Prescribing Scalar Curvature On Part 1: Apriori Estimates

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(1)j. differential geometry xx (2001) 1-105. PRESCRIBING SCALAR CURVATURE ON S N , PART 1: APRIORI ESTIMATES CHIUN-CHUAN CHEN & CHANG-SHOU LIN. Abstract In this paper, we describe great details of the bubbling behavior for a sequence of solutions wi of n+2. Lwi + Ri win−2 = 0 on S n , where L is the conformal Laplacian operator of (S n , g0 ) and Ri = n(n−2)+ ˆ R ˆ ∈ C 1 (S n ). As ti ↓ 0, we prove among other things the location of ti R, blowup points, the spherical Harnack inequality near each blowup point and the asymptotic formulas for the interaction of different blowup points. This is the first step toward computing the topological degree for the nonlinear PDE.. 1. Introduction This is the first of a series of papers to study the problem of prescribing scalar curvature on S n , the n-dimensional sphere with n ≥ 3. Let g0 be the metric on S n induced from the flat metric of Rn+1 , and R be a given C 1 positive function on S n . We are interested in the question whether there exists a metric g conformal to g0 such that R is the 4 scalar curvature of g. Set g = cn w n−2 g0 for a suitable positive constant cn . Then the question above is equivalent to finding a smooth positive solution of (1.1). n+2. Lw + Rw n−2 = 0. Received October 9, 2000. 1. on S n ,.

(2) 2. chiun-chuan chen & chang-shou lin. n(n − 2) is the conformal Laplacian operator of 4 n (S , g0 ). In general, the same question can be studied in any Riemannian manifold. For a compact Riemannian manifold and a constant R, this problem is called the Yamabe problem, which was solved in early 80s through the works by Trudinger [22], Aubin [1] and Schoen [19]. For a historic account, we refer the readers to Lee and Parker [14] and references therein. For the last three decades, Equation (1.1) has been continuing to be one of major subjects in nonlinear elliptic PDEs. For recent developments, see [1], [2], [3], [5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17], [18], [19], [20], [21] and the references therein. where L = ∆g0 −. In [5], Chang-Gursky-Yang considered Equation (1.1) when n = 3 and R is a positive Morse function on S 3 . Under some nondegenerate conditions on the critical points of R, Chang-Gursky-Yang were able to obtain the apriori bound for positive solutions of Equation (1.1). Furthermore, they computed the Leray-Schauder degree d for Equation (1.1) by the following formula  (1.2). d = − 1 +. .  (−1)ind(p)  ,. p∈Γ−. where Γ− = {p ∈ S 3 | p is a critical point of R satisfying ∆g0 R(p) < 0} and ind(p) is the Morse index of the Hessian of R at p. When the righthand side of (1.2) is assumed to be nonzero, the existence of positive solutions to Equation (1.1) was previously obtained by Bahri-Coron [3] and Schoen-Zhang [21]. However, the degree-counting formula (1.2) provides us more information about Equation (1.1). Particularly, it tells us when the concentration phenomenon for solutions of (1.1) could occur. Li [16] proved the apriori bound for Equation (1.1) on S 4 and derived the formula for the Leray-Schauder degree by adding the effect of the interaction of multiple blow-up points. In this series of papers, we will generalize the results of [5] and [16] on S 3 and S 4 to higher dimensions. As in our previous works [8], [9], it is more convenient for us to study (1.1) in Rn . Without loss of generality, we may assume that the north pole of S n is not a critical point of R. By using the stereographic n−2 2−n projection π from S n to Rn , we set u(x) = 2 2 (1 + |x|2 ) 2 w(π −1 (x)).

(3) prescribing scalar curvature on S n. 3. for x ∈ Rn . Then u(x) satisfies n+2 ∆u(x) + K(x)u n−2 = 0 in Rn , (1.3) at ∞, u(x) = O(|x|2−n ) where K(x) = R(π −1 (x)) for x ∈ Rn . When K(x) is a constant, solutions of (1.1) can be classified completely. See [13] and [4]. For nonconstant R(x), it is well-known that existence of solutions depends on K in a very subtle way. So, throughout the paper and [10], we always assume 0 < a ≤ K(x) ≤ b and K(x) has a finite set of critical points {q1 , . . . , qN }. Near each qj , by Taylor’s expansion, K(x) can be written as K(x) = K(qj ) + Qj (x − qj ) + Rj (x), where Qj (x) is a C 1 homogeneous function of degree βj > 1, i.e., Qj (λx) = λβj Qj (x) for λ > 0 and Rj satisfies lim |x − qj |−βj Rj (x) = lim |x − qj |1−βj | Rj |(x) = 0.. x→qj. x→qj. Here, βj is not necessarily an integer. Of course, if K(x) ∈ C ∞ , then βj must be an integer. (K0). | Qj (x)| ≥ c1 |x|βj −1 for some c1 > 0.. Let U1 (x) = (1 + |x|2 )−. n−2 2. .. (K1) At each critical point qj , according to βj , K satisfies one of the following conditions (i), (ii) and (iii): (i) If βj < n, Qj satisfies   2n n−2 0 (x)dx n Qj (x + ξ)U1 R . = , (1.4) 2n 0 n−2 Q (x + ξ)U (x)dx n j 1 R for any ξ ∈ Rn . (ii) If βj = n, then. (1.5). S n−1. Qj (x)dσ = 0. provided that there exists a vector ξ ∈ Rn satisfying. 2n Qj (x + ξ)U1n−2 (y)dy = 0. (1.6) Rn.

(4) chiun-chuan chen & chang-shou lin. 4. (iii) If βj > n, (1.7). Rn. x − qj , K

(5) |x − qj |−2n dx = 0.. We note that all integrals in (1.4)–(1.7) are L1 (Rn ). In [5], [9] and [16], we knew that only part of critical points of K might be blowup points for certain solutions. Denote by Γ− those critical points of K. More precisely: Definition 1.1. Assume that K satisfies (K0). We say qj ∈ Γ− if and only if K satisfies one of the following conditions (i), (ii) and (iii) at qj according to βj : (i) If βj < n, there exists ξ ∈ Rn such that . 2n   Qj (x + ξ)U1n−2 (x)dx = 0 and  n. R (1.8) 2n    Qj (x + ξ)U1n−2 (x)dx < 0. Rn. (ii) If βj = n, there exists ξ ∈ Rn satisfying . 2n  n−2   Qj (x + ξ)U1 (x)dx = 0 and. (1.9)    Qj (x)dσ < 0. S n−1. (iii) If βj > n, (1.10). Rn. x − qj , K

(6) |x − qj |−2n dx < 0.. Clearly, the notion qj ∈ Γ− and conditions (K0)–(K1) are invariant under the conformal transformations.We list several examples of Q to explain conditions (K0) and (K1). Example 1.2. 1. Q(y) = nj=1 aj yj2 . Clearly aj = 0 for all j iff (K0) holds. It is easy to see that ξ = 0 is the only vector satisfying Rn Q(y + 2n 2n ξ)U1n−2 (y)dy = 0 and Rn Q(y)U1n−2 (y)dy = cn nj=1 aj for some positive constant cn . Thus, (K0) and (K1) hold for a Morse function R on S n satisfying ∆R(q) = 0 for any critical point q of R. And q ∈ Γ− iff ∆R(q) < 0..

(7) prescribing scalar curvature on S n 2. Q(y) = satisfies. n . 3 j=1 aj yj , aj. 5. = 0, for j = 1, 2, . . . , n. Clearly, no ξ ∈ Rn 2n. Q(y + ξ)U1n−2 (y)dy = 0. 3 3. Q(y) = y13 − λy1 nj=2 yj2 . For λ > n−2 , Q(y) satisfies (K0) and (1.4). In fact, there are exactly two solutions ξ = ±ξ0 of 2n n−2 Q(y + ξ)U (y)dy = 0, where ξ0 = (ξ0,1 , 0, . . . , 0) for some n 1 R ξ0,1 > 0. Direct computations show. 2n 2n n−2 Q(y + ξ0 )U1 (y)dy = − Q(y − ξ0 )U1n−2 (y)dy < 0. Rn. The main purpose of our work is to show that homogeneous functions Qj (x) for qj ∈ Γ− completely determine the structure of solutions of (1.1). Conditions (K0) and (K1) are already enough for our purpose. However, in order to make our presentation transparent here, each Qj at qj ∈ Γ− is assumed to satisfy For each qj ∈ Γ− with βj < n, assume that. (K2). (1.11). 2n. Qj (x + ξ)U1n−2 (x)dx < 0 whenever n R. 2n Qj (x + ξ)U1n−2 (x)dx = 0. Rn. To state our main theorem, we introduce the notion Λ− . Assume (K0) and (K1). Let Λ− be a collection of subsets of Γ− such that a subset A of Γ− is an element in Λ− if and only if A satisfies the following conditions. 1. The number of the elements in A ≥ 2. 2. For any two elements qj = qk in A, the exponents βj and βk satisfies 1 2 1 , + > βj∗ βk∗ n−2 where (1.12). βj∗ = min(βj , n).. Now we can state a special case of the Main Theorem we are going to prove in this paper and the subsequent one [10]..

(8) 6. chiun-chuan chen & chang-shou lin. Theorem 1.1. Assume that K satisfies (K0) and (K1) such that βj n−2 of Qj at each critical point qj in Γ− satisfies βj > . In addition, 2 we assume 1 2 1 + ∗ = ∗ βj βk n−2. (1.13). for qj = qk ∈ Γ− . Then there exists a constant c > 0 such that for any solution w of (1.1), we have (1.14). c−1 ≤ w(y) ≤ c for y ∈ S n .. Let d denote the Leray-Schauder degree for the nonlinear map w + n+2 L−1 (Rw n−2 ) on C 2,α (S n ) with 0 < α < 1. Moreover, if (K2) holds additionally, then d satisfies   (1.15) d = − 1 + (−1)n+1 deg Fj + ((−1)n+1 deg Fk ) , A∈Λ− k∈A. j∈Γ−. where deg Fj denotes the standard topological degree of the mapping Fj (x) = Qj (x) from S n−1 to Rn \{0}, and Γ− and Λ− are defined as above. We remark that the assumption βj > n−2 2 in Theorem 1.1 is an also necessary condition for the existence of apriori bounds for solutions of Equation (1.1). In [11], we constructed blowing up solutions of (1.1) for some K satisfying (K0) and (K1) with βj < n−2 2 . To establish the apriori bound (1.14), the first step is to understand the details of blowing-up behavior of a sequence of solutions wi near each blow-up point. In [8], [9] for a sequence of local solutions ui of (1.16). n+2. ∆ui + Ki (x)uin−2 = 0 in B2 = {x | |x| < 2}. where 0 is assumed the only blowup point, we have completely classified types of concentrations of ui according to the flatness β of Q at the blowup point 0. In particular, if n−2 2 < β < n then (1.17). ui (x) ∼ Mi−γ. in any compact set of B 1 \{0}, where 2β −1 if β < n − 2 γ = n−2 1 if β ≥ n − 2,.

