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\markboth{ Gluing approximate solutions }
{ Yanyan Li \& Zhi-Qiang Wang }
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 215--223.\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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Gluing approximate solutions of minimum type on the Nehari manifold
%
\thanks{ {\em Mathematics Subject Classifications:} 35J20, 58E05.
\hfil\break\indent
{\em Key words:} luing, variational, minimax.
\hfil\break\indent
\copyright 2001 Southwest Texas State University.
\hfil\break\indent Published January 8, 2001. } }
\date{}
\author{ Yanyan Li \& Zhi-Qiang Wang }
\maketitle
\begin{abstract}
In the last decade or so, variational gluing methods have been
widely used to construct homoclinic and heteroclinic
type solutions of nonlinear
elliptic equations and Hamiltonian systems.
This note is concerned with the procedure of gluing
mountain-pass type solutions.
The first procedure to glue mountain-pass type solutions
was developed through the work of S\'er\'e, and Coti Zelati -
Rabinowitz. This procedure and its variants
have been extensively used in many problems by now for nonlinear
equations with superlinear nonlinearities.
In this note we provide an alternative device
to the by now standard procedure
which allows us to glue minimizers on the Nehari
manifold together as genuine, multi-bump type, solutions.
\end{abstract}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{theorem}[lemma]{Theorem}
\newtheorem{prop}[lemma]{Proposition}
\begin{section}{Introduction}
In the last decade or so, variational gluing methods have been
widely used to construct homoclinic and heteroclinic
type solutions of nonlinear
elliptic equations and Hamiltonian systems
(see, e.g. Rabinowitz \cite{paul} and
references therein).
The idea is to first
construct some basic solutions (or approximate
solutions) which are characterized by minimax
method and which are used as building blocks for construction of
multi-bump type solutions. These multi-bump type solutions then
are obtained by some gluing
procedures and look roughly like sums of the basic solutions.
The general idea
is clear by now,
though for different types of basic solutions one has to employ different
procedures for the concrete problems.
Different type of basic solutions have been glued together by
various authors, which include minimizers and
mountain-pass type solutions. In fact even $cat >1$ solutions
have been glued together, see for example
Giannoni and Rabinowitz \cite{gr}.
This note is concerned with the procedure of gluing
mountain-pass type solutions.
The first procedure to glue mountain-pass type solutions
was developed
through the work of S\'er\'e (\cite{sere} \cite{sere2}) and Coti Zelati -
Rabinowitz (\cite{cora1} \cite{cora2}), and this procedure and its
variants
have been extensively used in many problems by now for nonlinear
equations with superlinear nonlinearities
(see, e.g. Rabinowitz \cite{paul} and
references therein). In these papers the basic
solutions are mountain-pass type solutions.
On the other hand, under slightly stronger
conditions
these mountain-pass solutions can also be characterized as
minimizers of a constrained problem, namely, minimizers on the
Nehari manifold.
In this paper we provide an alternative device
to the by now standard procedure
which allows us to glue minimizers on the Nehari
manifold (or local minimizers,
approximate local minimizers) together as
genuine (multi-bump type) solutions.
Though the new procedure is somewhat parallel
to the original one for mountain-pass solutions there are still
technical complications needed to be fixed.
On the other hand, it seems the new device in gluing minimizers on Nehari
manifold
is simpler than those for gluing mountain-pass solutions
in the full space.
For instance,
one step involved in \cite{cora1} and \cite{cora2}
is to do a minimization problem on some annulus regions and to use
elliptic estimates to achieve the smallness of certain map. This step
has to be done on a case by case basis for ODEs, PDEs with
subcritical exponents and PDEs with critical exponents and
seems to be
somewhat laborsome for PDE problems, especially
for those involving critical exponents (\cite{l1} \cite{l2}).
Our device given here
will {\it avoid} this step
and treat all problems uniformly.
For simplicity we only present our device
for an ODE problem to demonstrate the procedure.
Although the results are not new, the procedure we use is
different from the known one and may prove to be of
advantage in dealing with some other problems with the presence
of a Nehari manifold.
