Yamabe type equations on three dimensional Riemannian manifolds

YanYan Li^{∗} Department of Mathematics Rutgers University New Brunswick, NJ 08903

Meijun Zhu Department of Mathematics University of British Columbia Vancouver, B.C. Canada, V6T 1Z2

Abstract

A theorem of Escobar and Schoen asserts that on a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere, a necessary and sufficient condition for a C^{2} function K to be the scalar curvature function of some conformal metric is that K is positive somewhere. We show that for any positive C^{2} function K, all such metrics stay in a compact set with respect to C^{3} norms and the total Leray-Schauder degree of all solutions is equal to −1. Such existence and compactness results no longer hold in such generality in higher dimensions or on manifolds conformally equivalent to standard three dimensional spheres. The results are also established for more general Yamabe type equations on three dimensional manifolds

∗Partially supported by the Alfred P. Sloan Foundation Research Fellowship, NSF grant DMS-9401815, and a Rutgers University Research Council grant.

0 Introduction

Let (M, g) be an n dimensional compact, smooth, Riemannian manifold. For n = 2, we know from the Uniformization Theorem of Poincaré that there exist constant Gauss curvature metrics which are pointwise conformal to g. Through the work of Yamabe [57], Trudinger [55], Aubin [2], and Schoen [50], the following Yamabe conjecture has been proved: For n ≥ 3, there exist constant scalar curvature metrics which are pointwise conformal to g. The Yamabe conjecture can be viewed as a generalization of the Uniformization Theorem to higher dimensions. See Lee and Parker [41] for a survey of the Yamabe conjecture. See also Bahri and Brezis [8] and Bahri [6] for works on the problem and its variants.

More recently, Schoen obtained compactness results for the Yamabe problem in [51] where he proved that when (M, g) is locally conformally flat but not conformally equivalent to the standard spheres, all solutions to the Yamabe problem stay in a compact set with respect to C^{3} norms and the total Leray-Schauder degree of all solutions is equal to −1. In the same paper, he also announced, with indication of the proof, the same result for general manifolds. See [52] for more details.

The heart of the proof of the above results of Schoen is some fine analysis of possible blowup behavior of solutions to the Yamabe problem. More specifically Schoen obtained in [51] energy independent estimates of solutions to

−L_{g}u = n(n − 2)u^{p},  u > 0,  in M,

where 1 < 1 + ϵ_{0} ≤ p ≤ (n + 2)/(n − 2), (M, g) is n dimensional (n ≥ 3) compact locally conformally flat Riemannian manifolds, L_{g} = Δ_{g} − \frac{n−2}{4(n−1)}R_{g} is the conformal Laplacian operator, and R_{g} is the scalar curvature function of g. For n = 3, Schoen and Zhang established in [53] such energy independent estimates for solutions to

−L_{g}u = Ku^{p},  u > 0,  in M,

where 1 < 1 + ϵ_{0} ≤ p ≤ 5, K > 0 is a positive function in C^{2}(M), and M is locally conformally flat.

Along the approach initiated by Schoen, the first author extended in [42]- [43] the above mentioned results of Schoen and Zhang to dimension n = 4, as well as to dimension n ≥ 5 under suitable (n − 2)−flatness hypothesis on K near critical points of K. He also established in [44] such energy independent estimates on three dimensional locally conformally flat Riemannian manifolds M with umbilic boundary ∂M for solutions to

\{ \begin{matrix}−L_{g}u = Ku^{p},  u > 0,  in M^{∘}, \\ B_{g}u = 0,  on ∂M, \end{matrix}

where 1 < 1+ϵ_{0} ≤ p ≤ 5, B_{g} = \frac{∂}{∂ν} + \frac{n−2}{2}h_{g}, h_{g} is the mean curvature function of ∂M, ν is the unit outer normal of ∂M, and K > 0 is some positive function in C^{2}(M). We recall that ∂M is called umbilic if at each point of ∂M the second fundamental form is a constant multiple of the metric g.

See Brezis, Li and Shafrir [15], Chang, Gursky and Yang [19], Chen and Lin [23]-[24], and Han and Li [35] for related results.

Instead of looking for conformal metrics with scalar curvature being a given constant as in the Yamabe problem, one may also look for conformal metrics with scalar curvature being a given function K. When dimension n ≥ 3, this is, up to some harmless constant, equivalent to solving the following semilinear elliptic equation with critical exponent:

−Δ_{g}u + \frac{n − 2}{4(n − 1)}R_{g}u = Ku^{\frac{n+2}{n−2}} , u > 0,  in M.  (1)

Extensive studies have been made on this prescribed scalar curvature problem following the pioneer work of Moser [46], see, for example, [4], [5], [7], [9], [10], [12]-[14], [18]-[29], [33], [34], [36], [38]-[40], [42], [43], [47], [52], [53], [56], and the references therein.

In this paper we give some compactness results on this prescribed scalar curvature problem on three dimensional manifolds. For n = 3, equation (1) takes the form

−Δ_{g}u + \frac{1}{8}R_{g}u = Ku^{5}, u > 0, on M.  (2)

In fact, we also consider, for 1 < p ≤ 5,

−Δ_{g}u + \frac{1}{8}R_{g}u = Ku^{p}, u > 0, on M,  (3)

which includes subcritical exponent p as well.

Definition 0.1 . A smooth compact Riemannian manifold (M, g) is said to be positive if the first eigenvalue of −L_{g} ≡ −Δ_{g} + \frac{n−2}{4(n−1)}R_{g} is positive.

It is well known that the Yamabe problem as well as the prescribed scalar curvature problem is much more difficult on positive manifolds. We define on a positive manifold a compact operator T : C^{2,α}(M)^{+} → C^{2,α}(M) by

T u = (−Δ_{g} + \frac{1}{8}R_{g})^{−1}(Ku^{5}),

where 0 < α < 1 and C^{2,α}(M)^{+} denotes the set of positive functions in C^{2,α}(M). If we can prove that all solutions of (2) stay in a bounded set of C^{2,α}(M)^{+}, then for Λ large enough 0 does not belong to (I − T)(∂D_{Λ}) where

D_{Λ} = {v ∈ C^{2,α}(M) : ∥v∥_{C^{2,α}(M)} < Λ, min_{M} v > 1/Λ}.

Consequently, the Leray-Schauder degree deg((I − T), D_{Λ}, 0) is well defined. Moreover deg((I−T), D_{Λ}, 0) ≠ 0 yields the existence of at least one solution to (2). See, for example, Nirenberg [48] for the definition of the Leray-Schauder degree and its various properties.

A theorem of Escobar and Schoen [29] asserts that on a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere, every positive C^{2} function K on M can be realized as the scalar curvature function of some metric conformal to g. Our first theorem says that for any positive C^{2} function K, all such conformal metrics stay bounded with respect to strong norms (say, C^{3}) and the Leray-Schauder degree deg((I − T), D_{Λ}, 0) = −1 for large Λ.

Theorem 0.1 Let (M, g) be a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere. Then for any 1 < p ≤ 5 and positive function K ∈ C^{2}(M), there exists some constant C depending only on M, g, ∥K∥_{C^{2}(M)}, and the positive lower bound of K and p − 1 such that

1/C ≤ u ≤ C and ∥u∥_{C^{3}(M)} ≤ C,

for all solution u of (3). Moreover deg((I − T), D_{Λ}, 0) = −1 for large Λ. Consequently, equation (2) has at least one solution.

We remark that the hypothesis (M, g) is not conformally equivalent to the standard three dimensional sphere is necessary since (2) may not have a solution in this case due to Kazdan and Warner′s condition for the solvability. We also remark that in dimension greater than three, results in such generality no longer hold. This can be seen as follows. First it has been shown in corollary 0.24 of [43] that, for all n ≥ 4, there exist C^{2} converging functions K_{i} ≥ 1 on ℝP^{n}, the real projective space, such that the corresponding solutions u_{i} of (1) blows up in L^{∞} norm. Next it follows from the work of Bianchi [12] and Bianchi-Egnell [13] that, for all n ≥ 4, there exist positive C^{2} functions K on ℝP^{n} such that (1) is not solvable.

Equations of Yamabe type have been studied by Brezis and Nirenberg in [16] where the framework for a functional analysis and variational attack, beyond the minimization techniques used in more geometric methods, is set. Much work follows. In particular, Bahri and Brezis considered in [8] the following:

−Δ_{g}u + ku = u^{\frac{n+2}{n−2}} , u > 0, on M,  (4)

where k ∈ L^{∞}(M) and Δ_{g} is the Laplace-Beltrami operator. They proved, among other things, that for 3 ≤ n ≤ 5 equation (4) has at least one solution under the necessary hypothesis that the first eigenvalue of −Δ_{g} +k is positive. For k = \frac{1}{8}R_{g}, equation (4) is the Yamabe equation.

It is natural to ask whether or not one can carry out energy independent estimates to solutions of (4). This is what we address in the first six sections of this paper. In section 7 and section 8, we treat more general equations −Δ_{g}u + ku = Ku^{5} which include prescribed scalar curvature equations. In particular, Theorem 0.1 will be established in section 7.

Let (M, g) be a three dimensional smooth compact Riemannian manifold. As before we use Δ_{g} to denote the Laplace-Beltrami operator with respect to the metric g and R_{g} to denote the scalar curvature of g. Consider, for k ∈ C^{1}(M) and 1 < p ≤ 5, the following semilinear equation

−Δ_{g}u + ku = u^{p}, u > 0, on M.  (5)

We use 𝕄_{k,p} to denote the set of solutions of (5) in C^{2}(M). Our next theorem gives a priori estimates of solutions of (5) in H^{1}(M) norm.

Theorem 0.2 Let (M, g) be a three dimensional smooth compact Riemannian manifold. Then for all ϵ_{0} > 0,

∥u∥_{H^{1}(M)} ≤ C ∀ u ∈ ∪_{1+ϵ_{0}≤p≤5}𝕄_{k,p},

where C depends only on M, g, ϵ_{0}, and ∥k∥_{C^{1}(M)}.

Though the above theorem asserts a priori estimates in H^{1} norm to solutions of (5), estimates in L^{∞} norm does not hold in such generality as will be seen shortly. To give a priori estimates in strong norms under suitable hypothesis, we introduce a function on M, denoted as A(y) ≡ A_{k,g}(y), defined as follows.

Suppose that the first eigenvalue λ_{1} of −Δ_{g} + k is positive. For y ∈ M, let G_{y} denote the Green′s function of −Δ_{g} + k with the pole at y, namely,

(−Δ_{g} + k)G_{y} = δ_{y} in M.

The existence and uniqueness of G_{y} is well known, see for example [3]. Let x = (x^{1}, x^{2}, x^{3}) be some geodesic normal coordinate system centered at y, so the metric g is locally given by g_{ij}(x)dx^{i}dx^{j} with g_{ij}(0) = δ_{ij} and ∂_{x^{l}}g_{ij}(0) = 0 for all i, j and l. It follows from Lemma 9.2 in the Appendix that, for some real number A_{k,g}(y),

G_{y}(x) = \frac{1}{3ω_{3}|x|} + A_{k,g}(y) + O(|x|^{α}), for |x| close to 0,

for all 0 < α < 1, where ω_{3} is the volume of the unit ball in ℝ^{3}. It is not difficult to check that the value of A_{k,g}(y) is independent of the choice of geodesic normal coordinate systems. It is also clear that A_{k,g}(y) thus defined is a continuous function on M.

As pointed out earlier, a necessary condition for (5) to have a solution is that the first eigenvalue λ_{1} of −Δ_{g} + k is positive.

Theorem 0.3 Let (M, g) be a three dimensional smooth compact Riemannian manifold and k ∈ C^{1}(M) with positive λ_{1}. Assume that

min_{y∈M} A_{k,g}(y) > 0.

Then for all ϵ_{0} > 0,

1/C ≤ u ≤ C in M and ∥u∥_{C^{3}(M)} ≤ C,  ∀ u ∈ ∪_{1+ϵ_{0}≤p≤5}𝕄_{k,p}, (6)

where C is some positive constant depending only on M, g, ϵ_{0}, ∥k∥_{C^{1}(M)}, and the positive lower bound of min_{M} A_{k,g} and λ_{1}.

The following corollary follows from the above theorem and the positive mass theorem of Schoen and Yau [54].

Corollary 0.1 Let (M, g) be a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere. Then for all ϵ_{0} > 0 and all k ∈ C^{1}(M) satisfying k ≤ \frac{1}{8}R_{g} on M, we have

1/C ≤ u ≤ C  and ∥u∥_{C^{3}(M)} ≤ C,  ∀ u ∈ ∪_{1+ϵ_{0}≤p≤5}𝕄_{k,p},

where C is some positive constant depending only on M, g, ϵ_{0}, and ∥k∥_{C^{1}(M)}.

To prove Corollary 0.1, we can assume that the first eigenvalue λ_{1} of −Δ_{g} + k is positive since otherwise 𝕄_{k,p} = ϕ. Making a conformal change of the metric using a positive eigenfunction associated with λ_{1}, we can assume without loss of generality that k > 0 on M. Applying the maximum principle we see that the Green′s function of −Δ_{g} + k is greater than or equal to the Green′s function of −Δ_{g} + \frac{1}{8}R_{g}. It follows that A_{k,g} ≥ A_{\frac{1}{8} R_{g},g} > 0, while the last inequality follows from the positive mass theorem of Schoen and Yau. Corollary 0.1 then follows from Theorem 0.3.

Once we have the compactness results in Theorem 0.3, we can derive existence results and multiplicity results (counting multiplicity).

Consider the following minimization problem

min{E(v) : v ∈ H^{1}(M), ∫_{M} v^{6} = 1},  (7)

where E(v) = ∫_{M}(|∇_{g}v|^{2} + kv^{2}).

Theorem 0.4 Under the hypothesis of Theorem 0.3, (7) is attained at some positive function v ∈ C^{2}(M).

Theorem 0.4 can be derived from Theorem 0.3 as follows. Since λ_{1} > 0, \sqrt{E(v)} is an equivalent H^{1}(M) norm. For 2 < q < 6, the embedding from H^{1}(M) to L^{q}(M) is compact, so we can find some nonnegative minimizer v_{q} to min{E(v) : v ∈ H^{1}(M), ∫_{M} |v|^{q} = 1}. The Euler-Lagrange equation of v_{q} is

−Δ_{g}v_{q} + kv_{q} = E(v_{q})v_{q}^{q−1}, in M.

It is easy to see that E(v_{q}) is bounded away from zero and infinity. We derive from the strong maximum principle that v_{q} is positive in M. Applying Theorem 0.3 and some standard elliptic estimates to E(v_{q})^{1/(q−2)}v_{q}, we have, along a subsequence, that v_{q} → v in C^{2}(M) as q → 6. It is then easy to see that v is a positive minimizer of (7).

Let v be the positive minimizer obtained in Theorem 0.4, then u = E(v)^{1/4}v ∈ 𝕄_{k,5}. The existence of u in 𝕄_{k,5} under the necessary hypothesis λ_{1} > 0 has been established in Bahri and Brezis [8]. It is interesting to note that (7) has no minimizer if k is sufficiently large, though 𝕄_{k,5} ≠ ϕ according to the result in [8]. The nonexistence of minimizers for large k can be seen as follows. We first observe that

min\{E(v) : v ∈ H^{1}(M), ∫_{M} v^{6} = 1\} ≤ \frac{1}{S_{1}} ,

where

\frac{1}{S_{1}} = inf \{\frac{∫_{ℝ^{3}} |∇u|^{2}}{(∫_{ℝ^{3}} u^{6})^{1/3}} : u ∈ L^{6}(ℝ^{3}) \ \{0\}, ∇u ∈ L^{2}(ℝ^{3})\} .

It follows from a result of Hebey and Vaugon [37] that for some constant C = C(M, g),

∫_{M} |∇_{g}v|^{2} + C ∫_{M} v^{2} ≥ \frac{1}{S_{1}} , ∀ v ∈ H^{1}(M), ∫_{M} v^{6} = 1.

Therefore if k > C on M, (7) has no minimizer. In view of the above derivation of Theorem 0.4 from Theorem 0.3, we also know that the compactness result (6) no longer holds if k > C on M. The reason is that the v_{q} constructed before does not have any subsequence converging in H^{1}(M) due to the nonexistence of minimizer of (7).

To state the multiplicity result, we introduce, as in Schoen [51], an operator F : C^{2,α}(M)^{+} → C^{2,α}(M) by

F (v) = v − ( − Δ_{g} + k)^{−1}(E(v)v^{5}),

where 0 < α < 1, and E(v) = ∫_{M}(|∇_{g}v|^{2} + kv^{2}).

