Alice Lebreton

Rafal Tomecki

Andrzej Dziembowski
email: andrzejd@ibb.waw.pl

Bertrand Séraphin
email: seraphin@igbmc.fr

Endonucleolytic RNA

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The exosome complex, first identified as a 3'→5' exonuclease necessary for the 3'-' end processing of the yeast 5.8S ribosomal RNA [START_REF] Mitchell | The exosome: a conserved eukaryotic RNA processing complex containing multiple 3'→5' exoribonucleases[END_REF] , was later implicated in the 3'-5' end processing of many stable RNA species [START_REF] Allmang | Functions of the exosome in rRNA, snoRNA and snRNA synthesis[END_REF] , in messenger RNA turnover [START_REF] Anderson | The 3' to 5' degradation of yeast mRNAs is a general mechanism for mRNA turnover that requires the SKI2 DEVH box protein and 3' to 5' exonucleases of the exosome complex[END_REF] , in various mRNA surveillance pathways [START_REF] Isken | Quality control of eukaryotic mRNA: safeguarding cells from abnormal mRNA function[END_REF] and in the degradation of cryptic unstable transcripts (CUTs) [START_REF] Wyers | Cryptic pol II transcripts are degraded by a nuclear quality control pathway involving a new poly(A) polymerase[END_REF] . The exosome is therefore a major eukaryotic 3'→5' exonuclease with pleiotropic functions [START_REF] Lebreton | Exosome-mediated quality control: Substrate recruitment and molecular activity[END_REF] .

Figure 1 | Endonucleolytic activity of the Dis3 PIN domain in vitro. a, Domain organization and amino-acid numbering of yeast Dis3 and E. coli RNase II: RNB domains have exoribonucleolytic activity, OB-folds (CSD1, CSD2 and S1) are RNA-binding domains, and the N-terminal PIN domain is unique to Dis3. Sites mutated in this study are indicated. b, Mutations of conserved aspartates of the Dis3 PIN domain abolish its endoribonucleolytic activity. Gel-electrophoresis analysis of products generated by the incubation of the 5'-labelled 17-A 14 oligoribonucleotide (5'-P; 3'-OH) with various Dis3 versions, in the presence of various divalent cations, for the durations indicated (minutes). Preferential cleavage at adenine residues was observed for this particular substrate but was not investigated further. c, The Dis3 PIN domain degrades circular RNA substrates. Wild-type (WT) Dis3 or PIN-domain mutants were incubated with a circularized oligoribonucleotide in the presence of 3 mM MnCl 2 for the durations indicated (minutes). Linear substrate molecules result from spontaneous cleavage or breakage during handling before the reaction.

Eukaryotic exosomes consist of a conserved core interacting with nuclear or cytoplasmic specific subunits, and pathway-specific cofactors. The core consists of a ring structure formed by nine subunits [START_REF] Liu | Reconstitution, activities, and structure of the eukaryotic RNA exosome[END_REF][START_REF] Hernandez | Subunit architecture of multimeric complexes isolated directly from cells[END_REF] , similar to the archaeal exosome and bacterial phosphorolytic nucleases, and the Dis3 protein (also called Rrp44), which contains a catalytic RNB domain responsible for the activity of hydrolytic RNase II/R family members [START_REF] Frazao | Unravelling the dynamics of RNA degradation by ribonuclease II and its RNA-bound complex[END_REF][START_REF] Lorentzen | Structure of the active subunit of the yeast exosome core, Rrp44: diverse modes of substrate recruitment in the RNase II nuclease family[END_REF][START_REF] Zuo | Structural basis for processivity and single-strand specificity of RNase II[END_REF] . Association of Dis3 with the ring structure seems stable in yeast but may be more labile in other species [START_REF] Lebreton | Exosome-mediated quality control: Substrate recruitment and molecular activity[END_REF] . Recent studies indicated that eukaryotic ring structures are devoid of nuclease activity and that only the Dis3 RNB domain is a functionally active 3'→5' exonuclease [START_REF] Dziembowski | A single subunit, Dis3, is essentially responsible for yeast exosome core activity[END_REF][START_REF] Liu | Reconstitution, activities, and structure of the eukaryotic RNA exosome[END_REF] . In yeast, the non-essential 3'→5' exoribonuclease nuclear-specific subunit Rrp6 cooperates with Dis3 [START_REF] Allmang | The yeast exosome and human PM-Scl are related complexes of 3'→5' exonucleases[END_REF] .

In addition to the RNB domain and three OB-fold nucleic-acid-binding domains present in all RNase II/R family members [START_REF] Frazao | Unravelling the dynamics of RNA degradation by ribonuclease II and its RNA-bound complex[END_REF][START_REF] Lorentzen | Structure of the active subunit of the yeast exosome core, Rrp44: diverse modes of substrate recruitment in the RNase II nuclease family[END_REF][START_REF] Zuo | Structural basis for processivity and single-strand specificity of RNase II[END_REF] , Dis3 harbours a highly conserved N-terminal PIN domain (Fig. 1a and Supplementary Fig. 1). PIN domains have been shown to adopt a RNase H fold, some of which are endowed with nuclease activity [START_REF] Arcus | Distant structural homology leads to the functional characterization of an archaeal PIN domain as an exonuclease[END_REF][START_REF] Glavan | Structures of the PIN domains of SMG6 and SMG5 reveal a nuclease within the mRNA surveillance complex[END_REF] . An alignment of Dis3 PIN domain sequences with those of PIN domains of known structure reveals that all acidic residues essential for the coordination of divalent cations, and thus putatively involved in catalysis, are conserved (Supplementary Fig. 1). To test whether the N-terminal PIN domain of Dis3 is endowed with ribonucleolytic activity, we assayed RNA degradation by Dis3 D551N , a recombinant Dis3 mutant of the RNB domain devoid of exoribonucleolytic activity [START_REF] Dziembowski | A single subunit, Dis3, is essentially responsible for yeast exosome core activity[END_REF] . Dis3 D551N slowly degraded a short 5'-end-labelled oligoribonucleotide 17-A 14 (CCCCACCACCAUCACUUA 14 ) in a divalent-cation-dependent manner (Mg 2+ , Mn 2+ or Zn 2+ ; Fig. 1b). Higher concentrations of divalent cations (1-3 mM) were required than were optimal for exoribonucleolytic activity [START_REF] Dziembowski | A single subunit, Dis3, is essentially responsible for yeast exosome core activity[END_REF] . To determine whether the PIN domain was responsible for the activity observed with Dis3 D551N , we purified mutant proteins (Dis3 D171N , D551N and Dis3 D198N , D551N ) with additional single asparagine substitutions of conserved aspartic residues in the predicted PINdomain catalytic centre (Fig. 1a and Supplementary Fig. 1). We obtained comparable quantities of the single-mutant and double-mutant proteins with equivalent degrees of purity (Supplementary Fig. 2a). The ability of Dis3 D551N to degrade the 5'-labelled RNA substrate in the presence of divalent cations was lost in the double mutants (Fig. 1b), showing that this RNase activity requires an intact Dis3 PIN domain. In addition, recombinant PIN domain alone (Dis3 amino-acid residues 1-241), but not D171N or D198N mutants of this domain, was able to degrade the same RNA substrate (Supplementary Fig. 3), showing that the PIN activity acts independently of the other Dis3 domains. The Dis3 D551N activity was highest with Mn 2+ and lowest with Mg 2+ , in good agreement with results obtained for another PIN domain [START_REF] Glavan | Structures of the PIN domains of SMG6 and SMG5 reveal a nuclease within the mRNA surveillance complex[END_REF] . Degradation intermediates observed with the 5'-end-labelled substrate were consistent with either a 3'→5' exonucleolytic activity or an endonucleolytic one. The degradation of a 3'-end-labelled substrate by Dis3 D551N , but not by the double mutants, revealed a complementary profile of intermediates (Supplementary Fig. 4), indicating that the activity was most probably endonucleolytic.

Comparison of the activities of the wild-type protein with single and double mutants of the PIN and RNB domains at different concentrations of divalent cations demonstrated endonucleolytic degradation in the context of an exonucleolytically active protein and indicated cooperation of the two activities (Supplementary Fig. 5). Moreover, the wild-type protein, but not the mutants of the PIN domain (Dis3 D171N or Dis3 D198N ), was able to digest a circular RNA molecule (Fig. 1c). Taken together, these results show that the Dis3 PIN domain is responsible for a previously unidentified endoribonucleolytic activity of the yeast exosome. The Dis3 protein therefore possesses two distinct nuclease domains that cooperate for RNA degradation, and their respective contributions in vitro depend on the assay conditions.

To test the physiological relevance of the PIN activity, we introduced wild-type or mutated dis3 versions, fused to the protein A tag, on centromeric plasmids in a strain in which the essential chromosomal DIS3 copy was under the control of a doxycyclin-repressible promoter [START_REF] Belli | An activator/repressor dual system allows tight tetracycline-regulated gene expression in budding yeast[END_REF] . We tested the growth phenotypes in the presence or absence of doxycyclin at various temperatures (Fig. 2). After repression of the wild-type DIS3 allele, the D171N PIN mutant showed a modest growth defect at elevated temperatures. Growth of the exonucleolytically inactive mutant D551N was impaired, as reported [START_REF] Dziembowski | A single subunit, Dis3, is essentially responsible for yeast exosome core activity[END_REF] . Finally, Dis3 D171N,D551N was much more profoundly affected than each single mutant. Mutant phenotypes did not result from protein instability or inability to assemble into exosome cores (Supplementary Fig. 6). As the growth phenotype paralleled the relative nuclease activities observed in vitro (Supplementary Fig. 5), this result indicates that both nuclease activities of Dis3 are physiologically relevant. The D171N mutation also exacerbated the growth phenotype of the rrp6D mutant, further supporting the functional implication of PIN in exosome function (Supplementary Fig. 7). In the absence of doxycyclin, the DIS3 D171N,D551N allele generated a weak dominant-negative phenotype at 37 °C, indicating that this inactive protein is expressed and probably blocks exosome function.

Analysis of total RNA by northern blotting with an oligonucleotide probe complementary to the 3' end of the 5'-external transcribed spacer (ETS) by-product of rRNA processing, a known exosome substrate 2 , revealed that both the D171N and D551N mutants accumulated this species, with a stronger effect for dis3 D551N (Fig. 3a, top panel). The double mutant accumulated even higher amounts of this RNA, its levels reaching those observed in the absence of Dis3. A series of discrete 5'-ETS degradation intermediates was detected in the D551N mutant. These species were no longer visible for dis3 D171N , D551N , suggesting that the PIN endonucleolytic activity was involved in their generation.

Further hybridizations performed with oligonucleotide or randomly primed probes spanning the 5'-ETS region revealed that the intermediates originated from various regions of this RNA, suggesting that they do not result from a simple 5'→3' or 3'→5' exonucleolytic decay (Supplementary Fig. 8). Deletion of RRP6 had modest effects on the degradation pattern, indicating that 5'-ETS digestion depends essentially on both endonuclease and exonuclease activity of Dis3. We used primer extension and circularization RT-PCR (cRT-PCR [START_REF] Couttet | Messenger RNA deadenylylation precedes decapping in mammalian cells[END_REF][START_REF] Mandl | Sequencing the termini of capped viral RNA by 5'-3' ligation and PCR[END_REF] ) to identify precisely the 5' and 3' ends of 5'-ETS intermediates accumulating in the Dis3 D551N mutant (Fig. 3b and Supplementary Fig. 9a). Intermediates, with sizes ranging from 79 to 462 nucleotides, spanned the complete 5'-ETS. Some of them contained short non-templated 3'-terminal heterogeneous oligo(A) tails, probably synthesized by the TRAMP (Trf4/Air2/Mtr4p polyadenylation) complex, a known exosome stimulator [START_REF] Wyers | Cryptic pol II transcripts are degraded by a nuclear quality control pathway involving a new poly(A) polymerase[END_REF][START_REF] Kadaba | Nuclear surveillance and degradation of hypomodified initiator tRNAMet in S. cerevisiae[END_REF][START_REF] Lacava | RNA degradation by the exosome is promoted by a nuclear polyadenylation complex[END_REF][START_REF] Vanacova | A new yeast poly(A) polymerase complex involved in RNA quality control[END_REF] . More importantly, the 3' end of some species precisely abutted the 5' end of other molecules, which is further evidence of their endonucleolytic origin (Fig. 3b). 5'-ETS cleavages formed in vivo mapped preferentially to internal and closing loops as well as to other single-stranded regions (Supplementary Fig. 9b), making an endonucleolytic rather than an exonucleolytic origin more likely. Similar 5'-ETS cleavages could be reproduced in vitro (Fig. 3c; compare with intermediate sizes observed with the random-primed probe shown in Supplementary Fig. 8).

Northern blot analyses targeted to other natural exosome substrates showed that the PIN domain endonuclease activity was not specific for the 5'-ETS (Fig. 3a, lower panels). D171N and D551N mutations in Dis3 had synergistic effects on the degradation of the CUT NEL025 and on the processing of the 7S precursor into 5.8S rRNA. These effects were further enhanced in an rrp6D Our results show that the yeast exosome, and probably exosomes from other eukaryotes, have endonuclease activity in addition to their long-known exonucleolytic activity. The current model of exosome action should therefore be revisited, because these observations have direct implications for the mechanism of recruitment and degradation of substrates. Inactivation of the Dis3 PIN domain by point mutation has limited effects on RNA processing and cell growth. This suggests that RNAs are still degraded in the absence of endonucleolytic cleavages, but probably more slowly or inefficiently. The PIN domain therefore seems to cooperate with the exonucleolytic activity of Dis3 and/or Rrp6, possibly by providing them with alternative 3'-end substrates, which is consistent with the additive phenotypes seen in multiple mutants. This feature could facilitate the degradation of structured RNA species by allowing degradation reinitiation in upstream regions when exosome progression is blocked. Dis3 is therefore a unique example of a RNase in which endoribonucleolytic and exoribonucleolytic catalytic sites are found encoded by different domains of a single protein.

Given the structure of related bacterial and archaeal enzymes, Dis3 is likely to originate from the fusion of a PIN domain to a RNase II/R enzyme during the early steps of eukaryotic evolution. In Escherichia coli, the main complex involved in RNA degradation, the degradosome, also has both endonucleolytic and exonucleolytic activities, even though the functions are performed by proteins with no similarity to Dis3, namely endoribonuclease E and polynucleotide phosphorylase [START_REF] Carpousis | The RNA degradosome of Escherichia coli: an mRNA-degrading machine assembled on RNase E[END_REF] . This evolutionary convergence supports the functional importance for cooperation between exonucleolytic and endonucleolytic activities. Further analysis should help us to understand the relative contributions and spatial arrangements of the two active sites of core eukaryotic exosomes.

