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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 131, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/131\hfil Multiple solutions]
{Multiple solutions for nonlinear elliptic equations on
 Riemannian manifolds}

\author[W. Chen, J. Yang \hfil EJDE-2009/131\hfilneg]
{Wenjing Chen, Jianfu Yang}  % in alphabetical order

\address{Wenjing Chen \newline
Department of Mathematics, Jiangxi Normal University, Nanchang,
Jiangxi, 330022, China}
\email{wjchen1102@yahoo.com.cn}

\address{Jianfu Yang \newline
Department of Mathematics, Jiangxi Normal University, Nanchang,
Jiangxi, 330022,  China}
\email{jfyang\_2000@yahoo.com}

\thanks{Submitted September 15, 2009. Published October 9, 2009.}
\subjclass[2000]{35J20, 35J61, 58J05} 
\keywords{Multiple solutions; Semilinear elliptic equation;
\hfill\break\indent Riemannian manifold; Ljusternik-Schnirelmann category}

\begin{abstract}
 Let $(\mathcal{M}, g)$ be a compact, connected, orientable,
 Riemannian $n$-manifold of class $C^{\infty}$
 with Riemannian metric $g$ $(n\geq 3)$. We study the
 existence of solutions to the  equation
 \[
 -\varepsilon^2\Delta_{g} u+V(x)u=K(x)|u|^{p-2}u
 \]
 on this Riemannian manifold. Here $2<p<2^{*}=2n/(n-2)$, $V(x)$
 and $K(x)$ are continuous functions. We show that the shape
 of $V(x)$ and $K(x)$ affects the  number of solutions,
 and then prove the existence of multiple solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

In this article, we consider the existence of solutions of the
 problem
\begin{equation}\label{eq:1.1}
-\varepsilon^2\Delta_{g} u+V(x)u=K(x)|u|^{p-2}u \quad \text{in }
\mathcal {M},
\end{equation}
where $(\mathcal {M}, g)$ is a compact, connected, orientable,
Riemannian manifold of class $C^{\infty}$ with Riemannian metric
$g$, $\dim\mathcal {M}=n\geq 3$, $2<p<2^{*}=\frac{2n}{n-2}$ and
$\Delta_{g}$ is the Laplace-Beltrami operator.

In the whole space $\mathbb{R}^n$, problem \eqref{eq:1.1} is the
so-called Schr\"{o}dinger equation. The existence of solutions of
Schr\"{o}dinger problem \eqref{eq:1.1} has been extensively
investigated, mainly in the semiclassical limit $\varepsilon\to 0$,
see for instance \cite{abc}, \cite{ams}, \cite{cl1}, \cite{cl2},
\cite{df}, \cite{R}, \cite{w}, \cite{wz}. In particular, it was
found in \cite{R} a mountain pass solution of problem \eqref{eq:1.1}
in the case $K(x)=1$. Later on, it was shown in \cite{w} that the
maximum point of the mountain pass solution concentrates at the
minimum point of $V$ as $\varepsilon\to 0$. In the case
$K(x)\not=$const., Wang and Zeng found in \cite{wz} a ground state
solution for $\varepsilon$ small. Furthermore, they studied the
concentration behavior of such a solution as $\varepsilon\to 0$. In
\cite{cl2}, it was shown that the number of solutions of problem
\eqref{eq:1.1} is affected by the shape of functions $V$ and $K$. In
fact, in \cite{cl2} the number of solutions of problem
\eqref{eq:1.1} was related to the topology of the set of global
minimum points of certain function.  On the other hand, for a
bounded domain $\Omega$ in $\mathbb{R}^N$ with rich topology, Benci
and Cerami proved that problem \eqref{eq:1.1} with $V=K=1$ has at
least $cat\,\,\Omega$ positive solutions. Such a result was recently
generalized to compact manifolds. In \cite{bbm}, the authors showed
that problem \eqref{eq:1.1} with $V=K=1$ and positive mass possesses
at least $\mathop{\rm cat}(\mathcal{M})+1$ solutions, while for the zero mass
case, similar results were obtained in \cite{v}. Inspired by
\cite{bbm}, \cite{cl2} and \cite{v}, we consider in this paper the
effect of coefficients $V, K$ on the existence of number of
solutions.


Problem \eqref{eq:1.1} is related to the problem
\begin{equation}\label{eq:1.4}
-\Delta u+V(\eta)u=K(\eta)|u|^{p-2}u \quad \text{in } \mathbb{R}^n
\end{equation}
for fixed $\eta\in \mathcal {M}$. It is well known that the problem
\begin{equation}\label{eq:1.3}
-\Delta u+u=|u|^{p-2}u \quad {\rm in}\ \ \mathbb{R}^n\ \ u>0,
\end{equation}
has a positive radial solution $U$; see for
instance \cite{bl1}. The function $U$ and its radial
derivatives satisfy the following decaying law
$$
U(r)\sim e^{-|r|}|r|^{-\frac{n-1}{2}}, \quad
\lim_{r\to \infty}\frac{U'(r)}{U(r)}=1, \quad r=|x|.
$$
By a result in \cite{k}, $U$ is the unique positive solution of
problem (\ref{eq:1.3}). We may verify that
$w(z):=\big(\frac{V(\eta)}{K(\eta)}\big)^{1/(p-2)}
U\Big(\big(V(\eta)\big)^{1/2}z\Big)$
with $K(\eta)>0$ is a ground state solution of problem
\eqref{eq:1.4}; that is, it is the minimizer of the variational
problem
$$
c_{\eta}:=\inf_{u\in N_{\eta}}E_{\eta}(u),
$$
where
$$
E_{\eta}(u)=\frac{1}{2}\int_{\mathbb{R}^n}(|\nabla u|^2+V(\eta)u^2)\,
dz-\frac{1}{p}\int_{\mathbb{R}^n}K(\eta)|u|^p\,dz
$$
is the associated energy functional of problem \eqref{eq:1.4} and
$$
N_{\eta}:=\big\{u\in H^1(\mathbb{R}^n)\backslash \{0\}:
\int_{\mathbb{R}^n}(|\nabla u|^2+V(\eta)u^2)\,dz=\int_{\mathbb{R}^n}K(\eta)|u|^p\,dz\big\}
$$
is the related Nehari manifold. In fact,
\[
  c_{\eta} = E_{\eta}(w)=\big(\frac{1}{2}-\frac{1}{p}\big)
\frac{V^{\frac{p}{p-2}-\frac{n}{2}}(\eta)}{K^{\frac{2}{p-2}}(\eta)}
\int_{\mathbb{R}^n}|U(z)|^p\,dz.
\]
Let
$$
c_0=\inf_{\eta\in \mathcal {M}}c_\eta \quad {\rm and}\quad
\Omega:=\{\eta\in \mathcal {M}:  c_{\eta}=c_0\}.
$$
For $\delta>0$ let
$$
\Omega_{\delta}:=\{\xi\in \mathcal {M}:  \inf_{\eta\in
\Omega}\|\xi-\eta\|_{g}\leq \delta\}.
$$
We assume in this paper that $V, K\in C(\mathcal{M},\mathbb{R})$
and there is a positive number $\nu>0$ such that $V, K\geq \nu>0$.
Denote by $\mathop{\rm cat}_{X}(A)$ the Ljusternik-Schirelmann category of $A$ in
$X$. Let
$$
K_{\rm max}= \max_{x\in\mathcal{M}} K(x), \quad
K_{\rm min}= \min_{x\in\mathcal{M}} K(x).
$$
Our main result is the following.

\begin{theorem}\label{th.1.1}
Problem \eqref{eq:1.1} has at least
$\mathop{\rm cat}_{\Omega_{\delta}}(\Omega)$
positive solutions for $\varepsilon>0$ small.
\end{theorem}


Solutions of problem \eqref{eq:1.1} will be found as critical points
of the associated functional
\[
I_{\varepsilon}(u)=\frac{1}{\varepsilon^n}
\Big(\frac{1}{2}\int_{\mathcal
{M}}\big(\varepsilon^2|\nabla_{g} u(x)|^2+V(x)u^2\big)\,
d\mu_{g}-\frac{1 }{p}\int_{\mathcal {M}}K(x)|u^{+}|^p\,
d\mu_{g}\Big),
\]
in the Hilbert space
$$
H_{g}^1(\mathcal {M}):=\big\{u: \mathcal
{M}\to \mathbb{R}: \int_{\mathcal {M}}(|\nabla_{g}u|^2
+u^2)\,d\mu_{g}<\infty\big\}
$$
with the norm
$$
\|u\|_{g}=\Big(\int_{\mathcal {M}}(|\nabla_{g}u|^2+u^2)\,d\mu_{g}\Big)^{1/2},
$$
where $d\mu_{g}=\sqrt{\det g}dz$ denotes the volume form on
$\mathcal {M}$ associated with the metric $g$. For $\sigma>0$, let
$$
\Sigma_{\varepsilon, \sigma}:=\{u\in \mathcal {N}_{\varepsilon}
: I_{\varepsilon}(u)< c_0+\sigma\}
$$
be a subset of the Nehari manifold
\[
    \mathcal {N}_{\varepsilon}:=\big\{u\in H_{g}^1(\mathcal {M})
\backslash \{0\}:\int_{\mathcal{M}}(\varepsilon^2|
\nabla_{g} u(x)|^2+V(x)u^2)\, d\mu_{g}
=\int_{\mathcal {M}}K(x)|u^{+}|^p\,d\mu_{g}\big\}
\]
related to the functional $I_{\varepsilon}$. To prove Theorem
\ref{th.1.1}, we first show that problem \eqref{eq:1.1} has at least
$\mathop{\rm cat}_{\Sigma_{\varepsilon, \sigma}}\Sigma_{\varepsilon, \sigma}$
solutions, then we need to relate $\mathop{\rm cat}_{\Sigma_{\varepsilon,
\sigma}}\Sigma_{\varepsilon, \sigma}$ with
$\mathop{\rm cat}_{\Omega_{\delta}}\Omega$. By a result in \cite{h}, we know that
$\mathcal {M}$ can be isometrically embedded in a Euclidean space
$\mathbb{R}^{N}$ as a regular sub-manifold with $N>2n$. For any set
$\omega\subset\mathcal {M}$ and $r>0$, we define
$$
[\omega]_r:=\{z\in \mathbb{R}^N :\mathop{\rm dist}(z, \omega)\leq r\}
$$
a subset of $\mathbb{R}^N$, where $\mathop{\rm dist}(z, \omega)$
denotes the distance between $z$ and $\omega$ with respect to
the Euclidian metric in $\mathbb{R}^N$. Let $r = r(\Omega_{\delta})$
be the radius of topological invariance of $\Omega_{\delta}$,
which is defined by
$$
r(\Omega_{\delta}):=\sup\{l>0 :
\mathop{\rm cat}([\Omega_{\delta}]_l)
=\mathop{\rm cat}(\Omega_{\delta})\}.
$$
We choose $r>0$ so small that the metric projection
\[
  \Pi : [\Omega_{\delta}]_{r}\subset   \mathbb{R}^N\to \Omega_{\delta}
\]
is well defined. We will construct a function $\phi_{\varepsilon}:
\Omega \to \Sigma_{\varepsilon, \sigma}$ and a function
$\beta: \Sigma_{\varepsilon, \sigma}\to
[\Omega_{\delta}]_{r}$ such that
\[
   \Omega\xrightarrow[]{\phi_{\varepsilon}}\Sigma_{\varepsilon, \sigma}
   \xrightarrow[]{\beta}[\Omega_{\delta}]_r\xrightarrow[]{\Pi}
   \Omega_{\delta},
\]
and $\Pi \circ \beta \circ \phi_{\varepsilon}$ is homotopic to the
identity on $\Omega_{\delta}$. It implies that
$\mathop{\rm cat}_{\Sigma_{\varepsilon, \sigma}}\Sigma_{\varepsilon, \sigma}\geq
\mathop{\rm cat}_{\Omega_\delta}\Omega$.