It is clear that the operator F is of the form I + compact and therefore the Leray-Schauder degree deg(F, D_{Λ}, 0) is well defined provided 0 does not belong to F (∂D_{Λ}).

Theorem 0.5 Under the hypothesis in Theorem 0.3, we have, for Λ large enough, that 0 does not belong to F (∂D_{Λ}) and

deg(F, D_{Λ}, 0) = −1.

Consequently, 𝕄_{k,5} ≠ ϕ.

Using Theorem 0.3, we can prove Theorem 0.5 in the same way as in Schoen [51]. We omit the details.

We will extend in section 7-8 Theorem 0.2-0.5 to include the following generalization of (5):

−Δ_{g}u + ku = Ku^{p}, u > 0, on M,  (8)

where 1 < p ≤ 5 and K ∈ C^{2}(M) is some positive function in M. In particular, Theorem 0.1 will be established in section 7.

The paper is organized as follows. In section 1, we give the definitions of isolated blowup points and isolated simple blowup points for a sequence of solutions. In section 2, we derive a Pohozaev type identity. In section 3, we give some properties of solutions near isolated and isolated simple blowup points. The main result in this section is Proposition 3.1 which gives sharp pointwise estimates to a sequence of solutions near isolated simple blowup points. In section 4, we prove that an isolated blowup point is in fact an isolated simple blowup point, which rules out the possibility of bubbles on top of bubbles. In section 5, we rule out the possibility of bubble accumulations. Theorem 0.2 is established by the end of section 5. The analysis in the first five sections is local in nature. In section 6, we study compactness of solutions and establish Theorem 0.3. Here, global properties are used. In section 7-8, we extend results in first six sections to cover more general equations (8) which include the prescribed scalar curvature equations. In particular, Theorem 0.1 follows from results in section 7. In the appendix, we provide, for readers′ convenience, some well known descriptions of singular behavior of positive solutions to linear elliptic equations in punctured balls.

1 Definitions and Notations

We start by introducing some notations and definitions. Let (M, g) be a three dimensional smooth compact Riemannian manifold. In local coordinates, the Laplace-Beltrami operator can be written as Δ_{g}u = g^{αβ}(∂_{αβ}u−Γ_{αβ}^{γ}∂_{γ}u) where Γ_{αβ}^{γ} denotes the Christopher symbol.

Let Ω ⊂ M be an open set, {k_{i}} be a sequence of functions converging to k in C^{1}(M), {p_{i}} be a sequence of numbers satisfying 2 ≤ p_{i} ≤ 5 and p_{i} → 5, and {u_{i}} be a sequence of C^{2} functions satisfying

−Δ_{g}u_{i} + k_{i}u_{i} = u_{i}^{p_{i}}, u_{i} > 0,  in Ω ⊂ M.  (9)

A point ȳ ∈ Ω is called a blowup point of {u_{i}} if u_{i}(y_{i}) → ∞ for some y_{i} → ȳ.

Following Schoen, we make the following definitions concerning blowup points of {u_{i}}.

Definition 1.1 . Let {u_{i}} satisfy (9). A point ȳ ∈ Ω is called an isolated blowup point of {u_{i}} if there exist 0 < r̄ < dist(ȳ, ∂Ω), C̃ > 0, and a sequence y_{i} tending to ȳ, such that, y_{i} is a local maximum of u_{i}, u_{i}(y_{i}) → +∞ and, for large i,

u_{i}(y) ≤ C̃d(y, y_{i})^{−2/(p_{i}−1)} ∀ d(y, y_{i}) < r̄,  (10)

where d(y, y_{i}) denotes the geodesic distance between y and y_{i}.

Let y_{i} → ȳ be an isolated blowup point of {u_{i}}, we set

ū_{i}(r) = \frac{1}{|∂B_{r}(y_{i})|} ∫_{∂B_{r}(y_{i})} u_{i}ds_{g}, 0 < r < r̄,

and

w̄_{i}(r) = r^{2/(p_{i}−1)}ū_{i}(r), 0 < r < r̄,

where B_{r}(y_{i}) denotes the geodesic ball of radius r centered at y_{i}, ds_{g} is the surface area element and |∂B_{r}(y_{i})| denotes the area of ∂B_{r}(y_{i}) with respect to g.

Definition 1.2 ȳ ∈ Ω is called an isolated simple blowup point, if ȳ is an isolated blowup point of {u_{i}}, such that, for some ρ ∈ (0, r̄) (independent of i),

w̄_{i} has precisely one critical point in (0, ρ)  (11)

for large i.

We use C, C_{0}, C_{1}, ⋯ to denote various constants in subsequent sections.

2 A Pohozaev type identity

In this section, we provide a Pohozaev type identity which plays important roles in subsequent estimates. Since we will only apply the Pohozaev type identity in small neighborhoods of some point, we write it in a local coordinate chart. Let Ω ⊂ M be some open set in one local coordinate chart x = (x^{1}, x^{2}, x^{3}) with g_{ij}(0) = δ_{ij} and Γ_{ij}^{k}(0) = 0. In this section we use notation Δ = ∑_{i=1}^{n} ∂_{ii}, ∇ = (∂_{1}, ∂_{2}, ∂_{3}), dv = dx^{1} ∧ dx^{2} ∧ dx^{3}, |x|^{2} = (x^{1})^{2} + (x^{2})^{2} + (x^{3})^{2}, B_{σ} = {(x^{1}, x^{2}, x^{3}) : |x| < σ}, and ds to denote the surface area element with respect to the flat metric.

Considering, for k ∈ C^{1}(Ω) and p > 0, the following equation

−Δ_{g}u + ku = u^{p}, u > 0, in Ω,  (12)

we have

Lemma 2.1 For any u ∈ C^{2}(Ω) satisfying (12) and B_{2σ} ⊂ Ω, we have

\begin{matrix}− ∫_{B_{σ}}(k(x) + \frac{x·∇k(x)}{2})u^{2}dv + (\frac{3}{p+1} − \frac{1}{2}) ∫_{B_{σ}} u^{p+1}dv − A(g, u) \\ = ∫_{∂B_{σ}} B(σ, x, u, ∇u)ds − ∫_{∂B_{σ}} σ(\frac{ku^{2}}{2} − \frac{u^{p+1}}{p+1} )ds, \end{matrix}  (13)

where

B(σ, x, u, ∇u) = σ(\frac{∂u}{∂ν})^{2} − \frac{σ}{2} |∇u|^{2} + \frac{1}{2}u\frac{∂u}{∂ν},

ν denotes the unit outer normal of ∂B_{σ} with respect to the flat metric, and

\begin{aligned}A(g, u) & = ∫_{B_{σ}}(x^{α}∂_{α}u)(g^{βγ} − δ^{βγ})∂_{βγ}udv − ∫_{B_{σ}}(x^{α}∂_{α}u)(g^{βγ}Γ_{βγ}^{μ})∂_{μ}udv \\ & +\frac{1}{2} ∫_{B_{σ}} u(g^{βγ} − δ^{βγ})∂_{βγ}udv − \frac{1}{2} ∫_{B_{σ}} u(g^{βγ}Γ_{βγ}^{μ})∂_{μ}udv. \end{aligned}

Proof. It is elementary to check (see e.g. [38]) that for any f ∈ C^{2}(\overline{B}_{σ}), we have

2Δu(∇u · ∇f) = div[2(∇u · ∇f)∇u − |∇u|^{2}∇f] + |∇u|^{2}Δf − 2∇u · ∇^{2}f · ∇u,

(14)

where ∇^{2}f = (\frac{∂^{2}f}{∂x^{α}∂x^{β}}), thus the last term is a quadratic form.

Let f = \frac{1}{2}|x|^{2}, then ∇f = x, Δf = 3, ∇u · ∇^{2}f · ∇u = |∇u|^{2}. Integrating both sides of (14) on B_{σ} , we have

∫_{B_{σ}} 2Δu(∇u · x) = ∫_{B_{σ}} div[2(∇u · x)∇u − |∇u|^{2}x] + ∫_{B_{σ}} |∇u|^{2}.

It is clear that

∫_{B_{σ}} div[2(∇u · x)∇u − |∇u|^{2}x] = σ ∫_{∂B_{σ}} (2|\frac{∂u}{∂ν}|^{2} − |∇u|^{2}) ,

and

∫_{B_{σ}} |∇u|^{2} = − ∫_{B_{σ}} Δuu + ∫_{∂B_{σ}} u\frac{∂u}{∂ν}.

Thus

∫_{B_{σ}} 2Δu∇u · x + ∫_{B_{σ}} Δuu = ∫_{∂B_{σ}}\{2|\frac{∂u}{∂ν}|^{2}σ − |∇u|^{2}σ + u\frac{∂u}{∂ν}\}.  (15)

Let g = g_{αβ}dx^{α}dx^{β} be the metric, we know

Δu = Δ_{g}u − (g^{αβ} − δ^{αβ})∂_{αβ}u + g^{αβ}Γ_{αβ}^{γ}∂_{γ}u.

Plugging the above into (15), we have

\begin{matrix}− ∫_{B_{σ}}(x^{α}∂_{α}u)Δ_{g}udv − \frac{1}{2} ∫_{B_{σ}} uΔ_{g}udv \\ + ∫_{B_{σ}}(x^{γ}∂_{γ}u)(g^{αβ} − δ^{αβ})∂_{αβ}udv − ∫_{B_{σ}}(x^{γ}∂_{γ}u)(g^{αβ}Γ_{αβ}^{μ})∂_{μ}udv \\ +\frac{1}{2} ∫_{B_{σ}} u(g^{αβ} − δ^{αβ})∂_{αβ}udv − \frac{1}{2} ∫_{B_{σ}} u(g^{αβ}Γ_{αβ}^{γ})∂_{γ}udv \\ = − ∫_{∂B_{σ}}[(\frac{∂u}{∂ν})^{2}σ − \frac{1}{2}|∇u|^{2}σ + \frac{1}{2}u\frac{∂u}{∂ν}]ds. \end{matrix}  (16)

Now, using the fact that u satisfies (12), we have

− ∫_{B_{σ}}(x^{α}∂_{α}u)Δ_{g}udv
= ∫_{B_{σ}}(x^{α}∂_{α}u)(u^{p} − ku)dv
= −\frac{3}{p+1} ∫_{B_{σ}} u^{p+1}dv + \frac{1}{p+1} ∫_{∂B_{σ}} u^{p+1}σds
+\frac{3}{2} ∫_{B_{σ}} ku^{2}dv + \frac{1}{2} ∫_{B_{σ}}(x · ∇k)u^{2}dv − \frac{1}{2} ∫_{∂B_{σ}} ku^{2}σds,

and

−\frac{1}{2} ∫_{B_{σ}} uΔ_{g}udv = \frac{1}{2} ∫_{B_{σ}} u(u^{p} − ku)dv.

Replacing the first line in (16) by using the above two identities, we have (13).

3 Properties of isolated and isolated simple blowup points

In this section, we give some properties of isolated and isolated simple blowup points. We follow [42] and [53] with refinement.

Lemma 3.1 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated blowup point. Then for any 0 < r < r̄/3, we have

max_{y∈B_{2r}(y_{i})\B_{\frac{r}{2}} (y_{i})} u_{i}(y) ≤ C_{1} min_{y∈B_{2r}(y_{i})\B_{\frac{r}{2}} (y_{i})} u_{i}(y),  (17)

where B_{s}(y_{i}) denotes the geodesic ball of radius s centered at y_{i}, and C_{1} is some positive constant independent of i and r.

Proof. Let x = (x^{1}, x^{2}, x^{3}) be some geodesic normal coordinates centered at y_{i} given by exp_{y_{i}}(x) and h = h_{αβ}(x)dx^{α}dx^{β} = g_{αβ}(rx)dx^{α}dx^{β} be the scaled

metric. Set v_{i}(x) = r^{\frac{2}{p_{i}−1}} u_{i}(exp_{y_{i}}(rx)) for |x| = \sqrt{(x^{1})^{2} + (x^{2})^{2} + (x^{3})^{2}} < 3, then v_{i}(x) satisfies

\{ \begin{matrix}−Δ_{h}v_{i}(x) & +r^{2}k̃_{i}(x)v_{i}(x) = v_{i}(x)^{p_{i}} & |x| < 3, \\ 0 < v_{i}(x) & ≤ C̃|x|^{− \frac{2}{p_{i}−1}} & |x| < 3, \end{matrix}

where k̃_{i}(x) = k_{i}(exp_{y_{i}}(rx)). Clearly v_{i}(x) ≤ C for all 1/4 ≤ |x| ≤ 9/4. It follows from the Harnack inequality (see for example [30]) that

max_{1/2≤|x|≤2} v_{i}(x) ≤ C_{1} min_{1/2≤|x|≤2} v_{i}(x),

which yields (17).

Lemma 3.2 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated blowup point. Then for any R_{i} → +∞, ϵ_{i} → 0^{+}, we have, after passing to a subsequence {u_{j_{i}}} (still denoted as {u_{i}}, {y_{i}}, etc...), that

\begin{matrix}||u_{i}(y_{i})^{−1}u_{i}(exp_{y_{i}}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)) − (1 + \frac{1}{3}|x|^{2})^{− \frac{1}{2}} ||_{C^{2}(B_{2R_{i}})} \\ +||u_{i}(y_{i})^{−1}u_{i}(exp_{y_{i}}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)) − (1 + \frac{1}{3}|x|^{2})^{− \frac{1}{2}} ||_{H^{1}(B_{2R_{i}})} & ≤ ϵ_{i}, \end{matrix} (18)

and

\frac{R_{i}}{log u_{i}(y_{i})} → 0 as i → ∞,  (19)

where x = (x^{1}, x^{2}, x^{3}) denotes some geodesic normal coordinates given by exp_{y_{i}}(x).

Proof. Let h = h_{αβ}(x)dx^{α}dx^{β} = g_{αβ}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)dx^{α}dx^{β} denote the scaled metric. Set

ξ_{i}(x) = u_{i}(y_{i})^{−1}u_{i}(exp_{y_{i}}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)), |x| < r̄u_{i}(y_{i})^{\frac{p_{i}−1}{2}} .

Then ξ_{i}(x) satisfies

\{ \begin{matrix}−Δ_{h}ξ_{i}(x) = ξ_{i}(x)^{p_{i}} − u_{i}(y_{i})^{1−p_{i}} k̃_{i}(x)ξ_{i}(x), & |x| < r̄u_{i}(y_{i})^{\frac{p_{i}−1}{2}} , \\ ξ_{i}(0) = 1, ∇ξ_{i}(0) = 0, \\ 0 < ξ_{i}(x) ≤ C̃|x|^{− \frac{2}{p_{i}−1}} & |x| < r̄u_{i}(y_{i})^{\frac{p_{i}−1}{2}} . \end{matrix} (20)

where k̃_{i}(x) = k_{i}(exp_{y_{i}}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)).

It follows from Lemma 3.1 that for all 0 < r ≤ 1,

max_{|x|=r} ξ_{i}(x) ≤ C min_{|x|=r} ξ_{i}(x),  (21)

where C is independent of i and r.

Since u_{i}(y_{i})^{1−p_{i}} k̃_{i}(x) → 0 uniformly in |x| ≤ 2, we have from (20) that

( − Δ_{h} + ∘(1))ξ_{i}(x) ≥ 0, ∀ |x| ≤ 2.

Consequently, η_{i}(x) := (1+|x|^{2})^{−1}ξ_{i}(x) satisfies, for some b_{α}(x), the following inequality for large i:

Δ_{h}η_{i}(x) + ∑_{α=1}^{n} b_{α}(x)∂_{α}η_{i}(x) ≤ 0 ∀ |x| ≤ 2.

It then follows from the maximum principle that

η_{i}(0) ≥ inf_{|x|=r} η_{i}(x) ∀ 0 < r ≤ 1

which, together with (21) and the fact ξ_{i}(0) = 1, yields

1 = ξ_{i}(0) = η_{i}(0) ≥ inf_{|x|=r} η_{i}(x) ≥ \frac{1}{2} inf_{|x|=r} ξ_{i}(x) ≥ \frac{1}{2C} max_{|x|=r} ξ_{i}(x) ∀ 0 < r ≤ 1.

Namely,

max_{|x|≤1} ξ_{i}(x) ≤ C.

Combining this with the third line in (20), we have

ξ_{i}(x) ≤ C ∀ |x| < r̄u_{i}(y_{i})^{\frac{p_{i}−1}{2}} ,

where C is independent of i and r.

