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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1999/33\hfil 
Uniqueness for a semilinear elliptic equation
\hfil\folio}
\def\leftheadline{\folio\hfil Kewei Zhang 
 \hfil EJDE--1999/33}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol. {\eightbf 1998}(1998), No.~33, pp.~1--10.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break 
ftp ejde.math.swt.edu (login: ftp)\bigskip} }

\topmatter
\title 
UNIQUENESS FOR A SEMILINEAR ELLIPTIC EQUATION\\
IN NON-CONTRACTIBLE DOMAINS\\
UNDER SUPERCRITICAL GROWTH CONDITIONS
\endtitle

\thanks 
{\it 1991 Mathematics Subject Classifications:} 35J65, 35B05, 58E05.\hfil\break\indent
{\it Key words and phrases:}  semilinear  elliptic equation,
supercritical growth, uniqueness, \hfil\break\indent
non-contractible domains, Pohozaev identity.  
\hfil\break\indent
\copyright 1999 Southwest Texas State University  and
University of North Texas.\hfil\break\indent
Submitted May 12, 1999. Published September 15, 1999.
\endthanks
\author  Kewei Zhang  \endauthor
\address Kewei Zhang \hfill\break
Department of Mathematics, Macquarie University \hfill\break
Sydney, Australia
\endaddress
\email kewei\@ics.mq.edu.au
\endemail

\abstract
We apply the Pohozaev identity to  sub-domains of a tubular neighbourhood of 
a closed or broken curve in $\Bbb R^n$ and establish uniqueness results for 
the smooth solutions of the Dirichlet problem for
$-\Delta u+|u|^{p-1}u=0$.
 We require the domain to be in $\Bbb R^n$ with $n\geq 4$ and with 
 $p> (n+1)/(n-3)$.
\endabstract
\endtopmatter

\document
\head 1. Introduction \endhead

In this note, we consider the uniqueness of smooth solutions for the Dirichlet problem 
$$
\gathered -\Delta u=|u|^{p-1}u\quad \text{ in $\Omega\subset R^n$},\\
 u=0\quad\text{ on $\partial\Omega.$}
\endgathered\tag 1$$
in some non-starshaped and non-contractible domains.
Since Pohozaev's work \cite{P}, there have been many uniqueness results for (1) and its generalizations
(see, for example \cite{PS, V, M}). These results are based on Pohozaev's identity \cite{P}
and are established on star-shaped domains. Under the critical growth condition $p=(n+2)/(n-2)$, it 
is known \cite{BC}  that (1) has nontrivial solutions when the topology of the domain is nontrivial.
For some simply connected domains, there are examples \cite{Da, Di} that (1) can have nontrivial 
solutions when $p=(n+2)/(n-2)$ is the critical Sobolev exponent.

Recently, possible generalizations have been considered
for `nearly star-shaped' domains \cite{DZ} and for carefully designed non-starshaped rotation
domains \cite{CZ} on which (1) does not have nontrivial smooth solutions. 

In \cite{CZ} a special class of
non-star shaped domains was constructed by rotating a two-dimensional graph designed by using inversions in 
Euclidean spaces. The first result of the present note is to generalize this result to domains
including all  rotation domains. Since there is much less restriction on the graph, we
have a weaker result, that is, when $n>3$ and $p\geq (n+1)/(n-3)$, the only smooth solution is
$u\equiv 0$. 
We also show that when $p> (n+1)/(n-3)$ the same result holds for  sufficiently small tubular neighbourhood of a given
closed, smooth embedded curve in $\Bbb R^n$. A simple example of such a non-contractible domain is the
solid torus in $\Bbb R^4$. In general, our non-contractible domains have the same homotopic type as
the unit circle $S^1$. When $p>(n+2)/(n-2)$, there are examples of non-starshaped
domains \cite{CZ, DZ} on which (1) has only trivial solutions.  However, for domains with
nontrivial topology,   examples I can find such that the same uniqueness result holds are in $\Bbb R^n$ with $n>3$ and with the growth condition 
$p>(n+1)/(n-3)$. 

The method we use is to apply the Pohozaev identity  \cite{P, PS} to certain sub-domains. We carefully
divide a tubular neighbourhood of a closed curve into sub-domains by using the normal planes
of the central curve, such that each sub-domain is star-shaped. We apply 
the Pohozaev identity on each of these sub-domains. Then we collect the resulting terms and pass to the 
limit by using the 
definition of Riemann integral. In the limit, we obtain quantities which are comparable. By 
adjusting the thickness of the tubular domain, we can show that, at least for
$n>3$ and $p > (n+1)/(n-3)$, the uniqueness result remains true.