Methods

RNase assays

In vitro enzymatic assays were performed in 20-µl reaction volumes containing 10 mM Tris-HCl pH 8.0, 75 mM NaCl, and 1 mM 2-mercaptoethanol, with MnCl 2 and/or MgCl 2 or ZnCl 2 at final concentrations ranging from 0 to 3 mM. Purified protein and substrate concentrations were 0.1 and 0.2 µM, respectively. Reaction mixtures were incubated at 30 °C for the indicated durations before the addition of 1 volume of formamide loading dye (90% formamide, 20 mM EDTA, 0.03% bromophenol blue, 0.03% xylene cyanol in 1 × TBE). Reaction products were resolved in denaturing gels containing 20% polyacrylamide, 8 M urea and 1 × TBE, and were detected with a Fuji PhosphorImager.

cRT-PCR

The circularization RT-PCR (cRT-PCR) procedure, adapted from previous studies [START_REF] Couttet | Messenger RNA deadenylylation precedes decapping in mammalian cells[END_REF][START_REF] Mandl | Sequencing the termini of capped viral RNA by 5'-3' ligation and PCR[END_REF] , permitted the mapping of 5' and 3' ends in non-polyadenylated, non-capped RNA intermediates. Total RNA from the D551N mutant strain BSY1668[pBS3270] was treated with DNase I. RNA (7 mg) was ligated in the presence of 1 ml of T4 RNA ligase for 2 h at 37 °C in a 100-ml final reaction volume. The circularized RNAs were extracted with phenol-chloroform, precipitated with ethanolammonium acetate, and resuspended in 20 ml of water. Reverse transcriptions were performed on 600 ng of circularized RNA with 10 pmol of primers OBS2999, OBS3001 or OBS3081 (Supplementary Table 1) and 1 ml of iScript reverse transcriptase (Bio-Rad). The cDNA junctions were then amplified by PCR with the cognate primer pair: either OBS2999-OBS3002 or OBS3001-OBS3080 or OBS3081-OBS3083. Products were cloned into pCR2.1-TOPO by TOPO-TA cloning (Invitrogen), and sequenced with the M13 reverse primer.

Oligonucleotides, plasmids, bacterial strains and yeast strains

Oligonucleotides used in this work are described in Supplementary Table 1, plasmids in Supplementary Table 2 and yeast strains in Supplementary Table 3.

Plasmids pAD260 and pAD262 were generated by site-directed mutagenesis with oligonucleotides RTADZ-32 to RTADZ-33, using pAD258 or pAD259 as templates, respectively. pAD261 and pAD263 were obtained by site-directed mutagenesis with oligonucleotides RTADZ-34 to RTADZ-35, employing pAD258 or pAD259 as templates. D171N and D198N mutations were confirmed by PCR with primers T7-long to RTADZ-10, followed by digestion with TaqI or BshTI, respectively. pAD264 expression vector, encoding the N-terminal fusion of wild-type PIN domain with 6×His-SUMOTag was constructed by cloning the sequence corresponding to residues 1-241 of Dis3 into BamHI and XhoI sites of pET-28M-SUMO Kan vector. pAD265, pAD266 and pAD267 were obtained by site-directed mutagenesis with oligonucleotides RTADZ-30 to RTADZ-31, RTADZ-32 to RTADZ-33 and RTADZ-34 to RTADZ-35, respectively, and pAD264 as a template. D171N and D198N mutations were confirmed as above, and D91N mutation in pAD265 was verified by PCR with primers T7-long to RTADZ-10, followed by digestion with DraI. pAD268 was generated by cloning of a PCR product corresponding to the 5 -ETS, made with RTADZ-70 to RTADZ-71 primers using genomic DNA from BMA64 strain as template, into pCR-Blunt II-TOPO. pBS3211 and pBS3210 were obtained by TOPO-blunt cloning (Invitrogen) of PCR products made with oligonucleotides OBS2638 and OBS2639 on DNA from yeast strains BSY1704 and BSY1708, respectively. The insert contained part of the LYS2 sequence, the complete DIS3 or dis3 D551N open reading frame in fusion with a tobacco etch virus (TEV) protease cleavage site-Protein A (PA) tag, and the TRP1 marker. The D171N mutation was introduced in these plasmids by site-directed mutagenesis with OBS2648 and OBS2649, giving pBS3241 and pBS3245, respectively. The four DIS3 alleles, together with their promoter sequences and the TEV-PA tag, were excised from the TOPO vectors by digestion with SpeI, and cloned into the yeast LEU2 centromeric plasmid pFL36 [START_REF] Bonneaud | A family of low and high copy replicative, integrative and single-stranded S. cerevisiae/E. coli shuttle vectors[END_REF] , giving pBS3269 to pBS3278. pBS3318 was obtained by TOPO-TA cloning of a PCR product corresponding to the 5'-ETS amplified with oligonucleotides OBS2616 and CS18 from genomic DNA.

For expression of the various DIS3 alleles, BSY1668, BSY1865 and BSY1883 were transformed with plasmids pBS3269 to pBS3278. Cultures in synthetic complete medium without leucine were grown at 28 °C. The chromosomal, wild-type DIS3 allele was repressed by the addition of doxycyclin (20 µg ml -1 ).

Heterologous expression and purification of recombinant proteins

For purification of full-length Dis3 versions, E. coli BL21-CodonPlus-RIL strain (Stratagene) was transformed with the appropriate plasmid (pAD258 to pAD263). Selected transformants were grown in 1 l of selective medium after inoculation from a pre-culture at D 600 ≈ 0.1. Growth was performed at 37 °C until D 600 reached 0.45-0.5. Cultures were cooled to 16 °C for 30 min. Protein expression was induced by the addition of isopropyl β-D-thiogalactoside to 0.5 mM and incubation was continued overnight at 16 °C. After centrifugation, bacterial pellets were resuspended in buffer A (20 mM Tris-HCl pH 8.0, 200 mM NaCl, 10 mM imidazole, 10 mM 2-mercaptoethanol) and lysed by sonication, followed by ultracentrifugation. Soluble fractions were loaded on a His-Trap FF Crude column on an Äkta FPLC purification system (GE Healthcare). The column was washed with buffer A and subsequently with buffer B (20 mM Tris-HCl pH 8.0; 1 M NaCl, 10 mM imidazole, 10 mM 2-mercaptoethanol). After re-equilibration with buffer A, bound proteins were eluted with buffer C (20 mM Tris-HCl pH 8.0, 200 mM NaCl, 600 mM imidazole, 10 mM 2mercaptoethanol). Proteins were further purified by ion-exchange chromatography with a MiniQ 4.6/50 PE column followed by size-exclusion chromatography on a Superdex 200 10/300 GL column (both from GE Healthcare). Final elution was performed in buffer containing 10 mM Tris-HCl pH 8.0, 150 mM NaCl. Finally, glycerol was added to the protein preparations (30% final concentration), which were stored at -80 °C. The purity and quantity of the proteins were estimated by analysis in NuPAGE gels (Invitrogen).

Various PIN versions were purified as follows. Transformation of E. coli BL21-CodonPlus-RIL with expression vectors (pAD264 to pAD267; see Supplementary Table 2), culture conditions, the induction of protein expression and extract preparation procedures were the same as for the fulllength Dis3 proteins (see above). Initial purification of 6×His-SUMOTag-PIN fusion proteins was performed as described above. Fractions eluted with buffer C (containing imidazole at high concentration) were combined and dialysed overnight at 4 °C against 2 l of buffer A in the presence of 6×His-SUMO protease. After dialysis, the protein mixture was subjected to the second round of purification on a His-Trap FF Crude column, encompassing the collection of purified PIN (lacking 6×His-SUMOTag) in the flowthrough (in buffer A) and elution of 6×His-SUMOTag together with 6×His-SUMO protease in buffer D (20 mM Tris-HCl pH 8.0, 1 M NaCl, 600 mM imidazole, 10 mM 2-mercaptoethanol). PIN versions thus obtained were further purified by size-exclusion chromatography on a Superdex 75 10/300 GL column (GE Healthcare) and stored at -80 °C in the presence of 30% glycerol.

Substrate preparation for biochemical assays

The 17-A 14 oligoribonucleotide (Invitrogen) was purified on a 10% denaturing polyacrylamide gel. Gel bands containing the oligoribonucleotide were detected by ultraviolet shadowing; they were then excised and the RNAs were eluted overnight at 25 °C in an equal volume of 100 mM Tris-HCl pH 8.0, 12.5 mM EDTA, 150 mM NaCl, 1% SDS and phenol. 5' ends labelling was performed with T4 polynucleotide kinase (NEB) and [γ-32 P]ATP (GE Healthcare) in accordance with the manufacturer's instructions.

3' ends labelling was performed with home-made [ 32 P]pCp (prepared by incubating cytidine 3'monophosphate with [γ-32 P]ATP in the presence of T4 polynucleotide kinase) and T4 RNA ligase (NEB). Circular 17-A 14 substrate was prepared by incubating 150 pmol of the oligoribonucleotide with 250 pmol of unlabelled ATP and 4 pmol of [γ-32 P]ATP in 50 mM Tris-HCl pH 8.0, 10 mM MgCl 2 , 10 mM dithiothreitol, using T4 polynucleotide kinase and T4 RNA ligase. 5'-ETS substrate was prepared with pAD268 plasmid (see Supplementary Table 1), digested with a combination of SapI and XhoI, as template for in vitro transcription, which was performed with T7 RNA polymerase (NEB) in the presence of [α-32 P]UTP (Amersham Biosciences), in accordance with the manufacturer's instructions.

All RNA substrates were purified from 6-20% denaturing polyacrylamide gels.

Exosome co-precipitation

Dis3-TEV-PA-associated complexes from 400 ml of BSY1865[pBS3269-78] culture at D 600 = 1 were affinity-purified on IgG-Sepharose resin, followed by TEV cleavage as described previously [START_REF] Rigaut | A generic protein purification method for protein complex characterization and proteome exploration[END_REF] . Input fraction (0.3 ml), or 7.5 ml of the eluates, was separated by gradient SDS-PAGE, and transferred to a nitrocellulose membrane for western blot analyses. A rabbit antibody recognizing the calmodulin-binding peptide tag and cross-reacting with the Protein A tag (anti-TAP antibody; Open Biosystems) was used at a dilution of 1:2,000, followed by incubation with a goat anti-rabbithorseradish peroxidase IgG secondary antibody (Pierce) at a dilution of 1:5,000. Signals generated by enhanced chemiluminescence (ECL; GE Healthcare) were detected with a Las3000 device (Fuji).

Northern blots and sequencing

Total RNA was extracted with a standard phenol procedure [START_REF] Daugeron | The yeast POP2 gene encodes a nuclease involved in mRNA deadenylation[END_REF] . RNA (3-10 mg) was fractionated by electrophoresis on a 6% polyacrylamide-urea gel before transfer to a Hybond N + membrane (GE Healthcare). The membrane was saturated and hybridizations were performed in Church buffer, as described previously [START_REF] Daugeron | The yeast POP2 gene encodes a nuclease involved in mRNA deadenylation[END_REF] , at 42 °C for 32 P-labelled oligonucleotide probes or at 55 °C for randomprimed PCR probes. OBS2998, OBS2999 and OBS3081 oligonucleotides hybridized in 5'-ETS. CS5 hybridized in the 5.8S rRNA [START_REF] Saveanu | Nog2p, a putative GTPase associated with pre-60S subunits and required for late 60S maturation steps[END_REF] . The PCR probes against scR1, 5'-ETS and NEL025 CUT were amplified from yeast genomic DNA with oligonucleotides FWSCR1D-R, OBS2616-CS18 and PP2-4, respectively. The three probes were labelled with [α-32 P]dCTP by using the MegaPrime DNA labelling system (GE Healthcare). Signals were detected by phosphorimaging on a Storm device (GE Healthcare).

Figure 2 |

 2 Figure 2 | Synthetic cell-growth phenotypes of PIN mutants. Growth phenotypes resulting from expression of plasmid-borne DIS3, DIS3 D551N , DIS3 D171N,D551N or DIS3 D171N alleles were assessed in the presence (repressed chromosomal DIS3) or absence (expressed chromosomal DIS3) of doxycyclin (DOX) after incubation for 50 h at 25 or 37 °C. -LEU, without leucine.

Figure 3 |

 3 Figure 3 | Dis3 PIN domain mediates cleavage of natural exosome substrates in vivo. a, Detection by northern blot of 5'-ETS, 5.8S rRNA and the NEL025 CUT in total RNA isolated from wild-type and mutant Dis3 strains. The top blot was probed with OBS3081 recognizing the 3' of 5'-ETS, the middle blots with a probe complementary to the 5.8S rRNA (two exposures), and the bottom blot with a random-primed probe overlapping the NEL025 CUT. Sizes in nucleotides of the denatured DNA ladder is specified on the left. 0 designates the empty vector. The values in the table represent the ratio of full-length 5'-ETS to control scR1 levels (means ± s.d.). b, cRT-PCR mapping of 5'-ETS intermediates accumulating in a dis3 D551N mutant strain. A bar indicates each intermediate identified, with grey areas highlighting the region required for PCR amplification by the selected primer pairs numbered in italics. Asterisks indicate fragments ending at A 0 . Dashed lines link 3' ends to 5' ends of contiguous fragments. Polyadenylated 3' ends are indicated between parentheses. c, In vitro cleavage of 5'-ETS by Dis3. 5'-ETS substrate was prepared by in vitro transcription and incubated alone or with various Dis3 versions for the durations indicated. Products resulting from endonucleolytic cleavages were separated by gel electrophoresis from short oligoribonucleotides resulting from exoribonucleolytic degradation that migrated at the bottom of the gel.

Acknowledgements

We thank E. Conti and J. Basquin for providing the Dis3 PIN domain expression construct and 6×His-SUMO protease, and E. Conti, E. Lorentzen, J. Kufel and members of our groups for insightful discussions. This work was supported by La Ligue contre le Cancer (Équipe Labellisée 2008), Agence Nationale de la Recherche project CUTs, CNRS, ESF RNA Quality program (project EUxosome) and the FP6 EU grant 3D repertoire, and an EMBO installation grant. R.T. is the recipient of the Stipend for Young Researchers from the Foundation for Polish Science and was supported through a Faculty of Biology, University of Warsaw intramural grant.

