# Multiplicity of 2-nodal solutions
of the Yamabe equation

Jorge Dávila<EMAIL_ADDRESS>,  Héctor Barrantes G
<EMAIL_ADDRESS>and  Isidro H. Munive
<EMAIL_ADDRESS>
###### Abstract.

Given any closed Riemannian manifold $(M,g)$, we use the gradient flow method
and Sign-Changing Critical Point Theory to prove multiplicity results for
2-nodal solutions of a subcritical non-linear equation on $(M,g)$, see Eq.
(1.1) below. If $(N,h)$ is a closed Riemannian manifold of constant positive
scalar curvature our result gives multiplicity results for the Yamabe-type
equation on the Riemannian product $(M\times N,g+\varepsilon h)$, for
$\varepsilon>0$ small.

##  1\. Introduction

On a compact Riemannian manifold $(M^{n},g)$ without boundary of dimension
$n\geq 3$, we consider the following equation

(1.1)
$-\varepsilon^{2}\Delta_{g}u+\left(\frac{s_{g}}{a_{m+n}}\varepsilon^{2}+1\right)u=|u|^{p_{m+n}-2}u,$

where $s_{g}$ is the scalar curvature of $g$, $\Delta_{g}$ is the is the
Laplace Beltrami operator associated to $g$, $a_{m+n}=\frac{4(n+m-1)}{n+m-2}$,
$p_{n+m}=\frac{2(n+m)}{n+m-2}$, with $m\in\mathbb{N}$. Moreover, we consider
$\varepsilon>0$ small enough so that

(1.2) $1+\frac{s_{g}}{a_{m+n}}\varepsilon^{2}>c_{\varepsilon}\quad\text{in
$M$},$

for some $c_{\varepsilon}>0$.

The study of this equation is motivated, on one hand, by the Yamabe problem on
products of Riemannian manifolds. If $u:M\rightarrow\mathbb{R}$ is a positive
solution of Eq. (1.1) then $u$ solves the Yamabe equation in the product
$(M^{n}\times N^{m},g+\varepsilon^{2}h)$, where $(N^{m},h)$ is a Riemannian
manifold with constant scalar curvature $s_{h}$ equal to $a_{m+n}$, see, for
instance, [30] for details.

There has also been interest in nodal solutions of non-linear equations of the
type (1.1) (i.e. solutions that change sign). See for instance the articles
[2, 11, 12, 16, 17, 20, 33] and, more recently, the paper [31]. Nodal
solutions $u$ of (1.1) do not give metrics of constant scalar curvature since
$u$ vanishes at some points and therefore $|u|^{p_{n}-2}g$ is not a Riemannian
metric, but they might have geometric interest. The existence of at least one
nodal solution is proved in general cases in [2], as minimizers for the second
Yamabe invariant. But there are not as many results about multiplicity of
nodal solutions as in the positive case.

In [9], M.Clapp and M. Micheletti considered the problem of obtain $2$-nodal
solutions to the equation

$-\varepsilon^{2}\Delta_{g}u+u=|u|^{p_{m+n}-2}u,$

over a closed Riemannian manifold $(M,h)$. In order to study this problem,
they used gradient flow techniques to prove the existence of $2$-nodal
solutions. In this work we obtain existence results for Eq. (1.1), see Theorem
1.1, using gradient flow techniques from [9] (see Chapter 1 of [38] for
details) for the functional

(1.3)
$J_{\varepsilon}(u)\doteq\frac{1}{\varepsilon^{n}}\int_{M}\left(\frac{1}{2}\varepsilon^{2}|\nabla_{g}u|_{g}^{2}+\frac{1}{2}\left(\frac{s_{g}}{a_{m+n}}\varepsilon^{2}+1\right)u^{2}-\frac{1}{p_{m+n}}|u|^{p_{m+n}}\right)d\mu_{g}.$

We recall here that (1.1) is the Euler-Lagrange equation of $J_{\varepsilon}$.
The Nehari manifold $\mathcal{N}_{\varepsilon}$ associate to the functional
$J_{\varepsilon}$ is the following set:

$\mathcal{N}_{\varepsilon}\doteq\Big{\\{}u\in
H_{\varepsilon}\setminus\\{0\\}:\mathcal{L}_{\varepsilon}(u,u)=|u|^{p}_{p,\varepsilon}\Big{\\}},$

where $|u|_{p,\varepsilon}$ and $\mathcal{L}_{\varepsilon}(u,u)$ are given by
(2.4) and (2.2), respectively. Notice that any sign changing solution belongs
to the set

(1.4) $\mathcal{E}_{\varepsilon}\doteq\Big{\\{}u\in
H_{\varepsilon}:u^{+},u^{-}\in\mathcal{N}_{\varepsilon}\Big{\\}}\subset\mathcal{N}_{\varepsilon}.$

Our first main result is the following.

######  Theorem 1.1.

(Existence) For every $\varepsilon>0$ there exists
$u_{\varepsilon}\in\mathcal{E}_{\varepsilon}$ such that
$J_{\varepsilon}(u_{\varepsilon})=\mathbf{d}_{\varepsilon}$, where
$\mathbf{d}_{\varepsilon}\doteq\inf_{\mathcal{E}_{\varepsilon}}J_{\varepsilon}$,
and $u_{\varepsilon}$ is a sign changing solutions of Eq. (1.1). Moreover,
$\textbf{d}_{\varepsilon}\geq 2\textbf{m}_{\varepsilon}$, where
$\textbf{m}_{\varepsilon}\doteq\inf_{\mathcal{N}_{\varepsilon}}J_{\varepsilon}$.

We consider the equivariant Lusternik-Schnirelmann category
$\mathrm{Cat}_{G}(X)$ of a $G$-space $X$ is the smallest integer $k$ such that
$X$ can be covered by $k$ locally closed $G$-invariant subsets
$X_{1},\ldots,X_{k}$, see Definition 5.5 in [4].

######  Theorem 1.2.

There exists $\varepsilon_{0}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0})$, $J_{\varepsilon}$ has at least
$\mathrm{Cat}(\mathcal{Z}_{\varepsilon}\cap J^{d_{\varepsilon}+\delta_{0}})$
critical points. Moreover $\mathrm{Cat}(\mathcal{Z}_{\varepsilon}\cap
J_{\varepsilon}^{d_{\varepsilon}+\delta_{0}})\geq\mathrm{Cupl}\
(\mathcal{Z}_{\varepsilon}\cap
J_{\varepsilon}^{d_{\varepsilon}+\delta_{0}})\geq Cupl(M)$.

We also obtain a multiplicity result, see Theorem 1.3 below, with the help of
the _center of mass_ of a function introduced by Petean in the paper [30].
This _center of mass_ plays the role of the _barycenter map_ , see for
instance [4], in the Riemannian setting. Given the set $F(M)\doteq\\{(x,y)\in
M\times M:x\neq y\\}$, we consider the quotient space $C(M)$ of $F(M)$, under
the free action $\theta(x,y)=(y,x)$, and define $\mathcal{H}^{*}$ for singular
cohomology with coefficients in $\mathbb{Z}_{2}$. Recall that the _cup-length_
of a topological space $X$, denoted by $\text{cupl}\,X$, is the smallest
integer $k\geq 1$ such that the cup-product of any $k$ cohomology classes in
$\widetilde{\mathcal{H}}^{*}(X)$ is zero, where $\widetilde{\mathcal{H}}^{*}$
is reduced cohomology.

######  Theorem 1.3.

(Multiplicity) There exists $\varepsilon_{0}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0})$, problem (1.1) has at least
$\mathrm{cupl}\,C(M)$ pairs of sign solutions $\pm u$ with
$J_{\varepsilon}(u)<d_{\varepsilon}+k_{0}$.

In [10] it is proved that

$\text{cupl}\,C(M)\geq n+1,$

and

######  Theorem 1.4.

If $\mathcal{H}^{i}(M)=0$ for all $0<i<m$ and if there are $k$ cohomology
classes $\xi_{1},\ldots,\xi_{k}\in\mathcal{H}^{m}(M)$ whose cup-product is
non-trivial, then

$\mathrm{cupl}\,C(M)\geq k+n.$

From Theorem 1.4 we get that if
$M=\mathbb{S}^{1}\times\cdots\times\mathbb{S}^{1}$ $n-$times, then
$\text{cupl}\,C(M)=2n$.

##  2\. Preliminaries

Let $H_{\varepsilon}$ be the Hilbert space $H^{1}_{g}(M)$ equipped with the
inner product

(2.1) $\langle
u,v\rangle_{\varepsilon}\doteq\frac{1}{\varepsilon^{n}}\int_{M}\left(\varepsilon^{2}\left\langle\nabla_{g}u,\nabla_{g}v\right\rangle_{g}+uv\right)d\mu_{g}$

and the induced norm

$\|u\|^{2}_{\varepsilon}\doteq\frac{1}{\varepsilon^{n}}\int_{M}\left(\varepsilon^{2}\left|\nabla_{g}u\right|_{g}^{2}+u^{2}\right)d\mu_{g}.$

Consider the bilinear form $\mathcal{L}_{\varepsilon}:H_{\varepsilon}\times
H_{\varepsilon}\rightarrow\mathbb{R}$ given by

(2.2)
$\mathcal{L}_{\varepsilon}(u,v)\>\doteq\frac{1}{\varepsilon^{n}}\int_{M}\Bigg{[}\varepsilon^{2}\left\langle\nabla_{g}u,\nabla_{g}v\right\rangle_{g}+\left(\frac{s_{g}}{a_{m+n}}\varepsilon^{2}+1\right)uv\Bigg{]}d\mu_{g},\quad
u,v\in H_{\varepsilon}.$

From (1.2) we have $\mathcal{L}_{\varepsilon}$ is coercive, meaning that

(2.3)
$c_{\varepsilon}\|u\|_{\varepsilon}\leq\mathcal{L}_{\varepsilon}(u,u)^{\frac{1}{2}}\leq
c^{-1}_{\varepsilon}\|u\|_{\varepsilon},\quad\forall\ u\in H_{\varepsilon},$

for some $c_{\varepsilon}>0$. This implies that
$\mathcal{L}_{\varepsilon}(\cdot,\cdot)$ and
$\langle\cdot,\cdot\rangle_{\varepsilon}$ are equivalent inner products in
$H_{\varepsilon}$. For simplicity, we set
$\mathcal{L}_{\varepsilon}(u)\doteq\mathcal{L}_{\varepsilon}(u,u)$, $u\in
H_{\varepsilon}$.