The same device clearly works
for analogous subcritical exponent periodic PDEs
$$
-\Delta u+ a(x) u = f(x,u), \;\; \hbox{in}\;\; \mathbb{R}^N,
$$
with suitable growth condition on $f$ and periodic
dependency in $x$;
and presumably should also work
for analogous critical exponent periodic PDEs.\end{section}
\begin{section}{An ODE problem}
Consider
\begin{equation}
\label{ode} -u''+a(t)u = f(t,u), \quad t \in \mathbb{R}
\end{equation}
We look for homoclinic solutions of this equation,
i.e.,solutions such that $\lim_{|t|\to \infty}
u(t) =0$ and $\lim_{|t|\to \infty} u'(t)=0$.
Assume
\begin{enumerate}
\item[(f1)] $a(t) \in C(\mathbb{R},\mathbb{R})$ is $T-$periodic and $\min_{\mathbb{R}} a(t) >0$.
\item[(f2)] $f(t,u)\in C(\mathbb{R}\times \mathbb{R},\mathbb{R})$ is $T-$periodic in $t$.
\item[(f3)] $f_u(t,0)=0$ and $|f_u(t,u)| \leq C(1+ |u|^p)$ for some $p>1$.
\item[(f4)] There is a $\theta> 1$ such that
$ f'(t,u)u^2 \geq \theta f(t,u)u$ for all $t$ and $u$.
\end{enumerate}
There is a variational formulation of the problem. Namely,
$$
I(u) =\frac{1}{2} \int_{\mathbb{R}}(|\dot{u}|^2 +au^2 ) dt -\int_{\mathbb{R}} F(t,u) dt
$$
for $u\in X:=H^1(\mathbb{R})$. Then critical points of $I$ are solutions of
(\ref{ode}).
We use $\|\cdot\|$ to denote the norm in $X$.
There is alternative approach to the above, namely the Nehari manifold.
Define
$$
\gamma(u):= \int_{\mathbb{R}} f(t,u)udt -\int_{\mathbb{R}}(|
\dot{u}|^2 +au^2 )dt,
$$ and
let
$$
V= \{u\in X\setminus \{0\}\;|\; \gamma(u)=0 \}.
$$
Then it is well known that
under conditions ($f_1$ - $f_4$), $V$ is a $C^1$ manifold and critical points of $I$ on $V$
are also critical points of $I$ in $X$ and therefore solutions of (\ref{ode}).
We use the usual notations. $I^c=\{u\in V\;|\; I(u)\leq c\}$,
$I_c=\{u\in V\;|\; I(u)\geq c\}$, $I^b_a= I^b\cap I_a$,
$K=\{u\in V\;|\; I'(u)=0\}$, $K^c= K\cap I^c$.
For an integer $j$, $\tau_j u = u(t-j)$ the translation of $u$.
Then for any $j$, $\tau_j w\in K^c$.
Let
$$
c:= \inf_{V} I(u),
$$
the ground state energy of $I$.
Using the following compactness results for (PS) sequences of $I$
one easily gets that
$c$ is always achieved at some $u$ which is a ground state solution of (\ref{ode}).
\begin{prop}
\label{comp}
Let $(u_n) \subset V$ be such that
$I(u_n) \to b$
and $(I_{|V})'(u_n) \to 0$. Then there is an $l\in N$ (depending on $b$),
$v_1, ...,v_l \in K\setminus \{0\}$, a subsequence of $u_n$ and corresponding
$(j_{i, n})_{i=1}^l\subset Z^l$ such that
$$
\|u_n- \sum_{i=1}^l \tau_{j_{i, n}} v_i\|\to 0, \quad
\sum_{i=1}^l I(v_i)=b,
$$
and for $i\neq \ell$, $|j_{i, n} - j_{\ell, n}|\to \infty$.
\end{prop}
This is just a reformulation of Prop. 2.31 in \cite{cora2}, since $V$ is a
natural
constraint of $I$ in the sense that $(I_{|V})'(u)=0$ iff $I'(u)=0$.
Due to the translation invariance of the problem, there may be many
solutions on the energy level $c$. We shall assume
$$K_c^c \mbox { has an isolated point, say, $w$}. \eqno(*)$$
For an integer $k\geq 2$,
let $\vec{j}=(j_1,\cdots,j_k)$, a $k$-tuples of integers.
We shall show that there are real solutions of (\ref{ode}) which roughly
look like $\sum_{i=1}^{k} \tau_{j_i} w$.