(9) prescribing scalar curvature on S n. 7. and Mi is the maximum of ui in B 1 . Hereafter, the notation ai ∼ bi for two sequences of positive numbers denotes that the ratio ai /bi is bounded above and below by two positive constants independent of i. 2 (B \{0}). The result (1.17) is important when Thus, ui (x) ↓ 0 in Cloc 1 global solutions ui of Equation (1.3) are considered, because those local maxima must satisfy certain rules according to (1.17). Together with 2 the Pohozaev identity, we must have β1∗ + β1∗ = n−2 for some blowup j. k. points qj and qk . The apriori bound (1.14) then follows from this. We will give a complete proof of this result in Section 10 of the paper. When n − 2 < βj < n for any critical point qj , the apriori bound was obtained previously in [15]. The degree counting formula (1.15) is more difficult to prove. Usually, there are two ways to establish the Leray-Schauder degree. One is to approach the nonlinear term in Equation (1.1) by subcritical exponents. Another one is to deform the curvature function R, e.g., replace R in Equation (1.1) by Rt = 1 + t(R − 1) for 0 ≤ t ≤ 1. For the latter case, if one can show for any ε > 0, solutions of (1.1) with R replaced by Rt are uniformly bounded for ε ≤ t ≤ 1, then the Leray-Schauder degree is the same for each t = 0. Thus, for our purpose, it suffices to compute the Leray-Schauder degree for small t > 0. In the situation when t is small enough, the degree theory developed by Chang-Yang [6] can be applied very well. But, Chang-Yang was only able to prove the degree counting formulas (1.2) for the class of Morse functions. More seriously, as we will see, the degree formula in [6] did not count all possible solutions. Roughly speaking, their results only covered the case when solutions of (1.1) possess at most one blow-up point as t tends to zero. Later in this paper, we will prove that under assumptions (K0) and (K1), if a sequence of solutions wi of (1.1) with Rti as the scalar curvature blows up as ti → 0, then the number of blow-up points must be greater than one. Therefore, solutions obtained in [6] only consist of bounded solutions as t → 0. We also remark that if the degree βj for each qj ∈ Γ− is no less than n−2, then any sequence of solutions of (1.1) with R replaced by Rti remains uniformly bounded as ti → 0. In this case, Λ− is an empty set and the degree-counting formula (1.15) reduces to d = −[1 + j∈Γ− (−1)n+1 deg Fj ]. When R is a Morse function on S 3 , this is the degree counting (1.2). In this paper, we consider a sequence of solutions ui of (1.3) with curvature functions Ki set by (1.18). ˆ Ki (x) = n(n − 2) + ti K(x),.

(10) 8. chiun-chuan chen & chang-shou lin. ˆ is a C 1 function satisfying the where we assume ti → 0. Here, K nondegenerate conditions (K0)–(K1). Solutions ui are always assumed to blow up at some points of Rn . The main purpose of this article is to study blowup behavior of ui near a blowup point and to study the effect due to the interaction between different blowup points. This is the first step for computing the degree-counting formula. Based on these, we will construct all possible blowup solutions of (1.1) as ti ↓ 0 in [10] and then we are able to compute the “local degree” for each blowup solution. In [10], we will give a complete proof of the degree formula. From the analytic point of view, the main difference between this paper and [9] are: First, we consider the degenerate case limi→∞ Ki = constant here, which can not be covered by the results for nondegenerate limi→∞ Ki in [9]. Second, we allow the number βj defined in (K0) to be greater than or equal to n in this paper, while we assume 1 < βj ≤ n − 2 in [9]. Third, we also consider the interaction between different blow-up points here, while we mainly study local behavior near a blow-up point in [9]. The first interesting question concerning a sequence of blowup solutions is to find the location of blowing up points. A general result states that if Ki converges to K in C 1 , then any blowup point must be a critical point (see [21], [16], [8]). Obviously, this result could not be of any help for our present situation because the limit function of Ki is identically a constant. Nevertheless, by using more delicate estimates than the nondegenerate case, we are still able to prove the following. ˆ satisfies (K0) and ui is a sequence of Theorem 1.2. Suppose K ˆ solutions of (1.3) with K = Ki given in (1.18). Then K(q) = 0 for any blowup point q of ui . Throughout the paper, we let {q1 , . . . , qm } be the set of blowup ˆ at qj . To analyze the points for {ui }, and βj be the degree of Qj of K blowup behavior of ui more accurately, the important step is to show the isolatedness of blowup points, that is, to prove the spherical Harnack inequality (1.19): (1.19). max ui (x) ≤ c. |x−qj |=r. min ui (x) for 0 ≤ r ≤ r0 .. |x−qj |=r. For nondegenerate case, the spherical Harnack inequality (1.19) was proved even for local solutions. See [8], [9] of the reference. For the degenerate case, we do not know whether the spherical Harnack inequality holds or not for local solutions. In Section 4, we study the.

(11) prescribing scalar curvature on S n. 9. situation when it fails. Due to the analysis there and the effect of interactions of different blowup points, nevertheless, the spherical Harnack inequaltiy is proved for global solutions. ˆ satisfies (K0) and (K1). Assume Theorem 1.3. Suppose that K 2(n − 2) βj ≥ for each qj ∈ Γ− . Then any blowup point is isolated. n Furthermore, if βj < n + 1 at a blowup point qj , then ui satisfies (1.20). ui (x) ≤ c |x − qj |−. n−2 2. for |x − qj | ≤ δ0 with some positive constants δ0 and c. By the theory of elliptic equations and the scaling property of Equation (1.3), inequality (1.20) implies (1.19). Hence, we also call (1.20) the spherical Harnack inequality. We note that in Theorem 1.3, (K1) is required only for those qj where βj < n − 2. For each blowup point qj , we let Mi,j and qi,j denote the local maximum and a local maximum point of ui near qj , that is, (1.21). Mi,j = ui (qi,j ) =. max ui (x),. |x−qj |≤δ0. where δ0 is a small positive number such that the distance of qj and qk are greater than 2δ0 . The following theorem is concerned with the asymptotic relations of Mi,j for different blowup points. Let l denote the nonnegative positive integer such that q1 , . . . , ql are simple blowup points and ql+1 , . . . , qm are not simple blowup points. For the notion of simple blowup points, we refer the reader to [8], [9] or Section 2 of this paper. ˆ satisfies (K0) and (K1) and assume Theorem 1.4. Assume that K n−2 − β of Q > 2 at any q ∈ Γ . Let {qj }m j=1 be the set of blowup points for ui , and Mi,j , qi,j and l be defined as above. Then m ≥ 2, l ≥ 1 and β1 = . . . = βl > βj for l + 1 ≤ j ≤ m. Furthemore, the following conclusions hold: (i) We have qj ∈ Γ− for 1 ≤ j ≤ m and there exists such that  2 − n−2   M if   i,j 2 − n−2 1 (1.22) |qi,j − qj | ≤ c Mi,j (log Mi,j ) n if  2 n    − n−2 βj −1 Mi,j if. a constant c > 0 βj < n + 1, βj = n + 1, βj > n + 1..

(12) chiun-chuan chen & chang-shou lin. 10. 2. n−2 Moreover, the limit vector ξ = limi→+∞ Mi,j (qi,j − qj ) satisfies (1.8) if βj < n, and satisfies (1.6) if n ≤ βj < n + 1. (ii) Assume that l = 1. We index qj according to the ordering of βj : β1 > β2 = . . . = βl1 > βl1 +1 ≥ . . . ≥ βm for some positive integer l1 . Then 1 2 1 + > , ∗ β1 β2 n−2. (1.23) Mi,j satisfies (1.24). 2β ∗. − 1 ti Mi,1n−2 − 2n ti Mi,1n−2 log Mi,1. if β1 = n if β1 = n.   . = (1 + o(1)). l1 j=2. −1 −1 η1,j Mi,j Mi,1 ,. and (1.25). −2βj. −1 −1 ti Mi,jn−2 = (1 + o(1))ηj,1 Mi,j Mi,1 for 2 ≤ j ≤ m,. where ηj,k =. (1.26) and. (1.27). n(n − 2)|S n−1 ||qj − qk |−n+2 , |bj |.  2n   βj Rn Qj (x + ξ)U1n−2 (x)dx   bj = with ξ = lim i→+∞ Mi,j (qi,j − qj )  n S n−1 Qj (x)dσ    ˆ x − qj , K

(13) |x − qj |−2n dx n R. if βj < n if βj = n if βj > n. (iii) Assume l ≥ 2. Then β1 = . . . = βl < n − 2 and Mi,j satisfies −. 2β1. (1.28) ti Mi,jn−2 = (1 + o(1)). l k=1,k =j. −1 −1 ηj,k Mi,j Mi,k for 1 ≤ j ≤ l,. and 2β. (1.29). − j ti Mi,jn−2. = (1+o(1)). l k=1. −1 −1 ηj,k Mi,k Mi,j for l +1 ≤ j ≤ m..

