\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 190, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/190\hfil Weak compactness of biharmonic maps]
{Weak compactness of biharmonic maps}

\author[S. Zheng \hfil EJDE-2012/190\hfilneg]
{Shenzhou Zheng}  % in alphabetical order

\address{Shenzhou Zheng \newline
 Department of Mathematics, Beijing Jiaotong University, 
 Beijing 100044, China}
\email{shzhzheng@bjtu.edu.cn, Tel: +1-859-257-4802}

\thanks{Submitted July 15,2012. Published October 31,2012.}
\subjclass[2000]{35J48, 35G50, 58J05}
\keywords{Biharmonic maps, conservation law, weak compactness}

\begin{abstract}
 This article shows that if a sequence of weak solutions of a perturbed
 biharmonic map satisfies $\Phi_k\to 0$ in $(W^{2,2})^*$
 and $u_k\rightharpoonup u$ weakly in $W^{2,2}$, then $u$ is a biharmonic map.
 In particular, we show that the space of biharmonic maps is sequentially 
 compact  under the weak-$W^{2,2}$ topology. 
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega\subset \mathbb{R}^4$ be a bounded smooth domain, 
and $(\mathcal{N},h)$ be a $C^3$-smooth compact $k$-dimensional Riemannian 
manifold without boundary, embedded into the Euclidean space $\mathbb R^{d}$.
For $1\le l<+\infty$ and $1\le p<+\infty$, define the Sobolev space 
$W^{l,p}(\Omega, \mathcal{N})$ by
$$
W^{l,p}(\Omega, \mathcal{N}) =\{v\in W^{l,p}(\Omega, \mathbb R^{d}): 
 v(x)\in \mathcal{N}  \text{ a.e. } x\in\Omega\}.
$$
In this article, we discuss the limiting behavior of weakly convergent
sequences of approximate (extrinsic) biharmonic maps 
$ \{u_m\}\subset W^{2,2}(\Omega,\mathcal{N})$ in dimension four,
especially weak compactness of sequence of biharmonic maps.
Before stating them, let us recall some related notions.

Recall that an (extrinsic) biharmonic map is the critical point 
of the  Hessian energy functional
\begin{equation}\label{energy-functional}
E_2(v) := \int_{\Omega}|\Delta v|^2, \quad\forall v\in W^{2,2}(\Omega, \mathcal{N}).
\end{equation}
More precisely; 

\begin{definition} \label{def1.1} \rm
A map $u\in  W^{2,2}(\Omega,\mathcal{N})$ is called weakly biharmonic if $u$ 
is a critical point of the Hessian energy functional \eqref{energy-functional} with respect to compactly supported variations on
$\mathcal{N}$. That is, if for all $\xi\in C^{\infty}_0 (\Omega,\mathbb{R}^d )$, we have
\begin{equation}\label{biharmonic}
\frac{d}{dt}\Big|_{t=0}E_2(\Pi(u+t\xi))=0
\end{equation}
where $\Pi$ denotes the nearest point projection from $\mathbb{R}^d$ to $\mathcal{N}$.
\end{definition}

Note that for the tubular neighborhood $V_\delta$ of $\mathcal{N}$ in $\mathbb{R}^d$ 
for $\delta > 0$ sufficiently small, we write the smooth nearest point projection
 $\Pi_\mathcal{N} : V_{\delta}\to \mathcal{N}$. For $u(x)\in \mathcal{N}$, let
$\mathbb P(u):\mathbb R^{d}\to T_u \mathcal{N}$ be the orthogonal projection 
from $\mathbb R^{d}$ onto the tangent space $T_u\mathcal{N}$ of $\mathcal{N}$ 
at $u\in \mathcal{N}$, and $\mathbb{P}^{\perp}(u)$ be the orthogonal 
projection onto the normal space with $\mathbb{P}^{\perp}(u)=Id-\mathbb P(u)$. 
Then, from the point of geometric view we readily have  
$\Delta^2 u \perp T_{u}\mathcal{N}$ a. e. $\Omega $, which implies the following 
Euler-Lagrange equation thanks to Wang's work \cite{W2,W3}.
\begin{equation}\label{biharm}
\Delta^2 u=\Delta(\mathbb B(u)(\nabla u,\nabla u))+2\nabla\cdot\langle\Delta u,
\nabla(\mathbb P(u))\rangle-\langle\Delta(\mathbb P(u)),\Delta u\rangle,
\end{equation}
where $\mathbb B(y)(X,Y)=-\nabla_X\mathbb P(y)(Y)$, \
 for $X, Y\in T_y \mathcal{N}$, is the second fundamental form on 
 $\mathcal{N}\subset\mathbb R^{d}$.

