ g ϕ + hϕ = f ′ (u 0 )ϕ}.
In the sequel, K 0 will be regarded as a subset of the Hilbert space H 2 1 (M ). It follows from Fredholm's theory for Hilbert spaces that K 0 is of finite dimension d ∈ N. We prove the following result in the spirit of Dancer [START_REF] Dancer | Real analyticity and non-degeneracy[END_REF]: Proposition 4.2. We let f ∈ C 1 (R) and u 0 ∈ C 2,θ (M ) be a solution to [START_REF] Musso | Nondegeneracy of nonradial nodal solutions to Yamabe problem[END_REF]. We let K 0 be as in [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF].Then there exist ν > 0 and φ

∈ C 1 (B ν (0) ⊂ K 0 , K ⊥ 0 ∩C 2,θ (M )) such that for all ϕ ∈ B ν (0) ⊂ K 0 and ψ ∈ B ν (0) ⊂ K ⊥ 0 , (22) Π K ⊥ 0 (u 0 + ϕ + ψ -(∆ g + h) -1 (f (u 0 + ϕ + ψ))) = 0 ⇔ ψ = φ(ϕ).
Moreover, if f is analytic on an open interval I and u 0 (x) ∈ I for all x ∈ M , then φ is analytic around 0.

As one checks, the function x → |x| 2 ⋆ -2 x is C 1 on R and analytic on (0, +∞). Therefore Proposition 4.1 is a direct consequence of Proposition 4.2.

Proof of Proposition 4.2. The first part of the statement is a direct application of the implicit function theorem and regularity theory. Since M is compact and u 0 is continuous, it follows from the analyticity of f that there exists A, B > 0 such that [START_REF] Rey | The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF] |a k (u 0 (x))| ≤ A • B k for all k ≥ 0 and x ∈ M, where

f (u 0 (x) + h) = ∞ k=0 a k (u 0 (x))h k for all x ∈ M and h ∈ (-B -1 , B -1 ).
Since φ is C ∞ its differential vanishes at 0, we write for any L ≥ 2 that

φ(ϕ) = L l=2 P l (ϕ) + o( ϕ L ) when ϕ → 0 ,
where for all l ≥ 2 and ϕ ∈ B ν (0) ⊂ K 0 , P l (ϕ) ∈ K ⊥ 0 is a homogeneous polynomial of degree l. We set P 1 (ϕ) := ϕ ∈ K 0 . Therefore, for any L ≥ 1, we have that

f (u 0 + ϕ + φ(ϕ)) = L k=0 a k (u 0 ) L l=1 P l (ϕ) k + o( ϕ L ) when ϕ → 0. We write that (24) L i=1 X i k = ∞ j=0 Q k,L,j (X 1 , ..., X L ),
where

Q k,L,j (X 1 , ..., X L ) := L 1 r l =k ; L 1 lr l =j k! L l=1 r l ! L l=1 X r l l .
Note that Q k,L,j (X 1 , ..., X L ) = 0 when j ∈ [k, Lk], so all the sums make sense. Therefore, for any L ≥ 2, the term of degree L in ( 22) is

Π K ⊥ 0 P L (ϕ) -(∆ g + h) -1 L k=0 a k (u 0 )Q k,L,L (P 1 (ϕ), ..., P L (ϕ)) = 0
for all L ≥ 2. In the sum, the term for k = 0 is 0, and the term for k = 1 is a 1 (u 0 )P L (ϕ) = f ′ (u 0 )P L (ϕ). Therefore, we have that

(25) P L (ϕ) = L -1 0 Π K ⊥ 0 (∆ g + h) -1 L k=2 a k (u 0 )Q k,L,L (P 1 (ϕ), ..., P L (ϕ))
for all L ≥ 2, where L 0 :

K ⊥ 0 → K ⊥ 0 is the isomorphism given by L 0 (ψ) = Π K ⊥ 0 ψ -(∆ g + h) -1 (f ′ (u 0 )ψ) for all ψ ∈ K ⊥ 0 .
Note that since k, L ≥ 2, the right-hand side of ( 25) is independent of P L (ϕ). We fix α ∈ (0, 1). It follows from elliptic theory that there exists C > 0 depending on (M, g), h and f ′ (u 0 ) such that (26)

P L (ϕ) C 1,α ≤ C L k=2 a k (u 0 )Q k,L,L (P 1 (ϕ), ..., P L (ϕ)) ∞
for all L ≥ 2. We fix K ≥ 2. Summing ( 26) from L = 2 to K, using ( 23), [START_REF] Robert | Sign-Changing Blow-Up for Scalar Curvature Type Equations[END_REF] and the nonnegativity of the coefficients of Q k,L,L , we get that

K L=2 P L (ϕ) C 1,α ≤ C • A K k=2 K L=k B k Q k,L,L ( P 1 (ϕ) ∞ , ..., P L (ϕ) ∞ ) (27) ≤ C • A K k=2 K L=k B k Q k,K,L ( P 1 (ϕ) ∞ , ..., P K (ϕ) ∞ ) ≤ C • A K k=2 B K l=1 P l (ϕ) ∞ k . We define h K (t) := sup ϕ ∞ ≤t K L=2 P L (ϕ) ∞ .

It follows from (27) that

t + h K (t) ≤ 1 2B ⇒ h K (t) ≤ 2C • A • B 2 • (t + h K (t)) 2 .
Therefore, since h K is continuous and non-decreasing, we get that

t < ε 0 := min 1 4B , 1 16AB 2 C ⇒ h K (t) ≤ ε 0 for all K ≥ 2 .
As a consequence, the series ( ∞ L=2 P L (ϕ)) converges uniformly on B ε0/2 (0) ⊂ K 0 in the C 0,α -norm. Inequality (27) yields the convergence in C 1,α (M ). The characterization [START_REF] Pistoia | Sign-changing bubble towers for asymptotically critical elliptic equations on Riemannian manifolds[END_REF] then yields

φ(ϕ) = ∞ l=2 P l (ϕ) for all ϕ ∈ B ε0 (0) ⊂ K 0 .
Elliptic theory yields convergence in C 2,θ (M ). This proves analyticity.

5.

Reduction of the problem to the analysis of J 0 (u 0 + ϕ + φ(ϕ))

From now on, we define:

u(ϕ) := u 0 + ϕ + φ(ϕ)
for all ϕ ∈ B ν0 (0) ⊂ K 0 , where φ(ϕ) is defined in Proposition 4.1. In particular, (28)

Π K ⊥ 0 u(ϕ) -(∆ g + h) -1 (F ′ 0 (u(ϕ)) = 0 for all ϕ ∈ B ν0 (0) ⊂ K 0 .
Since dφ 0 ≡ 0, it then follows from Proposition 4.1 that u satisfies the hypothesis [START_REF]Peak solutions without non-degeneracy conditions[END_REF]. For 0 < a < b to be fixed later, we define δ := tε Taking into account the expressions ( 13), ( 14), ( 18), [START_REF] Lee | The Yamabe problem[END_REF], and (28), we then get that

(29) J ε (u(ϕ) + W + φ) = J 0 (u(ϕ)) + εf 2 (ϕ) + K -n n n 1 -β n ε + n -2 2 ε ln 1 ε + ε • K -n n n • (n -2) 2 4 ln 1 t + F (ϕ, ξ)t n-2 2 + o(ε) when ε → 0 uniformly with respect to t ∈ [a, b].
Here, F ∈ C 1 (B ν0 (0) × M ) and we have that

F (0, ξ) = -κ 2 n ω n-1 u 0 (ξ) (n(n -2)) n-2 4 ω n +      2(n-1) (n-2)(n-4) (h -c n R g )(ξ) if n = 6 - 1 24(n-4)(n-6) |Weyl g (ξ)| 2 if n = 10 and h ≡ c n R g 0 otherwise.
The assumptions (4) (for κ = -1) and ( 5) (for κ = 1) then yield F (0, ξ) > 0 for all ξ ∈ M.