More precisely,
let
$$
2 r_0=\min\{\nu, \mu\}>0,
$$
where $\nu=\inf\{\|u\| \;|\; u\in K\setminus \{0\}\}$
and $ \mu=\inf\{\|u-w\|\;|\; u\in K\}$.
\begin{theorem}
\label{main}
Assume (f1 - f4) and $K_c^c$ has an isolated point.
For $0< \alpha < \frac{c}{2}$ and $0< r< r_0$
there is $j_0>0$ such that for all $k$-tuples of integers
$\vec{j}$ satisfying
$ \min_{i\neq \ell} |j_{i}-j_\ell|> j_0$
$$
K^{kc+\alpha}_{kc-\alpha} \cap N_{r}\Big( \sum_{i=1}^{k} \tau_{j_i} w
\Big) \neq \emptyset.
$$
\end{theorem}
Here, $N_r(\cdot)$ denotes the $r$-neighborhood
in $X$.
The proof of Theorem \ref{main} is based on
an indirect argument with the basic idea going
back to \cite{sere} \cite{cora1} \cite{cora2}.
Our procedure below is somewhat different
from the one used
in the original argument (\cite{sere} \cite{cora1} \cite{cora2}),
and in a way simpler.
\smallskip
\noindent{\bf Step 1}.
First, for $R>0$ we define a cut-off operator
$$
T_R(u)= \rho(2 R^{-1} |x|)u(x)
$$
where $\rho(t)= 1$ for $0\leq t\leq 1$ and $\rho(t)= 0$ for $t\geq 2$.
With $\vec{j}=(j_1,\cdots,j_k)$ satisfying
$
\inf_{i\neq \ell}| j_i - j_\ell|
> 2 R,
$
for $y=(y_1,...,y_k)$ with $y_i\geq 0$, $i=1,...,k$ and $\sum_{i=1}^k y_i=1$,
we define
$$
G_0(y) = b(y) \sum_{i=1}^k y_i \tau_{j_i} T_{R}(w)
$$
where
$b(y)> 0$ is such that $G_0(y) \in V$.
We fix a $\delta_0\in (0, 1/k)$ so that
$\max_{y} \gamma(\delta_0 b(y) w) <0$ (which can be done due to
($f_3$)) and
define
$$\Delta_k= \left
\{y=(y_1,...,y_k) \in \mathbb{R}^{k}| \sum_{i=1}^k y_i=1, y_i\geq \delta_0
\right \},
$$
a $(k-1)$-dimensional simplex.
Then $G_0\in C(\Delta_k,V)$.
By the explicit form of $G_0$ we have,
as $R\to \infty$,
$$
I(G_0(y)) = \sum_{i=1}^k I(b(y)y_i \tau_{j_i} T_R(w)) =
\sum_{i=1}^k I( b(y) y_i w ) + o(1)\leq k c +o(1).
$$
So we get
\begin{equation}
\label{uppe}
\lim_{R\to \infty} \max_{\Delta_k}I(G_0(y))\leq kc.
\end{equation}
Note that $I(G_0(y_c))\geq kc$, where $y_c =(\frac{1}{k}, ...,
\frac{1}{k})$ the center of $\Delta_{k}$.
Define
$$
\Gamma= \left \{ G
\in C(\Delta_k, V)\;|\;
G_{|\partial \Delta_k} = G_0\right \}.
$$
For $\vec{j}=(j_1,...,j_k)$ satisfying $\inf_{i\neq \ell}|j_i-j_{\ell}| > 2R$,
used for $G_0$, we define for any $u\in V$
$$
u^{(i)}(x) = \rho(R^{-1}|x-j_i|)u(x), \;\; i=1,...,k.
$$
\begin{lemma}
\label{inte}
Given $G_0$ as above with
$\vec{j}=(j_1,...,j_k) $ and $R$ fixed,
for any $G\in \Gamma$ there exists
$y_0\in \Delta_{k}$ such that
$$
\gamma(G(y_0)^{(i)}) =0,\;\; i=1,...,k.
$$
\end{lemma}
\noindent{\bf Proof.}
Regarding $\Delta_k$ as a part of an affine $(k-1)$-plane which we denote by $A^{k-1}$, we see
$A^{k-1}-(\frac{1}{k}, ...,\frac{1}{k})$ is a $(k-1)-$plane passing through the origin in
$\mathbb{R}^{k}$ which we denote by $\tilde{A}^{k-1}$.
