Pierre Cozannet

Yvan Primot

Cecile Gady

Jean-Paul Metayer

Michel ; Lessire

Fabien Skiba

Dr Jean Noblet
email: jean.noblet@rennes.inra.fr

J P Métayer

Standardized amino acids digestibility of wheat distillers dried grains with solubles in force-fed roosters

Keywords: Amino acids, Ileal digestibility, Lysine, Poultry, Wheat distillers

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

September 2007. First, 10 of them were chosen as representative of total variability according 73 to their major characteristics (nitrogen, starch, fat, crude fibre, ash) and were measured for 74 AA digestibility in pigs (Cozannet et al., 2010) and in cockerels (present study) and for 75 energy digestibility in pigs (Cozannet et al., 2010) and in broilers, cockerels, laying hens and 76 turkeys (Cozannet et al., 2010). Each plant was represented by at least one sample in the 10 77 selected samples. In the specific case of AA digestibility in cockerels, the 9 additional 78 samples were also measured in cockerels in the same conditions as those of the 10 samples. 79 These results will be used only for validation of the prediction equations. Experiments on 80 cockerels were carried out at Adisseo CERN, Commentry, France according to the Certificate 81 of Authorization to Experiment on Living Animals No. B0315901 (European Directive 82 24/11/86 86/609 CEE). 83

Diets, animals, housing, experimental design and samples collection 84

A total of 60 adult ISA Brown cockerels, surgically caecectomised according to the technique 85 described previously by [START_REF] Parsons | Influence of caecectomy on digestibility of amino acids by roosters 423 fed distillers' dried grains with solubles[END_REF], were used in a factorial block design. The birds,86 housed in an environmentally controlled room, were kept in individual cages with wire floors 87 that permitted collection of excreta on stainless trays. They had free access to water and were 88 subjected to a daily photoperiod of 16 h. Four birds in adjacent cages were considered as the 89 experimental unit. A maize-based standard maintenance diet was fed between assays. For the 90 present assay, 10 experimental diets were composed with one of the 10 DDGS mixed with 91 wheat starch and 40 g/kg premix in order to achieve a 180 g/kg as-fed CP diet. In addition, a 92 N-free diet (replacement of DDGS by starch) was also used. 93

The birds were assayed for 3 successive one-week periods. Over the first period, each 94 bird received the N-free diet in order to evaluate unspecific endogenous AA losses. For that 95 purpose, the birds, after a 24-h fasting period, received 100 g of glucose solution (800 g/kg 96 glucose) in water. Then, 24 h later, 50 g of N-free diet was force fed to each bird and a 4-h 97 collection period was carried out. Over the 2 following periods, each experimental diet was 98 measured on 3 experimental units. For this purpose, fasted birds for 24 h were force fed with 99 approximately 180 g of wet diet (500 g/kg DM) with a subsequent twice-daily total collection 100 of excreta over the following 48 h for each bird. Collected excreta were then immediately 101 frozen and freeze-dried at the end of the experiment. Freeze-dried individual samples from the 102 2-d collection were pooled and mixed per experimental unit before grinding through a Retsch 103 mill equipped with a 1-mm screen. 104

Chemical analyses 105

Samples of raw materials, diets and freeze-dried excreta were analysed for DM, N (according 106 to the methods of AOAC, 1990) and AA contents at Ajinomoto Eurolysine S.A.S., Amiens 107 (France). The AA contents were measured by cation exchange chromatography after acid 108 hydrolysis for 24 h (Directive 98/64/CE, 3/09/99 -Norm NF [START_REF] Nf | Aliment des animaux -Détermination de la teneur en acides 409 aminés[END_REF]. Analysis of 109 methionine was performed after initial oxidation of samples with performic acid. 110

Phenylalanine was analysed without oxidation, whereas tryptophan was analysed after 111 hydrolysis in 4 M barium hydroxide at 110°C for 16 h (AFNOR, 1998). The DDGS samples 112 were also analysed according to standard methods (AOAC., 1990) for ash, starch, crude fat 113 and crude fibre. Detergent fibre fractions (NDF, ADF and ADL) were determined by a 114 sequential procedure with prior amylolytic (thermamyl 120L) and proteolytic (extracted from 115

Streptomyces griseus) treatments [START_REF] Van Soest | Use of detergents in the analysis of fibrous feeds. IV. 452 Determination of plant cell wall constituents[END_REF] . Residual N in NDF and ADF 116 fractions was also measured. Because of incomplete removal of protein from NDF residue, a 117 sequential procedure that used both heat-stable α-amylase and sodium sulphite was developed 118 to measure insoluble dietary fibre in all feeds (values of NDF, ADF and ADL were the 119 average of both values using protease or sulphite). Protein solubilities in KOH and water as 120 defined by [START_REF] Araba | Evaluation of protein solubility as an indicator of 352 overprocessing of soybean meal[END_REF] and the American Oil Chemists [START_REF] Society | Official Methods and Recommended 350 Practices[END_REF] (Luminance, L; red index, a, and yellow index, b) were determined using a CR-410 124 colorimeter; the L, a and b ranged from 0 (black) to 100 (white), from -60 (green) to 60 (red) 125 and from -60 (blue) to 60 (yellow) respectively. 126

Total LYS content was dissociated in two categories: free LYS or LYS whose ε-amino 127 groups are bound to other groups. The free LYS (FLYS) estimate was based on free ε-NH 2 128 conversion to ε-dinitrophenyl-lysine through a reaction using 1-fluoro-2:4-dinitrobenzene 129 (FDNB) before the sample was acid-hydrolysed [START_REF] Carpenter | The estimation of the available lysine in animal protein foods[END_REF] for total AA hydrolysis. 130

The bound LYS (BLYS) content was indirectly evaluated from the furosine measurement. 131

Furosine is one of the compounds that are produced from Amadori products during the acid 132 hydrolysis step in amino acids analysis [START_REF] Finot | Identification of a new lysine 381 derivative obtained upon acid hydrolysis of heated milk[END_REF]. Amadori products in milk contain 133 320 g/kg furosine, 400 g/kg BLYS and 280 g/kg pyridosine [START_REF] Bujard | Mesure de la disponibilité & du blocage de la lysine 362 dans les laits industriels[END_REF]; these 134 ratios are assumed to be constant in any Amadori compound [START_REF] Guerra-Hernandez | Maillard 388 reaction evaluation by Furosine determination during infant cereal processing[END_REF]. 135 Chromatographic determination of furosine in wheat DDGS samples was performed by ion-136 pair RP-HPLC (hydrolysis with 8 N HCl under inert conditions at 110°C for 24 h. Finally, in 137 vitro digestibility of N in wheat DDGS was measured according to a method adapted from 138 [START_REF] Boisen | Prediction of the apparent ileal digestibility of 359 protein and amino acids in feedstuffs and feed mixtures for pigs by in vitro analyses[END_REF] with a 3 times higher pepsine level including addition of three 139 time pepsine activity. 140

Calculations and statistical analyses 141

Total LYS content corresponds to LYS measured after hydrolysis. Assuming constant 142 proportions of the three hydrolysis products of Amadori compounds, the BLYS content was 143 calculated from furosine content as: 144 with AAint and AAexc as the total AA intake and excreted (g DM) respectively, DMint as the 149 dry matter intake (kg) and endogenous AAloss corresponds to the basal endogenous losses 150 (g/kg DMint) as calculated for each experimental unit of birds fed the N-free diet and 151 assumed to be proportional to DM intake. 152 Data on each replicate over the 2 periods (n = 30) were subjected to an analysis of 153 variance with diet effect (n = 10) (SAS Institute Inc., Cary, NC). Linear regression equations 154 for predicting digestibility coefficients of nitrogen and LYS of wheat DDGS from chemical 155 characteristics were calculated according to a stepwise regression procedure including 156 chemical content, colour indicators or in vitro digestibility of nitrogen as explaining variables 157 using SAS regression procedures. Parameters with P value below 0.05 were considered in the 158 model. The equations with the lowest residual standard deviations (RSD) are presented. 159

RESULTS 160

Chemical composition and physical properties of wheat DDGS 161

The starch concentration was low in most DDGS samples (41g/kg DM) and varied between 162 25 and 95g/kg DM (Table 1). However, this variation was mainly due to two products with a 163 high starch content. Concentrations of CP, crude fat and crude fibre in the 10 samples 164 averaged 361, 46 and 83g/kg DM, respectively. One objective in the selection of DDGS to be 165 measured in the digestibility trials was to reduce correlations between chemical 166 characteristics; apart from the inevitable correlations between the different cell wall fractions 167 (Crude fibre, NDF, ADF and ADL), few correlations were significant (data not shown). The 168 N content measured on detergent fibre fractions indicates that a fraction of CP content was 169 included in the fibre fractions as indicated by the high CP content in NDF (91g/kg DM on 170 Table 1) 171 residues. In addition, CP in ADF and ADF content were positively correlated (r = 0.84; Table 172 2). Protein solubility measured in water and in KOH averaged 0.18 and 0.34 respectively. 173

Nevertheless, no significant correlation between the two criteria was obtained (r = 0.30; data 174 not shown). 175

The LYS and arginine levels in CP were highly variable (0.83 to 3.01 and 2.25 to 4.62 176 g/100g CP respectively), whereas contents in other AA were less variable. It should be noted 177 that all AA, except LYS and arginine, expressed in g/kg DM, were positively and 178 significantly correlated with CP content (g/kg DM) (data not shown). The FLYS content 179 averaged 1.53 g/100g CP and represented 0.76 of LYS in DDGS; this latter ratio varied from 180 0.52 to 0.89. Unavailable LYS determined from furosine content averaged 0.26 g/100g CP 181 (range: 0.13 to 0.54) and represented 0.16 of LYS (range: 0.05 to 0.37). Consequently, the 182 sum of unavailable and available LYS contents in g/100g of LYS (or recovery) averaged 0.93 183 with minimum and maximum values of 0.67 and 1.10. Finally, the correlation between LYS 184 and FLYS contents (g/100g CP) was high and positive (r = 0.99), whereas a poor correlation 185 was observed between LYS and BLYS (g/100g CP) content (r = 0.01) in connection with a 186 rather constant BLYS content and highly variable LYS values. 187

The colour, as estimated from values of L, ranged from 43 to 63. In fact, the L score 188 distribution revealed that, between the 10 DDGS samples, three products could be considered 189 as dark (L < 50) with low LYS levels (1.01 g/100g CP) and 7 as light (L > 50) with higher 190 and rather constant LYS levels (2.29 g/100g CP). Luminance values were correlated with 191 fibre fractions evaluated according to different chemical procedures and especially with ADF 192 content (r = -0.94), CP in ADF residue (r = -0.86), or FLYS content (r = 0.90) expressed in 193 g/100g of LYS (Table 2). In addition, light products might be subdivided in two additional 194 groups in connection with their starch content. Five products had starch content lower than 70 195 Tables 1 and2 g/kg and two had higher starch content. High starch content products had also higher furosine 196 content (1.4 vs 0.5 g/kg DM); the LYS recovery was also higher for these two high starch 197 samples (1.09 vs 0.92 and 0.82 for the light-low starch category and the dark category 198 respectively). 199

Digestibility of diets and wheat DDGS 200

All cockerels remained healthy throughout the duration of the experiment and no animal was 201 removed from the study. Endogenous AA losses are presented in Table 3. Aspartic and 202 glutamic acids represent the most important part of these losses (8.5 and 12.0 g/100g of total 203 losses respectively), while methionine and tryptophan each accounted for less than 2%. The 204 SDD of CP in DDGS varied between 0.55 and 0.85, with an average value of 0.74 (Table 4). 205

