ASYMPTOTIC ANALYSIS FOR FOURTH ORDER PANEITZ EQUATIONS WITH CRITICAL GROWTH

Transcription

1 ASYPTOTIC ANALYSIS FOR FOURTH ORDER PANEITZ EQUATIONS WITH CRITICAL GROWTH EANUEL HEBEY AND FRÉDÉRIC ROBERT Abstract. We investigate fourth order Paneitz equations of critical growth in the case of n-dimensional closed conformally flat manifolds, n 5. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the Q-curvature. We obtain sharp asymptotics for arbitrary bounded energy sequences of solutions of our equations from which we derive stability and compactness properties. In doing so we establish the criticality of the geometric equation with respect to the trace of its second order terms. In 1983, Paneitz [5] introduced a conformally covariant fourth order operator extending the conformal Laplacian. Branson and Ørsted [5], and Branson [, 3], introduced the associated notion of Q-curvature when n = 4 and in higher dimensions when dealing with the conformally covariant extensions of the Paneitz operator by Graham-Jenne-ason-Sparling. The scalar and the Q-curvatures are respectively, up to the conformally invariant Weyl s tensor in dimension four, the integrands in dimensions two and four for the Gauss-Bonnet formula for the Euler characteristic. The articles by Branson and Gover [4], Chang [6, 7], Chang and Yang [8], and Gursky [15] contain several references and many interesting material on the geometric and physics aspects associated to this notion of Q-curvature. In what follows we let, g be a smooth compact Riemannian manifold of dimension n 5 and consider the fourth order variational Paneitz equations of critical Sobolev growth which are written as gu + b g u + cu = u 1, 0.1 where g = div g u is the Laplace-Beltrami operator, b, c > 0 are positive real numbers such that c b 4 < 0, u is required to be positive, and = n n 4 is the critical Sobolev exponent. Equations like 0.1 are modeled on the conformal equation associated to the Paneitz operator when the background metric g is Einstein. In few words, the conformal equation associated to the Paneitz operator, relating the Q-curvatures Q g and Q g of conformal metrics on arbitrary manifolds, is written as gu div g A g du + n 4 Q g u = n 4 Q g u 1, 0. where g = u 4/n 4 g and, if Rc g and S g denote the Ricci and scalar curvature of g, A g is the smooth, 0-tensor field given by A g = n + 4 n 1n S gg 4 n Rc g. 0.3 Date: February 4, 010. Revised September 7, 010. The authors were partially supported by the ANR grant ANR-08-BLAN

2 EANUEL HEBEY AND FRÉDÉRIC ROBERT When g is Einstein, so that Rc g = λg for some λ R, equation 0. can be simplified and written as where b n and c n are given by gu + b nλ n 1 gu + c nλ n 1 u = n 4 Q g u 1, 0.4 b n = n n 4 and c n = nn 4n In particular, as mentioned, 0.1 is of the type 0.4. An important remark is that c n b n 4 = 1 is negative. Given Λ > 0 we let Sb,c Λ be the set S Λ b,c = { u C 4, u > 0, s.t. u H Λ and u solves 0.1 }, 0.6 where H is the Sobolev space of functions in L with two derivatives in L. Following standard terminology, we say that 0.1 is compact if for any Λ > 0, S Λ b,c is compact in the C 4 -topology we adopt here the bounded version of compactness, as first introduced by Schoen [3]. The stronger notion of stability we discuss in the sequel is defined as follows: Definition 0.1. Equation 0.1 is stable if it is compact and if for any Λ > 0, and any ε > 0, there exists δ > 0 such that for any b and c, it holds that d C 4SΛ b,c ; SΛ b,c < ε 0.7 as soon as b b + c c < δ, where Sb,c Λ and SΛ b,c are given by 0.6, and where for X, Y C 4, d C X; Y is the pointed distance defined as the sup over the u X 4 of the inf over the v Y of v u C 4. The meaning of 0.7 is that small perturbations of b and c in 0.1 do not create solutions which stand far from solutions of the original equation. Stability is an important notion in view of topological arguments and degree theory. Also it has a natural translation in terms of phase stability for solitons of the fourth order Schrödinger equations introduced by Karpman [] and Karpman and Shagalov [3] see the remark at the end of Section 5. The main questions we ask here are: Questions: Q1 describe and control the asymptotic behavior of arbitrary finite energy sequences of solutions of equations like 0.1. Q find conditions on b and c for 0.1 to be stable. By contradiction, 0.1 is stable if and only if for any sequences b and c of real numbers converging to b and c, and any sequence u of smooth positive solutions of gu + b g u + c u = u such that u is bounded in H, there holds that, up to a subsequence, u u in C 4, where u is a smooth positive solution of 0.1. In other words, 0.1 is stable if we can impede bubbling for arbitrary bounded sequences in H of solutions of arbitrary sequences of equations like 0.8, including 0.1 itself. In order to do so, we need sharp answers to Q1. As is well known, critical equations tend to be unstable precisely because of the bubbling which is usually associated with critical equations. A consequence of Theorem 0. below is that bubbling is not only associated with the criticality of

3 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 3 the equation but also with the geometry through the relation b = 1 n Tr ga g which, see 0.9 below, characterizes the middle term of the geometric equation 0.4. Concerning the bound on the energy we require in Definition 0.1, it should be noted that we cannot expect the existence of a priori H -bounds for arbitrary sequences of equations like 0.8 when dealing with large coefficents b and c like it is the case for Yamabe type equations associated with second order Schrödinger operators with large potentials. In parallel it is intuitively clear that bounded sequences in H of solutions of equations like 0.8 can develop an arbitrarily large number of peaks. Summing sphere singularities in a naive way we indeed can prove, see Hebey, Robert and Wen [0], that for any quotient of the n-sphere, n 1, there exist sequences u and v of smooth positive solutions of gu + b n g u + c u = u 1 such that u blows up with an arbitrarily large given number k of peaks and v H + as +, where c is a sequence of smooth functions converging in the C 1 -topology to c n, and b n and c n are as in 0.5. In other words, illustrating the above discussion, we see that equations like 0.1 create bubbling, even multiple of cluster type namely with blow-up points collapsing on a single point, and that there is no statement about universal a priori H -bounds for arbitrary solutions of arbitrary equations like 0.8. Also we see that an equation can be compact and unstable the geometric equation is compact on quotients of the sphere. Compactness for the geometric equation in the conformally flat case has been established by Hebey and Robert [19], and by Qing and Raske [30, 31]. The elegant geometric approach in Qing and Raske [30, 31] is based on the integral representation of the solutions through the developing map under the natural assumption that the Poincaré exponent is small. Recently, Wei and Zhao [34] constructed blow-up examples in the non conformally flat case when n 5. Let λ i A g x, i = 1,..., n, be the g-eigenvalues of A g x repeated with their multiplicity. Let λ 1 be the infimum over i and x, and λ be the supremum over i and x of the λ i A g x s. Following Hebey, Robert and Wen [0] we define the wild spectrum of A g to be the interval S w = [λ 1, λ ]. It was proved in [0] that 0.1 is stable on conformally flat manifolds when n = 6, 7, 8 and b < λ 1, or n 9 and b S w. We improve these results in different important significative directions in the present article: we add the case of dimension n = 5, we replace the condition b S w by the much weaker condition b 1 n Tr ga g, and we accept large values of b when n = 6, 7. On the other hand, we leave open the question of getting similar results in the nonconformally flat case. In the above discussion, and in what follows, Tr g A g = g ij A ij is the trace of A g with respect to g. There clearly holds that 1 n Tr ga g S w at any point in, and it is easily seen that Tr g A g = n n 4 S g. 0.9 n 1 Let u be a bounded sequence in H of solutions of 0.8. Up to a subsequence, u u weakly in H for some u H which solves 0.1. When c b 4 < 0, by the maximum principle, either u > 0 in or u 0. In the second order case, in low dimensions namely n = 3, 4, 5 we know from Druet [9] that we necessarily have that u 0 if the convergence of u to u is not strong but only weak and the u s solve Yamabe type equations. In the framework of question Q1, we also

