# Existence and concentration of semiclassical states for nonlinear
Schrödinger equations

Shaowei Chen

School of Mathematical Sciences, Capital Normal University,
Beijing 100048, P. R. China
E-mail adress<EMAIL_ADDRESS>(S. Chen),

Abstract: In this paper, we study the following semilinear Schrödinger
equation

$-\epsilon^{2}\triangle u+u+V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}),$

where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function $V$ is
bounded in $\mathbb{R}^{N},$ $\inf_{\mathbb{R}^{N}}(1+V(x))>0$ and it has a
possibly degenerate isolated critical point. Under some conditions on $f,$ we
prove that as $\epsilon\rightarrow 0,$ this equation has a solution which
concentrates at the critical point of $V$.

Key words: semilinear Schrödinger equation, variational reduction method.
2000 Mathematics Subject Classification: 35J20, 35J70

††††

##  1 Introduction and main result

In this paper, we are concerned with the following semilinear Schrödinger
equation

$-\epsilon^{2}\triangle u+u+V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}),$ (1.1)

where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function
$f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies

$(\bf{F_{1}}).$ $f\in C^{1}(\mathbb{R})$ and there exist $q\in(2,2^{*}),$
$2<p_{1}<p_{2}<2^{*}$ and a constant $C>0$ such that

$|f^{\prime}(t)|\leq C(|t|^{p_{1}-2}+|t|^{p_{2}-2}),\ t\in\mathbb{R}$

and for any $L>0$,

$\displaystyle\sup\\{|f^{\prime}(t)-f^{\prime}(s)|/|t-s|^{q-2}\ |\
t,s\in[-L,L],\ t\neq s\\}<\infty,$ (1.2)

where $2^{*}=2N/(N-2)$ if $N\geq 3$ and $2^{*}=\infty$ if $N=2$;

$(\bf{F_{2}}).$ there exists $\mu>2$ such that $f(t)t\geq\mu F(t)>0,$ $t\neq
0$, where $F(t)=\int^{t}_{0}f(s)ds$;

$(\bf{F_{3}}).$ $f(t)/|t|$ is an increasing function on
$\mathbb{R}\setminus\\{0\\}$;

######  Remark 1.1.

A typical function which satisfies $\bf(F_{1})-(F_{3})$ is

$f(t)=\sum^{m}_{i=1}a_{i}|t|^{\beta_{i}-2}t$

with $2<\beta_{1}<\cdots<\beta_{m}<2^{*}$ and $a_{i}>0,$ $1\leq i\leq m.$

The potential function $V$ satisfies the following conditions:

$(\bf{V_{0}}).$ $\inf_{x\in\mathbb{R}^{N}}(1+V(x))>0$ and
$\max_{x\in\mathbb{R}^{N}}|V(x)|<\infty;$

$(\bf{V_{1}}).$ $V\in C^{2}(\mathbb{R}^{N})$ has an isolated critical point
$x_{0}$ such that

$V(x)=Q_{n^{*}}(x-x_{0})+o(|x-x_{0}|^{n^{*}})$

in some neighborhood of $x_{0},$ where $n^{*}\geq 2$ is an even integer and
$Q_{n^{*}}$ is an $n^{*}$\- homogeneous polynomial in $\mathbb{R}^{N}$ which
satisfies that $\triangle Q_{n^{*}}\geq 0$ in $\mathbb{R}^{N}$ or $\triangle
Q_{n^{*}}\leq 0$ in $\mathbb{R}^{N}$ and $\triangle Q_{n^{*}}\not\equiv 0$ in
$\mathbb{R}^{N}$.

######  Remark 1.2.

Without loss of generality, in what follows, we always assume that $x_{0}=0.$
Typical examples for $Q_{n^{*}}$ are $\pm|x|^{n^{*}}$ $(n^{*}\geq 2).$

Our main result of this paper is the following theorem

######  Theorem 1.3.

Suppose that $f$ satisfies $\bf(F_{1})-\bf(F_{4})$ and $V$ satisfies
$\bf(V_{0})$ and $\bf(V_{1})$. Then there exist $\epsilon_{0}>0$ and a set
$\mathcal{K}$ whose elements are radially symmetric solutions of equation

$-\triangle u+u=f(u),\ u\in H^{1}(\mathbb{R}^{N})$ (1.3)

such that if $0<\epsilon<\epsilon_{0}$, then equation (1.1) has a solution
$u_{\epsilon}$ satisfying that

$\lim_{\epsilon\rightarrow
0}\mbox{dist}_{{}_{Y}}(v_{\epsilon},\mathcal{K})=0,$

where $v_{\epsilon}(x)=u_{\epsilon}(\epsilon x),$ $x\in\mathbb{R}^{N}$ and
$Y=H^{1}(\mathbb{R}^{N}).$

The analysis of the semilinear Schrödinger equation (1.1) has recently
attracted a lot of attention due to its many applications in mathematical
physics.

If $v$ is a solution of equation (1.1), then $v(\epsilon x)$ is a solution of
the following equation

$-\triangle u+u+V(\epsilon x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}).$ (1.4)

Equation (1.4) is a perturbation of the limit equation (1.3). If equation
(1.3) has a solution $w\in C^{2}(\mathbb{R}^{N})$ satisfying the non-
degeneracy condition:

$\ker L_{0}=\mbox{span}\left\\{\frac{\partial\omega}{\partial x_{i}}\ |\ 1\leq
i\leq N\right\\},$

where $L_{0}v=-\triangle v+v-f^{\prime}(\omega)v$, then in the celebrated
paper [1] (see also [2]), Ambrosetti, Badiale and Cingolani developed a kind
of variational reduction method and showed that if the potential function $V$
has a strictly local minimizer or maximizer $x_{0}$, then equation (1.4)
admits a solution $u_{\epsilon}$ which converges to $\omega(\cdot-x_{0})$ in
$H^{1}(\mathbb{R}^{N})$ as $\epsilon\rightarrow 0$. In their argument, the
non-degeneracy property of $\omega$ plays essential role. Using the non-
degeneracy condition and the reduction method, it was shown by Kang and Wei
[20] that, at a strict local maximum point $x_{0}$ of $V$ and for any positive
integer $k$, (1.1) has a positive solution with $k$ interacting bumps
concentrating near $x_{0}$, while at a non-degenerate local minimum point of
$V(x)$ such solutions do not exist. Moreover, under the assumption of the non-
degeneracy condition, multiplicity of solutions with one bump has also been
considered by Grossi [16].

However, for a general nonlinearity $f$, it is very difficult to verify the
non-degeneracy condition for a solution of (1.3). An effective method to
attack problem (1.1) without using the non-degeneracy condition is variational
method. In [21], Rabinowitz used a global variational method to show the
existence of least energy solutions for (1.1) when $\epsilon>0$ is small, and
the condition imposed on $V$ is a global one, namely

$0<\inf_{x\in\mathbb{R}^{N}}(1+V(x))<\liminf_{|x|\rightarrow\infty}(1+V(x)).$

In [12], [13], [14], [15] and [17], Del Pino, Felmer and Gui used different
variational methods to obtain nontrivial solution of (1.1) for small
$\epsilon>0$ under local conditions which can be roughly described as follows:
$V$ is local Hölder continuous on $\mathbb{R}^{N},$

$\displaystyle\inf_{x\in\mathbb{R}^{N}}(1+V(x))>0$ (1.5)

and there exists $k$ disjoint bounded regions $\Omega_{1},\cdots,\Omega_{k}$
in $\mathbb{R}^{N}$ such that

$\displaystyle\inf_{x\in\partial\Omega_{i}}V(x)>\inf_{x\in\Omega_{i}}V(x).$
(1.6)

Their methods involve the deformation of nonlinearity $f$ and some prior
estimates. Recently, Byeon, Jeanjean and Tanaka [5] [6] developed the
variational methods and made great advance in problem (1.1). Byeon and
Jeanjean showed in [5] that if $N\geq 3$, $V$ satisfies (1.5) and (1.6) with
$k=1$ and $f$ satisfies

$\bf(f_{1}).$ $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and
$\lim_{t\rightarrow 0+}f(t)/t=0;$

$\bf(f_{2}).$ there exists some $p\in(1,2^{*}-1)$ such that
$\lim_{t\rightarrow\infty}f(t)/t^{p}<\infty;$

$\bf(f_{3}).$ there exists $T>0$ such that $\frac{1}{2}mT^{2}<F(T)$, where
$F(t)=\int^{t}_{0}f(s)ds$ and $m=\inf_{x\in\Omega_{1}}V(x),$

then (1.1) exists positive solution $v_{\epsilon}$ concentrating in the
minimizers of $V$ in $\Omega_{1}$ as $\epsilon\rightarrow 0.$ And in [6],
Byeon, Jeanjean and Tanaka considered the case $N=1,2$ and obtained similar
results. Their conditions on the nonlinearity $f$ are almost optimal.
Moreover, when $V$ satisfies (1.5) and (1.6) with $k>1$ and $f$ satisfies
$\bf(f_{1})-(f_{3})$, in [10], Cingolani, Jeanjean and Secchi constructed
multi-bump solutions for magnetic nonlinear Schödinger equations which contain
equation (1.1) as a special case.

Comparing to the variational methods mentioned above, the Lyapunov reduction
method of Ambrosetti and Badiale, although it need the non-degeneracy
condition, has its advantages that their method can be used to deal with
elliptic equations involving critical Sobolev exponent (see, for example, [3])
and other problems involving concentration compactness (see, for example,
[18]).

In this paper, we indent to attack the problem (1.1) though a Lyapunov
reduction method, but avoiding the non-degeneracy condition for the solutions
of limit equation (1.3). In this paper, we develop a new reduction method for
an isolated critical set $\mathcal{K}$ of the functional corresponding to
(1.3). This method can be regarded as a generalization of Ambrosetti and
Badiale’s method. The non-degeneracy conditions for the solutions in this
critical set are no longer necessary and it does not involve the deformation
of nonlinearity. By combination of the new reduction method and Conley index
theory which was developed by Chang and Ghoussoub in [9](see also [8]), we
obtain a solution of (1.4) in a neighborhood of $\mathcal{K}$ for sufficiently
small $\epsilon>0.$ Our method is new and it can be used to other problems
which involve concentration compactness. In contrast with the results of
Byeon, Jeanjean and Tanaka, although the assumptions we imposed on the
nonlinearity $f$ are much stronger, the assumptions we made on $V$ seem weaker
in a sense, because by the assumption $\bf(V_{1})$, $x_{0}$ can be a local
maximum point of $V.$

This paper is organized as follows: In section 2, we obtain a critical set of
the functional corresponding to (1.3) with nontrivial Topology. In section 3
and section 4, a reduction for the function corresponding to (1.4) is
developed. In section 5, we give the proof of Theorem 1.3. Section 6 and 7 are
appendixes.

Notations. $\mathbb{R},\ \mathbb{Z}$ and $\mathbb{N}$ denote the sets of real
number, integer and positive integer respectively. Let $E$ be a metric space.
$B_{E}(a,\rho)$ denotes the open ball in $E$ centered at $a$ and having radius
$\rho$. The closure of a set $A\subset E$ is denoted by $\overline{A}$ or
$cl_{E}(A).$ $\mbox{dist}_{E}(a,A)$ denotes the distance from the point $a$ to
the set $A\subset E$. By $\rightarrow$ we denote the strong and by
$\rightharpoonup$ the weak convergence. By $\ker A$ denotes the null space of
the operator $A.$ If $g$ is a $C^{2}$ functional defined on a Hilbert space
$H$, $\nabla g$ (or $Dg$) and $\nabla^{2}g$ (or $D^{2}g$) denote the gradient
of $g$ and the second derivative of $g$ respectively. And for
$a,b\in\mathbb{R},$ we denote $g^{a}:=\\{u\in H\ |\ g(u)\leq a\\}$ and
$g_{b}:=\\{u\in H\ |\ g(u)\geq b\\}$ the sub- and super-level sets of the
functional $g,$ moreover, $g^{a}_{b}:=\\{u\in H\ |\ b\leq g(u)\leq a\\}.$
$\delta_{i,j}$ denotes the Kronecker notation, i.e., $\delta_{i,j}=1$ if $i=j$
and $0$ if $i\neq j.$ For a Banach space $E,$ denote $\mathcal{L}(E)$ the
Banach space consisting of all bounded linear operator from $E$ to $E.$ If $H$
is a Hilbert space and $W$ is a closed subspace of $H,$ we denote the
orthogonal complement space of $W$ in $H$ by $W^{\bot}.$ For a subset
$A\subset H,$ $\mbox{span}\\{A\\}$ denotes the subspace of $H$ generated by
$A.$ For a topology pair $(A,B)$ in metric space, $\check{H}^{*}(A,B)$ denotes
the $\check{\mbox{C}}\mbox{ech}$-Alexander-Spanier cohomology with coefficient
group $\mathbb{Z}_{2}$ (see [23]).

##  2 Critical sets of limit functional with nontrivial Topology

Throughout this paper, we denote the Sobolev space $H^{1}(\mathbb{R}^{N})$ and
the radially symmetric function space

$H^{1}_{r}(\mathbb{R}^{N}):=\\{u\in H^{1}(\mathbb{R}^{N})\ |\ u\ \mbox{is
radially symmetric}\\}$

by $Y$ and $X$ respectively. The inner product of $Y$ is

$\langle u,v\rangle=\int_{\mathbb{R}^{N}}(\nabla u\nabla v+uv)dx,$

and we use $||\cdot||$ to denote the norm of $Y$ corresponding to this inner
product. Define

$I(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+|u|^{2})dx-\int_{\mathbb{R}^{N}}F(u)dx,\ u\in X.$
$J(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+|u|^{2})dx-\int_{\mathbb{R}^{N}}F(u)dx,\ u\in Y,$
$E_{\epsilon}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+|u|^{2}+V(\epsilon x)|u|^{2})dx-\int_{\mathbb{R}^{N}}F(u)dx,\ u\in Y.$

For $h\in H^{-1}(\mathbb{R}^{N})$, let $(-\triangle+1)^{-1}h$ and
$(-\triangle+1+V(\epsilon x))^{-1}h$ be the solutions of

$\displaystyle-\triangle u+u=h,\ u\in H^{1}(\mathbb{R}^{N})$ (2.1)

and

$\displaystyle-\triangle u+u+V(\epsilon x)u=h,\ u\in H^{1}(\mathbb{R}^{N})$
(2.2)

respectively.

Under conditions $\bf(F_{1})-(F_{3})$, $I$ satisfies Palais-Smale condition
(see, for example, [24]) and has a mountain pass geometry, that is,

  * (i)

$I(0)=0$,

  * (ii)

there exist $\rho_{0}>0$ and $\delta_{0}>0$ such that $I(u)\geq\delta_{0}$ for
all $||u||=\rho_{0},$

  * (iii)

there exists $u_{0}\in X$ such that $||u_{0}||>\rho_{0}$ and $I(u_{0})<0.$

Thus the following minimax value is well defined and is larger than
$\delta_{0},$

$\displaystyle c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I(\gamma(t))$ (2.3)

where

$\displaystyle\Gamma=\\{\gamma\in C([0,1],X)\ |\ \gamma(0)=0,\
I(\gamma(1))<0\\}.$ (2.4)

######  Lemma 2.1.

For any $\sigma\in(0,\delta_{0}),$ if $a\in(c-\sigma,c)$ and
$b\in(c,c+\sigma)$ are regular values of $I$, then
$\check{H}^{1}(I^{b},I^{a})\neq 0.$

Proof. Since $b>c,$ by the definition of minimax value $c,$ there exists
$\gamma\in\Gamma$ such that

$\displaystyle\max_{t\in[0,1]}I(\gamma(t))<b.$ (2.5)

Let $u_{0}=\gamma(1).$ We infer that $0$ and $u_{0}$ lie in different
connected component of $I^{a}.$ It follows that the homomorphism

$\iota^{*}:\check{H}^{0}(I^{a})\rightarrow\check{H}^{0}(\\{0,u_{0}\\})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$

which is induced by the inclusion mapping $\iota:\\{0,u_{0}\\}\hookrightarrow
I^{a}$ is a surjection. Consider the following homomorphism which is induced
by the inclusion mapping $j:\\{0,u_{0}\\}\hookrightarrow I^{b},$

$j^{*}:\check{H}^{0}(I^{b})\rightarrow\check{H}^{0}(\\{0,u_{0}\\}).$

By (2.5), $0$ and $u_{0}$ lie in the same connected component of $I^{b}.$ It
follows that $j^{*}$ is not a surjection.

Consider the following communicative diagram

$\check{H}^{0}(I^{b})$$\check{H}^{1}(I^{b},I^{a})$$\check{H}^{0}(I^{a})$$\check{H}^{0}(\\{0,u_{0}\\})$$i^{*}$$\iota^{*}$$j^{*}$$\alpha^{*}$

Since $j^{*}$ is not a surjection and $\iota^{*}$ is a surjection, by this
communicative diagram, we deduce that
$\mbox{Image}(i^{*})\neq\check{H}^{0}(I^{a}).$ Moreover, by the property of
exact sequence, we have $\mbox{Image}(i^{*})=\ker\alpha^{*}.$ Thus
$\ker\alpha^{*}\neq\check{H}^{1}(I^{a})$. It follows that $\alpha^{*}\neq 0$.
Therefore, $\check{H}^{1}(I^{b},I^{a})\neq 0.$ $\Box$

From Chapter 4 of [24], we have the following lemma

######  Lemma 2.2.

If $\nabla I(u)=0$ and $I(u)<2c$, then $u$ does not change sign in
$\mathbb{R}^{N}$.