It was correlated with colour (r = 0.87) with the lightest samples having the greatest SDD 206 values (Table 2). Except for LYS, SDD, the SDD of each AA and the variability between 207 samples were in agreement with the SDD of CP. The average SDD of LYS was 0.46 and 208 varied between -0.04 and 0.71. Moreover, samples distribution of SDD also suggested a 209 division into two groups, as for chemical composition: a group with rather constant and high 210 SDD (0.61) with L values above 50 and a group with lower and variable SDD (-0.04 to 0.23) 211 and L values lower than 50. This observation could be repeated for all other AA but with 212 smaller differences between the two groups than for LYS. The difference in starch content for 213 the light samples did not affect the SDD of AA, except for LYS with a lower digestibility for 214 the high starch content DDGS (0.52 vs 0.64). These latter samples had also the highest 215 furosine levels (Table 1). 216

Prediction of nitrogen and LYS digestibility in wheat DDGS 217

Prediction equations were developed only for N and LYS digestibility whose values were the 218 most variable ( As reported by [START_REF] Stein | Amino acid and 445 energy digestibility in ten samples of distillers dried grains with solubles fed to growing 446 pigs[END_REF], DDGS chemical composition is highly related to the 233 process used for ethanol production. In connection with the process, most of the starch is 234 converted to ethanol and only small amounts remain in the DDGS. This was the case in the 235 present trial for wheat DDGS, except for the two highest values that originated from a process 236 with bran removal at the beginning of the process and a reintroduction of the bran at the end. 237

On the other hand, CP, crude fat and dietary fibre are not fermented and their levels in DDGS 238 are about three times higher than in wheat [START_REF] Sauvant | Tables of composition and nutritive value 432 of feed materials: Pigs, poultry, cattle, sheep, goats, rabbits, horses, fish[END_REF]. Overall, the average 239 nutrients composition of our DDGS samples was similar to previous reported values for 240 wheat DDGS [START_REF] Piron | Wheat-based dried 428 distillers' grains are variable in chemical composition[END_REF][START_REF] Lan | True ileal protein and amino acid 396 digestibilities in wheat dried distillers' grains with solubles fed to finishing pigs[END_REF]. 241

According to the process, most of the AA in the protein are from wheat (0.95) and the 242 rest comes from yeast growth [START_REF] Ingledew | Yeast for production of fuel ethanol[END_REF]. Therefore, it is logical to get a rather similar 243 AA profile in DDGS (present trial) and in wheat [START_REF] Sauvant | Tables of composition and nutritive value 432 of feed materials: Pigs, poultry, cattle, sheep, goats, rabbits, horses, fish[END_REF]. However, the 244 difference between wheat and wheat DDGS is high for LYS (2.9 vs 1.9 g/100g CP 245 wheat DDGS was measured by [START_REF] Piron | Wheat-based dried 428 distillers' grains are variable in chemical composition[END_REF]. This variability for LYS content in CP was 247 also observed in maize DDGS [START_REF] Cromwell | Physical, chemical and 372 nutritional characterisrics of distillers dried grains with solubles for chicks and pigs[END_REF][START_REF] Spiehs | Nutrient database for distiller's 442 dried grains with solubles produced from new ethanol plants in Minnesota and South 443 Dakota[END_REF] but to a smaller 248 extent than in the wheat DDGS of the present trial. 249

This relative disappearance and high variability of total LYS content in wheat DDGS 250 samples, as compared with LYS in wheat, would be associated with Maillard reactions 251 occurring during the the process of DDGS preparation [START_REF] Pedersen | Digestibility of energy and 425 phosphorus in ten samples of distillers dried grains with solubles fed to growing pigs[END_REF]van Boekel, 252 1998) and might explain the poor correlation between LYS expressed in g/kg DM and CP 253 content in wheat DDGS. These reactions produce brown-coloured compounds. the present 254 results indicate that samples with low lightness values (L < 50; n = 3) in connection with high 255 occurrence of these reactions have a specific nutrient profile with a high CP content in ADF, a 256 poor protein solubility in water and a poor LYS content. Based on reversible reaction between 257 the ε-amino group of LYS and reducing sugars under specific in vitro conditions, Maillard 258 reactions make LYS nutritionally unavailable for animals [START_REF] Hurrell | The estimation of available lysine in 391 foodstuffs after Maillard reactions[END_REF]. On 259 the other hand, high starch wheat DDGS had high luminance values (L > 60) and would not 260 be subject to Maillard reaction. In fact, these samples come from bio-ethanol plants where 261 wheat bran is removed at the beginning of the process and added back after fermentation. 262 Thus, these products have a specific nutrient profile with low ADF and CP in ADF. 263

Therefore, the colour score appears as a quick and reliable tool for wheat DDGS evaluation. 264

This also means that care must be taken in interpreting the results of LYS content in wheat 265 DDGS. Total LYS is considered to be a relevant chemical indicator of protein quality for 266 animal nutrition. However, complementary analyses indicate that in the case of wheat DDGS, 267 some LYS is potentially unavailable for the animal. Using the homoarginine method, Pahm et 268 al. (2008b) reported that the average FLYS concentration of 33 maize DDGS samples was 269 0.75 of LYS. This value is comparable to the average value obtained in the present study 270 with luminance below 50 should then be considered as abnormal and affected specific values. 297

In the present trial, ADF and CP in ADF appeared to be the best variables to predict 298 the variability in nitrogen or AA SDD among samples. In addition, the present study indicated 299 that samples with LYS content below 1.9 g/100g CP present low LYS SDD whereas constant 300 LYS SDD (0.63) could be applied for samples with LYS content above 1.9 g/100g CP. 301 Therefore, wheat DDGS with LYS content above 1.9 g/100g CP should be preferred for 302 poultry. A similar approach on maize DDGS in pigs indicates that only products with LYS 303 content above 3.2 g/100g CP should be considered as highly valuable for pigs, whereas 304 products with LYS below 2.3 g/100g in CP should be considered as badly heat-damaged 305 products [START_REF] Fontaine | Effect of 383 heat damage in an autoclave on the reactive lysine contents of soy products and corn 384 distillers dried grains with solubles. Use of the results to check on lysine damage in 385 common qualities of these ingredients[END_REF]. Wheat DDGS, when fed to pigs, had high and rather constant 306 LYS SDD values for more than 1.9 g/100g CP lys (Cozannet et al., 2010). This 1.9 g/100g CP 307 limit also corresponds with the previous limit defined between light and dark samples. 308 Therefore, DDGS colour may be a good indicator of AA SDD and particularly LYS SDD. 309

The prediction equations for LYS SDD obtained in the present trial were established 310 from measurements on 10 samples selected from a set of 17 samples. These 10 samples were 311 thoroughly analysed and measured for their AA standardised ileal digestibility (SID) in pigs 312 (Cozannet et al., 2010) and poultry (present trial). Wheat DDGS LYS SID in pigs averaged 313 0.56 (0.09 to 0.83), with a high correlation between SID of LYS in pigs and LYS SDD in 314 cockerels (r=0.93) and a very comparable ranking of LYS SID and LYS SDD values (Figure 315 1). Values for pigs could then be estimated from values in poultry (and vice versa). Chung 316 and Baker (1992) reported greater SID in pig compared with poultry, however, these authors 317 noted that their ranking of SID was not in agreement with that of [START_REF] Opapeju | Amino acid 411 digestibility in dry extruded-expelled soybean meal fed to pigs and poultry[END_REF] The results have then been used for validating the prediction equations established on the 10 321 samples using quadratic model with LYS content in g/100g CP as predictor (Figure 2). LYS 322 SDD of validation samples averaged 0.59 with an average predicted value of 0.56. In 323 addition, the correlation between the measured and predicted values was quite high (r = 0.98; 324 RSD=3.5). This indicates that the prediction equation based on LYS percentage in CP 325 represents an acceptable predictor of LYS SDD in wheat DDGS. 326

Conclusions 327

As reported for maize DDGS, the present results indicate that wheat DDGS have variable 328 nutrient composition and SDD of AA, with LYS content and digestibility exhibiting more 329 variability than other AA. In addition, the total CP content in wheat DDGS is relatively high 330 and DDGS can then be considered as a potential source of supplemental protein and AA for 331 poultry. Nevertheless, the protein quality of DDGS is rather low due to a deficiency in several 332 evaluated on cockerel and lysine standardised ileal digestibility (LYS SID) on pig (Cozannet 506 et al., 2010).

  19 samples of wheat DDGS from 7 European ethanol plants were collected in July-72

  not shown) and CP content in ADF (4 to 82; 28 g/kg on average;

  a highly variable LYS content in CP. A similar variability in LYS level of 246

  . In order to get more realistic nutritional values of wheat DDGS, samples 296

Figures

  Figures 1 and 2 near here

Figure 1 .

 1 Figure 1. Correspondence between wheat DDGS lysine standardised digestibility (LYS SDD) 505

Table 5 )

 5 . These equations indicate that N and LYS SDD might be predicted 219 from colour, in vitro and chemical characteristics. The strong relationship beteween in vitro N 220 or LYS SDD (r = 0.90 and 0.87, respectively; Table2) were notable. 221Nevertheless, the strong relationships previously reported between colour score and chemical 222 characteristics did not allow multivariate models involving both types of criteria. In addition, 223 observations on the darkest samples have a high influence on the coefficients. Equations 224 obtained according to a stepwise regression procedure indicated that nitrogen digestibility can 225 be evaluted from CP in ADF content. However, the only predictor for LYS SDD was LYS 226 concentration (g/100g CP) according to a linear (Table5, equation 5) or a quadratic regression 227 (Table5, equation 6) or a linear-plateau regression with a breakpoint for 1.9 g/100g LYS 228 content in CP and a plateau LYS SDD of 0.63 (Table5, equations 7 and 8). The breakpoint 229 defined for LYS content of 1.9 g/100g CP is close to the previous limit defined between light 230

	Tables 3, 4, 5 near here

Table 1 .

 1 Composition of wheat distillers' dried grains with solubles (DDGS)

	455 456
				Wheat DDGS
					(n=10)		Colour 1	Starch 2
							Dark
							(n =3	Light (n	< 7% DM (n	> 7% DM
				Mean Min Max	)	= 7)	= 5) 3	(n = 2)
	Dry matter		926 893 944	925	927	922	939
	Nutrient	content,
	g/kg DM
	Ash		52	43	67	57	49	50	48
	Protein
	(N*6.25)		361 326 389	358	362	366	353
	Crude fat		46	36	56	51	43	45	40
	Crude fibre		83	62	109	95	78	83	64
	NDF 4		292 251 338	293	292	304	262
	ADF 4		120	77	179	157	104	113	83
	ADL 4		48	21	105	84	33	36	24
	Starch Ewers	41	25	95	29	47	30	87
	Colour score
	L (luminance) 54	43	63	46	57	55	63
	a (red index)	6.2	4.4	7.3	5.3	6.7	6.4	7.2
	b	(yellow
	index)			13.4 5.3 19.0	8.0	15.8	14.5	19.0
	Protein
	solubility
	in water		0.18 0.09 0.30	0.22	0.16	0.19	0.09
	in KOH		0.34 0.27 0.43	0.36	0.33	0.34	0.32
	CP in ADFn (g/kg
	DM)			28	4	82	68	11	13	4
	Maillard reactions
	Furosine (g/kg DM) 0.71 0.36 1.50	0.63	0.74	0.50	1.40
	In	vitro	N
	digestibility 5		0.74 0.55 0.86	0.62	0.79	0.79	0.78
	Amino acid contents, g/100g CP
	Indispensable amino acids
	Arginine		3.76 2.25 4.62	2.47	4.30	4.23	4.49
	Histidine		1.95 1.66 2.18	1.71	2.06	2.04	2.11
	Isoleucine		3.42 3.28 3.54	3.31	3.47	3.47	3.48
	Leucine			6.41 5.83 6.77	6.10	6.54	6.50	6.62
	Lysine
	Total 6		1.91 0.83 3.01	1.01	2.29	2.35	2.12
	Free 7		1.53 0.50 2.59	0.57	1.95	2.00	1.82
	Bound 8		0.26 0.13 0.54	0.24	0.28	0.18	0.50
	Residual 9		0.11 -0.22 0.42	0.20	0.07	0.17	-0.20

Table 2 .