In this note all domains are open, bounded, and connected. Recall that a domain $\Omega$ is star-shaped if there is a point $x_0\in \Omega$ such that any 
line segment $\overline{x_0x}$ is contained in $\Omega$ when $x\in \Omega$. For convenience, 
we call $x_0$ a 
central point.  

We need the following  Pohozaev identity \cite{P, PS}. 

For the Dirichlet problem (1), the equation is the Euler-Lagrange equation for the energy density
$$F(u,Du)=\frac{1}{2}|Du|^2-\frac{|u|^{p+1}}{p+1}.\tag 2$$
Let $\Omega\subset \Bbb R^n$ be
a piecewise smooth domain. Let  $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ be a 
smooth solution of the Euler-Lagrange equation of the variational integral
$$I(u)=\int_\Omega F(u(x),Du(x))dx,\tag 3$$
 Then the identity
$$\aligned
\int_{\partial\Omega}\bigg[&\left(\frac{1}{2}|Du|^2
-\frac{|u|^{p+1}}{p+1}\right)\sum^n_{\alpha=1}(x-x^0)_\alpha\nu_\alpha \\
&-\bigg(\sum^n_{\alpha,\,\beta=1}h_\beta\nu_\alpha\frac{\partial u}
{\partial x_\beta}\frac{\partial u}{\partial x_\alpha}\bigg)
 -au\sum^n_{\alpha=1}\nu_\alpha \frac{\partial u}{\partial x_\alpha}
\bigg] dS\\
&\qquad =\int_\Omega\left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx
\endaligned
\tag 4$$
holds, where $a$ is any fixed constant and $h(x)=x-x^0$ with $x^0\in \Bbb R^n$ is a fixed vector. We use 
$\langle \cdot,\cdot\rangle$ to denote the inner product in $\Bbb R^n$. Then we can write (4) as
$$\aligned
&\int_{\partial\Omega}\left[ F(u,Du)\langle h,\nu\rangle -\langle Du,h\rangle\langle Du,\nu\rangle
-au\langle Du,\nu\rangle\right]dS\\
&=\int_{\Omega}\left[\left(\frac{n-2}{2}-a\right)|Du|^2+
\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx.
\endaligned
\tag 4'$$

If we further assume that $\Omega$ is star-shaped with  $x^0\in\bar\Omega$ a central point, and 
$u=0$ on a portion  $\Gamma$ of $\partial\Omega$, then on $\Gamma$ we have 
$\frac{\partial u}{\partial x_\alpha}=\frac{\partial u}{\partial\nu}\nu_\alpha$, so that
$$
\int_{\Gamma}\left[ F(u,Du)\langle h,\nu\rangle -\langle Du,h\rangle\langle Du,\nu\rangle
-au\langle Du,\nu\rangle\right]dS
=-\frac{1}{2}\int_{\Gamma}\left|\frac{\partial u}{\partial \nu}\right|^2\langle h,\nu\rangle dS
\leq 0,
\tag 5$$
because $\Omega$ is star-shaped and $x^0\in\bar\Omega$ is a central point. 

\bigskip
The following are the main results of this paper. Theorem 1 deals with general rotation-like domains
while Theorem 2 treats tubular neighbourhoods of a closed or broken curve.

\bigskip
\proclaim{Theorem 1} Suppose $\Omega\subset \Bbb R^n$ is a smooth domain with $n\geq 4$, and suppose
the orthogonal projection of the closure of the domain onto the first component is an interval $[a,b]$.
 We assume that there is a $\delta>0$, such that for all $a\leq t_1<t_2\leq b$, $|t_2-t_1|\leq \delta$,
the set 
$$\Omega_{t_1,t_2}=\{ x=(x_1,x_2,\dots, x_n)\in\Omega, \; t_1\leq x_1\leq t_2\}$$
 is star-shaped 
and there is some $t_0\in [t_1,t_2]$ such that    $x_0=(t_0,0,\dots,0)$ is a central point.
Let $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ be a smooth solution of (1) with $p\geq (n+1)/(n-3).$ Then $u\equiv 0$ in 
$\bar\Omega$.
\endproclaim