Let $L^{q}_{\varepsilon}$ be the Banach spaces $L^{q}_{g}(M)$ with the norm

(2.4)
$|u|_{q,\varepsilon}\doteq\left(\frac{1}{\varepsilon^{n}}\int_{M}|u|^{q}d\mu_{g}\right)^{\frac{1}{q}}.$

For $q\in(2,p_{n})$ if $n\geq 3$ or $q>2$ if $n=2$, the embedding
$i_{\varepsilon}:H_{\varepsilon}\hookrightarrow L^{q}_{\varepsilon}$ is a
continuous map. Moreover, one can easily check that there exists a constant
$c$ independent of $\varepsilon$ such that

$|i_{\varepsilon}(u)|_{q,\varepsilon}\leq c\|u\|_{\varepsilon},\quad\text{for
any $u\in H_{\varepsilon}$}.$

Let $q^{\prime}\doteq\frac{q}{q-1}$ so that
$\frac{1}{q}+\frac{1}{q^{\prime}}=1$. Notice that for $v\in
L^{q^{\prime}}_{\varepsilon}$, the map

$\varphi\rightarrow\langle
v,i_{\varepsilon}\left(\varphi\right)\rangle\doteq\frac{1}{\varepsilon^{n}}\int_{M}v\cdot
i_{\varepsilon}\left(\varphi\right)d\mu_{g},\quad\varphi\in H_{\varepsilon},$

is a continuous functional by the compact embedding
$i_{\varepsilon}:H_{\varepsilon}\hookrightarrow L^{q}_{\varepsilon}$. For each
$\varphi\in L^{q^{\prime}}_{\varepsilon}$, define the functional
$\mathcal{F}_{\varphi}:H_{\varepsilon}\rightarrow\mathbb{R}$ by

$\mathcal{F}_{\varphi}(v)\doteq\frac{1}{\varepsilon^{n}}\int_{M}\varphi\cdot
i_{\varepsilon}\left(v\right)d\mu_{g},\quad\forall\ v\in H_{\varepsilon}.$

By the Lax-Milgram Theorem, there exists $u\in H_{\varepsilon}$ such that
$\mathcal{L}_{\varepsilon}(u,v)=\mathcal{F}_{\varphi}(v)$ for all $v\in
H_{\varepsilon}$. In other words, such function $u\in H_{\varepsilon}$ is the
weak solution of

(2.5)
$-\varepsilon^{2}\Delta_{g}u+\left(\frac{s_{g}}{a_{m+n}}\varepsilon^{2}+1\right)u=\varphi\quad\text{in
}M,$

where $\varphi\in L^{q^{\prime}}_{\varepsilon}$. Recall that, by elliptic
regularity theory, if $\varphi\in C^{k,\alpha}(M)$, then $u\in
C^{k+2,\alpha}(M)$.

From now on we use the notation $p\doteq p_{m+n}$. Consider the functional
$J_{\varepsilon}\colon H_{\varepsilon}\to\mathbb{R}$ given by

$J_{\varepsilon}(u)\doteq\frac{1}{\varepsilon^{n}}\int_{M}\left(\frac{1}{2}\varepsilon^{2}|\nabla_{g}u|_{g}^{2}+\frac{1}{2}\left(\frac{s_{g}}{a_{m+n}}\varepsilon^{2}+1\right)u^{2}-\frac{1}{p}|u|^{p}\right)d\mu_{g}.$

Its gradient is given by $\nabla J_{\varepsilon}\colon H_{\varepsilon}\to
L(H_{\varepsilon},\mathbb{R})$, where

$\displaystyle\nabla J_{\varepsilon}(u)(v)$ $\displaystyle\doteq$
$\displaystyle\frac{1}{\varepsilon^{n}}\int_{M}\left(\varepsilon^{2}\left\langle\nabla_{g}u,\nabla_{g}v\right\rangle_{g}+\left(\frac{s_{g}}{a_{m+n}}\varepsilon^{2}+1\right)uv-|u|^{p-2}uv\right)d\mu_{g}$
$\displaystyle=$
$\displaystyle\mathcal{L}_{\varepsilon}(u,v)-\frac{1}{\varepsilon^{n}}\int_{M}|u|^{p-2}uv\;d\mu_{g}$

Consider the operator $J^{\prime}_{\varepsilon}\colon H_{\varepsilon}\to
H_{\varepsilon}$ given by $J^{\prime}_{\varepsilon}(u)\doteq
u-K_{\varepsilon}(u)$, where $K_{\varepsilon}(u)$ is the solution of (2.5)
with $\varphi=|u|^{p-2}u$ and $u\in H_{\varepsilon}$. Then,

(2.6) $\nabla
J_{\varepsilon}(u)(v)=\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon}(u),v),\quad\text{for
$u,v\in H_{\varepsilon}$}.$

The Nehari manifold $\mathcal{N}_{\varepsilon}$ associate to the functional
$J_{\varepsilon}$ is the following set:

$\mathcal{N}_{\varepsilon}\doteq\Big{\\{}u\in
H_{\varepsilon}\setminus\\{0\\}:\mathcal{L}_{\varepsilon}(u,u)=\bigl{|}u\bigr{|}^{p}_{p,\varepsilon}\Big{\\}}.$

######  Lemma 2.1.

The functional $J_{\varepsilon}:H_{\varepsilon}\rightarrow\mathbb{R}$
satisfies the Palais-Smale condition. Moreover, the functional
$J_{\varepsilon}$ restricted to $\mathcal{N}_{\varepsilon}$ is coercive.

###### Proof.

Suppose that $(u_{k})\subset H_{\varepsilon}$, with

(2.7) $(J_{\varepsilon}(u_{k}))\quad\text{bounded},$

and

(2.8) $J^{\prime}_{\varepsilon}(u_{k})\rightarrow 0\quad\text{in
$H_{\varepsilon}$}.$

Recall that (2.8) means that

(2.9) $u_{k}-K(u_{k})\rightarrow 0\quad\text{in $H_{\varepsilon}$}.$

Hence, for every $\delta>0$ we have,

$\displaystyle|\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon}(u_{k}),v)|$
$\displaystyle=$
$\displaystyle\Big{|}\mathcal{L}_{\varepsilon}(u_{k},v)-\frac{1}{\varepsilon^{n}}\int_{M}|u_{k}|^{p-2}u_{k}v\;d\mu_{g}\Big{|}<\delta\mathcal{L}_{\varepsilon}(v)^{\frac{1}{2}},$

for $k>0$ large enough and for every $v\in H_{\varepsilon}$. If we take
$v=u_{k}$ above we find

$\Big{|}\mathcal{L}_{\varepsilon}(u_{k})-\frac{1}{\varepsilon^{n}}\int_{M}|u_{k}|^{p}\;d\mu_{g}\Big{|}<\delta\mathcal{L}_{\varepsilon}(u_{k})^{\frac{1}{2}},$

for every $\delta>0$, and $k>0$ large enough. In particular, for $\delta=1$,

(2.10)
$|u|_{p,\varepsilon}^{p}\leq\mathcal{L}_{\varepsilon}(u_{k})+\mathcal{L}_{\varepsilon}(u_{k})^{\frac{1}{2}},$

for $k>0$ sufficiently large. Since (2.7) says that

$\frac{1}{2}\mathcal{L}_{\varepsilon}(u_{k})-\frac{1}{p}|u|_{p,\varepsilon}^{p}<C<\infty,$

for all $k$ and some constant $C>0$, we deduce from (2.10) that

$\displaystyle\mathcal{L}_{\varepsilon}(u_{k})\leq
2C+\frac{2}{p}\left(\mathcal{L}_{\varepsilon}(u_{k})+\mathcal{L}_{\varepsilon}(u_{k})^{\frac{1}{2}}\right).$

Given that $\mathcal{L}_{\varepsilon}$ is coercive, see (2.3), and that
$2/p<1$, we get that $(u_{k})$ is bounded in $H_{\varepsilon}$. Hence, there
exists a subsequence $(u_{k_{j}})$ and $u\in H_{\varepsilon}$, with
$u_{k_{j}}\rightharpoonup u$ weakly in $H_{\varepsilon}$, and
$u_{k_{j}}\rightarrow u$ in $L^{p}_{\varepsilon}$ by the compact embedding
$H_{g}^{1}(M)\hookrightarrow L_{g}^{p}(M)$. From this we get that
$|u_{k_{j}}|^{p-2}u_{k_{j}}\rightarrow|u|^{p-2}u$ in
$L^{p^{\prime}}_{\varepsilon}$. Therefore, $K(u_{k_{j}})\rightarrow K(u)$ in
$H_{\varepsilon}$. So, (2.9) implies

$u_{k_{j}}\rightarrow u\quad\text{in $H_{\varepsilon}$}.$

We now prove that $J_{\varepsilon}$ restricted to $\mathcal{N}_{\varepsilon}$
is coercive. By definition,

$J_{\varepsilon}(u)=\frac{1}{2}\mathcal{L}_{\varepsilon}(u)-\dfrac{1}{p}|u|_{p,\varepsilon}^{p}.$

Now, if $u\in\mathcal{N}_{\varepsilon}$, we have
$\mathcal{L}_{\varepsilon}(u)=\bigl{|}u\bigr{|}^{p}_{p,\varepsilon}$. So,

$J_{\varepsilon}(u)=\frac{1}{2}\mathcal{L}_{\varepsilon}(u)-\dfrac{1}{p}\mathcal{L}_{\varepsilon}(u)=\Big{(}\frac{1}{2}-\dfrac{1}{p}\Big{)}\mathcal{L}_{\varepsilon}(u)\geq\dfrac{p-2}{2p}c_{\varepsilon}\|u\|_{\varepsilon}^{2}.$

