\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
2007 Conference on Variational and Topological Methods: Theory, Applications,
Numerical Simulations, and Open Problems.
{\em Electronic Journal of Differential Equations},
Conference 18 (2010),  pp. 15--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{15}
\title[\hfilneg EJDE-2010/Conf/18/\hfil Nodal solutions]
{On the number of nodal solutions for a nonlinear elliptic problem on symmetric
Riemannian manifolds}

\author[M. Ghimenti, A. M. Micheletti\hfil EJDE/Conf/18 \hfilneg]
{Marco Ghimenti, Anna Maria Micheletti}  % in alphabetical order

\address{Marco Ghimenti \newline
Dipartimento di Matematica e Applicazioni,
Universit\`a di Milano Bicocca, via Cozzi 53, 20125, Milano, Italy}
\email{marco.ghimenti@unimib.it}

\address{Anna Maria Micheletti \newline
Dipartimento di Matematica Applicata,
Universit\`a di Pisa, via Buonarroti 1c, 56100, Pisa, Italy}
\email{a.micheletti@dma.it}


\thanks{Published July 10, 2010.}
\subjclass[2000]{35J60, 58G03}
\keywords{Riemannian manifolds; nodal solutions; topological methods}

\begin{abstract}
 We consider the problem
 $$
 -\varepsilon^2\Delta_g u+u=|u|^{p-2}u
 $$ 
 in a symmetric Riemannian manifold $(M,g)$.
  We give a multiplicity result for
 antisymmetric changing sign solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

Let $(M,g)$ be a smooth connected compact Riemannian manifold
of finite dimension $n\geq 2$ embedded in $\mathbb{R}^N$.
We consider the problem
\begin{equation}\label{P} %\tag{${\mathscr P}$}
-\varepsilon^2\Delta_g u+u=|u|^{p-2}u\text{ in } M, \quad
u\in H^1_g(M)
\end{equation}
where $2<p<2*=\frac {2N}{N-2}$, if $N\geq 3$.

Here $H^1_g(M)$ is the completion of $C^\infty(M)$ with respect to
\begin{equation}
\|u\|^2_g=\int_M |\nabla_g u|^2+u^2d\mu_g
\end{equation}

It is well known that the problem \eqref{P} has a mountain pass solution $u_\varepsilon$.
In \cite{BP05} the authors showed that $u_\varepsilon$ has a spike layer and its peak point
converges to the maximum point of the scalar curvature of $M$ as $\varepsilon$ goes to 0.

Recently there have been some results on the influence of the topology and
the geometry of $M$ on the number of solutions of the problem.
In \cite{BBM07} the authors proved that, if $M$ has a rich topology, problem \eqref{P} has
multiple solutions. More precisely they show that problem \eqref{P} has at least
$\operatorname{cat}(M)+1$ positive nontrivial solutions for $\varepsilon$ small enough. Here $\operatorname{cat}(M)$
is the Lusternik-Schnirelmann category of $M$.
In \cite{Vta} there is the same result for a more general nonlinearity. Furthermore in \cite{Hta}
it was shown that the number of solution is influenced by the topology of a suitable
subset of $M$ depending on the geometry of $M$. To point out the role of the geometry
in finding solutions
of problem \eqref{P}, in \cite{MP2ta}
it was shown that for any stable critical point of the scalar curvature
it is possible to build positive single peak solutions. The peak of these solutions approaches
such a critical point as $\varepsilon$ goes to zero.

Successively in \cite{DMP} the authors build positive $k$-peak solutions whose peaks collapse
to an isolated local minimum point of the scalar curvature as $\varepsilon$ goes to zero.