(14) prescribing scalar curvature on S n. 11. Theorem 1.4 gives us rather complete information about blowup solutions, that is, the local maxima of blowup solutions must satisfy the necessary conditions (1.24) and (1.25), or (1.28) and (1.29). Conversely, in [10] we will construct such blowup solutions satisfying these relations and compute the contribution of these solutions to the Leray-Schauder degree of Equation (1.1). We note that the third term of the right hand side of (1.15) corresponds to the effect of multiple blowup points. The paper is organized as follows: In Section 2–Section 9, we consider the degenerate case for Equation (1.3), that is, Ki (x) = n(n − 2) + ˆ ti K(x) with ti ↓ 0. In Section 2, main results for local solutions are stated and their proofs are given in the subsequent sections. There are two main issues in Section 2. The first one is the quantity Li , which is associated with each “good” local maximum point of solutions. The quantity Li is introduced in Sections 2 and will play an important role because it decides how large of the range where ui behaves “simply”. We will give its proof in Sections 3 and this is the major step where the method of moving planes is applied. Another important issue in Section 2 is the spherical Harnack inequality (1.20). We will see that when the flatness β ≥ n−2 2 , the spherical Harnack inequality always holds. See Theorem 2.4. The case β < n−2 2 is the difficult one for our analysis, even when the Harnack inequality holds. In the general principle, we can obtain the local bubbling informations through the Pohozaev identities. However, we have to compute each term in the identity very accurately and the Harnack inequality itself is not enough for us to achieve this goal. We need a sharper estimate for the error term of the solution and the approximation bubbles. This is a very delicate analysis because in general the solutions might lose the energy more than one bubble. In Section 5, we show that a method of ODE surprisingly gives us fine estimates when the spherical Harnack inequality is validated. Together with suitably chosen comparision functions, we complete the proof of our desired estimate in Sections 5. See Theorem 2.7. This is one of two difficult jobs in the paper. These estimates for the error term are required in the proof of Lemma 7.1 in Section 7. Lemma 7.1 exactly tells us how, through the Pohozaev identities, the local informations can be put together to obtain more global one. Section 4 will deal with the situation when the spherical Harnack inequaltiy (2.19) fails. Here, we employ a technique of Schoen to localize blowup points. Combined with the method of moving planes developed in Section 3, this provides a clear picture for the case when the Harnack inequality does not hold. Based on the analysis in Section 4 and Lemma 7.1, Theorem 1.3 and.

(15) 12. chiun-chuan chen & chang-shou lin. Theorem 1.4 are proved in Section 8 and Section 9, respectively. We will prove Theorem 1.2 in Section 6 as a direct consequence of results in Section 2. Finally, we will prove the apriori bound of Theorem 1.1 in Section 10. Acknowledgement Part of this work has been done while both authors visited the National Center for Theoretical Sciences of NSC in Taiwan. We want to thank NCTS for the warm hospitality and the stimulating environment. 2. Estimates for local solutions For the convenience of the reader, we briefly review some of previous results from [8] and [9], which would be useful later. Let ui be a solution of (2.1). n+2. ∆ui + Ki (x)uin−2 = 0 in Ω,. where Ω is an open set in Rn . Let x0 be a blowup point. Following Schoen’s idea, a blowup point x0 is called simple if there exists a constant c > 0 and a sequence of local maximum points xi of ui such that (2.2). x0 = lim xi , i→+∞. and (2.3). ui (xi + x) ≤ c Uλi (x) for |x| ≤ r0 , 2. where r0 > 0 is independent of i, λi = ui (xi )− n−2 tends to zero as i → +∞ and n−2 2 λ (2.4) Uλ (x) = for x ∈ Rn . 2 2 λ + |x| For any λ > 0, by elementary calculation, Uλ (x) satisfies n+2. ∆Uλ + n(n − 2)Uλn−2 (x) = 0 in Rn . We note that the definition of a simple blowup point is different from the original one given by Schoen. However, it is not difficult to prove that these two definitions are equivalent..

(16) prescribing scalar curvature on S n. 13. Instead of (2.3), the inequality (2.5). ui (xi + x) ≤ c ui (xi )−1 |x|−n+2. is often used when x0 is a simple blowup point. Also, by (2.4), we have (2.6). Uλ (x) ≤ (2|x|)−. n−2 2. for x = 0,. which implies that if x0 is a simple blowup point, then (2.7). ui (xi + x) ≤ c |x|−. n−2 2. for |x| ≤ r0 .. A blowup point x0 is called isolated if (2.7) holds for some c and r0 > 0. It is easy to see a simple blowup point must be isolated. The inequality (2.7) is important because it implies that the Harnack inequality holds for each sphere with center xi , i.e., there exists a positive constant c > 0 such that (2.8). max ui (x) ≤ c min ui (x). |x−xi |=r. |x−xi |=r. for 0 ≤ r ≤ r0 . Suppose that x0 is a blowup point of ui . Theorem 1.3 in [8] states that x0 is a simple blowup point if Ki (x) → K(x) in C 1 and Ki satisfies for some constant c either (i) | Ki (x)| ≤ c if n = 3 or (ii) (2.9). β−j. | j Ki (x)| ≤ c| Ki (x)| β−1. if n ≥ 4 in a neighborhood of x0 for 1 ≤ j ≤ β = n − 2. Also see [15] for the same conclusion when global solutions are considered. We make ˆ with K ˆ satisfying some remarks here. First, if Kj = n(n − 2) + ti K (2.9), then (2.9) holds for Ki also with the same constant c. Thus ˆ is smooth and Theorem 1.3 in [8] can apply to our case. Second, if K ˆ 0 )| ≥ c > 0, then obviously condition (2.9) holds for Ki also. | K(x Actually, from the first step of the proof of Theorem 1.3 in [8], the ˆ can be removed if x0 is not a critical point smoothness assumption of K ˆ of K. Even when x0 is a critical point, it is not necessary to assume ˆ is smooth. In this case, condition (2.9) can be replaced by that K ˆ ≤ c2 |x − x0 |β−1 c1 |x − x0 |β−1 ≤ | K(x)| (2.10). in a neighborhood of x0 for some constants c2 > c1 and β > 1..

(17) 14. chiun-chuan chen & chang-shou lin. Thus, Theorem 1.3 of [8] can be restated as follows: ˆ Theorem A. Let ui be a solutions of (2.1) with Ki = n(n − 2) + ti K and x0 ∈ Ω be a blowup point of ui . Assume that either x0 is not a ˆ or x0 is a critical point of K ˆ and K ˆ satisfies (2.10) critical point of K for some β ≥ n − 2. Then x0 is a simple blowup point. Obviously, if x0 is a simple blow-up point, then there are no blowup ˆ has a points in a small neighborhood of x0 . If we further assume that K discrete set of critical points in Ω, then by Theorem A, ui has a discrete set of blowup points at most. Hence, throughout Section 2 to Section 5, we always assume that ui is a solution of  n+2   ∆ui + Ki (x)uin−2 (x) = 0 on B 2 \{0}, (2.11) ui (x) is uniformly bounded in any compact   set of B 2 \{0}, ˆ where K ˆ satisfies where B2 = {x : |x| < 2}, and Ki (x) = n(n − 2) + ti K (2.10) with x0 = 0 for x ∈ B 2 and some β ≥ 1. Here, solutions ui is ˆ i denote the maximum of ui and xi be assumed to blow up at 0. Let M a maximum point of ui , i.e., (2.12). ˆ i = ui (xi ) = max ui (x) → +∞ M |x|≤2. as i → +∞. Clearly xi → 0. If β = 1 or β ≥ n − 2, by Theorem A, (2.3) holds for some constant c > 0. When 1 < β < n − 2, the situation is more complicated as shown in [9]. A solution ui may have local maximum points beside xi . Let zi be any local maximum point of ui with ui (zi ) → +∞. Then by assumption (2.11), limi→∞ zi = 0. Let vi (y) be the scaled function defined by (2.13). 2 − n−2. vi (y) = Mi−1 ui (zi + Mi. y) with Mi = ui (zi ). 2. Obviously, vi (y) is well-defined for |y| ≤ Min−2 when i is large. In the paper, we will always reduce the arguments to the situation when vi (y) is uniformly bounded in any compact set of (2.14). Rn , that is, for any ε > 0, there exists a sequence of Ri → +∞ such that |vi (y) − U1 (y)| ≤ εU1 (y) for |y| ≤ Ri ..

(18) prescribing scalar curvature on S n. 15. In this case, by passing to a subsequence, vi (y) converges to U1 (y) in 2 (Rn ), where U (y) is given in (2.4) with λ = 1. Cloc 1 For such “good” local maximum point zi , we set (2.15). ˆ 2β 2 1 1 n−2 |z |1−β ) n−2 , (t−1 u (z ) n−2 ) n−2 , u (z ) Li (zi ) = min (t−1 i i i i i i i. where βˆ = β if β < n and βˆ is any positive number in (n − 1, n) if β ≥ n. One of the main themes for local solutions is to know if the 2. scaled vector Min−2 zi is bounded. This is closely related to the quantity Li (zi ). To see this, let us assume β < n for simplicity. In this case, if n−2 lim ui (zi )|zi | 2 = +∞, then. i→+∞. 2. 2β. 2. 2β. ui (zi ) n−2 |zi |1−β = (ui (zi ) n−2 |zi |)1−β ui (zi ) n−2 = o(1)ui (zi ) n−2 and. 2. 1. n−2 |z |1−β ) n−2 . Li (zi ) = (t−1 i i ui (zi ). On the other hand, if lim ui (zi )|zi |. i→+∞. n−2 2. < +∞,. then it is easy to see 2β. 1. n−2 ) n−2 . Li (zi ) ∼ (t−1 i ui (zi ). The quantity Li (zi ) plays an important role for us to understand the bubbling profile of ui . Our first result concerns with Li (xi ) and the simple blowup at 0. We recall xi is a maximum point of ui and ˆ i = ui (xi ) is the maximum of ui . See (2.12). M ˆ satisfies Theorem 2.1. Suppose ui is a solution of (2.11) and K (2.10) for some β ≥ 1. Assume (1.4) in addition if β < n − 2. Then after passing to a subsequence, 0 is a simple blow-up point if and only if there exists a constant c > 0 independent of i such that 2. ˆ n−2 ≤ c Li (xi ) M i for all i. 2. ˆ − n−2 L(xi ) tends to +∞ as An interesting case is when the ratio M i i → +∞. If ui is a global solution of (1.3), by applying the method of.

