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\controldates{9-DEC-2004,9-DEC-2004,9-DEC-2004,9-DEC-2004}
 
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\copyrightinfo{2004}{American Mathematical Society} 
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\begin{document}
\title[The geometric Paneitz equation]
{Compactness and global estimates
for the geometric Paneitz equation in high dimensions}
\author{Emmanuel Hebey}
\address{Universit\'e de Cergy-Pontoise, 
D\'epartement de Math\'ematiques, Site de 
Saint-Martin, 2 avenue Adolphe Chauvin, 
95302 Cergy-Pontoise cedex, 
France}
\email{Emmanuel.Hebey@math.u-cergy.fr}

\author{Fr\'ed\'eric Robert}
\address{Laboratoire J.A.Dieudonn\'e, Universit\'e de 
Nice Sophia-Antipolis, 
Parc Valrose, 06108 Nice cedex 2, France}
\email{frobert@math.unice.fr}
\date{October 7, 2004}
\subjclass[2000]{Primary: 58E30, 58J05}
\keywords{Blow-up behavior, compactness, Paneitz operator}
\commby{Tobias Colding}
\begin{abstract} Given $(M,g)$, a smooth compact Riemannian manifold of
dimension $n \ge 5$,  we investigate compactness for the fourth order
geometric equation  $P_gu = u^{2^\sharp-1}$, where $P_g$ is the Paneitz
operator, and $2^\sharp = 2n/(n-4)$ is  critical from the Sobolev viewpoint. We
prove that the equation is compact when the  Paneitz operator is of strong
positive type. 
\end{abstract}
\maketitle

In 1983, Paneitz \cite{Pan} introduced a 
conformally fourth order operator defined on $4$-dimensional 
Riemannian manifolds. Branson \cite{Bra} 
generalized the definition to 
$n$-dimensional Riemannian manifolds, $n \ge 5$.  
While the conformal Laplacian is associated to the scalar curvature, 
the geometric 
Paneitz-Branson operator is associated to a notion of $Q$-curvature. The $Q$-curvature in dimension $4$, 
and for conformally flat manifolds, is the integrand in the Gauss-Bonnet formula for the Euler characteristic. 
In this article we let $(M,g)$ be a smooth compact conformally flat Riemannian $n$-manifold, $n \ge 5$, and 
consider the geometric Paneitz equation
\begin{equation}\label{GenEqtCompTh}
P_gu = u^{2^\sharp-1}\hskip.1cm ,
\end{equation}
where $P_g$ is the Paneitz operator in dimension $n \ge 5$, 
$u$ is required to be positive, and $2^\sharp = \frac{2n}{n-4}$ is the critical exponent for the Sobolev
embedding. The Paneitz operator in dimension $n \ge 5$ reads as
$$P_gu = \Delta_g^2u - \operatorname{div}_g\left(A_gdu\right) + \frac{n-4}{2}Q_gu\hskip.1cm ,$$ 
where $\Delta_g = -\hbox{div}_g\nabla$ is the Laplace-Beltrami operator, 
$Q_g$ is the $Q$-curvature of $g$, $A_g$ is the smooth symmetrical $(2,0)$-tensor
field given by
$$A_g = \frac{(n-2)^2 + 4}{2(n-1)(n-2)}S_gg - \frac{4}{n-2}Rc_g\hskip.1cm ,$$
and $Rc_g$ and $S_g$ are respectively the Ricci curvature and scalar curvature of $g$. The Paneitz operator 
is conformally invariant in the sense that if $\tilde g = u^{4/(n-4)}g$ is conformal to $g$, then
$P_{\tilde g}(f) = u^sP_g(uf)$ for all $f \in C^\infty(M)$, where $s = 1-2^\sharp$. 
From the viewpoint of conformal geometry, equation (\ref{GenEqtCompTh}) 
turns out to be the natural fourth order analogue of the 
second order Yamabe equation. We refer to 
Chang \cite{Cha} and Chang and Yang \cite{ChaYan} for more details on the above definitions.