Applying standard elliptic estimates to {ξ_{i}}, we have , after passing to a subsequence {ξ_{j_{i}}} (still denoted as {ξ_{i}}), ξ_{i} → ξ in both C_{loc}^{2}(ℝ^{3}) and H_{loc}^{1}(ℝ^{3}) for some ξ satisfying

\{ \begin{matrix}−Δξ(x) = ξ(x)^{5},  ξ(x) > 0 & in ℝ^{3}, \\ ξ(0) = 1, ∇ξ(0) = 0. \end{matrix}

It follows from [17] that

ξ(x) = (1 + \frac{1}{3}|x|^{2})^{− \frac{1}{2}} .

Lemma 3.2 follows from the above after passing to another subsequence.

Before stating our main estimate on isolated simple blowup points, we introduce the Green′s functions of −Δ_{g} + k in small balls. Let ȳ ∈ Ω and k ∈ C^{1}(Ω), it is well known (see Appendix A) that for δ_{0} > 0 small, there exists a unique G̃(·, ȳ) ∈ C^{2}(\overline{B_{δ_{0}}(ȳ)} \ \{ȳ\}) satisfying

\{ \begin{matrix}−Δ_{g}G̃ + kG̃ = 0 & in B_{δ_{0}}(ȳ) \ \{ȳ\}, \\ G̃ = 0, & on ∂B_{δ_{0}}(ȳ), \\ lim_{y→ȳ}d(y, ȳ)G̃(y) = 1. \end{matrix}  (22)

Proposition 3.1 Let {u_{i}} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point, with (10) and (11) for all i. Then for some constant C depending only on ρ, C̃, and ∥k_{i}∥_{C^{1}(Ω)}, we have

u_{i}(y) ≤ Cu_{i}(y_{i})^{−1}d(y, y_{i})^{−1}, ∀ d(y, y_{i}) ≤ ρ/2,  (23)

where ρ, C̃ are given in Definitions 1.1-1.2.

Furthermore, after passing to a subsequence, for some positive constant a > 0 and k(x) = lim_{i→∞} k_{i}(x),

u_{i}(y_{i})u_{i} → aG̃(·, ȳ) + b in C_{loc}^{2}(Bρ̃(\overline{y}) \ \{\overline{y}\}),

where ρ̃ = min\{δ_{0}, ρ/2\}, G̃ is given by (22) and b ∈ C^{2}(Bρ̃(\overline{y})) satisfies −Δ_{g}b + k(y)b = 0 in Bρ̃(\overline{y}).

We will establish this proposition through a series of lemmas.

Lemma 3.3 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (18) and (19) hold. Then for any given 0 < δ < 1/100, there exists ρ_{1} ∈ (0, ρ) which is independent of i (but depending on δ), such that

u_{i}(y) ≤ C_{2}u_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−1+δ}, ∀ R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1}, (24)
|∇_{g}u_{i}(y)| ≤ C_{2}u_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−2+δ}, ∀ R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1}, (25)

and

|∇_{g}^{2}u_{i}(y)| ≤ C_{2}u_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−3+δ}, ∀ R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1}, (26)

where λ_{i} = (1 − δ)\frac{p_{i}−1}{2} − 1 and C_{2} is some positive constant independent of i.

Proof. Let r_{i} = R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} , it follows from Lemma 3.2 that

u_{i}(y) ≤ Cu_{i}(y_{i})R_{i}^{−1} for d(y, y_{i}) = r_{i}.  (27)

We know from the assumption of isolated simple blowup and Lemma 3.2 that

r^{\frac{2}{p_{i}−1}} ū_{i}(r) is strictly decreasing for r_{i} < r < ρ.  (28)

We then derive from Lemma 3.1, (28) and (27) that for all r_{i} ≤ d(y, y_{i}) < ρ,

d(y, y_{i})^{\frac{2}{p_{i}−1}} u_{i}(y) ≤ Cd(y, y_{i})^{\frac{2}{p_{i}−1}} ū_{i}(d(y, y_{i}))

≤ Cri^{\frac{2}{p_{i}−1}} ū_{i}(r_{i})

≤ Cri^{\frac{2}{p_{i}−1}} u_{i}(y_{i})R_{i}^{−1}

= CR_{i}^{−1+ \frac{2}{p_{i}−1}} = CR_{i}^{− \frac{1}{2} +o(1)}.

Therefore, for all r_{i} ≤ d(y, y_{i}) ≤ ρ, we have

u_{i}(y)^{p_{i}−1} ≤ CR_{i}^{−2+o(1)}d(y, y_{i})^{−2}.

Now we consider the operator:

ℓ_{i}φ = Δ_{g}φ + u_{i}(y)^{p_{i}−1}φ − k_{i}(y)φ.

Since ℓ_{i}u_{i} = 0 and u_{i} > 0, the maximum principle holds for ℓ_{i} (see e.g. [11]). It follows from a straightforward calculation that Δ_{g}(d(y, y_{i})^{−μ}) = [−μ(1 − μ) + O(1)d(y, y_{i})^{2}]d(y, y_{i})^{−μ−2}. It follows that for any given 0 < δ < 1/100, we can choose ρ_{1} < ρ such that for i large enough and all r_{i} ≤ d(y, y_{i}) < ρ_{1}, we have

\begin{aligned}& ℓ_{i}(d(y, y_{i})^{−δ}) \\ = & −δ(1 − δ)d(y, y_{i})^{−δ−2} + u_{i}(y)^{p_{i}−1}d(y, y_{i})^{−δ} + O(1)d(y, y_{i})^{−δ} \\ ≤ & −δ(1 − δ)d(y, y_{i})^{−δ−2} + CR_{i}^{−2+o(1)}d(y, y_{i})^{−δ−2} + O(1)d(y, y_{i})^{−δ} \\ ≤ & −\frac{1}{2}δ(1 − δ)d(y, y_{i})^{−δ−2}, \end{aligned} (29)

and, similarly, that

ℓ_{i}(d(y, y_{i})^{−1+δ}) ≤ −\frac{1}{2}δ(1 − δ)d(y, y_{i})^{−3+δ}.  (30)

Let λ_{i} = (1 − δ)\frac{p_{i}−1}{2} − 1 and M_{i} = max_{∂B_{ρ_{1}}(y_{i})} u_{i}, where B_{ρ_{1}}(y_{i}) denotes the

geodesic ball of radius ρ_{1} centered at y_{i}. Set

φ_{i}(y) = M_{i}ρ_{1}^{δ}d(y, y_{i})^{−δ} + Au_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−1+δ} for r_{i} ≤ d(y, y_{i}) ≤ ρ_{1},

where A is some large constant to be chosen later. By (29) and (30), ℓ_{i}φ_{i}(y) ≤ 0 for r_{i} ≤ d(y, y_{i}) ≤ ρ_{1}. On ∂B_{ρ_{1}}(y_{i}),φ_{i}(y) ≥ M_{i} ≥ u_{i}(y); On ∂B_{r_{i}}(y_{i}), φ_{i}(y) ≥ Au_{i}(y_{i})^{−λ_{i}}r_{i}^{−1+δ} = Au_{i}(y_{i})R_{i}^{−1+δ}. Choosing A large enough, we know from (27) that φ_{i}(y) ≥ u_{i}(y) on ∂B_{r_{i}}(y_{i}). Thus, in view of ℓ_{i}u_{i} = 0, we have, using the maximum principle, that

φ_{i}(y) ≥ u_{i}(y) in r_{i} ≤ d(y, y_{i}) ≤ ρ_{1}.  (31)

In turn, we have, for all r_{i} < θ < ρ_{1} and large i, that

ρ_{1}^{\frac{2}{p_{i}−1}} M_{i} ≤ Cρ_{1}^{\frac{2}{p_{i}−1}} ū_{i}(ρ_{1})  (Lemma 3.1)
≤ Cθ^{\frac{2}{p_{i}−1}} ū_{i}(θ)  ( (28) )
≤ Cθ^{\frac{2}{p_{i}−1}} \{M_{i}ρ_{1}^{δ}θ^{−δ} + Au_{i}(y_{i})^{−λ_{i}}θ^{−1+δ}\} ( (31) ).

Since 2/(p_{i} − 1) − δ ≥ 1/5 for large i, we can fix θ to be small enough (independent of i) so that we can derive from the above that

M_{i} ≤ Cu_{i}(y_{i})^{−λ_{i}}.

Using (31) and the above, we have

u_{i}(y) ≤ Cu_{i}(y_{i})^{−λ_{i}}(d(y, y_{i})^{−δ} + d(y, y_{i})^{−1+δ}) ≤ C_{2}u_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−1+δ}.

Estimate (24) is established.

To derive (25) from (24), we assume, for simplicity, that g is the flat metric. The general case can be derived essentially in the same way. For any R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ |ỹ| ≤ ρ_{1}/2, we consider

v_{i}(z) = |ỹ|^{1−δ}u_{i}(y_{i})^{λ_{i}}u_{i}(|ỹ|z), \frac{1}{2} ≤ |z| ≤ 2.

It follows from (9) that v_{i} satisfies, for \frac{1}{2} ≤ |z| ≤ 2,

−Δv_{i}(z) + |ỹ|^{2}k_{i}(|ỹ|z)v_{i}(z) = |ỹ|^{3−δ−(1−δ)p_{i}}u_{i}(y_{i})^{(1−p_{i})λ_{i}}v_{i}(z)^{p_{i}}.  (32)

We derive from |ỹ| ≥ R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} and the definition of λ_{i} that

|ỹ|^{3−δ−(1−δ)p_{i}}u_{i}(y_{i})^{(1−p_{i})λ_{i}} ≤ R_{i}^{3−δ−(1−δ)p_{i}} = ∘(1).  (33)

In view of (24) (with R_{i} replaced by R_{i}/2), we have v_{i}(z) ≤ C ∀ \frac{1}{2} ≤ |z| ≤ 2. We can then derive from (32), (33), and standard elliptic estimates that

|∇v_{i}(z)| ≤ C ∀ |z| = 1,

which implies

|∇u_{i}(ỹ)| ≤ C|ỹ|^{−2+δ}u_{i}(y_{i})^{−λ_{i}}.

This establishes (25) in a smaller range 2R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ \frac{1}{2}ρ_{1}. However it is clear that we can actually establish (24) in a larger range and the above argument will yield (25). Estimate (26) can be derived in a similar way after differentiating (32) once. We omit the details. Lemma 3.3 is established.

Remark 3.1 Later on we will fix δ close to 0, hence fix ρ_{1}. Our goal is to obtain (24) with δ = 0 for r_{i} ≤ d(y, y_{i}) ≤ ρ_{1}, which, together with Lemma 3.1, yields Proposition 3.1.

Lemma 3.4 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (18) and (19) hold. Fixing 0 < δ < 1/100. Then for all fixed 0 < σ < ρ_{1}, we have for large i that

∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}u_{i}^{p_{i}+1} = O(u_{i}(y_{i})^{−4+∘(1)}),  (34)
∫_{d(y,y_{i})≤σ} u_{i}^{2} ≤ C_{3}σu_{i}(y_{i})^{−2λ_{i}},  (35)
∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}|∇u_{i}|^{2} ≤ O(u_{i}(y_{i})^{−4+∘(1)}) + C_{3}σu_{i}(y_{i})^{−2λ_{i}}, (36)

where C_{3} is some constant independent of i and σ.

Proof. To derive these estimates, we use (18) to estimate u in the region d(y, y_{i}) ≤ r_{i}, and to use Lemma 3.3 to estimate u in the region r_{i} ≤ d(y, y_{i}) ≤ σ.

We still use notation y = exp_{y_{i}}(u_{i}(y_{i})^{−(p_{i}−1)/2}x). It follows from (18) that

∫_{d(y,y_{i})≤r_{i}} d(y, y_{i})^{2}u_{i}^{p_{i}+1}
≤ Cu_{i}(y_{i})^{p_{i}+1} ∫_{|x|<R_{i}} |x|^{2}(1 + \frac{1}{3}|x|)^{− \frac{p_{i}+1}{2}} u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} dx
≤ Cu_{i}(y_{i})^{−4+ \frac{3τ_{i}}{2}} ,

Also, it follows from (24) that

∫_{r_{i}≤d(y,y_{i})≤σ} d(y, y_{i})^{2}u_{i}^{p_{i}+1}
≤ C ∫_{r_{i}≤d(y,y_{i})≤σ} d(y, y_{i})^{2}(u_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−1+δ})^{p_{i}+1}
≤ Cr_{i}^{5−(p_{i}+1)(1−δ)}u_{i}(y_{i})^{−λ_{i}(p_{i}+1)}
= CR_{i}^{5−(p_{i}+1)(1−δ)}u_{i}(y_{i})^{−4+ \frac{3τ_{i}}{2}} ,

where τ_{i} = 5 − p_{i} and r_{i} = R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} . Estimate (34) follows from the above immediately.

The proof of (35) is similar to that of (34). To prove (36), we multiply (9) by d(y, y_{i})^{2}u_{i} and integrate by parts, to obtain, in view of (34), that

∫_{d(y,y_{i})≤σ} ∇_{g}(d(y, y_{i})^{2}u_{i})∇_{g}u_{i}
= ∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}u_{i}^{p_{i}+1} − ∫_{d(y,y_{i})≤σ} k_{i}d(y, y_{i})^{2}u_{i}^{2} + O(σu_{i}(y_{i})^{−2λ_{i}})
= O(u_{i}(y_{i})^{−4+∘(1)}) + O(σu_{i}(y_{i})^{−2λ_{i}}).

It follows that

∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}|∇_{g}u_{i}|^{2}

= − ∫_{d(y,y_{i})≤σ} u_{i}∇_{g}(d(y, y_{i})^{2})∇_{g}u_{i} + O(σu_{i}(y_{i})^{−2λ_{i}})
≤ C ∫_{d(y,y_{i})≤σ} d(y, y_{i})u_{i}|∇_{g}u_{i}| + O(σu_{i}(y_{i})^{−2λ_{i}})
≤ \frac{1}{2} ∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}|∇_{g}u_{i}|^{2} + C ∫_{d(y,y_{i})≤σ} u_{i}^{2} + O(σu_{i}(y_{i})^{−2λ_{i}}).

This, together with (35), yields (36).

In the following, let (x^{1}, x^{2}, x^{3}) be some geodesic normal coordinate system centered at y_{i}. We use the notation explained at the beginning of section 2.

Lemma 3.5 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blow up point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (18) and (19) hold. Then for 0 < σ < ρ_{1}, we have

|A(g, u_{i})| ≤ C_{3}σu_{i}(y_{i})^{−2λ_{i}},

where C_{3} is some constant independent of i and σ.

Proof. Recall that

A(g, u_{i}) = ∫_{B_{σ}}(x^{α}∂_{α}u_{i})(g^{βγ} − δ^{βγ})∂_{βγ}u_{i}dv − ∫_{B_{σ}}(x^{α}∂_{α}u_{i})(g^{βγ}Γ_{βγ}^{μ})∂_{μ}u_{i}dv
+\frac{1}{2} ∫_{B_{σ}} u_{i}(g^{βγ} − δ^{βγ})∂_{βγ}u_{i}dv − \frac{1}{2} ∫_{B_{σ}} u_{i}(g^{βγ}Γ_{βγ}^{μ})∂_{μ}u_{i}dv
≡ I + II + III + IV.

It follows from Lemma 3.2 and Lemma 3.3 that

|I| ≤ C_{3} ∫_{B_{σ}} |x|^{3}|∇u_{i}||∇^{2}u_{i}|dv ≤ C_{3}σu_{i}(y_{i})^{−2λ_{i}},

|II| ≤ C_{3} ∫_{B_{σ}} |x|^{2}|∇u_{i}|^{2}dv ≤ C_{3}σu_{i}(y_{i})^{−2λ_{i}},

and

|III| + |IV | ≤ C_{3} ∫_{B_{σ}} |x|u_{i}|∇u_{i}|dv ≤ C_{3}σu_{i}(y_{i})^{−2λ_{i}}.

Lemma 3.5 is established.

Lemma 3.6 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (18) and (19) hold. Then

τ_{i} = O(u_{i}(y_{i})^{−2λ_{i}}).

Consequently,

u_{i}(y_{i})^{τ_{i}} → 1.

Proof. We derive from (18) that

∫_{d(y,y_{i})≤r_{i}} u_{i}^{p_{i}+1} ≥ C^{−1}u_{i}(y_{i})^{\frac{τ_{i}}{2}} ≥ C^{−1},  (37)

where r_{i} = R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} .

In view of the above estimate, we can derive from the Pohozaev identity of u_{i} given by Lemma 2.1, using Lemma 3.3 and Lemma 3.4, that

τ_{i} ≤ C(\frac{3}{p_{i} + 1} − \frac{1}{2}) ∫_{B_{σ}} u_{i}^{p_{i}+1} ≤ Cu_{i}(y_{i})^{−2λ_{i}}.