Here we have used again that $\mathcal{L}_{\varepsilon}$ is coercive. ∎

Now, if we define

(2.11)
$S_{\varepsilon}\doteq\inf\left\\{\dfrac{\mathcal{L}_{\varepsilon}(u)}{|u|^{2}_{q,\varepsilon}}:\quad
u\in H_{\varepsilon},u\neq 0\right\\},$

we get that

(2.12)
$\textbf{m}_{\varepsilon}=\frac{p-2}{2p}S_{\varepsilon}^{\frac{p}{p-2}},$

where
$\textbf{m}_{\varepsilon}\doteq\inf_{\mathcal{N}_{\varepsilon}}J_{\varepsilon}$.
Identity (2.12) follows from the fact that if $u\in
H_{\varepsilon}\setminus\\{0\\}$, then
$t_{\varepsilon}(u)u\in\mathcal{N}_{\varepsilon}$, where

(2.13)
$t^{p-2}_{\varepsilon}(u)\doteq\frac{\mathcal{L}_{\varepsilon}(u)}{|u|^{p}_{p,\varepsilon}}.$

We close this section with the following result from [30]. It is well known
that there exists a unique (up to translation) positive finite-energy solution
$U$ of the equation

(2.14) $-\Delta U+U=|U|^{q-2}U\quad\text{on $\mathbb{R}^{n}$}.$

Moreover, the function $U$ is radial around some chosen point, and it is
exponentially decreasing at infinity (see [18]):

$|U(x)|\leq Ce^{-c|x|},$

and

$|\nabla U(x)|\leq Ce^{-c|x|}.$

Consider the functional $E:H^{1}(\mathbb{R}^{n})\rightarrow\mathbb{R}$,

$E(f)\doteq\int_{\mathbb{R}^{n}}\left(\frac{1}{2}\|\nabla
f\|^{2}+\frac{1}{2}f^{2}-\frac{1}{q}|f|^{q}\right)dx,$

and the corresponding Nehari Manifold

$N(E)\doteq\left\\{u\in H^{1}(\mathbb{R}):\int_{\mathbb{R}^{n}}\big{(}\|\nabla
u\|^{2}+u^{2}\big{)}dx=\dfrac{1}{q}\int_{\mathbb{R}^{n}}|u|^{q}dx\right\\}.$

Note that $U$ is a critical point of $E$ and minimizer of the functional $E$
restricted to $N(E)$. The minimum is then

(2.15) $\textbf{m}(E)\doteq\min\left\\{E(u):u\in
N(E)\right\\}=\dfrac{q-2}{2q}\|U\|_{q}^{q}.$

######  Theorem 2.2.

We have that $\lim_{\varepsilon\rightarrow
0}\textbf{m}_{\varepsilon}=\textbf{m}(E)$, where $\textbf{m}(E)$ is given by
(2.15).

##  3\. Existence Of Nodal Solutions

Recall that for $u\in H_{\varepsilon}$, $J^{\prime}_{\varepsilon}(u)\doteq
u-K_{\varepsilon}(u)$, where $K_{\varepsilon}(u)$ is the solution of (2.5)
with $\varphi=|u|^{p-2}u$, is the gradient of $J_{\varepsilon}$ with respect
to the inner product $\mathcal{L}_{\varepsilon}(\cdot,\cdot)$. Consider the
negative gradient flow $\varphi_{\varepsilon}:\mathcal{G_{\varepsilon}}\to
H_{\varepsilon}$ defined by

$\begin{cases}\dfrac{d}{dt}\varphi_{\varepsilon}(t,u)=-J^{\prime}_{\varepsilon}(\varphi_{\varepsilon}(t,u)),\\\
\varphi_{\varepsilon}(0,u)=u,\end{cases}$

where $\mathcal{G_{\varepsilon}}=\\{(t,u):u\in H_{g}^{1}(M),\;0\leq t\leq
T^{\varepsilon}(u)\\}$ and $T^{\varepsilon}(u)\in(0,+\infty)$ is the maximal
existence time for $\varphi_{\varepsilon}$.

######  Definition 3.1.

A set $\mathcal{D}\subset H_{g}^{1}(M)$ is _strictly positively invariant_
under the flow $\varphi_{\varepsilon}$, if for every $u\in\mathcal{D}$ and
$t\in(0,T^{\varepsilon}(u))$,
$\varphi_{\varepsilon}(t,u)\in\overset{\circ}{\mathcal{D}}$, where
$\overset{\circ}{\mathcal{D}}$ denotes the interior of $\mathcal{D}$ in
$H_{\varepsilon}$.

If $\mathcal{D}$ is strictly positively invariant under the flow
$\varphi_{\varepsilon}$, the set

$\mathcal{A}_{\varepsilon}(\mathcal{D})\doteq\\{u\in
H_{g}^{1}(M):\varphi_{\varepsilon}(t,u)\in\mathcal{D}\text{ for some
}t\in(0,T^{\varepsilon}(u))\\}$

is an open subset of $H^{1}_{g}(M)$. We define the convex cone of non-negative
functions by $\mathcal{P}\doteq\\{u\in H_{\varepsilon}:u\geq 0\\}$. For
$\alpha>0$ define also the tubular neighborhood

$\mathcal{B}_{\alpha}(\varepsilon,\pm\mathcal{P})\doteq\Big{\\{}u\in
H_{\varepsilon}:\text{dist}_{\varepsilon}(u,\pm\mathcal{P})\leq\alpha\Big{\\}},$

where

$\text{dist}_{\varepsilon}(u,\pm\mathcal{P})\doteq\min_{v\in\pm\mathcal{P}}\mathcal{L}_{\varepsilon}(u-v,u-v)^{\frac{1}{2}}.$

For $a\in\mathbb{R}$, we consider the set $J_{\varepsilon}^{a}\doteq
J_{\varepsilon}^{-1}((-\infty,a])=\\{u\in
H_{\varepsilon}:J_{\varepsilon}(u)\leq a\\}$. Moreover, for $\varepsilon>0$ we
let

$\mathcal{D}_{\varepsilon}\doteq\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})\cup\mathcal{B}_{\alpha}(\varepsilon,-\mathcal{P})\cup
J_{\varepsilon}^{0},$

and

(3.1) $\mathcal{Z}_{\varepsilon}\doteq
H_{\varepsilon}\setminus\mathcal{A}_{\varepsilon}(\mathcal{D}_{\varepsilon}).$

Our first result is the following lemma.

######  Lemma 3.2.

If $\alpha\doteq\frac{1}{2}S_{\varepsilon}^{p/2(p-2)}$, then

  1. (1)

$\left(\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P}\right)\cup\mathcal{B}_{\alpha}(\varepsilon,-\mathcal{P}))\cap\mathcal{E}_{\varepsilon}=\emptyset$;

  2. (2)

${B}_{\alpha}(\varepsilon,\pm\mathcal{P})$ is strictly positive invariant for
the flow $\varphi_{\varepsilon}.$

###### Proof.

  1. (1)

First, note that

$\displaystyle|u^{-}|_{p,\varepsilon}=\min_{v\in\mathcal{P}}|u-v|_{p,\varepsilon}$
$\displaystyle\leq$ $\displaystyle
S_{\varepsilon}^{-1/2}\min_{v\in\mathcal{P}}\mathcal{L}_{\varepsilon}(u-v,u-v)^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
S_{\varepsilon}^{-1/2}\text{dist}_{\varepsilon}(u,\mathcal{P}).$

Then, if
$u\in\mathcal{E}_{\varepsilon}\cap\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})$,

$0<S^{p/p-2}_{\varepsilon}\leq\mathcal{L}_{\varepsilon}(u^{-})=|u^{-}|^{p}_{p,\varepsilon}\leq
S_{\varepsilon}^{-p/2}\text{dist}_{\varepsilon}(u,\mathcal{P})^{p}\leq\frac{1}{2^{p}}S^{p/p-2}_{\varepsilon}.$

This contradiction gives us that
$\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})\cap\mathcal{E}_{\varepsilon}=\emptyset$.
In similar fashion,
$\mathcal{B}_{\alpha}(\varepsilon,-\mathcal{P})\cap\mathcal{E}_{\varepsilon}=\emptyset$.
Hence, (1) is established.

  2. (2)

We prove the assertion for $\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})$. We
first show that if $u\in\mathcal{B}_{\alpha}(\varepsilon,\pm\mathcal{P})$,
then $K_{\varepsilon}(u)$ is in the interior of
$\mathcal{B}_{\alpha}(\varepsilon,\pm\mathcal{P})$. Observe that

$\displaystyle\text{dist}_{\varepsilon}(K_{\varepsilon}(u),\mathcal{P})\mathcal{L}_{\varepsilon}(K_{\varepsilon}(u)^{-})^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\mathcal{L}_{\varepsilon}(K_{\varepsilon}(u)^{-},K_{\varepsilon}(u)^{-})$
$\displaystyle=$
$\displaystyle\frac{1}{\varepsilon^{n}}\int_{M}|u|^{p-2}uK_{\varepsilon}(u)^{-}\;d\mu_{g}\;\;\;$
$\displaystyle\leq$
$\displaystyle|u^{-}|_{p}^{p-1}\left|K_{\varepsilon}(u)^{-}\right|_{p}$
$\displaystyle\leq$ $\displaystyle
S_{\varepsilon}^{-p/2}\text{dist}_{\varepsilon}(u,\mathcal{P})^{p-1}\mathcal{L}_{\varepsilon}(K_{\varepsilon}(u)^{-})^{\frac{1}{2}}\quad\text{(by
\eqref{umdis})}$ $\displaystyle\leq$ $\displaystyle
S_{\varepsilon}^{-p/2}\left(\frac{1}{2}S_{\varepsilon}^{p/2(p-2)}\right)^{p-1}\mathcal{L}_{\varepsilon}(K_{\varepsilon}(u)^{-})^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{p-1}}S_{\varepsilon}^{p/2(p-2)}\mathcal{L}_{\varepsilon}(K_{\varepsilon}(u)^{-})^{\frac{1}{2}}.$

Hence,

$\text{dist}_{\varepsilon}(K_{\varepsilon}(u),\mathcal{P})\leq\frac{1}{2^{p-1}}S_{\varepsilon}^{p/2(p-2)}.$

It follows that $K_{\varepsilon}(u)$ is in the interior of
$\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})$. Given that the set
$\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})$ is convex, we get that

(3.3) $u-\lambda(J^{\prime}_{\varepsilon}(u))=(1-\lambda)u-\lambda
K_{\varepsilon}(u)\in\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})$

for all $u\in\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})$ and
$\lambda\in[0,1]$. Then, we get from (3.3) that

(3.4) $\lim_{\lambda\rightarrow
0^{+}}\frac{\mathrm{dist}\,\left(u+\lambda(-J^{\prime}_{\varepsilon}(u))),\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})\right)}{\lambda}=0,\quad\text{for
every $u\in\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})$}.$

Hence, using (3.4), we get from Theorem 1.49 in [38] that

(3.5)
$\varphi_{\varepsilon}(u,t)\in\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P}),\quad\text{for
every $u\in\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P}),\,0\leq
t<T_{\varepsilon}(u)$}.$

Finally, using a convexity-type argument as in Proposition 3.1 in [4], we get
from (3.5) that
$\varphi_{\varepsilon}(u,t)\in\mathrm{int}\,\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})$,
for every $u\in\mathcal{B}_{\alpha}(\varepsilon,\mathcal{P})$ and
$0<t<T_{\varepsilon}(u)$.