Let $\mathcal{F}$ be a $C^{1}$ functional defined on a Hilbert space $M$ with
critical set $K_{\mathcal{F}}$. And let $V$ be a pesudo-gradient vector field
with respect to $D\mathcal{F}$ on $M$. A pesudo-gradient flow associated with
$V$ is the unique solution of the following ordinary differential equation in
$M:$

$\dot{\eta}=-V(\eta(x,t)),\ \eta(x,0)=x.$

A subset $W$ of $M$ is said to have the mean value property (for short (MVP))
if for any $x\in M$ and any $t_{0}<t_{1}$ we have
$\eta(x,[t_{0},t_{1}])\subset W$ whenever $\eta(x,t_{i})\in W,\ i=1,2.$

######  Definition 2.3.

(Definition I.10 of [9]) Let $\mathcal{F}$ be a $C^{1}$ functional on a
Hilbert space $M$. A subset $S$ of the critical set $K$ of $\mathcal{F}$ is
said to be a dynamically isolated critical set if there exist a closed
neighborhood $\mathcal{O}$ of $S$ and regular values $a<b$ of $\mathcal{F}$
such that

$\mathcal{O}\subset\mathcal{F}^{-1}[a,b]$ (2.6)

and

$cl(\widetilde{\mathcal{O}})\cap K\cap\mathcal{F}^{-1}[a,b]=S,$ (2.7)

where $\widetilde{\mathcal{O}}=\bigcup_{t\in\mathbb{R}}\eta(\mathcal{O},t)$.
$(\mathcal{O},a,b)$ is called an isolating triplet for $S.$

######  Definition 2.4.

(Definition III.1 of [9]) Let $\mathcal{F}$ be a $C^{1}$ functional on a
Hilbet space $M$ and let $S$ be a subset of the critical set $K_{\mathcal{F}}$
for $\mathcal{F}$. A pair $(W,W_{-})$ of subset is said to be a GM pair for
$S$ associated with a pesudo-gradient vector field $V$, if the following
conditions hold:

(1). $W$ is a closed (MVP) neighborhood of $S$ satisfying $W\cap K=S$ and
$W\cap\mathcal{F}_{\alpha}=\emptyset$ for some $\alpha.$

(2). $W_{-}$ is an exit set for $W,$ i.e., for each $x_{0}\in W$ and $t_{1}>0$
such that $\eta(x_{0},t_{1})\not\in W,$ there exists $t_{0}\in[0,t_{1})$ such
that $\eta(x_{0},[0,t_{0}])\subset W$ and $\eta(x_{0},t_{0})\in W_{-}.$

(3). $W_{-}$ is closed and is a union of a finite number of sub-manifolds that
transversal to the flow $\eta.$

For $\alpha,\beta\in\mathbb{R},$ define

$\mathcal{K}^{\beta}_{\alpha}:=\\{u\in X\ |\ \nabla I(u)=0,\ \alpha\leq
I(u)\leq\beta\\}.$

Let $a$ and $b$ are the regular values which come from Lemma 2.1. Then by
Definition 2.4, $\mathcal{K}^{b}_{a}$ is a dynamically isolated critical set
of $I$. By Lemma 2.1 and Theorem III.3 of [9], we have the following lemma

######  Lemma 2.5.

Let $\sigma>0$ be sufficiently small and $a\in(c-\sigma,c)$,
$b\in(c,c+\sigma)$ be regular values of $I$. If $(W,W_{-})$ is a GM pair of
$\mathcal{K}^{b}_{a}$ associated with some pseudo-gradient vector field of
$I$, then

$\check{H}^{1}(W,W_{-})\neq 0.$

######  Remark 2.6.

In this remark, we shall show that the set of regular values of $I$ is dense
in $\mathbb{R}.$ Therefore, for any $\sigma>0$, there always exist regular
values of $I$ in $(c-\sigma,c)$ and $(c,c+\sigma).$ In fact, we shall show
that $I(C)$ is of first category, where $C$ is the set of critical points of
$I$. It suffices to prove that for any $u\in C$, there exists $\delta_{u}>0$
such that $\overline{I(C\cap B_{X}(u,\delta_{u}))}$ does not contain interior
points.

Let $u\in C$. Since $u$ is radially symmetric, the dimension of the kernel
space of the following operator is at most one

$\displaystyle\nabla^{2}I(u):X\rightarrow X,\ h\in X\mapsto
h-(-\triangle+1)^{-1}f^{\prime}(u)h.$

If $\dim\nabla^{2}I(u)=0,$ then by Morse Lemma (see, e.g., Lemma 4.1 of [7]),
there exists $\delta_{u}>0$ such that $u$ is the unique critical point of $I$
in $B_{X}(u,\delta_{u}).$ Thus, in this case, $I(C\cap
B_{X}(u,\delta_{u}))=\\{I(u)\\}.$

If $\dim\nabla^{2}I(u)=1,$ let $N=\mbox{ker}\nabla^{2}I(u)$ and note that $I$
is a $C^{2}$ functional, then by Lemma 1 of [19] (see also Theorem 5.1 of
[7]), there exist an origin preserving $C^{1}$ diffeomorphism $\Phi$ of some
$B_{X}(0,\delta_{u})$ into $X$ and an an origin preserving $C^{1}$ map $h$
defined in $N\cap B_{X}(0,\delta_{u})$ into $X$ such that

$I\circ\Phi(z,y)=I(u)+||Pz||^{2}-||(\mbox{id}-P)z||^{2}+I(h(y)+y)$

where $P:N^{\bot}\rightarrow N^{\bot}$ is an orthogonal projection and
$N^{\bot}$ is the orthogonal complement of $N$ in $X$. Let $U=\\{y\in N\cap
B_{X}(0,\delta_{u})\ |\ h(y)+y\\}.$ Then $U$ is a $C^{1}$ one-dimensional
manifold. Let us restrict $I$ to $U$. Then $I:U\rightarrow\mathbb{R}$ is
$C^{1}$. Moreover, $C\cap B_{X}(0,\delta_{u})=C\cap U,$ so $I(C\cap
B_{X}(0,\delta_{u}))=I(C\cap U)$. Therefore, by classical Sard theorem,
$\overline{I(C\cap B_{X}(0,\delta_{u}))}$ does not contain interior points.

For $r>0,$ $A\subset X,$ let

$\displaystyle N_{r}(A):=\\{v\in X\ |\ \mbox{dist}_{X}(v,A)<r\\}.$ (2.8)

######  Lemma 2.7.

Let $c$ be the mountain pass value coming from Lemma 2.1. For any $r>0,$ there
exists $\sigma_{r}>0$ such that if $a\in(c-\sigma_{r},c)$ and
$b\in(c,c+\sigma_{r})$ are regular values of $I$, then there exists a GM pair
$(W,W_{-})$ of the critical set $\mathcal{K}^{b}_{a}$ of the functional $I$
associated with the negative gradient vector field of $I$ such that $W\subset
N_{r}(\mathcal{K}^{b}_{a}).$

Proof. By $\bf(F_{1})-(F_{3}),$ we know that $I$ satisfies the Palais-Smale
condition (see [24]). Therefore, for any $r>0,$ there exists $\kappa_{r}>0$
such that if $a\in(c-1,c)$ and $b\in(c,c+1)$, then

$\displaystyle||\nabla I(v)||\geq\kappa_{r},\ \forall v\in
I^{-1}[a,b]\setminus N_{r/3}(\mathcal{K}^{b}_{a}).$ (2.9)

Let

$\displaystyle 0<\sigma_{r}<\min\\{r\kappa_{r}/6,1\\}$ (2.10)

and $a\in(c-\sigma_{r},c)$ and $b\in(c,c+\sigma_{r})$ be regular values of
$I$. For

$\displaystyle u\in I^{-1}[a,b]\cap N_{r/3}(\mathcal{K}^{b}_{a}),$ (2.11)

consider the negative gradient flow:

$\displaystyle\dot{\eta}(t)=-\nabla I(\eta(t)),\ \eta(0)=u.$ (2.12)

Let

$T^{+}_{u}=\sup\\{t\geq 0\ |\ \mbox{for every}\ s\in[0,t],\ I(\eta(s))\geq
a\\}$

and

$T^{-}_{u}=\inf\\{t\leq 0\ |\ \mbox{for every}\ s\in[t,0],\ I(\eta(s))\leq
b\\}.$

Let

$U=\bigcup_{t\in[T^{-}_{u},T^{+}_{u}]}\\{\eta(t,u)\ |\ u\in I^{-1}[a,b]\cap
N_{r/3}(\mathcal{K}^{b}_{a})\\}.$

Then

$[\mathcal{K}^{b}_{a}]\subset U,$

where

$[\mathcal{K}^{b}_{a}]=\\{v\in X\ |\
\omega(v)\cup\omega^{*}(v)\in\mathcal{K}^{b}_{a}\\},$

$\omega(v)=\cap_{t>0}\overline{\eta(v,[t,+\infty))}$ is the $\omega-$limit set
of $v$ and $\omega^{*}(v)=\cap_{t>0}\overline{\eta(v,(-\infty,-t])}$ is the
$\omega^{*}-$limit set of $v$.

By [9, Proposition III.2], we deduce that there exists a GM pair $(W,W_{-})$
of $\mathcal{K}^{b}_{a}$ such that $W\subset U$. Thus, to prove this Lemma, it
suffices to prove that if $\sigma_{r}>0$ is small enough, then for $u$ which
satisfies (2.11),

$\displaystyle\sup_{t\in(T^{-}_{u},T^{+}_{u})}||\eta(t)-u||\leq\frac{2}{3}r.$
(2.13)

Since their arguments are similar, we only give the proof for

$\displaystyle\sup_{t\in[0,T^{+}_{u})}||\eta(t)-u||\leq\frac{2}{3}r.$ (2.14)

If (2.14) were not true, then there exist $0\leq t_{1}<t_{2}<T^{+}_{u}$ such
that

$r/3\leq||\eta(t)-u||\leq 2r/3,\ \forall t\in[t_{1},t_{2}]$
$\displaystyle||\eta(t_{1})-u||=r/3,\ ||\eta(t_{2})-u||=2r/3.$ (2.15)

According to (2.9), we have

$\displaystyle b-a\geq I(\eta(t_{1}))-I(\eta(t_{2}))$
$\displaystyle=\int^{t_{1}}_{t_{2}}\langle\nabla
I(\eta(t)),\dot{\eta}(t)\rangle dt=\int^{t_{2}}_{t_{1}}||\nabla
I(\eta(t))||^{2}dt\geq\kappa^{2}_{r}(t_{2}-t_{1}).$

It follows that

$\displaystyle t_{2}-t_{1}\leq(b-a)/\kappa^{2}_{r}.$ (2.16)

Combining (2.15) and (2.16) leads to

$\displaystyle\frac{r}{3}$ $\displaystyle\leq$
$\displaystyle||\eta(t_{2})-\eta(t_{1})||\leq\int^{t_{2}}_{t_{1}}||\dot{\eta}(t)||dt$
$\displaystyle\leq$
$\displaystyle(t_{2}-t_{1})^{1/2}(\int^{t_{2}}_{t_{1}}||\dot{\eta}(t)||^{2})^{1/2}=(t_{2}-t_{1})^{1/2}(\int^{t_{2}}_{t_{1}}||\nabla
I(\eta(t))||^{2})^{1/2}$ $\displaystyle\leq$
$\displaystyle(t_{2}-t_{1})^{1/2}(b-a)^{1/2}\leq(b-a)/\kappa_{r}<2\sigma_{r}/\kappa_{r}.$

It contradicts (2.10). Thus, (2.14) holds. $\Box$

##  3 A variational reduction for the limiting functional $I$

Let $\sigma>0$ be sufficiently small and $a\in(c-\sigma,c)$,
$b\in(c,c+\sigma)$ be regular values of $I$, where $c$ is defined by (2.3). In
what follows, for the sake of simplicity, we denote the critical set
$\mathcal{K}^{b}_{a}$ by $\mathcal{K}.$

By [4], if $u\in Y$ is a weak solution of

$-\triangle u+u=f(u),$ (3.1)

then $u$ and $\frac{\partial u}{\partial x_{i}},$ $1\leq i\leq N$ satisfy
exponential decay at infinity. As a consequence, $\mathcal{K}$ is a compact
subset of $W^{2,2}(\mathbb{R}^{N}).$ If $u\in Y$ is a solution of equation
(3.1), then $\frac{\partial u}{\partial x_{i}},$ $i=1,\cdots,N$ are the
eigenfunctions for the eigenvalue problem

$-\triangle h+h=f^{\prime}(u)h.$ (3.2)

######  Remark 3.1.

By [22, Theorem C. 3.4]), any eigenfunction of the eigenvalue problem (3.2)
satisfies exponential decay at infinity.

The argument in [11, Page 970-971] implies the following Lemma.

######  Lemma 3.2.

Suppose that $u\in X$ is a solution of equation (3.1) and it does not change
sign in $\mathbb{R}^{N}$. If $v\in Y$ is a solution of (3.2) and satisfies

$\left\langle v,\frac{\partial u}{\partial x_{i}}\right\rangle=0,\
i=1,\cdots,N,$

then $v\in X.$

######  Remark 3.3.

By Lemma 2.2, we infer that if $u\in\mathcal{K}$, then $u$ does not change
sign in $\mathbb{R}^{N}$.

As it has been mentioned above, $\mathcal{K}$ is a compact subset in
$W^{2,2}(\mathbb{R}^{N})$. Thus for any $u\in\mathcal{K}$ and any
$\varsigma>0$, there exists $\tau_{u}>0$ such that

$\displaystyle\sum^{N}_{j=1}\left|\left|\frac{\partial v}{\partial
x_{j}}-\frac{\partial u}{\partial x_{j}}\right|\right|<\varsigma,\ \forall
v\in\mathcal{K}\cap B_{X}(u,2\tau_{u}).$ (3.3)

Therefore, we can choose a finite open sub-covering of $\mathcal{K}$

$\displaystyle\mathcal{A}=\\{B_{X}(u_{i},\tau_{u_{i}})\ |\ i=1,\cdots,s\\}$
(3.4)

from the open covering $\\{B_{X}(u,\tau_{u})\ |\ u\in\mathcal{K}\\}$. Let
$\zeta\in C^{\infty}([0,+\infty))$ be such that $0\leq\zeta(t)\leq 1$ for all
$t,$ $\zeta(t)=1$ for $t\in[0,1/2]$ and $\zeta(t)=0$ for $t\in[1,\infty).$ Let

$\xi_{i}(u)=\frac{\zeta(||u-u_{i}||/\tau_{u_{i}})}{\sum^{s}_{i=1}\zeta(||u-u_{i}||/\tau_{u_{i}})},\
1\leq i\leq s.$

Then $\\{\xi_{i}\ |\ 1\leq i\leq s\\}$ is a $C^{\infty}$ partition of unity
corresponding to the covering $\mathcal{A}$.

For $u\in\mathcal{K},$ let

$Y_{u}:=\\{h\in X\ |\ \nabla^{2}I(u)h=0\\},\
Z_{u}:=\mbox{span}\\{\frac{\partial u}{\partial x_{i}}\ |\ 1\leq i\leq N\\}.$

Let

$\mathcal{Y}=\mbox{span}\\{\cup^{s}_{i=1}Y_{u_{i}}\\}.$ (3.5)

Let

$q=\dim\mathcal{Y}.$ (3.6)

Let $\\{e_{1},e_{2},\cdots,e_{q}\\}$ be an orthogonal normal base of
$\mathcal{Y}$. As mentioned in Remark 3.1, for every $1\leq n\leq q,$
$e_{n}\in W^{2,2}_{r}(\mathbb{R}^{N})$ and $e_{n}$ satisfies exponential decay
at infinity.

Let $\\{e^{\prime}_{1},e^{\prime}_{2}\cdots\\}$ be an orthogonal normal base
of $\mathcal{Y}^{\bot},$ where $\mathcal{Y}^{\bot}$ is the orthogonal
complement space of $\mathcal{Y}$ in $X$. From the appendix A of this paper,
for every $k\in\mathbb{N},$ there exists

$\displaystyle E_{k}:=\\{\tilde{e}_{j,k}\ |\ 1\leq j\leq k\\},$ (3.7)

such that

  * (i)

For every $k,$ $E_{k}\subset X\cap W^{2,2}_{r}(\mathbb{R}^{N})$ and
$E_{k}\bot\mathcal{Y}$;

  * (ii)

Every $\tilde{e}_{j,k}$ satisfies exponential decay at infinity,
$\langle\tilde{e}_{j,k},\tilde{e}_{j^{\prime},k}\rangle=\delta_{j,j^{\prime}}$
and

$\displaystyle\sup_{1\leq j\leq k}||\tilde{e}_{j,k}-e^{\prime}_{j}||\leq
1/2^{k}.$

For every $k,$ denote

$\displaystyle X_{k}:=\mbox{span}\\{E_{k}\\}\oplus\mathcal{Y}.$

Let $P_{k}:X\rightarrow X_{k}$ and $P^{\bot}_{k}:X\rightarrow X^{\bot}_{k}$ be
the orthogonal projections, where $X^{\bot}_{k}$ is the orthogonal complement
space of $X_{k}$ in $X.$ By the definition of $X_{k}$ and the properties
$\bf(i)$ and $\bf(ii)$ mentioned above, we have the following Lemma which is
easy to prove.

######  Lemma 3.4.

For every $h\in X,$
$\lim_{k\rightarrow\infty}||h-P_{k}h||=\lim_{k\rightarrow\infty}||P_{k}^{\bot}h||=0.$

######  Lemma 3.5.

For any $r>0,$ there exists $l_{r}\in\mathbb{N}$ such that if $k\geq l_{r}$,
then for every $v\in N_{r}(\mathcal{K})$,
$P^{\bot}_{k}\nabla^{2}I(v)|_{X^{\bot}_{k}}$ is invertible and

$\displaystyle||(P^{\bot}_{k}\nabla^{2}I(v)|_{X^{\bot}_{k}})^{-1}||_{\mathcal{L}(X^{\bot}_{k})}\leq
2.$

Proof. For $w\in X^{\bot}_{k}$,

$\displaystyle
P^{\bot}_{k}\nabla^{2}I(v)w=w-P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v)w.$

Denote the operator $w\mapsto P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v)w$
by $A_{v,k}.$ If we can prove that

$\limsup_{k\rightarrow\infty}\sup\\{||A_{v,k}||_{\mathcal{L}(X^{\bot}_{k})}\
|\ v\in N_{r}(\mathcal{K})\\}=0,$ (3.8)

then the conclusion of this Lemma follows. If (3.8) were not true, we can
choose $v_{k}\in N_{r}(\mathcal{K})$ and $w_{k}\in X_{k}^{\bot}$ with
$||w_{k}||=1$, $k=1,2,\cdots,$ such that

$\displaystyle\limsup_{k\rightarrow\infty}||A_{v_{k},k}w_{k}||>0.$ (3.9)

Without loss of generality, we assume that $v_{k}\rightharpoonup v_{0}$ in $X$
and $w_{k}\rightharpoonup w_{0}$ in $X$ as $k\rightarrow\infty.$ Since for any
$2\leq p<2^{*}$, $X$ can be compactly embedded into the radially symmetric
$L^{p}$ space (see, for example, [24, Corollary 1.26])

$L^{p}_{r}(\mathbb{R}^{N}):=\\{u\in L^{p}(\mathbb{R}^{N})\ |\ u\ \mbox{ is
radially symmetric}\\},$

combining the condition $\bf(F_{1})$, we can get that

$\lim_{k\rightarrow\infty}\sup\\{\int_{\mathbb{R}^{N}}|f^{\prime}(v_{k})w_{k}h-f^{\prime}(v_{0})w_{0}h|\
|\ h\in X,\ ||h||\leq 1\\}=0.$

It follows that

$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}(f^{\prime}(v_{k})w_{k}-f^{\prime}(v_{0})w_{0})||=0.$
(3.10)

By (3.10) and Lemma 3.4, we deduce that
$\lim_{k\rightarrow\infty}||A_{v_{k},k}w_{k}||=0.$ But this contradicts (3.9).
$\Box$

For $u\in\mathcal{K}$, denote $X_{k}\oplus Z_{u}$ by $W_{u,k}$ and let
$W_{u,k}^{\bot}$ be the orthogonal complement space of $W_{u,k}$ in $Y.$ Let
$P_{W_{u_{i},k}}:Y\rightarrow W_{u_{i},k}$ and
$P_{W^{\bot}_{u_{i},k}}:Y\rightarrow W^{\bot}_{u_{i},k}$ be the orthogonal
projections.