Lemma 3.6 follows from the above immediately.

Lemma 3.7 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (18) and (19) hold. Then for all 0 < σ < \overline{r}/2, we have

lim sup_{i→∞} max_{y∈∂B_{σ}(y_{i})} u_{i}(y)u_{i}(y_{i}) ≤ C(σ).

Proof. In view of Lemma 3.1, we only need to establish the lemma for some sufficiently small σ > 0. Without loss of generality, we assume \overline{r} = 2. We distinguish into two cases. Case 1: k_{i}(x) ≥ 0 ∀ x ∈ B_{1}(\overline{y}).

Pick any y_{σ} with d(y_{σ}, y_{i}) = σ and set ξ_{i}(y) = u_{i}(y_{σ})^{−1}u_{i}(y). Then ξ_{i}(y) satisfies

−Δ_{g}ξ_{i}(y) = u_{i}(y_{σ})^{p_{i}−1}ξ_{i}(y)^{p_{i}} − k_{i}(y)ξ_{i}(y).

It follows from Lemma 3.1 that for all compact set K ⊂ B_{1}(ȳ) \ {ȳ}, there exists some constant C(K) such that

C(K)^{−1} ≤ ξ_{i} ≤ C(K) on K.

We also know from (24) that u_{i}(y_{σ}) → 0 as i → ∞. Then by elliptic theories, we have, after passing to a subsequence,

ξ_{i} → ξ in C_{loc}^{2}(B_{1}(ȳ) \ {ȳ}),

where ξ satisfies

−Δ_{g}ξ + kξ = 0, ξ > 0, in B_{1}(ȳ) \ {ȳ}

with k(y) = lim_{i→∞} k_{i}(y). Notice that for any fixed 0 < r < σ,

lim_{i→∞} u_{i}(y_{σ})^{−1}r^{\frac{2}{p_{i}−1}} ū_{i}(r) = r^{\frac{1}{2}} ξ̄(r),

where ξ(r) = |∂B_{r}(\overline{y})|^{−1} ∫_{∂B_{r}(\overline{y})} ξ_{i}ds_{g}. ¿From the assumption that y_{i} → \overline{y} is an

isolated simple blowup point of \{u_{i}\}, we deduce that r^{\frac{1}{2}} ξ̄(r) is non-increasing for 0 < r < ρ, which is impossible if ξ were regular near {ȳ}. It follows from Corollary 9.1 in our appendix that as σ small enough, there exists some constant m > 0 independent of i such that for i large, we have

− ∫_{B_{σ}(y_{i})} Δ_{g}ξ_{i} = − ∫_{∂B_{σ}(y_{i})} ∇_{g}ξ_{i} · ν = − ∫_{∂B_{σ}(\overline{y})} ∇_{g}ξ · ν + ∘(1) > m. (38)

On the other hand, since k_{i}(x) ≥ 0, we have

\begin{matrix}− ∫_{B_{σ}(y_{i})} Δ_{g}ξ_{i}(y) = ∫_{B_{σ}(y_{i})}[u_{i}(y_{σ})^{−1}u_{i}(y)^{p_{i}} − k_{i}(y)ξ_{i}(y)]dy \\ ≤ u_{i}(y_{σ})^{−1} ∫_{B_{σ}(y_{i})} u_{i}(y)^{p_{i}}dy. \end{matrix}  (39)

Similar to (37), we have

∫_{d(y,y_{i})≤r_{i}} u_{i}^{p_{i}} ≤ Cu_{i}(y_{i})^{−1+ \frac{τ_{i}}{2}} .

Applying Lemma 3.3 and noticing 0 < δ < 1/100 , we have

∫_{r_{i}≤d(y,y_{i})≤σ} u_{i}^{p_{i}} ≤ C_{2} ∫_{r_{i}≤d(y,y_{i})≤σ}(u_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−1+δ})^{p_{i}}
≤ Cr_{i}^{3−p_{i}(1−δ)}u_{i}(y_{i})^{−λ_{i}p_{i}}
= CR_{i}^{3−p_{i}(1−δ)}u_{i}(y_{i})^{−1+ \frac{τ_{i}}{2}}
= o(1)u_{i}(y_{i})^{−1+ \frac{τ_{i}}{2}} .

Using Lemma 3.6, we have

∫_{B_{σ}(y_{i})} u_{i}^{p_{i}} ≤ Cu_{i}(y_{i})^{−1}.  (40)

Lemma 3.7 in this case follows from (38), (39) and (40). Case 2: k_{i}(x) < 0 for some x ∈ B_{1}(\overline{y}).

For σ_{1} > 0, let φ denote the first eigenfunction of −Δ_{g} in B_{2σ_{1}}(\overline{y}) with respect to the Dirichlet boundary condition, i.e.

\{ \begin{matrix}−Δ_{g}φ = λ_{1}φ, φ > 0, & in B_{2σ_{1}}(\overline{y}) \\ φ = 0 & on ∂B_{2σ_{1}}(\overline{y}), \end{matrix}

where λ_{1} denotes the first eigenvalue of −Δ_{g}. It is well known that we can choose σ_{1} > 0 small enough (independent of i), so that λ_{1} > ∥k_{i}∥_{L^{∞}(B_{2σ_{1}}(\overline{y}))}+1 for all i. Fix such a σ_{1}, we know that

−Δ_{g}φ + k_{i}φ > 0 in B_{σ_{1}}(\overline{y}).  (41)

Let g̃ = φ^{4}g, L_{g} and Lg̃ denote the conformal Laplacian operators of g and g̃ respectively . It is well known that

Lg̃ψ = φ^{−5}L_{g}(ψφ), ∀ ψ ∈ C^{∞}(B_{σ_{1}}(\overline{y})).  (42)

Set ũ_{i} = φ^{−1}u_{i}, we derive from (9) and the above with ψ = ũ_{i} that

−Δg̃ũ_{i} + k̃_{i}ũ_{i} = φ^{−τ_{i}}ũ_{i}^{p_{i}}

where k̃_{i} = \frac{1}{8}Rg̃ − (\frac{1}{8}R_{g} − k_{i})φ^{−4}. Taking ψ ≡ 1 in (42), we have

−\frac{1}{8}Rg̃ = (Δ_{g}φ − \frac{1}{8}R_{g}φ)φ^{−5}.

Using (41) and the above we have k̃_{i} = (−Δ_{g}φ+k_{i}φ)φ^{−5} > 0 in B_{σ_{1}}(\overline{y}). Now, we choose small σ ∈ (0, σ_{1}) and set ξ̃_{i}(y) = ũ_{i}(y_{σ})^{−1}ũ_{i}(y) = φ(y_{σ})φ^{−1}(y)ξ_{i}(y). Clearly (38) and (40) still hold when we substitute ξ̃_{i} for ξ_{i} and ũ_{i} for u_{i}. Using the positivity of k̃_{i}, we derive as in (39) that

− ∫_{B_{σ}} Δg̃ξ̃_{i} ≤ ũ_{i}(y_{σ})^{−1} ∫_{B_{σ}} ũ_{i}^{p_{i}}.

Lemma 3.7 in this case then follows as in Case 1.

Proof of Proposition 3.1. We first establish (23) by contradiction argument. Suppose the contrary, then after passing to a subsequence (still denoted as {u_{i}}) we can find {ỹ_{i}} such that d(ỹ_{i}, y_{i}) ≤ ρ/2 and

u_{i}(ỹ_{i})u_{i}(y_{i})d(ỹ_{i}, y_{i}) → ∞.  (43)

After passing to another subsequence {u_{j_{i}}} (still denoted as {u_{i}}, {y_{i}}, etc...), estimates (18) and (19) hold for some R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}}. It follows from (18) that r_{i} := R_{i}u_{i}(y_{i})^{−(p_{i}−1)/2} ≤ d(ỹ_{i}, y_{i}) ≤ ρ/2. Let (x^{1}, x^{2}, x^{3}) be some geodesic normal coordinate system centered at y_{i} and given by exp_{y_{i}}(x). Set r̃_{i} = d(ỹ_{i}, y_{i}) and

ũ_{i}(x) = (r̃_{i})^{2/(p_{i}−1)}u_{i}(exp_{y_{i}}(r̃_{i}x)), |x| < 2.

Clearly ũ_{i} satisfies

−Δ_{g_{i}} ũ_{i}(x) + k̃_{i}(x)ũ_{i}(x) = ũ_{i}(x)^{p_{i}}, |x| < 2,

where k̃_{i}(x) = r̃_{i}^{2}k_{i}(exp_{y_{i}}(r̃_{i}x) and (g_{i})_{αβ}(x) = g_{αβ}(r̃_{i}x)dx^{α}dx^{β}. Here the metrics g_{i} depend on i, but clearly all previous results still hold uniformly with respect to this sequence of metrics since the sequence stays in compact sets with respect to strong norms. It is easy to see that {ũ_{i}} satisfies the hypotheses of Lemma 3.7 and therefore

max_{|x|=1} ũ_{i}(0)ũ_{i}(x) ≤ C

from which we can deduce (using also Lemma 3.6) that

u_{i}(ỹ_{i})u_{i}(y_{i})d(ỹ_{i}, y_{i}) ≤ C.

This contradicts (43). We have thus established (23).

We see from (9) that u_{i}(y_{i})u_{i} satisfies

−Δ_{g}(u_{i}(y_{i})u_{i}) + k_{i}(u_{i}(y_{i})u_{i}) = u_{i}(y_{i})^{1−p_{i}}(u_{i}(y_{i})u_{i})^{p_{i}}.

In view of (23), we can apply Harnack inequality and standard elliptic estimates to obtain, after passing to a subsequence, that

u_{i}(y_{i})u_{i} → v in C_{loc}^{2}(Bρ̃(\overline{y}) \ \{\overline{y}\}),

where v > 0 satisfies

−Δ_{g}v + kv = 0 in Bρ̃(\overline{y}) \ \{\overline{y}\}

with k(x) = lim_{i→∞} k_{i}(x).

It is also clear that for all 0 < r < ρ,

lim_{i→∞} u_{i}(y_{i})r^{2/(p_{i}−1)}\overline{u}_{i}(r) = r^{1/2}\overline{v}(r),

where \overline{u}_{i}(r) is defined as in Definition 1.1 and \overline{v}(r) is defined similarly. Therefore it follows from Definition 1.2 (using also (18)) that r^{1/2}\overline{v}(r) is nonincreasing in the interval 0 < r < ρ, which implies that v is singular at the origin. In turn, it follows from Proposition 9.1 in our appendix that v = G(·, ȳ) + h as stated in Proposition 3.1. Proposition 3.1 is established. Another way to show that v is singular at the origin is to argue as in the derivation of (47).

Once we have established Proposition 3.1 we can strengthen Lemma 3.3, Lemma 3.4 and Lemma 3.5 by substituting (23) for (24) in their proofs. We will state these improved versions as three corollaries without repeating the proofs.

Corollary 3.1 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (18) and (19) hold. Then there exists ρ_{1} ∈ (0, ρ) which is independent of i such that

|∇_{g}u_{i}(y)| ≤ C_{2}u_{i}(y_{i})^{−1}d(y, y_{i})^{−2} for all R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1},

and

|∇_{g}^{2}u_{i}(y)| ≤ C_{2}u_{i}(y_{i})^{−1}d(y, y_{i})^{−3} for all R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1},

where C_{2} is some positive constant independent of i.

Corollary 3.2 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (18) and (19) hold. Then for all fixed 0 < σ < ρ_{1}, we have for large i that

∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}u_{i}^{p_{i}+1} = O(u_{i}(y_{i})^{−4+∘(1)}),

∫_{d(y,y_{i})≤σ} u_{i}^{2} ≤ C_{3}σu_{i}(y_{i})^{−2} + ∘(u_{i}(y_{i})^{−2}),

and

∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}|∇u_{i}|^{2} ≤ O(u_{i}(y_{i})^{−4+∘(1)}) + C_{3}σu_{i}(y_{i})^{−2},

where C_{3} is some constant independent of i and σ.

Corollary 3.3 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blow up point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (18) and (19) hold. Then

|A(g, u_{i})| ≤ C_{3}σu_{i}(y_{i})^{−2},

where A(g, u_{i}) is defined as in Lemma 2.1 and C_{3} is some constant independent of i and σ.

We derive easily from (23), Corollary 3.1, Corollary 3.2 and Corollary 3.3 the following

Corollary 3.4 Let u_{i} satisfy (9) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. After passing to some subsequence of {u_{i}}, the following estimates hold:

lim_{i→∞} u_{i}(y_{i})^{2} ∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}u_{i}^{p_{i}+1} = 0,

lim_{σ→0^{+}} lim sup_{i→∞} u_{i}(y_{i})^{2} ∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}|∇u_{i}|^{2} = 0,

lim_{σ→0^{+}} lim sup_{i→∞} u_{i}(y_{i})^{2} ∫_{d(y,y_{i})≤σ} u_{i}^{2} = 0,

and

lim_{σ→0^{+}} lim sup_{i→∞} u_{i}(y_{i})^{2}|A(g, u_{i})| = 0,

where A(g, u_{i}) is given in Lemma 2.1.

The following proposition will be used repeatedly in later sections.

Proposition 3.2 Let u_{i} satisfy (9), y_{i} → ȳ be an isolated simple blowup point with, for some ρ̃ > 0,

u_{i}(y_{i})u_{i} → h in C_{loc}^{2}(B_{ρ}(ȳ) \ {ȳ}).

Assume, for some a > 0, in some geodesic normal coordinate system x = (x^{1}, x^{2}, x^{3}), that

h(x) = \frac{a}{|x|} + A + ∘(1) as |x| → 0.

Then A ≤ 0.

Proof. For σ > 0 small, the Pohozaev identity of u_{i} is given by Lemma 2.1:

\begin{matrix}− ∫_{B_{σ}}(k_{i}(x) + \frac{x·∇k_{i}(x)}{2})u_{i}(x)^{2}dv + (\frac{3}{p_{i}+1} − \frac{1}{2}) ∫_{B_{σ}} u_{i}(x)^{p_{i}+1}dv \\ = ∫_{∂B_{σ}} B(x, σ, u_{i}, ∇u_{i})ds + A(g, u_{i}) − ∫_{∂B_{σ}} σ(\frac{k_{i}u_{i}^{2}}{2} − \frac{u_{i}^{p_{i}+1}}{p_{i}+1} )ds. \end{matrix}

Multiplying the above by u_{i}(0)^{2} and then applying lim_{σ→0^{+}} lim sup_{i→∞} to both sides, we have, using Corollary 3.4, that

lim_{σ→0^{+}} ∫_{∂B_{σ}} B(x, σ, h, ∇h) = lim_{σ→0^{+}} lim sup_{i→∞} u_{i}(0)^{2} ∫_{∂B_{σ}} B(x, σ, u_{i}, ∇u_{i}) ≥ 0.

On the other hand, a direct calculation yields

lim_{σ→0^{+}} ∫_{∂B_{σ}} B(x, σ, h, ∇h) = −\frac{aA}{2}|𝕊^{2}|,

where |𝕊^{2}| denotes the area of the standard two dimensional sphere. Proposition 3.2 follows from the above two formulas.

Remark 3.2 Proposition 3.2 still holds when substituting the metric g by a sequence of good enough metrics.

At the end of this section we point out that the analogue of (23) does not hold in higher dimensions. It was shown by Adimurthi and Yadava [1] that in dimension 4 ≤ n ≤ 6 there exists a sequence of radially symmetric solutions to the following Neumann problem:

\{ \begin{matrix}−Δu_{i} + \frac{1}{i}u_{i} = ui^{\frac{n+2}{n−2}} , u_{i} > 0,  in B_{1}(0), \\ \frac{∂u_{i}}{∂ν} = 0  on ∂B_{1}(0), \end{matrix}  (44)

with u_{i}(0) = max_{\overline{B}_{1}} u_{i} → ∞.

In connection with (44), we have Lemma 3.8 For n ≥ 3, let u_{i} be a radially symmetric solution of (44). Then, after passing to a subsequence, either u_{i} → 0 uniformly on \overline{B}_{1} or u_{i}(0) → ∞ and, for all 0 < |x| ≤ 1, u_{i}(0)u_{i}(x) → ∞.

Proof. As pointed out in remark 4.4 of [43], {u_{i}} can have at most one isolated blowup point at {0}. More precisely, we have

Step 1. For some constant C independent of i,

|x|^{\frac{n−2}{2}} u_{i}(x) ≤ C, ∀ |x| ≤ 1.