\medskip In what follows we let $H_2^2(M)$ be the Sobolev space 
consisting of functions in $L^2(M)$ with two derivatives in $L^2$. As shown in Hebey and Robert \cite{HebRob}, up to 
passing to a subsequence, bounded sequences $(u_\alpha)$ in $H_2^2(M)$ of 
nonnegative solutions of (\ref{GenEqtCompTh}) split into the sum of a nonnegative solution $u^0$,
namely the weak limit of the $u_\alpha$, 
a finite sum of $k$ bubbles $(B_\alpha^i)$, obtained by rescaling positive solutions of the 
Euclidean equation $\Delta^2u = u^{2^\sharp-1}$, 
and a remainder $R_\alpha$ which converges strongly to zero in $H_2^2(M)$ as $\alpha \to +\infty$. 
This splitting provides exact asymptotics for the $u_\alpha$ in the Sobolev setting. Following 
standard terminology, we say that equation (\ref{GenEqtCompTh}) is compact if for any 
bounded sequence $(u_\alpha)$ in $H_2^2(M)$ of nonnegative solutions of (\ref{GenEqtCompTh}), 
we necessarily have that $k = 0$ in such decompositions. Regularity theory 
(see for instance 
Esposito-Robert \cite{EspRob}) holds 
for (\ref{GenEqtCompTh}). Then, thanks to Agmon-Douglis-Nirenberg-type
estimates \cite{AgmDouNir1,AgmDouNir2}, 
and estimates like the ones developed in Hebey, Robert, and Wen \cite{HebRobWen}, an equivalent definition 
for compactness is that 
bounded sequences in $H_2^2(M)$ of nonnegative solutions of (\ref{GenEqtCompTh}) are actually bounded in 
$C^{4,\theta}(M)$. A major stress in studying compactness is to understand large solutions. 
Namely, solutions with large energies which, in studying their 
possible blow-up, involve multi-bubbles which may interact with each other 
on the pointwise level.

\medskip Compactness for fourth order equations like (\ref{GenEqtCompTh}) was
recently studied in  Hebey, Robert, and Wen \cite{HebRobWen}. It was shown in
\cite{HebRobWen} that equations like (\ref{GenEqtCompTh})  are compact as long
as they are not close to the geometric equation (\ref{GenEqtCompTh}).  In this
note we investigate compactness  for the geometric equation
(\ref{GenEqtCompTh}) and show how the analysis developed in  \cite{HebRobWen}
provides an answer to the question of whether (\ref{GenEqtCompTh}) is compact
or not.  The result we get is the fourth order analogue of the result proved 
in Schoen \cite{Sch}, where  the second order Yamabe equation was investigated.
As in  Hebey, Robert, and Wen \cite{HebRobWen}, and also Schoen \cite{Sch},  we
assume in what follows  that $(M,g)$ is conformally flat (and hence, since $n
\ge 5$, that the Weyl tensor of $g$ is zero).  We let $G_g$ be the Green's
function of $P_g$. The Green's function is unique if $P_g$ is positive.  By
conformal invariance of the Paneitz operator, if  $\tilde g = u^{4/(n-4)}g$ is
a conformal metric to $g$, then 
\begin{equation}\label{GrFct1} G_{\tilde
g}(x,y) = \frac{G_g(x,y)}{u(x)u(y)}\hskip.1cm . \end{equation} 
It is known that
if $\tilde g$ is a flat metric around some $x_0 \in M$, then
\begin{equation}\label{GrFct2} G_{\tilde g}(x_0,x) = \frac{\lambda_n}{d_{\tilde
g}(x_0,x)^{n-4}} + \mu_{\tilde g}(x_0,x)\hskip.1cm , 
\end{equation} 
where
$\lambda_n^{-1} = 2(n-2)(n-4)\omega_{n-1}$, $\omega_{n-1}$ is the volume  of
the unit $(n-1)$-sphere, and the  function $x \to \mu_{\tilde g}(x_0,x)$ is
smooth on $M$. Combining the above two equations,  noting that conformal
changes of metrics which leave a metric flat around one point come from 
conformal diffeomorphisms of the Euclidean space,  we easily get (as in Schoen
and Yau \cite{SchYau} for the conformal Laplacian)  that if $g$ and $\tilde g =
u^{4/(n-4)}g$ are conformal metrics, both being flat around $x_0$, then 
\begin{equation}\label{GrFct3} \mu_{\tilde g}(x_0,x_0) =
\frac{\mu_g(x_0,x_0)}{u(x_0)^2}\hskip.1cm . \end{equation} 
In particular, by
(\ref{GrFct3}), the sign of $\mu_g(x_0) = \mu_g(x_0,x_0)$  does not depend on
the choice of the metric in $[g]_{x_0}$, where $[g]_{x_0}$ stands for the set
of conformal metrics to $g$ which are flat around $x_0$.