\bigskip
\remark{Remark} A rotation domain is a special case of those treated in Theorem 1. More
precisely, suppose $x_2=f(x_1)>0$ is a smooth function defined in $[a,b]$. Then the rotation in $\Bbb R^{n-1}$ around the $x_1$-axis of the 
two-dimensional region bounded by $f$ and the $x_1$-axis  satisfies the hypotheses of Theorem 1. In particular,
the domains we treat are much more general than those in \cite{CZ}.
\endremark

\bigskip

 Theorem 2  below
deals with the uniqueness problem in general tubular neighbourhoods of embedded curves under
 a technical condition. We assume
that there is a smooth  orthogonal moving frame along the curve \cite{S, Ch 1}. Suppose that 
$\gamma:[0,l]\to \Bbb R^n$ is 
a smooth curve parameterized by its arc-length $s\in [0,l]$. Suppose that there is a smooth
orthogonal basis $e_2(s),\dots, e_n(s)$ on the normal hyperplane of $\gamma(s)$. Let
$\dot\gamma (s)=e_1(s)$. Then 
$$\aligned
&\dot e_1(s)=-k_1(s)e_2,\\
&\dot e_j(s)=k_{j-1}(s)e_{j-1}-k_j(s)e_{j+1},\quad 2\leq j\leq n-1,\\
&\dot e_n(s)=k_{n-1}e_{n-1}.
\endaligned
$$
We call $k_1(s)\geq 0$ \cite{S} the first curvature of $\gamma$ and
 $E(s):=\{ e_1(s),\, e_2(s),\dots, e_n(s)\}$, $0\leq s\leq l$
  a moving orthogonal frame along $\gamma$.

Notice that if $\gamma\subset \Bbb R^2$ is a  planar curve, such a moving frame always exists. 
Let $\gamma(s)=(x_1(s),x_2(s))$, $\alpha(s)=\dot\gamma(s)$, $\beta(s)=(-\dot x_2(s), \dot x_1(s))$,
and let $e_3,\dots e_n$ be the standard Euclidean basis for $\Bbb R^{n-2}$. Then
$\alpha(s),\beta(s),e_3,\dots,e_n$ form an orthogonal moving frame along $\gamma$.

Let $\gamma:[0,l]\to \Bbb R^n$ be a simple, smooth and closed curve with bounded curvatures.
Then it is easy to see that the $r$-neighbourhood 
$$\Omega_r=\{ x\in \Bbb R^n,\; \operatorname{dist}(x,\gamma)<r\}$$
is a tubular neighbourhood of $\gamma$ for $r>0$ small, with $(n-1)$-dimensional open balls of radius $r$ as its
fibres. If $\gamma$ is a broken curve, $\Omega_r$ is the union of a tubular neighbourhood
$\cup_{0<s<l}B_s$ and two half-balls at each end of the curve, where
$B_s$ is an $(n-1)$-dimensional open ball lying in the normal hyperplane of $\gamma(s)$ and centered at $\gamma(s)$.

We have

\proclaim{Theorem 2} Let $n\geq 4$, and let $\gamma$ be an embedded smooth ($C^2$) curve (closed or broken) 
in $\Bbb R^n$ with an associated smooth moving frame as defined above.  Let $p>(n+1)/(n-3)$.  Let $\Omega_r$ be the  
$r$-neighbourhood of $\gamma$. Then for 
sufficiently small $r>0$, the only smooth solution of (1) on $\Omega_r$ is
$u\equiv 0$.
\endproclaim 

\bigskip
\proclaim{Corollary 1} Let $\gamma$ be an embedded smooth ($C^2$)-planar curve (closed or broken) 
in $\Bbb R^2$. Let $\Omega_r$ be its 
$r$-neighbourhood in $\Bbb R^2\times \Bbb R^{n-2}$ with $n\geq 4$  and $p>(n+1)/(n-3)$.  Then for 
sufficiently small $r>0$ the only smooth solution of (1) on $\Omega_r$ is
$u\equiv 0$.
\endproclaim