Supplementary Information is linked to the online version of the paper at http://www.nature.com/nature/journal/v456/n7224/suppinfo/nature07480.html.

Author Contributions R.T. expressed and purified the recombinant proteins and performed all the in vitro experiments under the supervision of A.D. A.L. performed all in vivo experiments under the supervision of B.S. All authors discussed the results and wrote the paper.


Saikat Mazumdar
email: saikat.mazumdar@univ-lorraine.fr

GJMS-TYPE OPERATORS ON A COMPACT RIEMANNIAN MANIFOLD: BEST CONSTANTS AND CORON-TYPE SOLUTIONS

In this paper we investigate the existence of solutions to a nonlinear elliptic problem involving critical Sobolev exponent for a polyharmonic operator on a Riemannian manifold M . We first show that the best constant of the Sobolev embedding on a manifold can be chosen as close as one wants to the Euclidean one, and as a consequence derive the existence of minimizers when the energy functional goes below a quantified threshold. Next, higher energy solutions are obtained by Coron's topological method, provided that the minimizing solution does not exist. To perform this topological argument, we overcome the difficulty of dealing with polyharmonic operators on a Riemannian manifold and adapting Lions's concentration-compactness lemma. Unlike Coron's original argument for a bounded domain in R n , we need to do more than chopping out a small ball from the manifold M . Indeed, our topological assumption that a small sphere on M centred at a point p ∈ M does not retract to a point in M \{p} is necessary, as shown for the case of the canonical sphere where chopping out a small ball is not enough.

Introduction

Let M be a compact manifold of dimension n ≥ 3 without boundary. Let k be a positive integer such that 2k < n. Taking inspiration from the construction of the ambient metric of Fefferman-Graham [START_REF] Fefferman | Conformal invariants[END_REF] (see [START_REF]The ambient metric[END_REF] for an extended analysis of the ambient metric), Graham-Jenne-Mason-Sparling [START_REF] Graham | Conformally invariant powers of the Laplacian. I. Existence[END_REF] have defined a family of conformally invariant operators defined for any Riemannian metric. More precisely, for any Riemannian metric g on M , there exists a local differential operator P g : C ∞ (M ) → C ∞ (M ) such that P g = ∆ k g + lot where ∆ g := -div g (∇), and, given u ∈ C ∞ (M ) and defining ĝ = u 4 n-2k g, we have that [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF] P ĝ (ϕ) = u -n+2k n-2k P g (uϕ) for all ϕ ∈ C ∞ (M ).

Moreover, P g is self-adjoint with respect to the L 2 -scalar product. A scalar invariant is associated to this operator, namely the Q-curvature, denoted as Q g ∈ C ∞ (M ). When k = 1, P g is the conformal Laplacian and the Q-curvature is the scalar curvature multiplied by a constant. When k = 2, P g is the Paneitz operator introduced in [START_REF] Stephen | A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds[END_REF]. The Q-curvature was introduced by Branson and Ørsted [START_REF] Branson | Explicit functional determinants in four dimensions[END_REF]. The definition of Q g was then generalized by Branson [START_REF] Thomas | The functional determinant[END_REF][START_REF]Sharp inequalities, the functional determinant, and the complementary series[END_REF]. In the specific case n > 2k, we have that Q g := 2 n-2k P g [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF]. Then, taking ϕ ≡ 1 in (1), we get that

P g u = n-2k 2 Q ĝ u n+2k
n-2k on M . Therefore, prescribing the Q-curvature in a conformal class amounts to solving a nonlinear elliptic partial differential equation(PDE )of 2k th order. Results for the prescription of the Q-curvature problem for the Paneitz operator (namely k = 2) are in Djadli-Hebey-Ledoux [START_REF] Djadli | Paneitz-type operators and applications[END_REF], Robert [START_REF] Robert | Positive solutions for a fourth order equation invariant under isometries[END_REF], Esposito-Robert [START_REF] Esposito | Mountain pass critical points for Paneitz-Branson operators[END_REF]. Recently, Gursky-Malchiodi [START_REF] Gursky | A strong maximum principle for the Paneitz operator and a non-local flow for the Q-curvature[END_REF] proved the existence of a metric with constant Q-curvature (still for k = 2) provided certain geometric hypotheses on the manifold (M, g) holds. These hypotheses have been simplified by Hang-Yang [START_REF] Hang | Sign of Greens function of Paneitz operators and the Q curvature[END_REF] (see the lecture notes [START_REF]Lectures on the fourth order Q-curvature equation[END_REF])

In the present paper, we are interested in a generalization of the prescription of the Q-curvature problem. Namely, given f ∈ C ∞ (M ), we investigate the existence of u ∈ C ∞ (M ), u > 0, such that (2)

P u = f u 2 k -1 in M,
where 2 k := 2n n-2k and P : C ∞ (M ) → C ∞ (M ) is a smooth self-adjoint 2k th order partial differential operator defined by

P u = ∆ k g u + k-1 l=0
(-1) l ∇ j l ...j1 A l (g) i1...i l ,j1...j l ∇ i1...i l u (3) where the indices are raised via the musical isomorphism and for all l ∈ {0, . . . , k -1}, A l (g) is a smooth symmetric T 0 2l -tensor field on M (that is: A l (g)(X, Y ) = A l (g)(Y, X) for all T l 0 -tensors X, Y on M ). When P := P g , then (2) is equivalent to say that Q ĝ = 2 n-2k f with ĝ = u 4 n-2k g.

The conformal invariance (1) of the geometric operator P g yields obstruction to the existence of solutions to [START_REF] Michael | Convergence and rigidity of manifolds under Ricci curvature bounds[END_REF]. The historical reference here is Kazdan-Warner [START_REF] Kazdan | Scalar curvature and conformal deformation of Riemannian structure[END_REF]; for the general GJMS operators, we refer to Delanoë-Robert [START_REF] Delanoë | On the local Nirenberg problem for the Q-curvatures[END_REF]. In particular, it follows from [START_REF] Delanoë | On the local Nirenberg problem for the Q-curvatures[END_REF] that on the canonical sphere (S n , can), there is no positive solution u ∈ C ∞ (S n ) to P can u = (1 + ϕ)u 2 k -1 for all = 0 and all first spherical harmonic ϕ. For the conformal Laplacian (that is k = 1), Aubin [3] proved that the existence of solutions is guaranteed if a functional goes below a specific threshold. We generalize this result for any k ≥ 1 in Theorem 3. In the case of a smooth bounded domain, Coron [START_REF] Coron | Topologie et cas limite des injections de Sobolev[END_REF] introduced a variational method based on topological arguments, provided the minimizing solution does not exist. Our main theorem is in this spirit:

Theorem 1. Let (M, g) be a smooth, compact Riemannian manifold of dimension n and let k be a positive integer such that 2k < n. We let P be a coercive operator as in (3). Let ι g > 0 be the injectivity radius of the manifold M . Suppose that the manifold M contains a point x 0 such that the embedded (n-1)-dimensional sphere S x0 (ι g /2) := {x ∈ M/ d g (x, x 0 ) = ι g /2} is not contractible in M \{x 0 }. Then there exists 0 ∈ (0, ιg 2 ) such that the equation

P u = |u| 2 k -2 u in Ω M D α u = 0 on ∂Ω M for |α| ≤ k -1 (4)
has a non-trivial C 2k (Ω M ) solution for Ω M := M \B x0 ( 0 ). Moreover, if the Green's Kernel of P on Ω M is positive, then we can choose u > 0.

In the original result of Coron [START_REF] Coron | Topologie et cas limite des injections de Sobolev[END_REF] (see also Weth and al. [START_REF] Bartsch | A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator[END_REF] for the case k = 2), the authors work with a smooth domain of R n and assume that it has a small "hole".

In the context of a compact manifold, this assumption is not enough: indeed, the entire compact manifold minus a small hole might retract on a point. We discuss the example of the canonical sphere in Section 7, where the existence of a hole is not sufficient to get solutions to [START_REF] Michael | Convergence and rigidity of manifolds under Ricci curvature bounds[END_REF]. Concerning higher-order problems, we refer to Bartsch-Weth-Willem [START_REF] Bartsch | A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator[END_REF], Pucci-Serrin [START_REF] Pucci | Critical exponents and critical dimensions for polyharmonic operators[END_REF], Ge-Wei-Zhou [START_REF] Ge | A critical elliptic problem for polyharmonic operators[END_REF], the general monograph Gazzola-Grunau-Sweers [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF] and the references therein.

Among other tools, the proof of Theorem 1 uses a Lions-type Concentration Compactness Lemma adapted to the context of a Riemannian manifold: this will be the object of Theorem 4.

Equation [START_REF] Michael | Convergence and rigidity of manifolds under Ricci curvature bounds[END_REF] has a variational structure. Since P is self-adjoint in L 2 , we have that for all u, v ∈ C ∞ (M ).

M uP (v) dv g = M vP (u) dv g = M ∆ k/2 g u∆ k/2 g v dv g + k-1 l=0 M A l (g)(∇ l u, ∇ l v) dv g (5)
where

∆ l/2 g u := ∆ m g u if l = 2m is even ∇∆ m g u if l = 2m + 1 is odd and, when l = 2m + 1 is odd, ∆ k/2 g u ∆ k/2 g v = ∇∆ m g u, ∇∆ m g v g . If P is coercive and f > 0, then, up to multiplying by a constant, any solution u ∈ C ∞ (M ) to (2) is a critical point of the functional u → J P (u) := M uP (u) dv g M f |u| 2 k dv g 2/2 k . ( 6
)
It follows from (5) that J P makes sense in the Sobolev spaces H 2 k (M ), where for 1 ≤ l ≤ k, H 2 l (M ) which is the completion of C ∞ (M ) with respect to the u → l α=0 ∇ α u 2 . Equivalently (see Robert [START_REF]Admissible Q-curvatures under isometries for the conformal GJMS operators[END_REF]), H 2 l (M ) is also the completion of the space C ∞ (M ) with respect to the norm [START_REF] Thomas | The functional determinant[END_REF] u

2 H 2 l := l α=0 M (∆ α/2 g u) 2 dv g .
By the Sobolev embedding theorem we get a continuous but not compact embedding of

H 2 k (M ) into L 2 k (M ). The continuity of the embedding H 2 k (M ) → L 2 k (M ) yields a pair of real numbers A, B such that for all u ∈ H 2 k (M ) u 2 L 2 k ≤ A M (∆ k/2 g u) 2 dv g + B u 2 H 2 k-1 . ( 8
)
See for example Aubin [START_REF]Some nonlinear problems in Riemannian geometry[END_REF] or Hebey [START_REF] Hebey | Nonlinear analysis on manifolds: Sobolev spaces and inequalities[END_REF]. Following the terminology introduced by Hebey, we then define A(M ) := inf{A ∈ R : ∃ B ∈ R with the property that inequality (8) holds}. [START_REF] Branson | Explicit functional determinants in four dimensions[END_REF] As for the classical case k = 1 (see Aubin [START_REF]Some nonlinear problems in Riemannian geometry[END_REF]), the value of A(M ) depends only on k and the dimension n. More precisely, we let D k,2 (R n ) be the completion of

C ∞ c (R n ) for the norm u → ∆ k/2 u 2 , and we define K 0 (n, k) > 0 (10) 1 K 0 (n, k) := inf u∈D k,2 (R n )\{0} R n (∆ k/2 u) 2 dx R n |u| 2 k dx 2 2 k
.

as the best constant in the Sobolev's continuous embedding

D k,2 (R n ) → L 2 k (R n ).
Our second result is the following:

Theorem 2. Let (M, g) be a smooth, compact Riemannian manifold of dimension n and let k be a positive integer such that 2k < n. Then A(M ) = K 0 (n, k) > 0. In particular, for any > 0, there exists

B ∈ R such that for all u ∈ H 2 k (M ) one has M |u| 2 k dv g 2 2 k ≤ (K 0 (n, k) + ) M (∆ k/2 g u) 2 dv g + B u 2 H 2 k-1 . ( 11
)
As a consequence of this result, we will be able to prove the existence of solutions to (2) when the functional J P goes below a quantified threshold, see Theorem 3. This paper is organized as follows. In Section 2, we study the best-constant problem and prove Theorem 2. In Section 3, we prove Theorem 3 by classical minimizing method. In Section 4, we prove a Concentration-Compactness Lemma in the spirit of Lions. Section 5 is devoted to test-functions estimates and the proof of the existence of solutions to (4) via a Coron-type topological method. Section 6 deals with positive solutions, and Section 7 with the necessity of the topological assumption of Theorem 1. The appendices concern regularity and a general comparison between geometric norms.

Acknowledgements. I would like to express my deep gratitude to Professor Frédéric Robert and Professor Dong Ye, my thesis supervisors, for their patient guidance, enthusiastic encouragement and useful critiques of this work.

The Best Constant

It follows from Lions [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case. I and II[END_REF] and Swanson [START_REF] Swanson | The best Sobolev constant[END_REF] that the extremal functions for the Sobolev inequality [START_REF] Coron | Topologie et cas limite des injections de Sobolev[END_REF] exist and are exactly multiples of the functions

U a,λ = α n,k λ 1 + λ 2 |x -a| 2 n-2k 2 a ∈ R n , λ > 0 (12)
where the choice of α n,k 's are such that for all λ, U a,λ

2 k = 1 and U a,λ 2
D k,2 = 1 K0(n,k) . They satisfies the equation ∆ k u = 1 K0(n,k) |u| 2 k -2 u in R n
Next we consider the case of a compact Riemmanian manifold. The first result we have in this direction is the following.

Lemma 2.1. Let (M, g) be a smooth, compact Riemannian manifold of dimension n and let k be a positve integer such that 2k < n. Any constant A in inequality [START_REF]Sharp inequalities, the functional determinant, and the complementary series[END_REF] has to be greater than or equal to K 0 (n, k), whatever the constant B be.

