# Multiplicity results for constant $Q$-curvature conformal metrics

Salomón Alarcón  Universidad Técnica Federico Santa María, Av. España 1680,
2340000 Valparaíso, Valparaíso, Chile<EMAIL_ADDRESS>,  Jimmy Petean
Centro de Investigación en Matemáticas, CIMAT, Calle Jalisco s/n, 36023
Guanajuato, Guanajuato, México<EMAIL_ADDRESS>and  Carolina Rey  Universidad
Técnica Federico Santa María, Av. España 1680, 2340000 Valparaíso, Valparaíso,
Chile<EMAIL_ADDRESS>
###### Abstract.

We prove that certain subcritical Paneitz-Branson type equations on a closed
Riemannian manifold $(M,g)$ have at least $\mathrm{Cat}(M)$ positve solutions,
where $\mathrm{Cat}(M)$ is the Lusternik-Schnirelmann category of $M$. This
implies that if $(X,h)$ is a closed positive Einstein manifold then for
$\varepsilon>0$ small enough there are at least $\mathrm{Cat}(M)$ metrics of
constant $Q$-curvature in the conformal class of the Riemannian product
$g+\varepsilon h$.

2020 Mathematics Subject Classification: 35J35, 53C21.
Key words: Paneitz-Branson type equations, Q-curvature, Lusternik-Schnirelmann
category
S. Alarcón is partially supported by FONDECYT grant 1211766
J. Petean is supported by Fondo Sectorial SEP-CONACYT grant A1-S-45886
C. Rey is supported by FONDECYT grant 3200422.

##  1\. Introduction

Consider a closed Riemannian manifold $(\mathbf{M},\mathfrak{g})$ of dimension
$N\geq 5$. The Paneitz operator $P_{\mathfrak{g}}$ is defined using a local
$\mathfrak{g}$-orthonormal frame $\left(e_{i}\right)_{i=1}^{n}$, as

(1.1)
$P_{\mathfrak{g}}\psi:=\Delta_{\mathfrak{g}}^{2}\psi+\frac{4}{N-2}\operatorname{div}_{\mathfrak{g}}\left(\operatorname{Ric}_{\mathfrak{g}}\left(\nabla\psi,e_{i}\right)e_{i}\right)-\frac{N^{2}-4N+8}{2(N-1)(N-2)}\operatorname{div}_{\mathfrak{g}}\left(\operatorname{scal}_{\mathfrak{g}}\nabla\psi\right)+\frac{N-4}{2}Q_{\mathfrak{g}}\psi,$

where
$\Delta_{\mathfrak{g}}u=\operatorname{div}_{\mathfrak{g}}(\nabla_{\mathfrak{g}}u)$
is the (negative) Laplace operator on $(\mathbf{M},\mathfrak{g})$,
$\operatorname{div}_{\mathfrak{g}}$ is the divergence,
$\operatorname{scal}_{\mathfrak{g}}$ is the scalar curvature and
$\mathrm{Ric}_{\mathfrak{g}}$ is the Ricci tensor. $Q_{\mathfrak{g}}$ is the
$Q$-curvature of $\mathfrak{g}$, which is given by:

(1.2)
$Q_{\mathfrak{g}}:=-\frac{1}{2(N-1)}\Delta_{\mathfrak{g}}\operatorname{scal}_{\mathfrak{g}}-\frac{2}{(N-2)^{2}}\left|\operatorname{Ric}_{\mathfrak{g}}\right|^{2}+\frac{N^{3}-4N^{2}+16N-16}{8(N-1)^{2}(N-2)^{2}}\operatorname{scal}_{\mathfrak{g}}^{2}.$

The Paneitz operator and the related $Q$-curvature were introduced by Stephen
Paneitz [24] and by Thomas Branson [8]. We denote by
$\left[\mathfrak{g}_{0}\right]$ the conformal class of a given metric
$\mathfrak{g}_{0}$ in $\mathbf{M}$. For practical reasons, we usually write a
conformal metric $\mathfrak{g}\in\left[\mathfrak{g}_{0}\right]$ as
$\mathfrak{g}=u^{\frac{4}{N-4}}\mathfrak{g}_{0}$, for a positive function
$u:\mathbf{M}\rightarrow\mathbb{R}$. The Paneitz operator is conformally
invariant in the sense that for any $\psi:\mathbf{M}\rightarrow\mathbb{R}$,

$P_{\mathfrak{g}}\psi=u^{-\frac{N+4}{N-4}}P_{\mathfrak{g}_{0}}(u\psi).$

It follows that we can express the $Q$-curvature of $\mathfrak{g}$ in terms of
$\mathfrak{g}_{0}$ and $u$ by:

$Q_{\mathfrak{g}}=\frac{2}{N-4}P_{\mathfrak{g}}(1)=\frac{2}{N-4}u^{-\frac{N+4}{N-4}}P_{\mathfrak{g}_{0}}(u).$

Similarly to the Yamabe problem of finding metrics of constant scalar
curvature in a conformal class, there has been great interest in studying the
problem of finding metrics of constant $Q$-curvature in a conformal class.

It follows from the previous comments that finding a conformal metric
$\mathfrak{g}=u^{\frac{4}{n-4}}\mathfrak{g}_{0}$ with constant $Q$-curvature
amounts to finding a positive function $u$ which solves the fourth order
equation

(1.3) $P_{\mathfrak{g}_{0}}u=\lambda
u^{\frac{N+4}{N-4}},\quad\lambda\in\mathbb{R}.$

We call this equation the Paneitz-Branson Equation.

Note that the $Q$-curvature is defined if $\mathbf{M}$ has dimension greater
than 2. However, when the dimension is 3 or 4, the constant curvature equation
has a different form. Some significant results obtained in the $4$-dimensional
case are given, for instance, by Sun-Yung A. Chang and Paul C. Yang [10], by
Simon Brendle [9] and by Zindine Djadli and Andrea Malchiodi [12]. In this
work, we restrict our discussion to the case when the dimension is at least 5.

Previous research has considered the problem of the existence of metrics of
constant $Q$-curvature in dimension $N\geq 5$. For example, Jie Qing and David
Raske in [26] proved the existence of a positive solution of the Paneitz-
Branson equation on a locally conformally flat manifold of positive scalar
curvature and Poincaré exponent less than $\frac{N-4}{2}$. Meanwhile, Emmanuel
Hebey and Frédéric Robert in [11] also proved the existence of a positive
solution for locally conformally flat manifolds of positive scalar curvature,
assuming that the Paneitz operator and the corresponding Green function were
positive. Matthew J. Gursky and Andrea Malchiodi in [15] proved the existence
of a positive solution under the conditions that $Q_{\mathfrak{g}}$ is non-
negative and positive somewhere in $M$ and
$\operatorname{scal}_{\mathfrak{g}}\geq 0$.

Solutions of the Paneitz-Branson equation are the critical points of the
functional
$\mathcal{P}_{\mathfrak{g}}:H^{2}(\mathbf{M})\setminus\\{0\\}\rightarrow\mathbb{R}$,
given by

$\mathcal{P}_{\mathfrak{g}}(u):=\frac{2}{N-4}\frac{\int_{\mathbf{M}}u\,P_{\mathfrak{g}}(u)\
dv_{g}}{\left(\int_{\mathbf{M}}|u|^{\frac{2N}{N-4}}dv_{g}\right)^{\frac{N-4}{N}}}.$

Now consider the constant

$Y_{4}(\mathfrak{g})=\inf_{u\in
C^{\infty}(M),u>0}\mathcal{P}_{\mathfrak{g}}(u).$

Fengbo Hang and Paul C. Yang proved in [17] that if $Y_{4}(\mathfrak{g})$ is
positive, and also $Q_{\mathfrak{g}}>0$ and
$\operatorname{scal}_{\mathfrak{g}}>0$ (under some weaker hypotheses, it is
only needed that the Yamabe invariant of the conformal class of $\mathfrak{g}$
is positive and that $Q_{\mathfrak{g}}\geq 0$ and positive somewhere) then
$Y_{4}(\mathfrak{g})\leq Y_{4}(\mathfrak{g}^{N}_{0})$, where
$\mathfrak{g}_{0}^{N}$ is the round metric on the $N$-sphere, and the infimum
is always realized by a positive function, which is a solution of the Paneitz-
Branson equation. Other related existence results can be found in [11, 12, 13,
14, 16, 27].

In the case of the round sphere $(\mathbb{S}^{N},\mathfrak{g}_{0}$), there is
a non-compact family of solutions to the Paneitz-Branson equation that comes
from the family of conformal diffeomorphisms of the sphere. These are all the
solutions of the Paneitz-Branson equation on the sphere by a theorem of Chang-
Shou Lin [21].

There are also important results about the compactness of the space of
solutions, for instance see [18, 19, 20, 22]. Examples of non-compactness of
the space of positive solutions, different from the constant curvature metric
on the sphere, were constructed in high dimensions by Juncheng Wei and Chunyi
Zhao [30]. Naturally, if $\mathfrak{g}$ is Einstein it has constant
$Q$-curvature. Moreover, Jérôme Vétois [29] proved that if $\mathfrak{g}$ is
Einstein and it is not isometric to the constant curvature metric on the
sphere, then it is, up to multiplication by a constant, the only metric of
constant $Q$-curvature in its conformal class. This result is an analogue of
the classical due to Morio Obata [23] about constant scalar curvature.