The first result on sign changing solution is in \cite{MPta} where it is showed the
existence of a solution
with one positive peak $\eta_1^\varepsilon$ and one negative peak $\eta_2^\varepsilon$ such that, as $\varepsilon$ goes to zero,
the scalar curvature $S_g(\eta_1^\varepsilon)$ (respectively $S_g(\eta_2^\varepsilon)$)
goes to the minimum (resp. maximum)
of the scalar curvature when the scalar curvature of $(M,g)$ is non constant.
Here we give a multiplicity result for changing sign solutions when the Riemannian
manifold $(M,g)$ is symmetric.

We look for solutions of the problem
\begin{equation}\label{Ptau}
\begin{gathered} % \tag{$\mathscr P_\tau$}
-\varepsilon^2\Delta_g u+u=|u|^{p-2}u \quad u\in H^1_g(M);\\
u(\tau x)=-u(x)  \quad \forall x\in M,
\end{gathered}
\end{equation}
where $\tau:\mathbb{R}^N\to\mathbb{R}^N$ is an
orthogonal linear
transformation  such that $\tau\neq\operatorname{Id}$,
$\tau^2=\operatorname{Id}$, $\operatorname{Id}$ being the identity of $\mathbb{R}^N$. Here $M$ is a compact connected Riemannian manifold
of dimension $n\geq 2$ and $M$ is a regular submanifold of $\mathbb{R}^N$ which is invariant with respect to $\tau$.
Let $M_\tau:=\{x\in M:\tau x=x\}$ be the set of the fixed points with respect to the
involution $\tau$; in the case $M_\tau\neq \emptyset$ we assume that $M_\tau$ is a regular
submanifold of $M$.

We obtain the following result.

\begin{theorem}\label{mainteo}
The problem \ref{Ptau} has at least
$G_\tau-\operatorname{cat}(M-M_\tau)$ pairs of solutions $(u,-u)$
which change sign (exactly once) for $\varepsilon$ small enough
\end{theorem}

Here
$G_\tau-\operatorname{cat}$ is the
$G_\tau$-equivariant Lusternik Schnirelmann category for the group
$G_\tau=\{\operatorname{Id},\tau\}$.

In \cite{CC03} the authors prove a result of this type for
the Dirichlet problem

\begin{equation}\label{Plambda}
\begin{gathered}
%\tag{$\mathscr P_\lambda$}
-\Delta u-\lambda u-|u|^{2^*-2}u=0\quad  u\in H^1_0(\Omega);\\
u(\tau x)=-u(x).
\end{gathered}
\end{equation}
Here $\Omega$ is a bounded smooth domain invariant
with respect to $\tau$ and $\lambda$ is a positive parameter.

We point out that in the case of the unit sphere
$S^{N-1}\subset \mathbb{R}^N$ (with the metric $g$ induced
by the metric of $\mathbb{R}^N$) the theorem of existence of
changing sign solutions
of \cite{MPta} can not be used because it holds for manifold of
non constant curvature.
Instead, we can apply Theorem \ref{mainteo} to obtain sign changing
solutions because we can consider
$\tau=-\operatorname{Id}$, and we have $G_\tau-\operatorname{cat}
S^{N-1}=N$.

Equation like \eqref{P} has been extensively studied in a flat
bounded domain $\Omega\subset \mathbb{R}^N$.
In particular, we would like to compare problem \eqref{P} with
the following Neumann problem
\begin{equation}\label{pn} %\tag{$\mathscr P_N$}
\begin{gathered}
-\varepsilon^2\Delta u +u=|u|^{p-2}u\quad  \text{in }\Omega;\\
\frac{\partial u}{\partial \nu}=0\quad \text{in }\partial\Omega.
\end{gathered}
\end{equation}
Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$ and $\nu$
is the unit outer normal to $\Omega$.
Problems \eqref{P} and \eqref{pn} present many similarities.
We recall some classical results
about the Neumann problem.