Proof of Lemma 2.1: We fix > 0 small. It follows from Lemma 9.1 that there exists, δ 0 ∈ (0, ι g ) depending only on (M, g), , where ι g is the injectivity radius of M , such that for any point p ∈ M , any 0

< δ < δ 0 , l ≤ k and u ∈ C ∞ c (B 0 (δ)) M (∆ l/2 g (u • exp -1 p )) 2 g dv g ≤ (1 + ) R n (∆ l/2 u) 2 dx (13) and (1 -)   R n |u| 2 k dx   2/2 k ≤   M |u • exp -1 p | 2 k dv g   2/2 k (14)
Then plugging the above inequalities into [START_REF]Sharp inequalities, the functional determinant, and the complementary series[END_REF] we obtain that any

u ∈ C ∞ c (B 0 (δ)) satisfies   R n |u| 2 k dx   2/2 k ≤ 1 + 1 - A R n (∆ k/2 u) 2 dx + C k-1 l=0 R n |∇ l u| 2 dx. (15) Let v ∈ C ∞ c (R n ) with supp(v) ⊂ B 0 (R 0 ). For λ > 1 let v λ = v(λx). Then for λ large, supp(v λ ) ⊂ B 0 (δ). Taking u ≡ v λ in (15), a change of variable yields 1 λ n-2k   R n |v| 2 k dx   2/2 k ≤ 1 + 1 - • A λ n-2k R n (∆ k/2 v) 2 dx + C k-1 l=0 1 λ n-2l R n |∇ l v| 2 dx. (16)
Multiplying by λ n-2k and letting λ → +∞, we get that for all v ∈ D k,2 (R n ), we have

  R n |v| 2 k dx   2/2 k ≤ 1 + 1 - A R n (∆ k/2 v) 2 dx. ( 17
)
Therefore 1+ 1-A ≥ K 0 (n, k) for all > 0, and letting → 0 yields A ≥ K 0 (n, k). This ends the proof of Lemma 2.1.

2 We now prove [START_REF] Delanoë | On the local Nirenberg problem for the Q-curvatures[END_REF] to get Theorem 2.

Step 1: A local inequality. From a result of Anderson (Main lemma 2.2 of [START_REF] Michael | Convergence and rigidity of manifolds under Ricci curvature bounds[END_REF]) it follows that for any point p ∈ M there exists a harmonic coordinate chart ϕ around p. Then from Lemma 9.1, for any 0 < < 1, there exists τ > 0 small enough such that for any point p ∈ M and for any u ∈ C ∞ c (B p (τ )), one has

R n (∆ k/2 (u • ϕ -1 )) 2 dx ≤ 1 + 3K 0 (n, k) M (∆ k/2 g u) 2 dv g (18)
and

  M |u| 2 k dv g   2/2 k ≤ 1 + 3K 0 (n, k)   R n |u • ϕ -1 | 2 k dx   2/2 k . ( 19
)
The expression for the Laplacian ∆ g in the harmonic coordinates is ∆ g u = -g ij ∂ ij u. Then [START_REF] Coron | Topologie et cas limite des injections de Sobolev[END_REF] implies that for any u ∈ C ∞ c (B p (τ ))

  M |u| 2 k dv g   2/2 k ≤ (K 0 (n, k) + ) M (∆ k/2 g u) 2 dv g . ( 20
)
Step 2: Finite covering and proof of the global inequality. Since M is compact, it can be covered by a finite number of balls B pi (τ /2), i = 1, . . . , N . Let

α i ∈ C ∞ c (B pi (τ )) be such that 0 ≤ α i ≤ 1 and α i = 1 in B pi (τ /2).
We set

η i = α 2 i N j=1 α 2 j . (21)
Then (η i ) i=1,...,N is a partition of unity subordinate to the cover (B pi (τ )) i=1,...,N such that √ η i 's are smooth and

N i=1 η i = 1.
In the sequel, C denote any positive constant depending on k, n, the metric g on M and the functions (η i ) i=1,...,N . Now for any u ∈ C ∞ (M ), we have

u 2 2 k = u 2 2 k /2 = N i=1 η i u 2 2 k /2 ≤ N i=1 η i u 2 2 k /2 = N i=1 √ η i u 2 2 k . ( 22
)
So for any u ∈ C ∞ (M ), using inequality [START_REF] Hang | Sign of Greens function of Paneitz operators and the Q curvature[END_REF] we obtain that

  M |u| 2 k dv g   2/2 k ≤ (K 0 (n, k) + ) N i=1 M (∆ k/2 g ( √ η i u)) 2 g dv g . ( 23
)
Next we claim that there exists C > 0 such that

N i=1 M (∆ k/2 g ( √ η i u)) 2 dv g ≤ M (∆ k/2 g u) 2 dv g + C u 2 H 2 k-1 . ( 24
)
Assuming that (24) holds we have from ( 23)

  M |u| 2 k dv g   2/2 k ≤ (K 0 (n, k) + ) M (∆ k/2 g u) 2 g dv g + (K 0 (n, k) + )C u 2 H 2 k-1 . ( 25
)
This proves [START_REF] Delanoë | On the local Nirenberg problem for the Q-curvatures[END_REF], and therefore, with Lemma 2.1, this proves Theorem 8. We are now left with proving [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case. I and II[END_REF].

Step 3: Proof of (24): For any positive integer m, one can write that

∆ m g ( √ η i u) = √ η i ∆ m g u + P (2m-1,1) g (u, √ η i ) + L 2m-2 √ ηi,g (u) (26)
where

P (2m-1,1) g (u, √ η i ) = |l|=2m-1,|β|=1 (a l,β ∂ β √ η i )∇ l u, and L 2m-2 √ ηi,g (u) = 2m-2 |l|=0 a l ( √ η i ) ∇ l u (27)
the coefficients a l,β and a l ( √ η i ) are smooth functions on M . The a l,β 's depends only on the metric g and on the manifold M and a l ( √ η i )'s depends both on the metric g, the function √ η i and its derivatives upto order 2m. We shall use the same notations P

(2m-1,1) g (u, √ η i ), L 2m-2 √ ηi,g (u)
for any expression of the above form.

Step 3.1: k is even. We then write k = 2m, m ≥ 1, and then

N i=1 M ∆ m g ( √ η i u) 2 dv g = N i=1 M η i ∆ m g u 2 dv g + N i=1 M P (2m-1,1) g (u, √ η i ) 2 dv g + N i=1 M L 2m-2 √ ηi,g (u) 2 dv g +2 N i=1 M √ η i ∆ m g u P (2m-1,1) g (u, √ η i ) dv g + 2 N i=1 M √ η i ∆ m g u L 2m-2 √ ηi,g (u) dv g +2 N i=1 M P (2m-1,1) g (u, √ η i ) L 2m-2 √ ηi,g (u) dv g (28)
We note that

N i=1 M P (2m-1,1) g (u, √ η i ) 2 dv g ≤ C u 2 H 2 2m-1
. and

N i=1 M L 2m-2 √ ηi,g (u) 2 dv g ≤ C u 2 H 2 2m-2 . ( 29
)
On the other hand

N i=1 M √ η i ∆ m g u P (2m-1,1) g (u, √ η i ) dv g = N i=1 |l|=2m-1 |β|=1 M ( √ η i ∆ m g u)((a l,β ∂ β √ η i )∇ l u) dv g = 1 2 N i=1 |l|=2m-1 |β|=1 M (∆ m g u)((a l,β ∂ β η i )∇ l u) dv g = 1 2 |l|=2m-1 |β|=1 M (∆ m g u)((a l,β ∂ β ( N i=1 η i ))∇ l u) dv g = 0 ( 30
)
while using the integration by parts formula we obtain

N i=1 M √ η i ∆ m g u L 2m-2 √ ηi,g (u) dv g ≤ C u 2 H 2 2m-1 (31)
and by Hölder inequality

N i=1 M P (2m-1,1) g (u, √ η i ) L 2m-2 √ ηi,g (u) dv g ≤ C u 2 H 2 2m-1 (32)
Hence if k is even, then

N i=1 M ∆ m g ( √ η i u) 2 dv g ≤ M ∆ m g u 2 dv g + C u 2 H 2 2m-1 . ( 33
)
So we have the claim for k even.

Step 3.2: k is odd. We then write k = 2m + 1 with m ≥ 0. We have

∇ ∆ m g ( √ η i u) = √ η i ∇ ∆ m g u + (∆ m g u) ∇ √ η i + ∇ P (2m-1,1) g (u, √ η i ) + ∇ L 2m-2 √ ηi,g (u)
(34) and so

N i=1 M ∇ ∆ m g ( √ η i u) 2 dv g = N i=1 M η i ∇ ∆ m g u 2 dv g + N i=1 M (∆ m g u) 2 |∇ √ η i | 2 dv g (35) + N i=1 M ∇ P (2m-1,1) g (u, √ η i ) 2 dv g + N i=1 M ∇ L 2m-2 √ ηi,g (u) 2 dv g + 2 N i=1 M ( √ η i ∇ ∆ m g u , (∆ m g u) ∇ √ η i ) dv g + 2 N i=1 M ( √ η i ∇ ∆ m g u , ∇ P (2m-1,1) g (u, √ η i ) ) dv g + 2 N i=1 M ( √ η i ∇ ∆ m g u , ∇ L 2m-2 √ ηi,g (u) ) dv g + 2 N i=1 M ((∆ m g u) ∇ √ η i , ∇ P (2m-1,1) g (u, √ η i ) ) dv g + 2 N i=1 M ((∆ m g u) ∇ √ η i , ∇ L 2m-2 √ ηi,g (u) ) dv g + 2 N i=1 M (∇ P (2m-1,1) g (u, √ η i ) , ∇ L 2m-2 √ ηi,g (u) ) dv g (36)
We have that

N i=1 M ∇ P (2m-1,1) g (u, √ η i ) 2 dv g ≤ C u 2 H 2 2m and N i=1 M ∇ L 2m-2 √ ηi,g (u) 2 dv g ≤ C u 2 H 2 2m-1 (37)
while

N i=1 M ( √ η i ∇ ∆ m g u , (∆ m g u) ∇ √ η i ) dv g = N i=1 M (∇ ∆ m g u , (∆ m g u) ( √ η i ∇ √ η i )) dv g = 1 2 N i=1 M (∇ ∆ m g u , (∆ m g u) ∇η i ) dv g = 1 2 M (∇ ∆ m g u , (∆ m g u) ∇( N i=1 η i )) dv g = 0 (38)
And we obtain

N i=1 M ( √ η i ∇ ∆ m g u , ∇ P (2m-1,1) g (u, √ η i ) ) dv g = N i=1 |l|=2m-1 |β|=1 M ( √ η i ∇ ∆ m g u , ∇ (a l,β ∂ β √ η i )∇ l u ) dv g ≤ N i=1 |l|=2m |β|=1 M ( √ η i ∇ ∆ m g u , (a l,β ∂ β √ η i )∇ l u) dv g + N i=1 |l|=2m-1 |β|=1 M (∇ ∆ m g u , ( √ η i ∇(a l,β ∂ β √ η i )) ∇ l u) dv g ≤ N i=1 |l|=2m |β|=1 M ( √ η i ∇ ∆ m g u , (a l,β ∂ β √ η i )∇ l u) dv g + N i=1 |l|=2m-1 |β|=1 Bp i (τ ) (∇ ∆ m g u , ( √ η i ∇(a l,β ∂ β √ η i )) ∇ l u) dv g (39)
Then we apply the integration by parts formula on each of the domains ϕ

-1 (B p1 (τ )) ⊂ R n to obtain N i=1 |l|=2m |β|=1 M ( √ η i ∇ ∆ m g u , (a l,β ∂ β √ η i )∇ l u) dv g + N i=1 |l|=2m-1 |β|=1 Bp i (τ ) (∇ ∆ m g u , ( √ η i ∇(a l,β ∂ β √ η i )) ∇ l u) dv g ≤ N i=1 |l|=2m |β|=1 M ( √ η i ∇ ∆ m g u , (a l,β ∂ β √ η i )∇ l u) dv g + C u 2 H 2 2m ≤ 1 2 N i=1 |l|=2m |β|=1 M (∇ ∆ m g u , (a l,β ∂ β η i )∇ l u) dv g + C u 2 H 2 2m ≤ 1 2 |l|=2m |β|=1 M (∇ ∆ m g u , (a l,β ∂ β ( N i=1 η i ))∇ l u) dv g + C u 2 H 2 2m ≤C u 2 H 2 2m since N i=1 η i = 1 (40)
Similarly after integration by parts one obtains

N i=1 M ( √ η i ∇ ∆ m g u , ∇ L 2m-2 √ ηi,g (u) ) dv g ≤ C u 2 H 2 2m (41) N i=1 M ((∆ m g u) ∇ √ η i , ∇ P (2m-1,1) g (u, √ η i ) ) dv g ≤ C u 2 H 2 2m ( 42
)
and

N i=1 M ((∆ m g u) ∇ √ η i , ∇ L 2m-2 √ ηi,g (u) ) dv g + N i=1 M (∇ P (2m-1,1) g (u, √ η i ) , ∇ L 2m-2 √ ηi,g (u) ) dv g ≤ C u 2 H 2 2m ( 43
)
Hence for k odd, we also obtain that

N i=1 M ∇ ∆ m g ( √ η i u) 2 dv g ≤ M ∇ ∆ m g u 2 g dv g + C u 2 H 2 2m . (44)
Hence we have the claim and this completes the proof.