######  Lemma 3.6.

Suppose that $\kappa:=\max\\{\tau_{u_{i}}\ |\ 1\leq i\leq s\\}$ is
sufficiently small, where $\tau_{u_{i}}$ comes from (3.4). Then there exist
$C>0$ and $l_{\kappa}\in\mathbb{N}$ such that if $k\geq l_{\kappa}$ and $v\in
B_{X}(u_{i},\tau_{u_{i}})$ for some $1\leq i\leq s$, then
$P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)|_{W^{\bot}_{u_{i},k}}$ is invertible and

$\displaystyle||(P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)|_{W^{\bot}_{u_{i},k}})^{-1}||_{\mathcal{L}(W^{\bot}_{u_{i},k})}\leq
C.$ (3.11)

Proof. We note that for $w\in W^{\bot}_{u_{i},k}$,

$\displaystyle
P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)w=w-P_{W^{\bot}_{u_{i},k}}(-\triangle+1)^{-1}f^{\prime}(u)w.$

Since for any $p\in[2,2^{*})$, $X$ can be compactly embedded into the radially
symmetric $L^{p}$ space, by the condition $\bf(F_{1})$, we deduce that
$w\mapsto P_{W^{\bot}_{u_{i},k}}(-\triangle+1)^{-1}f^{\prime}(v)w$ is a
compact operator. It follows that
$P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)|_{W^{\bot}_{u_{i},k}}$ is a Fredholm
operator with index zero. Therefore, if we can prove that there exists $C>0$
which is independent of $k$ such that, for sufficiently large $k,$

$\displaystyle||P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)w||_{\mathcal{L}(W^{\bot}_{u_{i},k})}\geq\frac{1}{C}||w||,\
\forall w\in W^{\bot}_{u_{i},k},\ \forall v\in B_{X}(u_{i},\tau_{u_{i}})$

then the conclusion of this Lemma follows.

Without loss of generality, we assume that $u_{i}\equiv u_{1}$ and for the
sake of simplicity, we denote the operator
$P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)|_{W^{\bot}_{u_{1},k}}$ by $H_{v,k}.$ If
such $C>0$ does not exist, then there exist sequences
$\\{\tau^{k}_{u_{1}}\\}$, $\\{v_{k}\\}\subset X$ and $\\{w_{k}\\}\subset Y$
such that $\tau^{k}_{u_{1}}\rightarrow 0$ as $k\rightarrow\infty,$ $v_{k}\in
B_{X}(u_{1},\tau^{k}_{u_{1}})$, $w_{k}\in W^{\bot}_{u_{1},k}$, $||w_{k}||=1$,
$k=1,2,\cdots$ and

$\displaystyle\lim_{k\rightarrow\infty}||H_{v_{k},k}w_{k}||=0.$ (3.12)

Passing to a subsequence, we may assume that $w_{k}\rightharpoonup w_{0}$ in
$Y$ as $k\rightarrow\infty.$ By $\tau^{k}_{u_{1}}\rightarrow 0$ as
$k\rightarrow\infty$ and the assumption that $\\{v_{k}\\}\subset
B_{X}(u_{1},\tau^{k}_{u_{1}})$, we get that

$\displaystyle\lim_{k\rightarrow\infty}||v_{k}-u_{1}||=0.$ (3.13)

By $w_{k}\in W^{\bot}_{u_{1},k}$ and $w_{k}\rightharpoonup w_{0}$ in $Y$, we
get that $w_{0}\bot X\oplus Z_{u_{1}}$. Combining the condition $\bf(F_{1})$,
(3.13) and the fact that $w_{k}\rightharpoonup w_{0}$ in $Y$ leads to

$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}(f^{\prime}(v_{k})w_{k}-f^{\prime}(u_{1})w_{k})||=0$
(3.14)

and

$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}(f^{\prime}(u_{1})w_{k}-f^{\prime}(u_{1})w_{0})||=0.$
(3.15)

By (3.15) and (3.14), we get that

$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}(f^{\prime}(v_{k})w_{k}-f^{\prime}(u_{1})w_{0})||=0.$
(3.16)

By Lemma 3.4, we deduce that

$\displaystyle\lim_{k\rightarrow\infty}||P_{W^{\bot}_{u_{1},k}}h-P_{(X\oplus
Z_{u_{{}_{1}}})^{\bot}}h||=0,\ \forall h\in Y,$ (3.17)

where $P_{(X\oplus Z_{u_{1}})^{\bot}}:Y\rightarrow(X\oplus
Z_{u_{{}_{1}}})^{\bot}$ is the orthogonal projection. By (3.16) and (3.17), we
get that

$\displaystyle\lim_{k\rightarrow\infty}||P_{W^{\bot}_{u_{1},k}}((-\triangle+1)^{-1}f^{\prime}(v_{k})w_{k})-P_{(X\oplus
Z_{u_{{}_{1}}})^{\bot}}((-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0})||=0.$
(3.18)

By definition,

$\displaystyle
H_{v_{k},k}w_{k}=w_{k}-P_{W^{\bot}_{u_{1},k}}(-\triangle+1)^{-1}f^{\prime}(v_{k})w_{k}.$
(3.19)

By (3.18) and the assumption
$\lim_{k\rightarrow\infty}||H_{v_{k},k}w_{k}||=0$, we deduce that
$\\{w_{k}\\}$ is compact in $Y.$ Therefore, $||w_{k}-w_{0}||\rightarrow 0$ as
$k\rightarrow\infty.$ It follows that $||w_{0}||=1,$ since $||w_{k}||=1$ for
every $k.$

Sending $k$ into infinity in the equality (3.19), by $w_{0}\in(X\oplus
Z_{u_{{}_{1}}})^{\bot},$ (3.12) and (3.18), we get that

$\displaystyle P_{(X\oplus
Z_{u_{{}_{1}}})^{\bot}}(w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0})=0.$
(3.20)

By $w_{0}\bot X$ and $u_{1}\in X,$ we have

$\displaystyle\langle
w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0},h\rangle$ (3.21)
$\displaystyle=$ $\displaystyle\langle
w_{0},h\rangle-\langle(-\triangle+1)^{-1}f^{\prime}(u_{1})h,w_{0}\rangle=0,\
\forall h\in X.$

Since for any $h\in Z_{u_{1}},$

$h-(-\triangle+1)^{-1}f^{\prime}(u_{1})h=0,$

we get that

$\displaystyle\langle
w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0},h\rangle$ (3.22)
$\displaystyle=$ $\displaystyle\langle
h-(-\triangle+1)^{-1}f^{\prime}(u_{1})h,w_{0}\rangle=0,\ \forall h\in
Z_{u_{1}}.$

By (3.21) and (3.22), we get that

$\displaystyle P_{X\oplus
Z_{u_{{}_{1}}}}(w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0})=0.$ (3.23)

By (3.20) and (3.23), we obtain

$w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0}=0,$

that is, $w_{0}$ is an eigenfunction of (3.2) with $u=u_{1}\in\mathcal{K}.$
But $w_{0}$ satisfies $w_{0}\bot X\oplus Z_{u_{1}}$ and $||w_{0}||=1.$ This
contradicts Lemma 3.2. $\Box$

For $v\in\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}}),$ let

$\displaystyle\mathcal{T}_{v}=\mbox{span}\\{\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{i}}{\partial x_{j}}\ |\ 1\leq j\leq N\\}.$ (3.24)

The space $X_{k}\oplus\mathcal{T}_{v}$ is denoted by $E_{v,k}$. Let
$P_{E^{\bot}_{v,k}}:Y\rightarrow E^{\bot}_{v,k}$ be the orthogonal projection.

######  Lemma 3.7.

Suppose that $\kappa=\max\\{\tau_{u_{i}}\ |\ 1\leq i\leq s\\}$ is sufficiently
small. Then there exist $C^{\prime}>0$ and $l_{\kappa}\in\mathbb{N}$ such that
if $k\geq l_{\kappa}$, then for every
$v\in\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}})$, the operator
$P_{E^{\bot}_{v,k}}\nabla^{2}J(v)|_{E^{\bot}_{v,k}}$ is invertible and

$\displaystyle||(P_{E^{\bot}_{v,k}}\nabla^{2}J(v)|_{E^{\bot}_{v,k}})^{-1}||_{\mathcal{L}(E^{\bot}_{v,k})}\leq
C^{\prime}.$ (3.25)

Proof. As the proof of Lemma 3.6, it suffices to prove that there exists
$C^{\prime}>0$ which is independent of $k$ such that, for sufficiently large
$k,$

$\displaystyle||P_{E^{\bot}_{v,k}}\nabla^{2}J(v)w||_{\mathcal{L}(E^{\bot}_{v,k})}\geq\frac{1}{C^{\prime}}||w||,\
\forall w\in E^{\bot}_{v,k},\ \forall
v\in\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}}).$ (3.26)

Without loss of generality, we assume that $v\in B(u_{1},\tau_{u_{{}_{1}}})$.
Let $P_{X_{k}}:Y\rightarrow X_{k}$ and
$P_{\mathcal{T}_{v}}:Y\rightarrow\mathcal{T}_{v}$ be orthogonal projections.
For $h\in Y,$

$\displaystyle P_{E^{\bot}_{v,k}}h=h-P_{X_{k}}h-P_{\mathcal{T}_{v}}h,$ (3.27)

and

$\displaystyle
P_{\mathcal{T}_{v}}h=\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{i}}{\partial
x_{j}}\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{i}}{\partial x_{j}}}{||\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}.$ (3.28)

Since $\\{\xi_{i}\ |\ 1\leq i\leq s\\}$ is a partition of unity, we get that
for every $1\leq j\leq N,$

$\displaystyle||\frac{\partial u_{1}}{\partial
x_{j}}-\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial u_{i}}{\partial x_{j}}||$
$\displaystyle=$ $\displaystyle||\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{1}}{\partial x_{j}}-\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial u_{i}}{\partial
x_{j}}||$ (3.29) $\displaystyle\leq$
$\displaystyle\sum^{s}_{i=1}\xi_{i}(v)||\frac{\partial u_{1}}{\partial
x_{j}}-\frac{\partial u_{i}}{\partial x_{j}}||.$

If $\xi_{i}(v)\neq 0,$ then $v\in B_{X}(u_{i},\tau_{u_{i}})$. Combining the
assumption $v\in B_{X}(u_{1},\tau_{u_{1}})$, we get that $u_{1}\in
B_{X}(u_{i},2\tau_{u_{i}})\cap\mathcal{K}$. Therefore, by (3.3), we deduce
that

$\displaystyle\sum^{s}_{i=1}||\frac{\partial u_{1}}{\partial
x_{j}}-\frac{\partial u_{i}}{\partial x_{j}}||<\varsigma,\ \mbox{if}\
\xi_{i}(v)\neq 0.$ (3.30)

Combining (3.29) and (3.30) leads to

$\displaystyle||\frac{\partial u_{1}}{\partial
x_{j}}-\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial u_{i}}{\partial
x_{j}}||<\varsigma,\ \mbox{for every }\ 1\leq j\leq N.$ (3.31)

Thus, there exists $C>0$ which is independent of $k$ such that

$\displaystyle||P_{\mathcal{T}_{v}}h-P_{Z_{u_{1}}}h||\leq C\varsigma||h||,\
\forall h\in Y,$ (3.32)

where

$P_{Z_{u_{1}}}:Y\rightarrow Z_{u_{1}},\
h\mapsto\sum^{N}_{j=1}\Big{\langle}h,\frac{\partial u_{1}}{\partial
x_{j}}\Big{\rangle}\frac{\frac{\partial u_{1}}{\partial
x_{j}}}{||\frac{\partial u_{1}}{\partial x_{j}}||^{2}}$

is orthogonal projection. By (3.27) and (3.32), we have

$\displaystyle||P_{E^{\bot}_{v,k}}h-P_{W^{\bot}_{u_{{}_{1}},k}}h||\leq
C\varsigma||h||,\ \forall h\in Y.$ (3.33)

For $w\in E^{\bot}_{v,k},$ we have

$\displaystyle||P_{E^{\bot}_{v,k}}\nabla^{2}J(v)w||$ $\displaystyle\geq$
$\displaystyle||P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)w||-||(P_{E^{\bot}_{v,k}}-P_{W^{\bot}_{u_{1},k}})\nabla^{2}J(v)w||$
$\displaystyle\geq$
$\displaystyle||P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)w||-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||\
(\mbox{by}\ (\ref{gdvc66dtdfaa}))$ $\displaystyle\geq$
$\displaystyle||P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)(w-P_{Z_{u_{1}}}w)||-||P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)(P_{Z_{u_{1}}}w)||-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||$
$\displaystyle\geq$ $\displaystyle
C||w-P_{Z_{u_{1}}}w||-||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||P_{Z_{u_{1}}}w||$
$\displaystyle-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||\ (\mbox{by}\
w-P_{Z_{u_{1}}}w\in W^{\bot}_{u_{1},k}\ \mbox{and}\ (\ref{gdg5w5qdaxz}))$
$\displaystyle\geq$ $\displaystyle
C||w||-(C+||\nabla^{2}J(v)||_{\mathcal{L}(Y)})||P_{Z_{u_{1}}}w||-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||$
$\displaystyle=$ $\displaystyle
C||w||-(C+||\nabla^{2}J(v)||_{\mathcal{L}(Y)})||P_{\mathcal{T}_{v}}w-P_{Z_{u_{1}}}w||$
$\displaystyle-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||\
(\mbox{since}\ P_{\mathcal{T}_{v}}w=0)$ $\displaystyle\geq$ $\displaystyle
C||w||-\varsigma
C(C+||\nabla^{2}J(v)||_{\mathcal{L}(Y)})||w||-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||.\
(\mbox{by}\ (\ref{4gdvc66dtdfaa}))$

It follows that if $\kappa>0$ is sufficiently small, then there exist
$l_{\kappa}\in\mathbb{N}$ and $C^{\prime}>0$ such that for every $k\geq
l_{\kappa},$ (3.26) holds. $\Box$

Recall that $X^{\bot}_{k}$ is the orthogonal complement space of $X_{k}$ in
$X$ and $P_{k}:X\rightarrow X_{k}$, $P^{\bot}_{k}:X\rightarrow X^{\bot}_{k}$
are orthogonal projections. Let

$\mathcal{N}_{\delta,\tau,k}:=\\{u+v\in X\ |\ u\in X_{k},\
\mbox{dist}_{X}(u,P_{k}\mathcal{K})<\delta,\ v\in X^{\bot}_{k},\
||v||<\tau\\},$

where $P_{k}\mathcal{K}=\\{P_{k}v\ |\ v\in\mathcal{K}\\}.$ By Lemma 3.4 and
the fact that $\mathcal{K}$ is a compact subset of $X,$ we get that as
$k\rightarrow\infty$, the Hausdorff distance of $\mathcal{K}$ and
$P_{k}\mathcal{K}$,

$\displaystyle\sup_{v\in
P_{k}\mathcal{K}}\mbox{dist}_{X}(v,\mathcal{K})+\sup_{u\in\mathcal{K}}\mbox{dist}_{X}(u,P_{k}\mathcal{K})\rightarrow
0.$ (3.35)

Thus, for any $\delta>0$, $\tau>0$ and $0<r<\min\\{\delta,\tau\\}$, if $k$ is
sufficiently large, then

$N_{r}(\mathcal{K})\subset\mathcal{N}_{\delta,\tau,k},$ (3.36)

where $N_{r}(\mathcal{K})$ comes from (2.8). And for any $r>0$, if
$\delta,\tau\in(0,r/2)$, then for sufficiently large $k$,

$\mathcal{N}_{\delta,\tau,k}\subset N_{r}(\mathcal{K}).$ (3.37)

Let

$\displaystyle\mathcal{N}_{\delta,k}:=\\{u\in X_{k}\ |\
\mbox{dist}_{X}(u,P_{k}\mathcal{K})<\delta\\}.$ (3.38)

######  Lemma 3.8.

If $\delta>0$ is sufficient small and $k$ is sufficiently large, then there
exists a $C^{1}-$mapping

$\pi_{k}:\mathcal{N}_{\delta,k}\rightarrow X^{\bot}_{k},$

satisfying

  * (i)

$\langle\nabla I(v+\pi_{k}(v)),\phi\rangle=0,$ $\forall\phi\in X^{\bot}_{k};$

  * (ii)

$\lim_{k\rightarrow\infty}\sup\\{||\pi_{k}(v)||\ |\
v\in\mathcal{N}_{\delta,k}\\}=0;$

  * (iii)

$\lim_{k\rightarrow\infty}\sup\\{||D\pi_{k}(v)h||\ |\
v\in\mathcal{N}_{\delta,k},\ h\in X_{k},\ ||h||=1\\}=0;$

  * (iv)

If $v$ is a critical point of $I(v+\pi_{k}(v))$, then $v+\pi_{k}(v)$ is a
critical point of $I.$

Proof. By Lemma 3.5, if $r>0$ is small enough, then the operator

$L_{v,k}:=P^{\bot}_{k}\nabla^{2}I(v)|_{X^{\bot}_{k}}:X^{\bot}_{k}\rightarrow
X^{\bot}_{k}$

is invertible and if $k\geq l_{\kappa}$,

$\displaystyle||L_{v,k}^{-1}||_{\mathcal{L}(X_{k}^{\bot})}\leq 2,\ \forall
v\in N_{r}(\mathcal{K}).$ (3.39)

Assume that $0<\delta<r$, by (3.37), if $k$ is large enough, then
$\mathcal{N}_{\delta,k}\subset N_{r}(\mathcal{K})$.