 It is well known that biharmonic maps are  higher order extensions of harmonic maps. 
The study of biharmonic maps has generated considerable interests after the 
initial work by Chang-Wang-Yang \cite{CWY}, for further development
the readers can refer to Wang \cite{W2,W3}, Strzelecki \cite{St}, 
 Lamm-Rivi\`ere \cite{LamR}, Struwe \cite{Struwe},  Scheven \cite{scheven1, scheven2} 
etc.  Generally speaking, a singular phenomenon, which appears as various
 critical problems related nonlinear systems, especially in the investigation 
of harmonic, biharmonic and $p$-harmonic maps, leads to defects of 
strong convergence. Typically, some part energy of variation functional 
is lost in the  limit passage so that the sequence does not have to converge 
strongly. So it is an important and interesting problem to consider the 
compactness of the sequence of biharmonic maps in the weak topology.

\begin{definition} \label{def1.2} \rm
 A sequence of biharmonic  maps $\{u_m\}\subset W^{2,2}(\Omega,\mathcal{N})$ 
is called a Palais-Smale sequence for the
energy functional $E_2(u_m)$ on the admissible set $W^{2,2}(\Omega,\mathcal{N})$, 
if the following two conditions hold:
\begin{itemize}
\item[(a)] $u_m\rightharpoonup u $ weakly in $W^{2,2}(\Omega,\mathcal{N})$, and
\item[(b)] $E'_2 (u_m)\to 0$ in $(W^{2,2}(\Omega,\mathcal{N}))^*$,
\end{itemize}
where $(W^{2,2}(\Omega,\mathcal{N}))^*$ is the dual of $W^{2,2}(\Omega,\mathcal{N})$.
\end{definition}

In this note, we are mainly interested in the compactness problem 
in the weak topology for the sequence of biharmonic maps in dimension four.
 We show stability results of biharmonic maps in $\mathbb{R}^4$ under 
the weak convergence, which is done by a divergence structure of Euler-Lagrange's
 equations concerning biharmonic maps to general manifolds due to
 Lamm-Rivi\`ere (cf.\cite{LamR,Struwe,LauR2}). In this way we can avoid Lions' 
concentration compactness argument \cite{FMS,W1}. In order to further 
analyze the behaviour of weakly convergent Palais-Smale sequences for the 
variation functional, we are supposed that $u_m$ is a Palais-Smale sequence for 
Hessian energy functional with disturbance, which satisfies
\begin{equation}\label{PS-sequence-1}
-\Delta^2 u_m+\Phi_m\perp T_{u_m}\mathcal{N}
\end{equation}
and
\begin{equation}\label{assumptions}
u_m\text{ is bounded  in }  W^{2,2}(\Omega,\mathbb{R}^d),\quad
 \Phi_m\to 0 \text{ in }  (W^{2,2}(\Omega,\mathbb{R}^d))^*.
\end{equation}

\begin{theorem}\label{compactness}
Assume that $\{u_m\}\subset W^{2,2}(\Omega,\mathcal{N})$ is a Palais-Smale 
sequence of Hessian energy functionals \eqref{energy-functional}; i.e., 
they satisfy the relation \eqref{PS-sequence-1} with $\Phi_m\to 0$ 
in $(W^{2,2}(\Omega,\mathcal{N}))^*$, and $u_m\rightharpoonup u$ weakly 
in $W^{2,2}(\Omega,\mathcal{N})$.
Then $u \in W^{2,2}(\Omega,\mathcal{N})$ is a biharmonic map.
\end{theorem}