Best constant and direct Minimizaton

Let Ω M ⊂ M be any smooth n-dimensional submanifold of M , possibly with boundary. In the sequel, we will either take Ω M = M , or M \ B x0 ( 0 ) for some 0 > 0 small enough. We define

H 2 k,0 (Ω M ) ⊂ H 2 k (M ) as the completion of C ∞ c (Ω M ) for the norm • H 2 k .
In this section, we prove the following result in the spirit of Aubin [3]: Theorem 3. Let (M, g) be a compact Riemannian manifold of dimension n > 2k, with k ≥ 1. Ω M ⊂ M be any smooth n-dimensional submanifold of M as above. Let P be a differential operator as in (3) and let f ∈ C 0,θ (Ω M ) be a Hölder continuous positive function. Assume that P is coercive on

H 2 k,0 (Ω M ). Suppose that (45) inf u∈N f Ω M uP (u) dv g < 1 sup Ω M f 2 2 k K 0 (n, k) ,
where

N f := {u ∈ H 2 k,0 (Ω M ) : Ω M f |u| 2 k dv g = 1}. ( 46
)
Then there exists a minimizer u ∈ N f . Moreover, up to multiplication by a constant,

u ∈ C 2k (Ω M ) is a solution to P u = f |u| 2 k -2 u in Ω M D α u = 0 on ∂Ω M for |α| ≤ k -1.
In addition, if the Green's function of P on Ω M with Dirichlet boundary condition is positive, then any minimizer is either positive or negative. When Ω M = M , and the Green's function of P on M is positive, then up to changing sign, u > 0 is a solution to

P u = f u 2 k -1 in M.
Proof of Theorem 3: This type of result is classical. We only sketch the proof. For simplicity, we take Ω M = M . The proof of the general case is similar. Here and in the sequel, we define (see ( 5))

I P (u) := M uP (u) dv g for all u ∈ H 2 k (M ).
We start with the following lemma:

Lemma 3.1. Let (u i ) ∈ N f be a minimizing sequence for I P on N f . Then

(i) Either there exists u 0 ∈ N f such that u i → u 0 strongly in H 2 k (M )
, and u 0 is a minimizer of

I P on N f (ii) Or there exists x 0 ∈ Ω M such that f (x 0 ) = max Ω M f and |u i | 2 k dv g δ x0 as i → +∞ in the sense of measures. Moreover, inf u∈N f I P (u) = 1 K0(n,k)(max M f ) 2 2 k . Proof of Lemma 3.1: We define α := inf{I P (u)/ u ∈ N f }. As the functional I g is coercive so the sequence (u i ) is bounded in H 2 k (M ). We let u 0 ∈ H 2 k (M ) such that, up to a subsequence, u i u 0 weakly in H 2 k (M ) as i → +∞, and u i (x) → u 0 (x) as i → +∞ for a.e. x ∈ M . Therefore, u 0 2 k L 2 k ≤ lim inf i→+∞ u i 2 k L 2 k = 1. ( 47
)
We define v i := u i -u 0 . Up to extracting a subsequence, we have that

(v i ) i → 0 in H 2 k-1 (M ). We define µ i := (∆ k/2 g u i ) 2 dv g and νi = |u i | 2 k dv g and ν i = f |u i | 2 k dv g for all i.
Up to a subsequence, we denote respectively by µ, ν and ν their limits in the sense of measures. It follows from the concentration-compactness Theorem 4 that,

(48) ν = |u 0 | 2 k dv g + j∈J α j δ xj and µ ≥ (∆ k/2 g u 0 ) 2 dv g + j∈I β j δ xi
where J ⊂ N is at most countable, (x j ) j∈J ∈ M is a family of points, and (α j

) j∈J ∈ R ≥0 , (β j ) j∈J ∈ R ≥0 are such that α 2/2 k j ≤ K 0 (n, k) β j for all j ∈ J. ( 49
)
As a consequence, we get that

(50) ν = f |u 0 | 2 k dv g + j∈J f (x j )α j δ xj
Since (u i ) ∈ N f , and M is compact, we have that M dν = 1 and then

(51) 1 = M f |u 0 | 2 k dv g + j∈J f (x j )α j . Since (u i ) i → u 0 strongly in H 2 k-1 (M ), integrating (48) yields (52) α ≥ I P (u 0 ) + j∈J β j ≥ α u 0 2 2 k + K 0 (n, k) -1 j∈J α 2/2 k j . Since α ≤ K 0 (n, k) -1 (max M f ) -2/2 k
, we then get that (i) either u 0 2 k = 1 and α j = 0 for all j ∈ J , (ii) or u 0 ≡ 0, f (x j0 )α j0 = 1 for some j 0 ∈ J , f (x j0 ) = max M f and α j = 0 for all j = j 0 .

In case (i), we get from the strong convergence to 0 of

(v i ) i in H 2 k-1 (M ) that I P (u i ) = M (∆ k/2 g v i ) 2 dv g + I P (u 0 ) + o(1) as i → +∞. Since u 0 ∈ N f and (u i ) is
a minimizing sequence, we then get that (v i ) 0 goes to 0 strongly in H 2 k (M ), and therefore u i → u 0 strongly in H 2 k (M ).

In case (ii), (52

) yields α = K 0 (n, k) -1 (max M f ) -2/
2 k and I P (u 0 ) = 0, which yields u 0 ≡ 0 since the operator is coercive.

This completes the proof of Lemma 3.1. 2

We go back to the proof of Theorem 3. Let (u i ) i be a minimizing sequence for I P on N f . It follows from the assumption (45) that case (i) of Lemma 3.1 holds, and then, there exists a minimizer u 0 ∈ N f that is a minimizer. Therefore, it is a weak solution to P k g u 0 = αf |u 0 | 2 k -2 u 0 in M (see (145) for the definition). It then follows from the regularity Theorem 8.3 that u ∈ C 2k,θ (M ).

We let G : M ×M \{(x, x)/ x ∈ M } be the Green's function of P on M . We assume that G(x, y) > 0 for all x = y ∈ M . Green's representation formula yields (53) ϕ(x) = M G(x, y)(P ϕ)(y) dv g for all x ∈ M and all ϕ ∈ C 2k (M ).

It follows from Proposition 8.2 that there exists v ∈ H 2 k (M ) such that

P v = αf |u 0 | 2 k -1 in M. ( 54
)
Standard regularity (taking inspiration from Vand der Vorst [START_REF] Van Der | Best constant for the embedding of the space H 2 ∩ H 1 0 (Ω) into L 2N/(N -4) (Ω)[END_REF]) yields v ∈ C 2k (M ). We have that P (v ± u 0 ) ≥ 0. Since G > 0, it follows from Green's formula (53) that v ± u 0 ≥ 0. So v ≥ |u 0 | and therefore v = 0. Independently, since P v ≥ 0 and v ≡ 0, Green's formula (53) yields v > 0. Using Hölder's inequality and v ≥ |u 0 |, we get that

J P (u) = M vP (v) dv g M f |v| 2 k dv g 2/2 k = α M vf |u 0 | 2 k -1 dv g M f |v| 2 k dv g 2/2 k (55) ≤ α M f |v| 2 k dv g 1 2 k M f |u 0 | 2 k dv g 2 k -1 2 k M f |v| 2 k dv g 2/2 k (56) ≤ α M f |u 0 | 2 k dv g 2 k -1 2 k M f |u 0 | 2 k dv g 1/2 k ≤ α   M f |u 0 | 2 k dv g   2 k -2 2 k ≤ α (57) since M f |u 0 | 2 k dv g = 1.
Since α is the infimum of the functional, we get that J P (u) = α. Hence v attains the infimum and therefore it also solves the equation

P v = µf v 2 k -1 weakly in M , and v ∈ H 2 k,0 (M ).
Moreover, one has equality in all the inequalities above, and then |u 0 | = v > 0, and therefore either u 0 > 0 or u 0 < 0 in M . This ends the proof of Theorem 3. 2

Concentration Compactness Lemma

We now state and prove the concentration compactness lemma in the spirit of P.-L.Lions for the case of a closed manifold: Theorem 4 (Concentration-compactness). Let (M, g) be a smooth, compact Riemannian manifold of dimension n and let k be a positive integer such that 2k < n. Suppose (u m ) be a bounded sequence in H 2 k (M ). Up to extracting a subsequence, there exist two nonnegative Borel-regular measure µ, ν on M and u ∈ H

2 k (M ) such that (a) u m u weakly in H 2 k (M ) (b) µ m := (∆ k/2 g u m ) 2 dv g µ weakly in the sense of measures (c) ν m := |u m | 2 k dv g ν

weakly in the sense of measures

Then there exists an at most countable index set I, a family of distinct points {x i ∈ M : i ∈ I}, families of nonnegative weights {α i : i ∈ I} and {β i : i ∈ I} such that (i)

ν =|u| 2 k dv g + i∈I α i δ xi (58) µ ≥(∆ k/2 g u) 2 dv g + i∈I β i δ xi (59)
where δ x denotes the Dirac mass at x ∈ M with mass equal to 1.

(ii) for all i ∈ I, α 2/2 k i ≤ K 0 (n, k) β i . In particular i∈I α 2/2 k i < ∞.
Proof of Theorem 4: By the Riesz representation theorem (µ m ), and (ν m ) are sequences of Radon measures on M .

Step 1: First we assume that u ≡ 0. Let ϕ ∈ C ∞ (M ), then from (2) we have that, given any ε > 0 there exists

B ε ∈ R such that   M |ϕu m | 2 k dv g   2/2 k ≤ (K 0 (n, k) + ε) M (∆ k/2 g (ϕu m )) 2 dv g + B ε ||ϕu m || 2 H 2 k-1 . ( 60
)
Since u m 0 in H 2 k (M ), letting m → +∞ and then taking the limit ε → 0, it follows that

  M |ϕ| 2 k dν   2/2 k ≤ K 0 (n, k) M ϕ 2 dµ. (61)
By regularity of the Borel measure ν, (61) holds for any Borel measurable function ϕ and in particular for any Borel set E ⊂ M we have

ν(E) 2/2 k ≤ K 0 (n, k) µ(E). (62)
Therefore the measure ν is absolutely continuous with respect to the measure µ and hence by the Radon-Nikodyn theorem, we get By regularity of the Borel measures µ and ν (64) holds for any Borel measurable function ϕ. This implies that f g = 1 a.e with respect to ν. So, in particular g > 0 ν a.e in S. Let ψ ∈ C(M ), taking ϕ = ψχ S in (61) we have

  M |ψ| 2 k X S dν   2/2 k ≤ K 0 (n, k) M ψ 2 X S dµ = K 0 (n, k) M ψ 2 X S [gdν + dσ] = K 0 (n, k) M ψ 2 gX S dν (65)
Since dν⊥dσ and supp ν ⊂ S, we get that

  M |ψ| 2 k dν   2/2 k ≤ K 0 (n, k) M ψ 2 g dν (66)
By regularity of the Borel measure ν the above relation holds for any Borel measurable function ψ.

Let φ ∈ C(M ) and let ψ = φg 1 2 k -2 X {g≤N } , dν N = g 2 k 2 k -2 X {g≤N } dν. Then we have   M |φ| 2 k dν N   2/2 k ≤ K 0 (n, k) M φ 2 dν N . ( 67
M χ E g 2 k 2 k -2 dν = lim N →∞ M χ E dν N . (69) So g 2 k 2 k -2 dν = i∈I αi δ xi . Since g > 0 ν a.e , there exists α i > 0 such that we have dν = i∈I α i δ xi . Since µ = gdν + dσ ≥ gdν, we get that µ ≥ i∈I β i δ xi
where

β i = g(x i )α i (70) Taking ψ = X {xi} in (66) we have for all i ∈ I α 2/2 k i ≤ K 0 (n, k) gα i = K 0 (n, k) β i (71) and 1 K 0 (n, k) i∈I α 2/2 k i ≤ i∈I β i ≤ µ(M ) < +∞. ( 72
)
This proves the theorem for u ≡ 0. This ends Step 1.

Step 2: Assume u ≡ 0 and let v m := u m -u. Then v m 0 weakly in H 2 k (M ). Therefore, as one checks, μm := (∆

k/2 g v m ) 2 dv g µ -(∆ k/2
g u) 2 dv g and νm := |v m | 2 k dv g ν -|u| 2 k dv g weakly in the sense of measures. Applying Step 1 to the measures μm and νm yields Theorem 4.

2

We now prove the reversed Hölder inequality that was used in the proof.

Proposition 4.1. Let µ be a finite Borel measure on M and suppose that for any Borel measurable function ϕ one has

  M |ϕ| q dµ   1/q ≤ C   M |ϕ| p dµ   1/p ( 73
)
for some C > 0 and 1 ≤ p < q < +∞. Then there exists j points x 1 , . . . , x j ∈ M , and j positive real numbers c 1 , . . . , c j such that

µ = j i=1 c i δ xi (74)
where δ x denotes the Dirac measure concentrated at x ∈ M with mass equal to 1.

Moreover c i ≥ ( 1 C ) pq q-p .
Proof. Let E be a Borel set in M . Taking ϕ = χ E we obtain that, either

µ(E) = 0 or µ(E) ≥ ( 1 C ) pq q-p
We define O := {x ∈ M : for some r > 0 µ(B x (r)) = 0}. Then O is open. Now if K ⊂ O is compact, then K can be covered by a finite number of balls each of which has measure 0, therefore µ(K) = 0. By the regularity of the measure hence it follows that µ(O) = 0. If x ∈ M \O, then for all r > 0 one has µ(B x (r))

≥ ( 1 C ) pq q-p . Then µ({x}) = lim m→+∞ µ(B x (1/m)) ≥ 1 C pq q-p . ( 75
)
Since the measure µ is finite, this implies that that the set M \O is finite. So let

M \O = {x 1 , • • • , x j }, therefore for any borel set E in M µ(E) = µ(E ∩ {x 1 , • • • , x j }) = xi∈E µ({x i }) = j i=1 µ({x i })δ xi (E). ( 76
)
Hence the lemma follows with c i = µ({x i }).

Topological method of Coron

In this section we obtain higher energy solutions by Coron's topological method if the functional J P does not have a minimizer, for the case f ≡ 1. This will complete the proof of the first part of Theorem 1, that is the existence of solutions to (4) with no sign-restriction. For µ > 0 and y 0 ∈ R n , we define

(77) B y0,µ (y) = α n,k µ µ 2 + |y -y 0 | 2 n-2k 2
where the choice of α n,k 's are such that for all µ, B y0,µ L 2 k = 1 and

B y0,µ 2
D k,2 = 1 K0(n,k
) . These functions are the extremal functions of the Euclidean Sobolev Inequality [START_REF] Coron | Topologie et cas limite des injections de Sobolev[END_REF] and they satisfy the equation

∆ k B y0,µ = 1 K 0 (n, k) B 2 k -1 y0,µ in R n . (78) . Let ηr ∈ C ∞ c (R n
), 0 ≤ ηr ≤ 1 be a smooth cut-off function, such that ηr = 1 for x ∈ B 0 (r) and ηr = 1 for x ∈ R n \B 0 (2r). Let ι g > 0 be the injectivity radius of (M, g). For any p ∈ M , we let η p be a smooth cut-off function on M such that

η p (x) = η ιg 10 (exp -1 p (x)) for x ∈ B p (ι g ) ⊂ M 0 for x ∈ M \B p (ι g ). ( 79
)
For any x ∈ M , we define

B M p,µ (x) = η p (x) B 0,µ (exp -1 p (x)). (80) B M
p,µ is the standard bubble centered at the point p ∈ M and with radius µ

B M p,µ (x) = α n,k η p (x) µ µ 2 + d g (p, x) 2 n-2k 2 . ( 81
)
We have Proposition 5.1. Let (M, g) be a smooth, compact Riemannian manifold of dimension n and let k be a positve integer such that 2k < n. Consider the functional J P on the space H 2 k (M )\{0}. Then the sequence of functions B M p,µ ∈ C ∞ (M ) defined above is such that:

(a) lim µ→0 J P (B M p,µ ) = 1 K0(n,k) uniformly for p ∈ M (b) lim µ→0 B M p,µ L 2 k = 1 uniformly for p ∈ M (c) B M p,µ 0 weakly in H 2 k (M ), as µ → 0
Proof of Proposition 5.1: We claim that (c) holds. We first prove that B M p,µ is uniformly bounded in H 2 k (M ). Indeed,

α≤k M ∆ α/2 g B M p,µ 2
dv g ≤ α≤k Bp(ιg/5) ∆ α/2 g B M p,µ 2 dv g ≤ C l≤k B0(ιg/5) ∇ l B M p,µ • exp p 2 dx ≤ l≤k B0(ιg/5) ∇ l µ µ 2 + |x| 2 n-2k 2 2 dx ≤ l≤k B0(ιg/(5µ)) µ 2(k-l) ∇ l 1 + |x| 2 -n-2k 2 2
dx.