For $\rho>0$ and $v\in\mathcal{N}_{\delta,k},$ define

$\Psi_{v,k}:\overline{B_{X^{\bot}_{k}}(0,\rho)}\rightarrow X^{\bot}_{k},\
w\mapsto w-L^{-1}_{v,k}P^{\bot}_{k}\nabla I(v+w).$

For any $w_{i}\in\overline{B_{X^{\bot}_{k}}(0,\rho)},$ $i=1,2,$ by the
definition of $L_{v,k}$, we have
$w_{2}-w_{1}-L^{-1}_{v,k}P^{\bot}_{k}\nabla^{2}I(v)(w_{2}-w_{1})=0.$
Therefore,

$\displaystyle||\Psi_{v,k}(w_{2})-\Psi_{v,k}(w_{1})||$ (3.40) $\displaystyle=$
$\displaystyle||w_{2}-w_{1}-L^{-1}_{v,k}P^{\bot}_{k}\nabla^{2}I(v+\theta
w_{2}+(1-\theta)w_{1})(w_{2}-w_{1})||$ $\displaystyle(\mbox{by the mean value
theorem},\ 0<\theta=\theta(x)<1)$ $\displaystyle\leq$
$\displaystyle||w_{2}-w_{1}-L^{-1}_{v,k}P^{\bot}_{k}\nabla^{2}I(v)(w_{2}-w_{1})||$
$\displaystyle+||L^{-1}_{v,k}P^{\bot}_{k}(\nabla^{2}I(v+\theta
w_{2}+(1-\theta)w_{1})-\nabla^{2}I(v))(w_{2}-w_{1})||$ $\displaystyle=$
$\displaystyle||L^{-1}_{v,k}P^{\bot}_{k}(\nabla^{2}I(v+\theta
w_{2}+(1-\theta)w_{1})-\nabla^{2}I(v))(w_{2}-w_{1})||$ $\displaystyle\leq$
$\displaystyle 2||(\nabla^{2}I(v+\theta
w_{2}+(1-\theta)w_{1})-\nabla^{2}I(v))(w_{2}-w_{1})||\ (\mbox{by}\
(\ref{nnb99ifufjjj})).$

Since $I\in C^{2}(X,\mathbb{R})$ and $\mathcal{K}$ is compact in $X$, if
$\delta$ and $\rho$ are small enough, then for any
$v\in\mathcal{N}_{\delta,k}$ and $w\in\overline{B_{X^{\bot}_{k}}(0,\rho)},$

$||\nabla^{2}I(v+w)-\nabla^{2}I(v)||_{\mathcal{L}(X)}<1/4.$

Thus, by (3.40), we get that for any
$w_{i}\in\overline{B_{X^{\bot}_{k}}(0,\rho)},$ $i=1,2,$

$\displaystyle||\Psi_{v,k}(w_{2})-\Psi_{v,k}(w_{1})||\leq\frac{1}{2}||w_{2}-w_{1}||.$
(3.41)

If $\delta>0$ is small enough and $k$ is large enough, then for every
$v\in\mathcal{N}_{\delta,k}$,

$||\Psi_{v,k}(0)||\leq\rho/2.$

Then by (3.41), we get that for every
$w\in\overline{B_{X^{\bot}_{k}}(0,\rho)}$,

$\displaystyle||\Psi_{v,k}(w)||\leq||\Psi_{v,k}(w)-\Psi_{v,k}(0)||+||\Psi_{v,k}(0)||\leq\rho.$
(3.42)

By (3.41) and (3.42), $\Psi_{v,k}$ is a contractive mapping in
$\overline{B_{X^{\bot}_{k}}(0,\rho)}$ if $\delta$ and $\rho$ are small enough
and $k$ is large enough. Thus, by Banach fixed point theorem, there exists
unique fixed point $\pi_{k}(v)\in\overline{B_{X^{\bot}_{k}}(0,\rho)}.$ It is
easy to verify that $\pi_{k}$ is a $C^{1}-$mapping and it satisfies the result
$\bf(i)$.

Now, we give the proof of $\bf(ii).$ By $P_{k}^{\bot}\nabla I(v+\pi_{k}(v))=0$
and $\pi_{k}(v)\in X^{\bot}_{k}$, we get that

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\langle\nabla
I(v+\pi_{k}(v)),\pi_{k}(v)\rangle$ (3.43) $\displaystyle=$
$\displaystyle||\pi_{k}(v)||^{2}-\int_{\mathbb{R}^{N}}f(v+\pi_{k}(v))\cdot\pi_{k}(v).$

By Lemma 3.4, we deduce that for any sequence $\\{v_{k}\\}$ with
$v_{k}\in\mathcal{N}_{\delta,k}$, $\pi_{k}(v_{k})\rightharpoonup 0$ in $X$ as
$k\rightarrow\infty$. Combining the compact embedding $X\hookrightarrow
L^{p}_{r}(\mathbb{R}^{N})$, we obtain

$\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|f(v_{k}+\pi_{k}(v_{k}))|\cdot|\pi_{k}(v_{k})|=0.$

It follows that

$\displaystyle\lim_{k\rightarrow\infty}\sup\\{\int_{\mathbb{R}^{N}}f(v+\pi_{k}(v))\cdot\pi_{k}(v)\
|\ v\in\mathcal{N}_{\delta,k}\\}=0.$ (3.44)

The conclusion $\bf(ii)$ follows from (3.43) and (3.44).

Differentiating equation $P^{\bot}_{k}\nabla I(v+\pi_{k}(v))=0$ for the
variable $v$ in the direction $h\in X_{k}$, we get that

$\displaystyle
D\pi_{k}(v)h-P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v+\pi_{k}(v))(h+D\pi_{k}(v)h)=0.$
(3.45)

Note that $D\pi_{k}(v)h\in X^{\bot}_{k}$. By (3.39), (3.45) and
$\lim_{k\rightarrow\infty}||\pi_{k}(v)||=0$, we get that if $k$ is large
enough, then

$\displaystyle\frac{1}{2}||D\pi_{k}(v)h||$ $\displaystyle\leq$
$\displaystyle||D\pi_{k}(v)h-P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v+\pi_{k}(v))D\pi_{k}(v)h||$
$\displaystyle=$
$\displaystyle||P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v+\pi_{k}(v))h||$

It follows that for sufficiently large $k,$

$\displaystyle\sup\\{||D\pi_{k}(v)h||\ |\ v\in\mathcal{N}_{\delta,k},\ h\in
X_{k},\ ||h||\leq 1\\}<\infty.$ (3.47)

By (3.45), we get that

$\displaystyle||D\pi_{k}(v)h||^{2}=\int_{\mathbb{R}^{N}}f^{\prime}(v+\pi_{k}(v))\cdot(h+D\pi_{k}(v)h)\cdot
D\pi_{k}(v)h.$ (3.48)

(3.47) and the same argument as (3.44) yield

$\lim_{k\rightarrow\infty}\sup\\{\int_{\mathbb{R}^{N}}f^{\prime}(v+\pi_{k}(v))\cdot(h+D\pi_{k}(v)h)\cdot
D\pi_{k}(v)h\ |\ v\in\mathcal{N}_{\delta,k},\ h\in X_{k},\ ||h||\leq 1\\}=0.$

Combining (3.48), we get the conclusion $\bf(iii).$

By $\bf(iii),$ if $k$ is sufficiently large, then

$\\{h+D\pi_{k}(v)h\ |\ h\in X_{k}\\}+X^{\bot}_{k}=X.$

Combining the result $\bf(i)$, we get that if $v_{0}$ is a critical point of
$I(v+\pi_{k}(v))$, then $v_{0}+\pi_{k}(v_{0})$ is a critical point of $I$.
$\Box$

######  Remark 3.9.

By $\bf(ii)$ and $\bf(iv)$ of Lemma 3.8, $\mathcal{N}_{\delta,\tau,k}$ is a
neighborhood of $\mathcal{K}$ if

$\displaystyle\tau>\sup\\{||\pi_{k}(v)||\ |\ v\in\mathcal{N}_{\delta,k}\\}.$
(3.49)

######  Lemma 3.10.

Let
$\mathcal{I}_{k}(u)=\frac{1}{2}||P^{\bot}_{k}u||^{2}+I(P_{k}u+\pi_{k}(P_{k}u)).$
Then

$\lim_{k\rightarrow\infty}||\mathcal{I}_{k}-I||_{C^{1}(\overline{\mathcal{N}_{\delta,\tau,k}})}=0.$

Proof. By definition, we have

$\mathcal{I}_{k}(u)=\frac{1}{2}||u||^{2}+\frac{1}{2}||\pi_{k}(P_{k}u)||^{2}-\int_{\mathbb{R}^{N}}F(P_{k}u+\pi_{k}(P_{k}u)).$

For any sequence $\\{u_{k}\\}$ with
$u_{k}\in\overline{\mathcal{N}_{\delta,\tau,k}}$, by the mean value theorem,
we get that

$\displaystyle F(P_{k}u_{k}+\pi_{k}(P_{k}u_{k}))-F(u_{k})$ $\displaystyle=$
$\displaystyle\zeta(u_{k},\theta)(P_{k}u_{k}+\pi_{k}(P_{k}u_{k})-u_{k})$
$\displaystyle=$
$\displaystyle\zeta(u_{k},\theta)(\pi_{k}(P_{k}u_{k})-P_{k}^{\bot}u_{k})$

where

$\displaystyle\zeta(u_{k},\theta)=f^{\prime}(\theta
P_{k}u_{k}+\theta\pi_{k}(P_{k}u_{k})+(1-\theta)u_{k})$

with $0<\theta(x)<1,$ $x\in\mathbb{R}^{N}$. Then we have

$\displaystyle\int_{\mathbb{R}^{N}}\Big{|}F(P_{k}u_{k}+\pi_{k}(P_{k}u_{k}))-F(u_{k})\Big{|}=\int_{\mathbb{R}^{N}}|\zeta(u_{k},\theta)|\cdot|\pi_{k}(P_{k}u_{k})-P_{k}^{\bot}u_{k}|.$
(3.50)

By $\bf(ii)$ of Lemma 3.8, we get that for every $2\leq p<2^{*},$

$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|\pi_{k}(P_{k}u_{k})|^{p}=0.$
(3.51)

By Lemma 3.4,we have

$\displaystyle P^{\bot}_{k}u_{k}\rightharpoonup 0\ \mbox{in}\ X.$ (3.52)

Since $X$ can be compactly embedded into $L^{p}_{r}(\mathbb{R}^{N})$, by
(3.52), we get that for every $2\leq p<2^{*},$

$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|P^{\bot}_{k}u_{k}|^{p}=0.$
(3.53)

By (3.50), (3.51), (3.53) and the condition $\bf(F_{1})$, we obtain

$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}\Big{|}F(P_{k}u_{k}+\pi_{k}(P_{k}u_{k}))-F(u_{k})\Big{|}=0.$

Thus

$\displaystyle\lim_{k\rightarrow\infty}\sup\\{\int_{\mathbb{R}^{N}}\Big{|}F(P_{k}u+\pi_{k}(P_{k}u))-F(u)\Big{|}\
|\ u\in\overline{\mathcal{N}_{\delta,\tau,k}}\\}=0.$ (3.54)

By $\bf(ii)$ of Lemma 3.8 and (3.54), we get that

$\displaystyle\lim_{k\rightarrow\infty}||\mathcal{I}_{k}-I||_{C^{0}(\overline{\mathcal{N}_{\delta,\tau,k}})}=0.$
(3.55)

For $h\in X,$

$\displaystyle\langle\nabla\mathcal{I}_{k}(u),h\rangle$ $\displaystyle=$
$\displaystyle\langle
u,h\rangle+\langle\pi_{k}(P_{k}u),D\pi_{k}(P_{k}u)(P_{k}h)\rangle$
$\displaystyle-\int_{\mathbb{R}^{N}}f(P_{k}u+\pi_{k}(P_{k}u))\cdot(P_{k}h+D\pi_{k}(P_{k}u)(P_{k}h)).$

By $\bf(iii)$ of Lemma 3.8 and the same argument as above, we can get that

$\displaystyle\lim_{k\rightarrow\infty}\sup\\{\langle\nabla\mathcal{I}_{k}(u)-\nabla
I(u),h\rangle\ |\ u\in\overline{\mathcal{N}_{\delta,\tau,k}},\ ||h||\leq
1\\}=0.$ (3.56)

The result of this Lemma follows from (3.55) and (3.56). $\Box$

######  Remark 3.11.

For $r>0,$ let $\sigma\in(0,\sigma_{r/2})$, where $\sigma_{r/2}$ comes from
Lemma 2.7, and let $a\in(c-\sigma,c)$, $b\in(c,c+\sigma)$ be regular values of
$I$, where $c$ comes from (2.3). By Lemma 2.7, there exists a GM pair
$(W,W_{-})$ of $\mathcal{K}^{b}_{a}$ associated with some pseudo-gradient
vector field of $I$ such that $W\subset N_{r/2}(\mathcal{K}^{b}_{a}).$ By
(3.36), if $0<r<\min\\{\delta,\tau\\}$, then
$N_{r}(\mathcal{K})\subset\mathcal{N}_{\delta,\tau,k}$ if $k$ is sufficiently
large. Denote the critical set of $\mathcal{I}_{k}$ in
$\mathcal{N}_{\delta,\tau,k}$ by $\widehat{\mathcal{K}}_{k}$. By $\bf(i)$ and
$\bf(iv)$ of Lemma 3.8, we deduce that
$\widehat{\mathcal{K}}_{k}=P_{k}\mathcal{K}^{b}_{a}$. Then by (3.35),
$\widehat{\mathcal{K}}_{k}\subset\mbox{int}\ W$ if $k$ is large enough. By [9,
Theorem III.4] and Lemma 3.10, we infer that for sufficiently large $k$,
$(W,W_{-})$ is also a GM pair of $\mathcal{I}_{k}$ for
$\widehat{\mathcal{K}}_{k}$ associated with some pseudo-gradient vector filed
of $\mathcal{I}_{k}.$

For $v\in\mathcal{N}_{\delta,k},$ denote $I(v+\pi_{k}(v))$ by $g_{k}(v).$ And
denote the critical set of $g_{k}$ in $W$ by $\mathcal{K}_{k}$. By $\bf(i)$
and $\bf(iv)$ of Lemma 3.8, we deduce that
$\mathcal{K}_{k}=P_{k}\mathcal{K}^{b}_{a}=\widehat{\mathcal{K}}_{k}$. Let
$(W_{k},W^{-}_{k})$ be a GM pair of $g_{k}$ for $\mathcal{K}_{k}$. Note that
for $u=w+v\in\mathcal{N}_{\delta,\tau,k}$ with $w\in X^{\bot}_{k},$ $v\in
X_{k},$ $\mathcal{I}_{k}(u)=\frac{1}{2}||w||^{2}+g_{k}(v)$. By shifting
theorem (see Lemma 5.1 of [7]), we have

$\displaystyle\check{H}^{q}(W_{k},W^{-}_{k})=\check{H}^{q}(W,W^{-}),\
q=0,1,2,\cdots.$

Combining Lemma 2.5, we get that, for sufficiently large $k,$

$\displaystyle\check{H}^{1}(W_{k},W^{-}_{k})=\check{H}^{1}(W,W^{-})\neq 0.$
(3.57)

##  4 A variational reduction for the functional $E_{\epsilon}$

For $v\in\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}})$ and $y\in\mathbb{R}^{N}$,
denote the space

$\\{\zeta(\cdot-y)\ |\ \zeta\in X_{k}\\}\oplus\mathcal{T}_{v}(\cdot-y)$

by $T_{v,y,k}$, where $\mathcal{T}_{v}$ comes from (3.24). Denote the
orthogonal complemental space of $T_{v,y,k}$ in $Y$ by $T_{v,y,k}^{\bot}$.

Recall that (see (3.38))

$\displaystyle\mathcal{N}_{\delta,k}=\\{u\in X_{k}\ |\
\mbox{dist}_{X}(u,P_{k}\mathcal{K})<\delta\\}.$

For $v\in\mathcal{N}_{\delta,k},$ define

$L_{v,y,\epsilon,k}:T_{v,y,k}^{\bot}\rightarrow T_{v,y,k}^{\bot}$

by

$\displaystyle w\in T_{v,y,k}^{\bot}\mapsto
w-S_{v,y,k}(-\triangle+1+V(\epsilon x))^{-1}(f^{\prime}(v(\cdot-y))w)$ (4.1)

where $S_{v,y,k}:Y\rightarrow T_{v,y,k}^{\bot}$ is orthogonal projection and
the operator $(-\triangle+1+V(\epsilon x))^{-1}$ is defined by (2.2).