(19) chiun-chuan chen & chang-shou lin. 16. moving planes, we can prove that 0 is the only simple blowup point. See (6.8). 2. ˆ − n−2 L(xi ) is bounded, we have On the other hand, when the ratio M i the following result. ˆ satisfy the assumptions of Theorem 2.1 Theorem 2.2. Let ui and K ˆ i and Li (xi ) be defined in (2.12) and (2.15), respectively. and let xi , M Suppose that there is c > 0 such that 2. ˆ n−2 , Li (xi ) ≤ c M i n−2. ˆ i |xi | 2 is bounded and β < n − 2. Furthermore, if assume in then M ˆ satisfies (K0) with Q being the homogeneous function addition that K 2 ˆ n−2 xi , then ξ satisfies and lim ξi = ξ exists with ξi = M i. i→+∞. (2.16). Rn. 2n. Q(x + ξ)U1n−2 (x)dx = 0.. The following consequence of Theorem 2.2 is important when we come to determine the position of blowup points for global solutions of (1.3). Corollary 2.3. Let ui and Ki satisfy the assumptions of Theoˆ ˆ rem 2.1. Assume that either K(0). = 0 or K(0) = 0 with β ≥ n − 2, 2 − n−2 ˆ then lim Li (xi )M = +∞. i→+∞. i. Both proofs of Theorem 2.1 and 2.2 are given in Section 3, where the application of the reflection method are discussed. By Theorem A, the ˆ at 0 determines the bubbling behavior of ui . Convenflatness β of K tionally, ui is said to lose the energy of one bubble at 0 if ui converges 1 (B \ {0}) and to 0 in Cloc 2. (2.17). lim. i→+∞ |x|≤1. 2n n−2. ui. (x)dx =. Sn n(n − 2). n 2. ,. where Sn is the Sobolev best constant. Clearly, if ui blows up at 0 simply, then ui lost one bubble. ˆ satisfies (K0) and (K1) at 0 with Theorem 2.4. Assume that K ≤ β, and ui is a solution satisfying (2.11). Then ui loses the energy. n−2 2.

(20) prescribing scalar curvature on S n. 17 2. ˆ − n−2 < of only one bubble at 0. Suppose in addition that lim Li (xi )M i i→+∞. +∞. Then there exists a constant c > 0 such that (2.18). ui (x) ≤ c |x|. 2−n 2. for |x| ≤ 1.. 2. ˆ n−2 xi . Then after passing to a subsequence, the limit ξ = Set ξi = M i lim ξi satisfies (1.8). i→+∞. When β < n−2 2 , it is possible that (2.18) does not hold and it is also possible that ui loses energy of more than one bubble even (2.18) holds. We first consider the case when inequality (2.18) does not hold. There are two alternatives in this case. ˆ satisfies (K0) and (K1) at 0, and ui Theorem 2.5. Assume that K is a solution of (2.11). Suppose (2.19). lim sup(ui (x)|x|. i→+∞. n−2 2. ) = +∞.. Then one of the followings holds: (i) The origin is a simple blowup point and consequently, an isolated blowup point. More precisely, we have   ui (xi + x) ≤ c Uλi (x) for |x| ≤ 1, and (2.20) ˆ i |xi | n−2 2  lim M = +∞, i→+∞ 2. ˆ − n−2 . where λi = M i (ii) The origin is not a simple blowup point and is not an isolated n−2 blowup point. In this case, we have β < and there exists a 2 local maximum point zi of ui satisfying (2.21) n−2 n−2 ui (zi )|zi | 2 → ∞ and Li (zi )ui (zi )− 2 → ∞ as i → +∞ such that for any δ > 0, ui (x) is a simple blowup with center zi for x ∈ B(0, δ|zi |), i.e., (2.22). ui (x) ≤ c Uλi (x − zi ).

(21) chiun-chuan chen & chang-shou lin. 18. 2. for |x| ≥ δ|zi |, where λi = ui (zi )− n−2 . Also, for x ∈ B(zi , δ|zi |), we have (2.23). ui (x)|x|. n−2 2. ≤c. ˆ i , where with c = c(δ) independent of i. Moreover, ui (zi ) = o(1)M ˆ o(1) tends to 0 as i → +∞ and Mi = max ui (x). |x|≤2. Remark 2.6. Two consequences follow from Theorem 2.5. First, since (2.22) implies (2.24). min ui (x) ∼ ui (zi )−1 ,. |x|=1. the spherical Harnack inequality (2.18) holds if ui (x) ≥ c > 0 on B 2 for some c independent of i. Second, by (2.21), lim Li (zi )ui (zi )−. i→+∞. n−2 2. = +∞.. We will see later that this implies if ui is a sequence of global solutions, then the number of the type of blowup points described in (ii) of Theorem 2.5 is at most one. See (6.8). By using this fact, we then are able to apply Lemma 7.1 to get rid of the blowup point of the type of behavior in case (ii) of Theorem 2.5. This is indeed Theorem 1.3. 1 (B \{0}), we say u loses energy When ui converges to zero in Cloc 2 i of more than one bubble near 0 if n . 2n 2 Sn n−2 ui (x)dx > . (2.25) lim i→+∞ |x|≤1 n(n − 2) In this case, we have β < n−2 2 by Theorem A and Theorem 2.4. It is easy to see the blowup described in (ii) of Theorem 2.5 belongs to this case. Actually, when β < n−2 2 , it is possible for ui to lose infinite energy. See [11] for the existence for such solutions. To estimate ui more accurately when it satisfies (2.18) and loses energy of more than one bubble, let. 1 (2.26) ui (r) = ui dσ |∂Br | |x|=r be the spherical average of ui , and (2.27). wi (s) = ui (r)r. n−2 2. with r = es ..

(22) prescribing scalar curvature on S n. 19. Obviously, wi (s) is well-defined for s ≤ 0. Since 0 is a blowup point, wi has at least one maximum point. Let si ≤ 0 be the local maximum point of wi , which is nearest to zero. Set Mi = e−(. (2.28). (2.29). Li =. u i =. (2.31). 2β. n−2 t−1 i Mi. Ri = Lγi , γ =. (2.30). n−2 )si 2. , 1 n−2. 1 1−. Mi−1 ui. . 2β n−2 −2 n−2. Mi. ,. , and. x .. Then we have the following estimates: ˆ satisfies (K0) and (K1) at 0 with Theorem 2.7. Suppose that K n−2 , and ui is a solution of (2.11) which converges uniformly 1<β< 2 to zero in any compact set of B 2 \{0} and satisfies (2.18) and (2.25). i as above. Then lim Mi = +∞ and Define wi , si , Mi , Li , Ri and u i→+∞. there are c > 0, ai → 1, zi ∈ Rn and λi > 0 such that the following hold: (i) lim λi = λ and lim zi = z, where λ and z satisfy i→∞. (2.32) Set ξ =. i→+∞. √. 1 = λ2 + |z|2 . λz. Then ξ satisfies (1.8).. (ii) u i satisfies (2.33). | ui (x)| ≤ c |x|−. n−2 2. for |x| ≤ Ri−2 , and. (2.34) | ui (x) − ai Uλi (x − zi )| + Ri−n+2 |x|−n+2 + ≤ c (L−n+2 i 2. for Ri−2 ≤ |x| ≤ Min−2 .. max. 2. |y|=Min−2. | ui (y) − ai Uλi (y − zi )|),.

(23) 20. chiun-chuan chen & chang-shou lin −. 2. Remark 2.8. If Li Mi n−2 ≤ c for some constant c > 0, then from the proof of Theorem 2.7, we will see that u i (y) ≤ c1 L−n+2 for some i 2. constant c1 when |y| = Min−2 . Thus, the third term in the right hand 2 − n−2. side of (2.34) can be absorbed by L−n+2 when Li Mi i. ≤ c.. To extend the notion of simple blowup to cover the case when ui loses energy of more than one bubble, we modify (2.3) as follows. Let Br (y) denote {x : |x − y| < r}. Definition 2.9. Assume 0 is a blowup point. The blowup point 0 is called simple-like if there exist c > 0, r0 > 0, a sequence of numbers {λi }, a sequence of points {zi } and a sequence of balls {Bri (yi )} such that limi→∞ λi = 0, limi→∞ zi = limi→∞ yi = 0, limi→∞ ri λ−1 i = 0, and ui (x + zi ) ≤ c Uλi (x) on Br0 (0) \ Bri (yi ). According to the definition, it is not difficult to see that there are exactly three types of simple-like blowup point: simple blowup, the blowup described in (ii) of Theorem 2.5, and the blowup in Theorem 2.7 2. when Li ≥ cMin−2 for some constant c > 0. On the other hand, if 0 is non-simple-like, then by Theorem 2.5, inequality (2.18) holds and 0 must be isolated. Remark 2.10. When the assumption (K1) is concerned in the theorems of this section, (K1) is required only when β < n − 2. 3. Applications of the method of moving planes In this section, we will collect some well-known results and prove some lemmas which will be used in the proofs of the theorems in Section 2. In the proofs, we often assume there is a sequence of local maximum points zi of ui such that the scaled function vi in (2.13) satisfies (2.14). By applying the method of moving planes, we can improve the result of (2.14). When Ki satisfies if the nondegenerate conditions (K0) and (K1) with 1 < β ≤ n − 2, we proved that ui (zi + x) could 2. −. 2. be bounded by c Uλi (x) with λi = ui (zi )− n−2 for |x| ≤ Li Mi n−2 . See Lemma 3.1 in [9]. Actually the proof there can apply to the degenerate case. In the following, we give a brief sketch of the proof for the convenience of readers. In fact, Lemma 3.1 below deals with the case more general than the one considered in [9], namely, ui is allowed to.

(24) prescribing scalar curvature on S n. 21. have very large values, compared with ui (zi ), in some small region. Let d(B, 0) denote the distance from the origin to a ball B. Lemma 3.1. Suppose that ui is a solution of (2.11), zi is a local maximum point of ui and vi is given as in (2.13). Let B be a closed ball in Rn with d(B, 0) > 0 and ε be a positive (small ) number. Suppose that there is a sequence of Ri → +∞ as i → +∞ such that |vi (y) − U1 (y)| ≤ εU1 (y) for |y| ≤ Ri and y ∈ B. Then there exists δ = δ(ε, d(B, 0)) > 0 such that (3.1). min vi (y) ≤ (1 + 24)U1 (r). |y|≤r. 2. for 0 ≤ r ≤ L∗i (δ), where L∗i (δ) = min(δLi (zi ), Min−2 ). Proof. When B is an empty set and 1 ≤ β ≤ n−2, this is Lemma 3.1 in [9]. Thus, we only sketch the proof below. For the details, we refer the interested readers to [9]. Let e1 = (1, 0, · · · , 0) and τ = d(B, 0). We may assume the center of B is r0 e1 for some r0 > τ . Let τ 2x + τ e1 , |x|2 τ n−2 τ 2 x vi + τ e1 , v i (x) = 2 |x| |x| τ n−2 τ 2 x . U1 + τ e U 1 (x) = 1 |x| |x|2 F (x) =. (3.2). By a straighforward calculation, we have U 1 (x) =. λ 2 λ + |x − x0 |2. n−2 2. ,. τ 3 e1 τ2 and x = − . Also we have F −1 (B) = {x : 0 τ2 + 1 τ2 + 1 x = F −1 (y), y ∈ B} ⊂ {(x1 , x2 , · · · , xn ) : x1 > 0}, d(F (B), 0) > 0 and v i satisfies n+2 v i + K i (x)v i n−2 = 0 where λ =. 2 − n−2. for x ∈ / F −1 (B), where K i (x) = Ki (zi + Mi. F (x))..