As one checks, the right-hand-side is uniformly bounded wrt µ → 0, so (B M p,µ ) is uniformly bounded wrt p and µ → 0. Moreover, the above computations yield M (B M p,µ ) 2 dv g → 0 as µ → 0. Therefore, B M p,µ 0 as µ → 0 uniformly wrt p ∈ M . This proves the claim.

The space

H 2 k (M ) is compactly embedded in H 2 k-1 (M ). Therefore B M p,µ → 0 in H 2 k-1 (M ) as µ → 0. Hence lim µ→0   k-1 l=0 M A l (g)(∇ l B M p,µ , ∇ l B M p,µ ) dv g   = 0. ( 82
)
Now we estimate the term

M | B M p,µ | 2 k dv g . We claim that lim R→+∞ lim µ→0 M \Bp(µR) | B M p,µ | 2 k dv g = 0. (83)
We fix R. Now for µ sufficiently small

M \Bp(µR) | B M p,µ | 2 k dv g = Bp(ιg)\Bp(µR) | B M p,µ | 2 k dv g = B0(ιg)\B0(µR) | B M p,µ (exp p (y))| 2 k |g(exp p (y))| dy ≤ B0( ιg µ )\B0(R) |B 0,1 (y)| 2 k |g(exp p (µy))| dy. (84) Since B 0,1 ∈ L 2 k (R n ), this yields the claim.
Similarly, for µ sufficiently small

Bp(µR) | B M p,µ | 2 k dv g = B0(µR) | B M p,µ (exp p (y))| 2 k |g(exp p (y))| dy (85) = B0(R) | B 0,1 | 2 k |g(exp p (µy))| dy (86) = B0(R) | B 0,1 | 2 k dy + o B 0,1 L 2 k as µ → 0 (87) Therefore lim µ→0 M | B M p,µ | 2 k dv g = R n | B 0,1 | 2 k dv g . (88)

So we have (b).

Finally we estimate the term

M (∆ k/2 g B M
p,µ ) 2 dv g . We fix R > 0. By calculating in terms of the local coordinates given by exp p , we get for µ sufficiently small Bp(µR)

(∆ k/2 g B M p,µ ) 2 dv g = B0(R) (∆ k/2 B 0,1 ) 2 dy + o (1) as µ → 0. (89) We claim that lim R→+∞ lim µ→0 M \Bp(µR) (∆ k/2 g B M p,µ ) 2 dv g = 0. (90)
We prove the claim. Indeed, via the exponential map at p, we have that Since the manifold M is compact, we note that in the above calculations there is no dependence on the point p of the closed manifold M . So the convergence is uniform for all points p ∈ M . This ends the proof of Proposition 5.1.

M \Bp(µR) (∆ k/2 g B M p,µ ) 2 dv g = Bp(ιg)\Bp(µR) (∆ k/2 g B M p,µ ) 2 dv g (91) = B0(ιg)\B0(µR) (∆ k/2 exp * p g B 0,µ ) 2 dv exp * p g (92) ≤ C k |α|=0 B0(ιg)\B0(µR) |D α (η ιg 10 B 0,µ )| 2 dx (93) Since B 0,µ → 0 strongly in H 2 k-1,loc (R n ), then, as µ → 0, we have that M \Bp(µR) (∆ k/2 g B M p,µ ) 2 dv g ≤ C B0(ιg)\B0(µR) η2 ιg 10 |D k B 0,µ | 2 dx + o(1) (94) ≤ C B0(ιg/µ)\B0(R) |D k B 0,1 | 2 dx + o(1) ≤ C R n \B0(R) |D k B 0,1 | 2 dx + o(1). (95) Since D k B 0,1 ∈ L 2 (R n ),
2

Fix some θ such that 1 K0(n,k) + 4θ < 2 2k/n 1 K0(n,k) . Then from (5.1) it follows that, there exists µ 0 small, such that for all µ ∈ (0, ι g µ 0 ) and for all p ∈ M we have

J P (B M p,µ ) ≤ 1 K 0 (n, k) + θ. ( 96
)
We fix x 0 ∈ M , and we assimilate isometrically T x0 M to R n , and we define the sphere S n-1 := {x ∈ R n / x = 1}. For (σ, t) ∈ S n-1 × [0, ι g /2), we define σ M t := exp x0 (tσ) and

u σ t (x) = α n,k η σ M t (x) µ 0 (ι g /2 -t) (µ 0 (ι g /2 -t)) 2 + d g (σ M t , x) 2 n-2k 2 = B M σ M t ,µ0(ιg/2-t) (97)
It then follows from our previous step and the choice of µ 0 in (96)

J P (u σ t ) ≤ 1 K 0 (n, k) + θ ∀(σ, t) ∈ S n-1 × [0, ι g /2). (98) Let η ∈ C ∞ c (R n ) be a smooth, nonnegative, cut-off function such that η(x) = 1 for |x| ≥ 1/2 and η(x) = 0 for |x| < 1/4. For R ≥ 1, let η R be a smooth, nonnegative, cut-off function, such that η R (x) = 1 if d g (x 0 , x) ≥ ιg 10R η 10R ιg exp -1 x0 (x) if d g (x 0 , x) < ιg 10R (99)
Then the functions η R are such that η R (x) = 1 if d g (x 0 , x) ≥ ιg 20R and η R (x) = 0 if d g (x 0 , x) < ιg 40R . We define (100) v σ t,R (x) := η R (x) u σ t (x) for all x ∈ M. Then we have Proposition 5.2.

lim R→+∞ v σ t,R = u σ t in H 2 k (M ) uniformly ∀(σ, t) ∈ S n-1 × [0, ι g /2). (101)
Proof of Proposition 5.2: We first note that for all (σ, t) ∈ S n-1 × [0, ι g /2) the functions u σ t are uniformly bounded in C 2k -norm in the ball B x0 ( ιg 20 ) ⊂ M . And for any nonnegative integer α, one has

|∇ α g η R | g ≤ CR α . Therefore v σ t,R -u σ t 2 H 2 k = k α=0 M (∆ α/2 g (v σ t,R -u σ t )) 2 dv g (102) = k α=0 Bx 0 ( ιg 20R ) (∆ α/2 g (v σ t,R -u σ t )) 2 dv g (103) = k α=0 Bx 0 ( ιg 20R ) (∆ α/2 g ((η R -1)u σ t )) 2 dv g = O 1 R ( as n ≥ 2k + 1 ) (104)
The above convergence is uniform w.r.t (σ, t) ∈ S n-1 × [0, ι g /2). This proves Proposition 5.2.

2

So it follows that, there exists R 0 > 0, large, such that for all R ≥ R 0 one has

J P (v σ t,R ) ≤ J P (u σ t ) + 2θ < 2 2k/n 1 K 0 (n, k) ∀ (σ, t) ∈ S n-1 × [0, ι g /2). (105)
As one checks for any (σ, t) ∈ S n-1 × [0, ι g /2), the functions v σ t,R0 = 0, and has support in M \B x0 (ι g /40R 0 ). Let 0 > 0 be such that M \B x0 (ι g /40R 0 ) ⊂ M \B x0 ( 0 ) and we define ( 106)

Ω 0 := M \B x0 ( 0 ).
Then for any (σ, t) ∈ S n-1 × [0, ι g /2) the functions v σ t,R0 ∈ H 2 k,0 (Ω 0 )\{0}. Propositions 5.1 and 5.2 yield lim t→ιg/2

J P (v σ t,R0 ) = 1 K 0 (n, k)
uniformly for all σ ∈ S n-1 . (107) Also v σ 0,R0 is a fixed function independent of σ and

v σ t,R0 δ expx 0 ( ιg 2 σ)
weakly in the sense of measures as t → ι g /2. (108)

We define S k := K 0 (n, k) -1 . For any c ∈ R, we define the sublevel sets of the functional I P on N 0

I c := {u ∈ N 0 : I P (u) < c}. ( 109
)
where

N 0 := {u ∈ H 2 k,0 (Ω 0 )/ u 2 k = 1}.
Proposition 5.3. Suppose I P (u) > 1 K0(n,k) for all u ∈ N 0 , then there exists σ 0 > 0 for which there exists a continuous map

Γ : I S k +σ0 -→ Ω 0 (110) such that if (u i ) ∈ I S k +σ0 is a sequence such that |u i | 2 k dv g δ p0
weakly in the sense of measures, for some point p 0 ∈ Ω 0 , then

lim i→+∞ Γ(u i ) = p 0 . ( 111
)
Proof of Proposition 5.3: By the Whitney embedding theorem, the manifold M admits a smooth embedding into R 2n+1 . If we denote this embedding by

F : M → R 2n+1 , then M is diffeomorphic to F(M ) where F(M ) is an embedded submanifold of R 2n+1 . For u ∈ N 0 , we define Γ(u) := Ω M F(x) |u(x)| 2 k dv g (x). ( 112
)
Then Γ : N 0 → R 2n+1 is continuous. Next we claim that for every > 0 there exists a σ > 0 such that

u ∈ I S k +σ ⇒ dist Γ(u), F(Ω 0 ) < . (113)
Suppose that the claim is not true, then there exists an > 0 and a sequence (u i ) ∈ N 0 , such that lim i→+∞ I P (u i ) = S k and dist Γ(u), F(Ω 0 ) ≥ . Since there is no minimizer for I P on N 0 , it follows from Lemma 3.1 that for such a sequence (u i ) there exists a point p 0 ∈ Ω 0 such that |u i | 2 k dv g δ p0 weakly in the sense of measures.

So Γ(u i ) → F(p 0 ), a contradiction since dist Γ(u), F(Ω 0 ) ≥ . This proves our claim.

By the Tubular Neighbourhood Theorem, the embedded submanifold F(M ) has a tubular neighbourhood U in R 2n+1 and there exists a smooth retraction

π : U -→ F(M ). ( 114
)
Choose an 0 > 0 small so that {y ∈ R 2n+1 : dist (y, F(M )) < 0 } ⊂ U . Then from our previous claim it follows that, there exists σ 0 > 0 such that

u ∈ I S k +σ0 ⇒ Γ(u) ∈ U. ( 115
)
We define

Γ M (u) = F -1 • π   M F(x) |u(x)| 2 k dv g (x)   . ( 116
)
Then the map Γ M : I S k +σ0 → M is continuous. Similarly as in our previous claim we have: for every > 0 small there exists δ > 0 such that

u ∈ I S k +δ ⇒ d g Γ M (u), Ω 0 < . ( 117
)
Let π Ω 0 : M \B x0 ( 0 /2) -→ Ω 0 be a retraction. Choose an > 0 small so that {p ∈ M : d g p, Ω 0 < } ⊂ M \B x0 ( 0 /2) . Then from our claim it follows that there exists a δ 0 > 0 such that Γ M (u) ∈ M \B x0 ( 0 /2) for all u ∈ I S k +δ0 . So for u ∈ I S k +δ0 we define Γ(u) := π Ω 0 •Γ M (u). Then the map Γ satisfies the hypothesis of the proposition. This proves Proposition 5.3. This proposition is in the spirit of Proposition 4.4 of [START_REF] Malchiodi | Topological methods for an elliptic equation with exponential nonlinearities[END_REF] 2

Now we proceed to prove the first part of Theorem 1. By the regularity result obtained in Theorem 8.3, it is sufficient to show the existence of a non-trivial H 2 k,0 (Ω 0 ) weak solution to the equation (see (145) for the definition)

P u = |u| 2 k -2 u in Ω M D α u = 0 on ∂Ω M for |α| ≤ k -1 (118)
Suppose on the contrary the above equation only admits trivial solutions, we will show that this leads to a contradiction. Now suppose that the functional I P has no critical point in N 0 , that is there is not weak solution to (118). This is equivalent to the assertion that the functional (119)

F P (u) = 1 2 Ω 0 uP (u) dv g - 1 2 k Ω 0 |u| 2 k dv g .
does not admit a nontrivial critical point in H 2 k,0 (Ω 0 ). Proposition 5.4. If equation (118) admits only the trivial solution u ≡ 0, then the functional I P satisfies the (P.S) c condition for c ∈ (S k , 2

2k n S k ).
Proof of Proposition 5.4: Let (v i ) ∈ N 0 be a Palais-Smale sequence for the functional I P such that lim

i→+∞ I P (v i ) = c ∈ (S k , 2 2k n S k ), if this exists. Define u i := (I P (v i )) 1 2 k -2 v i . Then (u i
) is a Palais-Smale sequence for the functional F P on the space H 2 k,0 (Ω 0 ) such that lim

i→+∞ F P (u i ) ∈ ( k n S n/2k k , 2k n S n/2k k
). Since there is no nontrivial solution to (118), it follows from the Struwe-decomposition for polyharmonic operators by the author [START_REF] Mazumdar | Struwe decomposition for polyharmonic operators on compact manifolds with or without boundary[END_REF] that there exists d ∈ N non-trivial functions u j ∈ D k,2 (R n ), j = 1, . . . , d, such that upto a subsequence the following holds

F P (u i ) = d j=1 E(u j ) + o(1) as i → +∞ (120) where E(u) := 1 2 R n (∆ k/2 u) 2 dx -1 2 k R n |u| 2 k dx. The u j 's are nontrivial solutions in D k,2 (R n ) to ∆ k u = |u| 2 k -2 u on R n or on {x ∈ R n / x 1 < 0}
with Dirichlet boundary condition (we refer to [START_REF] Mazumdar | Struwe decomposition for polyharmonic operators on compact manifolds with or without boundary[END_REF] for details). It then follows from Lemma 3 and 5 of Ge-Wei-Zhou [START_REF] Ge | A critical elliptic problem for polyharmonic operators[END_REF] that for any j, either u j has fixed sign and

E(u) = k n S n/2k k
, or u j changes sign and E(u)

≥ 2k n S n/2k k , contradicting lim i→+∞ F P (u i ) ∈ ( k n S n/2k k , 2k n S n/2k k
). Therefore the Palais-Smale condition holds at level c ∈ (S k , 2 2k n S k ). More precisely, there is even no Palais-Smale sequence at this level. This ends the proof of Proposition 5.4.