######  Lemma 4.1.

Given $R>0,$ there exist $\delta_{0}>0$, $\epsilon_{0}>0$, $l^{*}>0$ and $C>0$
which are independent of $k,$ such that if $k\geq l^{*},$
$0<\delta\leq\delta_{0}$ and $0\leq\epsilon\leq\epsilon_{0},$ then for any
$v\in\overline{\mathcal{N}_{\delta,k}}$ and
$y\in\overline{B_{\mathbb{R}^{N}}(0,R)}$, $L_{v,y,\epsilon,k}$ is invertible
and

$\displaystyle||L_{v,y,\epsilon,k}w||\geq C||w||,\ \forall|y|\leq R,\ \forall
w\in T_{v,y,k}^{\bot}.$ (4.2)

Proof. Suppose $\kappa=\max\\{\tau_{u_{i}}\ |\ 1\leq i\leq s\\}$ is small
enough such that Lemma 3.7 holds. By (3.37), for sufficiently small
$\delta_{0}>0,$ there exists $l^{\prime}_{\kappa}>0$ such that
$\mathcal{N}_{\delta_{0},k}\subset\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}})$ if
$k\geq l^{\prime}_{\kappa}.$ Note that $L_{v,0,0,k}$ is exactly the operator
$P_{E^{\bot}_{v,k}}\nabla^{2}J(v)|_{E^{\bot}_{v,k}}$ which has been defined in
Lemma 3.7 and for every $w\in T^{\bot}_{v,y,k}$,

$L_{v,y,0,k}w=L_{v,0,0,k}w(\cdot-y).$

Thus, by Lemma 3.7, there exists $C^{\prime}>0$ such that if $k\geq
l^{*}:=\max\\{l_{\kappa},l^{\prime}_{\kappa}\\}$, then for any
$v\in\mathcal{N}_{\delta_{0},k}$,

$\displaystyle||L_{v,y,0,k}w||\geq C^{\prime}||w||,\ \forall|y|\leq R,\
\forall w\in T_{v,y,k}^{\bot},$

where $l_{\kappa}$ is the constant comes from Lemma 3.7. Therefore, to prove
(4.2), it suffices to prove that

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\Big{\\{}||L_{v,y,\epsilon,k}w-L_{v,y,0,k}w||\ |\ w\in
T_{v,y,k}^{\bot},\ ||w||\leq 1,$ (4.3) $\displaystyle\quad\quad\quad\quad\quad
v\in\overline{\mathcal{N}_{\delta_{0},k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ k\geq l^{*}\Big{\\}}=0.$

If we can prove that for any given sequences $\\{k_{n}\\}\subset\mathbb{N},$
$\\{\epsilon_{n}\\}\subset(0,+\infty),$
$\\{y_{n}\\}\subset\overline{B_{\mathbb{R}^{N}}(0,R)}$, $\\{v_{n}\\}$ and
$\\{w_{n}\\}$ which satisfy that $\epsilon_{n}\rightarrow 0$ as
$n\rightarrow\infty$, $v_{n}\in\overline{\mathcal{N}_{\delta_{0},k_{n}}}$,
$w_{n}\in T^{\bot}_{v_{n},y_{n},k_{n}}$ and $||w_{n}||\leq 1$, $n=1,2,\cdots$,

$\displaystyle\lim_{n\rightarrow\infty}||L_{v_{n},y_{n},\epsilon_{n},k_{n}}w_{n}-L_{v_{n},y_{n},0,k_{n}}w_{n}||=0,$
(4.4)

then (4.3) holds. We only give the proof of (4.4) in the case
$k_{n}\rightarrow\infty,$ $n\rightarrow\infty$, since the proofs in other
cases are similar. Without loss of generality, we assume that $\\{k_{n}\\}$ is
exactly the sequence $\\{k\\}$ and we shall denote $\epsilon_{n}$, $y_{n},$
$v_{n}$ and $w_{n}$ by $\epsilon_{k}$, $y_{k},$ $v_{k}$ and $w_{k}$
respectively, $k=1,2,\cdots.$

Passing to a subsequence, we may assume that as $k\rightarrow\infty$,
$y_{k}\rightarrow y_{0}$, $v_{k}\rightharpoonup v_{0}$ in $X$ and
$w_{k}\rightharpoonup w_{0}$ in $Y$.

Let

$\eta_{k}=(-\triangle+1+V(\epsilon_{k}x))^{-1}(f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}).$

It is easy to verify that $\\{\eta_{k}\\}$ is bounded in $Y$ and

$\displaystyle\eta_{k}=(-\triangle+1)^{-1}(f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k})-(-\triangle+1)^{-1}V(\epsilon_{k})\eta_{k}.$ (4.5)

Passing to a subsequence, we may assume that $\eta_{k}\rightharpoonup\eta_{0}$
in $Y$ as $k\rightarrow\infty.$

By definition of $L_{v,y,\epsilon,k}$ and (4.5), we get that

$\displaystyle
L_{v_{k},y_{k},\epsilon,k}w-L_{v_{k},y_{k},0,k}w=S_{v_{k},y_{k},k}(-\triangle+1)^{-1}V(\epsilon_{k}x)\eta_{k}.$
(4.6)

The condition $\bf(V_{1})$ implies that $V(0)=0$. It follows that for any
$h\in Y,$

$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}V(\epsilon_{k}x)\eta_{k}h=0.$
(4.7)

Since $\eta_{k}$ is a weak solution of the equation:

$\displaystyle-\triangle\eta_{k}+\eta_{k}+V(\epsilon_{k}x)\eta_{k}=f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k},$ (4.8)

by (4.7), $y_{k}\rightarrow y_{0},$ $\eta_{k}\rightharpoonup\eta_{0}$ and
$w_{k}\rightharpoonup w_{0}$ in $Y$, we get that $\eta_{0}$ is a weak solution
of the equation:

$\displaystyle-\triangle\eta_{0}+\eta_{0}=f^{\prime}(v_{0}(\cdot-
y_{0}))w_{0}.$ (4.9)

From (4.8) and (4.9), we obtain

$\displaystyle-\triangle(\eta_{k}-\eta_{0})+(\eta_{k}-\eta_{0})+V(\epsilon_{k}x)(\eta_{k}-\eta_{0})$
$\displaystyle=$ $\displaystyle(f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}-f^{\prime}(v_{0}(\cdot-y_{0}))w_{0})-V(\epsilon_{k}x)\eta_{0}.$

Multiplying the above equation by $\eta_{k}-\eta_{0}$ and integrating, we get
that there exists a constant $C>0$ such that

$\displaystyle C||\eta_{k}-\eta_{0}||^{2}$ (4.10) $\displaystyle\leq$
$\displaystyle||\eta_{k}-\eta_{0}||^{2}+\int_{\mathbb{R}^{N}}V(\epsilon_{k}x)(\eta_{k}-\eta_{0})^{2}\
(\mbox{by the condition }\ {\bf(V_{0})})$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\Big{(}f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}-f^{\prime}(v_{0}(\cdot-
y_{0}))w_{0}-V(\epsilon_{k}x)\eta_{0}\Big{)}\cdot(\eta_{k}-\eta_{0})$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{N}}\Big{|}f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}-f^{\prime}(v_{0}(\cdot-
y_{0}))w_{0}\Big{|}\cdot|\eta_{k}-\eta_{0}|$
$\displaystyle+(\int_{\mathbb{R}^{N}}V^{2}(\epsilon_{k}x)\eta^{2}_{0})^{\frac{1}{2}}\cdot||\eta_{k}-\eta_{0}||_{L^{2}(\mathbb{R}^{N})}.$

Since $v_{k}\rightharpoonup v_{0}$ in $X$ and $y_{k}\rightarrow y_{0}$ as
$k\rightarrow\infty$, by the fact that $X$ can be compactly embedding into
$L^{p}_{r}(\mathbb{R}^{N})$ ($\forall p\in[2,2^{*})$), we get that

$\displaystyle\lim_{k\rightarrow\infty}||v_{k}(\cdot-y_{k})-v_{0}(\cdot-
y_{0})||_{L^{p}(\mathbb{R}^{N})}=0,\ \forall p\in[2,2^{*}).$ (4.11)

By (4.11) and the condition $\bf(F_{1})$, we get that

$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}\Big{|}f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}-f^{\prime}(v_{0}(\cdot-
y_{0}))w_{0}\Big{|}\cdot|\eta_{k}-\eta_{0}|=0.$ (4.12)

By (4.10), (4.12) and

$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}V^{2}(\epsilon_{k}x)\eta^{2}_{0}=0,$
(4.13)

we get that

$\lim_{k\rightarrow\infty}||\eta_{k}-\eta_{0}||=0.$ (4.14)

(4.13) and (4.14) yield

$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}V^{2}(\epsilon_{k}x)\eta^{2}_{k}=0.$
(4.15)

It follows that

$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}V(\epsilon_{k}x)\eta_{k}||=0.$
(4.16)

Combining (4.16) and (4.6) leads to (4.4).

Finally, by definition, $L_{v,y,\epsilon,k}$ is a Fredholm operator with index
zero and by (4.2), it is an injection. Therefore, it is invertible. $\Box$

######  Theorem 4.2.

Given $R>0.$ There exist $\delta^{*}>0$ and $\epsilon^{*}>0$ such that if
$0<\delta\leq\delta^{*}$ and $0\leq\epsilon\leq\epsilon^{*}$, then there exist
$k(\delta)$ and a $C^{1}-$mapping

$w_{\delta,k}(\cdot,\cdot,\epsilon):\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\rightarrow
Y,\ (u,y)\mapsto w_{\delta,k}(u,y,\epsilon)$

for $k\geq k(\delta)$, satisfying

  * (i)

$w_{\delta,k}(u,y,\epsilon)\in T_{u,y,k}^{\bot},$
$\forall(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)};$

  * (ii)

$\langle\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),\phi\rangle=0,$
$\forall\phi\in T_{u,y,k}^{\bot};$

  * (iii)

$w_{\delta,k}(u,y,0)=(\pi_{k}(u))(\cdot-y),$
$\forall(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$;

  * (iv)

for any $r>0,$ there exists $\delta_{r}>0$ such that if
$0<\delta\leq\delta_{r}$, $u\in\overline{\mathcal{N}_{\delta,k}}$,
$y\in\overline{B_{\mathbb{R}^{N}}(0,R)}$ and $k\geq k(\delta)$, then
$||w_{\delta,k}(u,y,\epsilon)||\leq r;$

  * (v)

for any $n>0$,

$\displaystyle\sup\\{||(1+|x|)^{n}w_{\delta,k}(u,y,\epsilon)||_{L^{\infty}(\mathbb{R}^{N})}\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}$ $\displaystyle<\infty.$ (4.17)

Proof. By Lemma 4.1, we know that for any $R>0$, $L_{u,y,\epsilon,k}$ is
invertible if $0<\delta\leq\delta_{0}$, $0\leq\epsilon\leq\epsilon_{0}$ and
$k\geq l^{*}$. Moreover, the upper bound of $||L^{-1}_{u,y,\epsilon,k}||$ is
independent of $u,$ $y$, $\epsilon$ and $k.$ For
$u\in\overline{\mathcal{N}_{\delta,k}}$ and $r>0$, let

$\Phi_{u,y,\epsilon,k}:\overline{B_{T_{u,y,k}^{\bot}}(0,r)}\rightarrow
T_{u,y,k}^{\bot},$ $w\mapsto w-L^{-1}_{u,y,\epsilon,k}S_{u,y,k}\nabla
E_{\epsilon}(u(\cdot-y)+w).$

Now, we show that if $r$, $\delta$ and $\epsilon$ are small enough and $k$ is
large enough, then for any $u\in\overline{\mathcal{N}_{\delta,k}}$,
$\Phi_{u,y,\epsilon,k}$ is a contractive mapping in
$\overline{B_{T_{u,y,k}^{\bot}}(0,r)}.$

Using

$\displaystyle\nabla E_{\epsilon}(u(\cdot-y)+w)$ $\displaystyle=$
$\displaystyle u(\cdot-y)+w-(-\triangle+1+V(\epsilon x))^{-1}f(u(\cdot-y)+w)$

and the mean value theorem, we get that for any
$w_{1},w_{2}\in\overline{B_{T_{u,y,k}^{\bot}}(0,r)}$,
$\Phi_{u,y,\epsilon,k}(w_{1})-\Phi_{u,y,\epsilon,k}(w_{2})$ equals

$\displaystyle(w_{1}-w_{2})-L^{-1}_{u,y,\epsilon,k}S_{u,y,k}\Big{\\{}(w_{1}-w_{2})$
(4.18) $\displaystyle-(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})\cdot(w_{1}-w_{2}))\Big{\\}}$
$\displaystyle=$
$\displaystyle(w_{1}-w_{2})-L^{-1}_{u,y,\epsilon,k}S_{u,y,k}\Big{\\{}(w_{1}-w_{2})$
$\displaystyle-(-\triangle+1+V(\epsilon
x))^{-1}f^{\prime}(u(\cdot-y))(w_{1}-w_{2})$
$\displaystyle-(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))(w_{1}-w_{2})\Big{\\}}$

where $\tilde{w}=\theta w_{1}+(1-\theta)w_{2}$ for some $0<\theta<1.$ By the
condition $\bf(F_{1})$, we can prove that

$\displaystyle\lim_{r\rightarrow 0}\sup\\{||(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))\varphi||$
(4.19) $\displaystyle\quad\quad\quad\quad\ |\
u\in\overline{\mathcal{N}_{\delta,k}},\ |y|\leq R,\ \varphi\in Y,\
||\varphi||\leq 1,\ 0\leq\epsilon\leq\epsilon_{0}\\}=0.$

By $||L^{-1}_{u,y,\epsilon,k}||_{\mathcal{L}(Y)}\leq 1/C$ (see Lemma 4.1 ),
$||S_{u,y,k}||_{\mathcal{L}(Y)}\leq 1$ and (4.19), we deduce that if $r$ is
small enough, then

$\displaystyle||L^{-1}_{u,y,\epsilon,k}S_{u,y,k}(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))(w_{1}-w_{2})||$
(4.20) $\displaystyle\leq\frac{1}{C}||(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))(w_{1}-w_{2})||$
$\displaystyle\leq\frac{1}{2}||w_{1}-w_{2}||.$

By the definition of $L_{u,y,\epsilon,k}$,

$\displaystyle
L^{-1}_{u,y,\epsilon,k}S_{u,y,k}\Big{\\{}(w_{1}-w_{2})-(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y))(w_{1}-w_{2}))\Big{\\}}$ (4.21)
$\displaystyle=(w_{1}-w_{2}).$

Combining (4.20), (4.21) and (4.18), we deduce that there exists $r_{0}>0$
such that if $0<r\leq r_{0}$, $0<\delta\leq\delta_{0}$,
$0\leq\epsilon\leq\epsilon_{0}$ and $k\geq l^{*}$, then for any
$(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$
and $w_{1},w_{2}\in\overline{B_{T_{u,y,k}^{\bot}}(0,r)},$

$||\Phi_{u,y,\epsilon,k}(w_{1})-\Phi_{u,y,\epsilon,k}(w_{2})||\leq\frac{1}{2}||w_{1}-w_{2}||.$
(4.22)

Claim: For any $0<r\leq r_{0},$ there exist $\epsilon_{r}$, $\delta_{r}$ and
$k(\delta,r)$ such that if $0<\delta\leq\delta_{r}$,
$0\leq\epsilon\leq\epsilon_{r}$ and $k\geq k(\delta,r)$, then

$\displaystyle||\Phi_{u,y,\epsilon,k}(0)||\leq r/2,\
\forall(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}.$
(4.23)

Let $h_{u,y,\epsilon}=(-\triangle+1+V(\epsilon x))^{-1}f(u(\cdot-y)).$ It is
easy to verify

$\displaystyle
h_{u,y,\epsilon}=(-\triangle+1)^{-1}f(u(\cdot-y))-(-\triangle+1)^{-1}V(\epsilon
x)h_{u,y,\epsilon}.$ (4.24)

The same argument as (4.15) yields

$\lim_{\epsilon\rightarrow 0}\sup\\{\int_{\mathbb{R}^{N}}V^{2}(\epsilon
x)h^{2}_{u,y,\epsilon}\ |\ \ u\in\overline{\mathcal{N}_{\delta_{{}_{0}},k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ k\geq l^{*}\\}=0.$

Thus, by (4.24), as $\epsilon\rightarrow 0,$

$\displaystyle\sup\\{||(-\triangle+1+V(\epsilon x))^{-1}f(u(\cdot-y))$
$\displaystyle\quad\quad\quad-(-\triangle+1)^{-1}f(u(\cdot-y))||\ |\
u\in\overline{\mathcal{N}_{\delta_{{}_{0}},k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ k\geq l^{*}\\}$
$\displaystyle\rightarrow 0.$

It follows that as $\epsilon\rightarrow 0,$

$\displaystyle\sup\\{||\nabla E_{\epsilon}(u(\cdot-y))-\nabla J(u(\cdot-y))||\
|\ u\in\overline{\mathcal{N}_{\delta_{{}_{0}},k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ k\geq l^{*}\\}$ (4.25)
$\displaystyle\rightarrow 0.$

Therefore, for $0<r\leq r_{0}$, there exists $\epsilon_{r}>0$ such that for
any $u\in\overline{\mathcal{N}_{\delta_{{}_{0}},k}},$
$y\in\overline{B_{\mathbb{R}^{N}}(0,R)}$ and $k\geq l^{*}$,

$\displaystyle||\nabla E_{\epsilon}(u(\cdot-y))-\nabla
J(u(\cdot-y))||<\frac{C}{4}r\ \ \mbox{if}\ 0\leq\epsilon\leq\epsilon_{r},$
(4.26)

where the constant $C$ comes from Lemma 4.1. Since $\nabla
J(v(\cdot-y))=\nabla J(v)=0,$ $\forall v\in\mathcal{K},$ we get that for any
$0<r\leq r_{0}$, there exists $\delta_{r}$ such that for any
$0<\delta\leq\delta_{r}$ and any $u\in N_{2\delta}(\mathcal{K})$,

$\displaystyle||\nabla J(u(\cdot-y))||<\frac{C}{4}r.$ (4.27)

By (4.27) and the fact that (see (3.35))

$\lim_{k\rightarrow\infty}\overline{\mathcal{N}_{\delta,k}}\subset
N_{2\delta}(\mathcal{K}),$

we deduce that there exists $k(\delta,r)$ such that if $k\geq k(\delta,r)$,
then for any $0<\delta\leq\delta_{r}$ and any
$u\in\overline{\mathcal{N}_{\delta,k}}$,

$\displaystyle||\nabla J(u(\cdot-y))||<\frac{C}{4}r.$ (4.28)

Thus, the claim follows from (4.26), (4.28) and the fact that

$||\Phi_{u,y,\epsilon,k}(0)||\leq\frac{1}{C}||\nabla
E_{\epsilon}(u(\cdot-y))||.$

Combining (4.22) and (4.23) leads to

$||\Phi_{u,y,\epsilon,k}(w)||\leq r$

for every $w\in\overline{B_{T_{u,y,k}^{\bot}}(0,r)}$. Therefore,
$\Phi_{u,y,\epsilon,k}$ is a contractive mapping in
$\overline{B_{T_{u,y,k}^{\bot}}(0,r)}.$ By Banach fixed point theorem, there
exists unique fixed point $w_{\delta,k}(u,y,\epsilon)$ of
$\Phi_{u,y,\epsilon,k}$. Denote $\delta_{r_{{}_{0}}}$ by $\delta^{*}$,
$\epsilon_{r_{{}_{0}}}$ by $\epsilon^{*}$ and $k(\delta,r_{0})$ by
$k(\delta)$. It is easy to verify that the conclusions $\bf(i)-(iv)$ hold for
$w_{\delta,k}(u,y,\epsilon)$.