4 4 EANUEL HEBEY AND FRÉDÉRIC ROBERT address in this article the question of whether or not such type of results extend to the fourth order case when passing from Yamabe type equations to Paneitz equations like 0.1. We positively answer to this question in Theorem 0.1 below, the low dimensions being now 5, 6, 7. Theorem 0.1. Let, g be a smooth compact conformally flat Riemannian manifold of dimension n = 5, 6, 7 and b, c > 0 be positive real numbers such that c b 4 < 0. Let b and c be sequences of real numbers converging to b and c, and u be a bounded sequence in H of smooth positive solutions of 0.8 such that u u weakly in H as +. Then either u u strongly in any C k -topology, or u 0. Theorem 0.1 answers the above mentioned question of whether or not we can have a nontrivial limit profile for blowing-up sequences of solutions of 0.8. As a remark, the geometric equation on the sphere provides in any dimension n 5 an example of an equation like 0.1 with sequences u of solutions such that u u strongly and u 0. Now we return to the question of the stability of 0.1. When n = 5 we let G be the Green s function of the fourth order Paneitz type operator P g = g + b g + c. Then G x y = 1 6ω 4 d g x, y + µ xy, 0.10 where G x = Gx, is the Green s function at x of P g, ω 4 is the volume of the unit 5-sphere, and µ x is C 0,θ in for θ 0, 1. The mass at x of P g is µ x x. Our second result states as follows. Theorem 0.. Let, g be a smooth compact conformally flat Riemannian manifold of dimension n 5 and b, c > 0 be positive real numbers such that c b 4 < 0. Assume that one of the following conditions holds true: i n = 5 and µ x x > 0 for all x, ii n = 6 and b S w, iii n = 8 and b < 1 8 min Tr g A g, iv n = 7 or n 9 and b 1 n Tr ga g in. Then for any sequences b and c of real numbers converging to b and c, and any bounded sequence u in H of smooth positive solutions of 0.8 there holds that, up to a subsequence, u u in C 4 for some smooth positive solution u of 0.1. In particular, 0.1 is stable. As a direct consequence of Theorem 0., cluster solutions and bubble towers do not exist for 0.1 when one of the conditions i to iv is assumed to hold. This includes the existence of cluster solutions or bubble towers constructed by means of perturbing 0.1 such as in 0.8. Let G 0 be the Green s function of the geometric Paneitz operator P 0 in the left hand side of 0.. Humbert and Raulot [1] proved the very nice result that in the conformally flat case, assuming that the Yamabe invariant is positive, that P 0 is positive, and that G 0 > 0 outside the diagonal, then the mass of G 0 is nonnegative and equal to zero at one point if and only if the manifold is conformally diffeomorphic to the sphere. A similar result was previously established by Qing and Raske [30, 31] when the Poincaré exponent is small. By i in Theorem 0. we need to find conditions under which µ x x > 0 for all x, where µ x x is the mass of

5 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 5 our operator P g = g + b g + c. A third theorem we prove, based on the Humbert and Raulot [1] result, is as follows. Theorem 0.3. Let, g be a smooth compact conformally flat Riemannian manifold of dimension n = 5 with positive Yamabe invariant such that the Green s function of the geometric Paneitz operator P 0 is positive and let b, c > 0 be positive real numbers. We assume that bg A g in the sense of bilinear forms and c 1 Q g, and in case A g bg and c 1 Q g simultaneously, we assume in addition that, g is not conformally diffeomorphic to the standard sphere. Then the mass µ x x of P g is positive for all x. Theorem 0.3 raises the question of the existence of 5-dimensional compact conformally flat manifolds with positive Yamabe invariant and positive Green s function for P 0. By the analysis in Qing and Raske [31] compact conformally flat manifolds of positive Yamabe invariant and of Poincaré exponent less that n 4 = 1 are such that the Green s function of P 0 is positive. Explicit examples of such manifolds are in Qing-Raske [31]. The paper is organized as follows. In Section 1 we establish sharp pointwise estimates for arbitrary sequences of solutions of 0.8. This answers Q1. Thanks to these estimates we prove Theorem 0.1 in Section. Trace estimates are proved to hold in Section 3. By the estimates in Sections 1 and 3 we can prove Theorem 0. in Section 4 when n 6. In Section 5 we prove Theorem 0. in the specific case n = 5. Theorem 0. provides the answer to Q. We prove Theorem 0.3 in Section Pointwise estimates Let, g be a smooth compact Riemannian manifold of dimension n 5, and b, c > 0 be positive real numbers such that c b 4 < 0. We do not need in this section to assume that g is conformally flat. Let also b and c be converging sequences of real numbers with limits b and c as, and u be a bounded sequence in H of positive nontrivial solutions of 0.8. Up to a subsequence, u u weakly in H as +. By standard elliptic theory, either u u in C 4 or the u s blow up and u L as +. From now on we assume that 1.1 holds true. By Hebey and Robert [18] and Hebey, Robert and Wen [0], there holds that k u = u + B i + R, 1. i=1 where R 0 in H as, and the B s i are bubble singularities in H. Such B s i are given by B i = η d i, µ i, +, 1.3 d λn where η : R R is a smooth nonnegative cutoff function with small support less than the injectivity radius of g, d i, = d g x i,,, λ n = nn 4n 4, k 1 is an integer, and for any i, x i, is a converging sequence of points in and µ i, n 4