In the fundamental papers \cite{LNT,NT1,NT2}, Lin, Ni and Takagi established the existence of
least-energy solution to \eqref{pn} and showed that for $\varepsilon$ small enough the least energy
solution has a boundary spike, which approaches the maximum point of the mean curvature $H$ of
$\partial\Omega$, as $\varepsilon$ goes to zero. Later, in \cite{DFW,W1} it was proved that for any stable
critical point of the mean curvature of the boundary it is possible to construct single
boundary spike layer solutions, while in \cite{G,WW,Li} the authors construct multiple boundary
spike solutions at multiple stable critical points of $H$. Finally, in \cite{DY,GWW} the authors
proved that for any integer $K$ there exists a boundary $K$-peaks solutions, whose peaks
collapse to a local minimum point of $H$.
\section{Setting}


We consider the functional defined on $H^1_g(M)$
\begin{equation}
J_\varepsilon(u)=\frac1{\varepsilon^N}\int_M\Big(
\frac12 \varepsilon^2|\nabla_gu|^2+\frac12 |u|^2-\frac1p|u|^p
\Big)d\mu_g.
\end{equation}
It is well known that the
critical points of $J_\varepsilon(u)$
constrained on the Nehari manifold
\begin{equation}
{\mathcal{N}}_\varepsilon=\big\{
u\in H^1_g\setminus \{0\}:J'_\varepsilon(u)u=0 \big\}
\end{equation}
are non trivial solution of problem \eqref{P}.

The transformation $\tau:M\to M$ induces a transformation on $H^1_g$
we define the linear operator $\tau^*$ as
\begin{align*}
\tau^*:&H^1_g(M)\to H^1_g(M)\\
&\tau^*(u(x))=-u(\tau(x))
\end{align*}
and $\tau^*$ is a selfadjoint operator with respect to the scalar
product on $H^1_g(M)$
\begin{equation}
\langle u,v\rangle_\varepsilon=\frac1{\varepsilon^N}\int_M\big(
 \varepsilon^2\nabla_gu\cdot\nabla_gv+ u\cdot v \big) d\mu_g.
\end{equation}
Moreover, $\|\tau^*u\|_{L^p(M)}=\|u\|_{L^p(M)}$, and
$\|\tau^*u\|_\varepsilon=\|u\|_\varepsilon$, thus $J_\varepsilon(\tau^*u)=J_\varepsilon(u)$.
Then, for the Palais principle, the nontrivial solutions of \eqref{Ptau} are the critical points of the restriction
of $J_\varepsilon$ to the $\tau$-invariant Nehari manifold
\begin{equation}
{\mathcal{N}}_\varepsilon^\tau=\{u\in {\mathcal{N}}_\varepsilon :\tau^*u=u\}={\mathcal{N}}_\varepsilon\cap H^\tau.
\end{equation}
Here $H^\tau=\{u\in H^1_g:\tau^* u=u\}$.

In fact, since $J_\varepsilon(\tau^* u)=J_\varepsilon(u)$ and
$\tau^*$ is a selfadjoint operator we have
\begin{equation}
\langle \nabla J_\varepsilon(\tau^*u),\tau^*\varphi\rangle_\varepsilon=
\langle \nabla J_\varepsilon( u), \varphi\rangle_\varepsilon\ \ \forall\varphi\in H^1_g(M).
\end{equation}
Then $\nabla J_\varepsilon( u)=\tau^* \nabla J_\varepsilon(\tau^*u)=\tau^* \nabla J_\varepsilon(u)$ if $\tau^*u=u$.
We set
\begin{gather}
m_\infty=\inf_{\int_{\mathbb{R}^N} |\nabla u|^2+u^2
=\int_{\mathbb{R}^N}|u|^p}
\frac 12 \int_{\mathbb{R}^N} |\nabla u|^2+u^2
-\frac 1p\int_{\mathbb{R}^N}|u|^p;
\\
m_\varepsilon=\inf_{u\in {\mathcal{N}}_\varepsilon}J_\varepsilon;\\
m_\varepsilon^\tau=\inf_{u\in {\mathcal{N}}_\varepsilon^\tau}
J_\varepsilon.
\end{gather}