We define

a := 1 2 n -2 2 min ξ∈M F (0, ξ) 2 n-2 and b := 2 n -2 2 min ξ∈M F (0, ξ) 2 n-2 .
Since u 0 is a strict local minimizer of I 0 , it follows from Theorem 6.1 of next section that there exists ν 1 ∈ (0, ν 0 /2) such that (30) J 0 (u(ϕ)) > J 0 (u 0 ) for all ϕ ∈ B 2ν1 (0) \ {0}.

Due to compactness, for any ε > 0, there exists (ϕ

ε , t ε , ξ ε ) ∈ B ν1 (0) × [a, b] × M such that min (ϕ,t,ξ)∈Bν 1 (0)×[a,b]×M J ε (u(ϕ) + W κ,tε 2 n-2 ,ξ + φ ε (ϕ, tε 2 n-2 , ξ)) = J ε (u(ϕ ε ) + W κ,tεε 2 n-2 ,ξε + φ ε (ϕ ε , t ε ε 2 n-2 , ξ ε )).
It then follows from the Taylor expansion (29), the choice of 0 < a < b and (30) that t ε ∈ (a, b) and ϕ ε ∈ B ν1 (0) for small ε > 0. Moreover, we have that

lim ε→0 t ε = n -2 2 min ξ∈M F (0, ξ) 2 n-2
and lim ε→0 ϕ ε = 0 , and (ξ ε ) ε>0 approaches the set of minimizers of F (0, •) when ε > 0 is small. Therefore, since (ϕ ε , t ε , ξ ε ) lies in the interior of the domain, it is a critical point for the minimizing functional, and therefore, (ϕ ε , t ε ε

2 n-2 , ξ ε ) is a critical point for (ϕ, δ, ξ) → J ε (u(ϕ) + W κ,δ,ξ + φ ε (ϕ, δ, ξ)).
It then follows from Theorem 3.1 that

u ε := u(ϕ ε )+W κ,tεε 2 n-2 ,ξε +φ ε (ϕ ε , t ε ε 2 n-2 , ξ ε ) is a solution to ∆ g u ε + hu ε = |u ε | 2 ⋆ -2-ε u ε in M for ε > 0 small,
and in addition, due to [START_REF] Dancer | Real analyticity and non-degeneracy[END_REF] and the error control of φ ε in Theorem 3.1, we have that

u ε = u 0 + κB ε + o(1)
in H 2 1 (M ) when ε → 0, where B ε is as in ( 6) with µ ε := t ε ε 2 n-2 . This proves Theorems 2.1 and 2.2, and therefore Theorem 1.1.

We are now left with proving Theorem 6.1.

Equivalence of strict local minimizers

This section is devoted to the proof of the following: Theorem 6.1. The function u 0 is a strict local minimizer of I 0 iff 0 is a strict local minimizer of ϕ → J 0 (u 0 + ϕ + φ(ϕ)).

The proof goes through four claims and uses the analyticity of ϕ → φ(ϕ). Claim 6.1. There exists ν 0 > 0 such that

u 0 + ϕ + φ(ϕ) 2 h -u 0 + ϕ + φ(ϕ) 2 ⋆ 2 ⋆ = ∞ L=3 A L (ϕ)
and

u 0 + ϕ + φ(ϕ) 2 ⋆ 2 ⋆ = u 0 2 ⋆ 2 ⋆ - n 2 ∞ L=3 L -2 L A L (ϕ)
for ϕ ∈ B ν0 (0) ⊂ K 0 , where for any L ≥ 3, A L (ϕ) is a homogeneous polynomial of degree L.

Proof of Claim 6.1. We are going to compute the Taylor expansions of the two lefthand-sides and we will use the analyticity of ϕ → φ(ϕ) to prove Claim 6.1. We fix N ≥ 2. It follows from ( 24) that (31)

u 0 + ϕ + φ(ϕ) 2 ⋆ 2 ⋆ = M u 0 + N l=1 P l (ϕ) 2 ⋆ dv g + o ϕ N = u 0 2 ⋆ 2 ⋆ + N L=1 L j=1 L l=1 r l =j ; L l=1 lr l =L j-1 i=0 (2 ⋆ -i) L l=1 r l ! M u 2 ⋆ -j 0 L l=1 P l (ϕ) r l dv g + o ϕ N .
We claim that (32) u 0 ∈ K ⊥ 0 . We prove the claim. We let ϕ be in K 0 . The self-adjointness of the Laplacian yields

(u 0 , ϕ) h = M (∆ g u 0 + hu 0 )ϕ dv g = M (∆ g ϕ + hϕ)u 0 dv g .
It then follows from equation ( 2) and the definition (3) of K 0 that (u 0 , ϕ) h = 0. This proves the claim.

It follows from (32) that the term for

L = 1 in (31) is 2 ⋆ M u 2 ⋆ -1 0 ϕ dv g = 0.
Separating the cases j = 1 and j ≥ 2, we get that (33)

u 0 + ϕ + φ(ϕ) 2 ⋆ 2 ⋆ = u 0 2 ⋆ 2 ⋆ + N L=2 L j=2 l r l =j ; l lr l =L j-1 i=0 (2 ⋆ -i) L l=1 r l ! M u 2 ⋆ -j 0 L l=1 P l (ϕ) r l dv g + 2 ⋆ N L=2 M u 2 ⋆ -1 0 P L (ϕ) dv g + o ϕ N .
For L ≥ 2, it follows from the expression (25

) of P L (ϕ) that (34) (L 0 P L (ϕ), u 0 ) h = L j=2 l r l =j ; l lr l =L j i=1 (2 ⋆ -i) L l=1 r l ! M u 2 ⋆ -j 0 L l=1 P l (ϕ) r l dv g .
Since the operator L 0 is symmetric, we have that

(35) (L 0 P L (ϕ), u 0 ) h = (P L (ϕ), L 0 u 0 ) h = -(2 ⋆ -2) M u 2 ⋆ -1 0 P L (ϕ) dv g .
Plugging into (33) the expression of M u 2 ⋆ -1 0 P L (ϕ) dv g obtained by combining (35) and (34), we get that (36)

u 0 + ϕ + φ(ϕ) 2 ⋆ 2 ⋆ = u 0 2 ⋆ 2 ⋆ + n 2 N L=2 L j=2 l r l =j; l lr l =L (j -2) j-1 i=1 (2 ⋆ -i) L l=1 r l ! M u 2 ⋆ -j 0 L l=1 P l (ϕ) r l dv g +o ϕ N .
Note that the term in the above sum vanishes for j = 2. As one checks, for any 3 ≤ j ≤ L, we have that

L-1 q=1 L -2q L l s l =j-1 ; l ls l =L-q 1 l s l ! M u 2 ⋆ -j 0 l P l (ϕ) s l P q (ϕ) dv g = L-1 q=1 L -2q L l r l =j ; l lr l =L r q l r l ! M u 2 ⋆ -j 0 l P l (ϕ) r l dv g = l r l =j ; l lr l =L L-1 q=1 L -2q L r q 1 l r l ! M u 2 ⋆ -j 0 l P l (ϕ) r l dv g = (j -2) l r l =j ; l lr l =L 1 l r l ! M u 2 ⋆ -j 0 l P l (ϕ) r l dv g .
Plugging this identity into (36) yields (37)

u 0 + ϕ + φ(ϕ) 2 ⋆ 2 ⋆ = u 0 2 ⋆ 2 ⋆ + n 2 N L=3 L-2 q=1 L -2q L u L-q,q (ϕ) + o ϕ N ,
where

u k,q (ϕ) := k j=2 j i=1 (2 ⋆ -i) × l s l =j ; l ls l =k 1 l s l ! M u 2 ⋆ -1-j 0 l P l (ϕ) s l P q (ϕ) dv g .
For any L, q such that q ≥ 2 and L -q ≥ 2, the self-adjointness of L 0 yields (L 0 P q (ϕ), P L-q (ϕ)) h = (P q (ϕ), L 0 P L-q (ϕ)) h . Taking the explicit expression of ( 25) then yields u L-q,q (ϕ) = u q,L-q (ϕ) for 2 ≤ q ≤ L -2 .