6 6 EANUEL HEBEY AND FRÉDÉRIC ROBERT µ i, is a sequence of positive real numbers such that µ i, 0 as +. oreover, we also have that the following structure equation holds true: for any i j, d g x i,, x j, + µ i, + µ i, µ i, µ j, µ j, µ j, as +, and that there exists C > 0 such that for all and all x, where r x n 4 u x u x C 1.5 r x = min d gx i,, x. 1.6 i=1,...,k By 1. and 1.5 we have that u u in H and u u in C 0 loc \S, where S is the set consisting of the limits of the x i, s. We define µ by µ = max i=1,...,k µ i,. 1.7 There holds that µ 0 as + since µ i, 0 for all i as +. We aim here at proving the following sharp pointwise estimates. Proposition 1.1. Let, g be a smooth compact Riemannian manifold of dimension n 5, and b, c > 0 be positive real numbers such that c b 4 < 0. Let also b and c be converging sequences of real numbers with limits b and c as +, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. There exists C > 0 such that, up to a subsequence, j u C µ n 4 r 4 n j + u L 1.8 in, for all j = 0, 1,, 3 and all, where r is as in 1.6, and µ is as in 1.7. We split the proof of Proposition 1.1 into several lemmas. The first lemma we prove is a general basic result we will use further in the proof of Lemma 1.. Lemma 1.1. Let δ > 0 and g be a sequence of Riemannian metrics in the Euclidean ball B 0 δ such that g ξ in C 4 loc B 0δ as +, where ξ is the Euclidean metric. Let b and c be bounded sequences of positive real numbers such that c b 4 for all. Let w be a sequence of positive functions such that g w + b g w + c w = w in B 0 δ for all. Assume w L B 0δ + as + and that there exists C > 0 such that w L B 0δ\B 0δ/ C for all. Then w dx 1 + o1 Kn n/4, 1.10 B 0δ where o1 0 as + and K n is the sharp constant in the Euclidean Sobolev inequality u K n u, u Ḣ. Here Ḣ is the completion of C0 R n with respect to u u. Proof of Lemma 1.1. Let ν be given by ν n = max w B 0δ

7 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 7 and x be such that w x = ν n. By assumption, ν 0 as + and x B 0 δ/ for 1. Let w be given by w x = ν n 4 w x + ν x. For any R > 1, w is defined in B 0 R provided 1 is sufficiently large. Let d 1, and d, be given by d 1, = b b + 4 c and d, = b b 4 c Then, as is easily checked, for any R > 1, g w + b ν g w + c ν 4 w = g + d 1, ν g + d, ν w = w in B 0 R for all 1 sufficiently large, where g x = g x + ν x. Since 0 w 1, it follows from 1.1 and classical elliptic theory, as developped in Gilbarg-Trudinger [1], that the w s are bounded in C 4,θ loc Rn, θ 0, 1. In particular, there exists w such that, up to a subsequence, w w in Cloc 4 Rn as +. By rescaling invariance rules there also holds that w H. Passing to the limit as + in 1.1, we get that w = w 1. Since w0 = max R n w = 1, it follows from Lin s classification [4] that w is the ground state in 1.14 below. Noting that for any R > 1, w dv g = w dv g B 0R B x Rν 1 + o1 w dx, we get from the L loc -convergence w w that lim inf w dx + B 0δ B 0R B 0δ w dx Noting that w is an extremal function for the sharp inequality u K n u, it is easily seen that w dx = K n/4 R n n. Letting R + in 1.13, this ends the proof of the lemma. From now on we let B : R n R be the ground state given by Bx = n x λn, 1.14 where λ n is as in 1.3. Given x, µ > 0 and u : R, we also define the function R µ xu by R µ xuy = µ n 4 u expx µy, 1.15

8 8 EANUEL HEBEY AND FRÉDÉRIC ROBERT where y R n is such that y < ig µ, and i g > 0 is the injectivity radius of, g. For u as above, and i {1,..., k}, we define S i,r, S i,t R n by { } 1 S i,t = lim exp 1 x + µ i, x j,, j = 1,..., k i, { } S i,r = lim exp 1 x + µ i, x j,, j I i, i, where I i is the subset of {1,..., k} consisting in the j s which are such that d g x i,, x j, = O µ i, and µ j, = o µ i,. The second lemma we prove establishes local limits for the u s. Lemma 1.. Let, g be a smooth compact Riemannian manifold of dimension n 5, and b, c > 0 be positive real numbers such that c b 4 < 0. Let also b and c be converging sequences of real numbers with limits b and c as +, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. Then, up to a subsequence, R µi, x i, u B 1.17 in C 4 loc Rn \S i,r as + for all i, where S i,r is as in Proof of Lemma 1.. First we claim that for any i and any K R n \S i,t, there exists C K > 0 such that R µ i, x i, u CK 1.18 in K, for all 1 sufficiently large. Fix i and K. For any x K, and any j, d g exp xi, µ i, x, x j, C µ i, x exp 1 x i, x j, Cµ i, x 1 exp 1 x µ i, x j, i, Cµ i, by the definition of S i,t in Then 1.18 follows from 1.5. Also, by direct computations, using the structure equation 1.4, there holds that for any i j, R x µi, i, B j 0 in L loc Rn \S i,r as +, where the B s i are as in 1.3. Noting that R x µi, i, Bx i = ηµ i, xbx, where η is as in 1.3 and B is as in 1.14, we get that for any i, k B j B 1.19 R µi, x i, j=1 in L loc Rn \S i,r as +. Now we prove By 1., R µi, x i, u R µi, x i, k B j in L loc Rn as +. oreover, in any compact subset of R n, j=1 g ũ + b µ i, g ũ + c µ 4 i,ũ = ũ for 1 sufficiently large, where ũ = R µi, x i, u and g x = exp x i, g µ i, x. Since µ i, 0, we have that g ξ in C 4 loc Rn as +. By 1.18, the

9 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 9 sequence ũ is bounded in L loc Rn \S i,t. By 1.19 and 1.0, there also holds that lim lim sup ũ dx = 0 1. δ 0 + B xδ for all x R n \S i,r. By Lemma 1.1 we then get that the sequence ũ is actually bounded in L loc Rn \S i,r. By 1.1 and elliptic theory it follows that ũ is bounded in C 4,θ loc Rn \S i,r, θ 0, 1. This ends the proof of Lemma 1.. The following lemma establishes pointwise estimates for the u s. The estimates in Lemma 1.3 are a trace extension of the estimates 1.5. In particular, as is easily checked, 1.3 below implies 1.5. Lemma 1.3. Let, g be a smooth compact Riemannian manifold of dimension n 5, and b, c > 0 be positive real numbers such that c b 4 < 0. Let also b and c be converging sequences of real numbers with limits b and c as +, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. Then, up to a subsequence, r 4 k u u B i in L as +, where r is as in 1.6, and the B i s are as in 1.3. i=1 Proof of Lemma 1.3. Let D : R be given by D x = min d gx i,, x + µ i,. 1.4 i=1,...,k We prove sligthly more than 1.3, namely that D 4 k u u B i in L as +, where D is as in 1.4. Let x be such that Dx 4 k u x u x Bx i i=1 1.6 k = max x D4 x u x u x Bx i. i=1 First we claim that lim inf + D4 x u x u x = lim + D x 4 B i x = 0 i=1 k Bx i i=1 > for all i. In order to prove 1.7 we proceed by contradiction and assume that there exists ε 0 > 0 and i such that D x 4 Bx i ε 0, and thus such that 4 η d g x i,, x D x µ i, µ i, + ε dgxi,,x λn