In section 2, we outline our frame of work. The mappings
$\phi_{\varepsilon}$ and $\beta$ are constructed in section 3 and
section 4 respectively.


\section{The framework and preliminary results}

Let $\mathcal {M}$ be a compact Riemannian manifolds  of class
$C^{\infty}$.  On the tangent bundle of $\mathcal {M}$ we define the
exponential map $\exp: T\mathcal{M}\to \mathcal {M}$ which
has the following properties: (i) $\exp$ is of class $C^{\infty}$;
(ii) there exists a constant $R>0$ such that $\exp_x\big|_{B(0,R)}:
B(0,R)\to B_{g}(x,R)$ is a diffeomorphism for all
$x\in \mathcal {M}$. Fix such an $R$ in this paper and denote by $B(0,R)$
the ball in $\mathbb{R}^{n}$ centered at 0 with radius $R$ and
$B_{g}(x,R)$ the ball in $\mathcal {M}$ centered at $x$ with radius
$R$ with respect to the distance induced by the metric $g$. Let
$\mathcal {C}$ be the atlas on $\mathcal {M}$ whose charts are given
by the exponential map and $\mathcal {P}=\{\psi_{C}\}_{C\in \mathcal
{C} }$ be a partition of unity subordinate to the atlas $\mathcal
{C}$. For $u\in H_{g}^1(\mathcal {M})$, we have
\[
\int_{\mathcal {M}}|\nabla_{g}u|^2\,d\mu_{g}=\sum_{C\in
\mathcal {C}}\int_{C}\psi_{C}(x)|\nabla_{g}u|^2\,d\mu_{g}.
\]
Moreover, if $u$ has support inside one chart $C=B_{g}(\eta,R)$,
then
\begin{align*}
&\int_{\mathcal {M}}|\nabla_{g}u|^2\,d\mu_{g}\\
&=\int_{B(0,
R)}\psi_{C}(\exp_{x_0}(z))g_{x_0}^{ij}(z)\frac{\partial
u(\exp_{x_0}(z))}{\partial z_{i}}\frac{\partial
u(\exp_{x_0}(z))}{\partial z_{j}} |g_{x_0}(z)|^{1/2}\,dz,
\end{align*}
where $g_{x_0}$ denotes the Riemannian metric reading in $B(0,R)$
through the normal coordinates defined by the exponential map
$\exp_{x_0}$. In particular, $g_{x_0}(0)=Id$. We let
$|g_{x_0}(z)|:=det(g_{x_0}(z))$ and $(g^{ij}_{x_0})(z)$ is the
inverse matrix of $g_{x_0}(z)$. Since $\mathcal {M}$ is compact,
there are two strictly positive constants $h$ and $H$ such that
$$
\forall x\in \mathcal {M}, \quad \forall \upsilon\in T_{x}\mathcal
{M}, \quad h\|\upsilon\|^2\leq g_{x}(\upsilon, \upsilon)\leq
H\|\upsilon\|^2.
$$
Hence, we have
$$
\forall x\in \mathcal {M}, \quad h^n\leq |g_{x}|\leq H^n.
$$
Theorem \ref{th.1.1} will follow from the following result in
\cite{mw}.

\begin{proposition}\label{Prop.2.1}
Let $\mathcal {N}$ be  a $C^{1,1}$ complete Riemannian manifold
modeled on a Hilbert space and J be a $C^1$ functional on $\mathcal
{N}$ bounded from below. If there exists $b>\inf_{\mathcal
{N}}J$ such that $J$ satisfies the Palais-Smale condition on the
sublevel $J^{-1}(-\infty, b)$, then for any noncritical level a,
with $a<b$, there exist at least
$\mathop{\rm cat}_{J^a}(J^a)$ critical points of
$J$ in $J^a$, where $J^a:=\{u\in \mathcal {N}| J(u)\leq a\}$.
\end{proposition}

We need also the following Lemma.

\begin{lemma}\label{le:2.1}
Let $X$ and $Y$ be topological spaces, $Z\subset Y$ be a closed set
and $h_1\in C(Z,X)$, $h_2\in C(X,Y)$ with $h_2$ being a closed
mapping. Suppose that $h_2\circ h_1 : Z\to Y$ is homotopic
to the identity mapping $Id$ in $Y$, then $\mathop{\rm cat}_{X}(X)\geq
\mathop{\rm cat}_{Y}(Z)$.
\end{lemma}

\begin{proof}
Let $k=\mathop{\rm cat}_{X}(X)$, there exist closed sets $V_1, V_2, \cdots, V_k$
such that $X=\bigcup_{1\leq i\leq k}V_i$ and each $V_i$ is
contractible in $X$. Since $h_2\in C(X,Y)$ and $h_2$ being a closed
mapping, each $h_2(V_i)$ is closed and contractible in $Y$, then
\begin{equation} \label{eq:2.1}
   \mathop{\rm cat}\nolimits_{X}(X)\geq \mathop{\rm cat}\nolimits_{Y}(h_2(X)).
\end{equation}
Since $h_2\circ h_1(Z)\subset h_2(X)$, we have
\begin{equation} \label{eq:2.2}
   \mathop{\rm cat}\nolimits_{Y}(h_2(X))
\geq \mathop{\rm cat}\nolimits_{Y}(h_2\circ h_1(Z)).
\end{equation}
On the other hand, $h_2\circ h_1 : Z\to Y$ is homotopic to
the identity mapping $Id$ in $Y$, thus
\begin{equation}  \label{eq:2.3}
   \mathop{\rm cat}\nolimits_{Y}(h_2\circ h_1(Z))
\geq \mathop{\rm cat}\nolimits_{Y}(Z).
\end{equation}
By (\ref{eq:2.1})-(\ref{eq:2.3}),
$ \mathop{\rm cat}_{X}(X)\geq \mathop{\rm cat}_{Y}(Z)$.
\end{proof}


\section{The function $\phi_{\varepsilon}$}

We know that $\mathcal {N}_{\varepsilon}$ is a $C^{1,1}$ manifold.
If $u\in \mathcal {N}_{\varepsilon}$, we have $\|u\|_{g}\geq C>0$,
$C$ is independent of $u$. For $u\in H_{g}^1(\mathcal {M})$, there
exists a unique $t_{\varepsilon}(u)>0$, $t_{\varepsilon}:
H_{g}^1(\mathcal {M})\backslash\{0\}\to \mathbb{R}^{+}$,
such that $t_{\varepsilon}(u)u\in \mathcal {N}_{\varepsilon}$ and
$$
I_{\varepsilon}(t_{\varepsilon}(u)u)=\max_{t\geq0}I_{\varepsilon}(tu).
$$
More precisely,
\begin{equation}  \label{eq:3.1}
t_{\varepsilon}^{p-2}(u)=\frac{\int_{\mathcal
{M}}\left(\varepsilon^2|\nabla_{g} u(x)|^2+V(x)u^2\right)\,d\mu_{g}}{\int_{\mathcal {M}}K(x)|u^{+}|^p\,d\mu_{g}}.
\end{equation}
The function $t_{\varepsilon}(u)$ is $C^1$. Let us define a smooth
real function $\chi_{R}$ on $\mathbb{R}^{+}$ such that
\begin{equation}  \label{eq:3.2}
  \chi_{R}(t):=\begin{cases}
1  & \text{if }  0\leq t\leq \frac{R}{2};\\
0  & \text{if }  t\geq R\,.
\end{cases}
\end{equation}
and $|\chi'_{R}(t)|\leq \frac{2}{R}$. Fixing $\eta\in\Omega$ and
$\varepsilon>0$, we define
\begin{equation}  \label{eq:3.3}
  W_{\eta, \varepsilon}(x):= \begin{cases}
w_{\varepsilon}(\exp_{\eta}^{-1}(x))\chi_{R}(|\exp_{\eta}^{-1}(x)|)
   & \text{if } x\in B_{g}(\eta, R);\\
0  & \text{otherwise},
  \end{cases}
\end{equation}
where $w(z)$ is the ground state solution of problem \eqref{eq:1.4}
and $w_{\varepsilon}(z)=w(\frac{z}{\varepsilon})$. We define
$\phi_{\varepsilon}: \Omega\to \mathcal {N}_{\varepsilon}$
by
\begin{equation}\label{eq:3.4}
    \phi_{\varepsilon}(\eta) = t_{\varepsilon}(W_{\eta, \varepsilon}(x))W_{\eta,
    \varepsilon}(x).
\end{equation}

\begin{lemma}\label{le:3.2}
With the above notation, we have
\begin{gather} \label{eq:3.5}
  \frac{1}{\varepsilon^{n}}\int_{\mathcal {M}}\varepsilon^2|\nabla_{g}W_{\eta, \varepsilon}(x)|^2\,d\mu_{g} \to
  \int_{\mathbb{R}^n}|\nabla w|^2 dz \quad  \text{as } \varepsilon\to 0.
\\  \label{eq:3.6}
  \frac{1}{\varepsilon^{n}}\int_{\mathcal {M}}V(x)|W_{\eta, \varepsilon}(x)|^2\,d\mu_{g} \to
  \int_{\mathbb{R}^n}V(\eta)w^2(z) dz \quad \text{as } \varepsilon\to 0,
\\  \label{eq:3.7}
  \frac{1}{\varepsilon^{n}}\int_{\mathcal {M}}K(x)|W_{\eta, \varepsilon}(x)|^p\ \mu_{g} \to
  \int_{\mathbb{R}^n}K(\eta)w^p(z) dz \quad \text{as } \varepsilon\to 0.
\end{gather}
\end{lemma}