For any $G\in \Gamma$ we introduce a map
from
$\tilde{\Delta}_k = \Delta_k - (\frac{1}{k}, ...,\frac{1}{k})$ into $\tilde{A}^{k-1}$
(with $\tilde{y}= y -(\frac{1}{k}, ...,\frac{1}{k})$),
by
$$
h(\tilde{y}) = (h_1, ..., h_k):=(\gamma(G(y)^{(1)}), ..., \gamma(G(y)^{(k)})).
$$
Then the claim is proved if we can show that
$$
\deg (h, \tilde{\Delta}_k, 0) \neq 0.
$$
Note that $0$ is the center of $\tilde{\Delta}_k$.
But this degree only depends on $G_0$ because all $G$ agree with $G_0$
on the boundary of $\tilde{\Delta}_k$.
Finally, we compute the above degree for $G_0$ and we claim it is $1$.
We prove this by an induction in $k$. For $k=2$ it is easy to check it
by hand: when $y_1=\delta_0$ and $y_2= 1-\delta_0$ we have $\tilde{y}_1=
\delta_0 - \frac{1}{2} <0$ and $\tilde{y}_2= 1-\delta_0-\frac{1}{2} >0$;
and $h_1 = \gamma (b \delta_0 \tau_{j_1} T_R(w)) <0$
and therefore $h_2 = \gamma (b (1-\delta_0) \tau_{j_2} T_R(w)) >0$ for
$h_1+h_2=0$.
At the other end point of $\tilde{\Delta}_2$ we have similar computations,
which together shows that $h$ is homotopy to the identity map.
Now for $k\geq 3$,
$\tilde{\Delta}_k$ has $k$
faces (opposite to each vertex and denoted by $F_i$).
On the $i$th-face $F_i$, if $y=(y_1,...,y_k)\in F_i$ then
$y_i=\delta_0- \frac{1}{k}$. And we get $h_i = \gamma(b \delta_0
\tau_{j_i} T_R (w)) < 0$. Using this fact, we may first project
(by radial scalings on $\tilde{A}^{k-1}$) the image of $h$ to
$\epsilon \tilde{\Delta}_k$ for some $\epsilon >$ small. Then
using an expansion scaling we may have the image of $\partial
\tilde{\Delta}_k$ into itself. We denote this operation by $P$,
i.e., $P h$ is a map from $\tilde{\Delta}_k \to \tilde{A}^{k-1}$
such that $Ph(\partial \tilde{\Delta}_k) \subset \partial
\tilde{\Delta}_k$. By the homotopy property, $\deg (h,
\tilde{\Delta}_k, 0) = \deg (Ph, \tilde{\Delta}_k, 0)$. Note that
taking $y=0$ we see $\gamma(g_1(y))=\cdots = \gamma(g_k(y))=0$.
By some standard properties of the degree
(see, e.g. \cite{brni}),
$$
\deg(Ph, \tilde{\Delta}_k, 0) = \deg (Ph, \partial \tilde{\Delta}_k,
\partial \tilde{\Delta}_k).
$$
Now on $F_1$ the center $c_1$ has coordinates $y_1=\delta_0-\frac{1}{k}$
and for $i=2,...,k$, $y_i= \frac{1-\delta_0}{k-1} -\frac{1}{k}$.
Using this it is easy to see
that $c_1$ is not covered by $Ph( \cup_{i=2}^k F_i)$, for
if not
$c_1= Ph(y)$ for some $y\in F_i$ with $i\geq 2$, then we have $h_i(y) <0$ and
therefore $(Ph)_i(y) <0$, this is a contradiction with $y_i>0$ for $c_1$.
By the excision property
$$
\deg (Ph, \partial \tilde{\Delta}_k, \partial \tilde{\Delta}_k)=
\deg (Ph, F_1, c_1).
$$
However, this is what we would get from the $(k-1)$-map. The induction is
complete.
$\diamondsuit$
We need another technical result.
\begin{lemma}
\label{ener}
Let $u\in V$ be such that
$ u^{(i)}\in V$ for all $i=1,...,k$ (obtained by using
$\vec{j}=(j_{1}, ..., j_{k})$ satisfying $\inf_{i\neq \ell} |j_{i}-j_{\ell}|
> 2 R$). Then
$ I(u) \geq kc$.
\end{lemma}
\noindent{\bf Proof.}
First, we write $
W_R=\bigcup_{i=1}^{k} B_{R}(j_i)$.