10 10 EANUEL HEBEY AND FRÉDÉRIC ROBERT By 1.8 we get that d g x i,, x 0 as +, that there exists λ 0 such that, up to a subsequence, d g x i,, x µ i, λ 1.9 as +, and that for all and j. Let y be given by µ j, + d gx j,, x ε 1/ µ i, µ i, y = 1 µ i, exp 1 x i, x. By 1.30, dy, S i,r ε for all, where ε > 0 is independent of, while by 1.9 there holds that y C for all, where C > 0 is independent of. We have that D x µ i, by 1.9. By Lemma 1. we then get that D 4 x u x u x B i x as +. As in the proof of Lemma 1., using the structure equation 1.4, there also holds that D x 4 Bx j 0 as + for all j i. Coming back to 1.31, the contradiction follows with the assumption in 1.7. This proves 1.7. Now we prove 1.5. Here again we proceed by contradiction and assume that there exists ε 0 > 0 such that Dx 4 k u x u x Bx i ε 0, 1.3 where x is as in 1.6. We claim that i=1 u x as +. By 1.7 and 1.3, we get 1.33 if we prove that D x 0 as +. Suppose on the contrary that, up to a subsequence, D x δ as + for some δ > 0. By 1.7 and 1.3 there holds that u x u x C u x u x + o for all x B x δ/, and all 1. Assuming that 1.33 is false, it follows from 1.34 that the u s are uniformly bounded in a neighborhood of the x s. By elliptic theory we then get that u u in L Ω, where Ω is a neighborhood of the limit of the x s. Hence u x u x 0 and we get a contradiction with 1.7 and 1.3. This proves Now we let µ be given by µ n = u x and define ũ by ũ = R x µ u. Then g ũ + b µ g ũ + c µ 4 ũ = ũ 1, 1.35 where g x = exp x g µ x. By 1.33, µ 0 as +. It follows that g ξ in C 4 loc Rn as +. Also there holds that ũ 0 = 1 and that the ũ s are bounded in Ḣ. Up to a subsequence, ũ ũ in H loc and ũ = ũ 1, 1.36

11 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 11 where = div ξ is the Euclidean Laplacian. Let S be given by { } 1 S = lim exp 1 x + µ x i,, i I, where I consists of the indices i which are such that d g x i,, x = Oµ and µ i, = oµ. In what follows we let K R n \ S be compact, x K. By 1.6, 1.7 and 1.3, we have that k ũx µ n 4 u y µ n 4 By i i= D x 1 + o1 + o1, D y where y = exp x µ x. It can be checked that µ n 4 By i for all i, as +. By 1.33, 1.37 and 1.38 we then get that 4 ũ x D x 1 + o1 + o D y Since x K, and K R n \ S, there holds that D x = O D y. Hence, by 1.39, for any K R n \ S, there exists C > 0 such that ũ C in K. By elliptic theory and 1.35 we then get that ũ ũ 1.40 in Cloc 4 Rn \ S as +. It holds that 0 S since, if not the case, we can write that D x = oµ and we get a contradiction with 1.3. As a consequence, since ũ 0 = 1, it follows that ũ 0 = 1 and thus that ũ 0. By 1.36 and Lin s classification [4] we then get that ũ x = λ λ + x x0 λn n for some λ > 0 and x 0 R n such that λ n λλ 1 + x 0 = 0. Let K R n \ S, K. By 1. and 1.38 there holds that ũ 0 in L K. By 1.40 we should get that K dx = 0, a contradiction with This ends the proof of ũ Lemma 1.3. As already mentioned, it follows from 1.3 that there exists C > 0 such that r n 4/ u C. 1.4 Derivative companions to this estimate are as follows. Lemma 1.4. Let, g be a smooth compact Riemannian manifold of dimension n 5, and b, c > 0 be positive real numbers such that c b 4 < 0. Let also b and c be converging sequences of real numbers with limits b and c as +, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. There exists C > 0 such that, up to a subsequence, r n 4 +k k u C 1.43

12 1 EANUEL HEBEY AND FRÉDÉRIC ROBERT in for all, and all k = 1,, 3, where r is as in 1.6. Proof of Lemma 1.4. Lemma 1.4 follows from Green s representation formula and 1.3. Let G be the Green s function of g+b g +c. By Green s representation formula, u x = G x, yu y 1 dv g y 1.44 for all x. Then, by 1.44, k u x = k xg x, yu y 1 dv g y 1.45 for all x. There holds, see Grunau and Robert [13], that k y G x, y Cdg x, y 4 n k 1.46 for all, all x, y with x y, and all k {0, 1,, 3}. By 1.4, 1.45, 1.46 and Giraud s lemma, we then get that there exists C > 0 such that k u x k xg x, y u y 1 dv g y This proves C N i=1 Cr x n 4 k. d g x, y 4 n k d g x i,, y n+4/ dv g y Estimates such as in 1.5 and Lemma 1.4 are scale invariant estimates. When transposed to the Euclidean space, in the simple case of a single isolated blow-up point, we would get, for instance when k = 0 and k =, that x n 4/ ux C and x n/ ux C. These two estimates are invariant with respect to the scaling λ n 4/ uλx which leaves invariant both the equation u = u 1 and the Ḣ -norm. Scale invariant estimates, together with the Sobolev description 1., provide valuable informations on the u s. However they are still not strong enough to conclude to the theorem. Sharper estimates such as the ones in Proposition 1.1 are required. From now on we define the v s by v = g u + b u 1.47 for all. By 0.8 there holds that g v + b v = c u + u 1, 1.48 where c = b 4 c. In particular, when c b 4, we get by the maximum principle that either v > 0 in or v 0. When we also assume that b > 0, this implies that v > 0 when u is nontrivial. The following lemma is a key point toward the proof of Proposition 1.1. Lemma 1.5. Let, g be a smooth compact Riemannian manifold of dimension n 5. Let also b and c be converging sequences of positive real numbers satisfying that c b 4 0 for all, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. Let u be such that, up to a subsequence, u u a.e. in. There exists C 1 > 0 such that v C 1 u /

13 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 13 in for all, where the v s are as in Assuming that either u 0 or c b 4 < 0, where b and c are the limits of the sequences b and c, there also exists C > 0 such that v C u in for all. Proof of Lemma 1.5. We use twice the basic remark that if Ω is an open subset of, u, v are C -positive functions in Ω, and x 0 Ω is a point where v u achieves its supremum in Ω, then g vx 0 gux vx 0 ux 0 Indeed, v u = u v v u u g v u so that ux 0 vx 0 = vx 0 ux 0. Then, x 0 = ux 0 g vx 0 vx 0 g ux 0 u x 0 and we get 1.49 by writing that v g u x0 0. In what follows we let x be such that u x / u / = max v x x v Then, by 1.49, We compute g u = u g u x / u x / 1 It follows from 1.51 and 1.5 that By 1.48 and 1.53 we then get that where c = b 4 v x u x gv x v x g u u u. 1.5 g u x gv x u x v x = g u x + b u x gv x v x + b = gv x + b v x v x = u x 1 v x + c u x v x b b c. By assumption, b > 0 and c 0. In particular, we get that v x u x / for all. The estimate v Cu / follows from the definition 1.50 of x. This proves the first part of Lemma 1.5. In order to get the second part we let x be such that, u x v x = max u x x v x. 1.54