 2 Correlation coefficients 1 between wheat DDGS chemical characteristics and standardised digestibility (SDD) of nitrogen and lysine (n

	473
	474						= 10 ,2
			Ash	Protein	ADF	CP in ADF	L	Furosine	IDP 3	LYS 4	BLYS 5 FLYS 6	DNv 7 SDD N 8
		Protein	0.03
		ADF	0.34	0.05
		CP in ADF	0.49	0.12	0.84*
		L	-0.40	0.04	-0.94*	-0.86*
		Furosine	0.04	-0.11	-0.54	-0.26	0.54
		IDP 3	0.13	-0.52	0.57	0.30	-0.69*	-0.56
		LYS 4	-0.53	-0.04	-0.66	-0.91*	0.67	-0.06	0.07
		BLYS 5	0.57	-0.07	-0.06	0.46	-0.12	0.64*	-0.26	-0.69*
		FLYS 6	-0.35	0.01	-0.89*	-0.97*	0.87*	0.21	-0.37	0.87*	-0.44
		DNv 7	-0.50	-0.18	-0.73*	-0.91*	0.70*	-0.02	-0.15	0.89*	-0.66*	0.84*
		SID N 8	-0.50	0.00	-0.87*	-0.96*	0.87*	0.13	-0.38	0.84*	-0.51	0.96*	0.90*
		SID LYS 9	-0.58*	0.02	-0.58	-0.87*	0.64*	-0.14	-0.10	0.97*	-0.82*	0.85*	0.88*	0.87*
	475	1 Correlation whose absolute value is above 0.64 is significant (P < 0.05).
	476	2 Ash, Protein, ADF, CP in ADF, furosine and LYS expressed in g/100g CP. BLYS and FLYS in g/100g LYS. DNv, SDD N and SDD LYS (%).
	477 478 479 480 481 482 483	3 Protein Dispersibility Index. 4 Total LYS content. 5 Free LYS content. 6 Bound LYS content. 7 In vitro Digestibility Index. 8 N standardised ileal digestibility 9 Total LYS standardised ileal digestibility

Table 3 .

 3 Basal endogenous losses (g/kg DM intake)1, 2

	484
	485
			Mean	SD
		Nitrogen 3	1.99	0.19
		Indispensable amino acids
		Arginine	0.92	0.1
		Histidine	0.35	0.03
		Isoleucine	0.57	0.06
		Leucine	0.91	0.1
		Lysine	0.70	0.07
		Methionine	0.16	0.02
		Phenylalanine	0.51	0.06
		Threonine	1.21	0.13
		Tryptophane	0.24	0.02
		Valine	1.28	0.17
		Dispensable amino acids
		Alanine	0.84	0.07
		Aspartic acid	1.32	0.13
		Cystine	0.56	0.06
		Glutamic acid	1.92	0.19
		Glycine	1.27	0.14
		Proline	1.14	0.12
		Serine	1.08	0.12
	486 487 488 489	Tyrosine 1 Animals fed with 50g fresh N free diet 2 Endogenous amino acids losses measured on 15 experimental units 0.53 3 Nitrogen in digesta evaluated according to methodology described by Terpstra and De Hart 0.07 (1974)

Table 4 .

 4 Standardised ileal digestibilities of crude protein and amino acids in wheat DDGS Luminance ranged from 43 to 50 for the dark group and 52 to 64 for the light group. 4932 Starch content ranged from 2.5 to 4.5% DM for starch content < 7% and 8.6 to 10.1% DM for 494 starch content > 7% according to the classification of[START_REF] Sauvant | Tables of composition and nutritive value 432 of feed materials: Pigs, poultry, cattle, sheep, goats, rabbits, horses, fish[END_REF]. 4953 From the analysis of variance on 60 results with wheat DDGS sample (n=10; 6 measurements 496 per diet) as a fixed effect. 4974 Only wheat DDGS with luminance above 52 are considered for the low starch category.

	491
	492
		Wheat DDGS (n=10)	Colour 1	Starch 2	Statistics 3
				Dark	Light	< 7%	> 7%	DDGS
		Mean	Min Max	(n=3)	(n=7)	(n=5) 4	(n=2)	effect	RSD
	Nitrogen	0.74	0.55 0.85	0.58 0.81	0.81 0.81	< 0.001	0.02
	Indispensable amino acids
	Arginine	0.68	0.42 0.81	0.46 0.78	0.78 0.77	< 0.001	0.03
	Histidine	0.70	0.48 0.83	0.50 0.78	0.78 0.78	< 0.001	0.04
	Isoleucine	0.71	0.47 0.84	0.53 0.79	0.79 0.78	< 0.001	0.03
	Leucine	0.76	0.55 0.86	0.59 0.83	0.83 0.82	< 0.001	0.03
	Lysine	0.46	-0.04 0.71	0.12 0.61	0.64 0.52	< 0.001	0.03
	Methionine	0.75	0.57 0.84	0.60 0.81	0.81 0.81	< 0.001	0.02
	Phenylalanine	0.82	0.64 0.90	0.67 0.88	0.88 0.88	< 0.001	0.02
	Threonine	0.65	0.42 0.76	0.46 0.73	0.73 0.73	< 0.001	0.03
	Tryptophan	0.67	0.43 0.78	0.49 0.75	0.75 0.76	< 0.001	0.03
	Valine	0.72	0.47 0.85	0.52 0.81	0.81 0.80	< 0.001	0.03
	Dispensable amino acids
	Alanine	0.65	0.42 0.79	0.46 0.74	0.74 0.73	< 0.001	0.04
	Aspartic acid	0.54	0.27 0.68	0.31 0.65	0.64 0.66	< 0.001	0.03
	Cystine	0.60	0.30 0.77	0.37 0.71	0.70 0.71	< 0.001	0.03
	Glutamic acid	0.82	0.65 0.92	0.69 0.87	0.87 0.88	< 0.001	0.01
	Proline	0.81	0.64 0.91	0.67 0.87	0.87 0.88	< 0.001	0.01
	Serine	0.70	0.51 0.81	0.53 0.78	0.78 0.78	< 0.001	0.02
	Tyrosine	0.68	0.51 0.77	0.55 0.74	0.74 0.73	< 0.001	0.02
	1 498 499

Table 5 .

 5 Prediction equations of nitrogen (N) and lysine (LYS) standardised digestibility of

	500
	501	wheat DDGS in cockerels 1
	Equation number	Regression equations	R²	RSD
	Nitrogen standardised digestibility (n = 10)
	1	SDD N = -4.9 + 1.5×L	0.75	5.9
	2	SDD N = 85.4 -0.4×CP in ADF	0.92	3.4
	Lysine standardised digestibility (n =10)
	3	SDD LYS = -90.3 × 2.5×L	0.41	21.2
	4	SDD LYS = 67.5 -0.8×CP in ADF	0.75	13.7
	5	SDD LYS = -16.8 + 33.0 LYS	0.88	9.6
	6	SDD LYS = -56.6 + 82.7 LYS -13.4 LYS²	0.94	7.3
	7 8	SDD LYS (if LYS > 1.9) = 63.0 SDD LYS (if LYS < 1.9) = -42.4 + 55.5 LYS	0.90	9.3

 5) of lysine standardised 513 digestibility (LYS SDD) from an independent data set (n=9). 514


Emmanuel Hebey
email: emmanuel.hebey@math.u-cergy.fr

Trong Tuong Truong
email: tuong.truong@u-cergy.fr

STATIC KLEIN-GORDON-MAXWELL-PROCA SYSTEMS IN 4-DIMENSIONAL CLOSED MANIFOLDS

Static Klein-Gordon-Maxwell-Proca systems are massive versions of the electrostatic Klein-Gordon-Maxwell Systems. The vector field in these systems inherits a mass and is governed by the Proca action which generalizes that of Maxwell. Klein-Gordon-Maxwell systems are intended to provide a dualistic model for the description of the interaction between a charged relativistic matter scalar field and the electromagnetic field that it generates. The electromagnetic field is both generated by and drives the particle field. In the electrostatic form of the Klein-Gordon-Maxwell systems, looking for standing waves ue iωt , the matter field is characterized by the property that u, together with a gauge potential v, solve the electrostatic Klein-Gordon-Maxwell systems (0.3) with m 1 = 0. In the case of a closed manifold we discuss here the two equations in (0.3) are independent one of another when m 1 = 0 and the system reduces to the sole Schrödinger equation. The Proca formalism, for m 1 > 0, leads to a deeper phenomenon and is more appropriate to the closed case. The particle in this model interacts via the minimum coupling rule ∂ t → ∂ t + iqϕ and ∇ → ∇ -iqA (0.1) with an external massive vector field (ϕ, A) which is governed by the Maxwell-Proca Lagrangian. The Proca action is a gauge-fixed version of the Stueckelberg action in the Higgs mechanism (see Goldhaber and Nieto [START_REF]Photon and Graviton mass limits[END_REF], and Ruegg and Ruiz-Altaba [START_REF] Ruegg | The Stueckelberg field[END_REF]). In the Proca formalism, developped under the influence of de Broglie, the photon inherits a nonzero mass. This issue is of considerable importance and intensively studied in modern physics (see for instance Adelberger, Dvali and Gruzinov [START_REF] Adelberger | Photon-Mass Bound Destroyed by Vortices[END_REF], Byrne [START_REF] Byrne | Cosmic tests of Maxwell's equations. I -A photon rest mass[END_REF], Goldhaber and Nieto [START_REF] Goldhaber | Terrestrial and Extraterrestrial limits on the photon mass[END_REF][START_REF]Photon and Graviton mass limits[END_REF], Luo and Tu [START_REF] Luo | Experimental tests of Coulomb's Law and the photon rest mass[END_REF], Luo, Gillies and Tu [START_REF] Luo | The mass of the photon[END_REF] and the references in these papers). When n = 3, the KGMP equations consist in the nonlinear Klein-Gordon matter equation, the charge continuity equation and the massive modified Maxwell equations in SI units, which are hereafter explicitly written down:

∇.E = ρ/ε 0 -µ 2 ϕ , ∇ × H = µ 0 J + ε 0 ∂E ∂t -µ 2 A ,
∇ × E + ∂H ∂t = 0 and ∇.H = 0 .

(0.2)

Date: October 29, 2010.

These massive Maxwell equations, as modified to Proca form, appear to have been first written in modern format by Schrödinger [START_REF] Schrödinger | The Earth's and the Sun's permanent magnetic fields in the unitary field theory[END_REF]. The Proca formalism a priori breaks Gauge invariance. Gauge invariance can be restaured by the Stueckelberg trick, as pointed out by Pauli [START_REF] Pauli | Relativistic Field Theories of Elementary Particles[END_REF], and then by the Higgs mechanism. We refer to Goldhaber and Nieto [START_REF]Photon and Graviton mass limits[END_REF], Luo, Gillies and Tu [START_REF] Luo | The mass of the photon[END_REF], and Ruegg and Ruiz-Altaba [START_REF] Ruegg | The Stueckelberg field[END_REF] for very complete references on the Proca approach.

In what follows we let (M, g) be a smooth compact 3, 4-dimensional Riemannian manifold. We let also 2 = 2n n-2 be the critical Sobolev exponent, where n is the dimension of M . Given real numbers q > 0, m 0 , m 1 > 0, ω ∈ (-m 0 , m 0 ), and p ∈ (2, 2 ], the derivation of the Klein-Gordon-Maxwell-Proca system we investigate in this paper is written as

∆ g u + m 2 0 u = u p-1 + ω 2 (qv -1) 2 u ∆ g v + m 2 1 + q 2 u 2 v = qu 2 , (0.3)
where ∆ g = -div g ∇ is the Laplace-Beltrami operator. The system (0.3) corresponds to looking for standing waves ue iωt for the full KGMP system in the static case where the massive vector field (ϕ, A) depends on the sole spatial variable. The system is energy critical when n = 3 and p = 6 and when n = 4 and p = 4. It is subcritical otherwise, namely when n = 3 and p ∈ [START_REF] D'aprile | Layered solutions for a semilinear elliptic system in a ball[END_REF][START_REF] Berti | Non-compactness and multiplicity results for the Yamabe problem on S n[END_REF] or n = 4 and p ∈ (2, 4).