\medskip In what follows we say that $P_g$ is of {\it strong positive type} if
$P_g$ is positive, $G_g$ is positive,  and for any $x \in M$ there exists 
$\tilde g \in [g]_x$ such that  $\mu_{\tilde g}(x) > 0$. For example, the
Paneitz operator on quotients of the  unit sphere is of strong positive type.
Positivity of the Paneitz operator was studied in Xu and Yang \cite{XuYan}. The
main result of this note  is:

\begin{thm}\label{MainTh} The geometric equation \eqref{GenEqtCompTh} is
compact on compact conformally flat manifolds  of dimensions $n \ge 5$ with
Paneitz operator of strong positive type. \end{thm}

Let $(M,g)$ be a smooth compact conformally flat manifold of dimension $n \ge 5$ with positive Paneitz operator 
$P_g$, and positive Green's function $G_g$. 
Given $S \subset M$, let $[g]_S$ be the set of conformal metrics to $g$ 
which are flat in a neighborhood of $S$. We prove Theorem \ref{MainTh} by proving that if 
$(u_\alpha)$ is a bounded sequence in $H_2^2(M)$ of nonnegative solutions of (\ref{GenEqtCompTh}) which 
blows up with geometric blow-up points $S = \left\{x_1,\dots,x_N\right\}$, then 
$u_\alpha \rightharpoonup 0$ in $H_2^2(M)$ as $\alpha \to +\infty$, and for any $\tilde g \in [g]_S$, there 
exist $\lambda_{i,j} > 0$ such that for any $i = 1,\dots,N$,
\begin{equation}\label{MainEqtCompTh}
\mu_{\tilde g}(x_i) + \sum_{j \not= i}\lambda_{i,j}G_{\tilde g}(x_i,x_j) = 0\hskip.1cm .
\end{equation}
Theorem \ref{MainTh} clearly follows from (\ref{MainEqtCompTh}). 
We prove equation (\ref{MainEqtCompTh}) in the rest of this note, using the material proved in 
Hebey, Robert, and Wen \cite{HebRobWen}.

\section{Proof of the result}

\medskip In what follows we prove (\ref{MainEqtCompTh}), and thus Theorem \ref{MainTh}. 
For that purpose we let $(u_\alpha)$ be a bounded sequence
in $H_2^2(M)$ of nonnegative nontrivial solutions of (\ref{GenEqtCompTh}). 
Since the Green's function $G_g$ of $P_g$ is positive, the $u_\alpha$ are positive. In the sequel, everything 
is up to a subsequence. We know from Hebey and Robert \cite{HebRob} that 
there exist $k \in {\mathbb N}$, a nonnegative solution $u^0 \ge 0$ 
of (\ref{GenEqtCompTh}), and $k$ bubbles $(B_\alpha^i)$, 
$i = 1,\dots,k$, such that
\begin{equation}\label{SobDecCompTh}
u_\alpha = u^0 + \sum_{i=1}^kB_\alpha^i + R_\alpha\hskip.1cm,
\end{equation}
where $R_\alpha \to 0$ in $H_2^2(M)$ as $\alpha \to +\infty$. By contradiction, we assume that $k \ge 1$, and 
let $S = \left\{x_1,\dots,x_N\right\}$ be the geometric blow-up point
set consisting of the limits of the centers of the bubbles 
$(B_\alpha^i)$. Since bubbles may accumulate one on another (there are such examples for equations 
like (\ref{GenEqtCompTh}); we refer to Hebey, Robert, and Wen \cite{HebRobWen}), $N$ might be less than $k$. By conformal
invariance we may assume that $g = \tilde g$ is flat around the points in $S$. Then 
the geometric Paneitz equation (\ref{GenEqtCompTh}) reduces to $\Delta_g^2u_\alpha = u_\alpha^{2^\sharp-1}$ 
around the points in $S$.