\bigskip
\demo{Proof of Theorem 1} We divide $[a,b]$ evenly as $a=t_0<t_1<\cdots <t_N=b$, with
$t_{i+1}-t_i=(b-a)/N$, $i=0,1,2,\dots, N$ such that $(b-a)/N<\delta$. Let 
$$\Omega_i=\{ x\in\Omega, \; t_i\leq x_1\leq t_{i+1}\}$$
for $i=0,1,\dots, N-1$. From the property of $\Omega$, we see that $\Omega_i$ is star-shaped and
there is some $t_i^\prime\in
[t_i,t_{i+1}]$ such that $x^i=(t^\prime_i,0,\dots,0)$ is a central point of $\Omega_i$. We
divide the boundary of $\Omega_i$ into three parts: 
$$\partial\Omega_i=\Gamma_i\cup\Gamma_{i+1}\cup
S_i,$$
where $\Gamma_i=\{ x\in\bar\Omega,\, x_1=t_i\}$, and $S_i=\partial\Omega\cup\bar\Omega_i$.
Notice that both $\Gamma_0$ and $\Gamma_N$ are contained in $\partial\Omega$.

Now we apply (4') to $u$ over the sub-domain $\Omega_i$ for each fixed $i$ with $h^i=x-x^i$ to obtain
$$\aligned
&\int_{\partial\Omega_i}\left[ F(u,Du)\langle h^i,\nu\rangle
-\langle Du,h^i\rangle\langle Du,\nu\rangle -au\langle Du,\nu\rangle
\right] dS\\
&=\int_{\Omega_i}\left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx.
\endaligned
\tag 6$$
Now, let $I_i$ and $J_i$ be the left hand side and right hand side of (6),
respectively. If $0<i<N-1$, we have $\partial\Omega_i=\Gamma_i\cup\Gamma_{i+1}\cup
S_i,$ and on $S_i$, $u=0$ so that (5) implies
$$\aligned
&\int_{S_i}\left[F(u,Du)\langle h^i,\nu\rangle
-\langle Du,h^i\rangle\langle Du,\nu\rangle -au\langle Du,\nu\rangle\right]
 dS\\
&=-\frac{1}{2}\int_{S_i}\left|\frac{\partial u}{\partial \nu}\right|^2\langle h^i,\nu\rangle
dS\leq 0.
\endaligned
$$
Therefore 
$$ 
\aligned
&I_i\leq\int_{\Gamma_{i+1}}\left[F(u,Du)\langle h^i,\nu\rangle
-\langle Du,h^i\rangle\langle Du,\nu\rangle -au\langle Du,\nu\rangle
\right] dS\\
&-\int_{\Gamma_i}\left[F(u,Du)\langle h^i,\nu\rangle
-\langle Du,h^i\rangle\langle Du,\nu\rangle -au\langle Du,\nu\rangle
\right] dS,
\endaligned
\tag 7$$
where we have chosen the normal vector of $\Gamma_i$ as towards the positive direction
 of the $x_1$-axis.

If $i=0$, we have
$$
I_0\leq\int_{\Gamma_{1}}\left[F(u,Du)\langle h^0,\nu\rangle
-\langle Du,h^0\rangle\langle Du,\nu\rangle -au\langle Du,\nu\rangle
\right] dS.\tag 8$$
This is because that on $\Gamma_0\cup S_0$, $u=0$.
Similarly, When $i=N-1$, we have, 
$$I_{N-1}\leq -\int_{\Gamma_{N-1}}\left[F(u,Du)\langle h^{N-1},\nu\rangle
-\langle Du,h^{N-1}\rangle\langle Du,\nu\rangle -au\langle Du,\nu\rangle
\right] dS.\tag 9$$