In the above model, m 1 is a coupling constant which makes that the two equations in (0.3) are trully coupled (m 1 is the Proca mass in the Maxwell-Proca formalism) while m 0 is the mass of the particle, q is the charge of the particle, u is the amplitude in the writing of the particle, ω is its temporal frequency (referred to as the phase in the sequel), and v is the electric potential.

Let S g stand for the scalar curvature of g, and S p (ω) be the set consisting of the positive smooth solutions U = (u, v) of (0.3) with phase ω and nonlinear term u p-1 . Namely, S p (ω) = (u, v) smooth s.t. u > 0, v > 0, and (u, v) solve (0.3) .

(0.4)

Given ω ∈ [0, m 0 ), we let

K 0 (ω) = (-m 0 , -ω] [ω, m 0 ) . (0.5)
When ω = 0, K 0 (0) = (-m 0 , m 0 ) is the full admissible phase range. For θ ∈ (0, 1), and

U = (u, v), we let U C 2,θ = u C 2,θ + v C 2,θ .
The following result was proved in Druet and Hebey [START_REF]Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces[END_REF].

Theorem 0.1 (The 3-dimensional case -Druet and Hebey [START_REF]Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces[END_REF]). Let (M, g) be a smooth compact 3-dimensional Riemannian manifold m 0 , m 1 > 0, ω ∈ (-m 0 , m 0 ), and p ∈ [START_REF] D'aprile | Layered solutions for a semilinear elliptic system in a ball[END_REF][START_REF] Berti | Non-compactness and multiplicity results for the Yamabe problem on S n[END_REF]. When p = 6 assume

m 2 0 < ω 2 + 1 8 S g (x) (0.6)
for all x ∈ M . Then (0.3) possesses a smooth positive solution. Moreover, for any p ∈ (2, 6), and any θ ∈ (0, 1), there exists C > 0 such that for any ω ∈ K 0 (0), and any U ∈ S p (ω ), U C 2,θ ≤ C, where S p (ω ) is as in (0.4) and K 0 (0) is as in (0.5).

Assuming again (0.6), there also holds that for any θ ∈ (0, 1), U C 2,θ ≤ C for all U ∈ S 6 (ω ) and all ω ∈ K 0 (ω), where C > 0 does not depend on ω and U.

This result exhibits phase compensation in the 3-dimensional case. We aim in this paper in proving that a similar phenomenon holds true when n = 4. In this dimension the second equation in (0.3) becomes critical and this leads to serious difficulties. We prove below the existence of smooth positive solutions and the existence of uniform bounds for (0.3) in the subcritical cases p ∈ (2, 4) without any conditions, and in the critical case p = 4 assuming that the mass potential, balanced by the phase, is smaller than the geometric threshold potential of the conformal Laplacian. In doing so we prove that phase compensation still holds true for our systems when n = 4. Our result, in the subcritical case, is as follows.

Theorem 0.2 (The subcritical 4-dimensional case). Let (M, g) be a smooth compact 4-dimensional Riemannian manifold, q > 0, m 0 , m 1 > 0, ω ∈ (-m 0 , m 0 ), and p ∈ [START_REF] D'aprile | Layered solutions for a semilinear elliptic system in a ball[END_REF][START_REF] Aubin | Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF]. Then (0.3) possesses a smooth positive solution. Moreover, for any θ ∈ (0, 1), there exists C > 0 such that for any ω ∈ K 0 (0), and any U ∈ S p (ω ), U C 2,θ ≤ C, where S p (ω ) is as in (0.4) and K 0 (0) is as in (0.5).

In the critical case we prove the following result. The geometry of the ambiant inhomogeneous space, through the scalar curvature of g, comes to play a role as in the 3-dimensional case. However, the result now turns out to be local in its existence part.

Theorem 0.3 (The critical 4-dimensional case). Let (M, g) be a smooth compact 4-dimensional Riemannian manifold, q > 0, m 0 , m 1 > 0, ω ∈ (-m 0 , m 0 ), and p = 4. Assume

m 2 0 < ω 2 + 1 6 S g (x) (0.7)
for some x ∈ M . Then (0.3) possesses a smooth positive solution. Assuming that (0.7) holds true for all x ∈ M there also holds that for any θ ∈ (0, 1), U C 2,θ ≤ C for all U ∈ S 4 (ω ) and all ω ∈ K 0 (ω), where C > 0 does not depend on ω and U, S 4 (ω ) is as in (0.4), and K 0 (ω) is as in (0.5).

There are two consequences to Theorem 0.3. We list them in points (i)-(ii) below. In point (i) we illustrate the phase compensation effect associated with (0.3). There we always get existence and a priori bounds for all phases ω which are close to m 0 . Point (ii) concerns the full range of phases when we assume m 0 is not too large.

(i) Phase compensation in the critical case -Assume p = 4 and S g > 0 in M . Then there exists ε > 0 such that for any m 0 -ε < |ω| < m 0 , (0.3) possesses a smooth positive solution. Moreover, for any θ ∈ (0, 1), there exists C > 0 such that U C 2,θ ≤ C for all U ∈ S 4 (ω) and all ω a -ε < |ω| < ω a .

(iii) Full phase range in the critical case -Assume p = 4 and m 2 0 < 1 6 S g in M . For any ω ∈ (-m 0 , m 0 ), (0.3) possesses a smooth positive solution. Moreover, for any θ ∈ (0, 1), there exists C > 0 such that U C 2,θ ≤ C for all U ∈ S 4 (ω) and all ω ∈ (-ω a , ω a ).

As an immediate consequence of the C 2,θ -bounds in the above results we obtain phase stability for standing waves of the Klein-Gordon-Maxwell-Proca equations in electrostatic form. Standing waves for the Klein-Gordon-Maxwell-Proca equations in electrostatic form are written as S = ue iωt and they are coupled with a gauge potential v, where (u, v) solves (0.3). Roughly speaking, phase stability means that for any arbitrary sequence of standing waves u α e iωαt , with gauge potentials v α , the convergence of the phases ω α in R implies the convergence of the amplitudes u α and of the gauge potentials v α in the C 2 -topology. Phase stability prevents the existence of arbitrarily large amplitude standing waves.

High dimensional KGM systems in Coulomb gauge have been recently investigated by Rodnianski and Tao [START_REF] Rodnianski | Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions[END_REF] and with special emphasis in (1 + 4)-dimensions by Klainerman and Tataru [START_REF] Klainerman | On the optimal regularity for Yang-Mills equations in R 4+1[END_REF] and Selberg [START_REF] Selberg | Almost optimal local well-posedness of the Klein-Gordon-Maxwell system in 1+4 dimensions[END_REF]. Electrostatic KGM systems in the three dimensional case have been investigated by several authors. Possible references on the physics side are by Benci and Fortunato [START_REF] Benci | Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations[END_REF], Long [START_REF] Long | Existence and stability of solitary waves in non-linear Klein-Gordon-Maxwell equations[END_REF], Long and Stuart [START_REF] Long | Effective dynamics for solitons in the nonlinear Klein Gordon Maxwell system and the Lorentz force law[END_REF]. Blowing-up solutions to the electrostatic Schrödinger-Maxwell system, a cousin of the electrostaic KGM type systems that we consider here, have been constructed in D'Aprile and Wei [START_REF] D'aprile | Layered solutions for a semilinear elliptic system in a ball[END_REF][START_REF]Clustered solutions around harmonic centers to a coupled elliptic system[END_REF].

We briefly discuss in Section 1 the physics relevance of (0.3). We prove our theorem in Sections 2 to 4. The existence part in the theorem is proved in Section 2. The C 2,θ -bound in the subcritical case is established in Section 3. The more delicate C 2,θ -bound in the critical case is established in Sections 4 . The phase compensation phenomenon in the theorem holds true thanks to the 4-dimensional log effect µ 2 = o(µ 2 log µ) as µ → 0.

The physics origin of the system

The Klein-Gordon-Maxwell-Proca system discussed in this work describes an interacting field theory model in theoretical physics. Most electromagnetic phenomena are described by conventional electrodynamics, which is a theory of the coupling of electromagnetic fields to matter fields. Of prime importance for particle physics is fermion electrodynamics in which matter is represented by spinor fields. However one may have also boson electrodynamics in which matter is described by integer spin or bosonic fields. The simplest one is of course the complex scalar field, describing spinless particles having electric charges ±q. It gives rise to scalar electrodynamics, which describes in the non-relativistic limit the superconductivity of metals at very low temperatures. In the more general context of particle physics, a complex scalar field ψ may serve to describe scalar mesons in nuclear matter interacting via a massive vector boson field (ϕ, A).

The interaction in this model is described by the minimum substitution rule (0.1) in a nonlinear Klein-Gordon Lagrangian. As for the external massive vector field it is governed by the Maxwell-Proca Lagrangian. More precisely, assuming for short that the manifold is orientable, we define the Lagrangian densities L N KG and L M P of ψ, ϕ, and A by

L N KG (ψ, ϕ, A) = 1 2 ( ∂ ∂t + iqϕ)ψ 2 - 1 2 |(∇ -iqA)ψ| 2 + m 2 0 2 |ψ| 2 - 1 p |ψ| p , L M P (ϕ, A) = 1 2 ∂A ∂t + ∇ϕ 2 - 1 2 |∇ × A| 2 + m 2 1 2 |ϕ| 2 - m 2 1 2 |A| 2 , (1.1)
where ∇× = d, is the Hodge dual, ψ represents the matter complex scalar field, m 0 its mass, q its charge, (ϕ, A) the electromagnetic vector field, and m 1 its mass.

It can be noted that (ϕ, A) 2 L = |ϕ| 2 -|A| 2 is the square of the Lorentz norm of (ϕ, A) with respect to the Lorentz metric diag(1, -1, . . . , -1). The total action functional for ψ, φ, and A is then given by

S(ψ, ϕ, A) = (L N KG + L M P ) dv g dt . (1.2)
Writing ψ in polar form as ψ(x, t) = u(x, t)e iS(x,t) , taking the variation of S with respect to u, S, ϕ, and A, we get four equations which are written as

           ∂ 2 u ∂t 2 + ∆ g u + m 2 0 u = u p-1 + ∂S ∂t + qϕ 2 -|∇S -qA| 2 u ∂ ∂t ∂S ∂t + qϕ u 2 -∇. (∇S -qA) u 2 = 0 -∇. ∂A ∂t + ∇ϕ + m 2 1 ϕ + q ∂S ∂t + qϕ u 2 = 0 ∆ g A + ∂ ∂t ∂A ∂t + ∇ϕ + m 2 1 A = q (∇S -qA) u 2 , (1.3)
where ∆ g = -div g ∇ is the Laplace-Beltrami operator, ∆ g = δd is half the Laplacian acting on forms, and δ is the codifferential. We refer to this system as a nonlinear Klein-Gordon-Maxwell-Proca system. When n = 3, ∆ g A = ∇ × (∇ × A) and if we let

E = - ∂A ∂t + ∇ϕ , H = ∇ × A , ρ = - ∂S ∂t + qϕ qu 2 , and j = (∇S -qA) qu 2 , (1.4)
then the two last equations in (1.3) give rise to the first pair of the Maxwell-Proca equations (0.2) with 0 = µ 0 = 1 (units are chosen such that c = 1) and µ 2 = m 2 1 , while the two first equations in (1.4) give rise to the second pair of the equations. The first equation in (1.3) gives rise to the nonlinear Klein-Gordon matter equation. The second equation in (1.3) gives rise to the charge continuity equation ∂ρ ∂t + ∇.j = 0 which, thanks to (0.2), is equivalent to the Lorentz condition ∇.A + ∂ϕ ∂t = 0. We assume in what follows that u(x, t) = u(x) does not depend on t, S(x, t) = ωt does not depend on x, and ϕ(x, t) = ϕ(x), A(x, t) = A(x) do not depend on t. In other words, we look for standing waves solutions of (1.3) and assume that we are in the static case of the system where (ϕ, A) depends on the sole spatial variable. By the fourth equation in (1.3) we then get that

∆ g A + (q 2 u 2 + m 2 1 )A = 0 .
This clearly implies that, and is equivalent to, A ≡ 0 since

(∆ g A, A) = |dA| 2 .
As a remark, assuming that A ≡ 0, the Lorentz condition for the external Proca field (ϕ, A) would make ϕ dependent on the sole spatial variables. As for the second equation in (1.3) it reduces to ∂ 2 S ∂t 2 = 0. It is automatically satisfied when S(t) = ωt, and we are thus left with the first and third equations in (1.3). Letting S = -ωt, and ϕ = ωv, these equations are rewritten as

∆ g u + m 2 0 u = u p-1 + (qϕ -ω) 2 u ∆ g ϕ + m 2 1 ϕ + q(qϕ -ω)u 2 = 0 . (1.5)
In particular, letting ϕ = ωv, in (1.5), we recover our original system (0.3). In other words, our original system (0.3) corresponds to looking for standing waves solutions of the Klein-Gordon-Maxwell-Proca system (1.3) in static form.