\medskip As a preliminary step in the proof of (\ref{MainEqtCompTh}), 
we come back to the estimates proved in Hebey, Robert, and Wen \cite{HebRobWen} and explain 
why they are still valid in the present context. A rough argument would be that blow-up phenomena are local in nature,
while the Paneitz operator on conformally flat manifolds is 
locally, up to conformal changes of the metric, like the Paneitz operator on the sphere (and hence with 
positive constant 
coefficients as in \cite{HebRobWen}). More details follow. First we note that 
the standard procedure to get rescaling invariant pointwise estimates, as developed in 
\cite{HebRobWen} for fourth order operators, together with Agmon-Douglis-Nirenberg-type 
estimates for fourth order operators, give that there exists $C > 0$ such that, 
\begin{equation}\label{PointEstCompTh}
\Bigl(\min_{1 \le i \le k}d_g(x_\alpha^i,x)\Bigr)^{\frac{n-4}{2}}
\big\vert u_\alpha(x) - u^0(x)\big\vert \le C
\end{equation}
for all $\alpha$ and all $x$, where $u^0$, the weak limit of the $u_\alpha$, is as in (\ref{SobDecCompTh}), 
and where the $x_\alpha^i$, $i = 1,\dots,k$, are the centers of the bubbles $(B_\alpha^i)$ 
in (\ref{SobDecCompTh}). In particular, we get with (\ref{PointEstCompTh}) that $u_\alpha \to u^0$ in 
$C^4_{loc}(M\backslash S)$ as $\alpha \to +\infty$. In addition to (\ref{PointEstCompTh}), 
if $\Phi_\alpha(x)$ stands for the left hand side in (\ref{PointEstCompTh}), 
we also have that
\begin{equation}\label{PointEstBisCompTh}
\lim_{R\to+\infty}\lim_{\alpha\to+\infty}\sup_{x \in M\backslash\Omega_\alpha(R)}
\Phi_\alpha(x)^{\frac{n-4}{2}} = 0\hskip.1cm ,
\end{equation}
where, for $R > 0$, 
$\Omega_\alpha(R)$ is given by 
$\Omega_\alpha(R) = \bigcup_{i=1}^kB_{x_\alpha^i}(R\mu_\alpha^i)$, and the 
$\mu_\alpha^i$
are the weights of the bubbles $(B_\alpha^i)$ in (\ref{PointEstCompTh}). 
Going on with the estimates in Hebey, Robert, and Wen \cite{HebRobWen} we may assume, up to renumbering, 
and up to passing to a subsequence, that $(B_\alpha^1)$ 
is the bubble in (\ref{SobDecCompTh}) with the largest weight. Then we let 
the $x_\alpha$
and $\mu_\alpha$ 
be such that $x_\alpha = x_\alpha^1$ and 
$\mu_\alpha = \mu_\alpha^1$ for all $\alpha$. A preliminary 
remark is that the global splitting estimate
$\Vert u_\alpha\Vert_{p_1,p_2,\mu_\alpha^{-1}} \le C$ in \cite{HebRobWen} easily follows from the positivity of the 
Green's function with only slight modifications of the arguments in \cite{HebRobWen}. The arguments in \cite{HebRobWen} used 
the decomposition of the fourth order operator into the product of two second order operators. 
We may use instead Agmon-Douglis-Nirenberg-type
estimates and note that the positivity of the 
Green's function $G_g$ implies that $P_g$ satisfies the comparison principle. As an independent easy remark, 
since $\Delta_g(\Delta_gu_\alpha) \ge 0$ around the points in $S$, there exists $C > 0$ such that 
$\Delta_gu_\alpha \ge -C$ in $M$, for all $\alpha$. Then, with such an estimate, and the analysis developed 
in \cite{HebRobWen}, we easily get that 
the integral and asymptotic estimates in Hebey, Robert, and Wen \cite{HebRobWen} follow from 
(\ref{SobDecCompTh})--(\ref{PointEstBisCompTh}). In several places the 
computations in \cite{HebRobWen} simplify because of the simple nature of the geometric equation 
around the points in $S$. By the asymptotic estimates, 
if we let $\tilde u_\alpha$ be the rescaled function obtained from $u_\alpha$ by
$\tilde u_\alpha(x) = u_\alpha\left(\exp_{x_\alpha}(\sqrt{\mu_\alpha}x)\right)$, 
we get that there exist $\delta > 0$, $A > 0$, and a biharmonic 
function $\varphi \in C^4\left(B_0(2\delta)\right)$ such that, up to a subsequence,
\begin{equation}\label{AsyEstCompTh}
\tilde u_\alpha(x) \to \frac{A}{\vert x\vert^{n-4}} + \varphi(x)
\end{equation}
in $C^3_{loc}\left(B_0(2\delta)\backslash\{0\}\right)$ as $\alpha \to +\infty$. Moreover, since we assumed 
that $G_g > 0$, so that either $u^0 \equiv 0$ or $u^0 > 0$ everywhere, we also have 
the important information that 
$\varphi$ is positive in $B_0(2\delta)$ if $u^0 \not\equiv 0$. 