\begin{proof} We have
\begin{align*}
&\Big|\frac{1}{\varepsilon^{n}}\int_{\mathcal {M}}\varepsilon^2|\nabla_{g}W_{\eta, \varepsilon}(x)|^2\,d\mu_{g}
  -\int_{\mathbb{R}^n}|\nabla w|^2 dz\Big|\\
&= \Big|\frac{1}{\varepsilon^{n}}\int_{B_g(\eta, R)}\varepsilon^2
 \big|\nabla_{g}\left(w_{\varepsilon}(\exp_{\eta}^{-1}(x))
  \chi_{R}(|\exp_{\eta}^{-1}(x)|)\right)\big|^2\,d\mu_{g}
  -\int_{\mathbb{R}^n}|\nabla w|^2 dz\Big|\\
&= \Big|\frac{1}{\varepsilon^{n}}\int_{B(0, R)}\varepsilon^2
 \big|\nabla\left(w_{\varepsilon}(z)
  \chi_{R}(|z|)\right)\big|_{g}^2|g_{\eta}(z)|^{1/2}\,dz
  -\int_{\mathbb{R}^n}|\nabla w|^2 dz\Big|\\
&= \Big|\int_{B(0, \frac{R}{\varepsilon})}\left|\nabla\left(w(z)
  \chi_{\frac{R}{\varepsilon}}(|z|)\right)\right|_{g}^2\left|g_{\eta}(\varepsilon z)\right|^{1/2}\,dz
  -\int_{\mathbb{R}^n}|\nabla w|^2 dz\Big|\\
&\leq \int_{\mathbb{R}^n}\Big|\sum_{i,j=1}^{n}
  \frac{\partial w(z)}{\partial z_i}\frac{\partial w(z)}{\partial
  z_j}\left|\chi^2_{\frac{R}{\varepsilon}}
  (|z|)g^{ij}_{\eta}(\varepsilon z)|g_{\eta}(\varepsilon z)|^{1/2}
 -\delta_{ij}\right|\Big|\,  dz\\
&\quad +\int_{\mathbb{R}^n}\Big|\sum_{i,j=1}^{n}g^{ij}_{\eta}
 (\varepsilon z)  \chi_{\frac{R}{\varepsilon}}(|z|)w(z)
 \left(\frac{\partial w}{\partial z_i}
  \frac{\partial \chi_{\frac{R}{\varepsilon}}(|z|)}{\partial z_j}
 +\frac{\partial w}{\partial z_j}
  \frac{\partial \chi_{\frac{R}{\varepsilon}}(|z|)}{\partial z_i}
 \right)\Big||g_{\eta}(\varepsilon   z)|^{1/2}\,dz\\
&\quad +\int_{\mathbb{R}^n}\Big|\sum_{i,j=1}^{n}g^{ij}_{\eta}(\varepsilon z)
  w^2(z)\frac{\partial \chi_{\frac{R}{\varepsilon}}(|z|)}{\partial z_i}
  \frac{\partial \chi_{\frac{R}{\varepsilon}}(|z|)}{\partial z_j}\Big|
 |g_{\eta}(\varepsilon  z)|^{1/2}\,dz:=I_1+I_2+I_3.
\end{align*}
By the compactness of the manifold $\mathcal {M}$ and regularity of
the exponential map of the Riemannian metric $g$, we have
$$
\lim_{\varepsilon\to
0}\big|\chi^2_{\frac{R}{\varepsilon}}
  (|z|)g^{ij}_{\eta}(\varepsilon z)|g_{\eta}(\varepsilon
  z)|^{1/2}-\delta_{ij}\big|=0
$$
uniformly with respect to $\eta\in \Omega$, so $I_1\to 0$ as
$\varepsilon\to 0$. By the definition of $\chi_{R}(t)$,
\begin{align*}
  I_2
&\leq \frac{H^{n/2}}{h}\int_{\mathbb{R}^n}
 \Big|\sum_{i,j=1}^{n}
  w(z)\Big(\frac{\partial w}{\partial z_i}
  \frac{\partial \chi_{\frac{R}{\varepsilon}}(|z|)}{\partial z_j}+\frac{\partial w}{\partial z_j}
  \frac{\partial \chi_{\frac{R}{\varepsilon}}(|z|)}{\partial z_i}\Big)
 \Big|\,dz\\
&\leq \frac{4H^{n/2}\varepsilon}{Rh}\int_{\mathbb{R}^n}\left|w(z)
 \right|\left|\nabla w(z)\right|\,  dz\\
&= \frac{4H^{n/2}\varepsilon}{Rh}\left(\frac{V(\eta)}{K(\eta)}\right)
^{2/(p-2)}V(\eta)^{-n/2}\int_{\mathbb{R}^n}\left|U
  (z)\right|\left|\nabla U(z)\right|\,dz\\
&\leq \frac{2H^{n/2}\varepsilon}{Rh}\frac{V^{\frac{2}{p-2}
-\frac{n}{2}}(\eta)}{K^{\frac{2}{p-2}}(\eta)}
\int_{\mathbb{R}^n}(|\nabla U(z)|^2 +|U   (z)|^2)\,dz.
\end{align*}
Similarly,
\[
  I_3
  \leq\frac{H^{n/2}}{h}\frac{4\varepsilon^2}{R^2}
 \frac{V^{\frac{2}{p-2}-\frac{n}{2}}(\eta)}{K^{\frac{2}{p-2}}(\eta)}
  \int_{\mathbb{R}^n}U(z)^2\,dz.
\]
Hence, $I_2+I_3\to 0$ uniformly with respect to $\eta\in
\Omega$ as $\varepsilon\to 0$ and (\ref{eq:3.5}) follows.

Next, we prove (\ref{eq:3.6}). We have
\begin{align*}
 &\Big|\frac{1}{\varepsilon^{n}}\int_{\mathcal {M}}V(x)|W_{\eta, \varepsilon}(x)|^2\,d\mu_{g}-\int_{\mathbb{R}^n}V(\eta)w^2(z) dz\Big|\\
 &= \Big|\frac{1}{\varepsilon^{n}}\int_{B_g(\eta, R)}V(x)
 |w_{\varepsilon}(\exp_{\eta}^{-1}(x))\chi_{R}(|\exp_{\eta}^{-1}(x)|)|^2\,d\mu_{g}-\int_{\mathbb{R}^n}V(\eta)w^2(z) dz\Big|\\
 &= \Big|\frac{1}{\varepsilon^{n}}\int_{B(0, R)}V(\exp_{\eta}(z))|w_{\varepsilon}(z)\chi_{R}(|z|)|^2|g_{\eta}(z)
 |^{1/2}\,dz-\int_{\mathbb{R}^n}V(\eta)w^2(z) dz\Big|\\
 &= \Big|\int_{B(0, \frac{R}{\varepsilon})}V(\exp_{\eta}(\varepsilon z))|w(z)\chi_{R}(|\varepsilon z|)|^2
 |g_{\eta}(\varepsilon z)|^{1/2}\,dz-\int_{\mathbb{R}^n}V(\eta)w^2(z)\, dz
\Big|\\
&\leq \Big|\int_{\mathbb{R}^n}\left[V(\exp_{\eta}(\varepsilon z))|\chi_{R}(|\varepsilon z|)|^2
 |g_{\eta}(\varepsilon z)|^{1/2}-V(\eta)\right]w^2(z) dz\Big|\\
&\quad +\Big|\int_{\mathbb{R}^n\backslash B(0, \frac{R}{\varepsilon})}
\left[V(\exp_{\eta}(\varepsilon z))|\chi_{R}(|\varepsilon z|)|^2
 |g_{\eta}(\varepsilon z)|^{1/2}-V(\eta)\right]
 w^2(z) dz\Big|\\
 &:= I_4+I_5.
\end{align*}
We note that $\exp_{\eta}(\varepsilon z)\to \eta$ and
$g_{\eta}(\varepsilon z)\to \delta_{ij}$ as $\varepsilon
\to 0$, by the continuity of $V$,  $I_4\to 0$.
Obviously, $I_5\to 0$. So (\ref{eq:3.6}) holds. (\ref{eq:3.7}) can
be proved in the same way.
\end{proof}

\begin{proposition}\label{prop:3.1}
For $\varepsilon>0$, the map $\phi_{\varepsilon} : \Omega\to
\mathcal {N}_{\varepsilon}$ is continuous; and for any $\sigma>0$,
there exists $\varepsilon_0>0$ such that if
$\varepsilon<\varepsilon_0$ $\phi_{\varepsilon}(\eta)\in
\Sigma_{\varepsilon, \sigma}$ for all $\eta\in \Omega$.
\end{proposition}

\begin{proof}
The continuity of $\phi_{\varepsilon}$ can be proved as
\cite[Proposition 4.2]{bbm}, so we omit the details.
Now, we show $\phi_{\varepsilon}(\eta)\in \Sigma_{\varepsilon,
\sigma}$ for $\forall \eta\in \Omega$. By Lemma \ref{le:3.2},
\begin{align*}
t_{\varepsilon}^{p-2}(W_{\eta, \varepsilon}(x))
 &=  \frac{\frac{1}{\varepsilon^n}\int_{\mathcal
{M}}\varepsilon^2|\nabla_{g} W_{\eta, \varepsilon}(x)(x)|^2d
\mu_{g}+\frac{1}{\varepsilon^n}\int_{\mathcal {M}}V(x)\left(W_{\eta,
\varepsilon}(x)\right)^2\,d
\mu_{g}}{\frac{1}{\varepsilon^n}\int_{\mathcal {M}}K(x)|W^{+}_{\eta,
\varepsilon}(x)|^p\,d\mu_{g}}\\
& \to  \frac{\int_{\mathbb{R}^n}|\nabla w(z)|^2\,dz+\int_{\mathbb{R}^n}V(\eta)w^2(z)\,dz}{\int_{\mathbb{R}^n}K(\eta)w^p(z)\,dz}=1.
\end{align*}
Consequently,
\begin{align*}
 I_{\varepsilon}(\phi_{\varepsilon}(\eta))
&=  I_{\varepsilon}(t_{\varepsilon}(W_{\eta, \varepsilon}(x))W_{\eta,
    \varepsilon}(x))\\
&= \frac{1}{2}\int_{\mathbb{R}^n}(|\nabla w(z)|^2+V(\eta)w^2(z))\,
dz-\frac{1}{p}\int_{\mathbb{R}^n}K(\eta)w^p(z)\,dz+o(1)\\
&= c_\eta+o(1)=c_0+o(1)
\end{align*}
uniformly with respect to $\eta\in \Omega$ and the proof is
completed.
\end{proof}

\section{The function $\beta$}

Let us define the center of mass $\beta(u)\in \mathbb{R}^N$ for
$u\in \mathcal {N}_{\varepsilon}$ by
$$
\beta(u):=\frac{\int_{\mathcal {M}}x|u^{+}(x)|^p\,d\mu_{g}}{\int_{\mathcal {M}}|u^{+}(x)|^p\,d\mu_{g}}.
$$
The function $\beta$ is well defined on $u\in \mathcal
{N}_{\varepsilon}$ since $u^{+}\not\equiv 0$ if $u\in \mathcal
{N}_{\varepsilon}$. Let
\begin{equation}  \label{eq:4.a0}
m_{\varepsilon}:=\inf_{u\in \mathcal{N}_{\varepsilon}}I_{\varepsilon}(u),
\end{equation}
which is achieved as $\mathcal {M}$ is compact. Since $K(x), V(x)$
are bounded, we may show the following result as in
\cite[Lemma 5.1]{bbm}.

\begin{lemma}\label{le:4.1}
There exists a number $\alpha>0$ such that for any $\varepsilon>0$,
$m_{\varepsilon}\geq\alpha$.
\end{lemma}