Now we sum (7), (8) and (9) for $i=0,1,\dots, N-1$ to obtain
$$
\sum^{N-1}_{i=0}J_i
\leq 
\sum^{N-2}_{i=0}\bigg\{\int_{\Gamma_{i+1}}\left(F(u,Du)\langle x^{i+1}-x^i,\nu\rangle
-\langle Du,x^{i+1}-x^i\rangle\langle Du,\nu\rangle\right) dS\bigg\}.
\tag 10$$
Since $x^{i+1} -x^i=(t_{i+1}^\prime -t_i^\prime,0,\dots,0)$ and the normal vector $\nu$ on
every $\Gamma_i$ is $\nu=(1,0,\dots,0)$, we have in (10),
$$\gather
\sum^{N-2}_{i=0}\int_{\Gamma_{i+1}} \left[F(u,Du)\langle x^{i+1}-x^i,\nu\rangle
-\langle Du,x^{i+1}-x^i\rangle\langle Du,\nu\rangle -au\langle Du,\nu\rangle\right] dS\\
=\sum^{N-2}_{i=0}\int_{\Gamma_{i+1}}\left[F(u,Du)
-\left|\frac{\partial u}{\partial x_1}\right|^2\right] dS(t_{i+1}^\prime -t_i^\prime).
\tag 11
\endgather $$
We also see that 
$$\sum^{N-1}_{i=0}J_i=\int_{\Omega}
 \left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx.$$
Therefore we obtain
$$\aligned
&\int_{\Omega}
 \left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx\\
&\leq \sum^{N-2}_{i=0}\left[\int_{\Gamma_{i+1}}\left(F(u,Du)-
\left|\frac{\partial u}{\partial x_1}\right|^2\right) dS\right](t_{i+1}^\prime -t_i^\prime).
\endaligned
\tag 12
$$
Now we let $N\to\infty$ so that $\max_i\{ t_{i+1}^\prime -t_i^\prime\}\to 0$ in (12). We have,
 by the
definition of Riemann integral, 
$$\aligned
&\int_{\Omega}
 \left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx\\
&\leq \int_{\Omega} \left[\left(\frac{1}{2}|Du|^2
-\frac{|u|^{p+1}}{p+1}\right)-\left|\frac{\partial u}{\partial x_1}\right|^2\right]dx.
\endaligned
\tag 13
$$
Therefore,
$$\int_{\Omega}
 \left[\left(\frac{n-3}{2}-a\right)|Du|^2+\left( a-\frac{n-1}{p+1}\right)|u|^{p+1}\right] dx
\leq -\int_{\Omega}\left|\frac{\partial u}{\partial x_1}\right|^2 dx.\tag 14$$
If 
$$\frac{n-3}{2}>\frac{n-1}{p+1},\qquad \text{hence}\qquad p>\frac{n+1}{n-3}$$
we may find  a constant $a$ such that
$$\frac{n-3}{2}>a>\frac{n-1}{p+1}$$
and conclude from (14) that $u\equiv 0.$

If 
$$\frac{n-3}{2}=\frac{n-1}{p+1},\qquad \text{which implies}\qquad p=\frac{n+1}{n-3},$$
we can only choose $a=(n-3)/2$ and (14) is reduced to 
$$ \int_{\Omega}\left|\frac{\partial u}{\partial x_1}\right|^2 dx=0,$$
which gives that $\frac{\partial u}{\partial x_1}=0$ in $\Omega$. The zero boundary condition
implies that $u\equiv 0$.

\hfill ${\boxed\,}$ 

\enddemo

\bigskip
\demo{Proof of Theorem 2} Let $\gamma :[0,l]\to \Bbb R^n$ be a $C^2$ closed embedded  curve 
 parameterized by its arc-length, so that $\gamma(0)=\gamma(l)$. 
Define $k_0=\max_{0\leq s\leq l} k_1(s)$.
Let $\bar\Omega_r$ be the closed $r$-neighbourhood in $\Bbb R^n=\Bbb R^2\times \Bbb R^{n-2}$ with
$n\geq 4$, where $0<rk_0<1$.

We first  choose $r>0$ small enough 
so that the periodic mapping (in $s$ with period $l$)
$$F:(s, x_2, x_3, x_4,\dots, x_n)\to \gamma(s)+x_2e_2(s)+ x_3e_3(s)+\cdots +x_ne_n(s)$$
is one-to-one  from $[0,l]\times \bar B_r(0)$ to $\bar\Omega_r$ except at $0$ and $l$ where
$ F(0,\cdot)=F(l,\cdot)$,  with
$$\bar B_r(0)=\{ (x_2,x_3,\dots,x_n)\in
\Bbb R^{n-1}, \, x_2^2+x_3^2+\cdots x_n^2\leq r^2\}$$
the closed ball in $\Bbb R^{n-1}$. The Jacobian of this mapping is $ \pm(1+x_2k_1(s))$, where $k_1(s)$
 is the first 
curvature of $\gamma$.