(25) 22. chiun-chuan chen & chang-shou lin. Now assume that the conclusion of Lemma 3.1 does not hold. Then by passing to a subsequence, there is a sequence of positive number ri such that ri ≤ L∗i (δ) and (3.3). min vi (y) ≥ (1 + 24)U1 (ri ),. |y|≤ri. where δ = δ(ε) will be chosen later. By the assumptions, it is easy to see ri ≥ Ri → +∞ as i → +∞. Since by (3.2), v i (x) uniformly converges 2 (Rn \{0}), v has a local maximum at some point q near to U 1 (x) in Cloc i i x0 . Now we are going to apply the method of moving planes to obtain a contradiction. For any λ < 0, let Σλ = {x | x1 > λ}, Tλ = {x | x1 = λ} and xλ denote the reflection point of x with respect to Tλ . We also let Σ λ = Σλ ∩ {x | |x| ≥ τ 2 (ri − τ )−1 }. In the following, we will choose |x0 | and show that for λ ≤ λ0 , a number λ0 satisfying −|x0 | < λ0 < − 2 there exists i0 = i0 (λ0 ) such that (3.4). v i (xλ ) ≤ v i (x). for x ∈ Σ λ , λ ≤ λ0 and i ≥ i0 . This yields a contradiction to the fact that v i has a local maximum near x0 . Note that the local maximum point qi tends to x0 as i → ∞. Let wλ (x) = v i (x) − v i (xλ ). Then wλ satisfies (3.5) where. wλ + bλ (x)wλ (x) = Qλ (x). in Σ λ ,.  n+2 n+2  n−2 − (v (xλ ) n−2  i  b (x) = K (x) v i (x) i λ v i (x) − v i (xλ )  n+2   Qλ (x) = K i (xλ ) − K i (x) v i (xλ ) n−2 .. By (3.2) and (3.3), we have for |x| = τ 2 (ri − τ )−1 , ri − τ n−2 min vi ≥ (1 + ε)U 1 (0) (3.6) v i (x) ≥ τ |y|≤ri for i large. On the other hand, v i (x−|x0 | ) converges to U 1 (0−|x0 | ) = U 1 (0) uniformly for |x| = τ 2 ri−1 , where x−|x0 | and 0−|x0 | are the reflection points of x and 0 with respect to the hyperplane T−|x0 | . Hence −|x0 | such that there exists −|x0 | < λ0 < 2 ε v i (xλ ) ≤ (1 + )U 1 (0) 2.

(26) prescribing scalar curvature on S n. 23. for |x| = τ 2 (ri − τ )−1 , λ ≤ λ0 and large i. Together with (3.6), it implies for |x| = τ 2 (ri − τ )−1 , ε wλ (x) ≥ U 1 (0) 2 for λ ≤ λ0 and large i. In the following, we fix this λ0 . Then there is a small c0 such that (3.7). ε wλ (x) ≥ U (0) ≥ c0 ri−n+2 Gλ (x, 0) 2. holds for |x| = τ 2 (ri − τ )−1 , λ ≤ λ0 and large i, where Gλ (x, y) is Gλ (x, y) = cn. . 1 1 , − |y − x|n−2 |y λ − x|n−2. the Green function of − on Σλ = {x : x1 > λ}. If λ1 < 0 and |λ1 | is large, then we have (3.8). wλ (x) ≥. c0 −n+2 λ G (x, 0) r 2 i. for λ ≤ λ1 , x ∈ Σ λ and large i. For the details, see [9]. For λ > λ1 , let Q+ λ = max(0, Qλ ), Li = Li (zi ) and (3.9). Gλ (x, 0) − hλ (x) = aL−n+2 i. Σλ. Gλ (x, η)Q+ λ (η) dη,. where a is a positive number to be chosen later. Obviously, hλ satisfies hλ = Q+ λ ≥ Qλ. in Σ λ .. For λ ≤ λ0 and η ∈ Σλ , since |η λ | ≥ |η| and |η λ | ≥ |λ0 | ≥ one has by (3.2) |v i (η λ )| ≤ c1 (1 + |η λ |)−(n−2) .. |x0 | 2. > 0,. Here, we use F −1 (B) ⊂ Σλ also. For η ∈ Σ λ , we have |η| ≥ τ 2 (ri − τ )−1 ≥. τ2 ∗ τ2 − 2 Li (δ) ≥ Mi n−2 . 2 2. To estimate the integral term in (3.9), we note − 2 − 2 λ −(n+2) Q+ Ki (zi +Mi n−2 F (η λ ))−Ki (zi +Mi n−2 F (η)). λ (η) ≤ c2 (1+|η |).

(27) chiun-chuan chen & chang-shou lin. 24. By (2.10), when η ∈ Σ λ , (3.10). 2 − n−2. |Ki (zi + Mi. F (η)) − Ki (zi )| . 2 − n−2. ≤ cti Mi ≤. −. |F (η)| |zi |β−1 + Mi. − 2 c3 ti Mi n−2 (1. −1. + |η|. ˆ 2(β−1) n−2. −. ) |zi |β−1 + Mi. ˆ. |F (η)|β−1 ˆ 2(β−1) n−2. ˆ. (1 + |η|1−β ). ˆ. (1 + |η|−β ), ≤ c4 L2−n i τ2 − 2 where |η| ≥ Mi n−2 is used and βˆ is the number in (2.15). Thus, we 2 have (3.11). ˆ. −n+2 Q+ (1 + |η|−β )(1 + |η λ |)−(n+2) . λ (η) ≤ c5 Li. By (3.11), following the computation in the proof of Lemma 3.1 in [9], we obtain. −n+2 λ (3.12) Gλ (x, η)Q+ G (x, 0) λ (η)dη ≤ c6 Li Σλ. for x ∈ Σ λ , where c6 is a constant depending on the constants in (2.10), τ and n only. Set a = 2c6 in (3.9). Then a (3.13) 0 < [L(zi )]−n+2 Gλ (x, 0) ≤ hλ (x) ≤ a[L(zi )]−n+2 Gλ (x, 0). 2 Recall that ri ≤ δLi (zi ). Choose δ to be sufficiently small such that c0 δ −n+2 ≥ 2a. Then by (3.7) and (3.8), for i large, wλ (x) > hλ (x) holds for x ∈ Σ λ if λ = λ1 , and holds for |x| = τ 2 (ri − τ )−1 and λ ≤ λ0 . It follows that hλ satisfies the assumptions of Lemma 2.1 in [9] with λ1 ≤ λ ≤ λ0 when i is large. Applying Lemma 2.1 in [9], wλ (x) > hλ (x) > 0 for x ∈ Σ λ and λ ≤ λ0 . Hence, (3.4) is proved, and then the proof of Lemma 3.1 is finished. q.e.d. Note that if ui is a global solution defined in the whole space Rn , then we can choose 2. L∗i (δ) = min(L∗i (δ), λMin−2 ).

(28) prescribing scalar curvature on S n. 25. for any λ > 0. Inequality (3.1) is very useful when the Harnack inequality holds for vi on each sphere |y| = r. Actually, under some extra condition on ui , we can derive the spherical Harnack inequality from (3.1) itself by using the Green representation formula. We will explain this in Lemma 3.4, which tells us how to derive the Harnack inequality. Before that, we have to state two well-known lemmas. For their proofs, see [9]. Lemma 3.2. Suppose φ(x) satisfies 4. in Rn. φ(x) + n(n + 2)U1n−2 φ(x) = 0. with φ(x) → 0 as |y| → ∞. Then φ(x) can be written as φ(x) = c0 ψ0 (x) +. n . cj ψj (x). j=1. for some cj ∈ R, j = 0, 1, . . . , n, where ψj (x) =. ∂U1 for 1 ≤ j ≤ n and ∂xj. n−2 U1 + x · ∇U1 . 2 Lemma 3.3. Suppose that u is a positive smooth solution of. ψ0 (x) =. n+2. u + K(x)u n−2 = 0 in Br , where |K(x)| ≤ b. Then there exists a small 4o > 0, depending on b and n only, such that if ||u|| n−2 2n ≤ 4o , then the Harnack inequality L. u(x) ≤ c u(y) holds for |x|, |y| ≤ r/4, where c > 0 depends on b and n only. In Lemma 3.4, we consider a more general setting, which is needed later. Assume that 0 < a ≤ K(x) ≤ b, u is a solution of (3.14). n+2. u + K(x)u n−2 = 0, u > 0 for |x| ≤ l0 ,. and U is the solution of  n+2  U + K0 U n−2 = 0, U > 0 in Rn , (3.15)  U (0) = max U = 1, Rn.

(29) chiun-chuan chen & chang-shou lin. 26. where K0 is a positive constant. Let Br = {x : |x| < r}. Lemma 3.4. Let u, U and l0 be as above. Suppose 0 < σ < 1, l0 R ≤ , and E ⊆ BR/2 such that 8 (3.16). |u(x) − U (x)| ≤ σU (x). for x ∈ BR \E,. (3.17). n+2. |K(x) − K0 |U n−2 dx ≤ σ,. |x|≤R. (3.18). E. n+2. U n−2 dx < σ, and. min u(x) ≤ (1 + σ)U (r). (3.19). |x|=l. for some l ∈ [R, l40 ]. Then there is a constant c1 depending on n and b only such that. n+2 l (3.20) u n−2 dx ≤ c1 (R−2 + σ + ( )n−2 ), l0 R≤|x|≤l Furthermore, if (3.21). (3.22). l u(x) ≤ c2 (R−2 + σ + ( )n−2 )−1 , and l0 min u(x) ≤ c3 U (r). |x|=r. for R ≤ r ≤ l where c2 = c2 (n, a, b) is a small positive constant and c3 > 0, then (3.23) for |x| ≤. u(x) ≤ c4 U (x) l and x ∈ / E, where c4 depends on c2 and c3 . 2.