This can be proved by contradiction argument. Suppose the contrary, we have, for some 0 < |x_{i}| ≤ 1,

f_{i}(x_{i}) = max_{0≤|x|≤1} f_{i}(x) → ∞,

where f_{i}(x) := |x|^{\frac{n−2}{2}} u_{i}(x). Let σ_{i} = \frac{1}{2}|x_{i}|, clearly we have

σi^{\frac{n−2}{2}} max_{|x−x_{i}|≤σ_{i}} u_{i} ≥ 2^{− \frac{n−2}{2}} f_{i}(x_{i}) → ∞,

(2σ_{i})^{\frac{n−2}{2}} u_{i}(x_{i}) ≥ max_{|x−x_{i}|≤σ_{i}} f_{i} ≥ σi^{\frac{n−2}{2}} max_{|x−x_{i}|≤σ_{i}} u_{i}.

Consequently,

σi^{\frac{n−2}{2}} u_{i}(x_{i}) → ∞,

and

u_{i}(x_{i}) ≥ 2^{− \frac{n−2}{2}} max_{|x−x_{i}|≤σ_{i}} u_{i}.  (45)

Set

w_{i}(z) = u_{i}(x_{i})^{−1}u_{i}(u_{i}(x_{i})^{− \frac{2}{n−2}} z + x_{i}), |z| < u_{i}(x_{i})^{\frac{2}{n−2}} σ_{i} → ∞.

Clearly, w_{i} satisfies, in view of (45),

\{ \begin{matrix}−Δw_{i}(z) = w_{i}(z)^{\frac{n+2}{n−2}} − i^{−1}u_{i}(x_{i})^{− \frac{4}{n−2}} w_{i}(z), & u_{i}(x_{i})^{− \frac{2}{n−2}} z + x_{i} ∈ B_{1}(0), \\ 2^{− \frac{n−2}{2}} w_{i}(z) ≤ w_{i}(0) = 1, & u_{i}(x_{i})^{− \frac{2}{n−2}} z + x_{i} ∈ B_{1}(0), \end{matrix}

and also the zero Neumann boundary condition. After passing to a subsequence, w_{i} strongly converges to some w on any compact subset with w ≥ 0 satisfying w(0) = 1 and −Δw = w^{(n+2)/(n−2)} in either the whole space ℝ^{n} or some half space containing the origin. In the second case, we also know that w satisfies the zero Neumann boundary condition on the boundary of the half space. Due to the Liouville type theorem of Caffarelli, Gidas and Spruck, w is given explicitly. The explicit form of w and the strong convergence of w_{i} to w on any compact sets apparently violate the radial symmetry of u_{i}. Step 1 is thus established.

We know from Step 1 that {u_{i}} is uniformly bounded away from the origin, so we can derive from the Harnack inequalities (including the one with boundary conditions, see for example, Lemma A.1 in [35]) the following

Step 2. For any 0 < ϵ < 1, there exists C(ϵ) independent of i such that

max_{ϵ≤|x|≤1} u_{i} ≤ C(ϵ) min_{ϵ≤|x|≤1} u_{i}.

Next we show

Step 3. After passing to a subsequence, either u_{i} goes to zero uniformly on \overline{B}_{1}, or u_{i}(0) → ∞.

After passing to a subsequence, either {u_{i}(0)} stays bounded or u_{i}(0) → ∞. If {u_{i}(0)} stays bounded, then, with Step 1, we can show the same way as in the proof of Lemma 3.2 that \{u_{i}\} is uniformly bounded on \overline{B}_{1}. After passing to another subsequence, using standard elliptic estimates, u_{i} → u in C^{2}(\overline{B}_{1}) for some radially symmetric function u satisfying

\{ \begin{matrix}−Δu = u^{\frac{n+2}{n−2}} , u ≥ 0,  in B_{1}(0), \\ \frac{∂u}{∂ν} = 0  on ∂B_{1}(0). \end{matrix}

Since the only nonnegative radially symmetric solution is u ≡ 0, we have established Step 3. Now we are ready to show that

Step 4. After passing to a subsequence, either u_{i} goes to zero uniformly on \overline{B}_{1}, or

u_{i}(0)u_{i}(1) → ∞.  (46)

After passing to a subsequence, we have shown that either u_{i} goes to zero uniformly on \overline{B}_{1}, or u_{i}(0) → ∞. If u_{i}(0) → ∞, we see from the equation of u_{i} that Δu_{i} ≤ 0 near the origin which, together with the radial symmetry of u_{i}, implies that 0 is a local maximum point of u_{i}. Therefore, in view of Step 1, {0} is an isolated blowup point of {u_{i}}. Applying Lemma 3.2, we know, after passing to a subsequence, that

C^{−1}u_{i}(0) ≤ u_{i}(u_{i}(0)^{− \frac{2}{n−2}} ) ≤ Cu_{i}(0).

It is easy to see that

(−Δ + \frac{1}{i})(e^{−|x|}|x|^{2−n}) ≤ 0, ∀ |x| ≤ 1.

Applying the maximum principle in B_{1} \ B_{u_{i}(0)^{−2/(n−2)}}(0), we have, for some C independent of i, that

u_{i}(x) ≥ C^{−1}u_{i}(0)^{−1}[e^{−|x|}|x|^{2−n} − e^{−1}], ∀ u_{i}(0)^{− \frac{2}{n−2}} ≤ |x| ≤ 1.

Consequently, for all 0 < |x| < 1,

lim inf_{i→∞} u_{i}(0)u_{i}(x) ≥ C^{−1}[e^{−|x|}|x|^{2−n} − e^{−2}2^{2−n}].  (47)

Now we verify (46) by contradiction argument. Suppose the contrary, u_{i}(0)u_{i}(1) ≤ C along a subsequence. It follows from Step 2 that {u_{i}(0)u_{i}} is locally bounded in \overline{B}_{1} \ \{0\}. After passing to a subsequence,

u_{i}(0)u_{i}(x) → G_{0}(x) in C_{loc}^{2}(\overline{B}_{1} \ \{0\}),

where G_{0} satisfies

\{ \begin{matrix}−ΔG_{0} = 0, G_{0} > 0,  in B_{1}(0) \ \{0\}, \\ \frac{∂G_{0}}{∂ν} = 0  on ∂B_{1}(0). \end{matrix}

We see from (47) that G_{0} is singular near the origin, therefore, for some a_{0} > 0,

G_{0}(x) = \frac{a_{0}}{|x|^{n−2}} + H_{0}(x), in B_{2}(0) \ \{0\},

where H_{0} is a regular harmonic function in B_{1}. Apparently the harmonic function H_{0} is in C^{1}(\overline{B}_{1}) and satisfies \frac{∂H_{0}}{∂ν} = (n − 2)a_{0} > 0 on ∂B_{1}. This violates the maximum principle and the Hopf lemma. Lemma 3.8 follows from Step 1 to Step 4. We conclude from Lemma 3.8 and the result of Adimurthi and Yadava that the analogue of (23) does not hold in dimension 4 ≤ n ≤ 6.

4 An isolated blowup point is in fact an isolated simple blowup point

In this section, we prove that an isolated blowup point is in fact an isolated simple blowup point.

Proposition 4.1 Let u_{i} satisfy (9) and y_{i} → ȳ be an isolated blowup point. Then ȳ must be an isolated simple blowup point.

Proof. Due to our estimates in section 3, the proof is basically the same as that of proposition 3.1 of [42]. For readers′ convenience, we include the proof here. We prove it by contradiction argument. Suppose it is not an isolated simple blowup point, then after passing to a subsequence {u_{j_{i}}} (still

denoted as \{u_{i}\}, \{y_{i}\}, etc...), r^{\frac{2}{p_{i}−1}} ū_{i}(r) has at least two critical points in (0, \overline{μ}_{i}) with \overline{μ}_{i} → 0. By passing to another subsequence, we can assume that estimates (18) and (19) hold for some R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}}. It follows from (18) that r^{\frac{2}{p_{i}−1}} ū_{i}(r) has precisely one critical point in the interval 0 < r < r_{i} = R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} . Let μ_{i} be the second critical point of r^{\frac{2}{p_{i}−1}} ū_{i}(r). We have μ_{i} ≥ r_{i}, lim_{i→∞}μ_{i} = 0. As always we let x = (x^{1}, x^{2}, x^{3}) be some geodesic normal coordinate system centered at y_{i} given by exp_{y_{i}}(x), h = h_{αβ}(x)dx^{α}dx^{β} = g_{αβ}(μ_{i}x)dx^{α}dx^{β} be the scaled metric, and

ξ_{i}(x) = μi^{\frac{2}{p_{i}−1}} u_{i}(exp_{y_{i}}(μ_{i}x)), for |x| < 1/μ_{i}.

Then ξ_{i} satisfies

\{ \begin{matrix}−Δ_{h}ξ_{i}(x) + k̃_{i}(x)ξ_{i}(x) = ξ_{i}(x)^{p_{i}},  |x| < 1/μ_{i}, \\ |x|^{\frac{2}{p_{i}−1}} ξ_{i}(x) ≤ C,  |x| < 1/μ_{i}, \\ lim_{i→∞}ξ_{i}(0) = ∞, \\ r^{\frac{2}{p_{i}−1}} ξ̄_{i}(r) has precisely one critical point in 0 < r < 1, \\ \frac{d}{dr} \{r^{\frac{2}{p_{i}−1}} ξ̄_{i}(r)\}|_{r=1} = 0, \end{matrix}  (48)

where k̃_{i}(x) = μ_{i}^{2}k_{i}(exp_{y_{i}}(μ_{i}x)) and ξ̄_{i}(r) is the spherical average of ξ_{i} defined in the usual way.

It follows that {0} is an isolated simple blowup point of {ξ_{i}}. We know from Proposition 3.1 and the Harnack inequality that for all compact set K ⊂ ℝ^{3} \ {0},

C(K)^{−1} ≤ ξ_{i}(0)ξ_{i} ≤ C(K) on K,

and, for some constant a > 0, that

ξ_{i}(0)ξ_{i}(x) → h(x) := a|x|^{−1} + b(x) in C_{loc}^{2}(ℝ^{3} \ {0}),  (49)

where b(x) satisfies Δb = 0 in ℝ^{3}. Since h(x) is positive, we have lim inf_{|x|→∞} b(x) ≥ 0. It follows from the maximum principle that b(x) is nonnegative, hence a constant, denoted as b(x) ≡ b. We deduce from (48) and (49) that

\frac{d}{dr} \{r^{\frac{1}{2}} h(r)\}|_{r=1} = 0.

It follows that

b = a > 0.

This violates Proposition 3.2. Proposition 4.1 is established.

5 Ruling out bubble accumulations

For k(x) ∈ C^{1}(M), we start to analyze compactness properties of ∪_{2≤p≤5}𝕄_{k,p}. The following proposition gives preliminary description of large functions u ∈ ∪_{2≤p≤5}𝕄_{k,p}, which relies on two Liouville type theorems. One is due to Gidas and Spruck [32] which asserts that, for n ≥ 3 and 1 < p < \frac{n+2}{n−2}, there is no solution to the following subcritical exponent equation

−Δu = u^{p},  u > 0,  in ℝ^{n}.

The other is due to Caffarelli, Gidas and Spruck [17] which asserts that, for n ≥ 3, any solution of the following critical exponent equation

−Δu = u^{\frac{n+2}{n−2}} ,  u > 0,  in ℝ^{n}

is of the form

u(x) = (\frac{λ}{1 + \frac{1}{n(n−2)}λ^{2}|x − \overline{x}|^{2}} )^{\frac{n−2}{2}}

for some λ > 0 and \overline{x} ∈ ℝ^{n}. Under some additional decay hypothesis of u(x) at infinity, the above Liouville type theorems are obtained for the subcritical exponent equation in Gidas, Ni and Nirenberg [31], while for the critical exponent equation in Obata [49] and Gidas, Ni and Nirenberg [31].

Roughly speaking, the following proposition shows that for large u ∈ ∪_{2≤p≤5}𝕄_{k,p}, p has to be very close to 5 and one can find a finite collection of disjoint balls B_{\overline{r}_{1}}(y_{1}), ⋯, B_{\overline{r}_{N}} (y_{N}) (N may depend on u) inside which u is very well approximated in strong norms by standard bubbles. Furthermore u satisfies u(y) ≤ C_{1}d(y, \{y_{1}, ⋯, y_{N}\})^{− \frac{2}{p−1}} for all y in M, where C_{1} is some positive constant independent of u.

Proposition 5.1 Given any small ϵ > 0 and large R, there exist some positive constants C_{0} and C_{1} depending only on M, g, ∥k∥_{C^{1}(M)}, ϵ and R such that for all u in ∪_{2≤p≤5}𝕄_{k,p} with

max_{M} u > C_{0},

there exists some integer N = N(u) ≥ 1 and N local maximum points of u, denoted as y_{1}, ⋯, y_{N}, such that (i) 5 − ϵ < p ≤ 5. (ii)

\overline{B_{\overline{r}_{i}}(y_{i})} ∩ \overline{B_{\overline{r}_{j}}(y_{j})} = ϕ, for i ≠ j,

and, for each j,

∥u(y_{j})^{−1}u (exp_{y_{j}}(u(y_{j})^{− \frac{p−1}{2}} x)) − (1 + \frac{1}{3}|x|^{2})^{− \frac{1}{2}} ∥_{C^{2}(|x|<2R)} < ϵ,  (50)

where r̄_{j} = Ru(y_{j})^{−(p−1)/2}, B_{\overline{r}_{j}}(y_{j}) ⊂ M denotes the geodesic ball of radius \overline{r}_{j} and centered at y_{j}, x = (x^{1}, x^{2}, x^{3}) denotes some geodesic normal coordinates centered at y_{j}, and |x| = \sqrt{(x^{1})^{2} + (x^{2})^{2} + (x^{3})^{2}}.

(iii) d(y_{i}, y_{j})^{2/(p−1)}u(y_{j}) ≥ C_{0} for j > i, while d(y, {y_{1}, ⋯, y_{N}})^{2/(p−1)}u(y) ≤ C_{1} for all y in M.

Proposition 5.1 can be derived from the following lemma.

Lemma 5.1 Given any small ϵ > 0 and large R > 1. There exists some large positive constants C_{0}, depending only on M, g, ϵ, R, and ∥k∥_{C^{1}(M)} such that for any compact S ⊂ M and any u in ∪_{2≤p≤5}𝕄_{k,p} with

max_{y∈M\S} d(y, S)^{2/(p−1)}u(y) ≥ C_{0},

we have p > 5 − ϵ and for some local maximum point of u in M \ S, denoted as y_{0},

∥u(y_{0})^{−1}u (exp_{y_{0}}(u(y_{0})^{− \frac{p−1}{2}} x)) − (1 + \frac{1}{3}|x|^{2})^{− \frac{1}{2}} ∥_{C^{2}(|x|<2R)} < ϵ,  (51)

where d(y, S) denotes the distance of y to S, and d(y, S) = 1 if S = ϕ.

Proof. Suppose that for some ϵ and R such C_{0} does not exist. Then there exists compact S_{i} ⊂ M, 2 ≤ p_{i} ≤ 5, {∥k_{i}∥_{C^{1}(M)}} uniformly bounded, and u_{i} ∈ 𝕄_{k_{i},p_{i}}, such that max_{y∈M\S_{i}} d(y, S_{i})^{2/(p_{i}−1)}u_{i}(y) ≥ i, but no such y_{0} exists. Pick a point ỹ_{i} in M \ S_{i} which satisfies d(ỹ_{i}, S_{i})^{2/(p_{i}−1)}u_{i}(ỹ_{i}) = max_{y∈M\S_{i}} d(y, S_{i})^{2/(p_{i}−1)}u_{i}(y). Consider the following rescaled function

w_{i}(x) = u_{i}(ỹ_{i})^{−1}u_{i} (expỹ_{i}(u_{i}(ỹ_{i})^{−(p_{i}−1)/2}x)) , ∀ |x| < R_{i} := \frac{1}{4}u_{i}(ỹ_{i})^{(p_{i}−1)/2}d(ỹ_{i}, S_{i}).

We know that R_{i} ≥ i/4 → ∞ and, for i large, d(y, S_{i}) ≥ \frac{1}{2}d(ỹ_{i}, S_{i}) for all |x| ≤ R_{i} and y = expỹ_{i}(u_{i}(ỹ_{i})^{−(p_{i}−1)/2}x). It follows that

(\frac{1}{2}d(ỹ_{i}, S_{i}))^{\frac{2}{p_{i}−1}} u_{i}(y) ≤ d(y, S_{i})^{\frac{2}{p_{i}−1}} u_{i}(y) ≤ d(ỹ_{i}, S_{i})^{\frac{2}{p_{i}−1}} u_{i}(ỹ_{i}) ∀ |x| ≤ R_{i},

which implies

w_{i}(x) ≤ 2^{\frac{2}{p_{i}−1}} ≤ 4, ∀ |x| ≤ R_{i}.