2

Proof of Theorem 1: By the Deformation Lemma (see Theorem II.3.11 and Remark II.3.12 in the monograph by Struwe [START_REF] Struwe | Variational methods[END_REF]), there exists an retraction β :

I S k +4θ -→ I k S k +σ0
, where σ 0 is as given in Proposition 5.3. Let r N 0 : H 2 k,0 (Ω 0 )\{0} → N 0 be the projection given by u → u u

L 2 k
. Consider the map h :

S n-1 × [0, ι g /2] → Ω 0 given by h(σ, t) := Γ • β(r N 0 (v σ t,R0 )) for t < ι g /2 σ M ιg/2 for t = ι g /2 (121)
where σ M t := exp x0 (tσ). This map is well defined and continuous by Proposition 5.3 and there exists p 0 ∈ Ω 0 such that

h(σ, t) = p 0 for t = 0 exp x0 ( ιg 2 σ) for t = ι g /2 (122)
So we obtain a homotopy of the embedded (n-1)-dimensional sphere {exp x0 ( ιg 2 σ) : σ ∈ S n-1 }to a point in Ω 0 , which is a contradiction to our topological assumption. This proves Theorem 1 for potentially sign-changing solutions.

Positive solutions

This section is devoted to the second part of Theorem 1, that is the existence of positive solutions. The proof is very similar to the proof of Theorem 1 with no restriction on the sign. We just stress on the specificities and refer to the proof of Theorem 1 everytime it is possible. We let Ω M ⊂ M be any smooth n-dimensional submanifold of M , possibly with boundary. In the sequel, we will either take Ω M = M , or M \B x0 ( 0 ). For u ∈ H 2 k,0 (Ω M ), we define u + := max{u, 0}, u -:= max{-u, 0} and

N + := {u ∈ H 2 k,0 (Ω M ) : Ω M (u + ) 2 k dv g = 1}. (123) which is a codimension 1 submanifold of H 2 k,0 (Ω M ). Any critical point u ∈ H 2 k,0 (Ω M ) of I g on N + is a weak solution to (124) P u = u 2 k -1 + in Ω M ; D α u = 0 on ∂Ω M for |α| ≤ k -1.
Consider the Green's function G P associated to the operator P with Dirichlet boundary condition on the smooth domain Ω M M , which is a function

G P : Ω M × Ω M \{(x, x) : x ∈ Ω M } -→ R such that (i) For any x ∈ Ω M , the function G P (x, •) ∈ L 1 (Ω M ) (ii) For any ϕ ∈ C ∞ (Ω M ) such that D α ϕ = 0 on ∂Ω M for all |α| ≤ k -1, we have that ϕ(x) = Ω M
G P (x, y) P ϕ(y) dv g (y). (125) Lemma 6.1. Let (u i ) ∈ N + be a minimizing sequence for I k g on N + . Then (i) Either there exists u 0 ∈ N + such that u i → u 0 strongly in H 2 k,0 (Ω M ), and u 0 is a minimizer of

I P on N + (ii) Or there exists x 0 ∈ Ω M such that |u i | 2 k dv g δ x0 as i → +∞ in the sense of measures. Moreover, inf u∈N+ I P (u) = 1 K0(n,k) .
Proof of Lemma 6.1: As the functional I g is coercive so the sequence (u i ) is bounded in H 2 k,0 (Ω M ). We let u 0 ∈ H 2 k,0 (Ω M ) such that, up to a subsequence, u i u 0 weakly in H 2 k,0 (Ω M ) as i → +∞, and u i (x) → u 0 (x) as i → +∞ for a.e. x ∈ Ω M . As the sequences (u

+ i ), (u - i ) is bounded in L 2 k (Ω M ) and u + i (x) → u + 0 (x), u - i (x) → u - 0 (x) for a.e.
x ∈ Ω M , integration theory yields (126)

u + i u + 0 and u - i u - 0 weakly in L 2 k (Ω M ) as i → +∞.
Therefore,

u + 0 2 k L 2 k ≤ lim inf i→+∞ u + i 2 k L 2 k = 1 and u - 0 2 k L 2 k ≤ lim inf ı→+∞ u - i 2 k L 2 k . (127) We claim that (128) u - i → u - 0 strongly in L 2 k (Ω M ).
We prove the claim. We define v i := u i -u 0 . Up to extracting a subsequence, we have that (v i ) i → 0 in H 2 k-1 (M ). Therefore, as i → +∞,

I P (u i ) = Ω M (∆ k/2 g v i ) 2 dv g + I P (u 0 ) + o(1). (129)
And then, letting α := inf u∈N+ I P (u), we have that

α = I P (u i ) + o(1) ≥ Ω M (∆ k/2 g v i ) 2 dv g + α u + 0 2 L 2 k + o(1)
and then

(130) α 1 -u + 0 2 L 2 k ≥ Ω M (∆ k/2 g v i ) 2 dv g + o(1)
as i → +∞. We fix > 0. It then follows from [START_REF] Delanoë | On the local Nirenberg problem for the Q-curvatures[END_REF] and

(v i ) i → 0 in H 2 k-1 (M ) that α (K 0 (n, k) + ) 1 -u + 0 2 L 2 k ≥ v i 2 L 2 k + o(1). (131) Since 1 -a 2 k /2 ≥ (1 -a 2 ) 2 k /2 for 1 ≥ a ≥ 0, we get that (α (K 0 (n, k) + )) 2 k /2 1 -u + 0 2 k L 2 k ≥ v i 2 k L 2 k + o(1). (132) Integration theory yields u i 2 k L 2 k = v i 2 k L 2 k + u 0 2 k L 2 k +o(1) as i → +∞. Therefore (α (K 0 (n, k) + )) 2 k /2 1 -u + 0 2 k L 2 k + o(1) ≥ u i 2 k L 2 k -u 0 2 k L 2 k = u + i 2 k 2 k -u + 0 2 k 2 k + u - i 2 k 2 k -u - 0 2 k 2 k = 1 -u + 0 2 k 2 k + u - i 2 k 2 k -u - 0 2 k 2 k Then u - i 2 k L 2 k = u - i -u - 0 2 k L 2 k + u - 0 2 k L 2 k + o(1) as i → +∞ yields (µ (K 0 (n, k) + )) 2 k /2 -1 1 -u + 0 2 k L 2 k + o(1) ≥ u - i 2 k L 2 k -u - 0 2 k L 2 k (133) = u - i -u - 0 2 k L 2 k + o(1). (134)
Since αK 0 (n, k) ≤ 1 and > 0 is arbitrary small, we get (128). This proves the claim.

We define µ i := (∆ k/2 g u i ) 2 dv g and ν i = |u i | 2 k dv g for all i. Up to a subsequence, we denote respectively by µ and ν their limits in the sense of measures. It follows from the concentration-compactness Theorem 4 that,

(135) ν = |u 0 | 2 k dv g + j∈J α j δ xj and µ ≥ (∆ k/2 g u 0 ) 2 dv g + j∈I β j δ xi
where J ⊂ N is at most countable, (x j ) j∈J ∈ M is a family of points, and (α j

) j∈J ∈ R ≥0 , (β j ) j∈J ∈ R ≥0 are such that α 2/2 k j ≤ K 0 (n, k) β j for all j ∈ J. Since u - i → u - 0 strongly in L 2 k (M ), we then get that (136) |u + i | 2 k dv g |u + 0 | 2 k dv g + j∈J α j δ xj
as i → +∞ in the sense of measures. The sequel is similar to the proof of Lemma 3.1. We omit the details. This completes the proof of Lemma 6.1. 2 Lemma 6.2. We assume that there is no nontrivial solution to (124). Then the functional I P satisfies the (P.S) c condition on N + for c ∈ (S k , 2

2k n S k ) if the equation.

Proof of Lemma 6.2: This is equivalent to prove that the functional

F + P (u) = 1 2 Ω 0 uP (u) dv g - 1 2 k Ω 0 (u + ) 2 k dv g . (137) satisfies the (P.S) c condition on H 2 k,0 (Ω 0 ) for c ∈ ( k n S n/2k k , 2k n S n/2k k
). Let (u i ) be a Palais-Smale sequence for the functional F + P on the space H 2 k,0 (Ω 0 ). Then, as v ∈ H 2 k,0 (Ω 0 ) goes to 0, (138)

Ω 0 u i P k g (v) dv g - Ω 0 (u + i ) 2 k -1 v dv g = o v H 2 k .
Without loss of generality we can assume that u i ∈ C ∞ c (Ω 0 ) for all i. Let ϕ i ∈ C ∞ (Ω 0 ) be the unique solution of the equation

P k g ϕ i = (u + i ) 2 k -1 in Ω 0 D α ϕ i = 0 on ∂Ω 0 for |α| ≤ k -1. ( 139
)
The existence of such ϕ i is guaranteed by Theorem 8.2. It then follows from Green's representation formula that

ϕ i (x) = Ω M G P (x, y)(u + i (y)) 2 k -1 dv g (y) ≥ 0. ( 140
)
for all x ∈ Ω M . Note that the sequence (ϕ i ) is bounded in H 2 k,0 (Ω 0 ). It follows from (138) that ϕ i = u i + o(1), where o(1) → 0 in H 2 k,0 (Ω 0 ) as i → +∞. And so (ϕ i ) is Palais-Smale sequences for the functional F + P on the space H 2 k,0 (Ω 0 ). Therefore, since ϕ i ≥ 0, it is also a Palais-Smale sequence for F P defined in (119). Since there is no nontrivial critical point for F + P , using the Struwe decomposition [START_REF] Mazumdar | Struwe decomposition for polyharmonic operators on compact manifolds with or without boundary[END_REF] as in the proof of Proposition 5.4, we then get that (ϕ) i is relatively compact in H 2 k,0 (Ω 0 ), and so is (u i ). This ends the proof of Lemma 6.1. 2

Proof of Theorem 1, positive solutions: this goes essentially as in the proof of Theorem 1, the key remark being that the functions v σ t,R defined in (100) are nonnegative. We define

N 0 + = {u ∈ H 2 k,0 (Ω 0 ) : u + L 2 k = 1}
, where Ω 0 = M \ B 0 (x 0 ) and 0 > 0 was defined in (106). For c ∈ R we define the sublevel sets of the functional I P on N 0 + as I + c := {u ∈ N 0 + : I k g (u) < c}. Arguing as in the proof of Proposition 5.3, it follows from Lemma 6.1 that there exists a δ 0 > 0 such that there exists Γ :

I + S k +δ0 → Ω 0 a continous map such that: If (u i ) ∈ I + S k +δ0 is a sequence such that |u + i | 2 k dv g δ p0
weakly in the sense of measures, for some point

p 0 ∈ Ω 0 , then lim i→+∞ Γ(u i ) = p 0 . Let r N 0 + : H 2 k,0 (Ω 0 )\{ u + L 2 k = 0} → N 0 + be the map given by u → u u + L 2 k . Consider the map h : S n-1 × [0, ι g /2] → Ω 0 given by h(σ, t) = Γ • β(r N 0 + (v σ t,R0 )) for t < ι g /2 σ M ιg/2 for t = ι g /2 (141)
where β : I + S k +4θ → I + S k +δ0 is a retract (we have used Lemma 6.2) and σ M t = exp x0 (tσ). Note here that we use that v σ t,R0 ≥ 0. As in the proof of Theorem 1, h is an homotopy of the embedded (n-1)-dimensional sphere {exp x0 ( ιg 2 σ) : σ ∈ S n-1 } to a point in Ω 0 , which is a contradiction to our topological assumption. So there exists a nontrivial critical point u for the functional I P on N 0 + , which yields a weak solution to (124). It then follows from the regularity theorem 8.3 that u ∈ C ∞ (Ω 0 ), u > 0, is a solution to [START_REF] Michael | Convergence and rigidity of manifolds under Ricci curvature bounds[END_REF]. This ends the proof of Theorem 1 for positive solutions. 2

An Important Remark

We remark that the topological condition of Theorem 1 is in general a necessary condition. Consider the n-dimensional unit sphere S n endowed with its standard round metric h r and let P hr be the conformally invariant GJMS operator on S n . By the stereographic projection it follows that S n \{x 0 } is conformal to R n . Also one has that S n \{x 0 } is contractible to a point. Let Ω 0 be the domain in S n \{x 0 } constructed as earlier in [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF], and let u ∈ H 2 k,0 (Ω 0 ), u = 0 solve the equation

P hr u = (u + ) 2 k -1 in Ω 0 D α u = 0 on ∂Ω 0 for |α| ≤ k -1. (142)
Then by the stereographic projection it follows that there exists a ball of radius R, B 0 (R) such that there is a nontrivial solution v ∈ H 2 k,0 (B 0 (R)) to the equation

∆ k v = (v + ) 2 k -1 in B 0 (R) D α v = 0 on ∂B 0 (R) for |α| ≤ k -1. (143)
By a result of Boggio [START_REF] Boggio | Sulle funzioni di Green d'ordine m[END_REF], the Green's function for the Dirichlet problem above is positive. Therefore, we get that v > 0 is a smooth classical solution to

∆ k v = v 2 k -1 in B 0 (R) D α v = 0 on ∂B 0 (R) for |α| ≤ k -1. ( 144
)
This is impossible by Pohozaev identity, see Lemma 3 of Ge-Wei-Zhou [START_REF] Ge | A critical elliptic problem for polyharmonic operators[END_REF].

Appendix: Regularity

Let f ∈ L 1 (Ω M ). We say that u ∈ H 2 k,0 (Ω M ) is a weak solution of the equation

P u = f in Ω M and D α u = 0 on ∂Ω M for |α| ≤ k -1, if for all ϕ ∈ C ∞ c (Ω M ) (145) Ω M ∆ k/2 g u ∆ k/2 g ϕ dv g + k-1 α=0 Ω M A l (g)(∇ l u, ∇ l ϕ) dv g = Ω M f ϕ dv g .
Now let the operator P be coercive on the space H 2 k,0 (Ω M ), i.e there exists a constant C > 0 such that for all u ∈ H 2 k,0 (Ω M ) ( 146)

Ω M uP (u) dv g ≥ C u 2 H 2 k,0 (Ω M ) .
We then have

u i H p 2k (Ω M ) ≤ C p P u i L p + u i H p k = O(1). (148)
So there exists u 0 ∈ H p 2k,0 (Ω M ) such that upto a subsequence u i u 0 weakly in H p 2k,0 (Ω M ). Then u i → u 0 strongly in H p k,0 (Ω M ) and so u 0 H p k = 1. Also u 0 weakly solves the equation P u 0 = 0 in Ω M and D α u 0 = 0 on ∂Ω M for |α| ≤ k -1. It follows from standard elliptic estimates (see Agmon-Douglis-Nirenberg [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF]) that u 0 ∈ C ∞ (Ω M ). Then, multiplying the equation by u 0 and integrating over M , coercivity yields

C u 0 2 H 2 k (Ω M ) ≤ M u 0 P u 0 dv g = 0. ( 149
)
and hence u 0 ≡ 0, a contradiction since we have also obtained that u 0 H p k = 1. This proves Proposition 8.1.