\begin{remark}\label{remps} \rm
It is easy to verify that $J_\varepsilon$ satisfies the Palais Smale condition on
${\mathcal{N}}_\varepsilon^\tau$.
Then there exists $v_\varepsilon$ minimizer of $m_\varepsilon^\tau$ and $v_\varepsilon$ is a critical point for
$J_\varepsilon$ on $H^1_g(M)$. Thus $v_\varepsilon^+$ and $v_\varepsilon^-$
belong to ${\mathcal{N}}_\varepsilon$, then $J_\varepsilon(v_\varepsilon)\geq 2m_\varepsilon$.
\end{remark}

We recall some facts about equivariant Lusternik-Schnirelmann theory.
If $G$ is a compact Lie group, then a $G$-space is a topological
space $X$ with a continuous $G$-action $G \times X    \to X$,
$(g, x)\mapsto gx$. A $G$-map is a continuous function $f : X  \to Y$
between $G$-spaces $X$ and $Y$
which is compatible with the $G$-actions, i.e. $f (gx) = gf (x)$
for all $x \in X$, $g \in G$. Two
$G$-maps $f_0$, $f_1 : X \to Y$ are $G$-homotopic if there is a homotopy
$\theta : X \times [0, 1] \to Y$ such
that
$\theta(x, 0) = f_0(x)$,
$\theta(x, 1) = f_1(x)$ and
$\theta(gx, t) = g\theta (x, t)$ for all$x\in X$, $g \in G$,
$t \in [0, 1]$. A subset $A$ of a $X$ is $G$-invariant if
$ga \in A$ for every $a\in A$,
$g \in G$. The $G$-orbit of a point $x \in X$ is the set
$Gx = \{gx : g \in G\}$.

\begin{definition} \rm
The $G$-category of a $G$-map $f : X \to Y$ is the smallest number $k = G-\operatorname{cat}(f)$
of open $G$-invariant subsets $X_1, \dots , X_k$ of $X$ which cover $X$ and which have the
property that, for each $i = 1,\dots , k$, there is a point $y_i \in Y$ and a $G$-map
$\alpha_i : X_i \to Gy_i\subset Y$
such that the restriction of $f$ to $X_i$ is $G$-homotopic to $\alpha_i$.
If no such covering exists we define
$G-\operatorname{cat}(f)=\infty$.
\end{definition}

In our applications, $G$ will be the group with two elements,
acting as $G_\tau = \{\operatorname{Id}, \tau\}$
on $\Omega$, and as ${\mathbb Z}/2 = \{1,-1\}$ by multiplication
on the Nehari manifold
${\mathcal{N}}_\varepsilon^\tau$.
We remark the following result on the equivariant category.

\begin{theorem}\label{castroclapp}
Let $\phi : M \to \mathbb{R}$ be an even $C1$ functional on a
complete $C^{1,1}$ submanifold $M$
of a Banach space which is symmetric with respect to the origin.
Assume that $\phi$ is bounded
below and satisfies the Palais Smale condition $(PS)_c$ for every
$c\leq d$.
Then $\phi$ has at least ${\mathbb Z}/2-\operatorname{cat}(\phi^d)$
antipodal pairs $\{u,-u\}$ of critical points with critical values
$\phi(\pm u)\leq d$.
\end{theorem}