It then follows from standard elliptic estimates that after passing to a subsequence, still denoted as {w_{i}}, we have p_{i} → p ∈ [2, 5],

w_{i} → w in C_{loc}^{2}(ℝ^{3}),

where w satisfies

−Δw = w^{p},  w > 0,  in ℝ^{3}.

By the Liouville type theorem of Gidas and Spruck, we know that p = 5 and in particular p_{i} > 5 − ϵ for large i. In turn, it follows from the Liouville type

theorem of Caffarelli, Gidas and Spruck that w(x) = (_{\frac{λ}{1+ \frac{1}{3} λ^{2}|x−x̄|^{2}}} )^{1/2} for

some x̄ ∈ ℝ^{3}. Since 1 = w(0) = (_{\frac{λ}{1+ \frac{1}{3} λ^{2}|x̄|^{2}}} )^{1/2} and w(x) ≤ 4 for all x ∈ ℝ^{3}, we

see that 1 ≤ λ ≤ 2. We see from the explicit form of w that we can find x_{i} → x̄ which are local maximum points of w_{i}(x). Clearly w_{i}(x_{i}) → λ^{1/2} = max w. Defining y_{i} = expỹ_{i} (u_{i}(ỹ_{i})^{−(p_{i}−1)/2}x_{i}), then y_{i} ∈ M \ S_{i} is a local maximum point of u_{i}. Repeating the scaling with ỹ_{i} replaced by y_{i}, we will obtain a new

limit w(x) = (_{\frac{1}{1+ \frac{1}{3} |x|^{2}}} )^{1/2}. Consequently, for large i, we have

∥u_{i}(y_{i})^{−1}u_{i} (exp_{y_{i}}(u(y_{i})^{− \frac{p_{i}−1}{2}} x)) − (1 + \frac{1}{3}|x|^{2})^{− \frac{1}{2}} ∥_{C^{2}(|x|<2R)} < ϵ.

This shows that, for large i, u_{i} actually satisfies (51) which violates the contradiction hypothesis we start with. Lemma 5.1 is established.

Proof of Proposition 5.1. First we apply Lemma 5.1 with S = ϕ and d(y, S) ≡ 1. We have p > 5 − ϵ, y_{1} = y_{0} is a maximum point of u, and (51) holds. Next we take S = \overline{B_{\overline{r}_{1}}(y_{1})}, \overline{r}_{1} = Ru^{− \frac{p−1}{2}} (y_{1}). If max_{y∈M\S} d(y, S)^{2/(p−1)}u(y) ≤ C_{0}, we stop. Otherwise we obtain y_{2} = y_{0} given by Lemma 5.1. It is clear from the lemma that \overline{B_{\overline{r}_{1}}(y_{1})} ∩ \overline{B_{\overline{r}_{2}}(y_{2})} = ∅ since ϵ > 0 can be made very small from the beginning. We continue this process. The process has to stop after a finite number of times since each time ∫_{B_{\overline{r}_{i}}(y_{i})} |∇u|^{2} is greater than some positive universal constant. Thus we ob-

tain \{y_{1}, ⋯, y_{N}\} ⊂ M satisfying (ii), and d(y, ∪_{i=1}^{N}B_{\overline{r}_{i}}(y_{i}))^{2/(p−1)}u(y) ≤ C_{0}, for all y ∈ M \ ∪_{i=1}^{N}B_{\overline{r}_{i}}(y_{i}). Now for any y ∈ M, either y ∈ B_{2\overline{r}_{i}}(y_{i}) for some i, or d(y, y_{i}) > 2\overline{r}_{i}, ∀ 1 ≤ i ≤ N. In the first case, d(y, \{y_{1}, ⋯, y_{N}\}) ≤ d(y, y_{i}) < 2\overline{r}_{i}, so (50) implies u(y) ≤ 2u(y_{i}) = 2R^{\frac{2}{p−1}} \overline{r}_{i}^{−2/(p−1)}, and therefore d(y, {y_{1}, ⋯, y_{N}})^{2/(p−1)}u(y) ≤ 2(2R)^{2/(p−1)} ≤ 8R^{2}. In the second case, d(y, \{y_{1}, ⋯, y_{N}\}) ≤ 2d(y, ∪_{i=1}^{N}B_{\overline{r}_{i}}(y_{i})), therefore d(y, \{y_{1}, ⋯, y_{N}\})^{2/(p−1)}u(y) ≤ 2^{2/(p−1)}C_{0} ≤ 4C_{0}. Taking C_{1} = 8R^{2} + 4C_{0}, we have established Proposition 5.1.

We are ready to establish our main result in this section.

Proposition 5.2 Let k ∈ C^{1}(M), ϵ > 0 be sufficiently small and R > 1 be sufficiently large. Then there exists some positive constant δ^{∗} depending only on M, g, ϵ, R, and ∥k∥_{C^{1}(M)}, such that for all u ∈ ∪_{2≤p≤5}𝕄_{k,p} with max_{M} u ≥ C_{0} we have

d(y_{j}, y_{l}) ≥ δ^{∗}, for all 1 ≤ j ≠ l ≤ N,

where y_{j} = y_{j}(u), y_{l} = y_{l}(u), N = N(u) and C_{0} are the ones defined in Proposition 5.1.

Proof. The proof is similar to that of proposition 4.2 of [42]. Suppose the contrary: for some small ϵ > 0, large R, k_{i} → k in C^{1}(M), p_{i} ∈ [2, 5], p_{i} → p ∈ [2, 5], and u_{i} ∈ 𝕄_{k_{i},p_{i}} we have max_{M} u_{i} ≥ C_{0} and

lim_{i→∞} min_{j≠l}d(y_{j}(u_{i}), y_{l}(u_{i})) = 0.

Without loss of generality, we can assume that

σ_{i} := d(y_{1}(u_{i}), y_{2}(u_{i})) = min_{j≠l}d(y_{j}(u_{i}), y_{l}(u_{i})) → 0.

Since B_{Ru_{i}(y_{1})^{− \frac{p_{i}−1}{2}}} (y_{1})∩B_{Ru_{i}(y_{2})^{− \frac{p_{i}−1}{2}}} (y_{2}) = ∅, we have, in view of Propo-

sition 5.1 (i), that u_{i}(y_{1}), u_{i}(y_{2}) → ∞.

Let x = (x^{1}, x^{2}, x^{3}) be some geodesic geodesic normal coordinate system centered at y_{1}(u_{i}), exp_{y_{1}}(x) denote the exponential map, g = g_{αβ}(x)dx^{α}dx^{β} denote the metric in local coordinates. Let h_{αβ}(x) = g_{αβ}(σ_{i}x) and h = h_{αβ}(x)dx^{α}dx^{β} denote the scaled metric. Define the rescaled function

w_{i}(x) = σ_{i}^{2/(p_{i}−1)}u_{i}(exp_{y_{1}}(σ_{i}x)).

It follows that w_{i} satisfies

\{ \begin{matrix}−Δ_{h}w_{i}(x) + k̃_{i}(x)w_{i}(x) = w_{i}(x)^{p_{i}}, & in B_{\frac{1}{σ_{i}}} , \\ w_{i}(x) > 0 & in B_{\frac{1}{σ_{i}}} , \end{matrix}  (52)

where k̃_{i}(x) = σ_{i}^{2}k_{i}(exp_{y_{1}}(σ_{i}x)).

Let x_{j} = x_{j}(u_{i}) be the point so that exp_{y_{1}}(σ_{i}x_{j}) = y_{j} ∈ B\sqrt{σ_{i}}(y_{1}), j ∈ {1, ⋯, N}. It is clear that

x_{1} = 0,  min_{j≠l} |x_{j} − x_{l}| ≥ 1 + ∘(1),  |x_{2}| = 1 + ∘(1).

After passing to a subsequence, we have

x̄ = lim_{i→∞} x_{2}(u_{i}), |x̄| = 1.

It is also clear that

σ_{i} ≥ \frac{1}{C} max\{Ru_{i}(y_{1})^{− \frac{p_{i}−1}{2}} , Ru_{i}(y_{2})^{− \frac{p_{i}−1}{2}} \}.

We derive from from the above that

w_{i}(0), w_{i}(x_{2}) ≥ C′_{0},

Each x_{j} is a local maximum point of w_{i},

min_{1≤j≤N} |x − x_{j}|^{2/(p_{i}−1)}w_{i}(x) ≤ C_{1} ∀ |x| ≤ 1/σ_{i}.

We first show that

w_{i}(0) → ∞,  and w_{i}(x_{2}) → ∞.  (53)

If one of them tends to infinity along a subsequence, say w_{i}(0) → ∞. We know that {w_{i}} is an isolated blowup point, and therefore is an isolated simple blowup point. Then w_{i}(x_{2}) has to tend to infinity along the same sequence since otherwise, by an argument used in the proof of Lemma 3.2, {w_{i}} would be uniformly bounded near {x_{2}} along a further subsequence. In turn, using Proposition 3.1 and the Harnack inequality, we have {w_{i}} uniformly goes to zero near {x_{2}}, which violates w_{i}(x_{2}) ≥ C′_{0}. On the other hand, if both of {w_{i}(0)} and {w_{i}(x_{2})} stay bounded, we know from a similar argument as above that {w_{i}} is locally bounded. It then follows from standard elliptic estimates that after passing to a subsequence, w_{i} → w in C_{loc}^{3}(ℝ^{3}) where w satisfies −Δw = w^{5}, w > 0 in ℝ^{3} and ∇w(0) = ∇w(x̄) = 0. This is a contradiction since there is no positive solution to the equation with two distinct critical points according to the Liouville type theorem of Caffarelli, Gidas and Spruck. We have thus established (53).

Apparently {0} and x_{2} → x̄ are two isolated blowup points of {w_{i}}, and therefore according to Proposition 3.1 are isolated simple blowup points.

After passing to a subsequence, let S̃ denote the set of blowup points of {w_{i}}. Clearly 0, x̄ ∈ S̃ and d(x̂, x̃) ≥ 1 for any two distinct points x̂, x̃ in S̃. In view of Proposition 3.1, we know that {w_{i}(0)w_{i}} is locally bounded in ℝ^{3} \ S̃.

Multiplying (52) by w_{i}(0) and send i to infinity, we have, after passing to a subsequence, that

lim_{i→∞}w_{i}(0)w_{i} = h^{∗} in C_{loc}^{2}(ℝ^{3} \ S̃),

where h^{∗} is a regular harmonic function in ℝ^{3} \ S̃. Since all the blowup points of {w_{i}} are isolated simple blowup points, we know from Proposition 3.1 and Böcher′s theorem that

h^{∗}(x) = a_{1}|x|^{−1} + a_{2}|x − x̄|^{−1} + b^{∗}(x), ∀ x ∈ ℝ^{3} \ S̃,

where a_{1}, a_{2} > 0 are some positive constants, and b^{∗} is some regular harmonic function in ℝ^{3} \ {S̃ \ {0, x̄}}. We then derive from the maximum principle that b^{∗}(x) ≥ 0 for all x ∈ ℝ^{3} \ {S̃ \ {0, x̄}}. It follows that for some positive constant A > 0

h^{∗}(x) = a_{1}|x|^{−1} + A + O(|x|) for |x| close to 0.

This violates Proposition 3.2. Proposition 5.2 is established.

Proof of Theorem 0.2. Suppose the contrary, then there exist p_{i} → p ∈ (1, 5], k_{i} → k in C^{1}(M), u_{i} ∈ 𝕄_{k_{i},p_{i}} such that ∥u_{i}∥_{H^{1}(M)} → ∞, which, in view of standard elliptic estimates, implies that max_{M} u_{i} → ∞. In turn, it follows from the Liouville type theorem of Gidas and Spruck and some standard blow up argument as in the proof of Lemma 5.1 that p_{i} → 5. Applying Proposition 5.2, we know that for some small ϵ > 0, large R > 0, and some positive integer N, all independent of i, and y_{i}^{(1)} = y^{(1)}(u_{i}), ⋯, y_{i}^{(N)} = y^{(N)}(u_{i}) ∈ M such that (i)-(iii) in Proposition 5.1 hold. Arguing as in the proof of Proposition 5.2, we know that {y_{i}^{(1)}}, ⋯, {y_{i}^{(N)}} are isolated blowup points of {u_{i}}, and therefore, in view of Proposition 4.1, that they are isolated simp le blowup points of {u_{i}}. We conclude from (18) and Proposition 3.1 that {∥u_{i}∥_{H^{1}(M)}} is bounded. This is a contradiction. Theorem 0.2 is thus established.

6 Compactness properties of solutions

In this section, we study compactness properties of solutions of (5) and establish Theorem 0.3. Proof of Theorem 0.3. Due to standard elliptic estimates, we only need to establish the L^{∞} bound: u ≤ C. Suppose the contrary to the L^{∞} bound, then there exist p_{i} → p ∈ (1, 5], k_{i} → k in C^{1}(M) with

the first eigenvalue of (−Δ_{g} + k_{i}) ≥ c_{0} > 0, min_{y∈M} A_{k_{i},g}(y) ≥ c_{0} > 0,

for some positive constant c_{0} independent of i, and u_{i} ∈ 𝕄_{k_{i},g} such that max_{M} u_{i} → ∞. As in the proof of Theorem 0.2, we know, after passing to a subsequence, that p_{i} → 5 and {u_{i}} has N isolated simple blowup points y_{i}^{(1)} → y^{(1)}, ⋯, y_{i}^{(N)} → y^{(N)}, where N ≥ 1 is some positive integer independent of i. It follows from Proposition 3.1 that, after passing to a subsequence,

u_{i}(y_{i}^{(1)})u_{i}(y) → h(y) := ∑_{j=1}^{N} a_{j}G_{y^{(j)}}(y) + b(y) in C_{loc}^{2}(M \ {y^{(1)}, ⋯, y^{(N)}}),

where a_{1}, ⋯, a_{N} are some positive constants and b ∈ C^{2}(M) satisfies (−Δ_{g} + k)b = 0 in M. Since the first eigenvalue of k is positive, b is identically zero.

Let x = (x^{1}, x^{2}, x^{3}) be some geodesic normal coordinate system centered at y_{i}^{(1)}. It follows from the above discussion that for some positive constant A > 0 independent of i, we have

h(x) := h(exp_{y_{i}^{(1)}}(x)) = \frac{a_{1}}{3ω_{3}} |x|^{−1} + A_{i} + O(|x|^{α}) for |x| close to 0,

where A_{i} ≥ A > 0 for all i, ω_{3} is the volume of the unit ball in ℝ^{3} and 0 < α < 1. This violates Proposition 3.2. Theorem 0.3 is established.

7 More general equations

In this section we treat more general equations (8), still on a three dimensional smooth compact Riemannian manifold (M, g). Throughout this section we assume, unless otherwise stated, that k ∈ C^{1}(M), K ∈ C^{2}(M) and K > 0 on M. For 1 < p ≤ 5, we use 𝕄_{K,k,p} to denote the set of solutions of (8) in C^{2}(M). We first state the following generalization of Theorem 0.2 which gives a priori estimates in H^{1}(M) norm to solutions of (8).

Theorem 7.1 Let (M, g) be a three dimensional smooth compact Riemannian manifold. Then for all ϵ_{0} > 0,

∥u∥_{H^{1}(M)} ≤ C ∀ u ∈ ∪_{1+ϵ_{0}≤p≤5}𝕄_{K,k,p},

where C depends only on M, g, ϵ_{0}, ∥k∥_{C^{1}(M)}, ∥K∥_{C^{2}(M)}, and the positive lower bound of K on M.

We still use λ_{1} to denote the first eigenvalue of −Δ_{g} + k. Next theorem extends Theorem 0.3.

Theorem 7.2 Let (M, g) be a three dimensional smooth compact Riemannian manifold, k(y) ∈ C^{1}(M) with positive λ_{1}, and K(y) ∈ C^{2}(M) be a positive function. Assume that

min_{y∈M} A_{k,g}(y) > 0.

Then for all ϵ_{0} > 0,

1/C ≤ u ≤ C in M and ∥u∥_{C^{2}(M)} ≤ C, ∀ u ∈ ∪_{1+ϵ_{0}≤p≤5}𝕄_{K,k,p},

where C is some positive constant depending only on M, g, ϵ_{0}, ∥k∥_{C^{1}(M)}, ∥K∥_{C^{2}(M)}, and the positive lower bound of λ_{1}, A_{k,g}, and K.