Therefore, for L ≥ 4, we get that

L-2 q=2 L -2q L u L-q,q (ϕ) = 0 ,
and then (37) yields

u 0 + ϕ + φ(ϕ) 2 ⋆ 2 ⋆ = u 0 2 ⋆ 2 ⋆ + n 2 N L=3 L -2 L u L-1,1 (ϕ) + o ϕ N . ( 38
)
We now estimate u 0 + ϕ + φ(ϕ) 2 h -u 0 + ϕ + φ 2 ⋆ 2 ⋆ . Using [START_REF] Pistoia | Sign-changing bubble towers for asymptotically critical elliptic equations on Riemannian manifolds[END_REF] and that u 0 , φ(ϕ) ∈ K ⊥ 0 for all ϕ ∈ K 0 , we get that (writing φ = φ(ϕ) for simplicity)

u 0 + ϕ + φ 2 h -u 0 + ϕ + φ 2 ⋆ 2 ⋆ (39) = u 0 + ϕ + φ, u 0 + ϕ + φ -(∆ g + h) -1 (u 0 + ϕ + φ) 2 ⋆ -1 h = Π K0 (u 0 + ϕ + φ), Π K0 (u 0 + ϕ + φ -(∆ g + h) -1 (u 0 + ϕ + φ) 2 ⋆ -1 ) h = ϕ, ϕ -(∆ g + h) -1 (u 0 + ϕ + φ) 2 ⋆ -1 ) h = ϕ 2 h - M (u 0 + ϕ + φ) 2 ⋆ -1 ϕ dv g .
We fix N ≥ 3 and write φ(ϕ) =

N -1

L=2 P l (ϕ) + o ϕ N -1 when ϕ → 0.
A Taylor expansion and ( 24) yield ( 40)

M (u 0 + ϕ + φ(ϕ)) 2 ⋆ -1 ϕ dv g = M u 2 ⋆ -1 0 ϕ dv g + (2 ⋆ -1) M u 2 ⋆ -2 0 ϕ 2 dv g + N -1 l=2 (2 ⋆ -1) M u 2 ⋆ -2 0 ϕP l (ϕ) dv g + N L=3 L-1 j=2 j i=1 (2 ⋆ -i) l s l =j ; l ls l =L-1 1 l s l ! M u 2 ⋆ -1-j 0 l P l (ϕ) s l ϕ dv g + o ϕ N when ϕ → 0. The definition (3) of K 0 yields (41) (2 ⋆ -1) M u 2 ⋆ -2 0 ϕ 2 dv g = ϕ 2 h .
Moreover, since P l (ϕ) ∈ K ⊥ 0 for all l ≥ 2, we get that (42)

N -1 l=2 (2 ⋆ -1) M u 2 ⋆ -2 0 ϕP l (ϕ) dv g = N -1 l=2 P l (ϕ), ϕ h = 0 .
Plugging together (32) and ( 39)-( 42) yields (43)

u 0 + ϕ + φ(ϕ) 2 h -u 0 + ϕ + φ(ϕ) 2 ⋆ 2 ⋆ = - N L=3 u L-1,1 (ϕ) + o ϕ N when ϕ → 0. We define (44) A L (ϕ) := -u L-1,L = - L-1 j=2 j i=1 (2 ⋆ -i) l s l =j ; l ls l =L-1 1 l s l ! M u 2 ⋆ -1-j 0 l P l (ϕ) s l ϕ dv g
which is a homogenous polynomial of degree L. Claim 6.1 then follows from (38), (39), (43) and the analyticity of ϕ → φ(ϕ) (see Proposition 4.2).

We define

S K0 := {ϕ ∈ K 0 / ϕ h = 1}.
For any ϕ ∈ S K0 and any t ∈ (-ν 0 , ν 0 ), we define

f ϕ (t) := J 0 (u 0 + tϕ + φ(tϕ)) -J 0 (u 0 ) t 2 • u 0 2 h if t = 0 and f ϕ (0) = 0 .
It follows from Claim 6.1 that f ϕ is analytic on (-ν 0 , ν 0 ) and that

u 0 + tϕ + φ(tϕ) 2 ⋆ 2 ⋆ = u 0 2 ⋆ 2 ⋆ 1 - n 2 t 3 f ′ ϕ (t)
for |t| < ν 0 . Therefore, we have that (45) I 0 (u 0 + tϕ + φ(tϕ))

= I 0 (u 0 ) 1 + 2t 2 f ϕ (t) - n -2 2 t 3 f ′ ϕ (t) • 1 - n 2 t 3 f ′ ϕ (t) - 2 2 ⋆
. Claim 6.2. We assume that u 0 is a strict local minimizer of I 0 . Then there exists ν 1 ∈ (0, ν 0 ) such that for any ϕ ∈ S K0 and t ∈ (-ν 1 , ν 1 ) \ {0}, there holds

f ϕ (t) = 0 ⇒ f ′ ϕ (t) = 0 , f ′ ϕ (t) = 0 ⇒ f ϕ (t) > 0 . Proof of Claim 6.2. If f ϕ (t) = f ′ ϕ (t) = 0
, it then follows from (45) that u 0 + tϕ + φ(tϕ) is a minimizer for I 0 close to u 0 , and therefore there exists λ t > 0 such that u 0 + tϕ + φ(tϕ) = λ t • u 0 for t small. It then follows from the definition (22) of φ(tϕ) that λ t = 1 and that tϕ = 0, which is a contradiction since t = 0 and ϕ = 0. Therefore f ϕ (t) and f ′ ϕ (t) cannot vanish simultaneously for t = 0. Moreover, if f ′ ϕ (t) = 0, (45) yields f ϕ (t) ≥ 0. Combining these assertions yields Claim 6.2. Claim 6.3. We assume that u 0 is a strict local minimizer of I 0 . We claim that for all ϕ ∈ S K0 , there exists tϕ ∈ (0, ν 1 ) such that f ϕ (t) > 0 for all t ∈ (0, tϕ ).

Proof of Claim 6.3. It follows from Claim 6.2 that f ϕ does not vanish identically. Since it is analytic, there exists a = 0 and k ≥ 1 (both depending on ϕ) such that f ϕ (t) = at k + o(t k ) when t → 0. Obtaining from this the expansion of f ′ ϕ (t) and plugging these expressions into (45) yield

I 0 (u 0 + tϕ + φ(tϕ)) = I 0 (u 0 )(1 + 2at k+2 + o(t k+2 ))
when t → 0. Since u 0 is a local minimizer, we get that a ≥ 0, and then a > 0. This yields the existence of tϕ . This proves Claim 6.3.

It follows from Claims 6.2 and 6.3 that for any ϕ ∈ S K0 , there exists t ϕ ∈ (0, ν 1 ] such that f ϕ (t) > 0 for all t ∈ (0, t ϕ ), and in case t ϕ < ν 1 , we have that f ϕ (t) < 0 for all t ∈ (t ϕ , ν 1 ). Claim 6.4. We assume that u 0 is a strict local minimizer of I 0 . We claim that there exists ν 2 > 0 such that t ϕ > ν 2 for all ϕ ∈ S K0 .

Proof of Claim 6.4. We prove Claim 6.4 by contradiction. Indeed, otherwise, there exists a sequence (ϕ i ) ∈ S K0 such that t ϕi → 0 when i → +∞ and f ϕi (t ϕi ) = 0 for all i. Up to a subsequence, we can assume that ϕ i → ϕ ∈ S K0 when i → +∞. We fix t ∈ (0, ν 1 ). Then for i large enough, we have t ϕi < t, and therefore f ϕi (t) < 0. Passing to the limit when i → +∞ yields f ϕ (t) ≤ 0 for all t ∈ (0, ν 1 ). This is a contradiction with Claim 6.3. This proves Claim 6.4.

