\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 $20, has a positive radial solution$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$C=B_{g}(\eta,R), then \begin{align*} &\int_{\mathcal {M}}|\nabla_{g}u|^2\,d\mu_{g}\\ &=\int_{B(0, R)}\psi_{C}(\exp_{x_0}(z))g_{x_0}^{ij}(z)\frac{\partial u(\exp_{x_0}(z))}{\partial z_{i}}\frac{\partial u(\exp_{x_0}(z))}{\partial z_{j}} |g_{x_0}(z)|^{1/2}\,dz, \end{align*} whereg_{x_0}$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$a0$,$C$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, $$\label{eq:3.1} t_{\varepsilon}^{p-2}(u)=\frac{\int_{\mathcal {M}}\left(\varepsilon^2|\nabla_{g} u(x)|^2+V(x)u^2\right)\,d\mu_{g}}{\int_{\mathcal {M}}K(x)|u^{+}|^p\,d\mu_{g}}.$$ The function$t_{\varepsilon}(u)$is$C^1$. Let us define a smooth real function$\chi_{R}$on$\mathbb{R}^{+}$such that $$\label{eq:3.2} \chi_{R}(t):=\begin{cases} 1 & \text{if } 0\leq t\leq \frac{R}{2};\\ 0 & \text{if } t\geq R\,. \end{cases}$$ and$|\chi'_{R}(t)|\leq \frac{2}{R}$. Fixing$\eta\in\Omega$and$\varepsilon>0$, we define $$\label{eq:3.3} W_{\eta, \varepsilon}(x):= \begin{cases} w_{\varepsilon}(\exp_{\eta}^{-1}(x))\chi_{R}(|\exp_{\eta}^{-1}(x)|) & \text{if } x\in B_{g}(\eta, R);\\ 0 & \text{otherwise}, \end{cases}$$ 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 $$\label{eq:3.4} \phi_{\varepsilon}(\eta) = t_{\varepsilon}(W_{\eta, \varepsilon}(x))W_{\eta, \varepsilon}(x).$$ \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 0and (\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 $$\label{eq:4.a0} m_{\varepsilon}:=\inf_{u\in \mathcal{N}_{\varepsilon}}I_{\varepsilon}(u),$$ 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 \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} 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, wherer_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, \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} Moveover $$\label{eq:4.c} \int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash P_j^{\varepsilon}} |\tilde{u}_j|^2d\mu_g \leq\int_{B_g(q_j^{\varepsilon}, r_1(\varepsilon))\backslash P_j^{\varepsilon}} |u^{+}|^2d\mu_g,$$ and \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} 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*} whereC'=\max\{2, 2\Theta^2+1\}. Hence, \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} 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 $$\label{eq:4.8} I_{\varepsilon}(u_{\sigma})< I_{\varepsilon}(u), \quad \||u_{\sigma}-u|\|_{\varepsilon}<4\sqrt{\sigma},$$ where$\||u|\|_{\varepsilon}^2=\frac{1}{\varepsilon^n}\int_{\mathcal {M}}(\varepsilon^2|\nabla_g u|^2+u^2)\,d\mu_g$, and $$\label{eq:4.9} \big|\nabla|_{\mathcal {N}_{\varepsilon}}I_{\varepsilon}(u_{\sigma})\big| <\sqrt{\sigma}\||\xi|\|_{\varepsilon}.$$ \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$$$\label{eq:4.5} \min\big\{\psi_{B_g(\eta_k,R)}(x)| x\in B_g(\eta_k, \frac{R}{\rho})\big\}\geq \psi_0>0.$$ 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 $$\label{eq:A} -\Delta u+V(\eta)u=K(\eta)|u|^{p-2}u, \quad \text{on }\mathbb{R}^n.$$ \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_g0, 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. $$\end{proposition} \begin{proof} First, we show that the result holds for u\in \Sigma_{\varepsilon, \sigma}\cap I_{\varepsilon}^{m_{\varepsilon}+2\sigma}. Suppose by contradiction that there exists \theta\in (0, 1) such that we can find sequences \varepsilon_k and \sigma_k, which are positive and tending to zero as k\to \infty, and a sequence \{u_k\}\subset \Sigma_{\varepsilon_k, \sigma_k}\cap I_{\varepsilon_k}^{m_{\varepsilon_k}+2\sigma_k} such that for any \eta\in \Omega there holds $$\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.$$ By Lemma \ref{le:4.3}, we may assume that $$\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}).$$ Lemma \ref{le:4.2} implies that there exists a set P_k of the partition \mathcal {P}_{\varepsilon} such that$$ \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 \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} 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. $$Choosing \mu>K_{\rm max}/K_{\rm min} and \tau such that \frac{1-\theta}{1-\tau}<\frac{1-\theta}{1-\tau}\mu<1, we obtain $$\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$$ a contradiction to (\ref{eq:4.12}). Next, we show that \Sigma_{\varepsilon, \sigma}\cap I_{\varepsilon}^{m_{\varepsilon}+2\sigma}=\Sigma_{\varepsilon, \sigma}. In fact, for u\in \Sigma_{\varepsilon, \sigma}\cap I_{\varepsilon}^{m_{\varepsilon}+2\sigma}, we have I_{\varepsilon}(u)< c_0+\sigma and I_{\varepsilon}(u)< m_{\varepsilon}+2\sigma, which yield m_{\varepsilon}\geq(1-\theta)c_0 for any \theta\in (0,1). By Proposition \ref{prop:3.1}, \lim\sup_{\varepsilon\to 0}m_{\varepsilon}\leq c_0, and then \lim_{\varepsilon\to 0}m_{\varepsilon}=c_0, which implies \Sigma_{\varepsilon, \sigma}\subset I_{\varepsilon}^{m_{\varepsilon}+2\sigma} for \sigma, \varepsilon small enough. The proof is completed. \end{proof} \begin{proposition}\label{prop:4.2} There exists \sigma_0\in (0, c_0) such that for \sigma\in (0, \sigma_0), \varepsilon\in (0, \varepsilon_0) and u\in \Sigma_{\varepsilon,\sigma} there holds \beta(u)\in [\Omega_{\delta}]_r. \end{proposition} \begin{proof} By Proposition \ref{prop:4.1}, for \theta\in (0, 1) and u\in \Sigma_{\varepsilon,\sigma} with \varepsilon and \sigma suitably small, there exists \eta\in \Omega such that $$\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.$$ On the other hand, for u\in \Sigma_{\varepsilon,\sigma}, we have $I_{\varepsilon}(u) = \frac{1}{\varepsilon^n}\frac{p-2}{2p}\int_{\mathcal {M}}K(x)|u^{+}|^p\,d\mu_{g}< c_0+\sigma,$ therefore, $$\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).$$ Let$$ 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*} whereD$is the diameter of$\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$. 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