For a given $\varepsilon>0$, let $\mathcal
{P}_{\varepsilon}=\{P_j^{\varepsilon}\}_{j\in
\Lambda_{\varepsilon}}$ be a finite good partition of the manifold
$\mathcal {M}$ introduced in \cite{bbm}: if for any $j\in
\Lambda_{\varepsilon}$ the set partition $P_j^{\varepsilon}$ is
closed; $P_j^{\varepsilon}\cap P_i^{\varepsilon}\subseteq \partial
P_j^{\varepsilon}\cap \partial P_i^{\varepsilon}$ for any $i\neq j$;
there exist $r_1(\varepsilon)\geq r_2(\varepsilon)>0$ such that
there are points $q_j^{\varepsilon}\in P_j^{\varepsilon}$ for any
$j$, satisfying $B_g(q_j^{\varepsilon}, \varepsilon)\subset
P_j^{\varepsilon}\subset B_g(q_j^{\varepsilon},
r_2(\varepsilon))\subset B_g(q_j^{\varepsilon}, r_1(\varepsilon))$
and any point $x\in \mathcal {M}$ is contained in at most
$N_{\mathcal {M}}$ balls $B_g(q_j^{\varepsilon}, r_1(\varepsilon))$,
where $N_{\mathcal {M}}$ does not depend on $\varepsilon$. This last
condition can be satisfied for $\varepsilon$ small enough by the
compactness of $\mathcal {M}$, and $r_1(\varepsilon)$,
$r_2(\varepsilon)$ can be chosen so that $r_1(\varepsilon)\geq
r_2(\varepsilon)\geq (1+\frac{1}{\Theta})\varepsilon$ with a
constant $\Theta$ independent on $\varepsilon$. We may assume that
the value $\varepsilon_0$ of Proposition \ref{prop:3.1} is small
enough for the manifold $\mathcal {M} $ to have good partitions.

\begin{lemma}\label{le:4.2}
There exists a constant $\gamma>0$ such that for any fixed
$\sigma>0$, $\varepsilon\in (0, \varepsilon_0)$ and function $u\in
\Sigma_{\varepsilon, \sigma}$, there exists a set
$\tilde{P}_{\sigma}^{\varepsilon}\in \mathcal {P}_{\varepsilon}$
such that
$$
\frac{1}{\varepsilon^n}\int_{\tilde{P}_{\sigma}^{\varepsilon}}K(x)|u^{+}|^p\,d\mu_g\geq\gamma.
$$
\end{lemma}

\begin{proof}
Fixed $\sigma>0$ and $0<\varepsilon< \varepsilon_0$. Then for any
$u\in \mathcal {N}_{\varepsilon}$ and any good partition $\mathcal
{P}_{\varepsilon}=\{P_j^{\varepsilon}\}_{j\in
\Lambda_{\varepsilon}}$, let $u_j^{+}=u^{+}$ on the set
$P_j^{\varepsilon}$. Then
\begin{equation}  \label{eq:4.a}
\begin{aligned}
&\frac{1}{\varepsilon^n}\int_{\mathcal
{M}}(\varepsilon^2|\nabla_{g} u(x)|^2+V(x)u^2)\,d\mu_{g}\\
&=\frac{1}{\varepsilon^n}\int_{\mathcal {M}}K(x)|u^{+}|^p\,d\mu_{g}\\
&= \frac{1}{\varepsilon^n}\sum_{j\in
\Lambda_{\varepsilon}}\int_{P_j^{\varepsilon}}K(x)|u^{+}|^p\,d\mu_{g}\\
&\leq \max_j\Big(\frac{1}{\varepsilon^n}\int_{P_j^{\varepsilon}}K(x)
 |u_j^{+}|^p\,d\mu_{g}\Big)^{\frac{p-2}{p}}\sum_{j\in
\Lambda_{\varepsilon}}\Big(\frac{1}{\varepsilon^n}
 \int_{P_j^{\varepsilon}}K(x)|u_j^{+}|^p\,d\mu_{g}\Big)^{2/p}.
\end{aligned}
\end{equation}
Let
\[
  \chi_{\varepsilon}(t):=  \begin{cases}
1  & \text{if } t\leq r_2(\varepsilon);\\
0  & \text{if } t> r_1(\varepsilon)
\end{cases}
\]
be a smooth cutoff function, where $r_1(\varepsilon),
r_2(\varepsilon)$ are defined above for good partitions, and assume
that $|\chi'_{\varepsilon}|\leq \frac{\Theta}{\varepsilon}$
uniformly. Let
$$
\tilde{u}_j(x)=u^{+}(x)\chi_{\varepsilon}(|x-q_j^{\varepsilon}|).
$$
We know that $\tilde{u}_j(x)\in H_g^1(\mathcal {M})$, and
$supt(\tilde{u}_j(x))=B_g(q_j^{\varepsilon}, r_1(\varepsilon))$. By
the definition of $u_j^{+}$, we have $u_j^{+}=u^{+}$ on the set
$P_j^{\varepsilon}\subset B_g(q_j^{\varepsilon},
r_2(\varepsilon))\subset B_g(q_j^{\varepsilon}, r_1(\varepsilon))$.
By the Sobolev inequality there exists a positive constant $C$ such
that for any $j$,
\begin{equation}  \label{eq:4.b}
\begin{aligned}
&\Big(\frac{1}{\varepsilon^n}\int_{P_j^{\varepsilon}}K(x)|u_j^{+}
|^p\,d\mu_g\Big)^{2/p}\\
&=\Big(\frac{1}{\varepsilon^n}\int_{P_j^{\varepsilon}}K(x)|u^{+}|^p\,d\mu_g\Big)^{2/p} \\
&\leq  \Big(\frac{1}{\varepsilon^n}\int_{B_g(q_j^{\varepsilon}, r_2(\varepsilon))}K(x)|u^{+}\chi_{\varepsilon}(|x-q_j^{\varepsilon}|)|^p\,d\mu_g\Big)^{2/p} \\
&\leq \Big(\frac{1}{\varepsilon^n}\int_{B_g(q_j^{\varepsilon},
 r_1(\varepsilon))}K(x)|u^{+}\chi_{\varepsilon}(|x-q_j^{\varepsilon}|)|^p
\,  d\mu_g\Big)^{2/p}\\
&=\Big(\frac{1}{\varepsilon^n}\int_{\mathcal {M}}K(x)|\tilde{u}_j|^p\,d\mu_g\Big)^{2/p}\\
&\leq K_{\rm max}^{2/p}\Big(\frac{1}{\varepsilon^n}
 \int_{\mathcal {M}}|\tilde{u}_j|^p\,  d\mu_g\Big)^{2/p}\\
&\leq K_{\rm max}^{2/p}C\frac{1}{\varepsilon^n}\int_{\mathcal {M}}
\left(\varepsilon^2|\nabla_g  \tilde{u}_j|^2+|\tilde{u}_j|^2\right)\,
d\mu_g\\
&= K_{\rm max}^{2/p}C\frac{1}{\varepsilon^n}\int_{P_j^{\varepsilon}}
 \left(\varepsilon^2|\nabla_g \tilde{u}_j|^2+|\tilde{u}_j|^2\right)d\mu_g\\
&\quad+ K_{\rm max}^{2/p}C\frac{1}{\varepsilon^n}\int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash P_j^{\varepsilon}}
  \left(\varepsilon^2|\nabla_g
  \tilde{u}_j|^2+|\tilde{u}_j|^2\right)d\mu_g\\
&\leq K_{\rm max}^{2/p}C\frac{1}{\varepsilon^n}\int_{\mathcal {M}}
 \left(\varepsilon^2|\nabla_g u_j^{+}|^2+|u_j^{+}|^2\right)d\mu_g\\
&\quad  +K_{\rm max}^{2/p}C\frac{1}{\varepsilon^n}
 \int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash
 P_j^{\varepsilon}}
  \left(\varepsilon^2|\nabla_g \tilde{u}_j|^2+|\tilde{u}_j|^2\right)d\mu_g.
\end{aligned}
\end{equation}
Moveover
\begin{equation}  \label{eq:4.c}
\int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash
P_j^{\varepsilon}} |\tilde{u}_j|^2d\mu_g
\leq\int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash
P_j^{\varepsilon}} |u^{+}|^2d\mu_g,
\end{equation}
and
\begin{equation}  \label{eq:4.d}
\begin{aligned}
&\int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash
 P_j^{\varepsilon}}\varepsilon^2|\nabla_g  \tilde{u}_j|^2d\mu_g\\
&=\int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash P_j^{\varepsilon}}\varepsilon^2\left|\nabla_g
  \left(u^{+}(x)\chi_{\varepsilon}(|x-q_j^{\varepsilon}|)\right)\right|^2d\mu_g\\
&\leq 2\int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash P_j^{\varepsilon}}\varepsilon^2\left(|\nabla_g
  u^{+}|^2\chi^2_{\varepsilon}(|x-q_j^{\varepsilon}|)+\left(\chi'_{\varepsilon}(|x-q_j^{\varepsilon}|)\right)^2
  |u^{+}|^2\right)d \mu_g\\
&\leq 2\int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash P_j^{\varepsilon}}\left(\varepsilon^2|\nabla_g
  u^{+}|^2+\Theta^2|u^{+}|^2\right)d \mu_g.
\end{aligned}
\end{equation}
Substituting (\ref{eq:4.c}) and (\ref{eq:4.d}) into (\ref{eq:4.b}),
we get
\begin{align*}
 \Big((\frac{1}{\varepsilon^n}\int_{P_j^{\varepsilon}}K(x)|u_j^{+}|^p\,
d\mu_g\Big)^{2/p}
&\leq   K_{\rm max}^{2/p}C\frac{1}{\varepsilon^n}\int_{\mathcal {M}}
 \left(\varepsilon^2|\nabla_g  u_j^{+}|^2+|u_j^{+}|^2\right)d\mu_g\\
&\quad +K_{\rm max}^{2/p}CC'\frac{1}{\varepsilon^n}
 \int_{\mathcal {M}}\left(\varepsilon^2|\nabla_g
  u^{+}|^2+|u^{+}|^2\right)d\mu_g,
\end{align*}
where $C'=\max\{2, 2\Theta^2+1\}$. Hence,
\begin{equation}  \label{eq:4.e}
\begin{aligned}
&\sum_{j\in \Lambda_{\varepsilon}}
\Big(\frac{1}{\varepsilon^n}\int_{P_j^{\varepsilon}}K(x)|u_j^{+}|^p
 \,d\mu_g\Big)^{2/p}\\
&\leq  K_{\rm max}^{2/p}C\sum_{j\in \Lambda_{\varepsilon}}\frac{1}{\varepsilon^n}\int_{\mathcal {M}}\left(\varepsilon^2|\nabla_g
 u_j^{+}|^2+|u_j^{+}|^2\right)d\mu_g \\
&\quad +K_{\rm max}^{2/p}CC'N_{\mathcal {M}}\frac{1}{\varepsilon^n}\int_{\mathcal {M}}\left(\varepsilon^2|\nabla_g
  u^{+}|^2+|u^{+}|^2\right)d\mu_g\\
&\leq K_{\rm max}^{2/p}C(C'+1)N_{\mathcal {M}}\frac{1}{\varepsilon^n}\int_{\mathcal {M}}\left(\varepsilon^2|\nabla_g
  u^{+}|^2+|u^{+}|^2\right)d\mu_g\\
&\leq K_{\rm max}^{2/p}C(C'+1)N_{\mathcal {M}}\max\left\{1,\frac{1}{\nu}\right\}\frac{1}{\varepsilon^n}\int_{\mathcal {M}}
  \left(\varepsilon^2|\nabla_gu|^2+V(x)|u|^2\right)d\mu_g
\end{aligned}
\end{equation}
 From (\ref{eq:4.a}) and (\ref{eq:4.e}) we have
\begin{align*}
\max_{j}\Big\{\Big(\frac{1}{\varepsilon^n}\int_{P_j^{\varepsilon}}K(x)|u^{+}|^p\,d\mu_{g}\Big)^{\frac{p-2}{p}}\Big\}
&\geq \frac{\frac{1}{\varepsilon^n}\int_{\mathcal
{M}}(\varepsilon^2|\nabla_{g} u(x)|^2+V(x)u^2)\,d\mu_{g}}{\sum_{j\in \Lambda_{\varepsilon}}
\Big(\frac{1}{\varepsilon^n}\int_{P_j^{\varepsilon}}K(x)|u_j^{+}|^p\,d\mu_g\Big)^{2/p}}\\
&\geq \frac{1}{K^{2/p}_{\rm max}C(C'+1)N_{\mathcal
{M}}\max\{1,\frac{1}{\nu}\}}.
\end{align*}
Thus, the proof is completed.
\end{proof}