There are a few other cases where one can show the non-uniqueness of conformal
metrics of constant $Q$-curvature. One case, which also appears for the Yamabe
problem, is that of Riemannian products. If $(M,g)$ and $(X,h)$ are closed
Einstein manifolds with positive Einstein constant, of dimension at least $3$
then for any $\delta>0$, the Riemannian product $(M\times X,g+\delta h)$ has
constant positive $Q$-curvature and constant positive scalar curvature. It is
easy to compute that as $\delta\rightarrow 0$ or $\delta\rightarrow\infty$ we
have that $\mathcal{P}_{g+\delta h}(1)\rightarrow\infty$. We are in the
conditions of the result mentioned above [17], and therefore in the conformal
class of $g+\delta h$, there must be at least one other metric of constant
$Q$-curvature, which minimizes $\mathcal{P}_{g+\delta h}$. Note that a similar
argument works for the Yamabe problem, but in that case, one only needs that
the scalar curvatures are constant and positive.

Renato Bettiol, Paolo Piccione and Yannick Sire [6] considered appropriate
Riemannian submersions and reduce the equation to basic functions (functions
which are constant along the fibers of the submersion). Using bifurcation
theory for this reduced equation on the canonical variation obtained by
varying the size of the fibers (under certain conditions the total spaces of
these variations have constant $Q$-curvature), they prove the existence of
bifurcation instants, so proving the existence of metrics in the family for
which there are other conformal metrics of constant $Q$-curvature.

This paper aims to provide a lower bound for the number of metrics of constant
$Q$-curvature for a Riemannian product $(M\times X,g+\delta h)$ within the
family of conformal metrics to $g+\delta h$. Using the Lusternik-Schnirelmann
category, we prove the following multiplicity result.

######  Theorem 1.1.

Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold and $(X,h)$ a
closed $m$-dimensional Einstein manifold with Einstein constant
$\Lambda_{0}>0$. Consider the Riemannian product $(M\times
X,\mathfrak{g}_{\varepsilon}:=g+\varepsilon^{2}h)$. Assume that $m\geq 3$ or,
if $m=2$, that $n\geq 7$. Then there exists $\varepsilon_{0}>0$ such that for
any $\varepsilon\in(0,\varepsilon_{0})$ the Paneitz-Branson equation on
$(M\times X,\mathfrak{g}_{\varepsilon})$ has at least $\mathrm{Cat}(M)$
positive solutions.

Here $\mathrm{Cat}(M)$ denotes the Lusternick-Schnirelmann category of $M$, a
topological invariant defined as the minimum number of contractible open sets
needed to cover it. If $M$ is a manifold, $\mathrm{Cat}(M)$ coincides with the
minimal number of critical points among all smooth scalar maps
$M\rightarrow\mathbb{R}$ (see [28]). The solutions given in the theorem are
functions on $M$. We will see in Section 2 that a function
$u:M\rightarrow(0,\infty)$ solves the Paneitz-Branson equation on $(M\times
X,\mathfrak{g}_{\varepsilon})$ if and only if

(1.4)
$\Delta^{2}_{g}u-\frac{N^{2}-4N+8}{2(N-1)(N-2)}(m\Lambda_{0}\varepsilon^{-2})\Delta_{g}u+\frac{N-4}{2}\varepsilon^{-4}A_{N,m}u+\varphi(u)+\frac{N-4}{2}(f_{0}+\varepsilon^{-2}f_{2})u=u^{\frac{N+4}{N-4}}$

where $N=n+m$ is the dimension of the product manifold $M\times X$,
$\varphi(u)$ is a second order operator on $u$ which verifies

$\int_{M}\varphi(u)\,u\,dv_{g}\leq C\int_{M}|\nabla u|^{2}dv_{g},$

for some constant $C$, $A_{N,m}$ is a positive constant that depends on $N,m$,
and $f_{0}$ and $f_{2}$ are certain functions on $M$.

In general, for a closed $n$-dimensional Riemannian manifold $({M},g)$ we can
consider an equation of the form

$\Delta_{g}^{2}u+f(x)\Delta_{g}u+\phi(u)+g(x)u=h(x)u^{q},$

for functions $f,g,h$ on M, $q\in(1,\infty)$, and a second order operator
$\phi$. This equation is called critical if $q=\frac{n+4}{n-4}$, subcritical
if $q<\frac{n+4}{n-4}$ and supercritical if $q>\frac{n+4}{n-4}$ (if $n\leq 4$
it is considered subcritical for any $q>1$). For instance, in [11, Section 2]
the authors study the critical case when $f,g$ are constant, $\phi=0$.

Consider the equation

(1.5)
$\Delta_{g}^{2}u-b\varepsilon^{-2}\Delta_{g}u+a\varepsilon^{-4}u=u^{q}\quad\text{on
}M$

Assume that $a,b>0$. For $\varepsilon$ small Equation (1.4) is, in certain
sense, close to Equation 1.5. It will be easy to check along the proof of
Theorem 1.1 that it also gives a proof of

######  Theorem 1.2.

If $b^{2}\geq 4a$ and $q<\frac{n+4}{n-4}$ (if $n>4$), then there exists
$\varepsilon_{0}>0$ such that for any $\varepsilon\in(0,\varepsilon_{0})$,
Equation (1.5) has at least $\mathrm{Cat}(M)$ positive solutions.

In Section 2 we will give the explicit formula for the Paneitz operator
$P_{\mathfrak{g}_{\varepsilon}}$. In Section 3 we will see that the positive
solutions of the Paneitz-Branson equation for $\mathfrak{g}_{\varepsilon}$ are
precisely the critical points of the functional

$J_{\varepsilon,q}^{g,+}(u)=\frac{1}{\varepsilon^{n}}\int_{M}\\!\Big{(}\frac{\varepsilon^{4}}{2}(\Delta_{g}u)^{2}+\varepsilon^{2}\frac{b_{N,m}}{2}|\nabla
u|^{2}+\frac{a_{N,m}}{2}u^{2}+\varepsilon^{4}\varphi(u)u+\frac{\varepsilon^{2}}{2}(\varepsilon^{2}f_{0}(x)+f_{2}(x))u^{2}-\frac{(u^{+})^{q+1}}{q+1}\Big{)}dv_{g}\\!$

restricted to the associated Nehari manifold $\mathcal{N}^{g}_{\varepsilon}$.
In the functional $u^{+}=\max\\{u,0\\}$. We will also show that
$J_{\varepsilon,q}^{g,+}$ restricted to $\mathcal{N}^{g}_{\varepsilon}$ is
bounded from below and satisfies the Palais-Smale condition. By Lusternik-
Schnirelmann theory it follows that for any $d$ the number of positive
solutions of the Paneitz-Branson equation is at least
$\mathrm{Cat}(\Sigma_{\varepsilon,d})$, where
$\Sigma_{\varepsilon,d}=\\{u\in\mathcal{N}^{g}_{\varepsilon}:J^{g,+}_{\varepsilon,q}(u)\leq
d\\}$. Let

${\bf
m}_{\varepsilon}:=\inf_{u\in\mathcal{N}^{g}_{\varepsilon}}J^{g,+}_{\varepsilon,q}(u).$

The rest of the article is devoted to prove that for $\varepsilon,\delta$
small enough we have that $\mathrm{Cat}(\Sigma_{\varepsilon,{\bf
m}_{\varepsilon}+\delta})\geq\mathrm{Cat}(M)$. To prove this inequality we
will use a method introduced by Vieri Benci and Giovanna Cerami in [3] to
study multiplicity results for solutions of nonlinear elliptic problems in
Euclidean domains. The method was later used also to study similar problems in
Riemannian manifolds, for instance in [2]. It is sometimes called the
photography method, see for instance [5] where there is a very clear and
general description of the method.

In Section 4 we study the limit equation and functionals on $\mathbb{R}^{n}$.
A positive, minimizing, solution of this limit equation is used later to
construct, for any $x\in M$ a function in ${\bf
p}(x)\in\Sigma_{\varepsilon,{\bf m}_{\varepsilon}+\delta}$, which is
concentrated around $x$.

In Section 6 we show that for $\varepsilon,\delta$ small enough, functions
$u\in\Sigma_{\varepsilon,{\bf m}_{\varepsilon}+\delta}$ are concentrated
around some point. This is used in Section 7 to show that there is a well
defined center of mass function ${\bf b}:\Sigma_{\varepsilon,{\bf
m}_{\varepsilon}+\delta}\rightarrow M$. It is easily checked that ${\bf
b}\circ{\bf p}:M\rightarrow M$ is close, and therefore homotopic to the
identity. This implies the inequality $\mathrm{Cat}(\Sigma_{\varepsilon,{\bf
m}_{\varepsilon}+\delta})\geq\mathrm{Cat}(M)$.

##  2\. Constant $Q$-curvature on a product manifold

In this section, we study the Paneitz operator and the $Q$-curvature in the
case of a product manifold. Here we establish the necessary hypotheses to
apply the techniques mentioned above.

Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold and $(X,h)$ a
closed $m$-dimensional Einstein manifold with positive Einstein constant
$\Lambda_{0}$, i.e., its scalar curvature is
$\operatorname{scal}_{h}=m\Lambda_{0}$. We are interested in positive
solutions of the constant $Q$-curvature equation (1.3) for the product
manifold $(M\times X,\mathfrak{g}_{\varepsilon}:=g+\varepsilon^{2}h)$, which
is given by

$P_{\mathfrak{g}_{\varepsilon}}u=\lambda_{\varepsilon}u^{\frac{n+m+4}{n+m-4}},\quad\lambda_{\varepsilon}\in\mathbb{R},$

where $P_{\mathfrak{g}_{\varepsilon}}$ is the Paneitz operator defined in
(1.1). We consider the case $\lambda_{\varepsilon}>0$ and so we can
renormalize the previous equation as

$P_{\mathfrak{g}_{\varepsilon}}u=u^{\frac{n+m+4}{n+m-4}}.$

We put $N:=n+m$ and compute the $Q$-curvature for the product metric.