Proof of Theorem 6.1, first implication: We assume that u 0 is a strict local minimizer of I 0 . It follows from Claim 6.4 that J 0 (u 0 + ϕ + φ(ϕ)) > J 0 (u 0 ) for all ϕ ∈ B ν2 (0) \ {0}. This proves the first implication of Theorem 6.1.

Proof of Theorem 6.1, second implication: We assume that there exists ν 1 > 0 such that J 0 (u 0 + ϕ + φ(ϕ)) > J 0 (u 0 ) for all ϕ ∈ B ν1 (0) \ {0}. For ϕ ∈ B ν1 (0), we define δA(ϕ) and δB(ϕ) such that

u 0 + ϕ + φ(ϕ) 2 h = u 0 2 h • (1 + δA(ϕ)) and u 0 + ϕ + φ(ϕ) 2 ⋆ 2 ⋆ = u 0 2 h • (1 + δB(ϕ))
. Therefore, we have that

J 0 (u 0 + ϕ + φ(ϕ)) = J 0 (u 0 ) + u 0 2 h • 1 2 δA(ϕ) - 1 2 ⋆ δB(ϕ) , (46)
I 0 (u 0 + ϕ + φ(ϕ)) = I 0 (u 0 ) • (1 + δA(ϕ)) (1 + δB(ϕ)) -2/2 ⋆ (47)
for all ϕ ∈ B ν1 (0). It follows from our assumption and (46) that δA(ϕ) > 2 2 ⋆ δB(ϕ) for all ϕ ∈ B ν1 (0) \ {0}. It then follows from (47) that (48) I 0 (u 0 + ϕ + φ(ϕ)) > I 0 (u 0 ) for all ϕ ∈ B ν1 (0) \ {0}.

We now let (u i ) ∈ H 2 1 (M ) be minimizers for I 0 such that lim i→+∞ u i = u 0 . It follows from regularity theory that u i ∈ C 2,θ (M ) for all i and that the convergence holds in C 2,θ (M ). Without loss of generality, we can assume that u i is a solution to (2) for all i. It then follows from the definition of φ (see Proposition 4.1) that there exists ϕ i ∈ K 0 such that u i = u 0 + ϕ i + φ(ϕ i ) for all i. Since u i is a local minimizer, it then follows from (48) that ϕ i = 0 for i large, and thus u i = u 0 . Then u 0 is a strict local minimizer of I 0 . This proves the second implication of Theorem 6.1. as ϕ, φ → 0. Since I ′′ 0 (u 0 ) ≥ 0, it follows from (50) that there exists c 1 > 0 such that (58)

I ′′ 0 (u 0 )(u, u) ≥ 4c 1 Π (Ru0⊕K0) ⊥ (u) 2 h
for all u ∈ H 2 1 (M ). The positivity of A 4 yields the existence of c 2 > 0 such that (59) A 4 (ϕ) ≥ c 2 ϕ 4 for all ϕ ∈ K 0 .

Plugging (53), ( 56), ( 57), ( 58) and ( 59) into (55) yields the existence of c 3 > 0 such that (60)

I 0 (u 0 + ϕ + φ) ≥ I 0 (u 0 ) + c 3 ϕ 4 + c 1 Π (Ru0⊕K0) ⊥ (φ -φ(ϕ)) 2 h
as ϕ, φ → 0, where ϕ ∈ K 0 and φ ∈ (Ru 0 ⊕ K 0 ) ⊥ . This proves that u 0 is a strict local minimizer of I 0 .

For the second part, for any solution u ∈ B ν1 (u 0 ), we decompose u := u 0 + ϕ + ψ where ϕ ∈ K 0 and ψ ∈ K ⊥ 0 . We have that ϕ < ν 1 and ψ < ν 1 . It follows from Proposition 4.2 that if ν 1 > 0 is small enough, then ψ = φ(ϕ) and u = u(ϕ). The positivity of A 4 yields the existence of c > 0 such that A 4 (ϕ) ≥ 2c ϕ 4 for all ϕ ∈ K 0 . It then follows from Claim 6.1 that

u 2 h -u 2 ⋆ 2 ⋆ ≥ c ϕ 4 .
Since u is a solution to the equation, we then get that ϕ = 0 and then u = u 0 .

In this section, we exhibit situations in which the hypothesis of Proposition 7.1 hold, which yields strict local minimizers for I 0 .

7.1. The expression of A 4 when u 0 is constant. We assume here that h, u 0 > 0 are positive constants. In particular, we have that h = u 2 ⋆ -2 0 and that

K 0 = {ϕ ∈ C 2 (M )/ ∆ g ϕ = λϕ},
where λ := (2 ⋆ -2)u 2 ⋆ -2 0 > 0. In other words, u 0 is degenerate if and only if λ is an eigenvalue of ∆ g . As one checks, the operator

∆ g -λ : K ⊥ 0 → (K ⊥ 0 ) ′ φ → τ → M ((∇φ, ∇τ ) g -λφτ ) dv g
is a bi-continuous isomorphism and then definition (25) yields

P 2 (ϕ) = (2 ⋆ -1)(2 ⋆ -2) 2 (∆ g -λ) -1 (u 2 ⋆ -3 0 ϕ 2 )
for all ϕ ∈ K 0 . As a consequence, the expression (52) of A 4 can be rewritten

(61) A 4 (ϕ) = (2 ⋆ -1)(2 ⋆ -2)u 2 ⋆ -4 0 - (2 ⋆ -1)λ 2 M ϕ 2 (∆ g -λ) -1 (ϕ 2 ) dv g - 2 ⋆ -3 6 M ϕ 4 dv g for all ϕ ∈ K 0 .
7.2. The case of the Yamabe equation on the canonical sphere. In the case of the Yamabe equation on the sphere, the kernel K 0 parametrizes exactly the noncompact set of minimizers, which makes A 4 vanish. More precisely, Proposition 7.2. [Kobayashi [START_REF] Kobayashi | Yamabe metrics and conformal transformations[END_REF]] Assume that (M, g) = (S n , can) and that h ≡ c n R can . Then any solution u 0 to (2) is minimizing and A 4 ≡ 0 for all u 0 .

Proof of Proposition 7.2. This result is a consequence of Theorem 2.1 in Kobayashi [START_REF] Kobayashi | Yamabe metrics and conformal transformations[END_REF]. We give here an independent proof for the sake of self-content. The vanishing of A 4 is a consequence of the direct computation in the proof of (ii) of Proposition 7.3 below. We give here a shorter and less technical proof that stresses on properties of solutions to the scalar curvature equation on the sphere

(62) ∆ can + c n R can u = u 2 ⋆ -1 in S n .
The proof relies on two facts: first, the elements of the kernel K 0 satisfy a Bianchi-Egnell condition; second, all solutions to (62) minimize I 0 (see Obata [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF]).

We fix ϕ ∈ K 0 . It follows from properties of the canonical sphere (see below) that there exists t ∈ R → u(t) a smooth function such that u(t) ∈ C ∞ (S n ) is a solution to (62) for all t, u(0) = u 0 and u ′ (0) = ϕ. This is Bianchi-Egnell condition. Since u(t) is a positive solution to (62), it follows from Proposition 4.1 that for t small, there exists ϕ(t) ∈ K 0 such that u(t) = u 0 + ϕ(t) + φ(ϕ(t)). Moreover, t → ϕ(t) is smooth, ϕ(0) = 0 and ϕ ′ (0) = ϕ. It follows from (54) that A 3 ≡ 0 since u 0 minimizes I 0 . It then follows from the expansion (53) of A 4 that

A 4 (ϕ) 2 u 0 2 2 ⋆ = lim t→0 I 0 (u 0 + ϕ(t) + φ(ϕ(t))) -I 0 (u 0 ) t 4 I 0 (u 0 ) = lim t→0 I 0 (u(t)) -I 0 (u 0 ) t 4 I 0 (u 0 ) .
Moreover, it follows from Obata [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF] that positive solutions to (62) are all minimizing, and then I 0 (u(t)) = I 0 (u 0 ) for all small t. Therefore, we get that A 4 (ϕ) = 0 for all ϕ ∈ K 0 .