14 14 EANUEL HEBEY AND FRÉDÉRIC ROBERT By 1.49 and 1.54, and we get with 1.48 that As a consequence, and we get that v x b u x u x g u x u x gv x v x gv x + b v x v x c u x + u x 1 v x v x u x c u x v x + u x u x v x b b. v x u x c + u x Assuming that c b 4 < 0 there exists δ > 0 such that c δ for all. Similarly, let us assume that u 0. If G stands for the Green s function of g + b g + c, then u x = G x, u 1 dv g \ S G x, u 1 dv g. i Bx i δ By Lemma 1.3, u u uniformly in compact subsets of \ k i=1 {x i}. Letting +, and then δ 0, it follows that there exists δ > 0 such that u x δ for all. In particular, in both cases c b 4 < 0 and u 0, we get with 1.55 that v x Cu x for some C > 0 independent of. By the definition of x in 1.54 it follows that v Cu in for all. This ends the proof of the lemma. At that point, given δ > 0, we define η δ by η δ = max \ S v i Bx i, δ, 1.56 where v is as in Then we prove the following first set of pointwise ε-sharp estimates on the u s and v s. Lemma 1.6. Let, g be a smooth compact Riemannian manifold of dimension n 5, and b, c > 0 be positive real numbers such that c b 4 < 0. Let also b and c be converging sequences of real numbers with limits b and c as, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. Let 0 < ε < ε 0, where ε 0 > 0 is sufficiently small. There exist R ε > 0, δ ε > 0, and C ε > 0 such that u C ε µ n 4 n ε r 4 n+n ε + η δ ε, 1.57 v C ε µ n 4 n ε r n1 ε + η δ ε r nε in \ k i=1 B x i, R ε µ i, for all, where µ i, is as in 1.3, r is as in 1.6, µ is as in 1.7, v is as in 1.47, and η is as in 1.56.

15 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 15 Proof of Lemma 1.6. The first estimate we prove is the one on the v s from which we deduce then the estimate on the u s by using the Green s representation of u in terms of v. We establish the estimate on the v s for 0 < ε < 1, and the estimate on the u s for 0 < ε < 1 n min, n 4. 1 Proof of the estimate on the v s in We fix 0 < ε < 1. Let G 1 be the Green s function of g + 1 and let ψ,ε be given by ψ,ε x = µ n 4 n ε G 1x i,, x 1 ε + η δ ε G 1x i,, x ε. i i Given R > 0 we let Ω,R = i B x i, Rµ i,, and let x \Ω,R be such that max \Ω,R v = v x ψ,ε ψ,ε x. First we claim that for δ ε 1 and R ε 1 suitably chosen, x \Ω,R or r x δ ε We prove 1.58 by contradiction. We assume x \Ω,R and r x < δ ε. We have that g v x v x gψ,ε x ψ,ε x 1.59 and by direct computations, using standard properties of the Green s function G 1 such as its control by the distance to the pole, there also holds that since 0 < ε < 1, there exist C 0 ε, C 1 ε > 0 such that By 1.48 and Lemma 1.5, r x gv x v x r x gψ,ε x ψ,ε x C 0 ε C 1 εr x r x u x 1 + r x c u x v x v x 1.61 Cr x u x + Cr x for all, where c = b 4 c and C > 0 does not depend on. By Lemma 1.3, r x u x Cr x 4 + C r x 4 Bx i + o1 1.6 Cδ 4 ε + ε R, where ε R 0 as R +. By we get a contradiction by choosing δ ε 1 and R 1 sufficiently small. This proves Now that we have 1.58, up to increasing R, we claim that thanks to Lemma 1., Indeed, if r x δ ε, then max \Ω,R v x ψ,ε x v x η δ ε i v ψ,ε C ε G 1x i,, x ε C, i

16 16 EANUEL HEBEY AND FRÉDÉRIC ROBERT while if x \Ω,R, we get that v x = v exp xi, µ i, z = µ n i, v x ψ,ε x C µ g R x µi, i, u + b µ i, R x µi, i, u for some i, where z is such that x = exp xi, µ i, z, and g x = exp x i, g µ i, x. Then, by Lemma 1., and standard properties of G 1, n n 4 i, µn 1 ε i, µi, n ε C µ n 4 n ε up to choosing R 1 such that B 0 R S i,r =, where S i,r is as in In particular, we get that 1.63 holds true. Noting that G 1x i,, x Cr x n, this ends the proof of the estimate on the v s in Proof of the estimate on the u s in We fix 0 < ε < 1 n min, n 4. Let G be the Green s function of g + b 4. Let x be an arbitrary sequence of points such that x \Ω,R for all, where R > 0 is to be chosen later on. There holds that g u x = G x, x + b u xdv g x 4 G x, xv xdv g x 1.64 G x, xv xdv g x \Ω,Rε + G x, xv xdv g x i B xi, Rεµ i, for all, where R ε is the radius obtained when proving the estimate on the v s in We have that G x, x Cd g x, x n. Hence, by Giraud s lemma, G x, xr x n1 ε dv g x \Ω,Rε C d g x, x n d g x i,, x +n ε n dv g x 1.65 i C i C d g x i,, x 4 n+n ε since ε < n 4 n. Now we fix R > 0 such that R R ε. Then d g x, x 1 d gx i,, x for all x B xi, R ε µ i,, and we get that G x, xv xdv g x B xi, R ε Cd g x i, x n B xi, R ε v dv g µ Cd g x i, x n Vol g Bxi, R ε µ i, 1/ v L Cd g x i, x n µ n/ i, z 1.66

17 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 17 since the v s are bounded in L. At last, still by Giraud s lemma, G x, xr x nε dv g x \Ω,Rε C d g x, x n d g x i,, x nε dv g x i C 1.67 since 0 < ε < n. Combining with 1.64, thanks to the estimate on the v s in 1.57, we get that u x Cµ n 4 n ε d g x i,, x 4 n+n ε There holds that and + C i i µ n/ i, d gx i, x n + Cη δ ε µ n 4 n ε d g x i,, x 4 n+n ε + µ n/ i, d gx i, x n = µ n 4 n ε d g x i,, x 4 n+n ε 1 + d g x i,, x n ε µ n/ i, d g x i,, x n ε µ n/ i, n 4 µ +n ε C n 4 µ +n ε µi, µ n 4 n ε C 1.68 since d g x i,, x Rµ i,. Coming back to 1.68 we get that the estimate on the u s in 1.57 holds true. This ends the proof of the lemma. Thanks to the estimates in Lemma 1.6 we can now prove Proposition 1.1. Proof of Proposition 1.1. Consider the estimates: there exist C > 0, R > 0 and δ > 0 such that, up to a subsequence, j u C µ n 4 r 4 n j + η δ in \Ω,R for all j = 0, 1,, 3, and all, where Ω,R = i B x i, Rµ. We prove Proposition 1.1 by proving first these estimates, then by proving that we can replace η δ 1 by u L in 1.69, and at last by proving that the estimates hold in the whole of. 1 Proof of Let G be the Green s function of the fourth order operator g + b g + c. By Lemma 1.6, given 0 < ε 1, there exist R ε, C ε, δ ε > 0 such that u C ε µ n 4 n ε r 4 n+n ε + η δ ε 1.70 in \ k i=1 B x i, R ε µ i,. Since µ i, µ by the definition of µ, there holds that \Ω,R \ k i=1 B x i, Rµ i,. Let x be a sequence in \Ω,R. We aim at proving that there exists C, δ > 0 such that, up to a subsequence, j u x C µ n 4 r x 4 n j + η δ