∎

######  Remark 3.3.

We have that $\inf_{\mathcal{E}_{\varepsilon}}J_{\varepsilon}$ is attained and
any minimizer of $J_{\varepsilon}$ on $\mathcal{E}_{\varepsilon}$ is a sign
changing solution to Eq. (1.1). Hence, we set

(3.6)
$\mathbf{d}_{\varepsilon}\doteq\inf_{\mathcal{E}_{\varepsilon}}J_{\varepsilon}.$

By Lemma 3.2, we have that $\mathcal{D}_{\varepsilon}$ is strictly positive
invariant for the flow $\varphi_{\varepsilon}$. Therefore, the set
$\mathcal{Z}_{\varepsilon}$ is a closed subset of $H_{\varepsilon}$. Moreover,
every function in $\mathcal{Z}_{\varepsilon}$ is sign changing and every sign
changing solution for Eq. (1.1) lies in $\mathcal{Z}_{\varepsilon}$.
Therefore,

$\mathbf{d}_{\varepsilon}\geq\inf_{\mathcal{Z}_{\varepsilon}}J_{\varepsilon}.$

######  Lemma 3.4 (Ekeland’s variational principle).

Given $\varepsilon>0$, $\delta>0$ and $u\in\mathcal{Z}_{\varepsilon}$ such
that
$J_{\varepsilon}(u)\leq\inf_{\mathcal{Z}_{\varepsilon}}J_{\varepsilon}+\delta$,
there exists $v\in\mathcal{Z}_{\varepsilon}$ such that $J_{\varepsilon}(v)\leq
J_{\varepsilon}(u)$,
$\mathcal{L}_{\varepsilon}(u-v)^{\frac{1}{2}}\leq\sqrt{\delta}$ and
$\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon}(v))^{\frac{1}{2}}\leq\sqrt{\delta}$.

###### Proof.

Let
$t_{0}\doteq\inf\Big{\\{}t>0:\sqrt{\delta}\leq\mathcal{L}_{\varepsilon}(\varphi_{\varepsilon}(t,u)-\varphi_{\varepsilon}(0,u))^{\frac{1}{2}}\Big{\\}}\in(0,\infty]$.
Suppose that
$\sqrt{\delta}<\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon}(\varphi_{\varepsilon}(t,u)))^{\frac{1}{2}}\quad\text{
for all }t\in(0,t_{0}).$ This implies,

$\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon}(\varphi_{\varepsilon}(t,u)))^{\frac{1}{2}}\leq\dfrac{1}{\sqrt{\delta}}\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon}(\varphi_{\varepsilon}(t,u)))\text{
for all }t\in(0,t_{0}).$

Hence,

$\displaystyle\sqrt{\delta}=$
$\displaystyle\mathcal{L}_{\varepsilon}(\varphi_{\varepsilon}(t_{0},u)-\varphi_{\varepsilon}(0,u))^{\frac{1}{2}}=\mathcal{L}_{\varepsilon}\left(\int_{0}^{t_{0}}\dfrac{d}{dt}\varphi_{\varepsilon}(t,u)dt\right)^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\mathcal{L}_{\varepsilon}\left(\int_{0}^{t_{0}}-J^{\prime}_{\varepsilon}(\varphi_{\varepsilon}(t,u))\right)^{\frac{1}{2}}\leq\int_{t_{0}}^{0}\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon}(\varphi_{\varepsilon}(t,u)))^{\frac{1}{2}}dt$
$\displaystyle\leq$
$\displaystyle\dfrac{1}{\sqrt{\delta}}\int_{t_{0}}^{0}\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon}(\varphi_{\varepsilon}(t,u)))dt=\dfrac{1}{\sqrt{\delta}}\int_{t_{0}}^{0}\dfrac{d}{dt}J_{\varepsilon}(\varphi_{\varepsilon}(t,u))dt$
$\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{\delta}}(J_{\varepsilon}(u)-J_{\varepsilon}(\varphi_{\varepsilon}(t_{0},u))\leq\sqrt{\delta},$

given that
$J_{\varepsilon}(u)\leq\inf_{\mathcal{Z}_{\varepsilon}}J_{\varepsilon}+\delta$
and $\inf_{\mathcal{Z}_{\varepsilon}}J_{\varepsilon}\leq
J_{\varepsilon}(\varphi(t_{0},u))$. We have reached a contradiction, and,
therefore, the lemma follows. ∎

###### Proof of Theorem 1.1.

Let $u_{k}$ a minimizing sequences for $J_{\varepsilon}$ in
$\mathcal{Z}_{\varepsilon}$. By Lemma 3.4, we may assume that
$\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon}(u_{k}))\rightarrow 0$ when
$k\rightarrow\infty$. From Lemma 2.1,
$J_{\varepsilon}:H_{\varepsilon}^{1}(M)\rightarrow\mathbb{R}$ satifies the
Palais-Smale condition, and so there exists a subsequence
$u_{k_{j}}\rightarrow v_{\varepsilon}$ strongly in $H_{\varepsilon}^{1}(M)$
and
$J_{\varepsilon}(v_{\varepsilon})=\inf_{\mathcal{Z}_{\varepsilon}}J_{\varepsilon}$.
Since $\mathcal{Z}_{\varepsilon}$ is closed in $H_{\varepsilon}^{1}(M)$, we
get that $v_{\varepsilon}\in\mathcal{Z}_{\varepsilon}$. Finally,
$\mathcal{Z}_{\varepsilon}$ is invariant by negative flow, so,
$v_{\varepsilon}$ is fixed point of flow and, therefore, a solution for Eq.
(1.1). Now since every sign changins solution of (1.1) belongs to
$\mathcal{E}_{\varepsilon}$, we have that
$v_{\varepsilon}^{\pm}\in\mathcal{N}_{\varepsilon}$ and

$\textbf{d}_{\varepsilon}=\inf_{\mathcal{Z}_{\varepsilon}}J_{\varepsilon}\geq\inf_{\mathcal{E}_{\varepsilon}}J_{\varepsilon}\geq
2\textbf{m}_{\varepsilon}.$

∎

For any $\varepsilon>0$, we let

$E_{\varepsilon}(f):=\frac{1}{\varepsilon^{n}}\int_{\mathbb{R}^{n}}\left(\frac{\varepsilon^{2}}{2}|\nabla
f|^{2}+\frac{1}{2}f^{2}-\frac{1}{q}|f|^{q}\right)dx.$

Now, if we set $U_{\varepsilon}(x)\doteq U\left(\frac{x}{\varepsilon}\right)$,
then $U_{\varepsilon}$ is a critical point of $E_{\varepsilon}$, i.e.,
$U_{\varepsilon}$ is a solution of

(3.7) $-\varepsilon^{2}\Delta
U_{\varepsilon}+U_{\varepsilon}=U_{\varepsilon}^{q-1}.$

Let $x\in M$, since $M$ is closed we can fix $r_{0}>0$ such that
$\exp_{x}|_{B(0,r_{0})}:B(0,r_{0})\rightarrow B_{g}(x,r_{0})$ is a
diffeomorphism. Let $\chi_{r}$ be a smooth radial cut-off function. Let us
define on M the following function:

(3.8)
$u_{\varepsilon,x}(y):=\begin{cases}U_{\varepsilon}(\exp^{-1}_{x}(y))\chi_{r}(\exp^{-1}_{x}(y))&\text{if
$y\in B_{g}(x,r)$},\\\ 0&\text{otherwise}.\end{cases}$

Now, consider the set $F(M)=\\{(x,y)\in M\times M:x\neq y\\}$. We define

(3.9) $F_{\varepsilon}(M)\doteq\\{(x,y)\in M\times
M:\quad\mathrm{dist}_{g}(x,y)\geq 2\varepsilon R_{0}\\}\subset F(M),$

where $R_{0}=\text{diam}(M)$. Moreover, we define the function
$i_{\varepsilon}:F_{\varepsilon}(M)\rightarrow H_{\varepsilon}$ by

(3.10) $i_{\varepsilon}(x,y)\doteq
t_{\varepsilon}(u_{\varepsilon,x})u_{\varepsilon,x}-t_{\varepsilon}(u_{\varepsilon,y})u_{\varepsilon,y},$

where $u_{\varepsilon,x}$ and $u_{\varepsilon,y}$ are defined by (3.8). Recall
that for $u\in H_{\varepsilon}\setminus\\{0\\}$,
$t_{\varepsilon}(u)u\in\mathcal{N}_{\varepsilon}$ where $t_{\varepsilon}(u)$
is given by (2.13).