Existence Theory

We prove the existence part in Theorems 0.2 and 0.3. Formally, solutions of (0.3) are critical points of the functional S defined by

S(u, v) = 1 2 M |∇u| 2 dv g - ω 2 2 M |∇v| 2 dv g + m 2 0 2 M u 2 dv g - ω 2 m 2 1 2 M v 2 dv g - 1 p M u p dv g - ω 2 2 M u 2 (1 -qv) 2 dv g . (2.1)
The functional S is strongly indefinite because of the competition between u and v. Following a very nice idea going back to Benci-Fortunato [START_REF] Benci | Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations[END_REF], we introduce the auxiliary functional Φ given by

∆ g Φ(u) + (m 2 1 + q 2 u 2 )Φ(u) = qu 2 , (2.2)
and then consider that u in (0.3) can be seen as a critical point of

I p (u) = 1 2 M |∇u| 2 dv g + m 2 0 2 M u 2 dv g - 1 p M (u + ) p dv g - ω 2 2 M (1 -qΦ(u)) u 2 dv g , (2.3)
where

u + = max(u, 0). Let Ψ : H 1 (M ) → R be defined by Ψ(u) = 1 2 M (1 -qΦ(u)) u 2 dv g . (2.4)
The following lemma establishes the existence and differentiability of Φ, as well as the C 1 -smoothness of Ψ. Equation (2.2) is critical when n = 4 because of the term u 2 Φ(u).

Lemma 2.1. Let (M, g) be a smooth compact Riemannian 4-manifold and q > 0.

There exists Φ :

H 1 (M ) → H 1 (M ) such that (2.
2) holds true and 0 ≤ Φ(u) ≤ 1 q for all u ∈ H 1 (M ). Moreover, Φ is locally Lipschitz and differentiable. Its differential DΦ(u) = V u at u is given by

∆ g V u (ϕ) + (m 2 1 + q 2 u 2 )V u (ϕ) = 2qu (1 -qΦ(u)) ϕ (2.5) for all ϕ ∈ H 1 (M ). The functional Ψ : H 1 (M ) → R defined in (2.4) is C 1 in H 1 (M ) and DΨ(u).(ϕ) = M (1 -qΦ(u)) 2 uϕdv g (2.6) for all u, ϕ ∈ H 1 (M ).
Proof of Lemma 2.1. We briefly sketch the proof. Let u ∈ H 1 and H u : H 1 → R be defined by

H u (ϕ) = M |∇ϕ| 2 dv g + M (m 2 1 + q 2 u 2 )ϕ 2 dv g .
The functional is well defined since

H 1 ⊂ L 4 . Letting Φ(0) = 0 we can assume that u ≡ 0. Let µ = inf u∈H 1 , R u 2 ϕ=1
H u (ϕ) .

By standard minimization arguments there exists ϕ ∈ H 1 (M ) such that M u 2 ϕdv g = 1 and H u (ϕ) = µ. In particular, µ > 0. Letting Φ(u) = q µ ϕ we get that Φ(u) solves (2.2) in H 1 . It is easily seen that Φ(u) is unique. By the maximum principle, Φ(u) ≥ 0. Noting that

∆ g 1 q -Φ(u) + (m 2 1 + q 2 )u 2 1 q -Φ(u) ≥ 0
it also follows from the maximum principle that Φ(u) ≤ 1 q . Now we let u, v ∈ H 1 (M ). We have that

∆ g (Φ(v) -Φ(u)) + (m 2 1 + q 2 )u 2 (Φ(v) -Φ(u)) = q(v 2 -u 2 ) (1 -qΦ(v)) .
Multiplying the equation by Φ(v) -Φ(u), integrating over M , and by the Sobolev emedding theorem, we get that

Φ(v) -Φ(u) H 1 ≤ C ( u H 1 + v H 1 ) v -u H 1 .
(2.7)

In particular, Φ is locally Lipschitz continuous. We can prove the existence of V u (ϕ) in (2.5) as when proving the existence of Φ(u). Writing the equation satisfied by Φ(u + ϕ) -Φ(u) -V u (ϕ), multiplying the equation by Φ(u + ϕ) -Φ(u) -V u (ϕ) and integrating over M , we get that

Φ(u + ϕ) -Φ(u) -V u (ϕ) H 1 ≤ C ϕ H 1 ( ϕ H 1 + u H 1 Φ(u + ϕ) -Φ(u) H 1 )
Then the differentiability of Φ follows from the continuity of Φ. In particular, Ψ is differentiable. By (2.2),

Ψ(u) = 1 2 M |∇Φ(u)| 2 + m 2 1 Φ(u) 2 dv g + 1 2 M (1 -qΦ(u)) 2 u 2 dv g ,
and we also have that ∂H ∂Φ (u, Φ(u)) = 0, where

H(u, Φ) = 1 2 H u (Φ) -q M u 2 Φdv g . Noting that Ψ(u) = H (u, Φ(u)) + 1 2 M u 2 dv g ,
we get that (2.6) holds true. The continuity of DΨ can be proved directly from (2.6) and the continuity of Φ. This ends the proof of the lemma. Now we prove the subcritical existence of Theorem 0.2. We proceed by applying the mountain pass lemma to the functional I p in (2.3).

Proof of existence in Theorem 0.2. By Lemma 2.1,

I p is C 1 in H 1 . Let u 0 ∈ H 1 such that u + 0 ≡ 0. There holds I p (0) = 0 and I p (tu 0 ) → -∞ as t → +∞ since p > 2. Since 0 ≤ Φ(u) ≤ 1
q for all u, we also have that

I p (u) ≥ 1 2 M |∇u| 2 dv g + (m 2 0 -ω 2 ) M u 2 dv g - 1 p M |u| p dv g ≥ C 1 u 2 H 1 -C 2 u p H 1
for all u ∈ H 1 , where C 1 , C 2 > 0 do not depend on u. In particular, there exist δ, C > 0 such that

I p (u) ≥ C for all u ∈ H 1 such that u H 1 = δ. Let T 0 = T 0 (u 0 ), T 0
1, be such that I p (T 0 u 0 ) < 0, and c p = c p (u 0 ) be given by

c p = inf P ∈P max u∈P I p (u) , (2.8)
where P is the class of continuous paths joining 0 to T 0 u 0 . According to the above we can apply the mountain pass lemma and we get the existence of a sequence (u α ) α in H 1 such that I p (u α ) → c p and DI p (u α ) → 0 as α → +∞. Writing that I p (u α ) = c p + o(1) and that DI p (u α ).(u α ) = o( u α H 1 ), we get by Lemma 2.1 that

1 2 M |∇u α | 2 + m 2 0 u 2 α dv g = 1 p M (u + α ) p dv g + c p + ω 2 2 M (1 -qΦ(u α )) u 2 α dv g + o(1) 1 2 M |∇u α | 2 + m 2 0 u 2 α dv g = 1 2 M (u + α ) p dv g + ω 2 2 M (1 -qΦ(u α )) 2 u 2 α dv g + o ( u α H 1 ) (2.9) for all α. Writing that DI p (u α ).(u - α ) = o( u - α H 1 )
we get that u - α → 0 in H 1 as α → +∞. By (2.9) we then get that (u α ) α is bounded in H 1 . In particular, there exists u p ∈ H 1 (M ) such that, up to passing to a subsequence,

(i) u α u p weakly in H 1 , (ii) u α → u p in L p ,
and u α → u p a.e. as α → +∞. Substracting one equation to another in (2.9), letting α → +∞, and since c p = 0, we get that u p ≡ 0. Writing the equation satisfied by Φ(u α ) -Φ(u p ), multiplying the equation by Φ(u α ) -Φ(u p ) and integrating over M , we get that Φ(u α ) → Φ(u p ) in H 1 (2.10) as α → +∞. Now we let ϕ ∈ H 1 . There holds DI p (u α ).(ϕ) = o(1). Hence, by Lemma 2.1,

M ∇u α ∇ϕdv g + m 2 0 M u α ϕdv g = M (u + α ) p-1 ϕdv g + ω 2 M (1 -qΦ(u α )) 2 u α ϕdv g + o(1) .
(2.11)

Letting α → +∞ in (2.11) we then get by (2.10) that

∆ g u p + m 2 0 u p = (u + p ) p-1 + ω 2 (1 -qΦ(u p )) 2 u p in H 1 .
Multiplying the equation by u - p and integrating over M , it follows that u - p ≡ 0. In particular, u p ≥ 0, u p ≡ 0, and

∆ g u p + m 2 0 u p = u p-1 p + ω 2 (1 -qΦ(u p )) 2 u p (2.12)
in H 1 . By regularity results we get from (2.12) that u p ∈ H 2,s for all s. Then, by regularity results, Φ(u p ) ∈ H 2,s for all s. By the Sobolev embedding theorem, regularity theory, and the maximum principle, it follows that u p , Φ(u p ) ∈ C 2 (M ) and that u p , Φ(u p ) > 0 in M . Letting u = u p and v = Φ(u p ), this proves the existence part in Theorem 0.2.

An additional information we obtain is that u p realizes c p . Indeed, since u p ≥ 0, u - α → 0 in H 1 , and Φ(u α ) → Φ(u p ) in H 1 , there holds that

M (u + α ) p dv g → M u p p dv g , and
M (1 -qΦ(u α )) 2 u 2 α dv g → M (1 -qΦ(u p )) 2 u 2 p dv g
as α → +∞. The second equation in (2.9) together with (2.12) then give that

M |∇u α | 2 dv g → M |∇u p | 2 dv g .
It follows that u α → u p in H 1 as α → +∞. By the first equation in (2.9) we then get that I p (u p ) = c p . In other words, c p is realized by u p . Now, given x 0 ∈ M and ρ 0 > 0 small, sufficiently small, we define u ε by

u ε (x) = ε ε 2 +r 2 -ε ε 2 +ρ 2 0 if r ≤ ρ 0 , u ε (x) = 0 if r ≥ ρ 0 , (2.13)
where r = d g (x 0 , x). Then, see Aubin [START_REF] Aubin | Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF], for any λ ∈ R,

J λ (u ε ) = 1 K 2 4 1 + C 1 6 S g (x 0 ) -λ ε 2 ln ε + o(ε 2 ln ε) , (2.14)
where

J λ (u ε ) = M |∇u ε | 2 + λu 2 ε dv g M u 4 ε dv g 1/2
, and C > 0 is independent of α. Also there holds

M u 4 ε dv g = R 3 1 1 + |x| 2 4 dx + o(1) , M |∇u ε | 2 dv g = 8 M u 4 ε dv g + o(1) . (2.15)
In what follows we prove the existence part of Theorem 0.3.