We are now left with proving the existence of t → u(t). Up to conformal transformation (see Obata [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF]), we assume that u 0 is the sole positive constant solution to (62). In this case, K 0 = {ϕ ∈ C 2 (S n ) / ∆ can ϕ = nϕ} is the space of first spherical harmonics. We fix ϕ ∈ K 0 and we let Z := grad(ϕ) be the associated vector field. This is a conformal vector field and, denoting f t the associated flow, we have that f ⋆ t can = ω(t) 4/(n-2) can for some positive function t → ω(t) ∈ C ∞ (S n ) such that ω(0) = 1. It follows from the conformal invariance of the scalar curvature equation that u(t) := ω(t)u 0 is also a solution to (62) for all t. Moreover, since f ⋆ t can = ω(t) 4/(n-2) can, we have that ω

′ (0) = -n-2 2n ∆ can ϕ = n-2 2n div can (Z) = -n-2 2
ϕ, and then u ′ (0) = cϕ for some c = 0. This proves the result after rescaling. 7.3. Product of manifolds and examples of degenerate strict local minimizers. Let (M 1 , g 1 ) and (M 2 , g 2 ) be two compact manifolds of respective dimensions d ≥ 1 and n -d ≥ 1 with n ≥ 3. We consider the Riemannian manifold M := M 1 × M 2 endowed with the product metric g := g 1 ⊕ g 2 . For i = 1, 2, we let λ 1 (M i , g i ) > 0 be the first nonzero eigenvalue of ∆ gi on M i . We define for u ∈ H 2 1 (M 1 ) \ {0}. We prove the following: Proposition 7.3. Let (M 1 , g 1 ) and (M 2 , g 2 ) be two compact manifolds of respective dimensions d ≥ 1 and n -d ≥ 1 with n ≥ 3. We consider the Riemannian manifold M := M 1 × M 2 of dimension n ≥ 3 endowed with the product metric g := g 1 ⊕ g 2 . We let h, u 0 > 0 be as in (63). We assume that one of the following cases hold:

(i) d ≥ 3, λ 1 (M 1 , g 1 ) < λ 1 (M 2 , g 2 ), and ũ0 is a local minimizer of Ĩ0 , (ii) d ≥ 1 and (M 1 , g 1 ) = (S d (r), can) with r > d λ1(M2,g2) . Then u 0 is a degenerate solution to (2) and I ′′ 0 (u 0 ) ≥ 0. Moreover, we have that A 3 (ϕ) = 0 and A 4 (ϕ) > 0 for all ϕ ∈ K 0 \ {0}. In particular, u 0 is a strict local minimizer of I 0 .

In the case of the Yamabe equation on the product of spheres, this proposition is a consequence of Kobayashi [START_REF] Kobayashi | Yamabe metrics and conformal transformations[END_REF].

Proof of Proposition 7.3. We let (M 1 , g 1 ), (M 2 , g 2 ) be as in the proposition. Since λ 1 (S d (r), can) = dr -2 (see Berger-Gauduchon-Mazet [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF]), we have that (64) λ 1 (M 1 , g 1 ) < λ 1 (M 2 , g 2 ) in both Cases (i) and (ii). As one checks, K 0 = {ϕ ∈ C 2 (M )/ ∆ g ϕ = λ 1 (M 1 , g 1 )ϕ}.

It follows from spectral theory for products that K 0 is spanned by the functions (x, y) → u 1 (x)u 2 (y) where for i = 1, 2, u i : M i → R is an eigenfunction for the eigenvalue µ i for ∆ gi , where µ 1 + µ 2 = λ 1 (M 1 , g 1 ). It then follows from (64) that

K 0 = {(x, y) ∈ M → ϕ(x)/ ϕ ∈ Λ 1 (M 1 , g 1 )},
where Λ 1 (M 1 , g 1 ) := {ϕ ∈ C 2 (M 1 )/ ∆ g1 ϕ = λ 1 (M 1 , g 1 )ϕ} is the eigenspace associated to the first eigenvalue λ 1 (M 1 , g 1 ). We claim that (65) M1 ϕ 3 dv g1 = 0 for all ϕ ∈ Λ 1 (M 1 , g 1 ).

We prove the claim. In Case (i), since d ≥ 3 and ũ0 is a local minimizer, (65) follows from (51) and (54). In Case (ii), since (M 1 , g 1 ) = (S d (r), can), Λ 1 (M 1 , g 1 ) is the restriction to S d (r) of linear functions on R d+1 , and then (65) follows from symmetry. This proves the claim. Inequality (69) holds if g N is a Yamabe metric, that is a minimizer of the Yamabe functional. From the pde point of view, a metric g on M is a Yamabe metric iff R g is constant and the minimum of I 0 (with h ≡ c n R g ) is achieved by constants.

As a remark, Corollary 7.1 can be generalized by replacing the sphere by a manifold V of dimension d ≥ 3 with a degenerate Yamabe metric g V of positive scalar curvature satisfying R g N = dim(N )λ 1 (V, g V ) and λ 1 (V, g V ) < λ 1 (N, g N ). Note that the degeneracy of g V implies that R g V = (dim(V ) -1)λ 1 (V, g V ).

2 n- 2

 2 for t ∈ [a, b]. We assume that {3 ≤ n ≤ 6} or {h ≡ c n R g and 3 ≤ n ≤ 10} or {h ≡ c n R g and lcf}.

- 2 n-2 4 , 1 0in M 1 .M1 |u| 2

 24112 (63)h := λ 1 (M 1 , g 1 ) 2 ⋆ -2 and u 0 := λ 1 (M 1 , g 1 ) 2 ⋆so that u 0 is the only positive constant solution to ∆ g u 0 + hu 0 = u 2 ⋆ -In particular, ũ0 is a critical point for the functional Ĩ0 (u) := M1 |∇u| 2 g1 + hu 2 dv g1

Proof of Corollary 7 . 1 .= λ 1 (- 2 .

 7112 We fix r 0 := dim(N ) • d/R g N . It follows from inequality (69) that we are in Case (ii) of Proposition 7.3. With this choice of r 0 , we have thatc n R g = n -S d (r 0 ), can) 2 ⋆Therefore Proposition 7.3 applies. This proves Corollary 7.1.

Examples

In this section, we provide examples of strict local minimizers for the functional I 0 , and hence for J 0 by Theorem 6.1. We let u 0 ∈ C 2 (M ) be a solution to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF]. In particular I ′ 0 (u 0 ) = 0. As a preliminary remark, (49) if u 0 is a local minimizer of I 0 then I ′′ 0 (u 0 ) ≥ 0. Moreover, since u 0 is a solution to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF], the kernel of I ′′ 0 (u 0 ) is given as follows: for any f 0 ∈ H 2 1 (M ), (50) I ′′ 0 (u 0 )(f 0 , f ) = 0 for all f ∈ H 2 1 (M ) ⇔ {f 0 ∈ Ru 0 ⊕ K 0 }. Therefore, I ′′ 0 (u 0 ) cannot be positive definite, and a specific analysis along K 0 is necessary. It follows from the expression (44) of A L (ϕ) that

when ϕ → 0. Therefore, (54) if u 0 is a local minimizer of I 0 then A 3 ≡ 0 and A 4 (ϕ) ≥ 0 for all ϕ ∈ K 0 .

In the case of the Yamabe equation, this condition appeared in Kobayashi [START_REF] Kobayashi | Yamabe metrics and conformal transformations[END_REF]. Conversely, we have the following result:

Proposition 7.1. Assume that A 3 ≡ 0, I ′′ 0 (u 0 ) ≥ 0 and A 4 (ϕ) > 0 for all ϕ ∈ K 0 \ {0}. Then u 0 is a strict local minimizer for I 0 . Moreover, there exists ν 1 > 0 such that u 0 is the only solution to ∆ g u + hu = u 2 ⋆ -1 in B ν1 (u 0 ).