######  Lemma 3.5.

For every $\varepsilon>0$ the function
$i_{\varepsilon}:F_{\varepsilon}(M)\rightarrow H_{\varepsilon}$ is continuous.
For each $\delta>0$ there exists $\varepsilon_{0}$ such that, if
$\varepsilon\in(0,\varepsilon_{0})$ then

$i_{\varepsilon}(x,y)\in
J_{\varepsilon}^{2\mathbf{m}(E)+\delta}\cap\mathcal{E}_{\varepsilon}\quad\text{for
all }\quad(x,y)\in F_{\varepsilon}(M).$

###### Proof.

Let $x,y\in F_{\varepsilon}(M)$. From Proposition 4.2 in [6], we have that for
every $\delta>0$, there exists an $\varepsilon_{0}>0$ such that for
$\varepsilon<\varepsilon_{0}$,

(3.11)
$t_{\varepsilon}(u_{\varepsilon,x})u_{\varepsilon,x}\quad\text{and}\quad
t_{\varepsilon}(u_{\varepsilon,y})u_{\varepsilon,y}\in
J^{\mathbf{m}(E)+\frac{\delta}{2}}_{\varepsilon}.$

Observe that

$J(i_{\varepsilon}(x,y))=J(t_{\varepsilon}(u_{\varepsilon,x})u_{\varepsilon,x})+J(t_{\varepsilon}(u_{\varepsilon,x})u_{\varepsilon,y}),$

given that $u_{\varepsilon,x}$ and $u_{\varepsilon,y}$ have disjoint support.
From (3.11) we immediately obtain that $i_{\varepsilon}(x,y)\in
J_{\varepsilon}^{2\mathbf{m}(E)+\delta}$. Finally, using once again that
$u_{\varepsilon,x}$ and $u_{\varepsilon,y}$ have disjoint support we get that
$i_{\varepsilon}(x,y)^{+}=t_{\varepsilon}(u_{\varepsilon,x})u_{\varepsilon,x}\in\mathcal{N}_{\varepsilon}$
and
$i_{\varepsilon}(x,y)^{-}=-t_{\varepsilon}(u_{\varepsilon,y})u_{\varepsilon,y}\in\mathcal{N}_{\varepsilon}$,
therefore, $i_{\varepsilon}(x,y)\in\mathcal{E}_{\varepsilon}$. ∎

######  Proposition 3.6.

We have that

$\lim_{\varepsilon\rightarrow 0}\mathbf{d}_{\varepsilon}=2\mathbf{m}(E).$

###### Proof.

From Theorem 1.1 and Theorem 2.2 we have that

$\mathbf{d}_{\varepsilon}\geq 2\mathbf{m}_{\varepsilon}\quad\text{ and
}\quad\lim_{\varepsilon\rightarrow 0}\mathbf{m}_{\varepsilon}=\mathbf{m}(E).$

Moreover, from Lemma 3.5, we get for every $\delta>0$,

$d_{\varepsilon}\leq 2\mathbf{m}(E)+\delta,\quad\text{for $\varepsilon>0$
small enough}.$

Therefore,

$\lim_{\varepsilon\rightarrow 0}\mathbf{d}_{\varepsilon}=2\mathbf{m}(E),$

as claimed. ∎

##  4\. Concentration of sign changing function in $\mathcal{Z}_{\varepsilon}$

We begin this section with the following important result.

######  Lemma 4.1.

Let $u_{k}\in\mathcal{Z}_{\varepsilon_{k}}\cap
J_{\varepsilon_{k}}^{\textbf{d}_{\varepsilon_{k}}+\delta_{k}}$ where
$\varepsilon_{k},\delta_{k}>0$ are such that
$\varepsilon_{k},\delta_{k}\rightarrow 0$ as $k\rightarrow\infty$. Then,

$\mathrm{dist}_{\varepsilon_{k}}(u_{k}^{\pm},\mathcal{N}_{\varepsilon_{k}})\rightarrow
0\quad\text{ and }\quad
J_{\varepsilon_{k}}(u_{\varepsilon_{k}}^{\pm})\rightarrow\textbf{m}(E)\text{
as }\quad k\rightarrow\infty.$

###### Proof.

Observe that

(4.1) $\displaystyle\frac{p-2}{2p}\mathcal{L}_{\varepsilon_{k}}(u_{k},u_{k})$
$\displaystyle=J_{\varepsilon_{k}}(u_{k})-\frac{1}{p}\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon_{k}}(u_{k}),u_{k})$
$\displaystyle\leq
J_{\varepsilon_{k}}(u_{k})+\mathcal{L}_{\varepsilon_{k}}(J^{\prime}_{\varepsilon_{k}}(u_{k}),J^{\prime}_{\varepsilon_{k}}(u_{k}))^{\frac{1}{2}}\mathcal{L}_{\varepsilon_{k}}(u_{k},u_{k})^{\frac{1}{2}}.$

From Lemma 3.4 we may assume that
$\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon_{k}}(u_{k}))^{\frac{1}{2}}\rightarrow
0$. Therefore, from this fact and (4.1) we get that
$\mathcal{L}_{\varepsilon_{k}}(u_{k},u_{k})$ is uniformly bounded. Hence,

$\mathcal{L}_{\varepsilon}(J^{\prime}_{\varepsilon_{k}}(u^{\pm}_{k}),u^{\pm}_{k})=\Big{|}\mathcal{L}_{\varepsilon_{k}}(u^{\pm}_{k},u^{\pm}_{k})-|u^{\pm}_{k}|^{p}_{p,\varepsilon_{k}}\Big{|}\rightarrow
0.$

From this we get that $t_{\varepsilon_{k}}(u^{\pm}_{k})$, defined by (2.13),
tends to $1$ and, therefore,

$\mathrm{dist}_{\varepsilon_{k}}(u_{k}^{\pm},\mathcal{N}_{\varepsilon_{k}})\leq\mathcal{L}_{\varepsilon_{k}}(u^{\pm}_{k}-t_{\varepsilon_{k}}(u^{\pm}_{k})u^{\pm}_{k},u^{\pm}_{k}-t_{\varepsilon_{k}}(u^{\pm}_{k})u^{\pm}_{k})^{\frac{1}{2}}\rightarrow
0,\quad\text{as $k\rightarrow 0$}.$

If we use this, together with Theorem 2.2 and Proposition 3.6, we get

$\displaystyle 2m(E)$
$\displaystyle\leq\lim_{k\rightarrow\infty}J_{\varepsilon}(t_{\varepsilon_{k}}(u^{+}_{k})u^{+}_{k})+\lim_{k\rightarrow\infty}J_{\varepsilon}(t_{\varepsilon_{k}}(u^{-}_{k})u^{-}_{k})$
$\displaystyle=\lim_{k\rightarrow\infty}J_{\varepsilon}(u^{+}_{k})+\lim_{k\rightarrow\infty}J_{\varepsilon}(u^{-}_{k})$
$\displaystyle=\lim_{k\rightarrow\infty}J_{\varepsilon_{k}}(u_{k})$
$\displaystyle=2m(E).$

Therefore,

$\lim_{k\rightarrow\infty}J_{\varepsilon}(u^{\pm}_{k})=m(E).$

∎

######  Remark 4.2.

On any closed Riemannian manifold $M$ for any $\varepsilon>0$ there are points
$x_{j}\in M$, with $j=1,\ldots,K_{\varepsilon}$, such that the balls
$(B(x_{j},\varepsilon))$ are disjoint, and the set is maximal under this
condition. It follows that the ball $B(x_{j},2\varepsilon)$ cover $M$. It is
easy to construct closed sets $A_{j}$ such that $B(x_{j},\varepsilon)\subset
A_{j}\subset B(x_{j},2\varepsilon)$ which cover $M$ and only intersect in
their boundaries. Moreover, one can see by volume comparison argument that, if
$\varepsilon$ is small enough, there exists a constat $K>0$, independent of
$\varepsilon$, such that for any point in $M$ can be in at most $K$ of the
balls $B(x_{j},3\varepsilon)$.

######  Theorem 4.3.

For any $\eta\in(0,1)$ there exist $\varepsilon_{0},\delta_{0}>0$ such that
for any $\varepsilon\in(0,\varepsilon_{0})$, $\delta\in(0,\delta_{0})$ and
$u\in\mathcal{Z}_{\varepsilon}\cap
J_{\varepsilon}^{\textbf{d}_{\varepsilon}+\delta}$ there exist
$x^{+}=x^{+}(u)$ and $x^{-}=x^{-}(u)$ in M such that

$\int_{B(x^{\pm},r)}|u^{\pm}|^{p}d\mu_{g}\geq\eta\int_{M}|u^{\pm}|^{p}d\mu_{g}.$

###### Proof.

We do the proof only for $|u^{+}|^{p}$, the proof for $|u^{-}|^{p}$ is
similar. Assume the theorem is not true. Then there exist $0<\eta<1$ and
sequences $\varepsilon_{k}\rightarrow 0$, $\delta_{k}\rightarrow 0$ and
$u_{k}\in\mathcal{Z}_{\varepsilon_{k}}\cap
J_{\varepsilon_{k}}^{\textbf{d}_{\varepsilon_{k}}+\delta_{k}}$ such that for
all $x\in M$,

$\int_{B(x,r)}|u^{+}_{k}|^{p}d\mu_{g}<\eta\int_{M}|u^{+}_{k}|^{p}d\mu_{g}.$

We first show that there exist $\beta>0$, $k_{0}\in\mathbb{N}$ for each
$k>k_{0}$, a point $x_{k}\in M$ such that

(4.2)
$\dfrac{1}{\varepsilon^{n}_{k}}\int_{B(x_{k},2\varepsilon_{k})}|u^{+}_{k}|^{p}d\mu_{g}>\beta.$

Let us consider $\varepsilon_{0}>0$ such that for any
$0<\varepsilon<\varepsilon_{0}$ one can construct a set like in the Remark
4.2. Let $u_{k,j}\doteq u^{+}_{k}\chi_{A_{j}^{k}}$ be the restriction of
$u^{+}_{k}$ to $A_{j}^{k}$ and $0$ away from $A_{j}^{k}$. Then,

$\displaystyle|u^{+}_{k}|_{p,\varepsilon_{k}}^{p}$
$\displaystyle=\dfrac{1}{\varepsilon^{n}_{k}}\int_{M}|u^{+}_{k}|^{p}d\mu_{g}=\sum_{j}|u_{k,j}|_{p,\varepsilon_{k}}^{p}$
$\displaystyle=\sum_{j}\big{(}|u_{k,j}|_{p,\varepsilon_{k}}^{p-2}\big{)}\big{(}|u_{k,j}|_{p,\varepsilon_{k}}^{2}\big{)}\leq\Big{(}\max_{j}|u_{k,j}|_{p,\varepsilon_{k}}^{p-2}\Big{)}\sum_{j}|u_{k,j}|_{p,\varepsilon_{k}}^{2}.$