Then
\begin{eqnarray*}
I(u) & =&\frac{1}{2}\int |\nabla u|^2 + a |u|^2 - \int F(x,u)\\
&=& \sum_{i=1}^{k} \Big( \frac{1}{2}
\int_{ B_{2R}(j_{i})} |\nabla u^{(i)}|^2 +a |u^{(i)}|^2
-\int_{B_{2R}(j_{i})} F(x,u^{(i)})\Big) \\
&& + \frac{1}{2}
\int_{ \mathbb{R}\setminus W_{2R}}
|\nabla u|^2 +a |u|^2
-\int_{\mathbb{R}\setminus W_{2R}}
F(x,u) \\
&& + \sum_{i=1}^{k}\frac{1}{2} \int_{ B_{2R}(j_i)\setminus B_R(j_i)}
\Big( |\nabla ((1-\rho)u)|^2 +(1-\rho)^2 a u^2 \\
&& +2\nabla (\rho u)\nabla((1-\rho)u) +
2\rho(1-\rho) a u^2 \Big)\\
&& - \sum_{i=1}^{k} \int_{ B_{2R}(j_i)\setminus B_R(j_i)}
(F(x,u)-F(x,u^{(i)}))
\end{eqnarray*}
Using $u\in V$ and $u^{(i)}\in V$ for all $i=1,...,k$, we get
\begin{eqnarray*}
\int_{\mathbb{R}\setminus W_{2R}}
|\nabla u|^2 +a |u|^2-\int_{\mathbb{R}\setminus W_{2R}}f(x,u)u &&\\
+ \sum_{i=1}^{k} \int_{B_{2R}(j_i)\setminus B_R(j_i)}
\Big( |\nabla ((1-\rho)u)|^2 +(1-\rho)^2 a u^2 &&\\
+2\nabla (\rho u)\nabla((1-\rho)u) +2\rho(1-\rho) a u^2\Big)\\
-\sum_{i=1}^{k}\int_{B_{2R}(j_i)\setminus
B_R(j_i)} (f(x,u)u -f(x,u^{(i)}) u^{(i)}) &=& 0\,.\\
\end{eqnarray*}
Bringing this into the earlier formula we
have
\begin{eqnarray*}
I(u) &\geq& kc+ \int_{\mathbb{R}\setminus W_{2R}} (\frac{1}{2}
f(x,u)u -F(x,u)) \\
&&+ \sum_{i=1}^{k} \int_{B_{2R}(j_i)\setminus B_R(j_i)}
\Big(\frac{1}{2} f(x,u)u - \frac{1}{2}f(x,u^{(i)})u^{(i)}\\
&& - F(x,u) +F(x,u^{(i)})\Big)
\end{eqnarray*}
which implies
$
I(u) \geq kc$ since
the last two terms on the right
hand side
are both non-negative. Indeed, by ($f_4$)
we have
$$
\frac{1}{2}
f(x,u)u -F(x,u)\geq 0,
$$
and, writing $g(t)= \frac{1}{2} f(x,tu)tu -F(x,tu)$
by the mean value theorem,
we have for some $\xi \in (0,1)$,
\begin{eqnarray*}
\lefteqn{
\frac{1}{2} f(x,u)u -\frac{1}{2}f(x,u^{(i)})u^{(i)}-F(x,u)+F(x,u^{(i)}) }\\
&=&\frac{1}{2} f(x,u)u - \frac{1}{2}f(x,\rho(R^{-1}|x-j_i|)u)
\rho(R^{-1}|x-j_i|)u - F(x,u) \\
&&+F(x,\rho(R^{-1}|x-j_i|)u)\\
&=&
g(1) -g(\rho)\\
&=&g'(\rho+ \xi(1-\rho))(1-\rho)\\
&=& \frac{1}{2} \left \{f'(x, [\xi +\rho (1-\xi)] u) [\xi+\rho (1-\xi)] u^2 -
f(x, [\xi +\rho (1-\xi)] u) u\right \} (1-\rho)\\
&\geq& 0\,.
\end{eqnarray*}
which completes the present proof \hfill$\diamondsuit$\smallskip
Let $z_R=
b_R\sum_{i=1}^{k} \tau_{j_{i}} T_R(w)$ with $
\inf_{i\neq \ell}|j_{\ell}-j_{i}| > 2R$,
where $b_R>0$ is such that
$z_R\in V$. Note $b_R\to 1$ as $R\to \infty$.