Proof of existence in Theorem 0.3. As a preliminary remark, by standard arguments such as developed in Aubin [START_REF] Aubin | Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] and Brézis and Nirenberg [START_REF] Brézis | Positive solutions of nonlinear ellitpic equations involving critical Sobolev exponents[END_REF], we just need to prove that we can chose u 0 ∈ H 1 , u + 0 ≡ 0, such that

δ 0 ≤ c p ≤ 1 4K 4 4 -δ 0 (2.16)
for all p ∈ (4 -ε, 4) and some ε, δ 0 > 0, where c p = c p (u 0 ) is as in (2.8). Now we assume that (0.7) holds true for some x ∈ M , in particular for x ∈ M where S g is maximum. We let x 0 ∈ M be such that S g is maximum at x 0 , and (t ε ) ε be any family of positive real numbers such that the t ε 's are bounded. The first estimate we prove is that

M Φ(t ε u ε )u 2 ε dv g = O ε 2 , (2.17)
where the u ε 's are as in (2.13). By definition,

∆ g Φ(t ε u ε ) + (m 2 1 + q 2 )t 2 ε u 2 ε Φ(t ε u ε ) = qt 2 ε u 2 ε . (2.18)
Multiplying (2.18) by Φ(t ε u ε ) and integrating over M we get by Hölder's inequalities that

Φ(t ε u ε ) 2 H 1 = qt 2 ε M u 2 ε Φ(t ε u ε )dv g ≤ C M u 8/3 ε dv g 3/4 Φ(t ε u ε ) L 4
and it follows from the Sobolev inequality that

Φ(t ε u ε ) H 1 ≤ C M u 8/3 ε dv g 3/4 . (2.19)
Then, by (2.19),

M Φ(t ε u ε )u 2 ε dv g ≤ C M u 8/3 ε dv g 3/4 Φ(t ε u ε ) L 4 ≤ C M u 8/3 ε dv g 3/2
.

(2.20)

There holds,

M u 8/3 ε dv g ≤ ω 3 ρ0 0 ε ε 2 + r 2 8/3 r 3 dr = ω 3 ε 4/3 ρ0/ε 0 1 1 + r 2 8/3 r 3 dr = O(ε 4/3 ) .
(2.21) By (2.20) and (2.21), this proves (2.17). Let (ε α ) α be a sequence of positive real numbers such that ε α → 0 as α → +∞, u α = u εα , and F 4 be the functional defined in H 1 by

F 4 (u) = 1 2 M |∇u| 2 dv g + 1 2 (m 2 0 -ω 2 ) M u 2 dv g - 1 4 M |u| 4 dv g . (2.22)
By (2.15), there exists T 0 1 such that I 4 (T 0 u α ) < 0 for all α 1. There also holds that

max 0≤t≤T0 I 4 (tu α ) ≤ max 0≤t≤T0 F 4 (tu α ) + CT 2 0 max 0≤t≤T0 M Φ(tu α )u 2 α dv g ≤ 1 4 J λ (u α ) 2 + CT 2 0 max 0≤t≤T0 M Φ(tu α )u 2 α dv g
for all α, where λ = m 2 0 -ω 2 . By (2.14) and (2.17) we thus get that

max 0≤t≤T0 I 4 (tu α ) ≤ 1 K 4 4 1 + C 1 6 S g (x 0 ) -m 2 0 + ω 2 ε 2 α ln ε α + o(ε 2 α ln ε α ) ,
where C > 0 is independent of α. By assumption the ε 2 α ln ε α coefficient is positive. Let u 0 = u α , where α 1 is sufficiently large such that

max 0≤t≤T0 I 4 (tu α ) ≤ 1 4K 4 4 -δ 0
for some δ 0 > 0. Since u 0 is now fixed, we can write that

max 0≤t≤T0 I p (tu 0 ) ≤ (1 + δ ε ) max 0≤t≤T0 I 4 (tu 0 ) (2.23)
for all p ∈ (4 -ε, 4), where δ ε > 0 is such that δ ε → 0 as ε → 0. Noting that

I p (u) ≥ 1 2 M |∇u| 2 + (m 2 0 -ω 2 )u 2 dv g - 1 p M |u| p dv g , ≥ C 1 u 2 H 1 -C 2 u p H 1
where C 1 , C 2 > 0 are independent of u, there holds that there exist δ 1 , δ 2 > 0 such that δ 1 , δ 2 are as small as we want, and I p (u) ≥ δ 2 for all u such that u

H 1 = δ 1 .
As a conclusion, there exist δ 0 > 0 and ε > 0 such that (2.16) holds true for all p ∈ (4 -ε, 4). This ends the proof of the existence part in Theorem 0.3.

A priori bounds in the subcritical case

We prove the uniform bounds in the subcritical case of Theorem 0.2. In what follows p ∈ (2, 4).

Proof of the uniform bounds in Theorem 0.2. Let (ω α ) α be a sequence in (-m 0 , m 0 ) such that ω α → ω as α → +∞ for some ω ∈ [-m 0 , m 0 ]. Also let p ∈ (2, 4) and (u α , v α ) α be a sequence of smooth positive solutions of (0.3) with phases ω α . Then,

∆ g u α + m 2 0 u α = u p-1 α + ω 2 α (qv α -1) 2 u α ∆ g v α + m 2 1 + q 2 u 2 α v α = qu 2 α (3.1)
for all α. By the second equation in (3.1), 0 ≤ v α ≤ 1 q for all α. Assume by contradiction that max

M u α → +∞ (3.2)
as α → +∞. Let x α ∈ M and µ α > 0 be given by

u α (x α ) = max M u α = µ -2/(p-2) α . By (3.2), µ α → 0 as α → +∞. Define ũα by ũα (x) = µ 2 p-2 α u α exp xα (µ α x)
and g α by g α (x) = exp xα g (µ α x) for x ∈ B 0 (δµ -1 α ), where δ > 0 is small. Since µ α → 0, we get that g α → ξ in C 2 loc (R 3 ) as α → +∞. Moreover, by (3.1), ∆ gα ũα + µ 2 α m 2 0 ũα = ũp-1

α + ω 2 α µ 2 α (qv α -1) 2 ũα , (3.3)
where vα is given by vα (x) = v α exp xα (µ α x) . We have ũα (0) = 1 and 0 ≤ ũα ≤ 1. By (3.3) and standard elliptic theory arguments, we can write that, after passing to a subsequence, ũα → u in C 1,θ loc (R 4 ) as α → +∞, where u is such that u(0) = 1 and 0 ≤ u ≤ 1. Then ∆u = u p-1 in R 4 , where ∆ is the Euclidean Laplacian. It follows that u is actually smooth and positive, and, since 2 < p < 4, we get a contradiction with the Liouville result of Gidas and Spruck [START_REF] Gidas | A priori bounds for positive solutions of nonlinear elliptic equations[END_REF]. As a conclusion, (3.2) is not possible and there exists C > 0 such that

u α + v α ≤ C (3.4)
in M for all α. Coming back to (3.1) it follows that the sequences (u α ) α and (v α ) α are actually bounded in H 2,s for all s. Pushing one step further the regularity argument they turn out to be bounded in H 3,s for all s, and by the Sobolev embedding theorem we get that they are also bounded in C 2,θ , 0 < θ < 1. This ends the proof of the uniform bounds in Theorem 0.2 when p ∈ (2, 4).

If we assume that ω α → ω as α → +∞ for some ω ∈ (-m 0 , m 0 ), p ∈ (2, 4], and u α → u and v α → v in C 2 as α → +∞, then u > 0, v > 0, and u, v are smooth solutions of (0.3). Indeed, given ε > 0 sufficiently small, since m 2 0 -ω 2 > 0, ∆ g + (m 2 0 -ω 2 -ε) is coercive. There holds that 0 ≤ v α ≤ 1 q for all α. In particular, by (3.1) and the Sobolev inequality, for any α 1 sufficiently large,

M |∇u α | 2 + m 2 0 -ω 2 -ε u 2 α dv g ≤ M |∇u α | 2 dv g + m 2 0 M u 2 α dv g -ω 2 α M (qv 2 α -1) 2 u 2 α dv g = M u p α dv g ≤ C M |∇u α | 2 + m 2 0 -ω 2 -ε u 2 α dv g p/2
for some C > 0 independent of α. This implies u > 0 and then v > 0. Obviously the positivity of u and v does not hold anymore if we allow ω 2 = m 2 0 . Let (ε α ) α be a sequence of positive real numbers such that ε α → 0 as α → +∞. Let u α = ε α and

v α = qε 2 α m 2 1 + q 2 ε 2 α .
Then u α → 0 and v α → 0 in C 2 as α → +∞, and we do have that (u α , v α ) solves (3.1) , where

ω 2 α = 1 (qv α -1) 2 m 2 0 -ε p-2 α .
In this case ω 2 α → m 2 0 as α → +∞ and the construction provides a counter example to the above statement about the positivity of u and v.

A priori bounds in the critical case

In what follows we let (M, g) be a smooth compact 4-dimensional Riemannian manifold, m 0 , m 1 > 0, and (ω α ) α be a sequence in (-m 0 , m 0 ) such that ω α → ω as α → +∞ for some ω ∈ [-m 0 , m 0 ]. Also we let (u α , v α ) α be a sequence of smooth positive solutions of (0.3) with phases ω α and p = 4. Namely,

∆ g u α + m 2 0 u α = u 3 α + ω 2 α (qv α -1) 2 u α ∆ g v α + m 2 1 + q 2 u 2 α v α = qu 2 α (4.1)
for all α. By the second equation in (4.1), 0 ≤ v α ≤ 1 q for all α. In particular, if we let

h α = m 2 0 -ω 2 α (qv α -1) 2 , (4.2)
then h α L ∞ ≤ C for all α, where C > 0 is independent of α. Assume by contradiction that max

M u α → +∞ (4.3) as α → +∞.
In what follows we let (x α ) α be a sequence of points in M , and (ρ α ) α be a sequence of positive real numbers, 0 < ρ α < i g /7 for all α, where i g is the injectivity radius of (M, g). We assume that the x α 's and ρ α 's satisfy

     ∇u α (x α ) = 0 for all α, d g (x α , x)u α (x) ≤ C for all x ∈ B xα (7ρ α ) and all α , lim α→+∞ ρ α sup Bx α (6ρα) u α (x) = +∞ . (4.4)
We let µ α be given by

µ α = u α (x α ) -1 . (4.5)
Since the h α 's in (4.2) are L ∞ -bounded we can apply the asymptotic analysis in Druet and Hebey [START_REF]Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds[END_REF] and Druet, Hebey and Vétois [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF]. In particular, we get that ρα µα → +∞ as α → +∞ and that

µ α u α exp xα (µ α x) → 1 + |x| 2 8 -1 (4.6)
in C 1 loc (R 4 ) as α → +∞, where µ α is as in (4.5). As a consequence, µ α → 0 as α → +∞. Now we define ϕ α : (0, ρ α ) → R + by

ϕ α (r) = 1 |∂B xα (r)| g ∂Bx α (r) u α dσ g , (4.7)
where |∂B xα (r)| g is the volume of the sphere of center x α and radius r for the induced metric. Let Λ = 4 √ 2. We define r α ∈ [Λµ α , ρ α ] by

r α = sup r ∈ [Λµ α , ρ α ] s.t. (sϕ α (s)) ≤ 0 in [Λµ α , r] . ( 4
(r)) (r α ) = 0 if r α < ρ α . (4.11)
Let B α be defined in M by

B α (x) = µ α µ 2 α + dg(xα,x) 2 8
, (4.12)

where µ α is as in (4.5). The following sharp estimates, see Druet, Hebey and Robert [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF] and Druet, Hebey and Vétois [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF], hold true.