\section{Sketch of the proof of main theorem}

In our case we consider the even positive $C^2$ functional
$J_\varepsilon$ on the $C2$
Nehari manifold ${\mathcal{N}}_\varepsilon^\tau$ which is symmetric
with respect to the origin.
As claimed in Remark \ref{remps}, $J_\varepsilon$ satisfies Palais Smale
condition on ${\mathcal{N}}_\varepsilon^\tau$. Then we can apply
Theorem \ref{castroclapp} and our aim is to get an estimate of this lower bound for the number of solutions.
For $d>0$ we consider
\begin{gather*}
M_d = \{x\in \mathbb{R}^N:\operatorname{dist}(x,M)\leq d\};\\
M_d^- = \{x\in M:\operatorname{dist}(x,M_\tau)\geq d\}.
\end{gather*}
We choose $d$ small enough such that
\begin{gather*}
G_\tau-\operatorname{cat}_{M_d} M_d=G_\tau-\operatorname{cat}_M M\\
G_\tau-\operatorname{cat}_M M_d^-=G_\tau-\operatorname{cat}_M (M-M_\tau)
\end{gather*}
Now we build two continuous operator
\begin{gather*}
\Phi_\varepsilon^\tau:M_d^-\to {\mathcal{N}}_\varepsilon^\tau
 \cap J_\varepsilon^{2(m_\infty+\delta)};\\
\beta:{\mathcal{N}}_\varepsilon^\tau\cap
J_\varepsilon^{2(m_\infty+\delta)}\to M_d,
\end{gather*}
such that $\Phi_\varepsilon^\tau(\tau q)=-\Phi_\varepsilon^\tau(q)$,
$\tau\beta(u)=\beta(-u)$ and
$\beta\circ\Phi_\varepsilon^\tau$ is $G_\tau$ homotopic to the
inclusion $M_d^-\to M_d$.

By equivariant category theory we obtain
\begin{equation}
\begin{aligned}
G_\tau-\operatorname{cat}_M (M-M_\tau)
&=G_\tau-\operatorname{cat}(M_d^-\hookrightarrow M_d)\\
&=G_\tau-\operatorname{cat} \beta\circ\Phi_\varepsilon^\tau\\
&\leq {\mathbb Z}_2-\operatorname{cat} {\mathcal{N}}_\varepsilon^\tau
\cap J_\varepsilon^{2(m_\infty+\delta)}
\end{aligned}
\end{equation}

\section{Technical lemmas}

First of all, we recall that there exists a unique positive
spherically symmetric function $U\in H^1(\mathbb{R}^n)$
such that
\begin{equation}
-\Delta U+U=U^{p-1} \text{ in }\mathbb{R}^n
\end{equation}
It is well known that $U_\varepsilon(x)
=U\left(\frac x\varepsilon\right)$ is a solution of
\begin{equation}
-\varepsilon^2\Delta U_\varepsilon+U_\varepsilon=U_\varepsilon^{p-1}\text{ in }\mathbb{R}^n.
\end{equation}

Secondly, let us introduce the exponential map $\exp:TM\to M$ defined on the tangent
bundle $TM$ of $M$ which is a $C^\infty$ map. Then, for $\rho$ sufficiently small
(smaller than the injectivity radius of $M$ and smaller than $d/2$),
the Riemannian manifold $M$ has a special set of
charts $\{\exp_x:B(0,\rho)\to M\}$.
Throughout the paper we will use the following notation: $B_g(x,\rho)$ is the open ball in $M$
centered in $x$ with radius $\rho$ with respect to the distance given by the metric $g$.
Corresponding to this chart, by choosing an orthogonal coordinate system
$(x_1,\dots,x_n)\subset \mathbb{R}^n$ and identifying $T_xM$ with $\mathbb{R}^n$ for $x\in M$, we can define
a system of coordinates called {\em normal coordinates}.