######  Lemma 2.1.

Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold and let $(X,h)$ be
a closed $m$-dimensional Einstein manifold with positive Einstein constant
$\Lambda_{0}$. If $N=n+m$, then the $Q$-curvature for the product manifold
$(M\times X,\mathfrak{g}_{\varepsilon})$ is

$\displaystyle Q_{\mathfrak{g}_{\varepsilon}}$
$\displaystyle=-\frac{1}{2(N-1)}\Delta_{g}\operatorname{scal}_{g}-\frac{2}{(N-2)^{2}}\left|\mathrm{Ric}_{g}\right|^{2}+\frac{N^{3}-4N^{2}+16N-16}{8(N-1)^{2}(N-2)^{2}}\operatorname{scal}_{g}^{2}$
$\displaystyle\quad+\frac{1}{\varepsilon^{2}}\
\frac{N^{3}-4N^{2}+16N-16}{8(N-1)^{2}(N-2)^{2}}2m\operatorname{scal}_{g}\Lambda_{0}+\frac{1}{\varepsilon^{4}}\left(-\frac{2m\Lambda_{0}^{2}}{(N-2)^{2}}+\frac{N^{3}-4N^{2}+16N-16}{8(N-1)^{2}(N-2)^{2}}m^{2}\Lambda_{0}^{2}\right).$

###### Proof.

By using (1.2) and the fact that

$\left|\mathrm{Ric}_{\mathfrak{g}_{\varepsilon}}\right|^{2}=\left|\mathrm{Ric}_{g}\right|^{2}+\frac{1}{\varepsilon^{4}}\left|\mathrm{Ric}_{h}\right|^{2}=\left|\mathrm{Ric}_{g}\right|^{2}+\frac{1}{\varepsilon^{4}}m\Lambda_{0}^{2},$

straightforward calculations leads to the claim. ∎

Now, calling

(2.1) $\displaystyle f_{0}$
$\displaystyle:=-\frac{1}{2(N-1)}\Delta_{g}\operatorname{scal}_{g}-\frac{2}{(N-2)^{2}}\left|\mathrm{Ric}_{g}\right|^{2}+\frac{N^{3}-4N^{2}+16N-16}{8(N-1)^{2}(N-2)^{2}}\left(\operatorname{scal}_{g}^{2}\right),$
$\displaystyle f_{2}$
$\displaystyle:=\frac{N^{3}-4N^{2}+16N-16}{4(N-1)^{2}(N-2)^{2}}m\operatorname{scal}_{g}\Lambda_{0},$

and

(2.2)
$A_{N,m}:=\frac{m\Lambda_{0}^{2}}{(N-2)^{2}}\left(\frac{N^{3}-4N^{2}+16N-16}{8(N-1)^{2}}m-2\right),$

we get

$Q_{\mathfrak{g}_{\varepsilon}}=f_{0}+\varepsilon^{-2}f_{2}+\varepsilon^{-4}A_{N,m}.$

We get the following simple computation:

######  Lemma 2.2.

If $m=2$ then $A_{N,2}<0$ for $N=5,6,7,8$ and $A_{N,2}>0$ for $N\geq 9$. If
$m\geq 3$, then $A_{N,m}>0$.

###### Proof.

Since we have a closed $m$-dimensional Einstein manifold with positive
Einstein constant $\Lambda_{0}$, we must have $m\geq 2$.

Assume first that $m=2$. From (2.2), we have that $A_{N,2}>0$ if and only if

$N^{3}-4N^{2}+16N-16>8(N-1)^{2}.$

This inequality clearly holds if $N\geq 9$, whereas if $N=5,6,7,8$, we get
$A_{N,2}<0$.

Now, let $m\geq 3$. Then, $A_{N,m}>0$ if and only if

$m(N^{3}-4N^{2}+16N-16)>16(N-1)^{2}.$

The expression on the left is clearly increasing with $m$ and the lemma
follows from straightforward calculations. ∎

By restricting our study to functions that depend only on the first factor,
$u:M\rightarrow\mathbb{R}$, we obtain the following expression for the
restricted Paneitz operator:

$P_{\mathfrak{g}_{\varepsilon}}u=\Delta_{g}^{2}u+\frac{4}{N-2}\operatorname{div}_{g}\left(\operatorname{Ric}_{g}\left(\nabla
u,e_{i}\right)e_{i}\right)-\frac{N^{2}-4N+8}{2(N-1)(N-2)}\operatorname{div}_{\mathfrak{g}_{\varepsilon}}\left(\operatorname{scal}_{\mathfrak{g}_{\varepsilon}}\nabla
u\right)+\frac{N-4}{2}Q_{\mathfrak{g}_{\varepsilon}}u$

for a $g$-orthonormal frame $\left\\{e_{i}\right\\}_{i=1}^{n}$. Then, since
$\mathfrak{g}_{\varepsilon}=g+\varepsilon^{2}h$, we get

(2.3)
$\begin{array}[c]{ll}P_{g+\varepsilon^{2}h}u\displaystyle=\Delta_{g}^{2}u+\dfrac{4}{N-2}\operatorname{div}_{g}\left(\operatorname{Ric}_{g}\left(\nabla
u,e_{i}\right)e_{i}\right)-\dfrac{N^{2}-4N+8}{2(N-1)(N-2)}\Big{(}\operatorname{div}_{g}\left(\operatorname{scal}_{g}\nabla
u\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad\qquad\quad+\varepsilon^{-2}m\Lambda_{0}\Delta_{g}u\Big{)}+\dfrac{N-4}{2}(f_{0}+\varepsilon^{-2}f_{2}+\varepsilon^{-4}A_{N,m})u.\end{array}$

Finally, the constant $Q$-curvature equation (1.3) for the product manifold
$(M\times X,g+\varepsilon^{2}h)$ for a function $u:M\rightarrow\mathbb{R}$ is

(2.4)
$\Delta^{2}_{g}u-\frac{N^{2}-4N+8}{2(N-1)(N-2)}\frac{m\Lambda_{0}}{\varepsilon^{2}}\Delta_{g}u+\frac{N-4}{2}\frac{A_{N,m}}{\varepsilon^{4}}u+\varphi(u)+\frac{N-4}{2}(f_{0}+\varepsilon^{-2}f_{2})u=u^{\frac{N+4}{N-4}},$

where $\varphi$ is a linear second order operator given by

(2.5)
$\varphi(u):=\frac{4}{N-2}\operatorname{div}_{g}\left(\operatorname{Ric}_{g}\left(\nabla
u,e_{i}\right)e_{i}\right)-\frac{N^{2}-4N+8}{2(N-1)(N-2)}\operatorname{div}_{g}\left(\operatorname{scal}_{g}\nabla
u\right).$

Next we prove an elementary property of this operator.

######  Lemma 2.3.

Let $\varphi$ be the operator defined in (2.5). There exists a positive
constant $C$ such that

$\int_{M}\varphi(u)\,u\,dv_{g}\leq C\int_{M}|\nabla u|^{2}dv_{g}.$

###### Proof.

Using the divergence theorem, we have

$\int_{M}\varphi(u)\,u\,dv_{g}\begin{array}[t]{l}\displaystyle=\int_{M}\operatorname{div}_{g}\left(\frac{4}{N-2}\left(\operatorname{Ric}_{g}\left(\nabla
u,e_{i}\right)e_{i}\right)-\frac{N^{2}-4N+8}{2(N-1)(N-2)}\operatorname{scal}_{g}\nabla
u\right)u\ dv_{g}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\int_{M}\left(\frac{4}{N-2}g_{ij}\operatorname{Ric}_{g}\left(\nabla
u,e_{i}\right)(\nabla
u)^{i}-\frac{N^{2}-4N+8}{2(N-1)(N-2)}\operatorname{scal}_{g}|\nabla
u|^{2}\right)dv_{g}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle\leq
C\int_{M}|\nabla u|^{2}dv_{g},\end{array}$

where $C=C(m,n,g)$. ∎

Let

(2.6) $b_{N,m}:=\frac{N^{2}-4N+8}{2(N-1)(N-2)}m\Lambda_{0},\quad\text{and
}\quad a_{N,m}:=\frac{N-4}{2}A_{N,m},$

with $A_{N,m}$ as in (2.2). Then, from (2.3), we get

$P_{g+\varepsilon^{2}h}u=\Delta_{g}^{2}u-b_{N,m}\varepsilon^{-2}\Delta_{g}u+a_{N,m}\varepsilon^{-4}u+\varphi(u)+\frac{N-4}{2}(f_{0}+\varepsilon^{-2}f_{2})u,$

where $f_{0}$ and $f_{2}$ are given by (2.1). Consequently, after
renormalization, we can rewrite the equation (2.4) as

(2.7)
$\varepsilon^{4}\Delta^{2}_{g}u-\varepsilon^{2}b_{N,m}\Delta_{g}u+a_{N,m}u+\varepsilon^{4}\varphi(u)+\frac{N-4}{2}(\varepsilon^{4}f_{0}+\varepsilon^{2}f_{2})u=u^{\frac{N+4}{N-4}}.$

Observe that if $\Lambda_{0}>0$ then $b_{N,m}>0$, and by Lemma 2.2,
$a_{N,m}>0$ if $m\geq 3$ or $m=2$ and $N\geq 9$. We also get the following
result.