Lemma 4.1. Let (M, g) be a smooth compact Riemannian 4-dimensional manifold, and (u α , v α ) α be a sequence of smooth positive solutions of (4.1) such that (4.3) holds true. Let (x α ) α and (ρ α ) α be such that (4.4) hold true, and let R ≥ 6 be such that Rr α ≤ 6ρ α for all α 1. There exists C > 0 such that, after passing to a subsequence,

u α (x) + d g (x α , x) |∇u α (x)| ≤ Cµ α d g (x α , x) -2 (4.13)
for all x ∈ B xα ( R 2 r α )\ {x α } and all α, where µ α is as in (4.5), and where r α is as in (4.8). In addition, there also exist C > 0 and (ε α ) α such that

|u α -B α | ≤ Cµ α r -2 α + S α + ε α B α (4.14) in B xα (2r α )\{x α } for all α, where ε α → 0 as α → +∞ and S α (x) = d g (x α , x) -1 for x ∈ M \{x α }.
Lemma 4.1 provide a sharp control on the u α 's, but we need more to conclude. We prove that the following fundamental asymptotic estimate holds true. Lemma 4.2 is the key estimate we need to prove the a priori bounds in the critical case discussed in this section. Lemma 4.2. Let (M, g) be a smooth compact Riemannian 4-dimensional manifold and (u α , v α ) α be a sequence of smooth positive solutions of (4.1) such that (4.3) holds true. Let (x α ) α and (ρ α ) α be such that (4.4) holds true. Assume (0.7). There holds that r α → 0 as α → +∞, where r α is as in (4.8). Moreover ρ α = O (r α ) and

r 2 α µ -1 α u α exp xα (r α x) → 8 |x| 2 + H(x) (4.15)
in C 2 loc (B 0 (2)\{0}) as α → +∞, where µ α is as in (4.5), and H is a harmonic function in B 0 (2) which satisfies that H(0) ≤ 0.

Proof of Lemma 4.2. Let R ≥ 6 be such that Rr α ≤ 6ρ α for α 1. We assume first that r α → 0 as α → +∞. For x ∈ B 0 (3) we define

ũα (x) = r 2 α µ -1 α u α exp xα (r α x) , gα (x) = exp xα g (r α x) , and hα (x) = h α exp xα (r α x) ,
where h α is as in (4.2). Since r α → 0 as α → +∞, we have that gα → ξ in C 2 loc (R n ) as α → +∞, where ξ is the Euclidean metric. Thanks to Lemma 4.1,

|ũ α (x)| ≤ C |x| -2 (4.16) in B 0 ( R 2 )\{0}. By (4.1), ∆ gα ũα + r 2 α hα ũα = µ α r α 2 ũ3 α (4.17)
in B 0 ( R 2 ). Thanks to (4.9) and by standard elliptic theory, we then deduce that, after passing to a subsequence, ũα → ũ (4.18) in C 2 loc B 0 ( R 2 )\{0} as α → +∞, where W satisfies ∆ũ = 0 in B 0 ( R 2 )\{0} and ∆ is the Euclidean Laplace Beltrami operator. Moreover, thanks to (4.16), we know that

|ũ(x)| ≤ C |x| -2 (4.19) in B 0 ( R 2 )\{0}. Thus we can write that ũ(x) = Λ |x| 2 + H(x) (4.20)
where Λ ≥ 0 and H satisfies ∆H = 0 in B 0 ( R 2 ). In order to see that Λ = 8, it is sufficient to integrate (4.17) in B 0 (1) to get that

- ∂B0(1) ∂ ν ũα dσ gα = µ α r α 2 B0(1) ũ3 α dv gα -r 2 α B0 (1)
hα ũα dv gα .

By (4.16), B0

ũα dv gα ≤ C (4.22)

and by changing x into µα rα x, we can write that

B0(1) ũ3 α dv gα = r 2 α µ -2 α B0( rα µα ) û3 α dv ĝα ,
where ûα (x) = µ α u α exp xα (µ α x) and ĝα (x) = exp xα g (µ α x). By (4.6) and Lemma 4.1, we then get that

lim α→+∞ µ α r α 2 B0 (1)
∂ ν ũα dσ gα = -2ω 3 Λ , (4.24)
we get that Λ = 8 thanks to (4.22)-(4.24) by passing into the limit in (4.21) as α → +∞. At this point we claim that there exists β ∈ (0, 1] and C > 0 such that

v α ≤ Cu β α in M (4.25)
for all α. Let x α ∈ M be a point where vα

u β α is maximum. Then, ∆ g v α (x α ) v α (x α ) ≥ ∆ g u β α (x α ) u β α (x α )
and it follows from (4.1) that

q u α (x α ) 2 v α (x α ) -m 2 1 -q 2 u α (x α ) 2 ≥ -β(β -1) |∇u α (x α )| 2 u α (x α ) 2 + βu α (x α ) 2 -βm 2 0 + βω 2 α (qv α (x α ) -1) 2 . (4.26)
Choosing β ∈ (0, 1] such that m 2 1 -βm 2 0 > 0, since 0 < v α ≤ 1 q , we get that u β α (x α ) ≥ Cv α (x α ) for some C > 0 independent of α. This proves (4.25). In what follows we let X α be the 1-form given by

X α (x) = 1 - 1 18 Rc g (x). (∇f α (x), ∇f α (x)) ∇f α (x) , (4.27)
where f α (x) = 1 2 d g (x α , x) 2 , Rc g is the Ricci curvature of g, and is the musical isomorphism. We apply the Pohozaev identity in Druet-Hebey [START_REF]Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF] with the vector field X α to u α in B xα (r α ). We separate the regular part A α = m 2 0 -ω 2 α from the singular part in h α . Then, h α = A α + O (v α ) and we get that

Bx α (rα) A α u α X α (∇u α )dv g + 1 8 Bx α (rα) (∆ g div g X α ) u 2 α dv g + 1 4 Bx α (rα) (div g X α ) A α u 2 α dv g = Q 1,α + Q 2,α + Q 3,α + O Bx α (rα) v α u 2 α dv g + O Bx α (rα) v α u α |X α (∇u α )| dv g , (4.28)
where

Q 1,α = 1 4 ∂Bx α (rα) (div g X α ) u α ∂ ν u α dσ g - ∂Bx α (rα) 1 2 X α (ν)|∇u α | 2 -X α (∇u α )∂ ν u α dσ g , Q 2,α = - p i=1 Bx α (rα) ∇X α - 1 4 (div g X α ) g (∇u α , ∇u α ) dv g , Q 3,α = 1 4 ∂Bx α (rα) X α (ν) u 4 α dσ g - 1 8 ∂Bx α (rα) (∂ ν div g X α ) u 2 α dσ g ,
and ν is the unit outward normal derivative to B xα (r α ). We have that

|X α (x)| = O (d g (x α , x)) , div g X α (x) = n + O d g (x α , x) 2 ) , |∇ (div g X α ) (x)| = O (d g (x α , x)) , and ∆ g (div g X α ) (x) = 4 3 S g (x α ) + O (d g (x α , x)) . (4.29)
Following Druet, Hebey and Vétois [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF] we get from Lemma 4.1, (4.28) and (4.29) that

Q 1,α = -64ω 3 m 2 0 -ω 2 - 1 6 S g (x 0 ) µ 2 α ln r α µ α + o µ 2 α ln 1 µ α + o µ 2 α r -2 α + O Bx α (rα) v α u 2 α dv g + O Bx α (rα) v α u α |X α (∇u α )| dv g , (4.30)
where x α → x 0 as α → +∞. By Lemma 4.1 and (4.29) there also holds that

Q 1,α = O µ 2 α r -2 α . (4.31)
At this point we decompose v α into a quasi-harmonic part with nonzero Dirichlet boundary condition and a quasi-Poisson part with zero Dirichler boundary condition. More precisely, we write that

v α = w 1,α + w 2,α (4.32)
in B α = B xα (r α ), where rα = 5 2 r α , and w 1,α , w 2,α are given by

∆ g w 1,α + m 2 1 w 1,α = 0 in B α w 1,α = v α on ∂B α , (4.33)
and if

W α = ∆ g v α + m 2 1 v α , by ∆ g w 2,α + m 2 1 w 2,α = W α in B α w 2,α = 0 on ∂B α . (4.34)
Let G α be the Green's function of ∆ g + m 2 1 in B α with zero Dirichlet boundary condition on ∂B α . By the maximum principle, considering the Green's function on a larger ball of radius i g , we obtain by comparison of the two Green's functions that there exists C > 0 such that G α (x, y) ≤ Cd g (x, y) -2 for all x = y in B α . Writing that Independently, by the maximum principle, the w 1,α 's satisfy that 0 ≤ w 1,α ≤ 1 q . Let ĝα (x) = exp xα g (r α x) and ŵ1,α (x) = w 1,α exp xα (r α x) . There holds

w 2,α (x) = Bα G α (x,
∆ ĝα ŵ1,α + m 2 1 r2 α ŵ1,α = 0 in B w 1,α = vα on ∂B , (4.38)
where B = B 0 (1) ⊂ R 4 , and vα (x) = v α exp xα (r α x) . At this point we claim that

r α → 0 (4.39)
as α → +∞. In order to prove (4.39) we proceed by contradiction and assume that r α ≥ δ 0 > 0 for all α 1. By Lemma 4.1 and (4.25),

v α ≤ Cµ β α in M \B xα (r α ) , (4.40)
where C > 0 is independent of α since we assumed r α ≥ δ 0 > 0. In particular, v α L ∞ (∂Bα) → 0 as α → +∞. Then vα L ∞ (∂B) → 0 as α → +∞, and it follows from the maximum principle and (4.38) that ŵ1,α L ∞ (B) → 0 as α → +∞. In particular, w

v α u 2 α dv g = o µ 2 α ln 1 µ α Bx α (rα) u α v α |X α (∇u α )|dv g = o µ 2 α ln 1 µ α . ( 4
Q 1,α = -(128ω 3 H(0) + o(1)) µ 2 α r -2 α . (4.42)
Now we distinguish the two cases:

(i) r 2 α ln rα µα → 0 as α → +∞, and (ii) r 2 α ln rα µα ≥ δ 0 > 0 for all α.