Here, we would like to point out that if $\Phi_m$ is bounded in  
 $L^p$ for $p>\frac 43$. Theorem 1.3 is a consequence of the main theorem 
by Wang-Zheng\cite{WZ1,WZ2}.

\begin{remark} \label{rmk1.4} \rm
 While $u_k\in  W^{1,2}(\Omega,\mathcal{N})$ is a Palais-Smale sequence 
of harmonic maps, Bethuel \cite{Be} was the first to obtain the conclusion
 of its weak compactness. Later, Freire-M\"uller-Struwe \cite{FMS} 
simplified Bethuel's proof by way of Lions' concentration argument. 
With similar approach to the Freire-M\"uller-Struwe's approach, 
Strzelecki-Zatorska \cite{St-Z} and Wang \cite{W1} derived the conclusions
 of  compactness to higher dimensional H-system and n-harmonic maps into 
manifold in the critical dimension, respectively. Recently, Strzelecki 
\cite[Proposition 1.2]{St} and Goldstein-Strzelecki-Zatorska 
\cite[Theorem 1.2]{GSZ} also achieved the compactness for biharmonic maps
 and polyharmonic maps into sphere in the critical dimension respectively, 
who made use of transferring into the equation with divergence form on the
 basis of the geometric nature of sphere, which avoids to employ Lions' 
concentration argument.
\end{remark}

As an immediate corollary, we obtain a compactness conclusion of any weakly 
convergent sequence of weak solutions of biharmonic maps as follows.
More precisely, we have

\begin{corollary} \label{coro1.5}
Assume that $\{u_m\}\subset  W^{2,2}(\Omega,\mathcal{N})$ are a sequences of 
biharmonic maps converging weakly to $u$ in $W^{2,2}(\Omega,\mathcal{N})$. 
then $u$ is a biharmonic map.
\end{corollary}