\medskip As an important step in the proof of (\ref{MainEqtCompTh}), we claim that 
thanks to the asymptotics (\ref{AsyEstCompTh}), and thanks to the property that $\varphi$ is positive 
if $u^0$ is nonzero, 
we necessarily have that $u^0 \equiv 0$ when the $u_\alpha$ blow up. In order to 
do this, we use the Pohozaev [Pokhozhaev] identity as in 
\cite{HebRobWen}, and conformal invariance. The Pohozaev identity for fourth order equations reads as
\begin{equation}\label{PohTypeIdentity}
\begin{split}
&\int_\Omega\left(x^k\partial_ku\right)\Delta^2u dx 
+ \frac{n-4}{2} \int_\Omega u\Delta^2u dx\\
&\qquad= \frac{n-4}{2} \int_{\partial\Omega}\left(-u\frac{\partial\Delta u}{\partial\nu} + 
\frac{\partial u}{\partial\nu}\Delta u\right)d\sigma\\
&\qquad\quad+ \int_{\partial\Omega}\left(\frac{1}{2}(x,\nu)(\Delta u)^2 - (x,\nabla u)\frac{\partial\Delta u}{\partial\nu} 
+ \frac{\partial(x,\nabla u)}{\partial\nu}\Delta u\right)d\sigma
\end{split}
\end{equation}
for all smooth bounded domains $\Omega$ in ${\mathbb R}^n$ and 
all $u \in C^4\bigl(\overline{\Omega}\bigr)$, where 
$\Delta$ is the Euclidean Laplacian, $\nu$ is the outward unit normal of $\partial\Omega$, and $d\sigma$ 
is the Euclidean volume element on $\partial\Omega$.
We apply the Pohozaev identity 
(\ref{PohTypeIdentity}) to the $u_\alpha$ in the ball $\Omega = B_0(\delta\sqrt{\mu_\alpha})$. In the process 
we assimilate $x_\alpha$ and $0$ (thanks to the exponential map at $x_\alpha$), and regard $u_\alpha$ 
as a function in the Euclidean space. Noting that
$$\int_{B_\alpha}\left(x^k\partial_ku_\alpha\right)\Delta^2u_\alpha dx 
+ \frac{n-4}{2} \int_{B_\alpha}u_\alpha\Delta^2 u_\alpha dx 
= O\left(\int_{\partial B_\alpha}u_\alpha^{2^\sharp}d\sigma\right)\hskip.1cm ,$$
where $B_\alpha = B_0(\delta\sqrt{\mu_\alpha})$, and that 
$\int_{\partial B_\alpha}u_\alpha^{2^\sharp}d\sigma = o(\mu_\alpha^{(n-4)/2})$ by (\ref{AsyEstCompTh}), 
we get with 
(\ref{AsyEstCompTh}), the Pohozaev identity, and the computations developed in \cite{HebRobWen}, that
$$(n-2)(n-4)^2\omega_{n-1}A\varphi(0)\mu_\alpha^{\frac{n-4}{2}} + 
o\left(\mu_\alpha^{\frac{n-4}{2}}\right) = 0\hskip.1cm ,$$
where $\omega_{n-1}$ is the volume of the unit $(n-1)$-sphere, and $A$ and $\varphi$ are as in (\ref{AsyEstCompTh}). 
In particular, $\varphi(0) = 0$, and since 
$\varphi > 0$ if $u^0 \not\equiv 0$, this proves the above claim that we necessarily have 
that $u^0 \equiv 0$ when the $u_\alpha$ blow up. With respect to the 
terminology in Hebey, Robert, 
and Wen \cite{HebRobWen}, 
this amounts to saying that compactness 
reduces to pseudo-compactness for the geometric equation.