Now, let $\varphi_{\varepsilon_{k}}$ be the cut-off functions on
$\mathbb{R}^{n}$ which are $1$ in $B(0,2\varepsilon_{k})$ and vanish away from
$B(0,3\varepsilon_{k})$. Moreover,
$\|\nabla\varphi_{\varepsilon_{k}}\|=\dfrac{1}{\varepsilon_{k}}$ in the
intermediate annulus. Define for $j=1,\ldots,K_{\varepsilon}$,

$\widetilde{u}_{k,j}=u^{+}_{k}(x)\varphi_{\varepsilon_{k}}(d(x,x_{j})).$

Since $u_{k,j}\leq\widetilde{u}_{k,j}$, we have
$|u_{k,j}|_{p,\varepsilon_{k}}^{2}\leq|\widetilde{u}_{k,j}|_{p,\varepsilon_{k}}^{2}\leq
C\|\widetilde{u}_{k,j}\|_{\varepsilon_{k}}^{2}.$ Then, since we have that
$\widetilde{u}_{k,j}\leq u^{+}_{k}$ and on
$C_{k,j}=B(x_{j},3\varepsilon_{k})-A_{j}^{k}$,

$\varepsilon^{2}_{k}\|\nabla\widetilde{u}_{k,j}\|^{2}\leq
2\varepsilon^{2}_{k}\|\nabla u^{+}_{k}\|^{2}+2(u^{+}_{k})^{2}.$

We have that

(4.3) $|u_{k}^{+}|_{p,\varepsilon_{k}}^{p}\leq
c\|u_{k}^{+}\|_{\varepsilon_{k}}^{2}\max_{j}|u_{k,j}|_{p,\varepsilon_{k}}^{p-2}.$

Now, from Lemma 4.1,

(4.4)
$\lim_{k\rightarrow\infty}\|u_{k}^{\pm}\|_{\varepsilon_{k}}^{2}=\lim_{k\rightarrow\infty}|u_{k}^{\pm}|_{p,\varepsilon_{k}}^{p}=\dfrac{2p}{p-2}\textbf{m}(E).$

Therefore, there exists a $\beta>0$ such that for each $k$ large enough we can
find a $j\in\\{1,\ldots,K_{\varepsilon}\\}$ such that

$\beta<|u_{k,j}|_{p,\varepsilon_{k}}^{p}=\dfrac{1}{\varepsilon_{k}^{n}}\int_{A_{j}^{k}}|u_{k}^{+}|^{p}d\mu_{g}\leq\dfrac{1}{\varepsilon_{k}^{n}}\int_{B(x_{j},2\varepsilon_{k})}|u_{k}^{+}|^{p}d\mu_{g}.$

From this it follows that (4.2) is established.

Now, from (4.2) and given that $t_{\varepsilon_{k}}(u^{+}_{k})$ tends to $1$,
there is a $k^{\prime}_{0}\in\mathbb{N}$ such that for each $k>k^{\prime}_{0}$

(4.5)
$\dfrac{1}{\varepsilon^{n}_{k}}\int_{B(x_{k},2\varepsilon_{k})}|t_{\varepsilon_{k}}(u^{+}_{k})u^{+}_{k}|^{p}d\mu_{g}>\frac{\beta}{2}.$

From Lemma 4.1 we get that

$m_{\varepsilon_{k}}\leq
J_{\varepsilon_{k}}(t_{\varepsilon_{k}}(u^{+}_{k})u^{+}_{k})\leq
m_{\varepsilon_{k}}+\overline{\delta_{k}},$

for some sequence $\\{\overline{\delta_{k}}\\}$ such that
$\overline{\delta_{k}}\rightarrow 0$.

Set $v_{k}\doteq t_{\varepsilon_{k}}(u^{+}_{k})u^{+}_{k}$. Then, for each
$k\in\mathbb{N}$ we have that
$v_{k}\in\Sigma_{\varepsilon_{k},m_{\varepsilon}+\overline{\delta_{k}}}$,
where

$\Sigma_{\varepsilon_{k},m_{\varepsilon}+\overline{\delta_{k}}}\doteq\\{u\in
N_{\varepsilon_{k}}:J_{\varepsilon_{k}}(u)<m_{\varepsilon}+\overline{\delta_{k}}\\}.$

Moreover, from (4.5) we have that

$\dfrac{1}{\varepsilon^{n}_{k}}\int_{B(x_{k},2\varepsilon_{k})}|v_{k}|^{p}d\mu_{g}>\frac{\beta}{2}.$

Now, Lemma 3.4 in [30] gives a function $\bar{v}_{k}=v_{k,1}+v_{k,2}$ such
that
$\bar{v}_{k}\in\Sigma_{\varepsilon_{k},m_{\varepsilon}+\overline{\delta_{k}}}$,
and $v_{k,1}$ is supported inside a ball centered at $x_{k}$, $v_{k,1}$ and
$v_{k,2}$ have disjoint support and $v_{k}=\bar{v}_{k}$ in
$B(x_{k},2\varepsilon_{k})$ and outside $B(x_{k},r)$.

Then, we have that

$\dfrac{1}{\varepsilon^{n}_{k}}\int_{M}|v^{+}_{k,1}|^{p}d\mu_{g}>\frac{\beta}{2},$

and

$\dfrac{1}{\varepsilon^{n}_{k}}\int_{M}|v^{+}_{k,2}|^{p}d\mu_{g}\geq\dfrac{(1-\eta)}{\varepsilon^{n}_{k}}\int_{M}|v_{k}|^{p}d\mu_{g}\geq(1-\eta)\frac{2p}{p-2}\mathbf{m}_{\varepsilon}.$

Here $\eta\in(0,1)$ and it is chosen to be very close to $1$. Now, from
Corollary 3.3 in [30], there exists $\delta_{0}>0$, independent of $k$, such
that
$J_{\varepsilon_{k}}(\bar{v}_{k})\geq\Psi(\delta_{0})\mathbf{m}_{\varepsilon}$,
where $\Psi:(0,1)\rightarrow(1,\infty)$ is defined in Lemma 3.2 in [30].

On the other hand for $k$ large enough we have that
$J_{\varepsilon_{k}}(\bar{v}_{k})<\mathbf{m}_{\varepsilon}+2\delta_{k}<\Psi(\delta_{0})\mathbf{m}_{\varepsilon}$,
reaching a contradiction.

∎

##  5\. Multiplicity of Nodal Solutions

Recall that $F(M)=\\{(x,y)\in M\times M:x\neq y\\}$, and

(5.1) $F_{\varepsilon}(M)\doteq\\{(x,y)\in M\times
M:\quad\mathrm{dist}_{g}(x,y)\geq 2\varepsilon r_{0}\\}\subset F(M),$

where $R_{0}=\text{diam}(M)$. We define the function
$i_{\varepsilon}:F_{\varepsilon}\rightarrow H_{\varepsilon}$ by

(5.2)
$i_{\varepsilon}(x,y)=t_{\varepsilon}(u_{\varepsilon,x})u_{\varepsilon,x}-t_{\varepsilon}(u_{\varepsilon,y})u_{\varepsilon,y},$

where $t_{\varepsilon}(u)\in\mathbb{R}$ such that if $u\in
H_{\varepsilon}-\\{0\\}$ then
$t_{\varepsilon}(u)u\in\mathcal{N}_{\varepsilon}.$

######  Lemma 5.1.

For every $\varepsilon>0$ the function $i_{\varepsilon}$ is continuous. For
each $\delta>0$ there exists $\varepsilon_{0}$ such that, if
$\varepsilon\in(0,\varepsilon_{0})$ then

$i_{\varepsilon}(x,y)\in
J_{\varepsilon}^{2c_{\infty}+\delta}\cap\mathcal{E}_{\varepsilon}\quad\text{for
all }\quad(x,y)\in F_{\varepsilon}(M).$

###  5.1. Center of Mass.

In [19], H. Karcher and K. Grove define the center of mass of a function $u$
on a closed Riemannian manifold $(M,g)$, in the following form, since $M$ is
closed the exists $r_{0}>0$ such that for any $x\in M$ and $r\leq r_{0}$ the
geodesic ball of the radius $r$ center in at $x$, $B(x,r)$ is strongly convex
(see [30] and [19] for details). Let $u\in L^{1}(M)$ nonnegative. We consider
the function continuous $P_{u}:M\rightarrow\mathbb{R}$,

$P_{u}(x)\doteq\int_{M}(d(x,y))^{2}u(y)d\mu_{g}(y).$

Then, H. Karcher and K. Grove, proved that if $r>0$ is small enough such that
the support of $u$ is contained in $B(x,r)$, then $P_{u}$ as a unique global
minimum, which they defined as the center of mass of $u$ and denoted by
$\textbf{cm}(u).$

We consider now the center mass of a function introduced in Section 5 of [30].
For any function $u\in L^{1}(M)$ and positive $r$ let the
$(u,r)-$_concentration function_ defined by

(5.3)
$C_{u,r}(x)\doteq\dfrac{\int_{B(x,r)}|u|d\mu_{g}}{\|u\|_{{}_{L^{1}(M)}}}.$

We have that $C_{u,r}\colon M\rightarrow[0,1]$ it is a continuos function.
Where if $r\geq\mathrm{diam}(M)$, then $C_{u,r}\equiv 1$ and
$\lim_{r\rightarrow 0}C_{u,r}(x)=0$.

We define the $r-$_concentration coefficient_ of $u$, $C_{r}(u)$ be the
maximum of $C_{u,r}$,

(5.4) $C_{r}(u)\doteq\max_{x\in M}C_{u,r}(x).$

For any $\eta\in(0,1)$ let $L^{1}_{\varepsilon,\eta}(M,g)\doteq\\{u\in
L^{1}(M):C_{r}(u)>\eta\\}$. We will use the following construction, for any
$\eta\in(1/2,1)$ consider the piecewise linear continuous function
$\varphi_{\eta}:\mathbb{R}\rightarrow[0,1]$ defined by $\varphi_{\eta}(t)=0$
if $t\leq 1-\eta$ and $\varphi_{\eta}(t)=1$ if $t\geq\eta$ it is a linear and
increasing in $[1-\eta,\eta]$.