(30) prescribing scalar curvature on S n. 27. Proof. For r > 0, let Br = {x : |x| < r}. Let G(x, η) be the Green function of the Laplacian operator − on the ball Bl0 with zero boundary value. Let x0 be a point satisfy |x0 | = l and u(x0 ) = min|x|≤l u(x). By the Green identity and (3.19),. (3.24). (1 + σ)U (x0 ) ≥ u(x0 ) ≥. n+2. Bl0. G(x0 , η)K(η)u n−2 (η)dη,. and. U (x0 ) =. n+2. Bl0. G(x0 , η)K0 U n−2 dη + U (l0 ). (3.25). ≤. n+2. Bl0. G(x0 , η)K0 U n−2 dη + U (l0 ). Hence there is cn depending on n only such that. a cn. n+2. l R ≤η≤ 20 2. (l + |η|)−n+2 u n−2 dη. ≤ u(x0 ) −. (3.26). n+2. B R \E 2. G(x0 , η)K(η)u n−2 dη. ≤ (1 + σ)U (x0 ) −. n+2. B R \E. G(x0 , η)K(η)u n−2 dη.. 2. By the assumptions (3.16) and (3.17), there is c4 depending on n and b only such that. n+2. B R \E 2. ≥. G(x0 , η)K(η)u n−2 dη. B R \E. ! n+2 n+2 n+2 G(x0 , η) K0 U n−2 + K(η)(u n−2 − U n−2 ) n+2 " − |K(η) − K0 |U n−2 dη. 2. ≥. n+2. BR 2. G(x0 , η)K0 U n−2 dη − c4 l−n+2 σ..

(31) chiun-chuan chen & chang-shou lin. 28. Together with (3.25), it leads to. n+2 a cn (1 + l−1 |x|)−n+2 u n−2 dη l R ≤η≤ 20 2  . n+2 ≤ ln−2  G(x0 , η)K0 U n−2 dη + c4 l−n+2 σ  Bl0 \B R 2. n−2. [σU (x0 ) + U (l0 )] l ≤ c5 (σ + R−2 + ( )n−2 ), l0 +l. where c5 depends on n and b only. Obviously the inequality (3.20) follows immediately. Let 40 be the number in Lemma 3.3 and c2 be a small number such that c2 c5 (cn a)−1 < 40 . l If u(x) ≤ c2 (R−2 + σ + ( )n−2 )−1 for l0. R ≤|η|≤l 2. u. 2n n−2. dη <. R ≤|η|≤l 2. u. R 2. ≤ |x| ≤ l, then #. n+2 n−2. dη. $ max u. R ≤|η|≤l 2. < 40 .. By Lemma 3.3, the Harnack inequality holds for u on {x : |x| = r} with l R ≤ r ≤ . The inequality (3.23) then follows from it and (3.22) for 2 R ≤ r ≤ 2l . Together with (3.16), (3.23) holds for all |x| ≤ 2l and x ∈ E. q.e.d. Let zi be a local maximum point and vi be the scaled solution in (2.13) such that (2.14) holds and Ui (y) be the solution of (3.15) with K0 = Ki (zi ). In the next step, we are going to estimate the difference between vi and Ui (y). By (2.14), for any ε > 0, we have a sequence of Ri → +∞ such that |vi (y) − Ui (y)| ≤ εUi (y) for |y| ≤ Ri . By Lemma 3.1, there exists δ0 = δ0 (ε) > 0 such that (3.27). min vi (y) ≤ (1 + 2ε)Ui (r). |y|=r. for 0 ≤ r ≤ L∗i (δ0 ). Then Lemma 3.4 yields the following important result..

(32) prescribing scalar curvature on S n. 29. Lemma 3.5. Let vi and Ui be described as above. Suppose that there is a sequence of positive number li ≤ L∗i (δ0 ) such that (3.28). vi (y) ≤ c1 for |y| ≤ li .. Then there exists a small d > 0 such that (3.29). vi (y) ≤ c2 Ui (y), and. (3.30). |vi (y) − Ui (y)| ≤ c2 ri−n+2. for |y| ≤ ri = dli where d is a constant depending on n only. Further2 i (y) = Ki (zi ) − Ki (zi + M − n−2 y). Then for r ≤ ri , more, let Q i . n+2 n−2 (3.31) (y)ψ0 (y)dy ≤ c1 r−n+2 , Q(y)U i |y|≤r. and. . . (3.32). |y|≤r. n+2 n−2 (y)ψ (y)dy Q(y)U ≤ c1 r−n+1 j i. for 1 ≤ j ≤ n, where ψj (x) are given in Lemma 3.2. Proof. Without loss of generality, we might assume Ri << li . Otherwise, (3.29)–(3.30) hold automatically. By Lemma 3.1, (3.27) holds ˆ we have for 0 ≤ r ≤ li . Since Ki = n(n − 2) + ti K,. n+2 i (x) − Ki (zi )|U n−2 (x)dx ≤ c ti ≤ ε, |K |x|≤Ri. 2 i (x) = Ki zi + M − n−2 x . Thus, vi satisfies asfor ti small, where K i sumptions (3.16) ∼ (3.19) with an empty set E, R = Ri , l = dli and 2. l0 = Min−2 . Let d be small such that c1 (Ri−2 + ε + dn−2 ) < c2 , where c2 is the constant in (3.21). Then by (3.28), we have vi (y) ≤ c2 (Ri−2 + ε + dn−2 )−1 for |y| ≤ li . Then (3.29) follows immediately from Lemma 3.4. The inequality (3.30) can be proved by the same argument as in Lemma 3.3 of [9]. Hence, we omit the proof here..

(33) chiun-chuan chen & chang-shou lin. 30. To Prove (3.31) and (3.32), we let wi = vi (y) − Ui (y). Then wi satisfies (3.33) where. (3.34). n+2. i (y)U n−2 (y), ∆wi + bi (y)wi (y) = Q i    n+2 n+2  n−2 n−2  − Ui   bi (y) = K i (y)  vi ,    vi − Ui  2   i (y) = Ki zi + M − n−2 y , and K  i     Q i (y) = Ki (zi ) − K i (y).. Multiplying (3.33) by ψj , one has. . ∂ψj ∂wi − wi dσ ψj wi (∆ψj + bi ψj )dy + ∂ν ∂ν |y|≤r |y|=r. (3.35) n+2 i U n−2 ψj dy = Q i |y|≤r. for 0 ≤ j ≤ n. Let ri = dli . By (3.30), we have for |y| ≤ ri , |vi (y) − Ui (y)| ≤ c2 ri2−n .. (3.36). To estimate the first term of (3.35), we recall ∆ψj +. 4 n+2 Ki (zi )Uin−2 ψj = 0, n−2. and then n+2 n+2 i (y) − Ki (zi )) v n−2 − U n−2 ψj wi (∆ψj + bi ψj ) =(K i i n+2 n+2 4 n + 2 n−2 Ui wi ψj . + Ki (zi ) vin−2 − Uin−2 − n−2 Hence for j = 0, we have as in (3.10) |wi (∆ψ0 + bi ψ0 )| % & ˆ 2(2−n) ≤ c ri2−n Li (zi )−n+2 (1 + |y|)β−n−2 + ri (1 + |y|)−4 (3.37) 2(2−n). ≤ 2c ri. (1 + |y|)−2 ,.

(34) prescribing scalar curvature on S n. 31. where |ψ0 (y)| ≤ c(1 + |y|)2−n and βˆ < n are used. Similarly, by |ψj (y)| ≤ c(1 + |y|)1−n for 1 ≤ j ≤ n, we have 2(2−n). |wi (∆ψj + bi ψj )| ≤ c ri. (3.38). (1 + |y|)−3 .. By applying (3.37) and (3.38), we have . w (∆ψ + b ψ )dy = O(r2−n ) i j i j Br. for j = 0, and. . . Br. wi (∆ψj + bi ψj )dy = O(r1−n ). for 1 ≤ j ≤ n. When |y| = r, we have | vi (y)| ≤ c|y|−1 vi (y) = O(|y|1−n ) by the gradient estimate. Therefore, the boundary term of (3.35) is bounded by O(r2−n ) for j = 0 and is bounded by O(r1−n ) for 1 ≤ j ≤ n. Both (3.31) and (3.32) then follow from (3.35). q.e.d. Proof of Theorem 2.2. We prove Theorem 2.2 by contradiction. ˆ i |xi | n−2 2 Suppose lim M = +∞. If β ≥ n − 2, by the definition (2.15) i→+∞. 2 − n−2. ˆ and the assumption that Li (xi )M i (3.39). is bounded, we have. 1 2 ˆ n−2 |xi |1−β n−2 . Li (xi ) = t−1 M i i. If 1 ≤ β < n − 2, then 1−β 2β 2 2 ˆ n−2 |xi | ˆ n−2 |xi |1−β = t−1 M ˆ n−2 M M t−1 i i i i i 2β. ˆ n−2 , ≤ t−1 i Mi which implies (3.39) also. Let vi (y) be defined as (2.13) with zi = xi . Obviously, vi (y) ≤ 1 for 2 ˆ n−2 . By Lemma 3.5, there exists a δ2 > 0 such that (3.29)– |y| ≤ M i i in (3.32) hold with dli replaced by δ2 Li (xi ). Recall the quantity Q.

(35) chiun-chuan chen & chang-shou lin. 32. Lemma 3.5. We may assume limi→+∞ Then. ˆ i)

(36) K(x ˆ i )| |

(37) K(x. = e1 = (1, 0, . . . , 0).. −2 ˆ n−2 y − Ki (xi ) i = Ki xi + M −Q i −2. (3.40). 2 − n−2. ˆ i ), y) + c(δ, i)ti M ˆ n−2 ( K(x = ti M i i −2 n−2. ˆ = ti M i. ˆ i )||y| | K(x. 2 − n−2. ˆ i )|y1 + c(δ, i)ti M | K(x i. ˆ i )||y| | K(x. 2. ˆ n−2 |xi |, where c(δ, i) could be arbitrarily small if i is large for |y| ≤ δ M i and δ is small. Therefore, we can choose δ small enough so that. n+2 2 i )U n−2 (y)ψ1 (y)dy ≥ c ti M ˆ − n−2 |xi |β−1 (−Q i i |y|≤ri (3.41) = c (Li (xi ))2−n 2. ˆ n−2 |xi |. For the simplicity of notations, for some c > 0 where ri = δ M i we let li = δ2 Li (xi ). If ri ≥ li , then by (3.41), we have. (3.42). n+2. |y|≤li. i )U n−2 (y)|ψ1 (y)|dy ≥ c1 (Li (xi ))2−n . (−Q i. If li ≥ ri , as in (3.10), we have. n+2. ri ≤|y|≤li. ≤c. i |U n−2 (y)ψ1 (y)dy |Q i. ri ≤|y|≤li. . ˆ. 2β − n−2. ˆ Li (xi )−n+2 |y|−2n + ti M i. |y|−n−1 dy. = o(1)Li (xi )2−n . Together with (3.41), it implies that (3.42) holds also in the case of l i ≥ ri . On the other hand, by (3.32), we have . n+2 i U n−2 ψ1 dy ≤ c1 Li (xi )−n+1 . Q i |y|≤li 2. ˆ n−2 |xi | is bounded. This contradicts (3.42). Hence we conclude M i.