Let $\chi_\rho$ be a smooth cut off function such that
\begin{gather*}
\chi_\rho(z)=1\quad \text{if }z\in B(0,\rho/2);\\
\chi_\rho(z)=0\quad \text{if }z\in \mathbb{R}^n \setminus B(0,\rho);\\
|\nabla \chi_\rho(z)|\leq2\quad \text{for all } x.
\end{gather*}
Fixed a point $q\in M$ and $\varepsilon>0$,
let us define the function $w_{\varepsilon,q}(x)$ on $M$ as
\begin{equation}
w_{\varepsilon,q}(x)=
\begin{cases}
U_\varepsilon(\exp_q^{-1}(x))\chi_\rho(\exp_q^{-1}(x))
 &\text{if }x\in B_g(q,\rho)\\
0&\text{otherwise}
\end{cases}
\end{equation}
For each $\varepsilon>0$ we can define a positive number
$t(w_{\varepsilon,q})$ such that
\begin{equation}
\Phi_\varepsilon(q)=t(w_{\varepsilon,q})w_{\varepsilon,q}
\in H^1_g(M)\cap {\mathcal{N}_\varepsilon}\text{ for }q\in M.
\end{equation}
Namely, $t(w_{\varepsilon,q})$ turns out to verify
\begin{equation}
t(w_{\varepsilon,q})^{p-2}=\frac{\int_M \varepsilon^2|
\nabla_g w_{\varepsilon,q}|^2+|w_{\varepsilon,q}|^2d\mu_g}
{\int_M |w_{\varepsilon,q}|^pd\mu_g}
\end{equation}

\begin{lemma} \label{lem1}
Given $\varepsilon>0$ the application
$\Phi_\varepsilon(q):M\to H^1_g(M)\cap {\mathcal{N}}_\varepsilon$
is continuous. Moreover, given $\delta>0$ there exists
$\varepsilon_0=\varepsilon_0(\delta)$
such that, if $\varepsilon<\varepsilon_0(\delta)$
then $\Phi_\varepsilon(q)\in
{\mathcal{N}}_\varepsilon\cap J_\varepsilon^{m_\infty+\delta}$.
\end{lemma}

For the proof see \cite[ Proposition 4.2]{BBM07}.
Now, fixed a point $q\in M_d^-$, let us define the function
\begin{equation}
\Phi_\varepsilon^\tau(q)=t(w_{\varepsilon,q})w_{\varepsilon,q}-t(w_{\varepsilon,\tau q})w_{\varepsilon,\tau q}
\end{equation}

\begin{lemma}\label{lemma5}
Given $\varepsilon>0$ the application
$\Phi^\tau_\varepsilon(q):M_d^-\to H^1_g(M)\cap {\mathcal{N}}^\tau_\varepsilon$
is continuous. Moreover, given $\delta>0$ there exists $\varepsilon_0=\varepsilon_0(\delta)$
such that, if $\varepsilon<\varepsilon_0(\delta)$
then $\Phi^\tau_\varepsilon(q)\in {\mathcal{N}}^\tau_\varepsilon\cap J_\varepsilon^{2(m_\infty+\delta)}$.
\end{lemma}

\begin{proof}
Since $U_\varepsilon(z)\chi_\rho(z)$ is radially symmetric we set
$U_\varepsilon(z)\chi_\rho(z)=\tilde U_\varepsilon(|z|)$.
We recall that
\begin{gather*}
|\exp^{-1}_{\tau q}\tau x|=d_g(\tau x,\tau q)
=d_g(x,q)=|\exp^{-1}_{q} x|;\\
|\exp^{-1}_{q}\tau x|=d_g(\tau x, q)=d_g(x,\tau q).
\end{gather*}
We have
\begin{align*}
\tau^* \Phi_\varepsilon^\tau(q)(x)
&= -t(w_{\varepsilon,q})w_{\varepsilon,q}(\tau x)
 +t(w_{\varepsilon,\tau q})w_{\varepsilon,\tau q}(\tau x) \\
&=-t(w_{\varepsilon,q})\tilde U_\varepsilon(|\exp^{-1}_q(\tau x)|)
 +t(w_{\varepsilon,\tau q})
\tilde U_\varepsilon(|\exp^{-1}_{\tau q}(\tau x)|) \\
&=t(w_{\varepsilon,\tau q})\tilde U_\varepsilon(|\exp^{-1}_q(x)|)
 -t(w_{\varepsilon,q})
\tilde U_\varepsilon(|\exp^{-1}_{q}(\tau x)|) \\
&=t(w_{\varepsilon,q})\tilde U_\varepsilon(|\exp^{-1}_q(x)|)-
t(w_{\varepsilon,q})\tilde U_\varepsilon(|\exp^{-1}_{\tau q}(x)|),
\end{align*}
because by the definition we have $t(w_{\varepsilon,q})
=t(w_{\varepsilon,\tau q})$.