Proof of Proposition 7.1. We begin with proving the first part of the proposition. We let ϕ ∈ K 0 and φ ∈ (Ru 0 ⊕ K 0 ) ⊥ be in H 2 1 (M ). A Taylor expansion yields

h ) as ϕ, φ → 0. As one can check, for any u, v ∈ H 2 1 (M ), u ≡ 0, we have that

Therefore, since φ-φ(ϕ) ∈ K ⊥ 0 , it follows from the definition of φ(ϕ) in Proposition 4.1, Claim 6.1 and A 3 ≡ 0 that (57)

We claim that I ′′ 0 (u 0 ) ≥ 0. We prove the claim. Indeed, since I ′ 0 (u 0 ) = 0, straightforward computations yield

With the choice of u 0 and h in (63), we then get that

. Therefore, it follows from the above characterization of K 0 above that

0 (u 0 ), we then get that I ′′ 0 (u 0 ) ≥ 0. This proves the claim. Since the elements of K 0 are independent of the second variable, we get that

where the orthogonality in H 2 1 (M 1 ) is considered with respect to the standard L 2 -product. As a consequence, the expression (61) can be rewritten (66)

for all ϕ ∈ K 0 , where c 1 := (2 ⋆ -1)(2 ⋆ -2)u 2 ⋆ -4 0 and, for simplicity, we have written K 0 for Λ 1 (M 1 , g 1 ). We now distinguish Cases (i) and (ii) of Proposition 7.3:

We define Ã4 for ũ0 and therefore (61) yields 2

for all ϕ ∈ Λ 1 (M 1 , g 1 ). Plugging this expression into (66) yields

Vol g2 (M 2 ). In particular, if ũ0 is a local minimizer for Ĩ0 , then (54) yields Ã4 ≥ 0. Therefore, A 4 (ϕ) > 0 for all ϕ ∈ Λ 1 (M 1 , g 1 ) \ {0} since n -d > 0. This proves Proposition 7.3 in Case (i).

Case (ii): (M 1 , g 1 ) = (S d (r), can). The case d ≥ 3 is covered by Case (i), and only the cases d = 1, 2 remain to be covered. For simplicity, we assume that r = 1. It follows from Berger-Gauduchon-Mazet [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF] that the second positive eigenvalue λ 2 (S d , can) is 2(d + 1) and the eigenfunctions are the restrictions to S d of secondorder homogeneous harmonic polynomials on R d+1 . We let Eucl be the Euclidean metric on R d+1 . We claim that

where λ 1 = d and λ 2 = 2(d + 1). We prove the claim. We fix ϕ ∈ Λ 1 (S d , can).

In particular ϕ 2 is a second-order homogeneous polynomial on R d+1 , and

2(d+1) |x| 2 is a harmonic second-order homogenous polynomial, and therefore its restriction to S d is an eigenfunction for λ 2 . Since ∆ Eucl (ϕ 2 ) is constant and |x| 2 is constant on S d , (67) follows from a direct computation. This proves the claim.

We claim that (68)

We prove the claim. Since, up to homothetic transformation, ϕ is a coordinate function, proving (68) is equivalent to proving S d x 4 dv can = (3/(d+3)) S d x 2 dv can where x is the first coordinate in R d+1 . This latest identity follows from the change of variable (t, σ) → (t, √ 1 -t 2 σ) from (-1, 1) × S d-1 to S d \ {(±1, ..., 0)}. This proves the claim.

Plugging (67) and (68) into (66) yields

for all ϕ ∈ Λ 1 (S d , can). In particular, since d < n, we have that A 4 (ϕ) > 0 for all ϕ ∈ Λ 1 (S d , can) \ {0}. This proves Case (ii) of Proposition 7.3 when r = 1. The general case follows by rescaling. This proves Proposition 7.3.

As a remark, the computations made for Case (ii) are valid when d = n ≥ 3 (that is M = S d = S n ), and we get that A 4 ≡ 0, which has been obtained by another method in Proposition 7.2.

When h ≡ c n R g , an immediate consequence of Proposition 7.3 is the following: Corollary 7.1. Let (N, g N ) be a compact Riemannian manifold of positive constant scalar curvature. We choose d ≥ 1 and we assume that (69) R g N < dim(N )λ 1 (N, g N ) and n := d + dim(N ) ≥ 3 .

We endow the manifold M := S d dim(N ) • d/R g N × N with the product metric g := can ⊕g N . Then the positive constant solution to the scalar curvature equation ∆ g u + c n R g u = u 2 ⋆ -1 on M is a degenerate strict local minimizer.


Frédéric Robert
email: frederic.robert@univ-lorraine.fr

Jérôme Vétois
email: vetois@unice.fr

EXAMPLES OF NON-ISOLATED BLOW-UP FOR PERTURBATIONS OF THE SCALAR CURVATURE EQUATION ON NON LOCALLY CONFORMALLY FLAT MANIFOLDS

Keywords: nonlinear elliptic equations, blow-up, conformal invariance. 2010 Mathematics Subject Classification: 35J35, 35J60, 58J05, 35B44

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.

Introduction and statement of the results

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Given a sequence (h ε ) ε>0 ∈ C ∞ (M ), we are interested in the existence of multi peaks positive solutions (u ε ) ε>0 ∈ C ∞ (M ) to the family of critical equations [START_REF] Ambrosetti | Variational perturbative methods and bifurcation of bound states from the essential spectrum[END_REF] ∆ g u ε + h ε u ε = u 2 ⋆ -1 ε in M for all ε > 0, where ∆ g := -div g (∇) is the Laplace-Beltrami operator, and 2 ⋆ := 2n n-2

is the critical Sobolev exponent. We say that the family (u ε ) ε blows up as ε → 0 if lim ε→0 u ε ∞ = +∞. Blowing-up families to equations like (1) are described precisely by Struwe [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF] in the energy space H 2 1 (M) : namely, if the Dirichlet energy of u ε is uniformly bounded with respect to ε, then there exists u 0 ∈ C ∞ (M ), there exists k ∈ N, there exists k families (ξ i,ε ) ε ∈ M and (µ i,ε ) ε ∈ (0, +∞) such that [START_REF] Brendle | Blow-up phenomena for the Yamabe equation[END_REF]

u ε = u 0 + k i=1 n(n -2)µ i,ε µ 2 i,ε + d g (•, ξ i,ε ) 2 n-2 2 + o(1),
where lim ε→0 o(1) = 0 in H 2 1 (M ) and lim ε→0 µ i,ε = 0 for all i = 1, ..., k. In this situation, we say that u ε develops k peaks when ε → 0.

We say that ξ 0 ∈ M is a blow-up point for (u ε ) ε if lim ε→0 max Br(ξ 0 ) u ε = +∞ for all r > 0. It follows from elliptic theory that the blow-up points of a family of solutions (u ε ) ε to (1) satisfying ( 2) is exactly {lim ε→0 ξ i,ε / i = 1, .., k}.

Following the terminology introduced by Schoen

[17], ξ 0 ∈ M is an isolated point of blow-up for (u ε ) ε if there exists (ξ ε ) ε ∈ M such that • ξ ε is a local maximum point of u ε for all ε > 0, • lim ε→0 ξ ε = ξ 0 , • there exist C, r > 0 s.t. d g (x, ξ ε ) n-2 2 u ε (x) ≤ C for all x ∈ B r(ξ 0 ), • lim ε→0 max Br(ξ 0 ) u ε = +∞ for all r > 0.
The notion has proved to be very useful in the analysis of critical equations. Let c n := n-2 4(n-1) and R g be the scalar curvature of (M, g). Compactness for the Yamabe equation

(3) ∆ g u + c n R g u = u 2 ⋆ -1
when n ≤ 24 (the full result is due to Kuhri-Marques-Schoen [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF]) is established by proving first that the sole possible blow-up points for (3) are isolated, see Schoen [START_REF] Schoen | Notes from graduates lecture in Stanford University[END_REF][START_REF]On the number of constant scalar curvature metrics in a conformal class, Differential geometry[END_REF], Li-Zhu [START_REF] Li | Yamabe type equations on three-dimensional Riemannian manifolds[END_REF], Druet [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF], Marques [START_REF] Marques | A priori estimates for the Yamabe problem in the non-locally conformally flat case[END_REF], Li-Zhang (Theorem 1.1 in [START_REF] Li | Compactness of solutions to the Yamabe problem. III[END_REF]), and Kuhri-Marques-Schoen [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF]). When n ≥ 25, there are examples of non-compactness of equation (3) (Brendle [START_REF] Brendle | Blow-up phenomena for the Yamabe equation[END_REF] and Brendle-Marques [START_REF] Brendle | Blow-up phenomena for the Yamabe equation[END_REF]).