(38) prescribing scalar curvature on S n. 33. 2. ˆ n−2 |xi | is bounded, we have Suppose β ≥ n − 2. Since M i 2. ˆ n−2 |xi |1−β = M i. . 1−β 2β 2 n−2 ˆ ˆ n−2 Mi |xi | M i. ˆ i2 . ≥ c1 M Hence,. ˆ −2 ≥ c1 lim t−1 = +∞, lim Ln−2 (xi )M i i i. i→+∞. i→+∞. which yields a contradiction to our assumptions. Thus, β < n − 2 must hold. To prove (2.16), we let wi (y) = lin−2 (vi (y) − Ui (y)) where li = i (y) replaced ln−2 Q i in the δ2 Li (xi ). Then wi satisfies (3.33) with Q i right hand side. By (K0), 2 ˆ n−2 y i (y) = Ki (xi ) − Ki xi + M Q i ' ( 2 2 − n−2 − n−2 ˆ ˆ = −ti Q xi + M y + R xi + M y i i (3.43). + (Ki (xi ) − Ki (0)) * 2β ) ˆ − n−2 Q(ξi + y) + o(1)(|y|β + 1) = −ti M i + (Ki (xi ) − Ki (0)), 2. ˆ n−2 xi . By (3.30) of Lemma 3.5, wi (y) is uniformly where ξi = M i n bounded in R . After passing to a subsequence, we may assume that 2 ˆ n−2 |xi | is wi (y) converges to w(y) in C 2 (Rn ). Since β < n − 2 and M loc. bounded, we have. L−n+2 i. 2β. ˆ − n−2 . We may assume ∼ ti M i. i. −2β. ˆ n−2 > 0 c = lim ti lin−2 M i i→∞. i exists. Multiplying both sides of (3.33) by ψj = ∂U ∂yj , we have by integration by parts,. n+2 n−2 n−2 li Qi (y)Ui ψj (y)dy = wi (∆ψj + bi (y)ψj )dy Bli Bli. ∂w ∂ψj i − wi dσ. ψj + ∂ν ∂ν ∂Bl i.

(39) 34. chiun-chuan chen & chang-shou lin. By (3.30), the boundary term = O(li−1 ) → 0 as i → +∞, and 4 |∆ψj + bi (y)ψj | ≤ |bi (y)ψj (y)| + (n + 2)nU1n−2 (y)ψj (y) ≤ c(1 + |y|)−(n+2) . Thus, by Lebseque’s convergence theorem, the right hand side converges to. 4 n−2 w ∆ψj + n(n + 2)U1 ψj dy = 0. Rn. Together with (3.43), it implies. n+2 i (y)U n−2 ψj (y)dy lin−2 Q 0 = lim i i→+∞ B. li n+2 ∂U1 (y) Q(ξi + y)U1n−2 (y) dy = c lim i→+∞ B ∂yj li. 2n ∂ (n − 2)c Q(ξ + y) U1n−2 (y)dy = 2n ∂yj R n 2n −(n − 2)c ∂ = Q(ξ + y)U1n−2 (y)dy, 2n Rn ∂yj where U1 is defined in (1.4). Here, we have used the fact that ψj (y) is odd in yj , and. n+2 (Ki (xi ) − Ki (0))ψj (y)Uin−2 (y)dy = 0. Bli. The proof of Theorem 2.2 is complete.. q.e.d.. Proof of Theorem 2.1. Note that in Section 8, (2.16) is also proved when β < n + 1. This holds only for global solutions. See Lemma 8.1. ˆ i be the maximum point and the maximum of ui defined Let xi and M in (2.12). We first prove the “if” part. Assume there is a constant c > 0 such that (3.44). 2. ˆ n−2 . Li (xi ) ≥ cM i. Let vi (y) be the scaled solution defined in (2.13) with zi = xi . Obviously, 2 ˆ n−2 . By Lemma 3.1, Lemma 3.5 and (3.44), there vi (y) ≤ 1 for |y| ≤ M i exists a small positive number δ > 0 such that vi (y) ≤ c U1 (y) for 2 ˆ n−2 and for some c > 0. Therefore, 0 is a simple blow-up |y| ≤ δ M i point..

(40) prescribing scalar curvature on S n. 35. To prove the “only if” part, we assume 2. ˆ − n−2 = 0. lim Li (xi )M i. (3.45). i→+∞. Suppose that 0 is a simple blowup point. Then there exists positive constants c and δ0 < 1 such that vi (y) ≤ c U1 (y). (3.46) 2. ˆ n−2 . Following the notations of Lemma 3.5, we let wi (y) = for |y| ≤ δ0 M i vi (y) − Ui (y) and ψ0 (y) = n−2 2 Ui (y) + y · Ui (y). By the gradient estimate, we have by (3.46), | vi (y)| = O(|y|−n+1 ) for |y| ≥ 1. Thus,. ∂w ∂ψ0 i ˆ −2 ), − wi dσ = O(ˆ ri−n+2 ) = O(M (3.47) ψ0 i ∂ν ∂ν |x|=ˆ ri 2. ˆ n−2 . To estimate the first term of (3.35), we have by where rî = δ0 M i Lemma 3.5. wi (∆ψ0 + bi ψ0 )dy = wi (∆ψ0 + bi ψ0 )dy Brî. Bri. +. Brî \Bri. wi (∆ψ0 + bi ψ0 )dy,. where ri = δ0 Li (xi ). By Theorem 2.2, we have 1 ≤ β < n − 2. Similar to (3.37), we have by the fact β < n − 2 that 2(n−2) |wi (∆ψ0 + bi ψ0 )| ≤ c ri (1 + |y|)−4. for 1 ≤ r ≤ ri . Hence . wi (∆ψ0 + bi ψ0 )dy = O(ri−n+1 ). Br i. We note that Lemma 3.5 is crucial in the estimate above. By applying |vi (y)| + |Ui (y)| ≤ c|y|−n+2 and |ψ0 (y)| ≤ c|y|−n+2 for ri ≤ |y| ≤ rî , . w (∆ψ + b ψ )dy = O(ri−n+1 ). i 0 i 0 Brˆ \Br i. i.

(41) 36. chiun-chuan chen & chang-shou lin. Together with these two estimates, we have . (3.48) wi (∆ψ0 + bi ψ0 )dy = O(ri−n+1 ). Brˆ i. 2. ˆ n−2 xi is bounded. We may assume ξ = By Theorem 2.2, ξi = M i lim ξi . Then ξ satisfies. i→+∞. (3.49). Rn. 2n. Q(y + ξ)U1n−2 (y)dy = 0.. Also, the right hand side of (3.35) converges to 2β n−2. (3.50). ˆ lim t−1 M i i→+∞ i. #. Brî. =−. n+2 n−2. i U Q i. $ ψ0 (y)dy. n+2. Rn. Q(y + ξ)U1n−2 (y)ψ0 (y)dy.. Recall that ψ0 (y) = n−2 2 U1 (y) + y U1 (y). From integration by parts, (3.49) and y · Q(y) = βQ(y), we have. n+2 − Q(y + ξ)U1n−2 (y)ψ0 (y)dy Rn. 2n n−2 y · Q(y + ξ)U1n−2 (y)dy = (3.51) 2n Rn. 2n β(n − 2) Q(y + ξ)U1n−2 (y)dy = 0. = 2n Rn The last term does not vanish due to (K1). 2β. ˆ n−2 . Putting (3.50), (3.51) and (3.48) toRecall Li (xi )n−2 ∼ t−1 i Mi gether these estimates, we have . n+2 n−2 2−n ≤ c Li (xi ) Qi Ui ψ0 (y)dy Brˆ i. ˆ −2 ), ≤ c (Li (xi )1−n + M i which yields a contradiction to (3.45). Therefore, the proof of Theorem 2.1 is complete. q.e.d..

(42) prescribing scalar curvature on S n. 37. 4. The method of localizing blow-up points In this section, we will employ the method of localization of blow-up points to prove Theorem 2.4 and Theorem 2.5. This technique was due to R. Schoen. In the previous work [9], we have used this method to prove the isolatedness of blow-up points. For other applications of this method, see [17], [18]. We begin with the following lemma. Lemma 4.1. Let δ, σ and ε be small positive numbers and R > 1. Then there exist positive constants R = R(δ, σ) and C0 = C0 (δ, σ, R, ε) independent of i such that the following statements hold: n−2. (i) If ui (y0 )|y0 | 2 ≥ C0 , then there exists a local maximum point z ∈ B(y0 , 2δ|y0 |) of ui such that (4.1). ui (y0 ) ≤ ui (z). and the rescaled function 2. vi (y) = ui (z)−1 ui (ui (z)− n−2 y + z) satisfies +. the origin 0 is the only local maximum point of vi in B(0, 4R), and |vi − U1 |C 2 (B(0,4R)) ≤ σ(4R)2−n .. (4.2). i (ii) Let {zji }sj=1 denote all local maximum points of ui in the ball B 1 n−2. which satisfy ui (zji )|zji | 2 ≥ C0 and (4.2) with z = zji . Assume ui (z1i ) ≥ ui (z2i ) · · · ≥ ui (zsi i ). Then (a) ui (y) ≤ 2C0 |y|− Furthermore,. n−2 2. for y ∈ Ωi where Ωi = ∪j B(zji , 2δ|zji |). 2. |zji − zki | ≥ 4Rui (zji )− n−2 for j = k. (b) ui (x) ≤ 2ui (zji ) holds for x ∈ B(zji , 2δ|zji |) and (4.3). |zji | ≤ ε|zki | for j < k ≤ si ..