We can derive existence results and multiplicity results (counting multiplicity) from the compactness results given in Theorem 7.2 as in the introduction.

Consider the following minimization problem

min{E(v) : v ∈ H^{1}(M), ∫_{M} Kv^{6} = 1},  (54)

where E(v) is the same as in (7). The following theorem extends Theorem 0.4.

Theorem 7.3 Under the hypothesis in Theorem 7.2, (54) is attained at some positive function v ∈ C^{2}(M).

We can derive Theorem 7.3 from Theorem 7.2 in the same way as we derive Theorem 0.4 from Theorem 0.3.

We introduce, for 0 < α < 1, an operator F̃ : C^{2,α}(M)^{+} → C^{2,α}(M) by

F̃(v) = v − ( − Δ_{g} + k(y))^{−1}(E(v)Kv^{5}).

Our next theorem extends Theorem 0.5.

Theorem 7.4 Under the hypothesis in Theorem 7.2 we have, for Λ large enough, that 0 does not belong to F̃(∂Ω_{Λ}) and

deg(F̃, Ω_{Λ}, 0) = −1.

In particular, 𝕄_{K,k,5} ≠ ϕ.

Using Theorem 7.2, we can prove Theorem 7.4 in the same way as in Schoen [51]. We omit the details. In the rest of this section, we establish Theorem 7.1 and Theorem 7.2. The proof is a combination of arguments in earlier sections together with some arguments in [42].

Remark 7.1 Let (M, g) = (𝕊^{3}, g_{0}), the standard three dimensional sphere, k ∈ C^{1}(𝕊^{3}) satisfy k(y) < \frac{3}{4} ≡ \frac{1}{8}R_{g_{0}}, and K ∈ C^{2}(𝕊^{3}) be a positive function. Then the hypothesis in Theorem 7.2 holds. Consequently the conclusions given in Theorem 7.2, Theorem 7.3 and Theorem 7.4 all hold.

Remark 7.2 Let (M, g) be a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere. Then, in view of the positive mass theorem of Schoen and Yau, there exists ϵ = ϵ(M, g) > 0 such that for all k ∈ C^{1}(M) satisfying k ≤ \frac{1}{8}R_{g} + ϵ on M and all positive K ∈ C^{2}(M), the hypothesis in Theorem 7.2 holds. Consequently the conclusions given in Theorem 7.2, Theorem 7.3 and Theorem 7.3 all hold.

Let Ω ⊂ M be an open set, {k_{i}} be a sequence of functions converging to k in C^{1}(M), {K_{i}} be a sequence of functions converging to some positive function K in C^{2}(M), {p_{i}} be a sequence of numbers satisfying 2 ≤ p_{i} ≤ 5, p_{i} → 5, and {u_{i}} be a sequence of C^{2} functions satisfying

−Δ_{g}u_{i} + k_{i}u_{i} = K_{i}u_{i}^{p_{i}}, u_{i} > 0,  in Ω ⊂ M.  (55)

The definitions of isolated blowup points and isolated simple blowup points are the same as in Definition 1.1 and Definition 1.2.

The Pohozaev type identity for a solution of

−Δ_{g}u + ku = Ku^{p}, u > 0, in Ω

takes the form

\begin{aligned}& − ∫_{B_{σ}}(k(x) + \frac{x·∇k(x)}{2})u^{2}dv + (\frac{3}{p+1} − \frac{1}{2}) ∫_{B_{σ}} Ku^{p+1}dv − A(g, u) \\ & +\frac{1}{p+1} ∫_{B_{σ}}(x · ∇K(x))u^{p+1} \\ = & ∫_{∂B_{σ}} B(σ, x, u, ∇u)ds − ∫_{∂B_{σ}} σ(\frac{ku^{2}}{2} − \frac{Ku^{p+1}}{p+1} )ds, \end{aligned} (56)

where all the notations are the same as in (13).

Now we present properties of isolated and isolated simple blowup points which are parallel to that in section 3. Similar to Lemma 3.1 and Lemma 3.2, we have

Lemma 7.1 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated blowup point. Then for any 0 < r < r̄/3, we have

max_{y∈B_{2r}(y_{i})\B_{\frac{r}{2}} (y_{i})} u_{i}(y) ≤ C_{1} min_{y∈B_{2r}(y_{i})\B_{\frac{r}{2}} (y_{i})} u_{i}(y),

where B_{s}(y_{i}) denotes the geodesic ball of radius s centered at y_{i}, and C_{1} is some positive constant independent of i and r.

Lemma 7.2 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated blowup point. Then for any R_{i} → +∞, ϵ_{i} → 0^{+}, we have, after passing to a subsequence {u_{j_{i}}} (still denoted as {u_{i}}, {y_{i}}, etc...), that

\begin{matrix}||u_{i}(y_{i})^{−1}u_{i}(exp_{y_{i}}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)) − (1 + \frac{1}{3}K(y_{i})|x|^{2})^{− \frac{1}{2}} ||_{C^{2}(B_{2R_{i}})} \\ +||u_{i}(y_{i})^{−1}u_{i}(exp_{y_{i}}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)) − (1 + \frac{1}{3}K(y_{i})|x|^{2})^{− \frac{1}{2}} ||_{H^{1}(B_{2R_{i}})} & ≤ ϵ_{i}, \end{matrix}

(57)

and

\frac{R_{i}}{log u_{i}(y_{i})} → 0 as i → ∞,  (58)

where x = (x^{1}, x^{2}, x^{3}) denotes some geodesic normal coordinates given by exp_{y_{i}}(x).

The proof of the above two lemmas is very similar to that of their counter parts, we omit the details.

Proposition 7.1 Let {u_{i}} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point, with (10) and (11) for all i. Then for some constant C depending only on ρ, C̃, ∥k_{i}∥_{C^{1}(Ω)}, ∥K_{i}∥_{C^{2}(Ω)}, and the positive lower bound of inf_{i} inf_{y∈Ω} K_{i}(y), we have

u_{i}(y) ≤ Cu_{i}(y_{i})^{−1}d(y, y_{i})^{−1}, for all d(y, y_{i}) ≤ ρ/2,

where ρ, C̃ are given in Definitions 1.1-1.2.

Furthermore, after passing to a subsequence, for some positive constant a > 0 and k(x) = lim_{i→∞} k_{i}(x),

u_{i}(y_{i})u_{i} → aG̃(·, ȳ) + b in C_{loc}^{2}(Bρ̃(\overline{y}) \ \{\overline{y}\}),

where ρ̃ = min\{δ_{0}, ρ/2\}, G̃ is given by (22) and b ∈ C^{2}(Bρ̃(\overline{y})) satisfies −Δ_{g}b + k(y)b = 0 in Bρ̃(\overline{y}).

Similarly, the above proposition will be established through a series of lemmas.

Lemma 7.3 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (57) and (58) hold. Then for any given 0 < δ < 1/100, there exists ρ_{1} ∈ (0, ρ) which is independent of i (but depending on δ), such that

u_{i}(y) ≤ C_{2}u_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−1+δ} for all R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1},

|∇_{g}u_{i}(y)| ≤ C_{2}u_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−2+δ} for all R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1},

and

|∇_{g}^{2}u_{i}(y)| ≤ C_{2}u_{i}(y_{i})^{−λ_{i}}d(y, y_{i})^{−3+δ} for all R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1},

where λ_{i} = (1 − δ)\frac{p_{i}−1}{2} − 1 and C_{2} is some positive constant independent of i.

Proof. The proof is basically the same as that of Lemma 3.3. One difference is to replace the operator ℓ_{i} there by ℓ_{i}φ = Δ_{g}φ + K_{i}(y)u_{i}(y)^{p_{i}−1}φ − k_{i}(y)φ. Other modification is obvious, we omit the details.

Parallel to Lemma 3.4 we have

Lemma 7.4 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (57) and (58) hold. Fixing 0 < δ < 1/100. Then for all fixed 0 < σ < ρ_{1}, we have for large i that

∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}u_{i}^{p_{i}+1} = O(u_{i}(y_{i})^{−4+∘(1)}),
∫_{d(y,y_{i})≤σ} u_{i}^{2} ≤ C_{3}σu_{i}(y_{i})^{−2λ_{i}},
∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}|∇u_{i}|^{2} ≤ O(u_{i}(y_{i})^{−4+∘(1)}) + C_{3}σu_{i}(y_{i})^{−2λ_{i}},
∫_{d(y,y_{i})≤σ} d(y, y_{i})u_{i}^{p_{i}+1} = O(u_{i}(y_{i})^{−2λ_{i}}),

where C_{3} is some constant independent of i and σ.

Proof. The proof is basically the same as that of Lemma 3.4 with obvious modification, we omit the details.

In the following, let (x^{1}, x^{2}, x^{3}) be some geodesic normal coordinate system centered at y_{i}. We use the notation explained at the beginning of Section 2.

Lemma 7.5 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blow up point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (57) and (58) hold. Then for 0 < σ < ρ_{1}, we have

|A(g, u_{i})| ≤ C_{3}σu_{i}(y_{i})^{−2λ_{i}},

where C_{3} is some constant independent of i and σ.

Proof. The proof is the same as that of Lemma 3.5 with obvious modification, we omit the details.

Parallel to Lemma 3.6, we have

Lemma 7.6 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (57) and (58) hold. Then

τ_{i} = O(u_{i}(y_{i})^{−2λ_{i}}).

Consequently,

u_{i}(y_{i})^{τ_{i}} → 1.

Proof. The proof is a modification of that of Lemma 3.6. For instance, we use (57) instead of (18), and (56) instead of Lemma 2.1.

Lemma 7.7 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (57) and (58) hold. Then for all 0 < σ < \overline{r}/2, we have

lim sup_{i→∞} max_{y∈∂B_{σ}(y_{i})} u_{i}(y)u_{i}(y_{i}) ≤ C(σ).

Proof. The proof is basically the same as that of Lemma 3.7 with obvious modification, we omit the details.

Proof of Proposition 7.1. The proof is basically the same as that of Proposition 3.1 with obvious modification, we omit the details.

As in section 3 we have the following three corollaries and one proposition parallel to Corollary 3.1-3.4.

Corollary 7.1 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (57) and (58) hold. Then there exists ρ_{1} ∈ (0, ρ) which is independent of i such that

|∇_{g}u_{i}(y)| ≤ C_{2}u_{i}(y_{i})^{−1}d(y, y_{i})^{−2} for all R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1},

and

|∇_{g}^{2}u_{i}(y)| ≤ C_{2}u_{i}(y_{i})^{−1}d(y, y_{i})^{−3} for all R_{i}u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} ≤ d(y, y_{i}) ≤ ρ_{1},

where C_{2} is some positive constant independent of i.

Corollary 7.2 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (57) and (58) hold. Then for all fixed 0 < σ < ρ_{1}, we have for large i that

∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}u_{i}^{p_{i}+1} = O(u_{i}(y_{i})^{−4+∘(1)}),
∫_{d(y,y_{i})≤σ} u_{i}^{2} ≤ C_{3}σu_{i}(y_{i})^{−2},
∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}|∇u_{i}|^{2} ≤ O(u_{i}(y_{i})^{−4+∘(1)}) + C_{3}σu_{i}(y_{i})^{−2},
∫_{d(y,y_{i})≤σ} d(y, y_{i})u_{i}^{p_{i}+1} = O(u_{i}(y_{i})^{−2}),
∫_{d(y,y_{i})≤σ} d(y, y_{i})|∇u_{i}|^{2} = O(u_{i}(y_{i})^{−2} log u_{i}(y_{i})),

and

∫_{d(y,y_{i})≤σ} d(y, y_{i})^{3}|∇^{2}u_{i}|^{2} = O(u_{i}(y_{i})^{−2} log u_{i}(y_{i})),

where C_{3} is some constant independent of i and σ.

Corollary 7.3 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blow up point. Assume R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}} are sequences with which (57) and (58) hold. Then

|A(g, u_{i})| ≤ C_{3}σu_{i}(y_{i})^{−2},

where A(g, u_{i}) is defined as in Lemma 2.1 and C_{3} is some constant independent of i and σ.

Corollary 7.4 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. After passing to some subsequence of {u_{i}}, the following estimates hold:

lim_{i→∞} u_{i}(y_{i})^{2} ∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}u_{i}^{p_{i}+1} = 0,

lim_{σ→0^{+}} lim sup_{i→∞} u_{i}(y_{i})^{2} ∫_{d(y,y_{i})≤σ} d(y, y_{i})^{2}|∇u_{i}|^{2} = 0,

lim_{σ→0^{+}} lim sup_{i→∞} u_{i}(y_{i})^{2} ∫_{d(y,y_{i})≤σ} u_{i}^{2} = 0,

and

lim_{σ→0^{+}} lim sup_{i→∞} u_{i}(y_{i})^{2}|A(g, u_{i})| = 0,

where A(g, u_{i}) is given in Lemma 2.1.

Similar to lemma 2.5-2.6 in [42] we have

Lemma 7.8 Let u_{i} satisfy (55) and y_{i} → ȳ ∈ Ω be an isolated simple blowup point. Then

τ_{i} ≡ 5 − p_{i} = O(u_{i}(y_{i})^{−2}), |∇K_{i}(y_{i})| = O(u_{i}(y_{i})^{−2} log u_{i}(y_{i})).

Proof. The estimate of τ_{i} follows immediately from the Pohozaev identity of u_{i} given by (56) and the estimates we have obtained for u_{i} near isolated simple blowup points. Let x = (x^{1}, x^{2}, x^{3}) be some geodesic normal coordinates centered at y_{i}, and η be some smooth cutoff function with η(x) = 1 for |x| ≤ σ/2 and η(x) = 0 for |x| ≥ σ. Multiplying (55) by η∂u_{i}/∂x_{j} and integrating by parts on |x| ≤ σ, we have, by using our estimates for u_{i} near isolated simple blowup points, that

| ∫_{|x|≤σ} \frac{∂K_{i}}{∂x_{j}} (x)u_{i}(x)^{p_{i}+1}dx| = O(u_{i}(0)^{−2} log u_{i}(0)).

Since K_{i} is bounded in C^{2} norm, it follows that

|∇K_{i}(0)| ≤ O(u_{i}(0)^{−2} log u_{i}(0))+C ∫_{|x|≤σ} |x|u_{i}(x)^{p_{i}+1}dx = O(u_{i}(0)^{−2} log u_{i}(0)).

Lemma 7.8 is established.

Now we can state the analogue of Proposition 3.2.

Proposition 7.2 Let u_{i} satisfy (55) and y_{i} → ȳ be an isolated simple blowup point with, for some ρ̃ > 0,

u_{i}(y_{i})u_{i} → h in C_{loc}^{2}(B_{ρ}(ȳ) \ {ȳ}).

Assume, for some a > 0, in some geodesic normal coordinate system x = (x^{1}, x^{2}, x^{3}), that

h(x) = \frac{a}{|x|} + A + ∘(1) as |x| → 0.

Then A ≤ 0.

The proof of Proposition 7.2 is very similar to that of Proposition 3.2. The only main difference is, due to the difference in the Pohozaev identities, that we need to establish an extra estimate:

lim_{i→∞} u_{i}(0)| ∫_{B_{σ}}(x · ∇K_{i}(x))u_{i}(x)^{p_{i}+1}dv| = 0.  (59)

This can be obtained by writing

∫_{B_{σ}}(x·∇K_{i}(x))u_{i}(x)^{p_{i}+1}dv = ∇K_{i}(0)·∫_{B_{σ}} xu_{i}(x)^{p_{i}+1}dv+O(∫_{B_{σ}} |x|^{2}u_{i}(x)^{p_{i}+1}dv).

Estimate (59) can then be deduced from Lemma 7.8 and Corollary 7.4.

Next we assert, as in section 4, that an isolated blowup point is in fact an isolated simple blowup point.

Proposition 7.3 Let u_{i} satisfy (55) and y_{i} → ȳ be an isolated blowup point. Then ȳ must be an isolated simple blowup point.

With Proposition 7.2, the above proposition as well as Theorem 7.1 and Theorem 7.2 can be established with minor modification of the arguments in earlier sections. We omit the details. In view of the positive mass theorem, Theorem 0.1 follows from results in this section. Proof of Theorem 0.1. The a priori estimates follow from Theorem 7.2 and the positive mass theorem of Schoen and Yau. The degree counting of all solutions follows from the a priori estimates and arguments in [51].

8 Existence of minimizers

Let (M, g) be a n dimensional (n ≥ 3) smooth compact Riemannian manifold, k ∈ C^{1}(M) with positive λ_{1}, and K ∈ C^{2}(M) be a function which is positive somewhere.