Now, we prove that
$w_{\delta,k}:\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\rightarrow
Y$ is $C^{1}$. For any
$(u_{0},y_{0})\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$
and $(u,y)$ close to $(u_{0},y_{0})$, both
$S_{u_{0},y_{0},k}|_{T^{\perp}_{u,y,k}}:T^{\perp}_{u,y,k}\to
T^{\perp}_{u_{0},y_{0},k}$ and
$S_{u,y,k}|_{T^{\perp}_{u_{0},y_{0},k}}:T^{\perp}_{u_{0},y_{0},k}\to
T^{\perp}_{u,y,k}$ are isomorphisms, and finding a solution $w\in
T^{\perp}_{u,y,k}$ to the equation $S_{u,y,k}\nabla
E_{\epsilon}(u(\cdot-y)+w)=0$ is equivalent to finding a solution $w\in
T^{\bot}_{u_{0},y_{0},k}$ to the equation $S_{u_{0},y_{0},k}S_{u,y,k}\nabla
E_{\epsilon}(u(\cdot-y)+S_{u,y,k}w)=0$. Note that
$S_{u_{0},y_{0},k}S_{u,y,k}\nabla E_{\epsilon}(u(\cdot-y)+S_{u,y,k}w)$ is
$C^{1}$ near
$(u_{0},y_{0},w_{0})\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\times
T^{\perp}_{u_{0},y_{0},k}$ and the Fréchet partial derivative of
$S_{u_{0},y_{0},k}S_{u,y,k}\nabla E_{\epsilon}(u(\cdot-y)+S_{u,y,k}w)$ at
$(u_{0},y_{0},w_{0})$ with respect to $w$ is $L_{u_{0},y_{0},\epsilon,k}$
which is invertible. Therefore, the implicit functional theorem implies that

$w_{\delta,k}(\cdot,\cdot,\epsilon):\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\rightarrow
Y$

is $C^{1}$.

Finally, we give the proof of $\bf(v).$ Let

$\displaystyle\varphi_{u,y,\epsilon,k}=u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)-P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))),$ (4.29)

where $P_{T_{u,y,k}}:Y\rightarrow T_{u,y,k}$ is orthogonal projection. By the
conclusion $\bf(ii)$ of this Theorem, we get that

$\displaystyle P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))=\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)).$ (4.30)

Thus, by (4.29) and (4.30), $\varphi_{u,y,\epsilon,k}$ satisfies

$\displaystyle-\triangle\varphi_{u,y,\epsilon,k}+\varphi_{u,y,\epsilon,k}+V(\epsilon
x)\varphi_{u,y,\epsilon,k}=f(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)).$ (4.31)

By the definition of $T_{u,y,k}$, we have

$\displaystyle P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))$ (4.32) $\displaystyle=$
$\displaystyle\sum^{N}_{j=1}\Big{\langle}\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}||^{2}}$ $\displaystyle+\sum^{k}_{i=1}\langle\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),\tilde{e}_{i,k}(\cdot-y)\rangle\tilde{e}_{i,k}(\cdot-y)$
$\displaystyle+\sum^{q}_{i=1}\langle\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),e_{i}(\cdot-y)\rangle
e_{i}(\cdot-y).$

Since $\tilde{e}_{i,k}$, $e_{i}$, $u$ and $\frac{\partial u_{i}}{\partial
x_{j}}$ satisfy exponential decay at infinity, by (4.32), for any given $k\geq
k(\delta)$ and $n\geq 0$, there exists $C^{\prime}_{n,k}>0$ such that

$\displaystyle\sup\\{||(1+|x|)^{n}(P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))))||_{L^{\infty}(\mathbb{R}^{N})}$
$\displaystyle\quad\quad|\
u\in\overline{\mathcal{N}_{\delta,k}},y\in\overline{B_{\mathbb{R}^{N}}(0,R)},0\leq\epsilon\leq\epsilon^{*}\\}\leq
C^{\prime}_{k,n}$ (4.33)

and

$\displaystyle\sup_{u\in\overline{\mathcal{N}_{\delta,k}},y\in\overline{B_{\mathbb{R}^{N}}(0,R)}}||(1+|x|)^{n}u(\cdot-y)||_{L^{\infty}(\mathbb{R}^{N})}\leq
C^{\prime}_{k,n}.$ (4.34)

Note that $\varphi_{u,y,\epsilon,k}$ satisfies the elliptic equation (4.31).
Therefore, by the bootstrap argument and the fact that

$\\{w_{\delta,k}(u,y,\epsilon))\ |\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ 0\leq\epsilon\leq\epsilon^{*}\\}$

is compact in $Y$ (because for fixed $k$, $\overline{\mathcal{N}_{\delta,k}}$
is compact), we get that

$\displaystyle\sup\\{||\varphi_{u,y,\epsilon,k}||_{L^{\infty}(\mathbb{R}^{N})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty$ (4.35)

and

$\displaystyle\lim_{\rho\rightarrow\infty}\sup\\{||\varphi_{u,y,\epsilon,k}||_{L^{\infty}(\mathbb{R}^{N}\setminus\overline{B_{\mathbb{R}^{N}}(0,\rho)})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ 0\leq\epsilon\leq\epsilon^{*}\\}=0.$
(4.36)

By (4.35), (4.36) and (4.29), we get that

$\displaystyle\sup\\{||w_{\delta,k}(u,y,\epsilon)||_{L^{\infty}(\mathbb{R}^{N})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$ (4.37)

and

$\displaystyle\lim_{\rho\rightarrow\infty}\sup\\{||w_{\delta,k}(u,y,\epsilon)||_{L^{\infty}(\mathbb{R}^{N}\setminus\overline{B_{\mathbb{R}^{N}}(0,\rho)})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ 0\leq\epsilon\leq\epsilon^{*}\\}$
$\displaystyle=0.$

Let $d(t)=f(t)/t,$ $t\in\mathbb{R}.$ Then by (4.37), (4.34) and the condition
$\bf(F_{1})$, we have

$\displaystyle\sup\\{||d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))||_{L^{\infty}(\mathbb{R}^{N})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ 0\leq\epsilon\leq\epsilon^{*}\\}$
(4.39) $\displaystyle<\infty.$

By the condition $\bf(V_{0})$, the condition $\bf(F_{1})$ and (4), we deduce
that there exists $\rho_{0}$ such that

$\displaystyle\inf\\{1+V(\epsilon x)-d(u(x-y)+w_{\delta,k}(u,y,\epsilon))\ |\
|x|>\rho_{0},\ u\in\overline{\mathcal{N}_{\delta,k}},$
$\displaystyle\quad\quad\ y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}>0.$ (4.40)

Let $\eta$ be a cut-off function which satisfies that $\eta\equiv 1$ in
$B_{\mathbb{R}^{N}}(0,\rho_{0})$ and $\eta\equiv 0$ in
$\mathbb{R}^{N}\setminus\overline{B_{\mathbb{R}^{N}}(0,\rho_{0}+1)}$. We can
rewrite equation (4.31) as

$\displaystyle-\triangle\varphi_{u,y,\epsilon,k}+(1+V(\epsilon
x)-(1-\eta(x))d(u(x-y)+w_{\delta,k}(u,y,\epsilon)))\varphi_{u,y,\epsilon,k}$
$\displaystyle=$ $\displaystyle f_{u,y,\epsilon,k}$

with

$\displaystyle f_{u,y,\epsilon,k}$ $\displaystyle=$ $\displaystyle
d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))\cdot u(\cdot-y)$ (4.42)
$\displaystyle+\eta(x)\cdot d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))\cdot
w_{\delta,k}(u,y,\epsilon)$ $\displaystyle-(1-\eta(x))\cdot
d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$
$\displaystyle\quad\quad\quad\times(u(\cdot-y)-P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))).$

By (4.34), (4), (4.39) and the fact that

$\eta(x)d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))\cdot
w_{\delta,k}(u,y,\epsilon)$

has compact support, we deduce that there exists
$C^{\prime\prime\prime}_{n,k}>0$ such that

$\displaystyle\sup_{u\in\overline{\mathcal{N}_{\delta,k}},y\in\overline{B_{\mathbb{R}^{N}}(0,R)}}||(1+|x|)^{n}f_{u,y,\epsilon,k}||_{L^{\infty}(\mathbb{R}^{N})}\leq
C^{\prime\prime\prime}_{k,n}.$ (4.43)

By (4.43), (4), (4) and [25, Proposition 4.2], we get that there exists
$C^{\prime\prime}_{n,k}>0$ such that

$\displaystyle\sup_{u\in\overline{\mathcal{N}_{\delta,k}},y\in\overline{B_{\mathbb{R}^{N}}(0,R)}}||(1+|x|)^{n}\varphi_{u,y,\epsilon,k}||_{L^{\infty}(\mathbb{R}^{N})}\leq
C^{\prime\prime}_{k,n}.$ (4.44)

Then the conclusion $\bf(v)$ follows from (4.29), (4.44), (4) and (4.34).
$\Box$

By the conclusion $\bf(iii)$ of Theorem 4.2, we get that

$\displaystyle J(u(\cdot-y)+w_{\delta,k}(u,y,0))\equiv I(u+\pi_{k}(u)),\
\forall(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}.$
(4.45)

In what follows, for a $C^{1}$ mapping $f$ defined in
$\mathcal{N}_{\delta,k}\times B_{\mathbb{R}^{N}}(0,R)$, we use the the
notations $Df$, $D_{u}f$ and $D_{y}f$ to denote the derivatives of $f$ with
respect to $(u,y)$ variable, $u$ variable and $y$ variable respectively and
use $Df(u,y)[\bar{u},\bar{y}]$ to denote the derivative of $f$ at the point
$(u,y)$ along the vector $(\bar{u},\bar{y})\in X_{k}\times\mathbb{R}^{N}.$
Furthermore, we use $D_{u}f(u,y)[\bar{u}]$ and $D_{y}f(u,y)[\bar{y}]$ to
denote the Fréchet partial derivatives with respect to the $u$ and $y$
variables along the vectors $\bar{u}$ and $\bar{y}$ respectively.

The condition $\bf(V_{1})$ for the potential $V$ yields

$\displaystyle\lim_{\epsilon\rightarrow 0}\frac{V(\epsilon
x)}{\epsilon^{n^{*}}}=Q_{n^{*}}(x).$ (4.46)

The proof of the following proposition will be given in appendix.

######  Proposition 4.3.

Let $\delta>0$ be sufficiently small and $k\geq k(\delta)$. If $\iota<n^{*},$
then

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{\iota}}\Lambda_{k}(u,y,\epsilon)\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0$

where

$\displaystyle\Lambda_{k}(u,y,\epsilon)$ $\displaystyle=$
$\displaystyle||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||$
$\displaystyle+\sup_{\bar{y}\in\mathbb{R}^{N},|\bar{y}|\leq
1}||Dw_{\delta,k}(u,y,\epsilon)[0,\bar{y}]-D(\pi_{k}(u)(\cdot-y))[0,\bar{y}]||$
$\displaystyle+\sup_{v\in X_{k},||v||\leq
1}||Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]||.$

Moreover, there exists a constant $M>0$ which is independent of $(u,y)$ and
$\epsilon$ such that for every
$(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$
and $0\leq\epsilon\leq\epsilon^{*},$

$\displaystyle\Lambda_{k}(u,y,\epsilon)\leq M\epsilon^{n^{*}}.$

For $0<\delta\leq\delta^{*}$ and $0\leq\epsilon\leq\epsilon^{*},$ denote the
functional

$E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$
(4.47)

by $\Psi_{k}(u,y,\epsilon)$.

######  Theorem 4.4.

Suppose that $0<\delta\leq\delta^{*}$ and $k\geq k(\delta)$. Then there exists
$\epsilon_{k}>0$ such that if $0\leq\epsilon\leq\epsilon_{k}$ and
$(u_{\epsilon},y_{\epsilon})\in\mathcal{N}_{\delta,k}\times
B_{\mathbb{R}^{N}}(0,R)$ is a critical point of the functional
$\Psi_{k}(u,y,\epsilon)$, that is,

$\displaystyle D\Psi_{k}(u_{\epsilon},y_{\epsilon},\epsilon)[v,\bar{y}]=0,\
\forall(v,\bar{y})\in X_{k}\times\mathbb{R}^{N},$ (4.48)

then $u_{\epsilon}(\cdot-
y_{\epsilon})+w_{\delta,k}(u_{\epsilon},y_{\epsilon},\epsilon)$ is a critical
point of $E_{\epsilon}$.

Proof. By the conclusion $\bf(ii)$ of Theorem 4.2 and hypothesis (4.48), we
deduce that to prove $u_{\epsilon}(\cdot-
y_{\epsilon})+w_{\delta,k}(u_{\epsilon},y_{\epsilon},\epsilon)$ is a critical
point of $E_{\epsilon},$ it suffices to prove that for sufficiently small
$\epsilon>0,$

$\displaystyle\\{v(\cdot-
y_{\epsilon})-(\bar{y}\cdot\nabla_{x}u_{\epsilon})(\cdot-
y_{\epsilon})+Dw_{\delta,k}(u_{\epsilon},y_{\epsilon},\epsilon)[v,\bar{y}]\ |\
v\in X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}$
$\displaystyle+T^{\perp}_{u_{\epsilon},y_{\epsilon},k}=Y.$ (4.49)

If (4) were not true, then there exist $\epsilon_{n}\rightarrow 0$ as
$n\rightarrow\infty$ such that $Y_{n}\neq Y,$ where $Y_{n}$ denotes the space
appeared in the left side of (4) with $\epsilon=\epsilon_{n}.$ Passing to a
subsequence, we may assume that $y_{\epsilon_{n}}\rightarrow y_{k}$ and
$u_{\epsilon_{{}_{n}}}\rightarrow u_{k}$ in $Y$ as $n\rightarrow\infty,$ since
$\\{(u_{\epsilon_{n}},y_{\epsilon_{n}})\\}$ is a bounded sequence in the
finite dimensional space $X_{k}\times\mathbb{R}^{N}.$ By the hypothesis (4.48)
and Proposition 4.3, we deduce that $u_{k}$ is a critical point of
$I(v+\pi_{k}(v))$. Then by the conclusion $\bf(iv)$ of Lemma 3.8,
$u_{k}+\pi_{k}(u_{k})$ is a critical point of $I$. We denote it by
$\tilde{u}_{k}$. Since $D\pi_{k}(u_{k})v\in X$ and $\mathcal{T}_{u_{k}}\subset
X^{\bot}$, we get $D\pi_{k}(u_{k})v\bot\mathcal{T}_{u_{k}}$, where
$\mathcal{T}_{u_{k}}$ comes from (3.24). Moreover, by Lemma 3.8, we get that
$D\pi_{k}(u_{k})v\in X^{\bot}_{k}$. Thus,

$D\pi_{k}(u_{k})v\bot X_{k}\oplus\mathcal{T}_{u_{k}}=T_{u_{k},0,k}.$

It follows that the following subspace of $Y:$

$\\{v-\bar{y}\nabla_{x}u_{k}-\bar{y}\nabla_{x}\pi_{k}(u_{k})+D\pi_{k}(u_{k})v\
|\ v\in X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}+T^{\bot}_{u_{k},0,k}$ (4.50)

is equal to

$\displaystyle\\{v-\bar{y}\nabla_{x}u_{k}-\bar{y}\nabla_{x}\pi_{k}(u_{k})\ |\
v\in X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}+T^{\bot}_{u_{k},0,k}$ (4.51)
$\displaystyle=$ $\displaystyle\\{v-\bar{y}\nabla_{x}\tilde{u}_{k}\ |\ v\in
X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}+T^{\bot}_{u_{k},0,k}.$

As it has been mentioned above,
$\tilde{u}_{k}=u_{k}+\pi_{k}(u_{k})\in\mathcal{K}.$ Therefore, by (3.3), we
get that for every $1\leq j\leq N,$

$\displaystyle||\frac{\partial\tilde{u}_{k}}{\partial
x_{j}}-\sum^{s}_{i=1}\xi_{i}(\tilde{u}_{k})\frac{\partial u_{i}}{\partial
x_{j}}||\leq\sum^{s}_{i=1}\xi_{i}(\tilde{u}_{k})||\frac{\partial\tilde{u}_{k}}{\partial
x_{j}}-\frac{\partial u_{i}}{\partial x_{j}}||\leq\varsigma.$ (4.52)

By $\bf(ii)$ of Lemma 3.8 and the fact that every $\xi_{i}$ is a Lipschitz
function, we deduce that for every $1\leq j\leq N,$ as $k\rightarrow\infty$,

$\displaystyle||\sum^{s}_{i=1}\xi_{i}(\tilde{u}_{k})\frac{\partial
u_{i}}{\partial x_{j}}-\sum^{s}_{i=1}\xi_{i}(u_{k})\frac{\partial
u_{i}}{\partial x_{j}}||$ (4.53) $\displaystyle\leq$
$\displaystyle\sum^{s}_{i=1}|\xi_{i}(\tilde{u}_{k})-\xi_{i}(u_{k})|\cdot||\frac{\partial
u_{i}}{\partial x_{j}}||\leq
C\sum^{s}_{i=1}||\tilde{u}_{k}-u_{k}||\cdot||\frac{\partial u_{i}}{\partial
x_{j}}||\rightarrow 0,$

where $C$ is the the Lipschitz constant of $\xi_{i}.$ By (4.52) and (4.53), we
obtain that for every $1\leq j\leq N,$

$\limsup_{k\rightarrow\infty}||\frac{\partial\tilde{u}_{k}}{\partial
x_{j}}-\sum^{s}_{i=1}\xi_{i}(u_{k})\frac{\partial u_{i}}{\partial
x_{j}}||\leq\varsigma.$

It follows that

$\displaystyle\limsup_{k\rightarrow\infty}\sup_{|\bar{y}|\leq
1}||\bar{y}\nabla_{x}\tilde{u}_{k}-\sum^{N}_{j=1}\bar{y}_{j}\sum^{s}_{i=1}\xi_{i}(u_{k})\frac{\partial
u_{i}}{\partial x_{j}}||\leq\varsigma.$

Thus, when $\varsigma$ is sufficiently small and $k$ is sufficiently large,
the space defined by (4.51) is equal to $Y$. As a consequence, when
$\varsigma$ is sufficiently small and $k$ is sufficiently large, the space
defined by (4.50) is also $Y$. Therefore, the space