\begin{lemma}\label{le:4.3}
Let $\sigma$ and $\varepsilon$ be fixed, and
$I_{\varepsilon}^{m_{\varepsilon}+2\sigma}:=\{u\in \mathcal
{N}_{\varepsilon}| I_{\varepsilon}(u)< m_{\varepsilon}+2\sigma\}$,
where $ m_{\varepsilon}$ is defined in (\ref{eq:4.a0}). For any
$u\in \Sigma_{\varepsilon, \sigma}\cap
I_{\varepsilon}^{m_{\varepsilon}+2\sigma}$ there exists
$u_{\sigma}\in \mathcal {N}_{\varepsilon}$ such that
\begin{equation}  \label{eq:4.8}
I_{\varepsilon}(u_{\sigma})< I_{\varepsilon}(u),  \quad
\||u_{\sigma}-u|\|_{\varepsilon}<4\sqrt{\sigma},
\end{equation}
where $\||u|\|_{\varepsilon}^2=\frac{1}{\varepsilon^n}\int_{\mathcal
{M}}(\varepsilon^2|\nabla_g u|^2+u^2)\,d\mu_g$, and
\begin{equation}  \label{eq:4.9}
\big|\nabla|_{\mathcal
{N}_{\varepsilon}}I_{\varepsilon}(u_{\sigma})\big|
<\sqrt{\sigma}\||\xi|\|_{\varepsilon}.
\end{equation}
\end{lemma}

The above result follows by the Ekeland principle, also by the proof
in \cite[Lemma 5.4]{bbm}.

Let $u_k\in  \Sigma_{\varepsilon_k, \sigma_k}\cap
I_{\varepsilon_k}^{m_{\varepsilon_k}+2\sigma_k}$, where
$\varepsilon_k, \sigma_k\to 0$ as $k\to\infty$. For all $k$, the map
$\exp_{\eta_k}: T_{\eta_k}\mathcal {M}\to\mathcal {M}$ is a
diffeomorphism on the ball $B_g(\eta_k, R)$. Let $\{\psi_c\}$ be a
partition of unity induced on $\mathcal {M}$ by the cover of balls
of radius $R$. By the compactness of $\mathcal {M}$, we can assume
that there exists $\rho>0$ such that for all $k$
\begin{equation}  \label{eq:4.5}
\min\big\{\psi_{B_g(\eta_k,R)}(x)| x\in B_g(\eta_k,
\frac{R}{\rho})\big\}\geq \psi_0>0.
\end{equation}
Let
$$
\varphi_k:  B_g\big(\eta_k, \frac{R}{\rho}\big)\to
B\big(0, \frac{R}{\varepsilon_k\rho}\big)\subset \mathbb{R}^n, \quad
\varphi_k:=\frac{\exp_{\eta_k}^{-1}}{\varepsilon_k}
$$
and define $w_k: \mathbb{R}^n \to \mathbb{R}$ by
$$
w_k(z):=\chi_k(z)u_k(\varphi_k^{-1}(z))=\chi_{R}\left(\varepsilon_k|z|\rho\right)
u_k(\exp_{\eta_k}(\varepsilon_kz))
=\chi_{\frac{R}{\rho}}(|\exp_{\eta_k}^{-1}(x)|)u_k(x),
$$
where $x=\exp_{\eta_k}(\varepsilon_kz)\in \Omega$ and
$\chi_k(z):=\chi_{\frac{R}{\varepsilon_k\rho}}(|z|)$. Then, $w_k\in
H_0^1\left(B\left(0,\frac{R}{\varepsilon_k\rho}\right)\right)\subset
H^1(\mathbb{R}^n)$.

\begin{lemma}\label{le:4.5}
There exists $\tilde{w}\in H^1(\mathbb{R}^n)$ such that, up to a
subsequence, $w_k$ tends to $\tilde{w}$ weakly in
$H^1(\mathbb{R}^n)$ and strongly in $L_{loc}^p(\mathbb{R}^n)$. The
limit function $\tilde{w}$ is a ground state solution of the problem
\begin{equation}\label{eq:A}
-\Delta u+V(\eta)u=K(\eta)|u|^{p-2}u, \quad \text{on }\mathbb{R}^n.
\end{equation}
\end{lemma}

\begin{proof} We first show that $w_k$ is bounded in $H^1(\mathbb{R}^n)$.
There holds
$$
I_{\varepsilon_k}(u_k)=\big(\frac{1}{2}-\frac{1}{p}\big)\frac{1}{\varepsilon_k^n}
\int_{\mathcal {M}}\left(\varepsilon^2|u_k|^2+ V(x)u_k^2\right)\,d\mu_g<c_0+\sigma_k,
$$
which, together with the boundedness of $V(x)$, yield
\begin{align*}
\frac{1}{\varepsilon_k^n} \int_{\mathcal{M}}|u_k|^2\,d\mu_g
&\leq \frac{C}{\varepsilon_k^n} \int_{\mathcal{M}}V(x)|u_k|^2\,d\mu_g\\
&\leq \frac{C}{\varepsilon_k^n} \int_{\mathcal
{M}}\left(\varepsilon^2|\nabla_g u_k|^2+ V(x)u_k^2\right)\,d\mu_g\\
&\leq C\left(c_0+\sigma\right)
\end{align*}
and
\begin{align*}
& \frac{1}{\varepsilon_k^n} \int_{\mathcal {M}}|u_k(x)|^2\,
d\mu_g\geq \frac{1}{\varepsilon_k^n} \int_{B_g(\eta_k,
\frac{R}{\rho})}\chi_k^2(\varphi_k(x))|u_k(x)|^2\,d\mu_g\\
&= \frac{1}{\varepsilon_k^n} \int_{B(0,
\frac{R}{\rho})}\chi_k^2(\varphi_k(\exp_{\eta_k}(z)))|u_k(\exp_{\eta_k}(z))|^2\
|g_{\eta_k}(z)|^{1/2}\,dz\\
&= \int_{B(0,
\frac{R}{\varepsilon_k\rho})}\chi_k^2(z)|u_k(\varphi_k^{-1}(z))|^2\
|g_{\eta_k}(\varepsilon_kz)|^{1/2}\,dz\geq
h^{n/2}\int_{\mathbb{R}^n}|w_k|^2\,dz.
\end{align*}
Moreover,
\begin{align*}
&\int_{\mathbb{R}^n}|\nabla w_k|^2\,dz\\ &= \int_{B(0,
\frac{R}{\varepsilon_k\rho})}\sum_{i,j}\frac{\partial
\left(\chi_k(z)u_k(\varphi_k^{-1}(z))\right)}{\partial
z_i}\frac{\partial
\left(\chi_k(z)u_k(\varphi_k^{-1}(z))\right)}{\partial z_j}\,dz\\
&= \int_{B(0,
\frac{R}{\varepsilon_k\rho})}\sum_{i,j}\chi^2_k(z)\frac{\partial
\left(u_k(\varphi_k^{-1}(z))\right)}{\partial z_i}\frac{\partial
\left(u_k(\varphi_k^{-1}(z))\right)}{\partial z_j}\,dz\\
&\quad +\int_{B(0,
\frac{R}{\varepsilon_k\rho})}\sum_{i,j}u_k(\varphi_k^{-1}(z))
 \chi_k(z)\Big(\frac{\partial
\left(u_k(\varphi_k^{-1}(z))\right)}{\partial z_i}\frac{\partial
\big(\chi_k(z)\big)}{\partial z_j}\\
&\quad +\frac{\partial
\left(u_k(\varphi_k^{-1}(z))\right)}{\partial z_j}\frac{\partial
\big(\chi_k(z)\big)}{\partial z_i}\Big)\,dz\\
&\quad +\int_{B(0,
\frac{R}{\varepsilon_k\rho})}\sum_{i,j}u^2_k(\varphi_k^{-1}(z))
\frac{\partial
\big(\chi_k(z)\big)}{\partial z_i}\frac{\partial
\big(\chi_k(z)\big)}{\partial z_j}\,dz:=I_6+I_7+I_8.
\end{align*}
By the hypotheses on ${u_k}$, $\psi(x)$ denotes the functions of the
partition of unity associated to $B_g(\eta_k, R)$, using
(\ref{eq:4.5}), we obtain
\begin{align*}
&\frac{\varepsilon_k^2}{\varepsilon_k^n}
 \int_{\mathcal {M}}|\nabla_g u_k(x)|^2\, d\mu_g\\
&\geq\frac{\varepsilon_k^2}{\varepsilon_k^n}\int_{B_g(\eta_k, \frac{R}{\rho})}\psi(x)|\nabla_g u_k(x)|^2\,d\mu_g\\
&\geq \psi_0\int_{B(0,\frac{R}{\varepsilon_k\rho})}\left(\sum_{i,j}g_{\eta_k}^{ij}(\varepsilon_k z)
 \frac{\partial
\left(u_k(\varphi_k^{-1}(z))\right)}{\partial z_i}\frac{\partial
\left(u_k(\varphi_k^{-1}(z))\right)}{\partial
z_j}\right)|g_{\eta_k}(\varepsilon_kz)|^{1/2}\,dz\\
& \geq C(\mathcal {M})\psi_0 I_6
\end{align*}
for a positive constant $C(\mathcal {M})$  depending only on the
manifold. By the Minkowski and H\"{o}lder inequalities,
\begin{align*}
& |I_7|\\
&\leq  \Big|2\int_{B(0,
\frac{R}{\varepsilon_k\rho})}\sum_{i,j}u_k(\varphi_k^{-1}(z))\frac{\partial
\left(u_k(\varphi_k^{-1}(z))\right)}{\partial z_i}\frac{\partial
\big(\chi_k(z)\big)}{\partial z_j}\,dz\Big|\\
&\leq  2\sum_{i,j}\Big(\int_{B(0,
\frac{R}{\varepsilon_k\rho})}|u_k(\varphi_k^{-1}(z))|^2\,
dz\Big)^{1/2}\Big(\int_{B(0,
\frac{R}{\varepsilon_k\rho})}\frac{2\varepsilon_k\rho}{R}
\Big|\frac{\partial
\left(u_k(\varphi_k^{-1}(z))\right)}{\partial z_i}\Big|^2\,
dz\Big)^{1/2}
\end{align*}
and
\[
 |I_8| \leq \frac{4n\varepsilon_k^2\rho^2}{R^2}\int_{B(0,
\frac{R}{\varepsilon_k\rho})}\left|u_k(\varphi_k^{-1}(z))\right|^2\,dz.
\]
Hence, $w_k$ is uniformly bounded in $H^1(\mathbb{R}^n)$ since
$I_{\varepsilon_k}(u_k)\leq 2c_0$ for all $k$.