Consider, for 1 < p ≤ (n+2)/(n−2), the following minimization problem:

S_{p} = inf{∫_{M}(|∇_{g}u|^{2} + ku^{2}) | u ∈ H^{1}(M), ∫_{M} K|u|^{p+1} = 1}.

It is easy to see that for 1 < p < (n+2)/(n−2), S_{p} is achieved at some positive function, denoted as u_{p}, and S_{p} → S_{(n+2)/(n−2)} ∈ (0, ∞) as p → (n+2)/(n−2) from the left.

The Euler-Lagrange equation of u_{p} is

−Δ_{g}u_{p} + ku_{p} = S_{p}Ku_{p}^{p}, u_{p} > 0,  on M.  (60)

Lemma 8.1 Let k_{i} → k in C^{0}(M), K_{i} → K in C^{0}(M), p_{i} ≤ (n+2)/(n−2), p_{i} → (n + 2)/(n − 2), and u_{i} satisfy

−Δ_{g}u_{i} + k_{i}u_{i} = K_{i}u_{i}^{p_{i}}, u_{i} > 0,  on M,  (61)

and

{∥u_{i}∥_{H^{1}(M)}} be uniformly bounded.

Then for some positive numbers ϵ_{0} and C independent of i,

u_{i} ≤ C on {y ∈ M | K(y) ≤ ϵ_{0}}.

Proof. Suppose the contrary, then after passing to a subsequence, there exists ỹ_{i} → ỹ ∈ M such that u_{i}(ỹ_{i}) → ∞ and K(ỹ) ≤ 0. Pick ϵ_{i} → 0^{+} so that the set

O_{i} := {y ∈ M | K_{i}(y) < ϵ_{i}}

has the property that

d(ỹ_{i}, ∂O_{i})^{\frac{2}{p_{i}−1}} u_{i}(ỹ_{i}) → ∞.  (62)

Consider the following function on O_{i}:

f_{i}(y) = d(y, ∂O_{i})^{\frac{2}{p_{i}−1}} u_{i}(y),

and let y_{i} ∈ O_{i} denote a maximum point of f_{i} on O_{i}, namely,

f_{i}(y_{i}) = max_{O_{i}} f_{i}.

We know from (62) that f_{i}(y_{i}) → ∞.

Let σ_{i} = \frac{1}{2}d(y_{i}, ∂O_{i}) > 0, clearly we have

σi^{\frac{2}{p_{i}−1}} max_{B_{σ_{i}}(y_{i})} u_{i} ≥ 2^{− \frac{2}{p_{i}−1}} f_{i}(y_{i}) → ∞,

(2σ_{i})^{\frac{2}{p_{i}−1}} u_{i}(y_{i}) ≥ max_{B_{σ_{i}}(y_{i})} f_{i} ≥ σi^{\frac{2}{p_{i}−1}} max_{B_{σ_{i}}(y_{i})} u_{i}.

Consequently,

σi^{\frac{2}{p_{i}−1}} u_{i}(y_{i}) → ∞,

and

u_{i}(y_{i}) ≥ 2^{− \frac{2}{p_{i}−1}} max_{B_{σ_{i}}(y_{i})} u_{i}.  (63)

Let x = (x^{1}, ⋯, x^{n}) denote some geodesic normal coordinates centered at y_{i} and we set

w_{i}(x) = u_{i}(y_{i})^{−1}u_{i}(exp_{y_{i}}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)), |x| < u_{i}(y_{i})^{\frac{p_{i}−1}{2}} σ_{i} → ∞.

Clearly, w_{i} satisfies, in view of (63),

\{ \begin{matrix}−Δ_{h}w_{i}(x) = K̃_{i}(x)w_{i}(x)^{p_{i}} − u_{i}(y_{i})^{1−p_{i}} k̃_{i}(x)w_{i}(x), & |x| < σ_{i}u_{i}(y_{i})^{\frac{p_{i}−1}{2}} , \\ 2^{− \frac{2}{p_{i}−1}} w_{i}(x) ≤ w_{i}(0) = 1, & |x| < σ_{i}u_{i}(y_{i})^{\frac{p_{i}−1}{2}} . \end{matrix}

(64)

where K̃_{i}(x) = K_{i}(exp_{y_{i}}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)), k̃_{i}(x) = k_{i}(exp_{y_{i}}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)) and h = h_{αβ}(x)dx^{α}dx^{β} = g_{αβ}(u_{i}(y_{i})^{− \frac{p_{i}−1}{2}} x)dx^{α}dx^{β} denote the scaled metric.

On the other hand, we know from Sobolev embedding theorems that ∫_{M} u_{i}^{p_{i}+1} ≤ C∥u_{i}∥_{H^{1}(M)}^{p_{i}+1} ≤ C. It follows that

∫_{|x|≤σ_{i}u_{i}(y_{i})^{\frac{p_{i}−1}{2}}} w_{i}^{p_{i}+1} ≤ Cu_{i}(y_{i})^{\frac{(p_{i}−1)n}{2} −(p_{i}+1)} ∫_{M} u_{i}^{p_{i}+1} ≤ C.  (65)

Sending i to ∞, we derive from (64), (65), and standard elliptic theories that, after passing to a subsequence,

w_{i} → w in C_{loc}^{2}(ℝ^{n}),

where w ∈ L^{2n/(n−2)}(ℝ^{n}) satisfies, for some constant K̄ ≤ 0,

−Δw = K̄w^{\frac{n+2}{n−2}} , w > 0, in ℝ^{n}.

It is well known that the above equation has no solution for K̄ < 0; while for K̄ = 0, all solutions are positive constants and therefore can not be in L^{2n/(n−2)}(ℝ^{n}). This leads to a contradiction. Lemma 8.1 is established.

Next we state our main result in this section concerning the compactness of finite energy solutions on three dimensional manifolds when K is allowed to change signs.

Theorem 8.1 Let (M, g) be a three dimensional smooth compact Riemannian manifold, k in C^{1}(M) with positive λ_{1}, K ∈ C^{2}(M) be positive somewhere. Assume that

min_{y∈M} A_{k,g}(y) > 0.

Then for any p_{i} ≤ 5 with p_{i} → 5, k_{i} → k in C^{1}(M), K_{i} → K in C^{2}(M), and any solutions u_{i} of (61) with bounded energy {∥u_{i}∥_{H^{1}(M)}}, we have

1/C ≤ u_{i} ≤ C  and ∥u_{i}∥_{C^{3}(M)} ≤ C,  ∀ i,

where C is some constant independent of i.

Corollary 8.1 Let (M, g) be a three dimensional smooth compact Riemannian manifold, k ∈ C^{1}(M) with positive λ_{1}. Assume that

min_{y∈M} A_{k,g}(y) > 0.

Then for any K ∈ C^{2}(M) which is positive somewhere, S_{5} is attained at some positive function. In particular, 𝕄_{K,k,5} ≠ ϕ.

Remark 8.1 When (M, g) is a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere, and k = \frac{1}{8}R_{g}, it follows from the positive mass theorem that min_{M} A_{\frac{1}{8} R_{g},g} > 0, and Corollary 8.1 was already proved by Escobar and Schoen in [29].

Proof of Theorem 8.1. It follows from Lemma 8.1 that for some ϵ_{0} > 0, {u_{i}} is uniformly bounded in the region where K ≤ ϵ_{0}. Due to our results in section 7, {u_{i}} is either uniformly bounded on M, or, after passing to a subsequence, it has finitely many isolated simple blowup points in the region where K ≥ ϵ_{0}. If {u_{i}} is uniformly bounded, then higher derivative estimates follow from standard elliptic theories. Otherwise, let y_{i}^{(1)} → ȳ^{(1)}, ⋯, y_{i}^{(m)} → ȳ^{(m)} be all the isolated simple blowup points. We know from Proposition 7.1 that {u_{i}(y_{i}^{(1)})u_{i}} is uniformly bounded on any compact subset of {y ∈ M∥K(y) ≥ ϵ_{0}/2}\{ȳ^{(1)}, ⋯, ȳ^{(m)}}. Furthermore, using the fact {u_{i}} is uniformly bounded on {y ∈ M∥K(y) ≤ ϵ_{0}} and the Harnack inequality, we know that {u_{i}(y_{i}^{(1)})u_{i}} is uniformly bounded on any compact subset of M \ {ȳ^{(1)}, ⋯, ȳ^{(m)}}.

Now, as usual, we can write down the Pohozaev type identity of u_{i} in B_{σ}(y_{i}^{(1)}), and reach a contradiction. Here the global hypothesis min_{y∈M} A_{k,g}(y) > 0 is being used. This shows that the minimizing sequence {u_{i}} is in fact uniformly bounded. Theorem 8.1 is thus established.

9 Appendix: Positive solutions in punctured balls

In this appendix, we provide some well known descriptions on singular behaviors of positive solutions to some linear elliptic equations in punctured balls. For n ≥ 3, let B_{r} denote the ball in ℝ^{n} of radius r and centered at the origin. Throughout this appendix, g = g_{ij}dx^{i}dx^{j} denotes some smooth Riemannian metric in B_{1} and k(x) ∈ C^{1}(B_{1}).

Lemma 9.1 Suppose u ∈ C^{2}(B_{1} \ {0}) is a solution of

−Δ_{g}u + k(x)u = 0, in B_{1} \ {0},

and u(x) = o(|x|^{2−n}) as |x| → 0, then u ∈ C^{2,α}(B_{1/2}) for any 0 < α < 1.

Proof. We first show that −Δ_{g}u(x) + k(x)u(x) = 0 in B_{1} in the distribution sense. For any ϵ > 0, let ξ_{ϵ} be some cutoff function:

ξ_{ϵ}(x) = \{ \begin{matrix}1 & for |x| ≤ ϵ, \\ 0 & for |x| ≥ 2ϵ, \\ |∇_{g}ξ_{ϵ}| < \frac{C}{ϵ} , & |∇_{g}^{2}ξ_{ϵ}| < \frac{C}{ϵ^{2}} . \end{matrix}

Then for any ϕ ∈ C_{c}^{∞}(B_{1}) we have

− ∫_{B_{1}} Δ_{g}(ϕ(1 − ξ_{ϵ}))u + ∫_{B_{1}} kuϕ(1 − ξ_{ϵ}) = 0.

It follows that as ϵ tends to zero,

| − ∫_{B_{1}} Δ_{g}ϕu + ∫_{B_{1}} k(x)uϕ| ≤ Cϵ^{−2} ∫_{B_{2ϵ}\B_{ϵ}} |u| + C ∫_{B_{ϵ}} |u| = ∘(1),

where we have used u(x) = o(|x|^{2−n}) in the last step.

We know from u(x) = o(|x|^{2−n}) that u ∈ L_{loc}^{s}(B_{1}) for s < \frac{n}{n−2}. By W^{2,s} estimates we have u ∈ W_{loc}^{2,s}(B_{1}). The lemma then follows from standard bootstrap methods and elliptic estimates.

Lemma 9.2 There exists some constant δ_{0} > 0 depending on n, ∥g_{ij}∥_{C^{2}(B_{1})} and ||k(x)||_{L^{∞}}, such that the maximum principle holds for −Δ_{g} +k(x) on B_{δ_{0}}; and there exists a unique G(x) ∈ C^{2}(B_{δ_{0}} \ {0}) satisfying

−Δ_{g}G + k(x)G = 0 in B_{δ_{0}} \ {0}
G = 0,  on ∂B_{δ_{0}},

lim_{|x|→0}|x|^{n−2}G(x) = 1.

Furthermore, G(x) = |x|^{2−n} + R(x) where R(x) satisfies for all 0 < ϵ < 1 that

|x|^{n−4+ϵ}|R(x)| + |x|^{n−3+ϵ}|∇R(x)| ≤ C(ϵ), ∀ x ∈ B_{δ_{0}}, n ≥ 4,

and

|x|^{ϵ−1}|R(x) − R(0)| + |x|^{ϵ}|∇R(x)| ≤ C(ϵ), ∀ x ∈ B_{δ_{0}}, n = 3,

where C(ϵ) is some constant depending only on ϵ, n, ∥g_{ij}∥_{C^{2}(B_{1})}, and ∥k∥_{L^{∞}(B_{1})}.

Proof. See, e.g., Appendix B in [45].

Lemma 9.3 Assume u(x) ∈ C^{2}(B_{1} \ {0}) satisfies

−Δ_{g}u + k(x)u = 0, u > 0, in B_{1} \ {0},  (66)

then

a = \overline{lim_{r→0}} max_{|x|=r} u(x)|x|^{n−2} < +∞.

Proof. It follows from the Harnack inequality that there exists C > 0 such that for 0 < r < 1,

max_{|x|=r} u(x) ≤ C min_{|x|=r} u(x).

Suppose Lemma 9.3 were false, then a = +∞.Therefore for all A > 0, there exists r_{i} → 0^{+} such that

u(x) > A|x|^{2−n}, ∀ |x| = r_{i}.

Set v_{A}(x) = \frac{A}{2}G(x), where G(x) was defined in Lemma 9.2. It follows from the maximum principle that for large i

u(x) ≥ v_{A}(x), ∀ r_{i} ≤ |x| ≤ δ_{0}.

Sending i to ∞, we have

u(x) ≥ v_{A}(x) = \frac{A}{2}G(x), ∀ 0 < |x| < δ_{0}.

Sending A to ∞, we reach a contradiction.

Proposition 9.1 Suppose u ∈ C^{2}(B_{1} \ {0}) satisfies

−Δ_{g}u(x) + k(x)u(x) = 0, u(x) > 0, in B_{1} \ {0}.

Then there exists some constant b ≥ 0 such that

u(x) = bG(x) + E(x), in B_{δ_{0}} \ {0},

where G(x), δ_{0} are defined in Lemma 9.2, and E(x) ∈ C^{2}(B_{1}) satisfies

−Δ_{g}E(x) + k(x)E(x) = 0, in B_{1}.

Proof. Set

b = b(u) = sup{λ ≥ 0 | λG ≤ u in B_{δ_{0}} \ {0}}.  (67)

We know from Lemma 9.3 that 0 ≤ b ≤ a < ∞.

Case 1: b = 0.

In this case we claim: for any ϵ > 0, there exists r_{ϵ} ∈ (0, δ_{0}) such that

min_{|x|=r}{u(x) − ϵG(x)} ≤ 0 ∀ 0 < r < r_{ϵ}.

If the above claim were false, then there would exist some ϵ_{0} > 0 and r_{j} → 0^{+} such that

min_{|x|=r_{j}}{u(x) − ϵ_{0}G(x)} > 0.

Notice that u(x) − ϵ_{0}G(x) ≥ 0 for |x| = δ_{0}. We derive from the maximum principle that u(x)−ϵ_{0}G(x) ≥ 0 on B_{δ_{0}} \B_{r_{j}}. It follows that u(x)−ϵ_{0}G(x) ≥ 0 in B_{δ_{0}} \ {0} which implies that b ≥ ϵ_{0} > 0, a contradiction.

Therefore, for any ϵ > 0, and 0 < r < r_{ϵ}, there exists x_{ϵ} with |x_{ϵ}| = r, such that u(x_{ϵ}) ≤ ϵG(x_{ϵ}). By the Harnack inequality, we have

max_{|x|=r} u(x) ≤ Cu(x_{ϵ}) ≤ CϵG(x_{ϵ}).

It follows that

u(x) = o(|x|^{2−n}) as |x| → 0.

Setting E(x) = u(x), our result in this case follows from Lemma 9.1. Case 2: b > 0. We consider v(x) = u(x) − bG(x). From the definition of b(u), we know

that v(x) ≥ 0 . By the maximum principle, we know that either v(x) = 0 or v(x) > 0 in B_{δ_{0}} \ {0}. In the former case we are done by choosing E(x) = 0. In the latter case, v(x) satisfies (66). Set

b(v) = sup{μ ≥ 0 | μG ≤ v in B_{δ_{0}} \ {0}}.

It is easy to see that b(v) = 0. As in Case 1, we know v(x) = o(|x|^{2−n}). Letting E(x) = v(x), our result in this case follows from Lemma 9.1.

The following corollary, used in the proof of Lemma 3.7, is a simple consequence of Proposition 9.1 and Lemma 9.2.

Corollary 9.1 For n ≥ 3, let u be a solution of (66) which is singular near the origin. Then

lim_{r→0} ∫_{∂B_{r}} \frac{∂_{g}u}{∂ν} = b · lim_{r→0} ∫_{∂B_{r}} \frac{∂_{g}G}{∂ν} = −(n − 2)|𝕊^{n−1}|b,

where b > 0 was defined in (67) and ν denotes the unit outer normal.

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