$\displaystyle\\{v(\cdot-y_{k})-(\bar{y}\nabla_{x}u_{k})(\cdot-
y_{k})-(\bar{y}\nabla_{x}\pi_{k}(u_{k}))(\cdot-
y_{k})+(D\pi_{k}(u_{k})v)(\cdot-y_{k})$ $\displaystyle\quad\quad|\ v\in
X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}+T^{\bot}_{u_{k},y,k}$ (4.54)

is equal to $Y$. Then we can define a bounded linear operator

$\displaystyle H_{n}:Y\rightarrow Y,$ $\displaystyle w=v(\cdot-
y_{k})-(\bar{y}\nabla_{x}u_{k})(\cdot-
y_{k})-(\bar{y}\nabla_{x}\pi_{k}(u_{k}))(\cdot-
y_{k})+(D\pi_{k}(u_{k})v)(\cdot-y_{k})+\phi$ $\displaystyle\mapsto
H_{n}(w)=v(\cdot-
y_{\epsilon_{{}_{n}}})-(\bar{y}\nabla_{x}u_{\epsilon_{{}_{n}}})(\cdot-
y_{\epsilon_{{}_{n}}})+Dw_{\delta,k}(u_{\epsilon_{{}_{n}}},y_{\epsilon_{{}_{n}}},\epsilon_{n})[v,\bar{y}]+\phi,$

where $\phi\in T^{\bot}_{u_{k},y,k}.$ It satisfies $Y_{n}=H_{n}(Y)$, where
$Y_{n}$ denotes the space appeared in the left side of (4) with
$\epsilon=\epsilon_{n}.$ By $u_{\epsilon_{n}}\rightarrow u_{k}$,
$y_{\epsilon_{n}}\rightarrow y_{k}$ and Proposition 4.3, we get that as
$n\rightarrow\infty$,

$||H_{n}-id||_{\mathcal{L}(Y)}\rightarrow 0.$

Therefore, when $n$ is large enough, $H_{n}(Y)=Y$. It follows that $Y_{n}=Y,$
which contradicts the assumption. Thus, when $k(\delta)$ is large enough and
$k\geq k(\delta)$, there exists $\epsilon_{k}>0$ such that if
$0\leq\epsilon\leq\epsilon_{k},$ then (4) holds. $\Box$

##  5  Proof of Theorem 1.3

By the conclusions $\bf(iii)$ and $\bf(v)$ of Theorem 4.2, if
$u\in\overline{\mathcal{N}_{\delta,k}}$, then $\pi_{k}(u)$ decays
exponentially at infinity. Therefore, for
$u\in\overline{\mathcal{N}_{\delta,k}}$ and $y\in\mathbb{R}^{N}$, we can
define

$\Gamma_{k}(u,y)=\int_{\mathbb{R}^{N}}Q_{n^{*}}(x+y)(u+\pi_{k}(u))^{2}dx.$

By the same argument as Lemma 3.2 of [1] and by (4.46), (4.34) and the
Lebesgue Convergence Theorem, we can get the following Lemma:

######  Lemma 5.1.

For any given $k\geq k(\delta)$, as $\epsilon\rightarrow 0,$

$\sup\Big{\\{}\Big{|}\frac{1}{\epsilon^{n^{*}}}\int_{\mathbb{R}^{N}}V(\epsilon(x+y))(u+\pi_{k}(u))^{2}dx-\Gamma_{k}(u,y)\Big{|}\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\Big{\\}}\rightarrow
0$

and

$\displaystyle\sup\Big{\\{}\Big{|}D\Big{(}\frac{1}{\epsilon^{n^{*}}}\int_{\mathbb{R}^{N}}V(\epsilon(x+y))(u+\pi_{k}(u))^{2}dx-\Gamma_{k}(u,y)\Big{)}[v,\bar{y}]\Big{|}\
|\ v\in X_{k},\ ||v||\leq 1,$
$\displaystyle\quad\quad\quad\bar{y}\in\mathbb{R}^{N},\ |\bar{y}|\leq 1,\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\Big{\\}}\rightarrow
0.$

From now on, for the condition $\bf(V_{1})$, we always assume that $\triangle
Q_{n^{*}}\geq 0$ and $\triangle Q_{n^{*}}\not\equiv 0$ in $\mathbb{R}^{N}$,
since the proof for the other case is similar.

######  Lemma 5.2.

If $\delta>0$ is small enough, then for any
$u\in\overline{\mathcal{N}_{\delta,k}}$, $\Gamma_{k}(u,\cdot)$ has a strict
local minimum at $y=0$ and $D^{2}_{y}\Gamma_{k}(u,0)$ is a positive-definite
matrix. More precisely, there exists a constant $A_{k}>0$ such that

$\displaystyle D^{2}_{y}\Gamma_{k}(u,0)y\cdot y\geq A_{k}|y|^{2},\ \forall
u\in\overline{\mathcal{N}_{\delta,k}},\ \forall y\in\mathbb{R}^{N}.$ (5.1)

Proof. By Lemma 4.1 of [1], we know that $y=0$ is a critical point of
$\Gamma_{k}(u,\cdot)$ for every $u\in\overline{\mathcal{N}_{\delta,k}}$. If
(5.1) were not true, then there exist $\delta_{n}>0$,
$u_{n}\subset\overline{\mathcal{N}_{\delta_{n},k}},$ $n=1,2,\cdots$ and
$\\{y_{n}\\}\subset S^{N-1}$ such that $\delta_{n}\rightarrow 0$ as
$n\rightarrow\infty$ and

$\displaystyle\lim_{n\rightarrow\infty}|D^{2}_{y}\Gamma_{k}(u_{n},0)y_{n}\cdot
y_{n}|=0.$ (5.2)

Since $(u_{n},y_{n})$ is bounded in the finite dimensional space
$X_{k}\times\mathbb{R}^{N}$, passing to a subsequence, we may assume that
$u_{n}\rightarrow u_{0}$ in $X_{k}$, and $y_{n}\rightarrow y_{0}\in S^{N-1}$
as $n\rightarrow\infty$. Let $D_{ii}\Gamma_{k}(u_{n},y)$ be the second
derivative of $\Gamma_{k}(u_{n},y)$ with respect to the variable $y_{i}$ and
$\mbox{diag}\\{D_{11}\Gamma_{k}(u_{n},0),\cdots,D_{NN}\Gamma_{k}(u_{n},0)\\}$
be diagonal matrix with diagonal elements $D_{11}\Gamma_{k}(u_{n},0),$
$\cdots,$ $D_{NN}\Gamma_{k}(u_{n},0)$. By the appendix of [1], we get that

$\displaystyle
D_{ii}\Gamma_{k}(u_{n},0)=-\frac{2}{N}\int_{\mathbb{R}^{N}}(u_{n}+\pi_{k}(u_{n}))\nabla
Q_{n^{*}}(x)\cdot\nabla(u_{n}+\pi_{k}(u_{n}))dx,\ 1\leq i\leq N.$ (5.3)

Therefore,

$\displaystyle D^{2}_{y}\Gamma_{k}(u_{n},0)y_{n}\cdot y_{n}$ $\displaystyle=$
$\displaystyle
y^{T}_{n}\cdot\mbox{diag}\\{D_{11}\Gamma_{k}(u_{n},0),\cdots,D_{NN}\Gamma_{k}(u_{n},0)\\}\cdot
y_{n}$ (5.4) $\displaystyle=$
$\displaystyle-\frac{2}{N}|y_{n}|^{2}\int_{\mathbb{R}^{N}}(u_{n}+\pi_{k}(u_{n}))\nabla
Q_{n^{*}}(x)\cdot\nabla(u_{n}+\pi_{k}(u_{n}))dx$ $\displaystyle=$
$\displaystyle-\frac{1}{N}|y_{n}|^{2}\int_{\mathbb{R}^{N}}\nabla
Q_{n^{*}}(x)\cdot\nabla(u_{n}+\pi_{k}(u_{n}))^{2}dx$ $\displaystyle=$
$\displaystyle\frac{1}{N}|y_{n}|^{2}\int_{\mathbb{R}^{N}}\triangle
Q_{n^{*}}(x)\cdot(u_{n}+\pi_{k}(u_{n}))^{2}dx$

By (5.2) and (5.4), we infer that

$\lim_{n\rightarrow\infty}D^{2}_{y}\Gamma_{k}(u_{n},0)y_{n}\cdot
y_{n}=\frac{1}{N}|y_{0}|^{2}\int_{\mathbb{R}^{N}}\triangle
Q_{n^{*}}(x)\cdot(u_{0}+\pi_{k}(u_{0}))^{2}dx=0.$

It is a contradiction, since we have assumed that $\triangle Q_{n^{*}}(x)\geq
0$ and $\triangle Q_{n^{*}}\not\equiv 0$ in $\mathbb{R}^{N}$. $\Box$

In the rest of this section, we assume that $\delta>0$ is sufficiently small
and $k\geq k(\delta)$ is sufficiently large such that (3.57) holds, where the
constant $k(\delta)$ comes from Theorem 4.2.

Proof of Theorem 1.3:

By definition of $\Psi_{k}(u,y,\epsilon)$ (see (4.47)), for
$(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},$

$\displaystyle\Psi_{k}(u,y,\epsilon)$ (5.5) $\displaystyle=$
$\displaystyle\frac{1}{2}||u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)||^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)|^{2}dx$
$\displaystyle-\int_{\mathbb{R}^{N}}F(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))dx$
$\displaystyle=$
$\displaystyle\frac{1}{2}||u(\cdot-y)+w_{\delta,k}(u,y,0)||^{2}+\frac{1}{2}||w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)||^{2}$
$\displaystyle+\langle
u(\cdot-y)+w_{\delta,k}(u,y,0),w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)\rangle$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|u(\cdot-y)+w_{\delta,k}(u,y,0)|^{2}dx$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)|^{2}dx$
$\displaystyle+\int_{\mathbb{R}^{N}}V(\epsilon
x)(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx$
$\displaystyle-\int_{\mathbb{R}^{N}}F(u(\cdot-y)+w_{\delta,k}(u,y,0))dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx$
$\displaystyle-\eta_{1}(u,y,\epsilon),$

where

$\displaystyle\eta_{1}(u,y,\epsilon)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}F(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))dx-\int_{\mathbb{R}^{N}}F(u(\cdot-y)+w_{\delta,k}(u,y,0))dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx.$

By Taylor expansion, we deduce that there exists $0<\theta=\theta(x)<1,$
$\forall x\in\mathbb{R}^{N}$ such that

$\displaystyle\eta_{1}(u,y,\epsilon)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}f^{\prime}(u(\cdot-y)+\theta
w_{\delta,k}(u,y,0)+(1-\theta)w_{\delta,k}(u,y,\epsilon))$
$\displaystyle\quad\quad\quad\times(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))^{2}dx$

By the condition $\bf(F_{1})$, Proposition 4.3 and (5), we deduce that

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{n^{*}}}|\eta_{1}(u,y,\epsilon)|\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0.$
(5.7)

Note that for $v\in X_{k},$ $\bar{y}\in\mathbb{R}^{N}$,

$\displaystyle D\eta_{1}(u,y,\epsilon)[v,\bar{y}]$ (5.8) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$
$\displaystyle\quad\quad\times(v(\cdot-y)-\bar{y}(\nabla_{x}u)(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,\bar{y}])dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))$
$\displaystyle\quad\quad\times(v(\cdot-y)-\bar{y}(\nabla_{x}u)(\cdot-y)+Dw_{\delta,k}(u,y,0)[v,\bar{y}])dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))$
$\displaystyle\quad\quad\quad\quad\times(v(\cdot-y)-\bar{y}(\nabla_{x}u)(\cdot-y)+Dw_{\delta,k}(u,y,0)[v,\bar{y}])dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(Dw_{\delta,k}(u,y,\epsilon)[v,\bar{y}]-Dw_{\delta,k}(u,y,0)[v,\bar{y}])$

Then by the conclusion $\bf(iii)$ of Theorem 4.2, Proposition 4.3 and the
condition $\bf(F_{1})$, we deduce that

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{n^{*}}}||D\eta_{1}(u,y,\epsilon)||\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0.$
(5.9)

Combining (5.7) and (5.9) yields

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{n^{*}}}(|\eta_{1}(u,y,\epsilon)|+||D\eta_{1}(u,y,\epsilon)||)\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0.$
(5.10)

By the conclusion $\bf(ii)$ of Theorem 4.2 and the fact that

$w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)\in T^{\bot}_{u,y,k},$

we get

$\displaystyle\langle
u(\cdot-y)+w_{\delta,k}(u,y,0),w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)\rangle$
(5.11) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx.$

By Proposition 4.3, we deduce that

$\displaystyle\eta_{2}(u,y,\epsilon)$ $\displaystyle:=$
$\displaystyle\frac{1}{2}||w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)||^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)|^{2}dx$
$\displaystyle+\int_{\mathbb{R}^{N}}V(\epsilon
x)(u(\cdot-y)+w_{\delta,k}(u,y,0))(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx$

also satisfies (5.10). By the conclusion $\bf(iii)$ of Theorem 4.2, we infer
that

$\displaystyle
J(u(\cdot-y)+w_{\delta,k}(u,y,0))=J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))=I(u+\pi_{k}(u)).$
(5.12)

Finally, by the conclusions $\bf(iii)$ and $\bf(v)$ of Theorem 4.2 and (4.34),
we have

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|u(\cdot-y)+w_{\delta,k}(u,y,0)|^{2}dx$ (5.13) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)(u(\cdot-y)+\pi_{k}(u)(\cdot-y))^{2}dx$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{n^{*}}\Gamma_{k}(u,y)+\eta_{3}(u,y,\epsilon),$

where

$\displaystyle\Gamma_{k}(u,y)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}Q_{n^{*}}(x)(u(\cdot-y)+\pi_{k}(u)(\cdot-y))^{2}dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}Q_{n^{*}}(x+y)(u+\pi_{k}(u))^{2}dx.$

By Lemma 5.1, the conclusion $\bf(v)$ of Theorem 4.2 and (4.34), we deduce
that $\eta_{3}$ satisfies (5.10). By
$(\ref{vxvvdffrrfw44e})-(\ref{nv888rtr664})$, we get that

$\displaystyle\Psi_{k}(u,y,\epsilon)=I(u+\pi_{k}(u))+\frac{1}{2}\epsilon^{n^{*}}\Gamma_{k}(u,y)+\eta(u,y,\epsilon),$
(5.14)

where $\eta=\eta_{1}+\eta_{2}+\eta_{3}$ satisfies (5.10).

By Lemma 5.2, for every $u\in\overline{\mathcal{N}_{\delta,k}}$,
$\Gamma_{k}(u,y)$ has a strict local minimum at $y=0$ and there is a constant
$A_{k}>0$ such that

$\displaystyle D^{2}_{y}\Gamma_{k}(u,0)\geq A_{k}\mbox{Id}$ (5.15)

where Id denotes the $N\times N$ identity matrix. By (5.15) and (5.14), we
deduce that there exists $\epsilon^{\prime}_{k}>0$ such that if
$0\leq\epsilon\leq\epsilon^{\prime}_{k}$, then for every
$u\in\overline{\mathcal{N}_{\delta,k}}$, there exists $y_{\epsilon}(u)\in
B_{\mathbb{R}^{N}}(0,R/2)$ such that $y_{\epsilon}(u)$ is the unique minimizer
of $\Psi_{k}(u,\cdot,\epsilon)$ in $B_{\mathbb{R}^{N}}(0,R)$. Moreover, by
implicit functional theorem, $y_{\epsilon}(\cdot)\in
C^{1}(\overline{\mathcal{N}_{\delta,k}})$. By (5.14), we get that

$\lim_{\epsilon\rightarrow
0}||\Psi_{k}(u,y_{\epsilon}(u),\epsilon)-I(u+\pi_{k}(u))||_{C^{1}(\overline{\mathcal{N}_{\delta,k}})}=0.$
(5.16)

By [9, Theorem IV.3], a GM pair is a special kind of Conley index pair which
is associated with some pseudo-gradient flow of a functional. Therefore, the
GM pair $(W_{k},W_{k}^{-})$ which was defined in Remark 3.11 is a Conley index
pair associated with some pseudo-gradient flow of the functional
$g_{k}(u)=I(u+\pi_{k}(u)).$ Then by (5.16) and Theorem III.4 of [9], we deduce
that if $\epsilon$ is small enough, then $(W_{k},W_{k}^{-})$ is also a Conley
index pair associated with some pseudo-gradient flow of the functional
$\Psi_{k}(\cdot,y_{\epsilon}(\cdot),\epsilon).$ By (3.57) and Theorem 5.5.18
of [8], we infer that if $\epsilon$ is sufficiently small, then
$\Psi_{k}(\cdot,y_{\epsilon}(\cdot),\epsilon)$ has at least a critical point
$u_{\epsilon}\in\mathcal{N}_{\delta,k}$. Then by Theorem 4.4,
$\tilde{u}_{\epsilon}:=u_{\epsilon}(\cdot-
y_{\epsilon}(u_{\epsilon}))+w_{\delta,k}(u_{\epsilon},y_{\epsilon}(u_{\epsilon}),\epsilon)$
is a critical point of $E_{\epsilon}$. Moreover, by (5.16), we have

$\lim_{\epsilon\rightarrow
0}\mbox{dist}_{{}_{Y}}(\tilde{u}_{\epsilon},\mathcal{K})=0$

with $\mathcal{K}=\mathcal{K}^{b}_{a}$. This finishes the proof of Theorem
1.3. $\Box$

##  6 Appendix A

In this appendix, we shall give the proof of the existence of
$\\{\tilde{e}_{j,k}\\}$ which satisfies the conditions $\bf(i)$ and $\bf(ii)$
in Section 3.