\section{Conservation law for biharmonic maps}

In what follows, it is convenient to rewrite the geometric expression
 \eqref{PS-sequence-1} into the divergence structure, due to the initial work 
of Lamm-Rivi\`ere \cite{LamR} on biharmonic maps. To this end, we introduce 
some notation.
 Let $M(d)$ be the space of $ d\times d$ matrices and 
$so(d) := \{A \in M(d) : A^t + A = 0\}$,
 $\wedge^l\mathbb R^4$ be $l$-tuple exterior differential form on $\mathbb  R^4$, 
and $L^2\cdot W^{1,2}(B_1, M(d))$ be the space of linear combination  
of products of an $L^2$ map with an $W^{1,2}$ map form $B_1$ into the space $M(d)$. 
We observe the following fourth order elliptic systems in 4-dimension raised
 by Lamm-Rivi\`ere \cite{LamR} on biharmonic maps to Riemannian manifold 
$\mathcal{N}$.
\begin{equation}\label{fourth-systems-0}
\Delta^2 u=\Delta(V\cdot \nabla u)+\operatorname{div}w\nabla u)
+\nabla\omega\nabla u+F\nabla u.
\end{equation}
The study of this equation can be traced back to Rivi\`ere's work \cite{Ri} 
on the conservation law of conformally invariant variational problems
 including harmonic maps and H-surfaces, which makes the second order 
elliptic systems $ -\Delta u=\Omega\cdot \nabla u$ to be in divergence form,
 where $\Omega=\Omega_k\partial_{x_k}$ is a vectorfield tensored $d\times d$ 
antisymmetric matrices. Later, Lamm-Rivi\`ere \cite{LamR} and Struwe \cite{Struwe} 
discovered another conservation law obtained from \eqref{fourth-systems-0}
 by extending their original idea concerning harmonic maps to the fourth-order 
setting concerning biharmonic maps.
Now, let us proceed to a Hodge decomposition of 
$dPP^{\perp}-P^{\perp}dP=d\alpha+d^*\beta$ with 
$\alpha\in W^{2,2}(B_1,so(d)),\beta\in W^{2,2}_0(B_1,
\wedge^2(\mathbb{R}^4)\otimes M(d))$ (see \cite{IM}) so that
\begin{gather*}
\Delta\alpha=\Delta P P^{\perp}-P^{\perp}\Delta P \in L^2(B_1),\\
\Delta d^*\beta= \Delta P\wedge dP^{\perp}-\Delta P^{\perp}\wedge dP 
\in L^2\cdot W^{1,2}(B_1,\wedge^2(\mathbb{R}^4)\otimes M(d)).
\end{gather*}
Thanks to  \cite[Proposition 1.1]{LauR2}, we can rewrite the geometric
 expression \eqref{PS-sequence-1} as
\begin{equation}\label{fourth-systems}
\Delta^2 u_m=\Delta(V_m\cdot \nabla u_m)+\operatorname{div}w_m\nabla u_m)
+\nabla\omega_m\nabla u_m+F_m\nabla u_m+\Phi_m,
\end{equation}
in the distributional sense, where
\begin{equation}\label{representation}
\begin{gathered}
V_m=-2\nabla P_m^{\perp}-(\nabla P_mP_m^{\perp}-P_m^{\perp}\nabla P_m)  \\
w_m=\Delta P_m^{\perp}-2\nabla(\nabla P_mP_m^{\perp}-P_m^{\perp}\nabla P_m) \\
\omega_m =\Delta \alpha_m \\
F_m = \Delta d^*\beta_m+2 \nabla_y P_m^{\perp}\nabla(\nabla P_m^{\perp}\nabla u_m)+2\nabla_y\nabla P_m^{\perp}\Delta u_m,
\end{gathered}
\end{equation}
and
\begin{equation} \label{growth-est}
\begin{gathered}
|V_m|\le C|\nabla u_m|, \\
|w_m|+|\nabla V_m|+|\omega_m|\le C(|\nabla^2 u_m|+|\nabla u_m|^2),\\
|F_m|\le C(|\nabla^2 u_m||\nabla u_m|+|\nabla u_m|^3),
\end{gathered}
\end{equation}
furthermore, $V_m\in W^{1,2}(B_1,M(d)\otimes \wedge^1\mathbb R^4)$,
$w_m\in L^2(B_1,M(d))$, $\Phi_m\in L^p(B_1)$, $W_m=\nabla\omega_m +F_m$
such that $\omega_m\in L^2(B_1,so(d))$,
$F_m\in L^2\cdot W^{1,2}(B_1, M(d)\otimes \wedge^1\mathbb R^4)$ and
$\Phi_m\to 0$ in $(W^{2,2}(\Omega,\mathcal{N}))^*$.

First, recall a decomposition result to the fourth order systems 
from Lamm-Rivi\`ere's work (see \cite[Theorem 1.5]{LamR}), 
 which is related to biharmonic maps.

\begin{proposition}\label{fourth-Hodge}
Let $V\in W^{1,2}(B_1,M(d)\otimes \wedge^1\mathbb R^4)$, $w\in L^2(B_1,M(d))$, and 
$W=\nabla\omega +F$ with $\omega\in L^2(B_1,so(d))$, 
$F\in L^2\cdot W^{1,2}(B_1, M(d)\otimes \wedge^1\mathbb R^4)$. 
If the  smallness condition 
\begin{equation}\label{smallness-0}
\|V\|_{W^{1,2}}+\|w\|_{L^2}+\|\omega\|_{L^2}+\|F\|_{L^2\cdot W^{1,2}}<\varepsilon,
\end{equation}
is satisfied for some $\varepsilon>0$, 
then there exist $A\in L^{\infty}\cap W^{2,2}(B_1,GL(d))$ and
 $B\in W^{1,\frac 43}(B_1,M(d)\otimes \wedge^2\mathbb R^4)$ such that
\begin{equation}\label{decomposition}
\nabla\Delta A+\Delta AV-\nabla A w+AW=\operatorname{curl} B,\quad \text{in}\ B_1,
\end{equation}
with the estimate 
\begin{align*}
&\|A\|_{W^{2,2}}+\|\operatorname{dist}(A,SO(d))\|_{L^{\infty}}
+\|B\|_{W^{1,\frac 43}}\\
&\le C\big(\|V\|_{W^{1,2}}+\|w\|_{L^2}
+\|\omega\|_{L^2}+\|F\|_{L^2\cdot W^{1,2}}\big),
\end{align*}
see  \cite[Theorem 2.1]{LauR2}.
\end{proposition}