18 18 EANUEL HEBEY AND FRÉDÉRIC ROBERT for all. Let R = R ε + 1 and δ = δ ε. We have that u x = G x, yu y 1 dv g y 1.7 for all and x. By combining 1.46 and 1.7 we then get that j u x j x G x, x u x 1 dv g x C d g x, x 4 n j u x 1 dv g x d g x, x 4 n j u x 1 dv g x \Ω,Rε + d g x, x 4 n j u x 1 dv g x Ω,Rε Let k n = 1n. By 1.70, d g x, x 4 n j u x 1 dv g x \Ω,Rε Cµ n+4 knε A i + Cη δ ε 1, where A i = \B xi, R εµ i d g x, x 4 n j d g x i,, x n+4+knε dv g x. Let K i, = { x s.t. d g x, x 1 d gx i,, x }. Then A i d g x, x 4 n j d g x i,, x n+4+knε dv g x K i,\b xi, R εµ + d g x, x 4 n j d g x i,, x n+4+knε dv g x Ki, c \Bx i, Rεµ = A i 1, + A i, By the definition of K i,, there holds that d g x, x d g x i,, x in K i,. Hence, choosing ε 1 sufficiently small such that 4 k n ε > 0, we can write that d g x, x 4 n j d g x i,, x n+4+knε dv g x µ 4 knε K i,\b xi, R εµ K i,\b xi, R εµ θ d g x, x 4 n j θ d g x i,, x θ n dg x, x dv g x, d g x i,, x where 0 < θ 1 is chosen small, and by Giraud s lemma we get that A i 1, Cµ 4+knε d g x i,, x 4 n j In K c i, there holds that d gx, x 1 d gx i,, x, and we can directly write that A i, Cd g x i,, x 4 n j K ci, \Bxi, Rεµ d g x i,, x n+4+knε dv g x Cµ 4+knε d g x i,, x 4 n j. 1.77

19 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 19 At last, since R R ε + 1, we can write that d g x, x 1 d gx i,, x for all and all x B xi, R ε µ. Hence, by Hölder s inequality, for any i, d g x, x 4 n j u x 1 dv g x B xi, R εµ Cd g x, x i, 4 n j B xi, R εµ u x 1 dv g x Cd g x, x i, 4 n j Vol g Bxi, R ε µ 1 1 u 1 L Cµ n 4 r x 4 n j. Combining we then get that 1.71 holds true. This proves Proof that 1.69 holds true with u L instead of η δ 1. If u 0 there is nothing to do since, by Lemma 1.4, η δ C. Now we prove that η δ Cµ n 4/ 1.79 in case u 0. This is sufficient to conclude to the validity of 1.69 with u L instead of η δ 1. We assume in what follows that u 0 and we define Ω δ = i B x i, δ. By 1.69, while, by 1.48, we can write that max u Cµ n 4 + Cη δ 1, 1.80 \Ω δ/ max v C max u + C v L \Ω δ \Ω δ/ Let G 3 be the Green s function of g + b. There exists Λ > 0 such that G 3 Λ in and since v g + bu for 1, we get with Green s representation formula that Λ v L 1 G 3x, yv ydv g y u x for all and all x. Therefore, thanks to 1.81, η δ C max u. 1.8 \Ω δ/ By Lemma 1.3, there holds that max \Ωδ/ u 0 as +. In particular η δ 0 as +, and since by 1.80 and 1.8, we get that max u Cµ n 4 + Cη δ max u, \Ω δ/ \Ω δ/ max u Cµ n \Ω δ/ The existence of C > 0 such that 1.79 holds true follows from 1.8 and Proof that the estimates are global in. According to the preceding discussion, the estimates 1.8 hold in \Ω,R for some R > 0. We are left with the proof that they also hold in Ω,R. By Lemmas 1.3 and 1.4, r n 4 +j j u C

20 0 EANUEL HEBEY AND FRÉDÉRIC ROBERT in for all j = 0, 1,, 3. Noting that r n 4 j proof of the proposition. Cr 4 n j in Ω,R, this ends the. Proof of Theorem 0.1 We prove Theorem 0.1 by contradiction. We assume that, g is conformally flat of dimension n 5. We let b and c be converging sequences of real numbers with limits b and c as, c b 4 < 0, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. The Pohozaev identity for fourth order equations can be written as follows: for any smooth bounded domain Ω R n, and any u C 4 Ω, x k k u udx + n 4 u udx Ω Ω = n 4 u u Ω ν + u ν u dσ Ω x, ν u x, u u ν + x, u u dσ, ν where ν is the outward unit normal to Ω and dσ is the Euclidean volume element on Ω. A preliminary lemma we prove is concerned with the Pohozaev identity, applied to the u s, in balls of radii µ, where µ is as in 1.7. Without loss of generality, up to passing to a subsequence, we can suppose that µ = µ 1, for all. Then we let x = x 1, for all. We say x is the blow-up point associated with µ. The meaning of µ in this section is that it is precisely the distance up to which a bubble singularity like in 1.3, with x i, = x and µ i, = µ, interact in the L -topology. Namely, for such a B, lim + max \B x R B = ε R, µ where ε R 0 as R +. In particular, max Bx δ B 0 as + for δ any sequence δ of positive real numbers such that µ +. Lemma.1. Let, g be a smooth compact conformally flat Riemannian manifold of dimension n 5, and b, c > 0 be positive real numbers such that c b 4 < 0. Let also b and c be converging sequences of real numbers with limits b and c as, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. There exists δ > 0 and Ku 0 such that Ku > 0 if u 0, and B x δ µ = o Ag b g u, u dv g u B x δ dv g µ Ku + o1 µ n 4.. for all, where A g is as in 0.3, µ is as in 1.7, and x is the blow-up point associated with µ. Proof of Lemma.1. Let u be defined in bounded subsets of R n by u x = u expx µ x..3

21 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 1 By Hebey, Robert and Wen [0], there exist δ > 0, A > 0, and a biharmonic function ˆϕ C 4 B 0 δ such that, up to a subsequence, u x A + ˆϕx.4 x n 4 in Cloc 3 B 0δ\{0} as +, with the property that ˆϕ is nonnegative, and even positive in B 0 δ if u 0. oreover, there also holds that for any, u dv g = o1 u dv g,.5 B x δ µ B x δ µ where o1 0 as +. Now we let x be the limit of the x s and let δ 0 > 0 and ĝ be such that ĝ is flat in B x 4δ 0. We write that g = ϕ 4/n 4 ĝ with ϕx = 1, and let û = u ϕ. Define and B = 4b n 4 ϕ 8 n n 4 ĝ + ϕ 1 n n 4 Ag h = b ϕ n 4 ĝϕ n 4 n 4n 1 b ϕ 8 n 4 Sg + c ϕ 8 n 4 n 4 Q g ϕ 8 n+4 n 4 + ϕ n 4 divg A g dϕ 1, where Q g is the Q-curvature of g and A g is as in 0.3. By conformal invariance of the geometric Paneitz operator in the left hand side of 0., there holds that û + b ϕ 4 n 4 û B ϕ, û + h û + ϕ n+4 n 4 divg ϕ 1 A g dû = û 1.6 in B x 4δ, where A g is as in 0.3, B, and h are as above, and = ĝ is the Euclidean Laplacian. As a remark,.6 can be rewritten as û + ϕ 4 n 4 divξ A g b g dû + = û 1, where the dots represent lower order terms. Now we let δ > 0 be sufficiently small. We regard û as a function in the Euclidean space and assimilate x to 0 thanks to the exponential map exp x with respect to g. With an abusive use of notations, we still denote by ϕ the function ϕ exp x, by A g the tensor field exp x A g, and by ĝ the metric exp x ĝ. Applying the Pohozaev identity.1 to the û s in B 0 δ µ we get that x k B k û û dx + n 4 û 0δ µ B û dx 0δ µ = n 4 û û B + û 0δ µ ν ν û dσ B 0δ µ x, ν û x, û û + x, û û dσ. ν ν