Since $X\cap C^{\infty}_{0}(\mathbb{R}^{N})$ is dense in $X,$ for any
$\mu_{k}>0,$ we can choose $\\{\bar{e}_{j,k}\\}\subset X\cap
C^{\infty}_{0}(\mathbb{R}^{N})$ such that

$\displaystyle\sup_{1\leq j\leq k}||\bar{e}_{j,k}-e^{\prime}_{j}||\leq\mu_{k}\
\mbox{and}\ ||\bar{e}_{j,k}||=1,\ 1\leq j\leq k.$ (6.1)

We show that if $\mu_{k}$ is small enough, then $\\{\bar{e}_{j,k}\ |\ 1\leq
j\leq k\\}\cup\\{e_{j}\ |\ 1\leq j\leq q\\}$ is linearly independent. If it
were not true, without loss of generality, we may assume that

$\displaystyle\bar{e}_{k,k}=\sum^{k-1}_{j=1}\alpha_{j}\bar{e}_{j,k}+\sum^{q}_{j=1}\beta_{j}e_{j},$
(6.2)

then

$\bar{e}_{k,k}=\sum^{k-1}_{j=1}\alpha_{j}e^{\prime}_{j}+\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})+\sum^{q}_{j=1}\beta_{j}e_{j}.$

It follows that if $\mu_{k}<1/4\sqrt{2}$, then

$\displaystyle 1=||\bar{e}_{k,k}||^{2}$ $\displaystyle=$
$\displaystyle\sum^{k-1}_{j=1}\alpha^{2}_{j}+||\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})||^{2}+2\langle\sum^{k-1}_{j=1}\alpha_{j}e^{\prime}_{j},\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})\rangle$
(6.3)
$\displaystyle+\sum^{q}_{j=1}\beta^{2}_{j}+2\langle\sum^{q}_{j=1}\beta_{j}e_{j},\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})\rangle$
$\displaystyle\geq$
$\displaystyle\frac{3}{4}\sum^{k-1}_{j=1}\alpha^{2}_{j}+\frac{3}{4}\sum^{q}_{j=1}\beta^{2}_{j}+||\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})||^{2}-8\sum^{k-1}_{j=1}\alpha^{2}_{j}||\bar{e}_{j,k}-e^{\prime}_{j}||^{2}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\sum^{k-1}_{j=1}\alpha^{2}_{j}+\frac{1}{2}\sum^{q}_{j=1}\beta^{2}_{j}.$

By (6.2),

$e^{\prime}_{k}=\sum^{k-1}_{j=1}\alpha_{j}e^{\prime}_{j}+\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})+\sum^{q}_{j=1}\beta_{j}e_{j}+(e^{\prime}_{k}-\bar{e}_{k,k}),$

combining (6.3), we get that

$\displaystyle 1=||e^{\prime}_{k}||^{2}$ $\displaystyle=$
$\displaystyle\sum^{k-1}_{j=1}\alpha_{j}\langle\bar{e}_{j,k}-e^{\prime}_{j},e^{\prime}_{k}\rangle+\langle
e^{\prime}_{k}-\bar{e}_{k,k},e^{\prime}_{k}\rangle\leq\mu_{k}\sum^{k-1}_{j=1}|\alpha_{j}|+\mu_{k}$
$\displaystyle\leq$ $\displaystyle(\sqrt{2k}+1)\mu_{k}.$

This induces a contradiction if we assume $(\sqrt{2k}+1)\mu_{k}<1$. Thus,
$\\{\bar{e}_{j,k}\ |\ 1\leq j\leq k\\}\cup\\{e_{j}\ |\ 1\leq j\leq k\\}$ is
linearly independent if $\mu_{k}<\min\\{1/(\sqrt{2k}+1),1/4\sqrt{2}\\}$.

By (6.1) and

$\langle\bar{e}_{j,k},\bar{e}_{j^{\prime},k}\rangle=\langle
e^{\prime}_{j}+(\bar{e}_{j,k}-e^{\prime}_{j}),e^{\prime}_{j^{\prime}}+(\bar{e}_{j^{\prime},k}-e^{\prime}_{j^{\prime}})\rangle,\
\langle\bar{e}_{j,k},e_{j^{\prime}}\rangle=\langle
e^{\prime}_{j}+(\bar{e}_{j,k}-e^{\prime}_{j}),e_{j^{\prime}}\rangle,$

we get that

$\displaystyle\sup_{1\leq j,j^{\prime}\leq k,j\neq
j^{\prime}}|\langle\bar{e}_{j,k},\bar{e}_{j^{\prime},k}\rangle|\leq
2\mu_{k}+\mu^{2}_{k},\ \sup_{j\neq
j^{\prime}}|\langle\bar{e}_{j,k},e_{j^{\prime}}\rangle|\leq\mu_{k}.$ (6.4)

Therefore, if $\mu_{k}$ is sufficiently small, using Gram-Schmidt
orthogonalizing process to $\\{e_{j}\ |\ 1\leq j\leq q\\}\cup\\{\bar{e}_{j,k}\
|\ 1\leq j\leq k\\}$, we get $\\{\tilde{e}_{j,k}\ |\ 1\leq j\leq k\\}$ which
satisfies the conditions $\bf(i)$ and $\bf(ii)$ in Section 3.

##  7 Appendix B

In this appendix, we give the proof of Proposition 4.3.

Let

$\displaystyle\eta_{u,y,k}=(-\triangle+1)^{-1}f(u(\cdot-y)+\pi_{k}(u)(\cdot-y)).$

Then

$\displaystyle\eta_{u,y,k}$ $\displaystyle=$
$\displaystyle(-\triangle+1+V(\epsilon
x))^{-1}f(u(\cdot-y)+\pi_{k}(u)(\cdot-y)))$ (7.1)
$\displaystyle+(-\triangle+1+V(\epsilon x))^{-1}V(\epsilon x)\eta_{u,y,k}.$

Subtracting equation

$S_{u,y,k}\nabla E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))=0$

from equation

$S_{u,y,k}\nabla J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))=0,$

by (7.1) and the mean value theorem, we get that

$\displaystyle
L_{u,y,\epsilon,k}\Big{(}w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)\Big{)}$
(7.2) $\displaystyle=$ $\displaystyle-S_{u,y,k}(-\triangle+1+V(\epsilon
x))^{-1}V(\epsilon x)\eta_{u,y,k}$
$\displaystyle+S_{u,y,k}(-\triangle+1+V(\epsilon
x))^{-1}\Big{(}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))$
$\displaystyle\quad\quad\quad\times(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))\Big{)}$

where $\tilde{w}$ lies between $w_{\delta,k}(u,y,\epsilon)$ and
$\pi_{k}(u)(\cdot-y)$. By the conclusion $\bf(iv)$ of Theorem 4.2, we get that
$||w_{\delta,k}(u,y,\epsilon)||\leq r$ if $0<\delta\leq\delta_{r}$ and $k\geq
k(\delta)$. And by $\bf(ii)$ of Lemma 3.8, we deduce that if $k(\delta)$ is
large enough and $k\geq k(\delta),$ then $||\pi_{k}(u)(\cdot-y)||\leq r.$
Therefore, $||\tilde{w}||\leq r$ if $0<\delta\leq\delta_{r}$ and $k\geq
k(\delta)$. Moreover, by (4.19), we deduce that if $r$ is small enough,
$0<\delta\leq\delta_{r}$ and $k\geq k(\delta)$, then

$\displaystyle\Big{|}\Big{|}(-\triangle+1+V(\epsilon
x))^{-1}\Big{(}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y))\cdot(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))\Big{)}\Big{|}\Big{|}$
$\displaystyle\leq\frac{C}{2}||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||,$
(7.3)

where $C$ is the constant appeared in Lemma 4.1. By (7), (7.2) and Lemma 4.1,
we get that

$\displaystyle C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||\leq
2||(-\triangle+1)^{-1}V(\epsilon x)\eta_{u,y,k}||.$ (7.4)

By (4.34), the conclusion $\bf(v)$ of Theorem 4.2 and [25, Proposition 4.2],
we get that for any $n>0$,

$\displaystyle\sup\\{||(1+|x|)^{n}\eta_{u,y,k}||_{L^{\infty}(\mathbb{R}^{N})}\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}<\infty.$
(7.5)

By (7.5), using the same argument as Lemma 3.2 of [1], we can get that if
$\iota<n^{*},$

$\displaystyle\lim_{\epsilon\rightarrow
0}\\{\int_{\mathbb{R}^{N}}\frac{V^{2}(\epsilon
x)}{\epsilon^{2\iota}}\eta^{2}_{u,y,k}\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0$
(7.6)

and

$\sup\\{\int_{\mathbb{R}^{N}}\frac{V^{2}(\epsilon
x)}{\epsilon^{2n^{*}}}\eta^{2}_{u,y,k}\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$

Thus, for $\iota<n^{*},$

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{\iota}}||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0$
(7.7)

and

$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$ (7.8)

Combining (7.4), (7.7) and (7.8) yields that for $\iota<n^{*},$ if $\delta>0$
is small enough and $k\geq k(\delta),$ then

$\displaystyle\lim_{\epsilon\rightarrow
0}\\{\frac{1}{\epsilon^{\iota}}||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0$
(7.9)

and

$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}$ (7.10) $\displaystyle<\infty.$

Recall that $S_{u,y,k}:Y\rightarrow T^{\bot}_{u,y,k}$ is orthogonal
projection. Therefore, for $h\in Y,$

$\displaystyle S_{u,y,k}h$ $\displaystyle=$ $\displaystyle
h-\sum^{q}_{j=1}\langle h,e_{j}(\cdot-y)\rangle
e_{j}(\cdot-y)-\sum^{k}_{j=1}\langle
h,\tilde{e}_{j,k}(\cdot-y)\rangle\tilde{e}_{j,k}(\cdot-y)$
$\displaystyle-\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}.$

Thus, the Fréchet partial derivative of $S_{u,y,k}h$ with respect to $u$ along
the vector $v\in X_{k}$ is

$\displaystyle D_{u}(S_{u,y,k}h)[v]$ (7.11) $\displaystyle=$
$\displaystyle-\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}D\xi_{i}(u)[v]\cdot\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}$
$\displaystyle-\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}(D\xi_{i}(u)[v])\cdot\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}$
$\displaystyle+2\sum^{N}_{j=1}\Big{(}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\langle\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}},\sum^{s}_{i=1}(D\xi_{i}(u)[v])\frac{\partial
u_{i}}{\partial x_{j}}\rangle}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{4}}$
$\displaystyle\quad\quad\quad\times\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)\Big{)}$

and the Fréchet partial derivative of $S_{u,y,k}h$ with respect to $y$ along
the vector $\bar{y}\in\mathbb{R}^{N}$ is

$\displaystyle D_{y}(S_{u,y,k}h)[\bar{y}]$ (7.12) $\displaystyle=$
$\displaystyle\sum^{q}_{j=1}\langle h,(\bar{y}\nabla_{x}e_{j})(\cdot-y)\rangle
e_{j}(\cdot-y)+\sum^{k}_{j=1}\langle
h,(\bar{y}\nabla_{x}\tilde{e}_{j,k})(\cdot-y)\rangle\tilde{e}_{j,k}(\cdot-y)$
$\displaystyle+\sum^{q}_{j=1}\langle
h,e_{j}(\cdot-y)\rangle(\bar{y}\nabla_{x}e_{j})(\cdot-y)+\sum^{k}_{j=1}\langle
h,\tilde{e}_{j,k}(\cdot-y)\rangle(\bar{y}\nabla_{x}\tilde{e}_{j,k})(\cdot-y)$
$\displaystyle+\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\cdot(\bar{y}\nabla_{x}(\frac{\partial
u_{i}}{\partial
x_{j}}))(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}$
$\displaystyle+\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\cdot(\bar{y}\nabla_{x}(\frac{\partial
u_{i}}{\partial x_{j}}))(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}.$

Differentiating equations $S_{u,y,k}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))=0$ and
$S_{u,y,k}(\nabla J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))=0$ with respect to the
variable $u$ along the vector $v\in X_{k}$, we get that

$\displaystyle
S_{u,y,k}(\nabla^{2}E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0]))$
$\displaystyle+D_{u}(S_{u,y,k}h_{1})[v]=0$ (7.13)

and

$\displaystyle
S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(v(\cdot-y)+D\pi_{k}(u)(\cdot-y)[v,0]))$
$\displaystyle+D_{u}(S_{u,y,k}h_{2})[v]=0,$ (7.14)

where $h_{1}=\nabla E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$ and
$h_{2}=\nabla J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))$. By (7.1) and (7), it is easy
to verify that there exists a constant $C>0$ such that

$\displaystyle||h_{1}-h_{2}||\leq
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||.$ (7.15)

By (7.15) and (7.11), we get that for $||v||\leq 1,$ there exists a constant
$C>0$ such that

$\displaystyle||D_{u}(S_{u,y,k}h_{2})[v]-D_{u}(S_{u,y,k}h_{1})[v]||$ (7.16)
$\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||.$

A direct computation shows that

$\displaystyle
S_{u,y,k}(\nabla^{2}E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0]))$
(7.17) $\displaystyle-
S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(v(\cdot-y)+D\pi_{k}(u)(\cdot-y)[v,0]))$
$\displaystyle=$ $\displaystyle
S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))$
$\displaystyle-
S_{u,y,k}(-\triangle+1)^{-1}\Big{\\{}\Big{(}f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$
$\displaystyle-f^{\prime}(u(\cdot-y)+\pi_{k}(u)(\cdot-y))\Big{)}\times(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0])\Big{\\}}$
$\displaystyle+S_{u,y,k}(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)$

where

$\bar{\eta}_{u,y,\epsilon,k}(v)=(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))\cdot(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0])).$

By (4.34), the conclusion $\bf(v)$ of Theorem 4.2 and (1.2) in $\bf(F_{1})$,
we get that for any $v,h\in Y,$ $||v||=||h||=1,$

$\displaystyle\int_{\mathbb{R}^{N}}\Big{|}f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))-f^{\prime}(u(\cdot-y)+\pi_{k}(u)(\cdot-y))\Big{|}$
$\displaystyle\quad\quad\quad\times|v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0]|\cdot|h|dx$
$\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||.$

It follows that

$\displaystyle\Big{|}\Big{|}(-\triangle+1)^{-1}\Big{\\{}\Big{(}f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$
(7.18)
$\displaystyle\quad\quad-f^{\prime}(u(\cdot-y)+\pi_{k}(u)(\cdot-y))\Big{)}\times(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0])\Big{\\}}\Big{|}\Big{|}$
$\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||.$

By (7), (7) and $(\ref{gdttter00oqppp})-(\ref{gdhf66ey161994})$, we deduce
that

$\displaystyle||S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))||$
(7.19) $\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||$ $\displaystyle+C||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||.$

By the conclusion $\bf(ii)$ of Lemma 3.8 and (4.19), we deduce that

$\displaystyle\lim_{k\rightarrow\infty}\sup\Big{\\{}||\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))-\nabla^{2}J(u(\cdot-y))||_{\mathcal{L}(Y)}$
$\displaystyle\quad\quad\quad\quad\quad|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\Big{\\}}=0.$

Therefore, as $k\rightarrow\infty$,

$\displaystyle||S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))$
$\displaystyle-
S_{u,y,k}(\nabla^{2}J(u(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))||$
$\displaystyle=o(1)||Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]||.$
(7.20)

By (7.19) and (7), we get that as $k\rightarrow\infty$,

$\displaystyle||S_{u,y,k}(\nabla^{2}J(u(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))||$
(7.21) $\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||$ $\displaystyle+C||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||$
$\displaystyle+o(1)||Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]||.$

Let $\mathcal{T}_{u}(\cdot-y)=\\{h(\cdot-y)\ |\ h\in\mathcal{T}_{u}\\}$ and
$\mathcal{T}^{\bot}_{u}(\cdot-y)$ be the orthogonal complement space in $Y$,
where $\mathcal{T}_{u}$ is defined in (3.24). Let
$P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}:Y\rightarrow\mathcal{T}^{\bot}_{u}(\cdot-y)$
and $P_{\mathcal{T}_{u}(\cdot-y)}:Y\rightarrow\mathcal{T}_{u}(\cdot-y)$ be
orthogonal projections. Since $Dw_{\delta,k}(u,y,\epsilon)[v,0]\bot
X_{k}(\cdot-y)$ and $D(\pi_{k}(u)(\cdot-y))[v,0]\bot X_{k}(\cdot-y),$ where
$X_{k}(\cdot-y)=\\{v(\cdot-y)\ |\ v\in X_{k}\\}$, we deduce that

$P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0])\in
T^{\bot}_{u,y,k}.$

Therefore, by Lemma 4.1, we have

$\displaystyle||S_{u,y,k}(\nabla^{2}J(u(\cdot-y))P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))||$
(7.22) $\displaystyle=$
$\displaystyle||L_{u,y,0,k}P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D\pi_{k}(u)(\cdot-y))[v,0])||$
$\displaystyle\geq$ $\displaystyle
C||P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]||.$

Differentiating the following equation with respect to variable $u$ along the
vector $v,$

$\Big{\langle}w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y),\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}\Big{\rangle}=0$

we get that

$\displaystyle\Big{\langle}D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0],\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}\Big{\rangle}$ $\displaystyle=$
$\displaystyle-\Big{\langle}w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y),\sum^{s}_{i=1}(D\xi_{i}(u)[v])\frac{u_{i}(\cdot-y)}{\partial
x_{j}}\Big{\rangle}.$

It follows that there exists a constant $C>0$ such that

$\displaystyle||P_{\mathcal{T}_{u}(\cdot-y)}(D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0])||$
(7.23) $\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||.$

By $(\ref{vvcnbd7duqeeadyydtdtdt})-(\ref{gfbbv8uufy5534})$, we deduce that
when $k$ is large enough, then there exists a constant $C>0$ such that

$\displaystyle||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0]||$
$\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,\epsilon,k}||.$ $\displaystyle+C||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||.$

Then by $(\ref{baobaoainiiii})-(\ref{ncbcjdgftr745545})$ and the fact that for
$\iota<m,$

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\Big{\\{}\frac{1}{\epsilon^{\iota}}||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||$ $\displaystyle\quad\quad\quad\quad\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
v\in X_{k},\ ||v||\leq 1\Big{\\}}=0$

and

$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},$
$\displaystyle\quad\quad\ v\in X_{k},\ ||v||\leq 1,\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty,$

we get that for $\iota<n^{*},$

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\Big{\\{}\frac{1}{\epsilon^{\iota}}||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0]||$
$\displaystyle\quad\quad\quad\quad|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
v\in X_{k},\ ||v||\leq 1\Big{\\}}=0$ (7.24)

and

$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0]||\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},$
$\displaystyle\quad\quad\quad v\in X_{k},\ ||v||\leq 1,\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$ (7.25)

Differentiating the two equations $S_{u,y,k}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))=0$ and
$S_{u,y,k}(\nabla J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))=0$ with respect to the
variable $y$ along the vector $\bar{y}\in\mathbb{R}^{N}$, we get that

$\displaystyle
S_{u,y,k}(\nabla^{2}E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))(-\bar{y}\nabla_{x}u(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[0,\bar{y}]))$
$\displaystyle+D_{y}(S_{u,y,k}h_{1})[\bar{y}]=0$

and

$\displaystyle
S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(-\bar{y}\nabla_{x}u(\cdot-y)+D(\pi_{k}(u)(\cdot-y))[0,\bar{y}]))$
$\displaystyle+D_{y}(S_{u,y,k}h_{2})[\bar{y}]=0.$

The same arguments as (7) and (7) yield that for $\iota<n^{*},$

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\Big{\\{}\frac{1}{\epsilon^{\iota}}||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[0,\bar{y}]||$
$\displaystyle\quad\quad\quad\quad\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
\bar{y}\in\mathbb{R}^{N},\ |\bar{y}|\leq 1\Big{\\}}=0$

and

$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[0,\bar{y}]||\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},$
$\displaystyle\quad\quad\quad\bar{y}\in\mathbb{R}^{N},\ |\bar{y}|\leq 1,\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$

Acknowledgements The author would like to thank the referee for her or his
comments and suggestions on the manuscript. This work was supported by NSFC
(10901112) and BNSF (1102013).

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