Suppose now that $w_k\rightharpoonup \tilde{w}$ in
$H^1(\mathbb{R}^n)$. We show $\tilde{w}$ is a solution of problem
(\ref{eq:A}). Let $\omega_{\varepsilon_k}:=\{y\in \mathbb{R}^N|
\varepsilon_k y\in[\Omega]_r\}$ and denote by $\widetilde{\exp}$ the
exponential map associated to $\omega_{\varepsilon_k}$. We set
$v(y):=u(\varepsilon_ky)$ for $u\in H_g^1(\mathcal {M})$,
$y\in\omega_{\varepsilon_k}$ and let
$J_{\varepsilon_k}(v(y)):=I_{\varepsilon_k}(u(\varepsilon_ky))$. For
each $\eta_k\in \Omega$, we define
\begin{equation}  \label{eq:4.6}
\varphi_{k, \varepsilon_k}:
B_{g_{\varepsilon_k}}\left(\frac{\eta_k}{\varepsilon_k},
\frac{R}{\varepsilon_k\rho}\right)\to B\left(0,
\frac{R}{\varepsilon_k\rho}\right),\quad
\varphi_{k, \varepsilon_k}:=
\Big(\widetilde{\exp}_{\frac{\eta_k}{\varepsilon_k}}|_{B\left(0,
\frac{R}{\varepsilon_k\rho}\right)}\Big)^{-1}.
\end{equation}
For any $\xi\in C_0^{\infty}(\mathbb{R}^n)$, $\mathop{\rm supp}\xi\subset
\{\chi_k(z)=1\}$ for $k$ large enough. Hence,
$w_k(z)=u_k(\varphi^{-1}_{k, \varepsilon_k}(z))$ for
$z\in \mathop{\rm supp} \xi\subset B(0, \frac{R}{\varepsilon_k\rho})$
and $k$ large enough.
So we have
\begin{align*}
J'_{\varepsilon_k}\left(w_k(\varphi_{k,
\varepsilon_k}(y))\right)\left[\xi(\varphi_{k,
\varepsilon_k}(y))\right]
&=J'_{\varepsilon_k}\left(u_k\left(\varphi^{-1}_k(\varphi_{k,
\varepsilon_k}(y))\right)\right)\left[\xi(\varphi_{k,
\varepsilon_k}(y))\right]\\
&=I'_{\varepsilon_k}\left(u_k(x)\right)
\left[\xi\left(\varphi_{k,\varepsilon_k}
\big(\frac{x}{\varepsilon_k}\big)\right)\right]
\end{align*}
where if $y\in \omega_{\varepsilon_k}$ then $y\in
\frac{x}{\varepsilon_k}$ for a $x\in \Omega$. By the Ekeland
principle,
\[
\left|J'_{\varepsilon_k}\left(w_k(\varphi_{k,
\varepsilon_k}(y))\right)\left[\xi(\varphi_{k,
\varepsilon_k}(y))\right]\right|
<\sqrt{\sigma_k}|\|\xi\Big(\varphi_{k,
\varepsilon_k}\big(\frac{x}{\varepsilon_k}\big)\Big)|\|_{\varepsilon_k},
\]
while
\[
\|\|\xi\Big(\varphi_{k,
\varepsilon_k}\big(\frac{x}{\varepsilon_k}\big)\Big)\|\|_{\varepsilon_k}\to
\Big[\int_{\mathbb{R}^n}(|\nabla\xi|^2+|\xi|^2)\,dz\Big]^{1/2}
\]
as $k\to \infty$. Therefore,
\begin{equation}  \label{eq:4.7}
J'_{\varepsilon_k}\left(w_k(\varphi_{k,
\varepsilon_k}(y))\right)\left[\xi(\varphi_{k,
\varepsilon_k}(y))\right]\to 0
\end{equation}
for $\xi\in C_0^{\infty}(\mathbb{R}^n)$. Moreover,
\begin{align*}
&|J'_{\varepsilon_k}\left(w_k(\varphi_{k,
\varepsilon_k}(y))\right)\left[\xi(\varphi_{k,
\varepsilon_k}(y))\right]- J'(\tilde{w})[\xi]|
\\
&\leq \Big|\int_{B(0, \frac{R}{\varepsilon_k})\cap
\mathop{\rm supp}\xi}\sum_{i,j}g^{ij}_{\eta_k}(\varepsilon_kz)\frac{\partial
w_k(z)}{\partial z_i}\frac{\partial\xi(z)}{\partial
z_j}|g_{\eta_k}(\varepsilon_kz)|^{1/2}\,dz\\
&\quad -\int_{\mathbb{R}^n}\nabla \tilde{w}(z)\nabla\xi(z)\,dz\Big|\\
&\quad +\Big|\int_{B(0, \frac{R}{\varepsilon_k})\cap
\mathop{\rm supp}\xi}V(\exp_{\eta_k}(\varepsilon_kz))w_k(z)\xi(z)|
g_{\eta_k}(\varepsilon_kz)|^{1/2}\,dz\\
&\quad  -\int_{\mathbb{R}^n}V(\eta)\tilde{w}(z)\xi(z)\,dz\Big|\\
&\quad +\Big|\int_{B(0, \frac{R}{\varepsilon_k})\cap
\mathop{\rm supp}\xi}K(\exp_{\eta_k}(\varepsilon_kz))|w_k(z)
|^{p-1}\xi(z)|g_{\eta_k}(\varepsilon_kz)|^{1/2}\,dz\\
&\quad -\int_{\mathbb{R}^n}K(\eta)|\tilde{w}|^{p-1}(z)\xi(z)\,dz\Big|\\
&\leq \int_{\mathbb{R}^n}\sum_{i,j}\Big|g^{ij}_{\eta_k}(\varepsilon_kz)
\zeta_{B(0, \frac{R}{\varepsilon_k})}(z)\frac{\partial w_k(z)}{\partial
z_i}\frac{\partial\xi(z)}{\partial
z_j}|g_{\eta_k}(\varepsilon_kz)|^{1/2}-\delta_{ij}\frac{\partial
\tilde{w}(z)}{\partial z_i}\frac{\partial\xi(z)}{\partial
z_j}\Big|\,dz\\
&\quad +\int_{\mathbb{R}^n}\Big|\xi(z)\left(V(\exp_{\eta_k}
(\varepsilon_kz))\zeta_{B(0, \frac{R}{\varepsilon_k})}(z)w_k(z)
 |g_{\eta_k}(\varepsilon_kz)|^{1/2}-V(\eta)\tilde{w}(z)\right)\Big|\,dz\\
&\quad +\int_{\mathbb{R}^n}\Big|\xi(z)\Big(\zeta_{B(0,
\frac{R}{\varepsilon_k})}(z)|g_{\eta_k}(\varepsilon_kz)|^{1/2}
K(\exp_{\eta_k}(\varepsilon_kz))|w_k(z)|^{p-1}\\
&\quad -K(\eta)|\tilde{w}(z)|^{p-1}\Big)\Big|\,dz\\
&:=I_{9}+I_{10}+I_{11}
\end{align*}
where $\zeta_{B(0, \frac{R}{\varepsilon_k})}(z)$ denotes the
characteristic function of the set $B(0,
\frac{R}{\varepsilon_k})\subset \mathbb{R}^n$. We see that $I_{9},
I_{10}$ and $I_{11}$  tend to zero as $k\to\infty$. By the fact that
$$
\lim_{k\to\infty}|g^{ij}_{\eta_k}(\varepsilon_kz)\zeta_{B(0,
\frac{R}{\varepsilon_k})}(z)|g_{\eta_k}(\varepsilon_kz)|^{1/2}
-\delta_{ij}|=0
$$
and $\exp_{\eta_k}(\varepsilon_k z)-\eta_k\to 0$ as $k \to\infty$,
we obtain
\begin{equation}  \label{eq:4.8b}
J'_{\varepsilon_k}\left(w_k(\varphi_{k,
\varepsilon_k}(y))\right)\left[\xi(\varphi_{k,
\varepsilon_k}(y))\right]\to J'(\tilde{w})[\xi]\ \ \ {\rm
for}\ \ \forall \xi\in C_0^{\infty}(\mathbb{R}^n).
\end{equation}
Equations (\ref{eq:4.7}) and (\ref{eq:4.8b}) imply $\tilde{w}$ is a
solution of (\ref{eq:A}).

Finally, we show $\tilde{w}$ is a ground state solution of
(\ref{eq:A}). For $u_k\in \Sigma_{\varepsilon_k, \sigma_k}$ we have
\begin{align*}
(c_0+\sigma_k)
&\geq I_{\varepsilon_k}(u_k)
 =\big(\frac{1}{2}-\frac{1}{p}\big)\frac{1}{\varepsilon_k^n}\int_{\mathcal
{M}}K(x)|u_k^+|^p\,d\mu_g\\
& \geq \big(\frac{1}{2}-\frac{1}{p}\big)\frac{1}{\varepsilon_k^n}
 \int_{B_g(\eta_k,\frac{R}{\rho})}K(x)|u_k^+|^p\,d\mu_g\\
&= \big(\frac{1}{2}-\frac{1}{p}\big)
 \int_{B(0,\frac{R}{\varepsilon_k\rho})}K(\exp_{\eta_k}(\varepsilon_kz))
|u_k^+(\varphi_k^{-1}(z))|^p|g_{\eta_k}(\varepsilon_kz)|^{1/2}\,dz\,.
\end{align*}
The sequence of functions
$$
F_k(z):=\left(K(\exp_{\eta_k}(\varepsilon_kz))\right)^{1/p}u_k^+
(\varphi_k^{-1}(z))g^{1/(2p)}_{\eta_k}(\varepsilon_kz)\zeta_{B(0,
\frac{R}{\varepsilon_k\rho})}(z)\in L^p(\mathbb{R}^n),
$$
is bounded in $L^p(\mathbb{R}^n)$,  so there exists $F\in
L^p(\mathbb{R}^n)$ which is the $L^p-$ weak limit of the sequence
$F_k$. However, for $\xi\in C_0^{\infty}(\mathbb{R}^n)$, as  $w_k$
tends to $\tilde{w}$ weakly in $H^1(\mathbb{R}^n)$ and strongly in
$L_{loc}^p(\mathbb{R}^n)$, we get
\begin{align*}
\int_{\mathbb{R}^n}F_k(z)\xi(z)\,dz
&=\int_{\mathbb{R}^n}\left(K(\exp_{\eta_k}(\varepsilon_kz))
\right)^{1/p}w_k^+(z)g^{1/(2p)}_{\eta_k}(\varepsilon_kz)\xi(z)\,dz\\
&\to\int_{\mathbb{R}^n}K(\eta)^{1/p}\tilde{w}^+(z)\xi(z)\,dz
\quad\text{as }k\to\infty\,.
\end{align*}
 Hence, $F\equiv K^\frac{1}{p}(\eta)\tilde{w}^+
\equiv K^\frac{1}{p}(\eta)\tilde{w}$
and for any $k$,
$$
\big(\frac{1}{2}-\frac{1}{p}\big)
\int_{\mathbb{R}^n}K(\eta)|\tilde{w}|^p\,dz
\leq\lim\inf_{k\to\infty}\big(\frac{1}{2}
-\frac{1}{p}\big)\int_{\mathbb{R}^n}|F_k(z)|^p\,dz\leq c_0+\sigma_k,
$$
namely,
\begin{equation}  \label{eq:4.9b}
\int_{\mathbb{R}^n}K(\eta)|\tilde{w}|^p\,dz\leq
\frac{2p}{p-2}(c_0+\sigma_k).
\end{equation}
Hence, $\tilde{w}\in N_{\eta}\cup\{0\}$ and $J(\tilde{w})\leq c_0$.
If $\tilde{w}\not\equiv 0$, $\tilde{w}$ is a ground state solution.