Moreover by definition the support of the function
$\Phi_\varepsilon^\tau$ is
$B_g(q,\rho)\cup B_g(\tau q,\rho)$, and
$B_g(q,\rho)\cap B_g(\tau q,\rho)=\emptyset$
because $\rho<d/2$ and $q\in M_d^-$.
Finally, because
\begin{gather*}
\int_M|w_{\varepsilon,q}|^\alpha d\mu_g
=\int_M|w_{\varepsilon,\tau q}|^\alpha d\mu_g\ \text{ for }\alpha=2,p;\\
\int_M|\nabla w_{\varepsilon,q}|^2 d\mu_g
=\int_M|\nabla w_{\varepsilon,\tau q}|^2 d\mu_g,
\end{gather*}
we have
\begin{equation}
J_\varepsilon(\Phi_\varepsilon^\tau(q))
=\Big(\frac 12-\frac 1p\Big)\frac 1{\varepsilon^n}
\int_M |\Phi_\varepsilon^\tau(q)|^pd\mu_g
=2J_\varepsilon(\Phi_\varepsilon(q)).
\end{equation}
Then by previous lemma we have the claim.
\end{proof}

\begin{lemma}
We have $\lim_{\varepsilon\to0}m_\varepsilon^\tau=2m_\infty$
\end{lemma}

\begin{proof}
By the previous lemma and by Remark \ref{remps}
we have that for any $\delta>0$ there exists
$\varepsilon_0(\delta)$ such that, for
$\varepsilon<\varepsilon_0(\delta)$
\begin{equation}
2m_\varepsilon\leq m_\varepsilon^\tau
\leq 2J_\varepsilon(\Phi_\varepsilon(q))\leq 2(m_\infty+\delta).
\end{equation}
Since $\lim_{\varepsilon\to0}m_\varepsilon=m_\infty$
(see \cite[Remark 5.9]{BBM07})
we get the claim.
\end{proof}

For any function $u\in {\mathcal{N}}_\varepsilon^\tau$ we can
define a point $\beta(u)\in \mathbb{R}^N$ by
\begin{equation}
\beta(u)=\frac{\int_M x|u^+(x)|^pd\mu_g}{ \int_M |u^+(x)|^pd\mu_g}
\end{equation}

\begin{lemma}\label{lemma7}
There exists $\delta_0$ such that, for any $0<\delta<\delta_0$ and
any $0<\varepsilon<\varepsilon_0(\delta)$
(as in Lemma \ref{lemma5}) and for any function
$u\in {\mathcal{N}}_\varepsilon^\tau\cap J_\varepsilon^{2(m_\infty+\delta)}$, it holds $\beta(u)\in M_d$.
\end{lemma}

\begin{proof}
Since $\tau^* u=u$ we set
\[
M^+=\{x\in M:u(x)>0\},\quad M^-=\{x\in M:u(x)<0\}.
\]
It is easy to see that $\tau M^+=M^-$. Then we have
\begin{align*}
J_\varepsilon(u)
&=\Big(\frac 12-\frac 1p\Big)\frac 1{\varepsilon^n}
\int_M|u|^pd\mu_g \\
&=\Big(\frac 12-\frac 1p\Big)\frac 1{\varepsilon^n}
\Big[\int_{M^+}|u^+|^pd\mu_g+\int_{M^-}|u^-|^pd\mu_g
\Big]=2J_\varepsilon(u^+)
\end{align*}
By the assumption $J_\varepsilon(u)\leq 2(m_\infty+\delta)$ we have $J_\varepsilon(u^+)\leq m_\infty+\delta$
then by Proposition 5.10 of \cite{BBM07} we get the claim.
\end{proof}