Note that for  the Palais-Smale sequence $u_m\in W^{2,2}(B_1)$ of biharmonic 
maps in $\mathbb R^4$, by a simple computation from the estimates \eqref{growth-est} 
we obtain that
\begin{gather}\label{esti-V-w-F}
\|V_m\|^2_{W^{1,2}}+\|w_m\|^2_{L^2}+\|\omega_m\|^2_{L^2}
+\|F_m\|^2_{L^2\cdot W^{1,2}}\le C\int_{B_{1}}|\nabla^2 u_m|^2+ |\nabla u_m|^4,
\\ \label{esti-A-B}
\|A_m\|^2_{W^{2,2}}+\|\operatorname{dist}(A_m,SO(d))\|^2_{L^{\infty}}
+\|B_m\|^{\frac 43}_{W^{1,\frac 43}}\le  C\int_{B_{1}}|\nabla^2 u_m|^2
+ |\nabla u_m|^4.
\end{gather}
Therefore, under the smallness assumption 
$\int_{B_{1}}|\nabla^2 u_m|^2+ |\nabla u_m|^4\le \varepsilon$,
 we obtain from Proposition \ref{fourth-Hodge} and \eqref{fourth-systems} that
\begin{equation}\label{divergence}
\Delta(A_m\Delta u_m)=\operatorname{div} K_m+A_m\Phi_m, \quad x\in B_1,
\end{equation}
where
\begin{equation}\label{expression}
\begin{split}
K_m&=\Big (2\nabla A_m\Delta u_m-\Delta A_m\nabla u_m+A_mw_m\nabla u_m\\
&\quad-\nabla A_m(V_m\cdot \nabla u_m)+A_m\nabla(V_m\cdot\nabla u_m)
+B_m\cdot \nabla u_m\Big).
\end{split}
\end{equation}

\begin{lemma}[$\varepsilon$-weak compactness] \label{small-converg}
In dimension four, there exists an $\varepsilon_0 > 0 $ such that if 
$\{u_m\}\subset W^{2,2}(B_1,\mathcal{N})$ is a Palais-Smale sequence 
of (extrinsic) biharmonic maps \eqref{PS-sequence-1} 
and \eqref{assumptions} with $\Omega$ replaced by $B_1$, $u_m\rightharpoonup u$ 
weakly in $W^{2,2}(B_1,\mathcal{N})$, and satisfy
\begin{equation}\label{smallness}
\int_{B_{1}}|\nabla^2 u_m|^2+ |\nabla u_m|^4 \le \varepsilon^2;
\end{equation}
then $u\in W^{2,2}(B_1,\mathcal{N})$ is also a biharmonic map.
\end{lemma}