In case (i), since v α = O(1), we get from Lemma 4.1 and (4.29) that

Bx α (rα) v α u 2 α dv g = O µ 2 α ln r α µ α , Bx α (rα) v α u α |X α (∇u α )|dv g = O µ 2 α ln r α µ α . (4.43)
Since there also holds that r 2 α ln 1 µα → +∞ it follows from (4.30), (4.42) and (4.43) that H(0) = 0. Now we assume (ii). From (ii) we get that r α ≥ C(ln 1 µα ) -1/2 and by (4.25) we obtain that

v α ≤ C µ α ln 1 µ α β in M \B xα (r α ) .
In particular, v α L ∞ (∂Bα) → 0 as α → +∞. Then vα L ∞ (∂B) → 0 as α → +∞, and it follows from the maximum principle and (4.38) that ŵ1,α L ∞ (B) → 0 as α → +∞. In particular, w

v α u 2 α dv g = Bx α (rα) w 2,α u 2 α dv g + o µ 2 α ln r α µ α , Bx α (rα) v α u α |X α (∇u α )|dv g = Bx α (rα) w 2,α u α |X α (∇u α )|dv g + o µ 2 α ln r α µ α . (4.44)
There holds, ∆ g w 2,α + m 2 1 w 2,α = q (1 -qv α ) u 2 α . (4.45)

Let η : R n → R be such that η is smooth, 0 ≤ η ≤ 1, η = 1 in B 0 (1), and η = 0 in R n \B 0 (2). We define Proof of the uniform bounds in Theorem 0.3. Let (M, g) be a smooth compact Riemannian 4-dimensional manifold and (u α , v α ) α be a sequence of smooth positive solutions of (4.1) such that (0.7) holds true. By Druet, Hebey and Vétois [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF] there exists C > 0 such that for any α the following holds true: there exist N α ∈ N and N α critical points of u α , denoted by (x 1,α , x 2,α , . . . , x Nα,α ), such that

η α (x) = η d g (x α , x) r α ( 4
u α w 2,α |X α (∇u α )|dv g ≤ Bx α (rα) w 4 2,α dv g 1/4 Bx α (rα) |u α X α (∇u α )| 4/3 dv g 3/4 , (4.47
d g (x i,α , x j,α ) u α (x i,α ) ≥ 1 (4.55)
for all i, j ∈ {1, . . . , N α }, i = j, and min i=1,...,Nα

d g (x i,α , x) u α (x) ≤ C (4.56)
for all x ∈ M and all α. We define

d α = min 1≤i<j≤Nα d g (x i,α , x j,α ) . ( 4

.57)

If N α = 1, we set d α = 1 4 i g , where i g is the injectivity radius of (M, g). We claim that d α → 0 (4.58) as α → +∞. In order to prove this claim, we proceed by contradiction. Assuming on the contrary that d α → 0 as α → +∞, we see that N α ≥ 2 for α large, and we can thus assume that the concentration points are ordered in such a way that in B 0 (δd -1 α ), for all i. For any R > 0, we also let 1 ≤ N R,α ≤ N α be such that

d α = d g (x 1,α , x 2,α ) ≤ d g (x 1,α , x 3,α ) ≤ • • • ≤ d g (x 1,α , x Nα,α ) . (4.59) We set, for x ∈ B 0 (δd -1 α ), 0 < δ < 1 2 i g fixed, ûα (
d g (x 1,α , x i,α ) ≤ Rd α for 1 ≤ i ≤ N R,α , and
d g (x 1,α , x i,α ) > Rd α for N R,α + 1 ≤ i ≤ N α .
Such a N R,α does exist thanks to (4.59). We also have that N R,α ≥ 2 for all R > 1 and that (N R,α ) α is uniformly bounded for all R > 0 thanks to (4.57). In the sequel, we set

xi,α = d -1 α exp -1 x1,α (x i,α ) for all 1 ≤ i ≤ N α such that d g (x 1,α , x i,α ) ≤ 1 2 i g . Thanks to (4.56), for any R > 1, there exists C R > 0 such that sup B0(R)\ SN 2R,α i=1 B xi,α ( 1 R ) ûα ≤ C R . ( 4

.61)

By the Harnack inequality in Druet, Hebey and Vétois [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF], for any R > 1, there exists D R > 1 such that

∇û α L ∞ (Ω R,α ) ≤ D R sup Ω R,α ûα ≤ D 2 R inf Ω R,α ûα , (4.62)
where

Ω R,α = B 0 (R) \ N 2R,α i=1 B xi,α 1 R .
Assume first that, for some R > 0, there exists 1

≤ i ≤ N R,α such that ûα (x i,α ) = O(1) . ( 4

.63)

The two first equations in (4.4) are satisfied by the sequences x α = x i,α and ρ α = ). In particular, by standard elliptic theory, and thanks to (4.60), (û α ) α is uniformly bounded in C 1 B xi,α ( 12 ) . Since, by (4.55), we have that

|x i,α | n-2 2 |û α (x i,α )| ≥ 1 ,
we get the existence of some δ i > 0 such that for some δi > 0. Thus, using (4.62), we can deduce that these two situations are mutually exclusive in the sense that either (4.63) holds true for all i or (4.64) holds true for all i. Now we split the conclusion of the proof into two cases.

|û α | ≥ 1 2 |x i,α | 1-n 2 ≥ 1 2 R 1-
In the first case we assume that there exist R > 0 and 1 ≤ i ≤ N R,α such that ûα (x i,α ) = O(1). Then, thanks to the above discussion, we get that ûα (x j,α ) = O(1) for all 1 ≤ j ≤ N R,α and all R > 0. As above, we get that (û α ) α is uniformly bounded in C 1 loc (R 4 ). Thus, by standard elliptic theory, there exists a subsequence of (û α ) α which converges in C 1 loc (R 4 ) to some û solution of ∆û = û3 in R 4 . By the above discussion, |u| possesses at least two critical points, namely 0 and x2 , the limit of x2,α . This is absurd thanks to the classification of Caffarelli, Gidas and Spruck [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF].

In the second case we assume that there exist R > 0 and 1 ≤ i ≤ N R,α such that |û α (x i,α )| → +∞ as α → +∞. Then, thanks to the above discussion, ûα (x j,α ) → +∞ as α → +∞, for all 1 ≤ j ≤ N R,α and all R > 0. By (4.60) we have that where Ĥ(0) ≥ Λ 2 -Λ 1 R -2 -Λ 2 (R -1) -2 . Choosing R large enough, we can ensure that Ĥ(0) > 0 and this is in contradiction with Lemma 4.2.

By the above discussion we get that (4.58) holds true. Clearly, this implies that (N α ) α is uniformly bounded. Let (x α ) α be a sequence of maximal points of u α . Thanks to (4.3) and to (4.58), we clearly have that (4.4) holds true for the sequences (x α ) α and ρ α = δ for some δ > 0 fixed. This clearly contradicts Lemma 4.2 and thus concludes the proof of the uniform bounds in Theorem 0.3.

Existence and nonexistence of a priori estimates for critical elliptic Schrödinger type equations on manifolds have been investigated by Berti-Malchiodi [START_REF] Berti | Non-compactness and multiplicity results for the Yamabe problem on S n[END_REF], Brendle [START_REF] Brendle | Blow-up phenomena for the Yamabe equation[END_REF][START_REF]On the conformal scalar curvature equation and related problems[END_REF], Brendle and Marques [START_REF] Brendle | Blow-up phenomena for the Yamabe equation II[END_REF], Brézis and Li [START_REF] Brézis | Some nonlinear elliptic equations have only constant solutions[END_REF], Druet [START_REF] Druet | From one bubble to several bubbles: The low-dimensional case[END_REF][START_REF]Compactness for Yamabe metrics in low dimensions[END_REF], Druet and Hebey [START_REF] Druet | Elliptic equations of Yamabe type[END_REF][START_REF]Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds[END_REF][START_REF]Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF], Druet, Hebey, and Vétois [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF], Druet and Laurain [START_REF] Druet | Stability of the Pohozaev obstruction in dimension 3[END_REF], Hebey [START_REF] Hebey | Critical elliptic systems in potential form[END_REF][START_REF]Diagonal compactness for critical elliptic systems in potential form[END_REF], Khuri, Marques and Schoen [START_REF] Khuri | A compactness theorem for the Yamabe Problem[END_REF], Li and Zhang [START_REF]A Harnack type inequality for the Yamabe equations in low dimensions[END_REF][START_REF]Compactness of solutions to the Yamabe problem II[END_REF], Li and Zhu [START_REF]Yamabe type equations on three dimensional Riemannian manifolds[END_REF], Marques [START_REF] Marques | A priori estimates for the Yamabe problem in the non-locally conformally flat case[END_REF], Micheletti, Pistoia and Vétois [START_REF] Micheletti | Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds[END_REF], Schoen [START_REF] Schoen | Variational theory for the total scalar curvature functional for Riemannian metrics and related topics[END_REF][START_REF]On the number of constant scalar curvature metrics in a conformal class[END_REF], and Vétois [START_REF] Vétois | Multiple solutions for nonlinear elliptic equations on compact Riemannian manifolds[END_REF]. In the subcritical case, a priori estimates for Schrödinger equations go back to the seminal work by Gidas and Spruck [START_REF] Gidas | A priori bounds for positive solutions of nonlinear elliptic equations[END_REF]. The above list is not exhaustive.

ũ3 α dv gα = 16ω 3 .

 3

  ) while by Lemma 4.1 and (4.29) there holds that Bx α (rα) u 8/3 α dv g = O µ 4/3 α and Bx α (rα) |u α X α (∇u α )| 4/3 dv g = O µ 4

  x) = d α u α exp x1,α (d α x) , ĥα (x) = h α exp x1,α (d α x) , and ĝα (x) = exp x1,α g (d α x) . It is clear that ĝα → ξ in C 2 loc (R n ) as α → +∞ since d α → 0 as α → +∞. Thanks to (4.1) we have that ∆ ĝα ûα + d 2 α ĥα ûα = û3 α (4.60)

1 8

 1 d α . Then it follows from (4.6) that the last equation in (4.4) cannot hold and thus that (û α ) α is uniformly bounded in B xi,α ( 3 4

n 2 in

 2 B xi,α (δ i ). Assume now that, for some R > 0, there exists 1≤ i ≤ N R,α such that |û α (x i,α )| → +∞ (4.64) as α → +∞. Since (4.4) is satisfied by the sequences x α = x i,α and ρ α = 1 8 d α , it follows from Lemma 4.2 that the sequence (|û α (x i,α )|×|û α |) α is uniformly bounded in Ωα = B xi,α ( δi )\B xi,α ( δi 2 )

∆ ĝα vα + d 2 α

 2 ûα (0)û α . Applying Lemma 4.2 and standard elliptic theory, and thanks to (4.62) and to the above discussion, one easily checks that, after passing to a subsequence, ûα (0)û α → Ĝ in C 1 loc (R n \{x i } i∈I ) as α → +∞, where I = {1, . . . , lim R→+∞ lim α→+∞ N R,α } and, for any R > 0, Ĝ(x) = ÑR i=1 Λ i |x -xi | 2 + ĤR (x) in B 0 (R), where 2 ≤ ÑR ≤ N 2R is such that |x ÑR | ≤ R and |x ÑR +1 | > R,where N 2R,α → N 2R as α → +∞, where λ i > 0, and where ĤR is a harmonic function in B 0 (R). Since Ĝ ≥ 0, we can write thanks to the maximum principle that, in a neighbourhood of the origin, Ĝ(x) = Λ 1 |x| n-2 + Ĥ(x) ,

  y)W α (y)dv g (y)

	it follows that
	|w 2,α (x)| ≤ C	Bα	u 2 α (y)dv g (y) d g (x, y) 2 .	(4.35)
	By (4.6) and Lemma 4.1 we can write that
	u α (x) ≤	Cµ α α + d g (x α , x) 2 µ 2	(4.36)
	in B α . Combining (4.35) and (4.36) we then get that
	|w 2,α (x)| ≤ C	µ 2 α ln 2 + µ 2 α + d g (x α , x) 2 dg(xα,x) 2 µ 2 α	.	(4.37)

  1,α L ∞ (Bα) → 0 as α → +∞. By (4.32) and(4.37), thanks to what we just obtained about the w 1,α 's, we get that v α L ∞ (Bα) → 0 as α → +∞. Then, by Lemma 4.1 and (4.29) we get that

	Bx α (rα)

  1,α L ∞ (Bα) → 0 as α → +∞ and we get with (4.32), Lemma 4.1, and (4.29), that

Bx α (rα)

Acknowledgments: The first author was partially supported by the ANR grant ANR-08-BLAN-0335-01. The authors are indebted to Olivier Druet for a very valuable idea which has helped to improve the manuscript. They are also indebted to Valdimir Georgescu for several interesting discussions on this work.

Multiplying (4.45)

and by (4.9) it follows that

Coming back to (4.51), it follows that

By (4.54) we get that H(0) ≤ 0. At this point it remains to prove that ρ α = O (r α ). We prove that ρ α = r α . If not the case, then r α < ρ α and we get with (4.11) that (rϕ(r)) (1) = 0, where

Hence H(0) = 8 and we get a contradiction with H(0) ≤ 0. In other words, ρ α = r α for all α 1. This ends the proof of the lemma.

Thanks to Lemma 4.2 we can now prove the uniform bounds in Theorem 0.3. This is the subject of what follows.