######  Lemma 2.4.

Let $b_{N,m}$ and $a_{N,m}$ be the constants defined in (2.6). Then $m\geq 3$,
or $m=2$ and $N\geq 9$, imply that $b_{N,m}>2\sqrt{a_{N,m}}$.

###### Proof.

We have

$\begin{array}[t]{ll}b_{N,m}^{2}-4{a_{N,m}}\displaystyle\frac{16N\Lambda_{0}^{2}m^{2}}{4(N-1)^{2}(N-2)^{2}}>0.\end{array}$

The proof finishes from the previous calculation and bearing in mind that
$b_{N,m}>0$ and $a_{N,m}>0$. ∎

######  Remark 2.5.

It is crucial for the computations throughout the rest of this paper to
highlight that

$\begin{array}[t]{ll}\displaystyle\varepsilon^{4}\int_{M}u\,P_{g+\varepsilon^{2}h}u\
dv_{g}\\!&\\!\displaystyle=\varepsilon^{4}\int_{M}(\Delta_{g}u)^{2}dv_{g}+\varepsilon^{2}\int_{M}b_{N,m}|\nabla
u|^{2}dv_{g}+\int_{M}a_{N,m}u^{2}\ dv_{g}\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ &\qquad\qquad\displaystyle+\varepsilon^{4}\int_{M}u\,\varphi(u)\
dv_{g}+\frac{N-4}{2}\int_{M}(\varepsilon^{4}f_{0}+\varepsilon^{2}f_{2})u^{2}dv_{g}.\end{array}$

It is easy to see then that for any $\delta>0$, we can pick
$\varepsilon_{0}>0$ such that for any $\varepsilon\in(0,\varepsilon_{0})$ and
for any $u\in H^{2}(M)$ we have that

$\begin{array}[t]{ll}\displaystyle(1+\delta)^{-1}\left(\varepsilon^{4}\int_{M}(\Delta_{g}u)^{2}dv_{g}+\varepsilon^{2}\int_{M}b_{N,m}|\nabla
u|^{2}dv_{g}+\int_{M}a_{N,m}u^{2}\ dv_{g}\right)\vskip 6.0pt plus 2.0pt minus
2.0pt\\\
\displaystyle\qquad\leq\varepsilon^{4}\int_{M}u\,P_{g+\varepsilon^{2}h}u\,dv_{g}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad\leq(1+\delta)\left(\varepsilon^{4}\int_{M}(\Delta_{g}u)^{2}dv_{g}+\varepsilon^{2}\int_{M}b_{N,m}|\nabla
u|^{2}dv_{g}+\int_{M}a_{N,m}u^{2}\ dv_{g}\right).\end{array}$

We will use the previous remark in next sections to obtain estimates for
$\varepsilon^{4}\int_{M}u\,P_{g+\varepsilon^{2}h}u\ dv_{g}$ by using estimates
for the simpler expression
$\varepsilon^{4}\int_{M}(\Delta_{g}u)^{2}dv_{g}+\varepsilon^{2}\int_{M}b_{N,m}|\nabla
u|^{2}dv_{g}+\int_{M}a_{N,m}u^{2}\ dv_{g}$.

##  3\. Paneitz functional and the variational setting

In this section we establish some assumptions in order to study the problem of
constant $Q$-curvature on a product manifold. Again, we consider a closed
$n$-dimensional Riemannian manifold $(M,g)$ and a closed $m$-dimensional
Einstein manifold $(X,h)$ with positive Einstein constant $\Lambda_{0}$, and
we denote by $P_{\mathfrak{g}}$ to the Paneitz operator defined in (1.1) and
by $Q_{\mathfrak{g}}$ to the $Q$-curvature of $\mathfrak{g}$ given by (1.2).

We denote by $\|\cdot\|_{p}$ the standard $L^{p}(M)$-norm for $1\leq
p\leq\infty$. We consider $H^{2}(M)$ endowed with the usual norm

$\|u\|_{H^{2}(M)}^{2}=\|\Delta_{g}u\|_{2}^{2}+\|\nabla
u\|_{2}^{2}+\|u\|_{2}^{2},$

which is induced by the inner product

$\langle u,v\rangle=\int_{M}\Delta_{g}u\,\Delta_{g}v\,dv_{g}+\int_{M}\nabla
u\nabla v\,dv_{g}+\int_{M}u\,v\,dv_{g}.$

A maximum principle for the Paneitz operator and an affirmative answer to the
problem of finding metrics of constant $Q$-curvature in a closed Riemannian
manifold was given by M. Gursky and A. Malchiodi in [15, Theorems A, D], and
it reads as

######  Theorem [Gursky and Malchiodi].

Let $\left(M,\mathfrak{g}\right)$ be a closed Riemannian manifold of dimension
$n\geq 5$. Assume

  * (i)

$Q_{\mathfrak{g}}$ is semi-positive,

  * (ii)

$\operatorname{scal}_{\mathfrak{g}}\geq 0$.

If $u\in C^{4}$ satisfies

$P_{\mathfrak{g}}u\geq 0,$

then either $u>0$ or $u\equiv 0$ on $M$. There exists a conformal metric
$h=u^{\frac{4}{n-4}}\mathfrak{g}$ with constant positive $Q$-curvature and
positive scalar curvature.

Consider the normalized total $Q$-curvature functional defined on the space of
Riemannian metrics $g$ on the $n$-dimensional manifold $M$, which is given by

$\mathcal{Q}(g):=\frac{\int_{M}Q_{g}\,dv_{g}}{\mathrm{Vol}(M,g)^{\frac{n-4}{n}}}.$

Consider the restriction of the functional to the conformal class $[g]$ and
the Paneitz functional $\mathcal{P}_{g}$ given by

$\mathcal{P}_{g}(u):=\mathcal{Q}(u^{\frac{4}{n-4}}g)=\frac{2}{n-4}\frac{\int_{M}u\,P_{g}(u)\
dv_{g}}{\left(\int_{M}|u|^{\frac{2n}{n-4}}dv_{g}\right)^{\frac{n-4}{n}}}.$

Then we define the following constants as in [14]:

$Y_{4}(M,g)=\inf_{u\in C^{\infty}(M)}\mathcal{P}_{g}(u)\quad\text{ and }\quad
Y_{4}^{+}(M,g)=\inf_{h\in[g]}\mathcal{Q}(h)=\inf_{u>0}\mathcal{P}_{g}(u).$

We consider $m\geq 3$ and $n\geq 2$, or $m=2$ and $n\geq 7$, so the Riemannian
product $(X\times M,\mathfrak{g}_{\varepsilon}:=g+\varepsilon^{2}h)$ satisfies
the conditions of Lemma 2.2. It follows from Lemma 2.1 that:

######  Lemma 3.1.

There exists $\varepsilon_{0}>0$ such that if
$\varepsilon\in(0,\varepsilon_{0})$ then
$\operatorname{scal}_{\mathfrak{g}_{\varepsilon}}>0$ and
${Q}_{\mathfrak{g}_{\varepsilon}}>0$.

In particular, for $\varepsilon\in(0,\varepsilon_{0})$, we are under the
conditions of the Gursky-Malchiodi Theorem, and then the equality
$Y_{4}(M,\mathfrak{g}_{\varepsilon})=Y_{4}^{+}(M,\mathfrak{g}_{\varepsilon})$
is achieved by a metric of constant $Q$-curvature.

For $q>1$ consider the functional
$J_{\varepsilon,q}^{g,+}:H^{2}(M)\rightarrow\mathbb{R}$ defined by

(3.1)
$\\!J_{\varepsilon,q}^{g,+}(u)\\!=\\!\frac{1}{\varepsilon^{n}}\\!\int_{M}\\!\Big{(}\varepsilon^{4}\frac{1}{2}(\Delta_{g}u)^{2}+\varepsilon^{2}\frac{b_{N,m}}{2}|\nabla
u|^{2}+\frac{a_{N,m}}{2}u^{2}+\varepsilon^{4}\Phi(u)+\frac{\varepsilon^{2}}{2}F(x,\varepsilon)u^{2}-\frac{1}{q+1}(u^{+})^{q+1}\Big{)}dv_{g}\\!$

where

$\Phi(u)=\varphi(u)u\quad\text{and}\quad
F(x,\varepsilon)=\varepsilon^{2}f_{0}(x)+f_{2}(x),$

and $f_{0}$ and $f_{2}$ are defined in (2.1).

Also consider the associated Nehari manifold $\mathcal{N}^{g}_{\varepsilon}$,
given by

(3.2)
$\begin{array}[t]{lll}\\!\\!\mathcal{N}^{g}_{\varepsilon}:=\\!\bigg{\\{}u\\!\in\\!H^{2}(M)\setminus\\{0\\}\\!:\\!\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\quad\quad\,\,\int_{M}\\!\\!\big{(}\varepsilon^{4}(\Delta
u)^{2}+\varepsilon^{2}{b_{N,m}}|\nabla
u|^{2}+a_{N,m}u^{2}+\varepsilon^{4}\Phi(u)+\varepsilon^{2}F(x,\varepsilon)u^{2}\big{)}\,dv_{g}=\int_{M}(u^{+})^{q+1}dv_{g}\bigg{\\}}\\!\\!\end{array}$

######  Lemma 3.2.

Assume that $q>1$ and if $n>4$ that $q<\frac{n+4}{n-4}$. The functional
$J_{\varepsilon,q}^{g,+}\in C^{1}\left(H^{2}(M),\mathbb{R}\right)$ and

(3.3)
$\displaystyle\left\langle(J_{\varepsilon,q}^{g,+})^{\prime}(u),v\right\rangle$
$\displaystyle=\\!\frac{1}{\varepsilon^{n}}\left(\\!\varepsilon^{4}\\!\int_{M}\Delta_{g}u\Delta_{g}v\,dv_{g}+b_{N,m}\varepsilon^{2}\\!\int_{M}\nabla
u\nabla v\,dv_{g}+a_{N,m}\\!\int_{M}\\!u\,v\,dv_{g}\right)$
$\displaystyle\quad\,+\frac{1}{\varepsilon^{n}}\\!\left(\\!\varepsilon^{4}\int_{M}\varphi(u)v\,dv_{g}+\varepsilon^{2}\\!\int_{M}\left(\varepsilon^{2}f_{0}+f_{2}\right)u\,v\,dv_{g}-\int_{M}(u^{+})^{q-1}u\,v\,dv_{g}\\!\right),$

for all $u,v\in H^{2}(M)$. Moreover, if $u\in H^{2}(M)$, then
$(J^{g,+}_{\varepsilon,q})^{\prime}(u)\in(H^{2}(M))^{*}$, where
$(H^{2}(M))^{*}$ denotes the dual space of $H^{2}(M)$.