22 EANUEL HEBEY AND FRÉDÉRIC ROBERT Integrating by parts, using.6, we can also write that x k B k û û dx + n 4 û 0δ µ B û dx 0δ µ = b ϕ 4 n 4 û dx ϕ 8 n 4 Ag û, û dx B 0δ µ B 0δ µ + o û B dx + O û dx 0δ µ B 0δ µ + O û 1 + û B dx + O û 0δ µ B dx, 0δ µ.8 where, in this equation, as already mentioned, we regard ϕ and A g as defined in the Euclidean space. The proof of.8 involves only straightforward computations. By.4, û 1 + û B dx = o µ n 4, and 0δ µ.9 û dx = o µ n 4 while, by.5, B 0δ µ B 0δ µ û dx = o û B dx..10 0δ µ Independently, we can also write with the change of variables x = µ y and.4 that if R stands for the right hand side in.7, then µ n 4 R n 4 ũ ũ B 0δ ν + ũ ν ũ dσ 1 + x, ν ũ x, ũ ũ ν + x, ũ.11 ũ dσ ν as +, where B 0δ ũx = A + ˆϕx.1 x n 4 is given by.4 so that ˆϕ = 0. Coming back to the Pohozaev identity.1, taking Ω = B 0 δ\b 0 r, and since ũ = 0 in Ω, it comes that n 4 ũ ũ B 0δ ν + ũ ν ũ dσ 1 + B 0δ x, ν ũ x, ũ ũ x, ũ + ũ dσ ν ν = n 4 ũ ũ B 0r ν + ũ.13 ν ũ dσ 1 + x, ν ũ x, ũ ũ ν + x, ũ ũ dσ ν B 0r

23 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 3 for all r > 0. Combining.11,.1, and.13, letting r 0, we then get that µ n 4 R Ku.14 as +, where Ku = n n 4 ω n 1 A ˆϕ0. We have that A > 0 and we know that ˆϕ0 > 0 if u 0. It follows that Ku > 0 if u 0. By combining.7.10, and.14, we can write that b ϕ 4 n 4 û dx ϕ 8 n 4 Ag û, û dx B 0δ µ = o û B dx 0δ µ B 0δ µ + Ku + o1 µ n 4,.15 where o1 0 as +. The norm of û in the first term of.15 is with respect to the Euclidean metric ĝ = ξ. Noting that u ĝ = ϕ4/n 4 u g, it follows from.15 that ϕ 8 n 4 Ag b g û, û dx B 0δ µ = o û B dx 0δ µ an equation from which we easily get with.5 that Ag b g u, u dv g Ku + o1 µ n 4 B x δ µ = o u B dv g 0δ µ This ends the proof of the lemma. Ku + o1 µ n Thanks to Lemma.1, and to the estimates in Section 1, we can prove Theorem 0.1. Proof of Theorem 0.1. By Lemma.1 it suffices to prove that when n = 5, 6, 7, u dv g = oµ n 4,.17 B x δ µ where δ > 0 is as in Lemma.1. For 1 sufficiently large, we write that k u B x δ dv g u dv g µ i=1 B xi, Rµ.18 + B x δ µ S u dv g, k \ i=1 Bx i, Rµ where R > 0 is as in Proposition 1.1. By the embedding H H 1,, where = n n, the functions u are bounded in L. Using Hölder s inequalities it follows that for any i, u dv g Cµ n1 = Cµ..19 B xi, Rµ

24 4 EANUEL HEBEY AND FRÉDÉRIC ROBERT There holds that µ = oµ n 4 when n = 5, 6, 7. Independently, thanks to Proposition 1.1, we can write that B x δ µ S u dv g k \ i=1 Bx i, Rµ k Cµ n + Cµ n 4 i=1 B x δ µ \B xi, Rµ d g x i,, 6 n dv g..0 There holds that d B x δ g x i,, 6 n dv g CS.1 µ \B xi, Rµ for all, where S = 1 when n = 5, S = ln 1 µ when n = 6, and S = 1 µ when n 7. Combining.18.1 we get.17. This ends the proof of Theorem Trace estimates We prove trace estimates in this section. Such estimates are required to prove Theorem 0.. As in Section 1 we do not need to assume here that g is conformally flat. We let, g be a compact Riemannian manifold and let b and c be converging sequences of real numbers with limits b and c as, where c b 4 < 0. We let also u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. We aim at proving that if A is a smooth, 0- tensor field, the the integral of A u, u around the maximum blow-up point x behaves like the trace of A at x times µ, where x x as +. In what follows we define Ĩ1 and Ĩ to be the subsets of {1,..., k} given by { } Ĩ 1 = Ĩ = i = 1,..., k s.t. d g x i,, x = o1 { i = 1,..., k s.t. d g x i,, x = o µ, and }, 3.1 where the x i, s and k are given by the decomposition 1., µ is as in 1.7, and x is the blow-up point associated with µ. Namely, assuming that, up to a subsequence, µ = µ i0, for some i 0 and all, then x = x i0,. Proposition 3.1. Let, g be a smooth compact Riemannian manifold of dimension n 7, and b, c > 0 be positive real numbers such that c b 4 < 0. Let also b and c be converging sequences of real numbers with limits b and c as +, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. Let A be a smooth, 0-tensor field. Let δ > 0 be such that d g x i,, x δ µ for all and all i Ĩ, where µ is as in 1.7, x is the blow-up point associated with µ, and Ĩ is as in 3.1. Then there exists β > 0 such that, up to a subsequence, 1 A u, u dv g = βtr g Ax, 3. lim + µ B x δ µ where x is the limit of the x s. Similarly, if u 0, and δ > 0 is such that d g x i,, x δ for all and all i Ĩ1, where x is the blow-up point associated