\begin{lemma}
There exists $\varepsilon_0>0$ such that for any
$0<\varepsilon<\varepsilon_0$ the composition
\begin{equation}
I_\varepsilon=\beta\circ\Phi_\varepsilon^\tau:M_d^-\to M_d
\subset\mathbb{R}^N
\end{equation}
is well defined, continuous, homotopic to the identity and $I_\varepsilon(\tau q)= \tau I_\varepsilon(q)$.
\end{lemma}

\begin{proof}
It is easy to check that
\[
\Phi_\varepsilon^\tau(\tau q)=-\Phi_\varepsilon^\tau(q),\quad
\beta(-u)=\tau\beta(u).
\]
Moreover, by Lemma \ref{lemma5} and by Lemma \ref{lemma7}, for any
$q\in M_d^-$ we have $\beta\circ\Phi_\varepsilon^\tau(q)=\beta(\Phi_\varepsilon(q))\in M_d$,
and $I_\varepsilon$ is well defined.

In order to show that $I_\varepsilon$ is homotopic to identity,
we evaluate the difference between $I_\varepsilon$ and the identity as follows.
\begin{align*}
I_\varepsilon(q)-q
&=\frac{\int_M(x-q)|w^+_{\varepsilon,q}|^pd\mu_g}
{\int_M|w^+_{\varepsilon,q}|^pd\mu_g}\\
&=\frac{\int_{B(0,\rho)}
z\left|U\left(\frac z\varepsilon\right)\chi_\rho(|z|)\right|^p
\big|g_q(z)\big|^{1/2}}
{ \int_{B(0,\rho)}
\left|U\left(\frac z\varepsilon\right)\chi_\rho(|z|)\right|^p
\big|g_q(z)\big|^{1/2}} \\
&= \frac{ \varepsilon\int_{B(0,\rho/\varepsilon)}z
\big|U(z)\chi_\rho(|\varepsilon z|)\big|^p
\big|g_q(\varepsilon z)\big|^{1/2}}
 {\int_{B(0,\rho/\varepsilon)}
\big|U(z)\chi_\rho(|\varepsilon z|)\big|^p
\big|g_q(\varepsilon z)\big|^{1/2}},
\end{align*}
hence $|I_\varepsilon(q)-q| <\varepsilon c(M)$ for a constant
$c(M)$ that does not depend on $q$.
\end{proof}

Now, by previous lemma and by Theorem \ref{castroclapp} we can
prove Theorem \ref{mainteo}.
In fact, we know that, if $\varepsilon$ is small enough, there exist
$G_\tau-\operatorname{cat}(M-M_\tau)$
minimizers which change sign,
because they are antisymmetric.
We have only to prove that
any minimizer changes sign exactly once.
Let us call $\omega=\omega_\varepsilon$ one of these minimizers.
Suppose that the set $\{x\in M:\omega_\varepsilon(x)>0 \}$ has
$k$ connected
components $M_1, \dots, M_k$. Set
\begin{equation}
\omega_i=\begin{cases}
\omega_\varepsilon(x) &  x\in M_i\cup\tau M_i;\\
0 &\text{elsewhere}
\end{cases}
\end{equation}
For all $i$, $\omega_i\in {\mathcal{N}}^\tau_\varepsilon$.
Furthermore we have
\begin{equation}
J_\varepsilon(\omega)=\sum_iJ_\varepsilon(\omega_i),
\end{equation}
thus
\begin{equation}
m_\varepsilon^\tau=J_\varepsilon(\omega)
=\sum_{i=1}^k J_\varepsilon(\omega_i)\geq k \cdot m_\varepsilon^\tau,
\end{equation}
so $k=1$, that concludes the proof.

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\end{document}