Now we  divide $[0,l]$ evenly as 
$$0=s_0<s_1<\cdots <s_{N-1}<s_N=l,\qquad s_{i+1}-s_i=\frac{l}{N},\; i=0,1,\dots N-1$$
and let $s^\prime_i$ be the midpoint of $[s_i,s_{i+1}]$. We let $\Gamma_i$ be the intersection of
the normal hyperplane of $\gamma$ at $s=s_i$ and $\Omega_r$ and define $\bar \Omega_i$ to be the
closed sub-domain of $\Omega_r$ bounded by $\Gamma_i$ and $\Gamma_{i+1}$. Notice that $\gamma$
is a closed  curve so that $\Gamma_N=\Gamma_0$ and $\Omega_N=\Omega_0$. 

As in the proof of Theorem 1, we apply (4') to each $\Omega_i$ with 
$h^i(x)=x-\gamma(s^\prime_i)$. We have
$$\aligned
&\int_{\partial\Omega_i}\left[F(u,Du)\langle h^i,\nu\rangle
-\langle Du,h^i\rangle\langle Du,\nu\rangle -au\langle Du,\nu\rangle
\right] dS\\
&=\int_{\Omega_i}\left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx.
\endaligned
\tag 15$$
As in the proof of Theorem 1, we let $I_i$ and $J_i$ be the left and right hand sides of (15), 
respectively, and
let $\partial\Omega_i=\Gamma_i\cup\Gamma_{i+1}\cup S_i$, where $S_i=\partial\Omega_i\cap \partial
\Omega_r$. 

Let us first consider the surface integral over $S_i\subset \partial\Omega_r$.
Notice that 
$u=0$ on $S_i$, so that (5) gives
$$\aligned
&\int_{S_i}\left[F(u,Du)\langle h^i,\nu\rangle
-\langle Du,h^i\rangle\langle Du,\nu\rangle -au\langle Du,\nu\rangle\right]
 dS\\
&=-\frac{1}{2}\int_{S_i}\left|\frac{\partial u}{\partial \nu}\right|^2\langle h^i,\nu\rangle dS.
\endaligned
\tag 16$$
We claim that for sufficiently large $N>0$,  $\langle h^i,\nu\rangle\geq 0$ on $S_i$.
A general point $x\in S_i$ can be written as
$$x=\gamma(s)+x_2e_2(s)+x_3e_3(s)+\cdots +x_ne_n(s)$$
with $x_2^2+x_3^2+\cdots +x_n^2=r^2$, for some $s\in [s_i,s_{i+1}]$, and the outward normal vector at
$x$ is
$$\nu=[x_2e_2(s)+x_3e_3(s)+\cdots +x_ne_n(s)]/r.$$ 
We  have
$$\aligned
&r\langle h^i,\nu\rangle =r\langle x-\gamma(s^\prime_i),\nu\rangle\\
&=\langle \gamma(s)+x_2e_2(s)+x_3e_3(s)+\cdots x_ne_n(s) -\gamma(s^\prime_i),
x_2e_2(s)+x_3e_3(s)+\cdots x_ne_n(s)\rangle\\
&=\langle \gamma(s)-\gamma(s^\prime_i), x_2e_2(s)+x_3e_3(s)+\cdots +x_ne_n(s)\rangle +r^2\\
&\geq r^2-|\gamma(s)-\gamma(s^\prime_i)|r\geq r^2-r|s-s^\prime_i|>0,
\endaligned
$$
when $|s-s^\prime_i|\leq l/N$ is sufficiently small.