25 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 5 with µ, and Ĩ1 is as in 3.1, then 1 A u, u dv g = βtr g Ax 3.3 lim + µ B x δ for some β > 0, where, here again, x is the limit of the x s. Proof of Proposition 3.1. Let I be the subset of { 1,..., k } consisting of the i s such that µ = Oµ i, and, for i given, let Îi be the subset of { 1,..., k } consisting of the j s such that d g x i,, x j, = O µ i,. Given R > 0 we define A i,,r to be the annuli type sets A i,,r = B 1 xi,rµ i,\ R µ i,. j Îi B xj, We claim that for any sequences Ω of domains in, Ω S u dv g u dv g + o \ i I A i,,r Ω Ω \ S k i=1 Bx i, Rµ u dv g O Vol g Ω + ε R µ Vol g Ω n + ε R µ, 3.4 for all, where ε R 0 as R +. First we prove 3.4, then we prove 3. and at last we prove Proof of the first estimate in 3.4. We use the Sobolev decomposition 1.. Thanks to 1., by the Sobolev embedding H H 1,, where = n n, by Höder s inequality, and since n 7 so that there holds µ n 4 = oµ, we can write that Ω S u dv g \ i I A i,,r u dv g Ω k B j dv g + o Vol g Ω n. j=1 Ω \ S i I A i,,r Independently, for any j, Ω S B j dv g Cµ x \ i I A j, i,,r R n \ 1 µ K 1 + x n dx + oµ, 3.6 j, where K = exp 1 x j, i I A i,,r. In case j I, then µ j, = oµ and B j dv g = oµ. 3.7 In case j I, then Ω \ S i I A i,,r Cµ j, Cµ j, Ω \ S i I A i,,r B j dv g R n \ 1 µ j, exp 1 x j, A j,,r Rn\KR x 1 + x n dx + oµ x 1 + x n dx + oµ 3.8

26 6 EANUEL HEBEY AND FRÉDÉRIC ROBERT where K R = B 0 R\ i Îj B y i R and y i is the limit of the 1 µ j, exp 1 x j, x i, s. The first estimate in 3.4 clearly follows from Proof of the second estimate in 3.4. Here we use Proposition 1.1. By 1.8 we can write that Ω S u dv g k \ i=1 Bx i, Rµ C 1 + µ n 4 Ω S k \ i=1 Bx i, Rµ CVol g Ω + Cµ n 4 k j=1 r 6 n Ω \B xj, Rµ dvg and there holds that d g x j,, x 6 n dv g C 1 + C µ 6 n Ω \B xj, Rµ d g x j,, x 6 n dv g, R n \B 0R 3.9 x 6 n dx Since n 7, the second estimate in 3.4 follows from 3.9 and This proves Proof of 3.. Let K 1 = i I A i,,r and K = k i=1 B x i, Rµ. By the structure equation, 1.4, A i,,r Aj,,R = for all i j in I. We start by writing that A u B x δ, u dv g µ = A u, u dv g + B x δ µ T K 1 = A u, u dv g + i J A i,,r B x δ µ \K 1 B x δ µ \K 1 where J = I Ĩ and Ĩ is as in 3.1, and that A u B x δ, u dv g µ \K 1 A u, u dv g + By 3.4, B x δ µ \K B x δ µ \K K \K1 K \K1 A u, u dv g A u, u dv g, A u, u dv g. A u, u dv g oµ + Cε R µ, A u, u dv g oµ + Cε R µ, where ε R 0 as R +. Combining , it follows that B x δ µ A u, u dv g = i J A i,,r A u, u dv g + oµ + ε R µ,

27 ASYPTOTIC ANALYSIS FOR PANEITZ EQUATIONS 7 where lim R + lim + ε R = 0. We fix i J and define g to be the metric in Euclidean space given by g x = exp x i, g µ i, x. For j Îi, we let also a j, be the point in R n given by a j, = µ 1 i, exp 1 x i, x j,. Since j Îi there holds that, up to a subsequence, a j, a j in R n. We define ũ i, to be the function defined in the Euclidean space by ũ = R x µi, i, u, where the R x µ action is as in In other words, Then A i,,r A u, u dv g = µ i, ũ x = µ n 4 i, u exp xi, µ i, x. B 0R\W A ũ, ũ dv g, 3.15 where A x = exp x i, A µ i, x, W = Baj, 1 j Îi R, and B aj, 1 R is the ball of center a j, and radius 1/R with respect to g. Since g ξ in the C 4 -topology, where ξ is the Euclidean metric, it follows from Lemma 1. and 3.15 that A u, u dv g A i,,r = µ i, B S Â 0 B, Bdx + oµ i, 0R\ i Îi Ba j 1 R = µ i, Â 0 B, Bdx + oµ i, + ε R µ, R n 3.16 where Â0 = exp x A 0, a j, a j as +, ε R 0 as R +, and B is as in Since B is radially symmetrical, we get from 3.16 that A u, u dv g A i,,r 3.17 = µ 1 i, n Tr gax B dx + oµ i, + ε R µ, R n and 3. follows from 3.14 and 3.17 with β = i J µ i B dx, where R n µ i is the limit of µi, µ as +. Assuming µ = µ 1, for all, there holds that 1 J and β > 0. This ends the proof of Proof of 3.3. We take advantage of u 0. By 1.8 in Proposition 1.1, and since n 7, we get that u dv g Cµ n 4 r x 6 n dv g x B x δ\k Cµ n 4 Cµ n 4 B x δ\k k i=1 B x δ\b xi, Rµ k 1 + µ 6 n i=1 = oµ + ε R µ, R n \B 0R d g x i,, x 6 n dv g x 6 n dx 3.18

28 8 EANUEL HEBEY AND FRÉDÉRIC ROBERT where ε R 0 as R +. Then, we can write that A u, u dv g B x δ = A u, u dv g + A u, u dv g, i J A i,,r where J = I Ĩ 1 and Ĩ1 is as in 3.1, while A u, u dv g B x δ\k 1 A u, u dv g + B x δ\k B x δ\k 1 K \K1 A u, u dv g By 3.4 and 3.18 we then get from 3.19 and 3.0 that A u, u dv g = A u, u dv g + oµ i J A + ε R µ, 3.1 i,,r B x δ where ε R 0 as R +, and 3.3 follow from 3.17 and 3.1. This ends the proof of the proposition. 4. Proof of Theorem 0. when n 6 We prove Theorem 0. by contradiction. We assume that, g is conformally flat of dimension n 6. We let b and c be converging sequences of real numbers with limits b and c as, and u be a bounded sequence in H of positive nontrivial solutions of 0.8 satisfying 1.1. We split the proof in the two cases n = 6, 7 and n 8. First we assume n = 6, 7. By Theorem 0.1 we know that u 0 in 1.. Let S = {x 1,..., x N } be the geometric blow-up set consisting of the limits of the x i, s, where the x i, s are as in Section 1. Let x and µ be as in Sections 1 and, µ being as in 1.7. We may assume x = x 1, for all. Given x i S, since g is conformally flat, there exists up to the assimilation of x i with 0 a smooth positive function ϕ > 0 in a neighborhood U of x i such that ϕ 4/n 4 ξ = g in U = B 0 δ 0, where ξ is the Euclidean metric. We may also assume U S = {x i }. We define û = ϕu and apply the Pohozaev identity.1 to η δ û in B 0 δ for δ 0, δ 0, where η δ x = η δ x and η is such that η 1 in B 01 and η 0 in R n \B 0 4/3. By Hebey, Robert and Wen [0], there holds that k û dx = o1 u dx 4.1 B 0δ\B 0δ/ for all k = 0, 1,, where o1 0 as +, and there also holds since u 0 that B δ u dv g u dv g 1 4. as +, where B δ = N i=1 B x i δ. These estimates may be proved directly from Proposition 1.1. By.6 and 4.1, following the computations in Hebey,