In this note, we address the questions to know whether or not blowup solutions for (1) do exist, and whether or not they necessarily have isolated blow-up points. When h ε ≤ c n R g , blow-up does not occur for n ≤ 5 as shown by Druet [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF] (except for the conformal class of the round sphere). When the potential is allowed to be above the scalar curvature, blow-up is possible: we refer to Druet-Hebey [START_REF] Druet | Blow-up examples for second order elliptic PDEs of critical Sobolev growth[END_REF] for examples of nonisolated blow-up on the sphere with C 1 -perturbations of the scalar curvature term in (3), and to Esposito-Pistoia-Vétois [START_REF] Esposito | The effect of linear perturbations on the Yamabe problem[END_REF] for examples of isolated blow-up on general compact manifolds with arbitrary smooth perturbations of the scalar curvature. We present in this note examples of non-isolated blow-up points for smooth perturbations of the scalar curvature term in (3). This is the subject of the following theorem.

Theorem 1.1. Let (M, g) be a non-locally-conformally flat compact Riemannian manifold of dimension n ≥ 6 with positive Yamabe invariant. We fix ξ 0 ∈ M such that the Weyl tensor at ξ 0 is such that Weyl g (ξ 0 ) = 0. We let k ≥ 1 and r ≥ 0 be two integers. Then there exists

(h ε ) ε>0 ∈ C ∞ (M ) such that lim ε→0 h ε = c n R g in C r (M ), and there exists (u ε ) ε>0 ∈ C ∞ (M ) a family of solutions to ∆ g u ε + h ε u ε = u 2 ⋆ -1 ε in M for all ε > 0,
such that (u ε ) ε develops k peaks at the blow-up point ξ 0 . Moreover, ξ 0 is an isolated blow-up point if and only if k = 1.

In particular, Theorem 1.1 applies for M := S p × S q (p, q ≥ 3) endowed with the product metric. In this case, any point can be a blow-up point since the Weyl tensor never vanishes on S p × S q . As a consequence, when dealing with general perturbed equations like [START_REF] Ambrosetti | Variational perturbative methods and bifurcation of bound states from the essential spectrum[END_REF], one has to deal with the delicate situation of the accumulation of peaks at a single point. The C 0 -theory by Druet-Hebey-Robert [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF] addresses this question in the a priori setting and L ∞ -norm. We refer also to Druet [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF] and Druet-Hebey [START_REF]Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF] where the analysis of the radii of interaction of multi peaks solutions is performed.

The choice of this note is to perturb the potential c n R g of the equation. Alternatively, one can fix the potential c n R g and multiply the nonlinearity u 2 ⋆ -1 by smooth functions then leading to consider Kazdan-Warner type equations: in this slightly different context, Chen-Lin [START_REF] Chen | Blowing up with infinite energy of conformal metrics on S n[END_REF] and Brendle (private communication) have constructed non-isolated local blowup respectively in the flat case and in the Riemannian case.

Acknowledgements: the authors express their deep thanks to E. Hebey for stimulating discussions and constant support for this project. The first author thanks C.-S. Lin for stimulating discussions and S. Brendle for communicating his unpublished result.

Proofs

The proof of Theorem 1.1 relies on a Lyapunov-Schmidt reduction. We fix ξ 0 ∈ M such that Weyl g (ξ 0 ) = 0. It follows from the classical conformal normal coordinates theorem of Lee-Parker [START_REF] Lee | The Yamabe problem[END_REF] that there exists Λ ∈ C ∞ (M × M ) such that for any ξ ∈ M ,

R g ξ (ξ) = 0, ∇R g ξ (ξ) = 0, and ∆ g ξ R g ξ (ξ) = 1 6 Weyl g (ξ) 2 g ,
where Λ ξ := Λ(ξ, •) and g ξ := Λ 4/(n-2) ξ g. Without loss of generality, up to a conformal change of metric, we assume that g ξ 0 = g. We let r 0 > 0 be such that r 0 < i g ξ (M ) for all ξ ∈ M compact, where i g ξ (M ) is the injectivity radius of M with respect to the metric g ξ . We let χ ∈ C ∞ (R) be such that χ(t) = 1 for t ≤ r 0 /2 and χ(t) = 0 for t ≥ r 0 . We define a bubble centered at ξ with parameter δ as:

W δ,ξ := χ(d g (•, ξ))Λ ξ n(n -2)δ δ 2 + d g ξ (•, ξ) 2 n-2 2
.

We fix an integer k ≥ 1. Given α > 1 and K > 0, we define the set

D (k) α,K (δ) := ((δ i ) i , (ξ i ) i ) ∈ (0, δ) k × M k 1 α < δ i δ j < α ; d g (ξ i , ξ j ) 2 δ i δ j > K for i = j .
For any h ∈ C 0 (M ), we define the functional:

J h (u) := 1 2 M (|∇u| 2 g + hu 2 ) dv g - 1 2 ⋆ M u 2 ⋆ + dv g for all u ∈ H 2 1 (M ). For ((δ i ) i , (ξ i ) i ) ∈ D (k)
α,K , we define the error

R (δ i ) i ,(ξ i ) := (∆ g + h) k i=1 W δ i ,ξ i - k i=1 W δ i ,ξ i 2 ⋆ -1 2n n+2
The classical Lyapunov-Schmidt finite-dimensional reduction yields the following:

Proposition 2.1. We fix α > 1, η > 0, C 0 > 0 such that h ∞ ≤ C 0 and λ 1 (∆ g +h) ≥ C -1 0 . Then there exists K 0 = K 0 ((M, g), α, C 0 , η) > 0, δ 0 = δ 0 ((M, g), α, C 0 , η) > 0 and φ ∈ C 1 (D (k) α,K 0 (δ 0 ), H 2 1 (M )) such that • R (δ i ) i ,(ξ i ) < η for all (δ i ) i , (ξ i ) ∈ D (k) α,K 0 (δ 0 ), • u((δ i ) i , (ξ i )) := k i=1 W δ i ,ξ i + φ((δ i ) i , (ξ i )) is a critical point of J h iff ((δ i ) i , (ξ i )) is a critical point of ((δ i ) i , (ξ i )) → J h (u((δ i ) i , (ξ i ))) in D (k) α,K 0 (δ 0 ), • φ((δ i ) i , (ξ i )) H 2 1 = O(R (δ i ) i ,(ξ i ) ), • J h (u((δ i ) i , (ξ i ) i )) = J h ( k i=1 W δ i ,ξ i ) + O(R 2 (δ i ) i ,(ξ i ) ). Here, |O(1)| ≤ C((M, g), α, C 0 ) uniformly in D (k) α,K 0 (δ 0 ).
This result is essentially contained in the existing litterature. It is a particular case of the general reduction theorem in Robert-Vétois [START_REF] Robert | A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations via Lyapunov-Schmidt's reduction[END_REF]. We also refer to Esposito-Pistoia-Vétois [START_REF] Esposito | The effect of linear perturbations on the Yamabe problem[END_REF] and to the general framework by Ambrosetti-Badiale [START_REF] Ambrosetti | Variational perturbative methods and bifurcation of bound states from the essential spectrum[END_REF] for nondegenerate critical manifolds.