Now we show that $\tilde{w}\not\equiv 0$. Given $T>0$, we can choose
$\eta_k\in\mathcal {M}$ such that for $k$ big enough $\eta_k\in
\tilde{P}^{\varepsilon_k}_{\sigma}\subset B_g(\eta_k,
\varepsilon_kT), \varepsilon_k<\frac{R}{\rho}$. By Lemma
\ref{le:4.2},
\begin{align*}
 \|w_k^+\|^p_{L^p(B(0,T))}
&=  \int_{B(0, T)}\chi^p_k(z)\left|u^{+}_k(\varphi_k^{-1}(z))\right|^p\,dz\\
&=\frac{1}{\varepsilon_k^n}\int_{B(0,
 \varepsilon_kT)}\Big|u^+_k\Big(\varphi_k^{-1}(\frac{z}{\varepsilon_k})
 \Big)\Big|^p\,dz\\
&\geq \frac{1}{H^{n/2}}\frac{1}{\varepsilon_k^n}\int_{B(0,
 \varepsilon_kT)}\Big|u^+_k\Big(\varphi_k^{-1}(\frac{z}{\varepsilon_k})
 \Big)\Big|^p|g_{\eta_k}(\varepsilon_k z)|^{1/2}\,dz\\
&\geq \frac{1}{K_{\rm max}H^{n/2}}\frac{1}{\varepsilon_k^n}\int_{B_g(\eta_k,
 \varepsilon_kT)}K(x)\left|u^+_k(x)\right|^p\,d\mu_g\\
&\geq \frac{1}{K_{\rm max}H^{n/2}}\frac{1}{\varepsilon_k^n}
 \int_{\tilde{P}^{\varepsilon_k}_{\sigma}}K(x)\left|u^+_k(x)\right|^p\,d\mu_g\\
&\geq \frac{\gamma}{K_{\rm max}H^{n/2}}
\end{align*}
This implies $\tilde{w}\not\equiv  0$ because $w_k$ converges
strongly to $\tilde{w}$ in
$L^p(B(0,T))$. The assertion then follows.
\end{proof}

\begin{proposition}\label{prop:4.1}
For $\theta\in (0,1)$ there exists $\sigma_0< c_0$ such that for
$\sigma\in (0, \sigma_0)$, $\varepsilon\in (0,\varepsilon_0)$ and
$u=u_{\varepsilon,\sigma}\in \Sigma_{\varepsilon, \sigma}$ we can
find  $\eta=\eta(u)\in \Omega$ such that
$$
\frac{1}{\varepsilon^n}\int_{B_g(\eta,\frac{R}{2})}K(x)|u^+|^{p}\,d\mu_g
>\frac{2p(1-\theta)}{p-2}c_0.
$$
\end{proposition}

\begin{proof}
First, we show that the result holds for $u\in \Sigma_{\varepsilon,
\sigma}\cap I_{\varepsilon}^{m_{\varepsilon}+2\sigma}$. Suppose by
contradiction that there exists $\theta\in (0, 1)$ such that we can
find sequences $\varepsilon_k$ and $\sigma_k$, which are positive
and tending to zero as $k\to \infty$, and a sequence
$\{u_k\}\subset \Sigma_{\varepsilon_k, \sigma_k}\cap
I_{\varepsilon_k}^{m_{\varepsilon_k}+2\sigma_k}$ such that for any
$\eta\in \Omega$ there holds
\begin{equation}  \label{eq:4.10}
\frac{1}{\varepsilon^n}\int_{B_g(\eta,\frac{R}{2})}K(x)|u_k^{+}|^p
\,d\mu_g\leq \frac{2p(1-\theta)}{p-2}c_0.
\end{equation}
By Lemma \ref{le:4.3}, we may assume that
\begin{equation}  \label{eq:4.11}
\left|\nabla|_{\mathcal
{N}_{\varepsilon_k}}I_{\varepsilon_k}(u_{k})\right|
<\sqrt{\sigma_k}\||\xi|\|_{\varepsilon_k}\
\ \ \forall \xi\in H^1_g(\mathcal {M}).
\end{equation}
Lemma \ref{le:4.2} implies that there exists a set $P_k$ of the
partition $\mathcal {P}_{\varepsilon}$ such that
$$
\frac{1}{\varepsilon_{k}^n}\int_{P_k}K(x)|u_k^{+}|^p\,d\mu_g>\gamma,
$$
and we may choose  $\eta_k\in P_k$. By the compactness of
$\mathcal{M}$, we may assume that $\eta_k\to \eta \in\mathcal {M}$ as
$k\to\infty$.

By the hypothesis on $K$, $K_{\rm min}>0$. We claim that for any $T>0$
and $\tau\in (0,1)$ it holds
\[
 |w_k^+|_{L^p(B(0,T))}^p\leq\frac{1}{K_{\rm min}}
\frac{1}{1-\tau}(1-\theta)\frac{2p}{p-2}c_0
\]
for $k$ large enough. Indeed, we note
$|g_{\eta_k}(\varepsilon_kz)|\to|g_{\eta}(0)|=1$ for all
$z\in B(0,R)$ and fixed $\tau\in (0, 1)$. For $k$ large enough,
$|g_{\eta_k}(z)|>(1-\tau)$ if $z\in B(0,\varepsilon_kT)$. By this
fact and (\ref{eq:4.10}) we have
\begin{equation}  \label{eq:4.12}
\begin{aligned}
|w_k^+|_{L^p(B(0,T))}^p
&= \int_{B(0,T)}\chi^p_k(z)\left|u_k^{+}(\varphi_k^{-1}(z))\right|^p\,dz\\
&=\frac{1}{\varepsilon_k^n}\int_{B(0,\varepsilon_kT)}
 \chi^p_{\frac{R}{\rho}}(z) \left|u_k^{+}(\exp_{\eta_k}(z))\right|^p\,dz \\
&\leq  \frac{1}{\varepsilon_k^n}\int_{B(0,\varepsilon_kT)}
 \frac{|g_{\eta_k}(z)|^{1/2}}{1-\tau}
 \left|u_k^{+}(\exp_{\eta_k}(z))\right|^p\,dz\\
&=\frac{1}{1-\tau}\frac{1}{\varepsilon_k^n}\int_{B_g(\eta_k,
 \varepsilon_kT)}|u_k^{+}|^p\,d\mu_g \\
&\leq  \frac{1}{(1-\tau)\varepsilon_k^n K_{\rm min}}\int_{B_g(\eta_k,
 \frac{R}{2})}K(x)|u_k^{+}|^p\,d\mu_g\\
&\leq\frac{1}{K_{\rm min}}\frac{1-\theta}{1-\tau}\frac{2p}{p-2}c_0.
\end{aligned}
\end{equation}
We know from Lemma \ref{le:4.5} that $\tilde{w}$ is a ground state
solution of problem (\ref{eq:A}); that is,
$$
E_{\eta}(\tilde{w})=\big(\frac{1}{2}-\frac{1}{p}\big)
\int_{\mathbb{R}^n}K(\eta)|\tilde{w}^+|^p\,dz=c_0.
$$
By Lemma \ref{le:4.5}, there exists $T>0$ such that for $k$ large
enough
$$
\frac{2p}{p-2}c_0=\int_{\mathbb{R}^n}K(\eta)|\tilde{w}^+|^p\,dz
\leq\int_{B(0, T)}K(\eta)|w_k^+|^p\,dz \leq K_{\rm max}\int_{B(0,
T)}|w_k^+|^p\,dz.
$$
Choosing $\mu>K_{\rm max}/K_{\rm min}$ and  $\tau$ such that
$\frac{1-\theta}{1-\tau}<\frac{1-\theta}{1-\tau}\mu<1$, we obtain
\begin{equation}  \label{eq:4.13}
\frac{1}{K_{\rm min}}\frac{1-\theta}{1-\tau}\frac{2p}{p-2}c_0
<\frac{\mu}{K_{\rm max}}\frac{1-\theta}{1-\tau}\frac{2p}{p-2}c_0
<\int_{B(0, T)}|w_k^+|^p\,dz
\end{equation}
a contradiction to (\ref{eq:4.12}).

Next, we show that $\Sigma_{\varepsilon, \sigma}\cap
I_{\varepsilon}^{m_{\varepsilon}+2\sigma}=\Sigma_{\varepsilon,
\sigma}$. In fact, for $u\in \Sigma_{\varepsilon, \sigma}\cap
I_{\varepsilon}^{m_{\varepsilon}+2\sigma}$, we have
$I_{\varepsilon}(u)< c_0+\sigma$ and $I_{\varepsilon}(u)<
m_{\varepsilon}+2\sigma$, which yield
$m_{\varepsilon}\geq(1-\theta)c_0$ for any $\theta\in (0,1)$. By
Proposition \ref{prop:3.1}, $\lim\sup_{\varepsilon\to
0}m_{\varepsilon}\leq c_0$, and then
$\lim_{\varepsilon\to 0}m_{\varepsilon}=c_0$, which
implies $\Sigma_{\varepsilon, \sigma}\subset
I_{\varepsilon}^{m_{\varepsilon}+2\sigma}$ for $\sigma, \varepsilon$
small enough. The proof is completed.
\end{proof}

\begin{proposition}\label{prop:4.2}
There exists $\sigma_0\in (0, c_0)$ such that for $\sigma\in (0,
\sigma_0)$, $\varepsilon\in (0, \varepsilon_0)$ and $u\in
\Sigma_{\varepsilon,\sigma}$ there holds $\beta(u)\in
[\Omega_{\delta}]_r$.
\end{proposition}