(43) chiun-chuan chen & chang-shou lin. 38. Lemma 4.1 can be proved by the blow-up method of Schoen and the method of moving planes, Lemma 3.1. See Lemma 4.1–Lemma 4.4 in [9]. In fact, we can prove more in Lemma 4.2 below. In the following, zji is indexed by the ordering ui (z1i ) ≥ . . . ≥ ui (zsi i ). i be the local maximum points in Lemma 4.1 Lemma 4.2. Let {zji }sj=1 and δ > 0 be a small number. Then we have the following statements if the positive constant C0 in Lemma 4.1 is large enough.. (i) The inequality n−1. 2. Li (zji ) ≥ (δui (zji ) n−2 |zji |) n−2 holds for 1 ≤ j ≤ si . (ii) Let. 2. L∗i (zji ) = min(Li (zji ), ui (zji ) n−2 ) and. 2. Dji = {y : |y − zji | ≤ cL∗i (zji )ui (zji )− n−2 }. with c small. Then. / Dji zki ∈. when k > j. Proof. We follow notations in Section 3. Let vi be defined in (2.13) with zi = zji and Ui be the solution to (3.15) with K0 = Ki (zi ). We may assume C0 is very large. If 1 < β < n, by (2.15) and n−2 ui (zji )|zji | 2 ≥ C0 , we have 2. 2β. 2. 2β. ui (zji ) n−2 |zji |1−β = (ui (zji ) n−2 |zji |)1−β ui (zji ) n−2 < ui (zji ) n−2 and (4.4). 2. 1. i n−2 i 1−β n−2 |zj | ) . Li (zji ) = (t−1 i ui (zj ) 2. 1. i n−2 |z i |1−β ) n−2 , then by (2.15), If β ≥ n and Li (zji ) = (t−1 j i ui (zj ). ˆ 1 2β n−1 2 i n−2 n−2 Li (zji ) = t−1 u (z ) ≥ (ui (zji ) n−2 |zji |) n−2 i j i for large i since βˆ > n − 1, that is, (i) holds in this case. Hence in 2 1 i n−2 |z i |1−β ) n−2 order to prove (i), we may assume Li (zji ) = (t−1 j i ui (zj ) and 1 < β < n..

(44) prescribing scalar curvature on S n. 39. Let δ be small enough, Mi = ui (zji ), Li = Li (zji ) and 2. ri = δ min(Li , Min−2 |zji |). . Then by (b) of part (ii) in Lemma 4.1, vi (y) ≤ 2 for |y| ≤ ri . By Lemma 3.1, Lemma 3.4 and Lemma 3.5, we have vi (x) ≤ c Ui (x) and. |vi (x) − Ui (x)| ≤ c ri−2+n. for |x| ≤ ri , where c is a constant independent of δ and i. For the sake of simplicity, δ always denotes a small positive number, but could change ˆ i ) is in the direction e1 = (1, 0, · · · , 0). from line to line. Assume ∇K(z j Let ψ1 = (4.5). ∂Ui . By (3.32) of Lemma 3.5, ∂y1 . n+2 i U n−2 ψ1 dx ≤ c1 r−n+1 . Q i i |x|≤ri . By (3.40), we have 2 − n−2. i (y) = ti M −Q i. ˆ i )|(y1 + o(1)|y|) | K(z j. for |y| ≤ ri , where o(1) could be arbitrarily small if δ is small. Since ψ1 (y)y1 ≥ 0, we have . n+2 − 2 n−2 Qi Ui ψ1 dy ≥ c2 ti Mi n−2 |zji |β−1 (4.6) Br i. 2. 1. i n−2 |z i |1−β ) n−2 , it for some c2 > 0. Since we assume Li (zji ) = (t−1 j i ui (zj ) follows from (4.5) and (4.6) that. (4.7). L−n+2 ≤ c3 ri−n+1 . i. Since ri ≤ Li , C0 is large and Li → +∞ as i → +∞, we conclude ri /Li 2 is small from (4.7). Thus ri = δui (zji ) n−2 |zji |. Since both c1 and c2 are.

(45) 40. chiun-chuan chen & chang-shou lin. independent of δ and i, part (i) of Lemma 4.2 follows from (4.7) if δ is chosen to be small enough. We prove (ii) by contradiction. Assume that after passing to a sen−2 quence, there exists ji < ki such that zki i ∈ Djii and both ui (zjii )|zjii | 2 n−2. and ui (zki i )|zki i | 2 tend to +∞. For simplicity of notations, we let zi = zjii and wi = zki i . Recall that ui satisfies ui (wi ) ≤ ui (zi ).. (4.8). Let Mi = ui (zi ) and vi (y) be the solution in (2.13) scaled with respect 2. to the local maximum point zi . Since Min−2 |zi | → +∞ and (4.2) holds, we have for any σ > 0, by Lemma 3.1 min vi (y) ≤ (1 + 2σ)U1 (r). (4.9). |y|≤r. if i is large and 0 ≤ r ≤ 4d0 L∗i (zi ) with some d0 = d0 (σ) > 0. Let 2. li = d0 L∗i (zi ). Applying Lemma 3.5 with an empty set E, l0 = Min−2 and l = li , there is a constant c1 independent of σ and i. n+2 vin−2 (y)dy ≤ c1 σ, (4.10) R≤|y|≤li. 1. 1. provided that d0 < σ 2 and R ≥ σ − 2 . Set 2 Bi = {x | |x − wi | ≤ ui (wi )− n−2 } and. 2. î = {y | M − n−2 (y + zi ) ∈ Bi }. B i. By (ii) of Lemma 4.1 and (4.8), 2. 2. 4R ≤ ui (wi ) n−2 |zi − wi | ≤ Min−2 |zi − wi | ≤ cL∗i (zi ) because wi ∈ Di . By (ii) of Lemma 4.1, we have |zi | = o(1)|wi | and 2. 2. 2. Min−2 ui (wi )− n−2 << Min−2 |wi | 2. = (1 + o(1))Min−2 |zi − wi | ≤ cL∗i (zi ). Thus, Bi ⊆ 2Di . Since ui (x) ≤ ui (wi ) ≤ ui (zi ) for x ∈ Bi , we have vi (y) ≤ 1 for î . Since by Lemma 4.1, 0 is the unique local maximum of vi (y) y ∈B.

(46) prescribing scalar curvature on S n. 41. î ⊆ {y | R ≤ |y| ≤ li } if the constant c in Di is for |y| ≤ 4R, we have B j small. Again by (i) of Lemma 4.1, we have for some constant c2 > 0,. n+2 2n 2n n−2 n−2 0 < c2 ≤ ui (x)dx = vi dy ≤ vin−2 (y)dy î î B B. Bi n+2 n−2 ≤ vi (y)dy ≤ c1 σ, R≤|y|≤li. which yields a contradiction if σ is small enough. Therefore, (ii) is proved. q.e.d. ˆ i = ui (xi ). Suppose Proof of Theorem 2.4. Let Li = Li (xi ) and M 2. ˆ − n−2 → +∞, then by Theorem 2.1, 0 is a simple blowup point that Li M i and ui loses the energy of one bubble at 0. Therefore, we suppose that 2 ˆ − n−2 < +∞. lim Li M i→+∞. i. 2. ˆ n−2 |xi | n−2 2 is bounded, β < n − 2 and ξ = By Theorem 2.2, M i 2 n−2 ˆ lim M xi satisfies (2.16). From the definition (2.15) of Li , we have i i→+∞ 2β 1 ˆ n−2 n−2 . Applying Lemma 3.1 and Lemma 3.5, ui Li (xi ) ∼ t−1 M i i satisfies (4.11) for. ˆ −1 |x|2−n ≤ ui (x) ≤ c2 M ˆ −1 |x|2−n c1 M i i 2. 2β. ˆ − n−2 ≤ |x| ≤ δ(t−1 M ˆ n−2 −2 ) n−2 M i i i 1 2β ˆ n−2 −2 n−2 . Then, we have with a small δ > 0. Let ri = δ t−1 M i i (4.12) Now suppose. 1. 2β. ˆ 1− n−2 . min ui (x) ∼ ti M i. |x|=ri. n−2 = +∞. lim sup ui (x)|x| 2 i→+∞ B2. Let zi = z1i , where z1i is the local maximum point in Lemma 4.2. Let 2. Mi = ui (zi ). Since Min−2 |zi | ≥ C0 is very large, we have (4.13). 2β 1 n−2 n−2 Li (zi ) ≤ t−1 M , i i.

(47) chiun-chuan chen & chang-shou lin. 42. and by (i) of Lemma 4.2, 2. (4.14). Min−2 |zi | << Li (zi ).. Since ui (x) is a positive superharmonic function, there exists a small constant c > 0 such that ui (zi + x) ≥ c Mi−1 (|x|2−n − (3/2)2−n ). (4.15) 2 − n−2. for Mi (4.16). ≤ |x| ≤ 32 . In particular, we have 2β 1− n−2. min. |x−zi |≤min(ˆ ri ,1). ui (x) ≥ cMi−1 rî2−n ≥ cti Mi. ,. 1 2β −2 n−2 n−2 M . Since ui (x) has the only maximum point where rî = t−1 i i xi in the region {x | |x| ≤ ri }, we have by (4.14) ri ≤ |zi | << rî , namely, the ball Bri (0) is contained inside of the ball B(zi , rî ). Hence, if rî is bounded, by (4.12), (4.16) and the maximum principle, we have 2β 1− n−2. ˆ ti M i. ∼ min ui ≥ |x|=ri. ≥. min ui |x−zi |≤ˆ ri 2β 1− n−2 cti Mi .. n−2 ˆ First we consider the case when β > n−2 2 . Since β > 2 and Mi is the ˆ maximum of ui , it implies Mi ∼ Mi . Hence, the function vi (y) rescaled with respect to the center zi satisfies. vi (y) ≤ c 2. for some constant c > 0 and |y| ≤ Min−2 . Thus, vi (y) ∼ U1 (y) for |y| ≤ δLi (zi ) by Lemma 3.4. Particularly, we have 2. −2 ˆ i ∼ Mi U1 (|zi |M n−2 ) = Mi (Mi |zi | n−2 2 ) = o(1)Mi , M i. which obviously yields a contradiction..