From now on, we fix ((

δ i ) i , (ξ i ) i ) ∈ D (k) α,K 0 (δ 0 ). Standard computations yield J h k i=1 W δ i ,ξ i = k i=1 J h (W δ i ,ξ i ) + i =j M (∇W δ i ,ξ i , ∇W δ j ,ξ j ) g + hW δ i ,ξ i W δ j ,ξ j dv g - 1 2 ⋆ M k i=1 W δ i ,ξ i 2 ⋆ - k i=1 W 2 ⋆ δ i ,ξ i dv g and M k i=1 W δ i ,ξ i 2 ⋆ - k i=1 W 2 ⋆ δ i ,ξ i dv g = O i =j W δ i ,ξ i ≤W δ j ,ξ j W δ i ,ξ i W 2 ⋆ -1 δ j ,ξ j dv g .
Choosing K 0 larger if necessary, there exists c 1 = c 1 (α, K 0 ) > 0 such that for any i = j and x ∈ M such that W δ i ,ξ i (x) ≤ W δ j ,ξ j (x), we have that d g ξ i (x, ξ i ) ≥ c 1 (d g (ξ i , ξ j ) + d g (x, ξ j )). Therefore, we get that

W δ i ,ξ i (x) ≤ c 2 δ
(n-2)/2 j d g (ξ i , ξ j ) 2-n for all such x, for some constant c 2 = c 2 (α, K 0 ) > 0. Consequently, a rough upper bound yields

J h k i=1 W δ i ,ξ i = k i=1 J h (W δ i ,ξ i ) + O i =j δ i δ j d g (ξ i , ξ j ) 2 n-2 2
and

R (δ i ) i ,(ξ i ) ≤ k i=1 (∆ g +h)W δ i ,ξ i -W 2 ⋆ -1 δ i ,ξ i 2n n+2 +O i =j δ i δ j d g (ξ i , ξ j ) 2 n-2 4 uniformly in D (k)
α,K 0 (δ 0 ). Moreover, see Proposition 2.3 in Esposito-Pistoia-Vétois [START_REF] Esposito | The effect of linear perturbations on the Yamabe problem[END_REF], we have that

J h (W δ,ξ ) = K -n n n 1 + 2(n -1) (n -2)(n -4) (h -c n R g )(ξ)δ 2 + O( h -c n R g C 1 )δ 3 -|W eyl g (ξ)| 2 g 1 64 δ 4 ln 1 δ + O(δ 4 ) when n = 6 1 24(n-4)(n-6) δ 4 + O(δ 5 ) when n ≥ 7 and (∆ g + h)W δ,ξ -W 2 ⋆ -1 δ,ξ 2n n+2 ≤ Cδ 2 1 + h -c n R g C 0 ln 1 δ 2/3 when n = 6 √ δ + h -c n R g C 0 when n ≥ 7.
Here again, |O(1)| ≤ C((M, g), α, C 0 ) uniformly in D

α,K 0 (δ 0 ). We now choose the (δ i ), (ξ i ) ′ s and the function h. For any ε > 0, we let δ ε > 0 be such that

δ 2 ε ln 1 δ ε = ε when n = 6 and δ 2 ε = ε when n ≥ 7.
We let

H ∈ C ∞ (R n ) be such that • H(x) = -1 for all |x| > 2,
• H admits k distinct strict local maxima at p i,0 ∈ B 1 (0) for i = 1, ..., k, • H(p i,0 ) > 0 for all i = 1, ..., k. We let r > 0 be such that for any i ∈ {1, ..., k}, the maximum of H on B 2r (p i,0 ) is achieved exactly at p i,0 and such that |p i,0 -p j,0 | ≥ 3r for all i = j. We let (µ ε ) ε ∈ (0, +∞) be such that lim ε→0 µ ε = 0 and (| ln ε|) -1/4 = o(µ ε ) when n = 6 and ε n-6

2(n-2) = o(µ ε ) when n ≥ 7,
where both limits are taken when ε → 0. As one can check, δ ε = o(µ ε ) when ε → 0. We define

h ε (x) := c n R g (x) + εH µ -1 ε exp -1 ξ 0 (x) for all x ∈ M.
Here, the exponential map is taken with respect to the metric g and after assimilation to R n of the tangent space at ξ 0 : this definition makes sense for ε > 0 small enough. For (t i ) i ∈ (0, +∞) k and (p i

) i ∈ (R n ) k , we define ũε ((t i ) i , (p i ) i ) := u (t i δ ε ) i , (exp ξ 0 (µ ε p i )) i with h ≡ h ε .
The above estimates and the choice of the parameters yield

(4) lim ε→0 J hε (ũ ε ((t i ) i , (p i ) i )) -k K -n n n εδ 2 ε = k i=1 F n (t i , p i ) in C 0 loc ((0, +∞) k × k i=1 B r (p i,0
)), where

F n (t, p) := 2(n -1) (n -2)(n -4) H(p)t 2 -d n |W eyl g (ξ 0 )| 2 g t 4
for (t, p) ∈ (0, +∞) × R n , with d 6 = 1 64 and d n := 1 24(n-4)(n-6) for n ≥ 7. As easily checked, up to choosing the t ′ i s in suitable compact intervals I 1 , ..., I k , the right-hand-side of (4) has a unique maximum point in the interior of k i=1 I i × k i=1 B r(p i,0 ). As a consequence, for ε > 0 small enough, J hε (ũ ε ((t i ) i , (p i ) i )) admits a critical point, ((t i,ε ) i , (p i,ε ) i ) ∈ (α, β) k × k i=1 B r(p i,0 ) for some 0 < α < β independent of ε. Defining ξ i,ε := exp ξ 0 (µ ε p i,ε ) for all i = 1, ..., k, there exists c 0 > 0 such that d(ξ i,ε , ξ i,ε ) ≥ c 0 µ ε for all i = j ∈ {1, ..., k} and all ε > 0 small enough. Defining u ε := ũε ((t i,ε ) i , (p i,ε ) i ), it follows from Proposition 2.1 and the strong maximum principle that

∆ g u ε + h ε u ε = u 2 ⋆ -1 ε in M
for ε > 0 small enough. In addition to the hypotheses above, we require that ε = o(µ r ε ) when ε → 0, which yields lim ε→0 h ε = c n R g in C r (M ). We prove that (u ε ) ε develops no isolated blow-up point when k ≥ 2. We argue by contradiction. Moser's iterative scheme yields the convergence of u ε to 0 in C 2 loc (M \ {ξ 0 }). We then get that the isolated blow-up point is ξ 0 , and thus that there exists r 1 > 0 and (ξ ε ) ε ∈ M such that lim ε→0 ξ ε = ξ 0 and there exists C > 0 such that [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF] d g (x, ξ ε ) n-2 2 u ε (x) ≤ C for all ε > 0 and x ∈ B r 1 (ξ 0 ).

For any i = 1, .., k, we recall that ξ i,ε := exp ξ 0 (µ ε p i,ε ) and we define ũi,ε (x) := (δ ε t i,ε ) n-2 2 u ε (exp ξ i,ε (δ ε t i,ε x)) for all |x| < r 0 /(2δ ε t i,ε ). It follows from standard elliptic theory that (6) lim ε→0 ũi,ε = n(n -2)

1 + | • | 2 n-2 2 in C 2 loc (R n ).
Moreover, if δ ε = o(d g (ξ i,ε , ξ ε )) when ε → 0, inequality (5) yields the convergence of ũi,ε to 0 in C 0 loc (R n ): a contradiction to [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF]. Therefore, d g (ξ ε , ξ i,ε ) = O(δ ε ) when ε → 0 for all i = 1, ..., k, and then d g (ξ i,ε , ξ j,ε ) = O(δ ε ) = o(µ ε ) when ε → 0 for all i = j. This contradicts the fact that d g (ξ i,ε , ξ j,ε ) ≥ c 0 µ ε when k ≥ 2. This proves the non-simpleness when k ≥ 2.

The authors are partially supported by the ANR grant ANR-08-BLAN-0335-01.