###### Proof.

First, we write
$J^{g,+}_{\varepsilon,q}=\dfrac{1}{\varepsilon^{n}}\left(I^{1}_{\varepsilon}-I^{2}_{\varepsilon}\right)$,
where

$I_{\varepsilon}^{2}(u)=\frac{1}{q+1}\int_{M}(u^{+})^{q+1}\,dv_{g},$

and $I^{1}_{\varepsilon}=J^{g,+}_{\varepsilon,q}+I^{2}_{\varepsilon}$. It is
easy to see that for all $u,v\in H^{2}(M)$, (3.3) holds and that
$I^{1}_{\varepsilon}\in C^{1}\left(H^{2}(M),\mathbb{R}\right)$.

Next, we show $I^{2}_{\varepsilon}\in C^{1}(H^{2}(M),\mathbb{R})$. Let
$\left\\{u_{j}\right\\}\subset H^{2}(M)$ a sequence such that
$u_{j}\rightarrow u$ for some $u\in H^{2}(M)$, strongly in $H^{2}(M)$ as
$j\rightarrow\infty$. Since $1<q+1<2_{n}^{\sharp}$, there exists a subsequence
of $\left\\{u_{j}\right\\}$ still denoted by $\left\\{u_{j}\right\\}$ such
that $u_{j}\to u$ a.e. in $M$. By Hölder’s inequality, we get

$\displaystyle\left|\left\langle{I^{2}_{\varepsilon}}^{\prime}(u_{j}),v\right\rangle\right|$
$\displaystyle\leq$
$\displaystyle\displaystyle\int_{M}\left|(u_{j}^{+})^{q-1}u_{j}v\right|\
dv_{g}$ $\displaystyle\leq$ $\displaystyle\displaystyle\int_{M}|u_{j}|^{q}|v|\
dv_{g}$ $\displaystyle\leq$
$\displaystyle\|u_{j}\|_{L^{q}(M)}^{1-1/q}\|v\|_{L^{q}(M)}^{q}$
$\displaystyle\leq$ $\displaystyle
C\left\|u_{j}\right\|_{H^{2}(M)}\left\|v\right\|_{H^{2}(M)}$

since $H^{2}(M)\hookrightarrow L^{q}(M)$. Additionally, the fact that
$u_{j}\to u$ strongly in $H^{2}(M)$ implies

$\lim_{n\rightarrow\infty}\|u_{j}-u\|_{H^{2}(M)}=0.$

Then we have

$\left\|{I^{2}_{\varepsilon}}^{\prime}(u_{j})-{I^{2}_{\varepsilon}}^{\prime}(u)\right\|=\sup_{v\in
H^{2}(M),\|v\|_{H^{2}(M)}\leq
1}\left|\left\langle{I_{\varepsilon}^{2}}^{\prime}\left(u_{j}\right)-{I_{\varepsilon}^{2}}^{\prime}(u),v\right\rangle\right|\rightarrow
0\qquad\text{ as }j\rightarrow\infty.$

∎

From the previous lemma, it follows that positive solutions of (2.7) belong to
$\mathcal{N}^{g}_{\varepsilon}$ and are critical points for
$J^{g,+}_{\varepsilon,q}$. Moreover, we get the following result.

######  Lemma 3.3.

For $\varepsilon$ small enough, the critical points of
$J^{g,+}_{\varepsilon,q}$ restricted to $\mathcal{N}^{g}_{\varepsilon}$ are
the positive solutions of Equation (2.7).

###### Proof.

Let $u\in\mathcal{N}^{g}_{\varepsilon}$ be a critical point of
$J^{g,+}_{\varepsilon,q}$ restricted to $\mathcal{N}^{g}_{\varepsilon}$. Then
$u$ must be nonnegative and a weak solution of (2.7). Since the embedding of
$H^{2}(M)$ in $L^{q}(M)$ is compact, we know from the classical bootstrap
argument that $u\in L^{s}(M)$ for all $s\geq 1$. By regularity theory (see
[27]), we get $u\in C^{4}(M)$.

Finally, by the theorem of Gursky and Malchiodi [15, Theorem A] that we
mentioned above we know that for $\varepsilon$ small enough, if $u\in
C^{4}(M\times X)$ satisfies

$P_{\mathfrak{g}_{\varepsilon}}u\geq 0,$

then either $u>0$ or $u\equiv 0$ on $M\times X$. In consequence, $u$ is
positive and smooth. ∎

By definition, a sequence $\left(u_{i}\right)$ of functions in
$\mathcal{N}^{g}_{\varepsilon}$ is said to be a Palais-Smale sequence for the
functional $J^{g,+}_{\varepsilon,q}$ in (3.1) if:

  * (PS1)

$J^{g,+}_{\varepsilon,q}\left(u_{i}\right)=O(1)$, when $i\to\infty$, and

  * (PS2)

$(J^{g,+}_{\varepsilon,q})^{\prime}\left(u_{i}\right)\rightarrow 0$ in
$(H^{2}(M))^{*}$ as $i\to+\infty$, where $(H^{2}(M))^{*}$ denotes the dual
space of $H^{2}(M)$.

We say that $J^{g,+}_{\varepsilon,q}$ satisfies Palais-Smale condition in
$\mathcal{N}^{g}_{\varepsilon}$, if for any Palais-Smale sequence
$\left\\{u_{i}\right\\}\subset\mathcal{N}^{g}_{\varepsilon}$, there exists a
convergent subsequence of $\left\\{u_{i}\right\\}$.

######  Theorem 3.4.

Assume that $q>1$ and if $n>4$ that $q<\frac{n+4}{n-4}$. Then,
$J^{g,+}_{\varepsilon,q}$ restricted to $\mathcal{N}^{g}_{\varepsilon}$ is
bounded below and it satisfies the Palais-Smale condition for $\varepsilon$
small enough.

###### Proof.

Note that if $u\in\mathcal{N}^{g}_{\varepsilon}$ then
$J^{g,+}_{\varepsilon,q}(u)\geq 0$. Let $\left\\{u_{i}\right\\}$ be a Palais-
Smale sequence in $\mathcal{N}^{g}_{\varepsilon}$ for
$J^{g,+}_{\varepsilon,q}$. Then, for $\varepsilon$ small enough, we have

$O(1)+o(1)\|u_{i}\|_{H^{2}(M)}\begin{array}[t]{ll}=(q+1)J^{g,+}_{\varepsilon,q}(u_{i})-\langle(J^{g,+}_{\varepsilon,q})^{\prime}(u_{i}),u_{i}\rangle\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\frac{1}{\varepsilon^{n}}\bigg{(}\frac{q-1}{2}\int_{M}\big{(}\varepsilon^{4}(\Delta
u)^{2}+\varepsilon^{2}b_{N,m}|\nabla u|^{2}+a_{N,m}u^{2}\big{)}\,dv_{g}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad\qquad+\varepsilon^{4}q\int_{M}\varphi(u)\,u\,dv_{g}+\frac{q-1}{2}\int_{M}\varepsilon^{2}F(\varepsilon,x)u^{2}dv_{g}\bigg{)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle\geq
C\|u_{i}\|^{2}_{H^{2}(M)},\end{array}$

where $C=C(N,m,q,\varepsilon)$, due to Remark 2.5. It follows that
$\|u_{i}\|_{H^{2}(M)}$ is bounded in $H^{2}(M)$ and in consequence, there is a
subsequence, still denoted $(u_{i})$ which converges weakly to a function
$u\in H^{2}(M)$ and strongly in $H^{1}(M)$. Set $v_{i}=u-u_{i}\in H^{2}(M)$.
We have

(3.4)
$\displaystyle\varepsilon^{4}\int_{M}|\Delta_{g}u_{i}|^{2}\,dv_{g}+b_{N,m}\
\varepsilon^{2}\int_{M}|\nabla
u_{i}|^{2}\,dv_{g}+a_{N,m}\int_{M}u_{i}^{2}\,dv_{g}\vskip 6.0pt plus 2.0pt
minus 2.0pt$
$\displaystyle\quad=\varepsilon^{4}\int_{M}(\Delta_{g}u)^{2}dv_{g}+b_{N,m}\
\varepsilon^{2}\int_{M}|\nabla u|^{2}dv_{g}+a_{N,m}\int_{M}u^{2}\,dv_{g}\vskip
6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle\quad\qquad-2\left(\varepsilon^{4}\int_{M}\Delta_{g}u\Delta_{g}v_{i}\,dv_{g}+b_{N,m}\
\varepsilon^{2}\int_{M}\nabla u\nabla
v_{i}\,dv_{g}+a_{N,m}\int_{M}uv_{i}\,dv_{g}\right)\vskip 6.0pt plus 2.0pt
minus 2.0pt$
$\displaystyle\quad\qquad+\varepsilon^{4}\int_{M}(\Delta_{g}v_{i})^{2}\,dv_{g}+b_{N,m}\
\varepsilon^{2}\int_{M}|\nabla
v_{i}|^{2}\,dv_{g}+a_{N,m}\int_{M}v_{i}^{2}\,dv_{g}$