\medskip Going on with the proof of (\ref{MainEqtCompTh}), and 
now that we know that $u^0 \equiv 0$, we need to add one important estimate to the
estimates listed above, which we proved in \cite{HebRobWen}. We claim here that
\begin{equation}\label{ConvGrFct}
\lambda_\alpha u_\alpha(x) \to \sum_{i=1}^N\lambda_iG_g(x_i,x)
\end{equation}
in $C^4_{loc}(M\backslash S)$ as $\alpha \to +\infty$, 
where $\lambda_\alpha \to +\infty$ as $\alpha \to +\infty$, 
$S = \left\{x_1,\dots,x_N\right\}$ is the geometric blow-up point
set of the $u_\alpha$, $G_g$ is the Green's function 
of $P_g$, and the $\lambda_i \ge 0$, 
$i = 1,\dots,N$, are such that $\sum_{i=1}^N\lambda_i = 1$. In order to prove 
(\ref{ConvGrFct}), we use the positivity of $P_g$ and $G_g$ as 
follows. By the positivity of $P_g$, the lowest eigenvalue 
$\lambda$ of $P_g$ is positive. If $\psi$ is an eigenfunction for $\lambda$, letting $P_gu = \vert P_g\psi\vert$, 
and writing that $P_gu \ge P_g\psi$ and $P_gu \ge -P_g\psi$, we get with the positivity of $G_g$ that 
$u \ge \vert \psi\vert$. Noting that $u > 0$ since $P_gu = \lambda\vert \psi\vert$ and $G_g > 0$, 
plugging $u$ into the Rayleigh characterization of $\lambda$, it follows that 
either $\psi < 0$ or $\psi > 0$. Without loss of generality we may assume that $\psi > 0$. Then the conformal 
metric $\tilde g = \psi^{4/(n-4)}g$ is such that $Q_{\tilde g} > 0$, where $Q_{\tilde g}$ is the $Q$-curvature 
of $\tilde g$. By conformal invariance, $u = \psi^{-1}u_\alpha$ solves the equation 
$P_{\tilde g}u = u^{2^\sharp-1}$. 
Integrating over $M$, since $Q_{\tilde g} > 0$, we get that there exists $C > 0$ such that 
$\Vert u_\alpha\Vert_{L^1(M)} \le C\Vert u_\alpha\Vert_{L^s(M)}^s$ where $s = 2^\sharp-1$.
By Agmon-Douglis-Nirenberg-type estimates, noting that (\ref{PointEstCompTh}) 
gives that the $u_\alpha$ are bounded in $C^0_{loc}(M\backslash S)$ as $\alpha\to+\infty$, we get that 
for any $p > 1$, and any $\delta > 0$, the $L^\infty$-norm of the $u_\alpha$ 
in sets like $M\backslash B_\delta$ is controled by the $L^p$-norm of the 
$u_\alpha$
 in $M$, where 
$B_\delta$ is the union over the $x \in S$ of the geodesic balls $B_x(\delta)$. By 
the above estimate, using H\"older's inequality with $1 \le p \le 2^\sharp$, choosing $p > 1$ close to 
$1$, and since $u_\alpha \to 0$ in $L^q(M)$ for $q < 2^\sharp$, it easily follows that 
$\Vert u_\alpha\Vert_{L^s(M\backslash B_\delta)} = o\left(\Vert u_\alpha\Vert_{L^s(M)}\right)$ 
for all $\delta > 0$, where $s$ is as above.  
In particular, the $\lambda_i$ given by
$$\lambda_i = \lim_{\alpha \to +\infty} 
\frac{\int_{B_{x_i}(\delta)}u_\alpha^{2^\sharp-1}dv_g}{\int_Mu_\alpha^{2^\sharp-1}dv_g}$$
are nonnegative, independent of $\delta > 0$ small, and such that $\sum\lambda_i = 1$. In what follows we 
let $\lambda_\alpha = \Vert u_\alpha\Vert_{2^\sharp-1}^{1-2^\sharp}$. Then $\lambda_\alpha \to +\infty$ 
as $\alpha \to +\infty$, while we can write with the Green's representation formula that for 
$x \in M\backslash B_\delta$, and $0 < \delta^\prime \ll \delta$,
\begin{eqnarray*} u_\alpha(x)
& = & \int_{B_{\delta^\prime}}G_g(x,y)u_\alpha^{2^\sharp-1}(y)dv_g(y) 
+ \int_{M\backslash B_{\delta^\prime}}G_g(x,y)u_\alpha^{2^\sharp-1}(y)dv_g(y)\\
& = & \left(\sum_{i=1}^N\lambda_iG_g(x_i,x) + o_{\delta^\prime}(1)\right)\lambda_\alpha^{-1}\hskip.1cm ,
\end{eqnarray*}
where $\lim_{\delta^\prime \to 0}\lim_{\alpha\to +\infty}o_{\delta^\prime}(1) = 0$. In particular, 
$\lambda_\alpha u_\alpha(x) \to \sum_{i=1}^N\lambda_iG_g(x_i,x)$ in $C^0_{loc}(M\backslash S)$ 
as $\alpha \to +\infty$, an equation from which we easily get that (\ref{ConvGrFct}) is true.