Now we sum up $I_i$'s as in the proof of Theorem 1 to obtain
$$\aligned
&\sum^{N-1}_{i=0}I_i
\leq \\
&\sum^{N-1}_{i=0}
\int_{\Gamma_{i+1}}\left[F(u,Du)\langle\gamma(s^\prime_{i+1})-\gamma(s^\prime_{i}),\nu\rangle
-\langle Du,\gamma(s^\prime_{i+1})-\gamma(s^\prime_{i})\rangle\langle Du,\nu\rangle\right] dS\\
&=\sum^{N-1}_{i=0}\left[\int_{\Gamma_{i+1}}\left(\frac{1}{2}|Du|^2
-\frac{|u|^{p+1}}{p+1}\right)\langle \gamma(s^\prime_{i+1})-\gamma(s^\prime_{i}) ,\, \nu\rangle
dS\right.\\
&\left. -\int_{\Gamma_{i+1}}
\langle Du,\, \gamma(s^\prime_{i+1})-\gamma(s^\prime_{i})\rangle\langle Du,\nu\rangle dS\right]
\\
&=A_N.
\endaligned
\tag 17$$
Notice that $\Gamma_N=\Gamma_0$, $\nu =\dot\gamma(s_{i+1})$,
$$\aligned
&\langle \gamma(s^\prime_{i+1})-\gamma(s^\prime_{i}),\nu\rangle\\
&=\langle\dot \gamma(s_{i+1})(s^\prime_{i+1}-s^\prime_{i}),\dot \gamma(s_{i+1})\rangle\\
&+\langle \frac{1}{2}\ddot \gamma(\xi_{i+1})(s^\prime_{i+1}-s_{i+1})^2
-\frac{1}{2}\ddot \gamma(\eta_{i+1})(s_{i+1}-s^\prime_{i})^2,\dot \gamma(s_{i+1})\rangle,
\endaligned
$$
where $\xi_{i+1}$ and $\eta_{i+1}$ are two points in $(s_{i+1}, s^\prime_{i+1})$ and 
$(s^\prime_{i}, s_{i+1})$ respectively. Now we have 
$$\aligned
&\langle\dot \gamma(s_{i+1})(s^\prime_{i+1}-s^\prime_{i}),\dot \gamma(s_{i+1})\rangle\\
&= s^\prime_{i+1}-s^\prime_{i}.
\endaligned
\tag 18$$
Since $\gamma$ is of class $C^2$, there is a constant $C_0>0$ such that $|\ddot\gamma(s)|\leq C_0$ for
all $s\in [0.l]$. Therefore we  also have
$$\aligned
&\left|\langle \frac{1}{2}\ddot \gamma(\xi_{i+1})(s^\prime_{i+1}-s_{i+1})^2
-\frac{1}{2}\ddot \gamma(\eta_{i+1})(s_{i+1}-s^\prime_{i})^2,\,\dot \gamma(s_{i+1}) \rangle\right|\\
&\leq \frac{1}{2}C_0\left[ (s^\prime_{i+1}-s_{i+1})^2+(s_{i+1}-s^\prime_{i})^2\right]\\
&\leq  C_0(s^\prime_{i+1}-s^\prime_{i})^2.
\endaligned
\tag 19$$
Similarly, we have
$$\aligned
&\langle \gamma(s^\prime_{i+1})-\gamma(s^\prime_{i}),Du\rangle\\
&=\langle \dot \gamma (s_{i+1}),Du\rangle (s^\prime_{i+1}-s^\prime_{i})\\
&+\langle \frac{1}{2}\ddot \gamma(\xi^\prime_{i+1})(s^\prime_{i+1}-s_{i+1})^2
-\frac{1}{2}\ddot \gamma(\eta^\prime_{i+1})(s_{i+1}-s^\prime_{i})^2,Du\rangle,
\endaligned
\tag 20$$
with 
$$\aligned
&\left| \langle \frac{1}{2}\ddot \gamma(\xi^\prime_{i+1})(s^\prime_{i+1}-s_{i+1})^2
-\frac{1}{2}\ddot \gamma_r(\eta^\prime_{i+1})(s_{i+1}-s^\prime_{i})^2,Du\rangle\right| \\
&\leq C_0|Du|(s^\prime_{i+1}-s^\prime_{i})^2.
\endaligned
\tag 21
$$
Now we can estimate the  sum $A_N$ in (17):
$$\aligned 
&A_N\leq\sum^{N-1}_{i=0}\int_{\Gamma_{i+1}}\left[F(u,Du)- \langle Du,\dot\gamma(s_{i+1})\rangle^2\right] dS
(s^\prime_{i+1}-s^\prime_{i})\\
&+C_0\sum^{N-1}_{i=0}\frac{l}{N}\left[\int_{\Gamma_{i+1}}\left|
\frac{1}{2}|Du|^2
-\frac{|u|^{p+1}}{p+1}\right|+|Du|^2dS\right](s^\prime_{i+1}-s^\prime_{i})\\
&=B_1(N)+B_2(N),
\endaligned
$$
where
$$\aligned
&B_1(N)=\sum^{N-1}_{i=0}\int_{\Gamma_{i+1}}\left[F(u,Du)- \langle Du,\dot\gamma(s_{i+1})\rangle^2\right] dS
(s^\prime_{i+1}-s^\prime_{i})\\
&\to \int_0^l\int_{\Gamma_s}\left[\left(\frac{1}{2}|Du|^2
-\frac{|u|^{p+1}}{p+1}\right)- \langle Du,\dot\gamma(s)\rangle^2\right]dS ds,\endaligned
$$
as $N\to\infty$, where 
$$\Gamma_s=\{ \gamma(s)+x_2e_2(s)+x_3e_3(s)+\cdots +x_ne_n(s),\; x_2^2+x_3^2+\cdots+x_n^2\leq r^2\}
.\tag22$$
We also have
$$
B_2=C_0\sum^{N-1}_{i=0}\frac{l}{N}\int_{\Gamma_{i+1}}\left[\left|
\frac{1}{2}|Du|^2
-\frac{|u|^{p+1}}{p+1}\right|+|Du|^2\right]dS(s^\prime_{i+1}-s^\prime_{i})
\to 0$$
as $N\to 0$ because 
$$\sum^{N-1}_{i=0}\int_{\Gamma_{i+1}}\left[\left|
\frac{1}{2}|Du|^2
-\frac{|u|^{p+1}}{p+1}\right|+|Du|^2\right]dS(s^\prime_{i+1}-s^\prime_{i})$$
converges to an integral.