For any $\epsilon >0$, by choosing $R>0$ large
we may get, by (\ref{uppe}),
$\max_{\Delta_k} I(G_0(y))< kc+\epsilon$.
Also we remark that
when $r_0>r>0$ is fixed, for all small $\epsilon$ and large
$R$ it holds
that $I(G_0(y)) \geq kc -\epsilon$ implies
$G_0(y) \in N_\frac{r}{8}(z_R)$.
We fix $r>0$ now such that
for all $R\geq 1$ if
$G\in \Gamma$ satisfying $\| G(y)-G_0(y)\|\leq r$ then
$ G(y)^{(i)}\neq 0$ for all $y$ and $i=1,...,k$.
\smallskip
\noindent{\bf Step 2}.
If we assume the conclusion of Theorem \ref{main} is not true,
using a deformation argument from a pseudo-negative gradient flow
we deform $G_0$ to a map $G_1 : \Delta_k\to V$ such that
$\max_{\Delta_k} I(G_1(y))\leq kc-\epsilon$,
$\|G_1(y)-G_0(y)\|\leq r$ and $G_1|_{\partial \Delta_k}=G_0$. Then
using Lemmas \ref{inte} and \ref{ener} we will have a
contradiction. We need the following lemma.
\begin{lemma}
\label{lowe}
There exist
$\delta_r>0$ and $R_r>0$
such that for all $R\geq R_r$ and
for all $u\in N_{r}(z_R)\setminus N_{\frac{r}{8}} (z_R)$
$$
\|I'(u)\| \geq \delta_r.
$$
\end{lemma}
\noindent{\bf Proof.}
If the conclusion is not true, we would have a sequence
$R_n\to \infty$ and
$u_n\in N_{r}(z_R)\setminus N_{\frac{r}{8}} (z_R)$
such that
$I'(u_n) \to 0$.
Then $(u_n)$ is a (PS)$_b$ sequence for $I$ with some $b$.
By Proposition \ref{comp}
$$\|u_n-\sum_{i=1}^l \tau_{j_{i,n}} v_i\| \to 0$$
for some integer $l$ and $v_1,..,v_{l} \in K$ and
$|j_{i,n} - j_{\ell,n}|\to \infty$ for $i\neq \ell$.
Since as $R_n\to \infty$,
$\|z_{R_n} - \sum_{i=1}^k \tau_{j_{i,R_n}} w\| \to 0$,
we get
$$
\| \sum_{i=1}^l \tau_{j_{i,n}} v_i - \sum_{i=1}^k \tau_{j_{i,R_n}} w\| \to 0.
$$
From this it is easy to argue by using (*) that
$l=k$, $v_i=w$ for all $i$ and for $n$ large $j_{i,n}= j_{i,R_n}$.
This is a contradiction to $ \|u_n- z_{R_n}\| \geq \frac{r}{8}$.
$\diamondsuit$
\smallskip
Now we can finish the proof of our
main theorem.
We take $0< \epsilon < \frac{3 r\delta_r}{8}$ and $R\geq R_r$
so that
$\max_{\Delta_k} I(G_0(y))< kc+\epsilon$ and that
$I(G_0(y)) \geq kc -\epsilon$ implies
$G_0(y) \in N_\frac{r}{8}(z_R)$.
\smallskip
Next, choose $\epsilon < \epsilon_1 < c$.
%Let
%$$
%\psi(u) =\frac{\|u- (I^{kc-\epsilon_1}\cup I_{kc+\epsilon_1})\|}
%{\|u-(I^{kc-\epsilon_1}\cup I_{kc+\epsilon_1})\| + \| u- I^{kc+
%\epsilon}_{kc-\epsilon}\|}
%$$
%and
Let
$$
\phi(u) = \frac{\|u- X\setminus N_{r}(z_R) \|}{\|u- N_{\frac{r}{2}}(z_R)\|+
\|u-X\setminus N_{r}(z_R)\|}.
$$
and let $U$ be a locally Lipschitz pseudo-gradient vector
field for $I$ on $V\setminus K$ such that
\begin{tabular}{cl}
(i) & $\|U(u)\|\leq \frac{4\epsilon_1}{\|I'(u)\|}$,\\ (ii) &
$I'(u)U(u) \geq 2\epsilon_1$.\\
%(iii)& $U$ is equivariant with respect to the translations.