\begin{proof}
Under the smallness condition  
$\int_{B_{1}}|\nabla^2 u_m|^2+ |\nabla u_m|^4 \le \varepsilon^2$,
 it follows from Proposition \ref{fourth-Hodge} that the divergent structure 
\eqref{divergence} is valid. Now we analyze the convergence of sequence 
to any weak solutions of equation \eqref{divergence} under the disturbance
 conditions \eqref{assumptions}.
Thanks to
$$
u_m\rightharpoonup u \text{ in } W^{2,2}(B_1,\mathbb{R}^d),\quad 
\Phi_m\to 0\text{ in }   (W^{2,2}(B_1,\mathbb{R}^d))^*,
$$
up to taking subsequences, we derive from the estimates \eqref{esti-V-w-F}
 and \eqref{esti-A-B} that
$V_m\rightharpoonup V$ weakly in ${W^{1,2}}(B_1)$, $w_m\rightharpoonup w$ 
weakly in  $ {L^2}(B_1)$,$F_m\rightharpoonup F$ weakly in
 $ L^2\cdot W^{1,2}(B_1)$, $A_m\rightharpoonup A$ weakly in 
 $W^{2,2}(B_1)$ and $B_m\rightharpoonup B$ weakly in  $W^{1,\frac 43}(B_1)$. 
By Sobolev's compact imbedding theorem,  it follows that $\nabla u_m\to \nabla u$  
strongly in  $L^2(B_1)$, $\nabla A_m\to \nabla A$  strongly in $L^2(B_1)$ 
and $A_m\to A$  strongly in $L^2(B_1)$. Hence, it gives
\begin{gather}\label{1-converg}
\Delta (A_m\Delta u_m) \to \Delta( A\Delta u), \quad \mathcal{D}'(B_1),
 \\
\big|A_m\Phi_m\big |_{\{W^{2,2},(W^{2,2})^*\}} 
\le \|A_{m}\|_{W^{2,2}(B)}\|\Phi_m\|_{(W^{2,2}(B))^*}\to 0;
\end{gather}
where $\mathcal{D}'$ is Schwartz distributional spaces of $\mathcal{D}$.

For $K_m$, we readily see that $K_m\in L^{\frac 43}(B_1)$. Then we have
\begin{equation} \label{2-converg}
\begin{aligned}
&2\nabla A_m\Delta u_m-\Delta A_m\nabla u_m+A_mw_m\nabla u_m-\nabla A_m(V_m\cdot \nabla u_m)+A_m\nabla(V_m\cdot\nabla u_m) \\
&\rightharpoonup 2\nabla A\Delta u-\Delta A\nabla u+Aw\nabla u
-\nabla A(V\cdot \nabla u)+A\nabla(V\cdot\nabla u)
\end{aligned}
\end{equation}
weakly $L^1(B_1)$. In addition, since $B_m\in W^{1,\frac 43}(B_1)$,
it implies $B_m\rightharpoonup B$ weakly in $L^2(B_1)$.
Then, for $B_m\cdot \nabla u_m$ we have
$$
B_m\cdot \nabla u_m\rightharpoonup B\cdot \nabla u
$$
weakly in $L^1(B_1)$, and
\begin{equation}\label{3-converg}
\operatorname{div} K_m\to \operatorname{div} K,\quad \mathcal{D}'(B_1).
\end{equation}
Combining \eqref{1-converg},\eqref{2-converg} and \eqref{3-converg}, we obtain
\begin{equation}
\Delta(A\Delta u)=\operatorname{div} K, \quad x\in B_1,
\end{equation}
in $\mathcal{D}'(B_1)$, where
$$
K=\Big (2\nabla A\Delta u-\Delta A\nabla u+Aw\nabla u
 -\nabla A(V\cdot \nabla u)+A\nabla(V\cdot\nabla u)+B\cdot \nabla u\Big).
$$
That is,  $u$ is a biharmonic map of $B_1$.
\end{proof}