Because of the weakly convergence in $H^{2}(M)$, we know that

$o(1)=\int_{M}\Delta_{g}w\Delta_{g}v_{i}\,dv_{g}+\int_{M}\nabla w\nabla
v_{i}\,dv_{g}+\int_{M}wv_{i}\,dv_{g},$

for all $w\in H^{2}(M)$, and the strong convergence in $H^{1}(M)$ implies

$\int_{M}(|\nabla v_{i}|^{2}+v_{i}^{2})\,dv_{g}=o(1).$

Then, from (3.4). we get

$\displaystyle\varepsilon^{4}\int_{M}(\Delta_{g}u_{i})^{2}\,dv_{g}+b_{N,m}\
\varepsilon^{2}\int_{M}|\nabla
u_{i}|^{2}\,dv_{g}+a_{N,m}\int_{M}u_{i}^{2}\,dv_{g}$
$\displaystyle=\varepsilon^{4}\int_{M}(\Delta_{g}u)^{2}\,dv_{g}+b_{N,m}\
\varepsilon^{2}\int_{M}|\nabla
u|^{2}\,dv_{g}+a_{N,m}\int_{M}u^{2}\,dv_{g}+\varepsilon^{4}\int_{M}(\Delta_{g}v_{i})^{2}\,dv_{g}+o(1).$

Moreover, since $2\leq q+1<2_{n}^{\sharp}$ and the embedding
$H^{2}(M)\hookrightarrow L^{q}(M)$ is compact, we can assume that
$\lim_{i\to+\infty}u_{i}=u$ in $L^{q}(M)$. Consequently, we get that

$\lim_{i\rightarrow+\infty}\int_{M}(u_{i}^{+})^{q+1}dv_{g}=\int_{M}(u^{+})^{q+1}dv_{g}.$

Plugging this equalities in
$\langle(J^{g,+}_{\varepsilon,q})^{\prime}(u_{i}),v_{i}\rangle$, one gets that

$\int_{M}(\Delta_{g}v_{i})^{2}dv_{g}=o(1)$

which means, that the convergence in $H^{2}(M)$ is strong. ∎

We are now in the position to apply the well known result from Lusternik-
Schnirelmann theory, which says that a $C^{1}$-functional $J$ on a Banach
manifold $N$ which is bounded form below and satisfies the Palais-Smale
condition has at least $\mathrm{Cat}(\\{x\in N:J(x)\leq a\\})$ critical
points, for any $a$ (see for instance [1, Chapter 9] or [4]).

Let
$\Sigma_{\varepsilon,d}=\\{u\in\mathcal{N}^{g}_{\varepsilon}:J^{g,+}_{\varepsilon,q}(u)\leq
d\\}$. Then Lemma 3.3 and Theorem 3.4 imply:

######  Corollary 3.5.

There exists $\varepsilon_{0}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0})$ and any $d>0$ Equation (2.7) has at least
$\mathrm{Cat}(\Sigma_{\varepsilon,d})$ positive solutions.

##  4\. The limit equation and functionals on $\mathbb{R}^{n}$

In this section we introduce the limit problem. Recall that in the equation
$2<q+1<2_{n}^{\sharp}$, where $2_{n}^{\sharp}=\infty$ if $n\leq 4$ and
$2_{n}^{\sharp}=\frac{2n}{n-4}$ if $n>4$. For $\alpha,\beta>0$, consider
$J_{\alpha,\beta}:H^{2}(\mathbb{R}^{n})\rightarrow\mathbb{R}$, given by

$J_{\alpha,\beta}(u):=\frac{1}{2}\int_{\mathbb{R}^{n}}\left((\Delta
u)^{2}+\beta|\nabla u|^{2}+\alpha u^{2}\right)dx.$

Let $E_{\alpha,\beta,q}:H^{2}(\mathbb{R}^{n})\rightarrow\mathbb{R}$, defined
by

$E_{\alpha,\beta,q}(u):=\frac{1}{2}\int_{\mathbb{R}^{n}}\left((\Delta
u)^{2}+\beta|\nabla u|^{2}+\alpha
u^{2}\right)dx-\frac{1}{q+1}\int_{\mathbb{R}^{n}}|u|^{q+1}dx.$

Also let
$Y_{\alpha,\beta,q}:H^{2}(\mathbb{R}^{n})\setminus\\{0\\}\rightarrow\mathbb{R}$,

$Y_{\alpha,\beta,q}(u):=\frac{J_{\alpha,\beta}(u)}{\|u\|_{q+1}^{2}}.$

Consider

$\mathcal{N}(\alpha,\beta,q):=\bigg{\\{}u\in
H^{2}(\mathbb{R}^{n})\setminus\\{0\\}:2J_{\alpha,\beta}(u)=\int_{\mathbb{R}^{n}}|u|^{q+1}dx\bigg{\\}}.$

Let

$m_{\alpha,\beta}:=\inf_{u\in\mathcal{N}(\alpha,\beta,q)}E_{\alpha,\beta,q}(u)=\inf_{u\in\mathcal{N}(\alpha,\beta,q)}\frac{q-1}{2(q+1)}\|u\|_{q+1}^{q+1}\geq
0.$

Note that for any $u\in H^{2}(\mathbb{R}^{n})\setminus\\{0\\}$ there exists
exactly one $\lambda=\lambda(u)>0$ such that
$\lambda(u)u\in\mathcal{N}(\alpha,\beta,q)$ and that for any constant $c\neq
0$, $Y_{\alpha,\beta,q}(cu)=Y_{\alpha,\beta,q}(u)$. It follows that

(4.1) $\displaystyle m_{\alpha,\beta}$
$\displaystyle=\inf_{u\in\mathcal{N}(\alpha,\beta,q)}\frac{q-1}{2(q+1)}\|u\|_{q+1}^{q+1}$
$\displaystyle=\frac{q-1}{2(q+1)}\left(\inf_{u\in\mathcal{N}(\alpha,\beta,q)}\|u\|_{q+1}^{q-1}\right)^{\frac{q+1}{q-1}}$
$\displaystyle=\frac{q-1}{2(q+1)}\left(\inf_{u\in\mathcal{N}(\alpha,\beta,q)}2Y_{\alpha,\beta,q}(u)\right)^{\frac{q+1}{q-1}}$
$\displaystyle=\frac{q-1}{q+1}\ 2^{\frac{2}{q-1}}\left(\inf_{u\in
H^{2}(\mathbb{R}^{n})\setminus\\{0\\}}Y_{\alpha,\beta,q}(u)\right)^{\frac{q+1}{q-1}}$
$\displaystyle=\frac{q-1}{q+1}\ 2^{\frac{2}{q-1}}\left(\inf_{u\in
H^{2}(\mathbb{R}^{n}),\|u\|_{q+1}=1}J_{\alpha,\beta}(u)\right)^{\frac{q+1}{q-1}}.$

Let us recall an important result from [7, Theorem 1.1] about this infimum.

######  Theorem [Bonheure and Nascimento].

If $\beta\geq 2\sqrt{\alpha}$ then $m_{\alpha,\beta}$ is realized by a
positive, radial (around some point), radially decreasing function.

Assume from now on that $\beta\geq 2\sqrt{\alpha}$ and let
$U_{\alpha,\beta}\in\mathcal{N}(\alpha,\beta,q)$ be a minimizer like in the
theorem. Similarly consider

$E_{\alpha,\beta,q}^{+}(u)=\frac{1}{2}\int_{\mathbb{R}^{n}}\left((\Delta
u)^{2}+\beta|\nabla u|^{2}+\alpha
u^{2}\right)dx-\frac{1}{q+1}\int_{\mathbb{R}^{n}}(u^{+})^{q+1}dx$

and $Y_{\alpha,\beta,q}^{+}:\\{u\in H^{2}(\mathbb{R}^{n}):u^{+}\neq
0\\}\rightarrow\mathbb{R}$,

$Y_{\alpha,\beta,q}^{+}(u)=\frac{J_{\alpha,\beta}(u)}{\|u^{+}\|_{q+1}^{2}}.$

Note that if $u\in H^{2}(\mathbb{R}^{n})$ and $u^{+}\neq 0$, then
$Y_{\alpha,\beta,q}^{+}(u)\geq Y_{\alpha,\beta,q}(u)$.