\end{tabular}
Let $\eta$ be the flow given by the solution of
$$
\frac{d\eta}{dt}= -\phi(\eta)
%\psi(\eta)
U(\eta), \eta(0,u)=u.
$$
Let $u=G_0(y)$ be such that $I(u)\geq kc -\epsilon $ so that
$u\in N_\frac{r}{8}(z_R)$.
Using Proposition \ref{comp} we can show
either (i) $\eta(t,u)$ reaches $\partial B_r(z_R)$ for some $t\leq 1$ or
(ii) $\eta(t,u)$ remains in $B_r(z_R)$ for $t\in [0,1]$.
If (i) occurs, in some time interval $[t_1,t_2]$,
$\eta(t,u)$ reaches from $\partial B_{\frac{r}{8}}(z_R)$ to
$\partial B_r(z_R)$.
Then it must reach $I^{kc-\epsilon}$ already in the time interval.
Otherwise,
$$
\frac{7r}{8}= \|\eta(t_2,u)-\eta(t_1,u)\| \leq \int_{t_1}^{t_2} \phi(\eta)
%\psi(\eta)
\|U(\eta(t,u)\|dt \leq
\frac{4\epsilon_1}{\delta_r}\int_{t_1}^{t_2} \phi(\eta)
%\psi(\eta)
dt, $$ and $$2\epsilon \geq I(\eta(t_2,u))-I(\eta(t_1,u)) =
\int_{t_1}^{t_2} \frac{dI}{dt}(\eta(t,u))dt \geq
2\epsilon_1\int_{t_1}^{t_2} \phi(\eta)
%\psi(\eta)
dt.
$$
This implies $\epsilon \geq \frac{7 r\delta_r}{8}$, a contradiction.
Thus if (i) occurs there is a unique
$\sigma(u)\leq 1$ such that $I(\eta(\sigma(u),u))=kc-\epsilon$.
If (ii) occurs we may have either $\eta(t,u)$ has to go from
$B_{\frac{r}{8}}(z_R) $ to the boundary of
$B_{\frac{r}{2}}(z_R) $ and similar argument shows that
there is a unique
$\sigma(u)\leq 1$ such that $I(\eta(\sigma(u),u))=kc-\epsilon$, or
$\eta(t,u)$ stays in $B_{\frac{r}{2}}(z_R) $ for $t\in [0,1]$. In the
latter case
if $\eta(t,u)$ does not reach $I^{kc-\epsilon}$ we would have
%$\psi $ and
$\phi$ equal to 1 along $\eta(t,u)$ and we have $ 2\epsilon \geq
I(\eta(0,u)) - I(\eta(1,u)) \geq 2 \epsilon_1$, a contradiction.
In both cases, we have
$
\|\eta(\sigma(u),u)-u\| \leq r$.
We get $G_1(y) = \eta(\sigma(G_0(y)),G_0(y))$ which
is a continuous map from $\Delta_k$ into $V$ and agrees with
$G_0(y)$ on $\partial \Delta_k$.
Moreover,
\begin{equation}
\label{rsmall}
\|G_1(y)-G_0(y)\|\leq r.
\end{equation}
To finish the proof of Theorem \ref{main},
let us produce a contradiction as follows.
Applying Lemma \ref{inte} to $G_1(y)$ we conclude that
there exists $y\in \Delta_k$ such that
$$
\gamma(G_1(y)^{(i)})=0,\;\; i=1,...,k.
$$
Due to (\ref{rsmall}), we obtain $G_1(y)^{(i)}\neq 0$ for $i=1,...,k$,
i.e., $G_1(y)^{(i)}\in V$ for $i=1,...,k$.
Applying Lemma \ref{ener}, we get a contradiction with $\max I(G_1(y))\leq kc-\epsilon$.
\end{section}
\paragraph{Acknowledgement.} The authors
want to thank P. Rabinowitz for his helpful comments.
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\noindent{\sc Yanyan Li }\\
Department of Mathematics,
Rutgers University \\
New Brunswick, NJ 08903 USA\\
e-mail: yyli@math.rutgers.edu \smallskip
\noindent{\sc Zhi-Qiang Wang} \\
Department of Mathematics,
Utah State University \\
Logan, UT 84322 USA \\
e-mail: wang@sunfs.math.usu.edu
\end{document}