\medskip With (\ref{ConvGrFct}) we can now end the proof of (\ref{MainEqtCompTh}). 
If $f$ stands for the function on
the right hand side of (\ref{ConvGrFct}), then $\Delta^2f = 0$ 
in a set like $\Omega = \bigcup_{i=1}^NB_{x_i}(\delta_0)\backslash S$, where $\delta_0 > 0$.
We apply the Pohozaev identity 
(\ref{PohTypeIdentity}) to the $u_\alpha$ in $B_{x_i}(\delta)$ for $\delta > 0$ small and 
$i$ in $\left\{1,\dots,N\right\}$. In the process 
we assimilate $x_i$ and $0$ (thanks to the exponential map at $x_i$), and regard $u_\alpha$ 
as a function in the Euclidean space. Noting that
$$\int_B\left(x^k\partial_ku_\alpha\right)\Delta^2u_\alpha dx 
+ \frac{n-4}{2} \int_Bu_\alpha\Delta^2u_\alpha dx 
= O\left(\int_{\partial B}u_\alpha^{2^\sharp}d\sigma\right)\hskip.1cm ,$$
where $B = B_0(\delta)$, it follows from (\ref{ConvGrFct}) and the Pohozaev identity that
\begin{equation}\label{PohConcl1}
\begin{split}
&\frac{n-4}{2} \int_{\partial B_0(\delta)}\left(-f\frac{\partial\Delta f}{\partial\nu} + 
\frac{\partial f}{\partial\nu}\Delta f\right)d\sigma\\
&\qquad+ \int_{\partial B_0(\delta)}\left(\frac{1}{2}(x,\nu)(\Delta f)^2 - 
(x,\nabla f)\frac{\partial\Delta f}{\partial\nu} 
+ \frac{\partial(x,\nabla f)}{\partial\nu}\Delta f\right)d\sigma = 0 \hskip.1cm .
\end{split}
\end{equation}
By (\ref{GrFct2}), and (\ref{ConvGrFct}), we can write that
$$f(x) =\frac{\hat\lambda_i}{\vert x-x_i\vert^{n-4}} + R_i(x)$$
for $x \not= x_i$ close to $x_i$, where $\lambda_i$ is as in (\ref{ConvGrFct}), $\hat\lambda_i = \lambda_n\lambda_i$, 
$\lambda_n$ is as in (\ref{GrFct2}), $R_i$ is smooth around $x_i$, and
$$R_i(x_i) = \lambda_i\mu_g(x_i) 
+ \sum_{j\not= i}\lambda_jG_g(x_j,x_i)\hskip.1cm .$$
Plugging these equations into (\ref{PohConcl1}) and letting $\delta \to 0$, we get that 
$R_i(x_i) = 0$. This equation holds for all $i = 1,\dots,N$, and we assumed that $G_g > 0$. It follows 
that $\lambda_i > 0$ for all $i$. 
This ends the proof of (\ref{MainEqtCompTh}) and of Theorem \ref{MainTh}.

\medskip As a remark, Theorem \ref{MainTh} still holds if we replace 
the critical exponent $2^\sharp$ in (\ref{GenEqtCompTh}) 
by $2^\sharp-p_\alpha$, where $p_\alpha \ge 0$ is 
such that $p_\alpha \to 0$ as $\alpha \to +\infty$. In this case the 
left hand side in (\ref{MainEqtCompTh}) is not zero anymore, but is 
nonpositive. Needless to say, 
compactness of the subcritical equations provides a minimizing solution of (\ref{GenEqtCompTh}).

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