Let

$\mathcal{N}^{+}(\alpha,\beta,q)=\bigg{\\{}u\in
H^{2}(\mathbb{R}^{n})\setminus\\{0\\}:2J_{\alpha,\beta}(u)=\int_{\mathbb{R}^{n}}(u^{+})^{q+1}dx\bigg{\\}}.$

In the same way as before, we can see that for any $u\in
H^{2}(\mathbb{R}^{n})$ such that $u^{+}\neq 0$ there exists exactly one
$\lambda^{+}=\lambda^{+}(u)>0$ such that
$\lambda^{+}(u)u\in\mathcal{N}^{+}(\alpha,\beta,q)$ and that for any positive
constant $c$, $Y^{+}_{\alpha,\beta,q}(cu)=Y^{+}_{\alpha,\beta,q}(u)$. If we
define

$m^{+}_{\alpha,\beta}:=\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q)}E_{\alpha,\beta,q}^{+}(u)=\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q)}\frac{q-1}{2(q+1)}\|u^{+}\|_{q+1}^{q+1}\geq
0,$

then

$\displaystyle m^{+}_{\alpha,\beta}$
$\displaystyle=\frac{q-1}{2(q+1)}\left(\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q)}\|u^{+}\|_{q+1}^{q-1}\right)^{\frac{q+1}{q-1}}$
$\displaystyle=\frac{q-1}{q+1}\ 2^{\frac{2}{q-1}}\left(\inf_{u\in
H^{2}(\mathbb{R}^{n}),\ u^{+}\neq
0}Y^{+}_{\alpha,\beta,q}(u)\right)^{\frac{q+1}{q-1}}$
$\displaystyle=\frac{q-1}{q+1}\ 2^{\frac{2}{q-1}}\left(\inf_{u\in
H^{2}(\mathbb{R}^{n}),\|u^{+}\|_{q+1}=1}J_{\alpha,\beta}(u)\right)^{\frac{q+1}{q-1}}.$

######  Remark 4.1.

It follows easily from the previous observations that $m_{\alpha,\beta}\leq
m^{+}_{\alpha,\beta}$. But since we have a minimizer for $m_{\alpha,\beta}$
which is a positive function it follows that actually
$m_{\alpha,\beta}=m^{+}_{\alpha,\beta}$.

Throughout the paper, we will use the following notation: $B(0,r)$ is the ball
in $\mathbb{R}^{N}$ centered at $0$ with radius $r$.

Given $\varepsilon>0$ and $u:\mathbb{R}^{n}\rightarrow\mathbb{R}$, the
function $u_{\varepsilon}(x)=u(\varepsilon^{-1}x)$ satisfies the following
properties:

$\Delta u_{\varepsilon}(x)=\varepsilon^{-2}\Delta
u(\varepsilon^{-1}x),\quad\text{and}\quad|\nabla
u_{\varepsilon}(x)|^{2}=\varepsilon^{-2}|\nabla u(\varepsilon^{-1}x)|^{2}.$

Note also that for any function $f$ we have for any $r>0$,

$\int_{B(0,\varepsilon
r)}f_{\varepsilon}(x)\,dx=\varepsilon^{n}\int_{B(0.r)}f(x)\,dx,$

and, in consequence,

$\int_{\mathbb{R}^{n}}f_{\varepsilon}(x)\,dx=\varepsilon^{n}\int_{\mathbb{R}^{n}}f(x)\,dx.$

Therefore

$\varepsilon^{-n}\int_{\mathbb{R}^{n}}\left(\varepsilon^{4}(\Delta
u_{\varepsilon})^{2}+\varepsilon^{2}\beta|\nabla u_{\varepsilon}|^{2}+\alpha
u_{\varepsilon}^{2}\right)dx=\int_{\mathbb{R}^{n}}\left((\Delta
u)^{2}+\beta|\nabla u|^{2}+\alpha u^{2}\right)dx.$

Now, consider the functionals

$J^{\varepsilon}_{\alpha,\beta}(u):=\frac{\varepsilon^{-n}}{2}\int_{\mathbb{R}^{n}}\left(\varepsilon^{4}(\Delta
u)^{2}+\beta\varepsilon^{2}|\nabla u|^{2}+\alpha u^{2}\right)dx$

and

$E^{\varepsilon}_{\alpha,\beta,q}(u):=\frac{\varepsilon^{-n}}{2}\int_{\mathbb{R}^{n}}\left(\varepsilon^{4}(\Delta
u)^{2}+\varepsilon^{2}\beta|\nabla u|^{2}+\alpha
u^{2}\right)dx-\frac{\varepsilon^{-n}}{q+1}\int_{\mathbb{R}^{n}}|u|^{q+1}dx.$

Consider

$\mathcal{N}(\alpha,\beta,q,\varepsilon):=\bigg{\\{}u\in
H^{2}(\mathbb{R}^{n})\setminus\\{0\\}:2J^{\varepsilon}_{\alpha,\beta}(u)=\varepsilon^{-n}\int_{\mathbb{R}^{n}}|u|^{q+1}dx\bigg{\\}}.$

Note that $u\in\mathcal{N}(\alpha,\beta,q)$ if and only if
$u_{\varepsilon}\in\mathcal{N}(\alpha,\beta,q,\varepsilon)$ and
$J_{\alpha,\beta}(u)=J^{\varepsilon}_{\alpha,\beta}(u_{\varepsilon}),\
E_{\alpha,\beta,q}(u)=E^{\varepsilon}_{\alpha,\beta,q}(u_{\varepsilon})$.
Therefore, from (4.1), for any $\varepsilon>0$,

(4.2)
$m_{\alpha,\beta}=\inf_{u\in\mathcal{N}(\alpha,\beta,q,\varepsilon)}E^{\varepsilon}_{\alpha,\beta,q}(u).$

Let

(4.3)
$U^{\varepsilon}_{\alpha,\beta}(x)=U_{\alpha,\beta}({\varepsilon}^{-1}x),$

where $U_{\alpha,\beta}\in\mathcal{N}(\alpha,\beta,q)$ is a minimizer like in
the Theorem of Bonheure and Nascimento. Hence,
$U^{\varepsilon}_{\alpha,\beta}\in\mathcal{N}(\alpha,\beta,q,\varepsilon)$ and
realizes
$\inf_{u\in\mathcal{N}(\alpha,\beta,q,\varepsilon)}E^{\varepsilon}_{\alpha,\beta,q}(u)$.

Define now
$Y_{\alpha,\beta,q}^{\varepsilon}:H^{2}(\mathbb{R}^{n})\setminus\\{0\\}\rightarrow\mathbb{R}$,

$Y_{\alpha,\beta,q}^{\varepsilon}(u):=\frac{J^{\varepsilon}_{\alpha,\beta}(u)}{\varepsilon^{\frac{-2n}{q+1}}\|u\|_{q+1}^{2}}.$

and note that from (4.1) and (4.2) we get

$m_{\alpha,\beta}\begin{array}[t]{ll}\displaystyle=\inf_{u\in\mathcal{N}(\alpha,\beta,q,\varepsilon)}\frac{q-1}{2q+2}\varepsilon^{-n}\|u\|^{q+1}_{q+1}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\frac{q-1}{2(q+1)}\varepsilon^{-n}\left(\inf_{u\in\mathcal{N}(\alpha,\beta,q,\varepsilon)}\|u\|_{q+1}^{q-1}\right)^{\frac{q+1}{q-1}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\frac{q-1}{q+1}2^{\frac{2}{q-1}}\left(\inf_{u\in\mathcal{N}(\alpha,\beta,q,\varepsilon)}Y^{\varepsilon}_{\alpha,\beta,q}(u)\right)^{\frac{q+1}{q-1}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\frac{q-1}{q+1}2^{\frac{2}{q-1}}\left(\inf_{u\in
H^{2}(\mathbb{R}^{n})\setminus\\{0\\}}Y^{\varepsilon}_{\alpha,\beta,q}(u)\right)^{\frac{q+1}{q-1}}.\end{array}$

######  Definition 4.2.

Let $\displaystyle{\bf Y}_{\alpha,\beta,q}^{\varepsilon}:=\inf_{u\in
H^{2}(\mathbb{R}^{n})\setminus\\{0\\}}Y_{\alpha,\beta,q}^{\varepsilon}(u)$.

Then we have that

(4.4) ${\bf
Y}_{\alpha,\beta,q}^{\varepsilon}=2^{\frac{-2}{q+1}}\left(\frac{q+1}{q-1}m_{\alpha,\beta}\right)^{\frac{q-1}{q+1}},$

and, for any $u\in H^{2}(\mathbb{R}^{n})$,

(4.5) $J^{\varepsilon}_{\alpha,\beta}(u)\geq{\bf
Y}_{\alpha,\beta,q}^{\varepsilon}\
\varepsilon^{\frac{-2n}{q+1}}\|u\|_{q+1}^{2}.$

Similarly, let

$E^{\varepsilon,+}_{\alpha,\beta,q}(u)=\frac{\varepsilon^{-n}}{2}\int_{\mathbb{R}^{n}}\left(\varepsilon^{4}(\Delta
u)^{2}+\varepsilon^{2}\beta|\nabla u|^{2}+\alpha
u^{2}\right)dx-\frac{\varepsilon^{-n}}{q+1}\int_{\mathbb{R}^{n}}(u^{+})^{q+1}dx.$

Consider

$\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)=\bigg{\\{}u\in
H^{2}(\mathbb{R}^{n})\setminus\\{0\\}:2J_{\alpha,\beta,\varepsilon}(u)=\varepsilon^{-n}\int_{\mathbb{R}^{n}}(u^{+})^{q+1}dx\bigg{\\}}.$

Note that $u\in\mathcal{N}^{+}(\alpha,\beta,q)$ if and only if
$u_{\varepsilon}\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)$ and
$J_{\alpha,\beta}(u)=J^{\varepsilon}_{\alpha,\beta}(u_{\varepsilon}),\ \
E^{+}_{\alpha,\beta,q}(u)=E^{\varepsilon,+}_{\alpha,\beta,q}(u_{\varepsilon})$,
where

$E^{+}_{\alpha,\beta,q}(u):=\frac{1}{2}\int_{\mathbb{R}^{n}}\left((\Delta
u)^{2}+\beta|\nabla u|^{2}+\alpha
u^{2}\right)dx-\frac{1}{q+1}\int_{\mathbb{R}^{n}}(u^{+})^{q+1}dx.$

Therefore, we also have that for any $\varepsilon>0$,

$m_{\alpha,\beta}=\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)}E^{\varepsilon,+}_{\alpha,\beta,q}(u).$

Note that if $u\in H^{2}(\mathbb{R}^{n})$, $u^{+}\neq 0$, there exists a
unique $\lambda^{+,\varepsilon}(u)>0$ such that
$\lambda^{+,\varepsilon}(u)\,u\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)$.