For $r>0$ such that $2r\leq r_{0}$, we let

$\Phi_{r,\eta}(u)(x)\doteq\varphi_{\eta}(C_{u,r}(x))u(x),\quad\text{where
$u\in L_{r,\eta}^{1}(M)$ and $x\in M$}.$

For the proof of the following results, namely Lemma 5.2 and Theorem 5.3, see
Pag. 15 of the already mentioned paper [30].

######  Lemma 5.2.

For any $u\in L_{r,\eta}^{1}(M)$ the support of $\Phi_{r,\eta}(u)$ is
contained in a geodesic ball of radius $2r.$ (centered at a point of maximal
r-concentration)

######  Theorem 5.3.

For any $0<r<1/2r_{0}$ and $\eta>1/2$ there exists continuos function
$\textbf{Cm}(r,\eta):L_{r,\eta}^{1}\rightarrow M$ , such that if $x\in M$
verifies that $C_{r,u}(x)>\eta$ then $\textbf{Cm}(r,\eta)(u)\in B(x,2r).$
Where

(5.5) $\textbf{Cm}(r,\eta)(u)=\textbf{cm}(\Phi_{r,\eta}(u)).$

######  Definition 5.4.

For any function $u$ as in Theorem 5.3, $\textbf{Cm}(r,\eta)(u)$ will be
called a $(r,\eta)-$_Riemannian center of mass_ of $u$.

######  Proposition 5.5.

Let $0<r<1/2r_{0}$. Then, there exist $\delta_{0}>0$ and $\varepsilon_{0}>0$
such that, for any $u\in\mathcal{Z}_{\varepsilon}\cap
J_{\varepsilon}^{\textbf{d}_{\varepsilon}+\delta}$ with
$\varepsilon\in(0,\varepsilon_{0})$ and $\delta\in(0,\delta_{0}]$,

$\textbf{Cm}(\varepsilon r,\eta)((u^{+})^{p})\neq\textbf{Cm}(\varepsilon
r,\eta)((u^{-})^{p}).$

###### Proof.

Let $\varepsilon_{k},\delta_{k}>0$ and
$u_{k}\in\mathcal{Z}_{\varepsilon_{k}}\cap
J_{\varepsilon_{k}}^{\textbf{d}_{\varepsilon}+\delta}$ be such that
$\varepsilon_{k}\rightarrow 0,$ $\delta_{k}\rightarrow 0$ and for each $k$,
$\textbf{Cm}(\varepsilon_{k}r,\eta)((u^{+}_{k})^{p})=\textbf{Cm}(\varepsilon_{k}r,\eta)((u^{-}_{k})^{p}).$
From Theorem 4.3, there exist sequences $q^{+}_{k},q^{-}_{k}\in M$ such that

(5.6)
$\int_{B(q^{\pm}_{k},\varepsilon_{k}r)}|u^{\pm}_{k}|^{p}d\mu_{g}\geq\eta\int_{M}|u^{\pm}_{k}|^{p}d\mu_{g}.$

From (5.3) and (5.6),

$C_{u^{+},\varepsilon_{k}r}(q_{k}^{+})\doteq\dfrac{\int_{B(q_{k}^{+},\varepsilon_{k}r)}|(u^{+})^{p}|d\mu_{g}}{\|(u^{+})^{p}\|_{{}_{L^{1}(M)}}}\geq\eta.$

Moreover, if we use Theorem 5.3, we get

$\textbf{Cm}(\varepsilon_{k}r,\eta)((u^{+}_{k})^{p})\in
B(q^{+}_{k},2\varepsilon_{k}r).$

Hence,

$\lim_{k\rightarrow\infty}\|\textbf{Cm}(\varepsilon_{k}r,\eta)((u^{+}_{k})^{p})-q^{+}_{k}\|=0.$

In similar fashion, we also have

$\lim_{k\rightarrow\infty}\|\textbf{Cm}(\varepsilon_{k}r,\eta)((u^{-}_{k})^{p})-q^{-}_{k}\|=0.$

Now, given that
$\textbf{Cm}(\varepsilon_{k}r,\eta)((u^{+}_{k})^{p})=\textbf{Cm}(\varepsilon_{k}r,\eta)((u^{-}_{k})^{p})$
and $M$ is compact, we have that

(5.7) $q^{+}_{k}\rightarrow q\quad\text{ and}\quad q^{-}_{k}\rightarrow q.$

We now define $w_{k}^{1},w^{2}_{k}\in H^{1}(\mathbb{R}^{n})$ by

(5.8)
$w_{k}^{1}(x)\doteq\chi(\varepsilon_{k}\|x\|)u_{k}(exp_{c_{k}^{+}}(\varepsilon_{k}x))\quad\text{and}\quad
w_{k}^{2}(x)\doteq\chi(\varepsilon_{k}\|x\|)u_{k}(exp_{c_{k}^{-}}(\varepsilon_{k}x)),$

where $c_{k}^{+}\doteq\textbf{Cm}(\varepsilon_{k}r,\eta)((u^{+}_{k})^{p})$ and
$c_{k}^{-}\doteq\textbf{Cm}(\varepsilon_{k}r,\eta)((u^{-}_{k})^{p}.$ Note
that, since we are considering the centers of mass $c_{k}^{\pm}$, then
$w_{k}^{1}\neq 0$ and $w_{k}^{2}\neq 0$.

By (4.4), the sequence $\|u_{k}\|_{\varepsilon_{k}}$ is bounded, so
$w_{k}^{1},w_{k}^{2}$ are bounded in $\in H^{1}(\mathbb{R}^{n})$ (see Lemma
5.6 in [6]). Therefore, we have that, up to a subsequence,
$w_{k}^{i}\rightharpoonup w^{i}$ weakly in $H^{1}(\mathbb{R}^{n})$,
$w_{k}^{i}\rightarrow w^{i}$ a.e in $\mathbb{R}^{n}$, and
$w_{k}^{i}\rightarrow w^{i}$ strongly in $L_{loc}^{p}(\mathbb{R}^{n})$ for
$i=1,2$.

Now, from Theorem 4.3, $(w^{1})^{+}\neq 0$, then $w^{1}>0.$ Analogously
$w^{2}<0$. Both $w^{1}$ and $w^{2}$ are weak solutions of equation $-\Delta
w+w=|w|^{p-2}w$ and $J_{\infty}(w^{i})\leq 2c_{\infty}$. Since in our setting
we still have Ekeland’s Lemma, see Lemma 3.4, the proof follows the argument
of Lemma 5.7 in [6].

We consider the function

(5.9)
$w_{k}(x)\doteq\chi(\varepsilon_{k}x)u_{k}(exp_{q}(\varepsilon_{k}x)),\quad\text{where
$q$ is as in \eqref{Pointq}}.$

Once more, up to a subsequence $w_{k}\rightharpoonup w$ weakly in
$H^{1}(\mathbb{R}^{n}),$ $w_{k}\rightarrow w$ a.e in $\mathbb{R}^{n}$, and
$w_{k}\rightarrow w$ strongly in $L_{loc}^{p}(\mathbb{R}^{n})$, and $w\neq 0.$
In order to see this, we notice that for every $\varphi\in
C_{c}^{\infty}(\mathbb{R}^{n})$ and $k$ large enough,

$\int_{\mathbb{R}^{n}}w_{k}(x)\varphi(x)dx=\int_{\mathbb{R}^{n}}w_{k}^{1}(\psi_{k}(x))\varphi(x)dx=\int_{\mathbb{R}^{n}}w_{k}^{1}(x)\varphi(\psi_{k}^{-1}(x))\|\det\psi_{k}^{{}^{\prime}}(x)\|dx,$

where

$\psi_{k}(x)=\varepsilon_{k}^{-1}\exp_{c_{k}^{+}}^{-1}(\exp_{q}(\varepsilon_{k}x))$

Now, if $k\rightarrow\infty$, we have

$\int_{\mathbb{R}^{n}}w(x)\varphi(x)dx=\int_{\mathbb{R}^{n}}w^{1}(x)\varphi(x)dx\qquad\text{for
all }\varphi\in C_{c}^{\infty}(\mathbb{R}^{n}).$

So, $w=w^{1}$. Following a similar argument, we also have $w=w^{2}$.
Therefore, $w=w^{1}>0$ and $w=w^{2}<0$. This is a contradiction. ∎

For $\delta_{0}>0$ and $\varepsilon_{0}>0$ as in Proposition 5.5, we define
the map $\textbf{c}_{\varepsilon}:\mathcal{Z}_{\varepsilon}\cap
J_{\varepsilon}^{\textbf{d}_{\varepsilon}+\delta_{0}}\rightarrow F(M)$ by

(5.10)
$\textbf{c}_{\varepsilon}(u)\doteq(\textbf{Cm}(r,\eta)((u^{+})^{p}),\textbf{Cm}(r,\eta)((u^{-})^{p}))$

######  Remark 5.6.

The group $\mathbb{Z}_{2}=\\{-1,1\\}$ acts in $F(M)$ by
$\theta\cdot(x,y)=(y,x)$. This action is free, moreover, the maps
$\textbf{c}_{\varepsilon}$ and $i_{\varepsilon}$ are
$\mathbb{Z}_{2}-$invariants. In this way we have defined the following
function

$\widehat{\textbf{c}_{\varepsilon}}:\Big{(}\mathcal{Z}_{\varepsilon}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}\Big{)}/\mathbb{Z}_{2}\rightarrow
C(M)=F(M)/\mathbb{Z}_{2}.$

######  Remark 5.7.

Let $\check{\mathcal{H}}$ be Čech cohomology with $\mathbb{Z}_{2}$
coefficients. This cohomology coincides with singular cohomology
$\mathcal{H}^{*}$ on manifolds.

For $C_{\varepsilon}\doteq F_{\varepsilon}/\mathbb{Z}_{2}$, we have the
following result.