\section{Proof of main result}


\begin{proof}[Proof of Theorem \ref{compactness}]
 Assume that $\{u_m\}^{\infty}_{m=1} \in W^{2,2}(\Omega,\mathcal{N})$ 
is a Palais-Smale sequence of biharmonic maps with disturbance \eqref{assumptions}, 
and $u_m\rightharpoonup u $ weakly in the space $W^{2,2}(\Omega,\mathcal{N})$. 
Since $ u_m$ is a bounded sequence in $W^{2,2}(\Omega,\mathcal{N})$, 
we have that  $\mu_m:= \int_{\Omega}|\nabla^2 u_m|^2+ |\nabla u_m|^4$ 
is a family of nonnegative Radon measures with $M = \sup_m \mu_m(\Omega) < \infty$. 
Therefore, we can assume, after passing up to subsequences, that there is a
 nonnegative Radon measure $\mu$ on $\Omega$ such that
$$
\mu_m:= \int_{\Omega}|\nabla^2 u_m|^2+ |\nabla u_m|^4 \to \mu,
$$
as a convergence of Radon measures. Let $\varepsilon_0 > 0$ be the 
same constant as in Lemma \ref{small-converg} and
define singular set $\Sigma$ by
\begin{equation}
\Sigma:=\{x\in \Omega:\mu(\{x\})\ge \varepsilon_0^2\}.
\end{equation}
Then, by a simple covering argument we have that $\Sigma$ is a finite set, and
$$
H^0(\Sigma)\le\frac{M}{\epsilon_0^2},\quad
 M=\sup_{m}\int_{\Omega}|\nabla^2 u_m|^2+ |\nabla u_m|^4<+\infty.
$$
Here $H^0$ is 0-dimensional Hausdorff measure. In fact, 
let $\Sigma' = \{x_1,\dots, x_m\}\subset \Sigma$ be any finite subset of $\Sigma$,
then there is a small $\delta_0 > 0$ such that $\{B_{\delta_0} (x_i)\}^s_{i=1}$ 
are mutually disjoint balls such that
$$
\liminf_{m\to\infty} \int_{B_{\delta_0} (x_i)}|\nabla^2 u_m|^2
+ |\nabla u_m|^4\ge \varepsilon^2_0,\quad i=1,2,\dots,s;
$$
which implies that there is a natural number $K_s \in \mathbf{N}$ 
such that for any $m \ge K_s$ satisfying
$$
\int_{B_{\delta_0} (x_i)}|\nabla^2 u_m|^2+ |\nabla u_m|^4
\ge \varepsilon^2_0,\quad i=1,2,\dots,s.
$$
Therefore, for any $m \ge K_s$ we have
\begin{align*}
s\varepsilon^2_0 
&\le \sum_{i=1}^s\int_{B_{\delta_0} (x_i)}|\nabla^2 u_m|^2+ |\nabla u_m|^4 \\
&= \int_{\bigcup_{i=1}^s B_{\delta_0} (x_i)}|\nabla^2 u_m|^2+ |\nabla u_m|^4\\
&\le  \int_{\Omega}|\nabla^2 u_m|^2+ |\nabla u_m|^4\le M<\infty.
\end{align*}
This implies $s\le M \varepsilon^{-2}_0$.

Therefore, for any $x_0\in \Omega\backslash\Sigma$ there exists an
 $r_0 > 0$ such that $\mu(B_{2r_0 }(x_0)) < \varepsilon^2_0$. On account of
$$
\limsup_{m\to\infty}\int_{B_{r_0} (x_0)}|\nabla^2 u_m|^2+ |\nabla u_m|^4
\le \mu(B_{2r_0 }(x_0)),
$$
which states clearly that we may assume that there exists $K_0\ge 1$ 
such that while $m>K_0$, it follows that
\begin{equation}
\int_{B_{r_0} (x_0)}|\nabla^2 u_m|^2+ |\nabla u_m|^4\le \varepsilon^2_0.
\end{equation}
Therefore, from Lemma \ref{small-converg} it implies that $u$ is a biharmonic 
map in $B_{r_0} (x_0)$. Since $x_0\in \Omega\backslash\Sigma$ is
arbitrary, we conclude that $u$ is a biharmonic map in $\Omega\backslash\Sigma$.
 It is standard argument to show that $u$ is
biharmonic map in $\Omega$ (see \cite{W2, W3}). 
The proof of Theorem 1.3 is complete.
\end{proof}

\subsection*{Acknowledgements}
The work is partially supported by grant 11071012 from the NSFC . 
The author would like to thank Professor Changyou Wang for his constructive
and interesting discussions, and  to the Department of Mathematics at the
University of Kentucky for their kind hospitality while visiting them.


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