\begin{proof}
By Proposition \ref{prop:4.1}, for $\theta\in (0, 1)$ and $u\in
\Sigma_{\varepsilon,\sigma}$ with $\varepsilon$ and $\sigma$
suitably small, there exists $\eta\in \Omega$ such that
\begin{equation}  \label{eq:4.14}
(1-\theta)\frac{2p}{p-2}c_0<\frac{1}{\varepsilon^n}\int_{B_g(\eta,
\frac{R}{2})}K(x)|u^+|^p\,d\mu_g.
\end{equation}
On the other hand, for $u\in \Sigma_{\varepsilon,\sigma}$, we have
\[
 I_{\varepsilon}(u)
= \frac{1}{\varepsilon^n}\frac{p-2}{2p}\int_{\mathcal
{M}}K(x)|u^{+}|^p\,d\mu_{g}< c_0+\sigma,
\]
therefore,
\begin{equation}  \label{eq:4.15}
\frac{1}{\varepsilon^n}\int_{\mathcal {M}}|u^{+}|^p\,d\mu_{g}\leq\frac{1}{K_{\rm min}}\frac{1}{\varepsilon^n}\int_{\mathcal
{M}}K(x)|u^{+}|^p\,d\mu_{g}<
\frac{1}{K_{\rm min}}\frac{2p}{p-2}\left(c_0+\sigma\right).
\end{equation}
Let
$$
f\left(u(x)\right):=\frac{|u^+(x)|^p}{\int_{\mathcal {M}}|u^+(x)|^p\,d\mu_g}.
$$
By (\ref{eq:4.14}) and (\ref{eq:4.15}),
\[
\int_{B_g(\eta, \frac{R}{2})}f\left(u(x)\right)\,d\mu_g
\geq\frac{\frac{1}{K_{\rm max}}\frac{1}{\varepsilon^n}\int_{B_g(\eta,
\frac{R}{2})}K(x)|u^+(x)|^p\,d
\mu_g}{\frac{1}{\varepsilon^n}\int_{\mathcal {M}}|u^+(x)|^p\,d\mu_g}>
 \frac{K_{\rm min}(1-\theta)c_0}{K_{\rm max}(c_0+\sigma)}.
\]
Therefore,
\begin{align*}
|\beta(u)- \eta|
&\leq  \Big|\int_{B_g(\eta, \frac{R}{2})}(x-\eta)f\left(u(x)\right)
\,d\mu_g\Big|+
\Big|\int_{\mathcal {M}\backslash B_g(\eta, \frac{R}{2})}
(x-\eta)f\left(u(x)\right)\,d\mu_g\Big|\\
&\leq \frac{r(\Omega_{\delta})}{2}+D\Big(1-
\frac{K_{\rm min}(1-\theta)c_0}{K_{\rm max}(c_0+\sigma)}\Big),
\end{align*}
where $D$ is the diameter of  $\Omega_{\delta}$ as a subset of
$\mathcal {M}$. The assertion follows by choosing $\theta$ and
$\sigma$ suitably small.
\end{proof}

\begin{proof}[Proof of Theorem \ref{th.1.1}]
 We know that $I_{\varepsilon}\in C^1$ and $\mathcal {N}_{\varepsilon}$
is a $C^{1,1}$ complete Riemannian manifold. Also $I_{\varepsilon}$ is
bounded from below on $\mathcal {N}_{\varepsilon}$ and satisfies the
$(PS)$ condition. By Proposition \ref{Prop.2.1}, $I_{\varepsilon}$
has at least $\mathop{\rm cat}_{\Sigma_{\varepsilon,
\sigma}}(\Sigma_{\varepsilon, \sigma})$ critical points.

By Propositions \ref{prop:3.1} and \ref{prop:4.1}, $\beta \circ
\phi_{\varepsilon}: \Omega\to [\Omega_{\delta}]_r$ is well
defined and $\beta \circ \phi_{\varepsilon}(\eta)\in
[\Omega_{\delta}]_{r}\subset \mathbb{R}^N$ for $\eta\in \Omega$. Now
we show that $\Pi \circ \beta \circ \phi_{\varepsilon}$ is homotopic
to the identity on $\Omega_{\delta}$. Indeed,
\begin{align*}
\Pi \circ \beta \circ \phi_{\varepsilon}(\eta)-\eta
& = \int_{\mathcal {M}}(x-\eta)
 f\left(\phi_{\varepsilon}(\eta)\right)\,d\mu_g\\
&=  \int_{\mathcal {M}}(x-\eta)
 f\Big(t_{\varepsilon}(w_{\varepsilon}(\exp_{\eta}^{-1}(x))
 \chi_{R}(|\exp_{\eta}^{-1}(x)|))\\
&\quad\times
w_{\varepsilon}(\exp_{\eta}^{-1}(x))\chi_{R}(|\exp_{\eta}^{-1}(x)|)\Big)
\,d\mu_g\\
&= \frac{\int_{\mathcal
{M}}(x-\eta)w^p_{\varepsilon}(\exp_{\eta}^{-1}(x))
\chi^p_{R}(|\exp_{\eta}^{-1}(x)|)\,d\mu_g}{\int_{\mathcal
{M}}w^p_{\varepsilon}(\exp_{\eta}^{-1}(x))\chi^p_{R}
(|\exp_{\eta}^{-1}(x)|)\,d\mu_g}\\
&=\frac{\int_{B_g(\eta,R)}(x-\eta)w^p_{\varepsilon}
(\exp_{\eta}^{-1}(x))\chi^p_{R}(|\exp_{\eta}^{-1}(x)|)\,d\mu_g}
{\int_{B_g(\eta,R)}w^p_{\varepsilon}(\exp_{\eta}^{-1}(x))
\chi^p_{R}(|\exp_{\eta}^{-1}(x)|)\,d \mu_g}\\
&= \frac{\int_{B(0,R)}zw^p_{\varepsilon}(z)
 \chi^p_{R}(|z|)|g_{\eta}(z)|^{1/2}\,dz}
 {\int_{B(0,R)}w^p_{\varepsilon}(z)\chi^p_{R}(|z|)|g_{\eta}(z)|^{1/2}\,dz}\\
&=\frac{\varepsilon\int_{B(0, \frac{R}{\varepsilon})}zw^p(z)\chi^p_{R}
 (|\varepsilon z|)|g_{\eta}(\varepsilon z)|^{1/2}\,dz}
 {\int_{B(0,\frac{R}{\varepsilon})}w^p(z)\chi^p_{R}(|\varepsilon
z|)|g_{\eta}(\varepsilon z)|^{1/2}\,dz}.
\end{align*}
Hence,  $|\Pi \circ \beta \circ \phi_{\varepsilon}(\eta)-\eta|\leq
\varepsilon C\to 0$, where $C>0$ does not depend on $\eta$.
Applying Lemma \ref{le:2.1} with $X=\Sigma_{\varepsilon,\sigma}$,
$Y=\Omega_{\delta}$, $Z=\Omega$ and $h_1=\phi_{\varepsilon}$,
$h_2=\Pi \circ \beta$, we obtain $\mathop{\rm cat}_{\Sigma_{\varepsilon,
\sigma}}(\Sigma_{\varepsilon, \sigma})\geq
\mathop{\rm cat}_{\Omega_{\delta}}(\Omega)$. The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
The authors would like to thank the anonymous
referee for reading carefully the paper and for giving us valuable
comments. This work was  supported by grants  N10961016 and 10631030
from the NNSF of China.

\begin{thebibliography}{00}

\bibitem{abc} A. Ambrosetti, M. Badiale and S. Cingolani;
 Semiclassical states of
nonlinear Schr\"{o}dinger equations, \emph{Arch. Rational Mech.
Anal.}, 140(1997), 285-300.

\bibitem{ams}
A. Ambrosetti, A. Malchiodi, S. Secchi; Multiplicity results for
some nonlinear Schr\"{o}dinger equations with potentials, \emph{Arch.
Rational Mech. Anal.}, 159(2001), 253-271.

\bibitem{bbm}
V. Benci, C. Bonanno, A. M. Micheletti; On the multiplicity of
solutions of a nonlinear elliptic problem on Riemannian manifolds,
\emph{J. Funct. Anal.}, 252 (2007), 464-489.

\bibitem{bc}
V. Benci, G. Cerami; The effect of the domain topology on the number
of positive solutions of nonlinear elliptic problems, \emph{Arch.
Rational Mech. Anal.}, 114 (1991), 79-93.

\bibitem{bl1}
H. Berestycki, P. L. Lions; Nonlinear scalar field equations. I.
Existence of a ground state, \emph{Arch. Rational Mech. Anal.}, 82
(1983), 313-345.

\bibitem{bl2}
H. Berestycki, P. L. Lions; Nonlinear scalar field equations. II.
Existence of infinitely many solutions, \emph{Arch. Rational Mech.
Anal.}, 82 (1983), 347-375.

\bibitem{cl1}
S. Cingolani, M. Lazzo;  Multiple semiclassical standing waves for a
class of nonlinear SchrĄ§odinger equations, \emph{Topol. Methods
Nonlinear Anal.}, 10 (1997), 1-13.

\bibitem{cl2}
S. Cingolani, M. Lazzo; Multiple positive solutions to nonlinear
SchrĄ§odinger equations with competing potential functions, \emph{
Journal of Different. Equations}, 160 (2000), 118-138.

\bibitem{gnn}
B. Gidas, W. M. Ni, and L. Nirenberg; Symmetry of positive solutions
of nonlinear elliptics in $\mathbb{R}^N$, \emph{Math. Anal. Appl.
Part A}, 7 (1981), 369-402.

\bibitem{df}
M. Del Pino and P. L. Felmer; Local mountain pass for semilinear
elliptic problem in unbounded domains, \emph{Calc. Var. PDE}, 4
(1996), 121-137.

\bibitem{h}
E. Hebey; Nonlinear Analysis on Manifolds: Sobolev Spaces and
Inequalities, Courant Lect. Notes Math., Vol. 5, Courant Institute
of Mathematical Sciences, New York University, 1999.

\bibitem{hi}
N. Hirano; Multiple existence of solutions for a nonlinear elliptic
problem on a Riemannian manifold, \emph{Nonlinear Analysis: TMA}, 70
(2009), 671-692.

\bibitem{k}
M. A. Kwong; Uniqueness of positive solutions of $\Delta u-u+u^p=0$
in $\mathbb{R}^N$, \emph{Arch. Rat. Mech. Anal.}, 105 (1989),
243-266.


\bibitem{mw}
J. Mawhin, M. Willem; Critical Point theory and Hamiltonian Systems.
Applied Mathematical Sciences, 74. Springer-Verlag, New York, 1989.


\bibitem{R}
P. H. Rabinowitz; On a class of nonlinear Schr\"{o}dinger equations,
\emph{Z. Angew Math. Phys.}, 43(1992), 27-42.

\bibitem{v}
D. Visetti; Multiplicity of solutions of a zero mass nonlinear
equation on a Riemannian manifold. \emph{J. Differential Equations},
245(2008), 2397-2439.


\bibitem{w}
X. F. Wang; On a concentration of positive bound states of nonlinear
Schr\"{o}dinger equations, \emph{Comm. Math. Phys.}, 153 (1993),
223-243.

\bibitem{wz}
X. F. Wang, B. Zeng; On concentration of positive bound states of
nonlinear Schr\"{o}dinger equations with competing potential
functions, \emph{SIAM J. Math. Anal.}, 28(1997), 633-655.

\bibitem{wi}
M. Willem; Minimax Theorems. Progress in Nonlinear Differential
Equations and their Applications, 24. Birkhauser Boston, Inc.,
Boston, MA, 1996.

\end{thebibliography}

\end{document}