2 Proposition 8.2 (Existence and Uniqueness). Let the operator P be coercive. Then given any f ∈ L p (Ω M ), 1 < p < +∞, there exists a unique weak solution u ∈

H p k,0 (Ω M ) ∩ H p 2k (Ω M ) to P u = f in Ω M D α u = 0 on ∂Ω M for |α| ≤ k -1. ( 150
)
The proof is classical and we only sketch it here. For p = 2, existence and uniqueness follows from the Riesz representation theorem in Hilbert spaces. For arbitrary p > 1, we approximate f in L p by smooth compactly supported function on Ω M . For each of these smooth functions, there exists a solution to the pde with the approximation as a right-hand-side. The coercivity and the Agmon-Douglis-Nirenberg estimates yield convergence of these solutions to a solution of the original equation. Coercivity yields uniqueness.

We now proceed to prove our regularity results. The proof is based on ideas developed by Van der Vorst [START_REF] Van Der | Best constant for the embedding of the space H 2 ∩ H 1 0 (Ω) into L 2N/(N -4) (Ω)[END_REF], and also employed by Djadli-Hebey-Ledoux [START_REF] Djadli | Paneitz-type operators and applications[END_REF] for the case k = 2. Theorem 5. Let (M, g) be a smooth, compact Riemannian manifold of dimension n and let k be a positve integer such that 2k < n. Let Ω M be a smooth domain in M and suppose u ∈ H 2 k,0 (Ω M ) be a weak solution of the equation

P u = f (x, u) in Ω M D α u = 0 on ∂Ω M for |α| ≤ k -1 (151) where |f (x, u)| ≤ C|u|(1 + |u| 2 k -2 ) for some positive constant C, then u ∈ L p (Ω M ) for all 1 < p < +∞. (152) Proof of 5: We write f (x, u) = bu where |b| ≤ C(1 + |u| 2 k -2 ). Then b ∈ L n/2k
(Ω M ) and u solves weakly the equation

P u = bu in Ω M D α u = 0 on ∂Ω M for |α| ≤ k -1. ( 153
)
Step 1: We claim that for any > 0 there exists q ∈ L n/2k (Ω M ) and

f ∈ L ∞ (Ω M ) such that bu = q u + f , q L n/2k (Ω M ) < . (154) Now lim i→+∞ {|u|≥i}
|b| n/2k dv g = 0, so given any > 0 we can choose i 0 such that {|u|≥i0} |b| n/2k dv g < n/2k .

We define q := χ {|u|≥i0} b and f :

= (b -q )u = χ {|u|<i0} b. Then, since |b| ≤ C(1 + |u| 2 k -2
), we have that q L n/2k (Ω M ) < and f ∈ L ∞ (M ). This proves our claim and ends Step 1.

We rewrite (153) as

P u = q u + f in Ω M D α u = 0 on ∂Ω M for |α| ≤ k -1. (155)
Let H be the operator defined formally as

H u = (P k g ) -1 (q u). (156)
Then P u = q u + f becomes u -H u = (P k g ) -1 (f ).

Step 2: we claim that for any s > 1, H maps L s (Ω M ) to L s (Ω M ).

We prove the claim. Let v ∈ L s (Ω M ), s ≥ 2 k , then q v ∈ L ŝ(Ω M ) where ŝ = ns n + 2ks , and we have by Hölder inequality

q v L ŝ(Ω M ) ≤ q L n/2k (Ω M ) v L s (Ω M ) (157) Since q L n/2k (Ω M ) < , so we have q v L ŝ(Ω M ) ≤ v L s (Ω M ) (158) From (8.2) it follows that there exists a unique v ∈ H ŝ 2k (Ω M ) such that P v = q v in Ω M D α v = 0 on ∂Ω M for |α| ≤ k -1 (159)
weakly. Further we have for a positive constant C(s)

v H ŝ 2k (Ω M ) ≤ C(s) q v L ŝ(Ω M ) (160)
So we obtained that

v H ŝ 2k (Ω M ) ≤ C(s) v L s (Ω M ) . (161)
By Sobolev embedding theorem

H ŝ 2k (Ω M ) is continuously imbedded in L s (Ω M ) so v ∈ L s (Ω M ) and we have v L s (Ω M ) ≤ C(s) v L s (Ω M ) . (162)
In other words, for any s ≥ 2 k the operator H acts from L s (Ω M ) into L s (Ω M ), and its norm H L s →L s ≤ C(s) . This proves the claim and ends Step 2.

Step 3: Now let s ≥ 2 k be given, then for > 0 sufficiently small one has

H L s →L s ≤ 1 2 (163)
and so the operator

I -H : L s (Ω M ) -→ L s (Ω M ) is invertible. We have u -H u = (P k g ) -1 (f ) (164) Since u ∈ L 2 k (Ω M ) and f ∈ L ∞ (Ω M ), so u ∈ L p (Ω M ) for all 1 < p < +∞.
This ends the proof of Theorem 5.

2

Proposition 8.3. Let (M, g) be a smooth, compact Riemannian manifold of dimension n and let k be a positive integer such that 2k < n. Let f ∈ C 0,θ (Ω M ) a Hölder continuous function. Let Ω M be a smooth domain in M and suppose u ∈ H 2 k,0 (Ω M ) be a weak solution of the equation

P u = f |u| 2 k -2 u or f (u + ) 2 k -1 in Ω M D α u = 0 on ∂Ω M for |α| ≤ k -1. (165) Then u ∈ C 2k (Ω M ), and is a classical solution of the above equation. Further if u > 0 and f ∈ C ∞ (Ω M ), then u ∈ C ∞ (Ω M ).
Proof of Proposition 8.3: It follows from (5) that u ∈ H p 2k (Ω M ) for all 1 < p < +∞. By Sobolev imbedding theorem this implies u ∈ C 2k-1,γ (Ω M ) for all 0 < γ < 1.

|u| 2 k -2 u, (u + ) 2 k -1 ∈ C 1 (Ω M ).
The Schauder estimates (here again, we refer to Agmon-Douglis-Nirenberg [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF]) then yield u ∈ C 2k,γ (Ω M ) for all γ ∈ (0, 1), and u is a classical solution.

If u > 0, then the right-hand-side is u 2 k -1 and has the same regularity as u. Therefore, iterating the Schauder estimates yields u ∈ C ∞ (Ω M ). This ends the proof of Proposition 8. Here we have identified T p M ∼ = R n for any point p ∈ M . For example, one can take the exponetial map at p : exp p , which is normal at p. We will let ι g be the injectivity radius of M . Using the above local comparison of the Riemannian metric with the Euclidean metric one obtains Lemma 9.1. Let (M, g) be a smooth, compact Riemannian manifold of dimension n and let k be positive integer such that 2k < n. We fix s ≥ 1. Let ϕ -1 p : Ω ⊂ R n → M , ϕ(p) = 0 be a local coordinate around p with the above mentioned properties. Then given any 0 > 0 there exists τ ∈ (0, ι g ), such that for any point p ∈ M , and u ∈ C ∞ c (B 0 (τ )) one has

(1 -0 ) R n (∆ k/2 u) 2 dx ≤ M (∆ k/2 g (u • ϕ p )) 2 dv g ≤ (1 + 0 ) R n (∆ k/2 u) 2 dx (166)
and

(1 -0 ) R n |u| s dx ≤ M |u • ϕ p | s dv g ≤ (1 + 0 ) R n |u| s dx (167)
Proof of Lemma 9.1: In terms of the coordinate map ϕ -1 p : Ω ⊂ R n → M , for any f ∈ C 2 (M ) we have

(168) ∆ g f (x) = -g ij (x) ∂ 2 (f • ϕ -1 ) ∂x i ∂x j (x) -Γ k ij (ϕ(x)) ∂(f • ϕ -1 ) ∂x k (x) .
Since the manifold M is compact, then given any > 0 there exists a τ ∈ (0, ι g ) depending only on (M, g), such that for any point p ∈ M and for any x ∈ B 0 (τ ) ⊂ R n one has for all indices i, j, k = 1, . . . , n

(1 -)δ ij ≤ g ij (x) ≤ (1 + )δ ij as bilinear forms.

|g ij (x) -δ ij | ≤
Without loss of generality we can assume that τ < 1. We let u ∈ C ∞ c (R n ) be such that supp(u) ⊂ B 0 (τ ). In the sequel, the constant C will denote any positive constant depending only on (M, g) and τ : the same notation C may apply to different constants from line to line, and even in the same line. All integrals below are taken over B 0 (τ ), and we will therefore omit to write the domain for the sake of clearness. And then by calculations similar to the even case, along with the integration by parts formula, we obtain

|∇(∆ m g (u • ϕ p ))| 2 (ϕ -1 p (x)) dx -|∇(∆ m u)| 2 (x) dx ≤ Cg + √ τ |∇(∆ m u)| 2 dx. (184)
Now given an 0 > 0 small, we first choose small and then choose a sufficiently small τ , so that for any u ∈ C ∞ c (B 0 (τ )) we have

M |∇(∆ m g (u • ϕ p ))| 2 dv g -|∇(∆ m u)| 2 dx ≤ 0 |∇(∆ m u)| 2 dx. (185)
Then one has the lemma for k odd. This ends the proof of Lemma 9.1.

2

  (63) dν = f dµ and dµ = gdν + dσ where f ∈ L 1 (M, µ) and g ∈ L 1 (M, ν) are nonnegative functions, σ is a positive Borel measure on M and dν⊥dσ. Let S = M \(supp σ). Then for any ϕ ∈ C(M ) with support supp(ϕ) ⊂ S one has M

)I

  By regularity of the Borel measure ν the above relation holds for any Borel measurable function φ. It follows from Proposition 4.1 below that for each N there exist a finite set I N , a finite set of distinct points {x i : i ∈ I N }and a finite set of weights {α i : i ∈ I N } such that N . Then I is a countable set. For a Borel set E, then one has by monotone convergence theorem

  this yields (90). This proves the claim. Equations (89) and (90) yield (a) and (b) of Proposition 5.1 for any fixed p ∈ M .

Proposition 8 . 1 ( 1 :

 811 We proceed by contradiction. If not, then there exists a sequence (u i ) ∈ C ∞ c (Ω M ) such that u i H p k = 1 and lim i→+∞ P u i p = 0. It follows from classical estimates (see Agmon-Douglis-Nirenberg [1]) that

3 . 2 9.

 32 Appendix: Local Comparison of the Riemannian norm with the Euclidean norm Let (M, g) be a smooth, compact Riemannian manifold of dimension n ≥ 1. For any point p ∈ M there exists a local coordinate around p, ϕ -1 p: Ω ⊂ R n → M , ϕ(p) = 0, such that in these local coordinates one has for all indices i, j, k = 1, . . . , n(1 -)δ ij ≤ g ij (x) ≤ (1 + )δ ij as bilinear forms. |g ij (x) -δ ij | ≤

Case 1 : 1 β=1∇ 2 dx + 2 ∇ 1 β=1 1 |β|=1∇

 112211 k is even. We then write k = 2m, m ≥ 1. Then calculating in terms of local coordinates we obtain(169) ∆ m g (u • ϕ p )(ϕ -1 p (x)) -∆ m u(x) ≤ ∇ 2m u(x) + C 2m-(2m-β) u(x)where C g is a constant depending only on the metric g on M . Then we have∆ m g (u • ϕ p )(ϕ -1 p (x)) 2 dx -(∆ m u) 2m u |∆ m u| dx + C 2m-|∆ m u| ∇ (2m-β) u dx. β such that β ≤ 2m -1 we have ∇ (2m-β) u ∈ D β,2 (R n ) and by Sobolev embedding theorem this implies that ∇ (2m-β) u 2 ∈ L 2 β /2 (R n )Applying the integration by parts formula, we obtain(174) ∇ 2m u 2 dx = (∆ m u) 2 dx.So we have, since τ < 12m-2m-β u 2 dx ≤ Cτ (∆ m u) 2 dx. (175) Therefore, we get that ∆ m g (u • ϕ p )(ϕ -1 p (x)) 2 dx -(∆ m u) 2 dx ≤ C ( + τ ) (∆ m u) 2 dx. (176)Now in these local coordinates one has(1 -) n/2 ∆ m g (u • ϕ p )(ϕ -1 p (x)) 2 dx ≤ M ∆ m g (u • ϕ p ) 2 dv g (177) ≤ (1 + ) n/2 ∆ m g (u • ϕ p )(ϕ -1 p (x)) 2 dx. (178)So given an 0 > 0 small, we first choose small and then choose a sufficiently small τ , so that for any u ∈ C ∞ c (B 0 (τ )) we have ∆ m g (u • ϕ p ) 2 dv g -(∆ m u) 2 dx ≤ 0 (∆ m u) 2 dx. (179) So we have the lemma for k even.

Case 2 :

 2 k is odd. We then write k = 2m + 1 with m ≥ 0. Calculating in terms of local coordinates, like in the even case, we obtain|∇(∆ m g (u • ϕ p ))| 2 (ϕ -1 p (x)) -|∇(∆ m u)| 2 (x) ≤ |∇(∆ m u)| 2 ∇ 2m+1 u (x) |∇(∆ m u)|(x) + C 2m β=1 ∇ (2m+1-β) u (x) |∇(∆ m u)|(x) (182) for all x ∈ B 0 (τ ). Therefore |∇(∆ m g (u • ϕ p ))| 2 (ϕ -1 p (x)) dx -|∇(∆ m u)| 2 (x) dx ≤ |∇(∆ m u)| 2 dx 2m+1 u |∇(∆ m u)| dx + C 2m β=1 ∇ (2m+1-β) u |∇(∆ m u)| dx(183)

author, funded by "Fédération Charles Hermite" (FR3198 du CNRS) and "Région Lorraine". The author acknowledges these two institutions for their supports.