Now we sum up the right hand side of (15):
$$\aligned
&\sum^{N-1}_{i=0}J_i=
\sum^{N-1}_{i=0}\int_{\Omega_i}\left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx
\\
&=\int_{\Omega_r}
\left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx.
\endaligned
$$
We now change variables
$$x=\gamma(s)+x_2e_2(s)+x_3e_3(s)+\cdots + x_ne_n(s),$$
to obtain
$$\aligned
&\int_{\Omega_r}
\left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right] dx\\
&=\int^l_0\int_{\Gamma_s}
\left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right]
(1+x_2k_1(s))dS \, ds,
\endaligned
$$
when $rk_0<1$. Finally we obtain
$$\aligned
&\int^l_0\int_{\Gamma_s}
\left[\left(\frac{n-2}{2}-a\right)|Du|^2+\left( a-\frac{n}{p+1}\right)|u|^{p+1}\right]
(1+x_2k_1(s))dS \, ds\\
&\leq \int_0^l\int_{\Gamma_s}\left[\left(\frac{1}{2}|Du|^2
-\frac{|u|^{p+1}}{p+1}\right)- \langle Du,\dot\gamma(s)\rangle^2\right]dS ds.
\endaligned
\tag 23$$
Now, we deduce from (23) that
$$\aligned
&\int^l_0\int_{\Gamma_s}
\left[\left(\frac{n-2}{2}-a\right)\phi -\frac{1}{2}\right]|Du|^2+
\left[\left( a-\frac{n}{p+1}\right)\phi +\frac{1}{p+1}\right]|u|^{p+1}dS \, ds\\
&-\int^l_0\int_{\Gamma_s}\langle Du,\dot\gamma(s)\rangle^2dS ds\leq 0,
\endaligned
\tag 24
$$
where $\phi:=1+x_2k_1(s).$ Now, $|\phi-1|\leq rk_0\to 0$ as $r\to 0.$ 
Therefore
$$\left(\frac{n-2}{2}-a\right)\phi-\frac{1}{2}\to \frac{n-3}{2}-a,
\quad\text{and}\quad
\left( a-\frac{n}{p+1}\right)\phi +\frac{1}{p+1}\to a-\frac{n-1}{p+1}$$
uniformly on $[0,\, l]\times \bar B_r(0)$ as $r\to 0$. Because $p>(n+1)/(n-3)$,
 it is possible to find some
$a\in\Bbb R$ and $c>0$ such that 
$$\left(\frac{n-2}{2}-a\right)\phi-\frac{1}{2}\geq c,\qquad 
\left( a-\frac{n}{p+1}\right)\phi +\frac{1}{p+1}\geq c$$
on $[0,\, l]\times \bar B_r(0)$ as $r>0$ sufficiently small. Thus (24) 
implies that $u=0$ on 
$\Omega_r.$

\medskip
If $\gamma$ is not a closed curve, the proof is similar. We need to extend 
the curve at the two end points
$\gamma(0)$ and $\gamma(l)$ along the tangent directions as straight line 
segments so that the extended curve reaches the boundary of $\Omega_r$ at
two points. Then the proof proceeds as in the case of closed curves.
\hfill ${\boxed \,}$
\enddemo

\bigskip
{\bf Acknowledgement} I would like to thank Professor K. J. Brown and the 
referee for their helpful suggestions.

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\enddocument 

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