######  Proposition 5.8.

There exists a homomorphism

$\tau_{\varepsilon}:\check{\mathcal{H}}\Big{(}\Big{(}\mathcal{Z}_{\varepsilon}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}\Big{)}/\mathbb{Z}_{2}\Big{)}\rightarrow\mathcal{H}^{*}(C_{\varepsilon}(M))$

such that the composition

$\tau_{\varepsilon}\circ\widehat{\textbf{c}_{\varepsilon}}^{*}:\mathcal{H}^{*}(C(M))\rightarrow\mathcal{H}^{*}(C_{\varepsilon}(M))$

is the homomorphism induced by the inclusion
$C_{\varepsilon}(M)\hookrightarrow C(M)$, wich is an isomorphism for
$\varepsilon>0$ small enough.

Recall that the cup-length of a topological space $X$, denote it by
$\text{cupl}\,(X)$, is the smaller integer $k\geq 1$ such that the cup-product
of any $k$ cohomology class in $\widetilde{\mathcal{H}}^{*}(X)$ is zero, where
$\widetilde{\mathcal{H}}^{*}(X)$ is the reduced cohomology.

###### Proof of Theorem 1.2.

From Lemma 2.1 we have that $J_{\varepsilon}$ satisfies the Palais-Smale
condition in $\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}$. Suppose that contains
$k$ pairs $\pm u_{1},\ldots,\pm u_{k}$ critical points of $J_{\varepsilon}$
and $J_{\varepsilon}(u_{1})\leq J_{\varepsilon}(u_{2})\leq\cdots\leq
J_{\varepsilon}(u_{k}).$ From Lemma 3.4, we have that
$\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}$ is positively invariant
for the negative gradient flow $\varphi_{\varepsilon}$ of $\nabla
J_{\varepsilon}$. Hence, for all $u\in\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}$ there exists $j$ with
$\varphi_{\varepsilon}(t,u)\rightarrow\pm u_{j}$ as $t\rightarrow\infty.$ Let
us consider the sets

$X_{j}\doteq\\{u\in\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}:\varphi_{\varepsilon}(t,u)\rightarrow\pm
u_{j}\text{ as }t\rightarrow\infty\\}.$

The sets $X_{j}$ are pairwise disjoint and cover
$\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}$. Now, by the Palais-
Smale condition for $J_{\varepsilon}$ in $\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}$, we have that the union
$X_{1}\cup\cdots\cup X_{j}$ for every $j=1,\ldots,k,$ is an open set of
$\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}$, therefore $X_{j}$ is a
locally closed subset of $\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}$. Using the flow
$\varphi_{\varepsilon}$, $X_{j}$ can be deformated to $\pm u_{j}$ in
$\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}}$. Hence,

$\mathrm{Cat}_{\mathbb{Z}_{2}}(\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}})\leq k.$

∎

###### Proof of Theorem 1.3.

We have by Theorem 1.2 that $J_{\varepsilon}$ has at least
$\mathrm{Cat}_{\mathbb{Z}_{2}}(\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}})$ sign changing
solutions. Moreover, we have that
$\mathrm{Cat}_{\mathbb{Z}_{2}}(\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}})\geq\mathrm{Cupl}\Big{(}\Big{(}\mathcal{Z}_{\varepsilon}\cap
J_{\varepsilon}^{d_{\varepsilon}+\delta_{0}}\Big{)}/\mathbb{Z}_{2}\Big{)}\Big{)}$,
see [4] sections 5.2. The inclusion
$i_{\varepsilon}:F_{\varepsilon}(M)\hookrightarrow F(M)$ is a homotopy
equivalent for all $\varepsilon\in(0,\varepsilon_{0})$, therefore, from the
Proposition 5.8 it follows that

$\mathrm{Cupl}((\mathcal{Z_{\varepsilon}}\cap
J_{\varepsilon}^{\mathbf{d}_{\varepsilon}+\delta_{0}})/\mathbb{Z}_{2})\geq\mathrm{Cupl}\,C(M).$

∎

## References

  * [1] B. Ammann, M. Dahl & E. Humbert, Smooth Yamabe invariant and surgery, J. Differential Geometry 94 (2013), 1–58.
  * [2] B. Ammann & E. Humbert, The second Yamabe invariant, J. Funct. Anal. 235 (2006), 377–412.
  * [3] T. Aubin, Equations differentielles non-lineaires et probleme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296.
  * [4] T. Bartsch, M. Clapp & T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schodinger equation Math. Ann. (2005) 338:147-187.
  * [5] B. Asari, Riemannian $L^{p}$ Center of Mass: Existence, Uniqueness, and Convexity Proceedings of the American Mathematical Society. Volume 139, Number 2, February 2011, pages 655-673.
  * [6] V. Benci, C. Bonanno & M. Micheletti, On a multiplicity of solutions of a nonlinear elliptic problem on a Riemannian manifolds J. Funct. Anal. 252(2007), no. 1, 147-185.
  * [7] R. Bettiol & P. Piccione, Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions, Pacific J. Math. 266, 1–21 (2013).
  * [8] A. Castro, J. Cossio & J.M. Neuberger, A sign-changing solution for a superliner Dirichlet problem Rocky Mountain J. Math. 27 (2005), no. 4, 1041-1053.
  * [9] M. Clapp & M. Micheletti, Asymptotic profile of $2$-nodal solutions to a semilinear elliptic problem on a Riemannian manifold, Adv. Differential Equations 19(3/4): 225-244 (March/April 2014). DOI: 10.57262/ade/1391109085.
  * [10] M. Clapp & M. Micheletti, Multiplicity and asymptotic profile of 2-nodal solutions to a semilinear elliptic problem on a Riemannian manifold, arXiv:1301.0143 (2013).
  * [11] M. Clapp, J. Faya & A. Pistoia, Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents. _Calc. Var. Partial Differ. Equ._ 48 (2013), 611–623
  * [12] M. Clapp & J.C. Fernández, Multiplicity of nodal solution to the Yamabe problem. _Calc. Var. Partial Differ. Equ._ 56:145 (2017), 611–623
  * [13] E. N. Dancer, A. M. Micheletti & A. Pistoia,  Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds, Manuscripta Math. 128 (2009), 163-193.
  * [14] L.L. de Lima, P. Piccione & M. Zedda, On bifurcation of solutions of the Yamabe problem in product manifolds, Annales de L’institute Henri Poincare (C) Non Linear Analysis 29, 261-277 (2012).
  * [15] Deng, S., Khemiri, Z. & Mahmoudi, F.: On spike solutions for a singularly perturbed problem in a compact Riemannian manifold. Commun. Pure Appl. Anal. 17(5), 2063–2084 (2018).
  * [16] Y. Ding, On a conformally invariant elliptic equation on $\mathbb{R}^{n}$ _Comm. Math. Phys._ 107 (1986), no. 2, 331–335.
  * [17] J. C. Fernandez & J. Petean, Low energy nodal solutions to the Yamabe equation, arXiv:1807.06114.
  * [18] B. Gidas, W-M. Ni & L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${\bf R}^{n}$, Adv. Math. Suppl. Stud. A, 369–402 (1981).
  * [19] K. Grove & H. Karcher, How to Conjugate $C^{1}$-close Group Actions, Math. Z. 132, 11-20 (1973).
  * [20] G. Henry & F. Madani, Equivariant Second Yamabe constant, The Journal of Geometric Analysis 28, 3747-3774 (2018). https://doi.org/10.1007/s12220-017-9978-x
  * [21] G. Henry & J. Petean, Isoparametric hypersurfaces and metrics of constant scalar curvature, Asian J. Math. 18, 53-68 (2014).
  * [22] E. Hebey, NonLinear Analysis on Manifolds: Sobolov Spaces and inequalities. Courant Lecture Notes in Mathematics, 5. New York University, Courant Institute of Mathematics Sciences, New York; American Mathematical Society, Province, Rhode Island, 1999.
  * [23] N. Hirano, Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold Noonlinear Analysis 70 (2009) 671-692.
  * [24] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p}=0$ in ${\bf R}^{n}$, Arch. Rational. Mech. Anal. 105, 243-266 (1989).
  * [25] K.-C.-Chang, Methods in Nonlinear Analysis, Springer . (2005).
  * [26] Kung-Ching-Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Basel . (1993).
  * [27] J. M. Lee & T. H. Parker, The Yamabe problem. Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91.
  * [28] A. M. Micheletti & A. Pistoia,  The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds, Calc. Var. 34 (2009), 233-265.
  * [29] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Diff. Geom., No. 6, 247-258, 1971.
  * [30] J. Petean, Multiplicity results for the Yamabe equation by Lusternik-Schnirelamm theory, Journal of Functional Analysis 276 1788-1805 (2019).
  * [31] B. Premoselli & J. Vétois,  Sign-changing blow-up for the Yamabe equation at the lowest energy level, Advances in Mathematics, 410, Part B, 2022, https://doi.org/10.1016/j.aim.2022.108769.
  * [32] C. Rey & J. M. Ruiz, Multipeak solutions for the Yamabe equation, arXiv:1807.08385.
  * [33] F. Robert & J. Vétois. Sign-Changing Blow-Up for Scalar Curvature Type Equations. _Comm. Partial Differential Equations_ 38 (2013), 1437–1465.
  * [34] Martin Schechter & Wenming Zou: Critical Point Theory and its applications. Springer Verlang 2006.
  * [35] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geometry 20 (1984), 479-495.
  * [36] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Lecture Notes in Math. 1365, Springer-Verlag, Berlin, 1989, 120-154.
  * [37] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265–274.
  * [38] W. Zou: Sign-Changing Critical Point Theory. Springer Verlang 2008.
  * [39] Z. Zhang: Variational, Topological and Partial Order Methods with Their Applications. Developments in Matematics VOLUME 29 Springer Verlang 2013.
  * [40] J. Wei & M. Winter: Mathematics Aspects of Pattern Formations in Biological System., AMS 189, Springer Verlang 2014.
  * [41] T. Weth, Energy bounds for entire nodal solutions of autonomus superlinear equations. Calc. Var. Partila Differential Equations 27 (2006), no. 4, 421-437.
  * [42] H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21-37.