Let $Y_{\alpha,\beta,q}^{\varepsilon,+}:\\{u\in
H^{2}(\mathbb{R}^{n}):u^{+}\neq 0\\}\rightarrow\mathbb{R}$,

$Y_{\alpha,\beta,q}^{\varepsilon,+}(u):=\frac{J^{\varepsilon}_{\alpha,\beta}(u)}{\varepsilon^{\frac{-2n}{q+1}}\|u^{+}\|_{q+1}^{2}}.$

Then we have that

$m_{\alpha,\beta}\begin{array}[t]{ll}\displaystyle=\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)}E^{\varepsilon,+}_{\alpha,\beta,q}(u)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)}\frac{q-1}{2(q+1)}\varepsilon^{-n}\|u^{+}\|_{q+1}^{q+1}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\frac{q-1}{2(q+1)}\varepsilon^{-n}\left(\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)}\|u^{+}\|_{q+1}^{q-1}\right)^{\frac{q+1}{q-1}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle=\frac{q-1}{q+1}\
2^{\frac{2}{q-1}}\left(\inf_{u\in H^{2}(\mathbb{R}^{n}),\ u^{+}\neq
0}Y^{\varepsilon,+}_{\alpha,\beta,q}(u)\right)^{\frac{q+1}{q-1}}\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \displaystyle=\frac{q-1}{q+1}\
2^{\frac{2}{q-1}}\left(\inf_{u\in
H^{2}(\mathbb{R}^{n}),\|u^{+}\|_{q+1}=1}J^{\varepsilon}_{\alpha,\beta}(u)\right)^{\frac{q+1}{q-1}}.\end{array}$

On $\mathbb{R}^{n}$ consider a smooth radial cut-off function
$\varphi_{s}:\mathbb{R}^{n}\rightarrow[0,1]$ supported in $B(0,s/2)$ such that
it is equal to 1 on $B(0,s/4)$. Assume that
$|\nabla\varphi_{s}|^{2},(\Delta\varphi_{s})^{2}\leq D$, for some positive
constant $D$ (which depends on $s$). Then, for $\varepsilon>0$ small, we
define

(4.6)
$U_{\alpha,\beta}^{\varepsilon,s}:=\,\varphi_{s}\,U_{\alpha,\beta}^{\varepsilon}$

where $U_{\alpha,\beta}^{\varepsilon}$ is given by (4.3).

######  Lemma 4.3.

$\lim_{\varepsilon\rightarrow
0}\lambda^{+,\varepsilon}(U_{\alpha,\beta}^{\varepsilon,s})=1$ and
$\lim_{\varepsilon\rightarrow
0}E_{\alpha,\beta,q}^{\varepsilon,+}(\lambda^{+,\varepsilon}(U_{\alpha,\beta}^{\varepsilon,s})U_{\alpha,\beta}^{\varepsilon,s})=m_{\alpha,\beta}$.

###### Proof.

Note that

$\lambda^{+,\varepsilon}(u)=\left(\frac{\int_{\mathbb{R}^{n}}\left(\varepsilon^{4}(\Delta
u)^{2}+\varepsilon^{2}\beta|\nabla u|^{2}+\alpha
u^{2}\right)dx}{\int_{\mathbb{R}^{n}}(u^{+})^{q+1}dx}\right)^{\frac{1}{q-1}}$

We have that $\Delta
U_{\alpha,\beta}^{\varepsilon,s}=\Delta(U_{\alpha,\beta}^{\varepsilon}\,\varphi_{s})=\varphi_{s}\,\Delta
U_{\alpha,\beta}^{\varepsilon}+2\langle\nabla
U_{\alpha,\beta}^{\varepsilon},\nabla\varphi_{s}\rangle+U_{\alpha,\beta}^{\varepsilon}\Delta\varphi_{s}$.

Then

$\left|\Delta
U_{\alpha,\beta}^{\varepsilon,s}\right|^{2}\begin{array}[t]{ll}\displaystyle=\left|\varphi_{s}\Delta
U_{\alpha,\beta}^{\varepsilon}\right|^{2}+\left|2\langle\nabla
U_{\alpha,\beta}^{\varepsilon},\nabla\varphi_{s}\rangle+U_{\alpha,\beta}^{\varepsilon}\Delta\varphi_{s}\right|^{2}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle\qquad+2\varphi_{s}\Delta
U_{\alpha,\beta}^{\varepsilon}\left(2\langle\nabla
U_{\alpha,\beta}^{\varepsilon},\nabla\varphi_{s}\rangle+U_{\alpha,\beta}^{\varepsilon}\Delta\varphi_{s}\right).\end{array}$

Note that for any $\delta>0$ we have that

$|2\varphi_{s}\Delta U_{\alpha,\beta}^{\varepsilon}(2\langle\nabla
U_{\alpha,\beta}^{\varepsilon},\nabla\varphi_{s}\rangle+U_{\alpha,\beta}^{\varepsilon}\Delta\varphi_{s})|\leq\delta\left|\varphi_{s}^{2}\Delta
U_{\alpha,\beta}^{\varepsilon}\right|^{2}+\delta^{-1}\left|2\langle\nabla
U_{\alpha,\beta}^{\varepsilon},\nabla\varphi_{s}\rangle+U_{\alpha,\beta}^{\varepsilon}\Delta\varphi_{s}\right|^{2}.$

It follows that we can pick $\delta$ small and write

$\begin{array}[t]{ll}\displaystyle\varphi^{2}_{s}(1-\delta)\left(\varepsilon^{4}(\Delta
U_{\alpha,\beta}^{\varepsilon})^{2}+O(\varepsilon^{4})(|\nabla
U_{\alpha,\beta}^{\varepsilon}|^{2}+(U_{\alpha,\beta}^{\varepsilon})^{2})\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle\quad\leq\varepsilon^{4}(\Delta
U_{\alpha,\beta}^{\varepsilon,s})^{2}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\quad\leq\varphi^{2}_{s}(1+\delta)(\varepsilon^{4}(\Delta
U_{\alpha,\beta}^{\varepsilon})^{2}+O(\varepsilon^{4})(|\nabla
U_{\alpha,\beta}^{\varepsilon}|^{2}+(U_{\alpha,\beta}^{\varepsilon})^{2}).\end{array}$

Similarly

$|\nabla U_{\alpha,\beta}^{\varepsilon,s}|^{2}=|\varphi_{s}\nabla
U_{\alpha,\beta}^{\varepsilon}+U_{\alpha,\beta}^{\varepsilon}\nabla\varphi_{s}|^{2}\leq(1+\delta)\varphi_{s}^{2}|\nabla
U_{\alpha,\beta}^{\varepsilon}|^{2}+(1+\delta^{-1})|\nabla\varphi_{s}|^{2}(U_{\alpha,\beta}^{\varepsilon})^{2}$

and

$|\nabla
U_{\alpha,\beta}^{\varepsilon,s}|^{2}\geq(1-\delta)\varphi_{s}^{2}|\nabla
U_{\alpha,\beta}^{\varepsilon}|^{2}+(1-\delta^{-1})|\nabla\varphi_{s}|^{2}(U_{\alpha,\beta}^{\varepsilon})^{2}.$

It follows that

$\begin{array}[t]{lll}\displaystyle\int_{\mathbb{R}^{n}}\big{(}\varepsilon^{4}(\Delta
U_{\alpha,\beta}^{\varepsilon,s})^{2}+\varepsilon^{2}\beta|\nabla
U_{\alpha,\beta}^{\varepsilon,s}|^{2}+\alpha(U_{\alpha,\beta}^{\varepsilon,s})^{2}\big{)}\,dx\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\quad\leq\int_{\mathbb{R}^{n}}\big{(}\varepsilon^{4}(1+\delta)\varphi_{s}^{2}(\Delta
U_{\alpha,\beta}^{\varepsilon})^{2}+\varepsilon^{2}(1+\delta)\beta\varphi_{s}^{2}(1+o(\varepsilon^{2}))|\nabla
U_{\alpha,\beta}^{\varepsilon}|^{2}+\alpha\varphi_{s}^{2}(1+o(\varepsilon^{2}))(U_{\alpha,\beta}^{\varepsilon})^{2}\big{)}\,dx.\end{array}$

Therefore for any $\delta>0$ we can choose $\varepsilon_{0}>0$ such that for
every $\varepsilon\in(0,\varepsilon_{0})$ we have that

(4.7)
$\begin{array}[t]{lll}\displaystyle\varepsilon^{-n}\int_{\mathbb{R}^{n}}\big{(}\varepsilon^{4}(\Delta
U_{\alpha,\beta}^{\varepsilon,s})^{2}+\varepsilon^{2}\beta|\nabla
U_{\alpha,\beta}^{\varepsilon,s}|^{2}+\alpha(U_{\alpha,\beta}^{\varepsilon,s})^{2}\big{)}\,dx\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad\leq\varepsilon^{-n}(1+2\delta)\int_{\mathbb{R}^{n}}\big{(}\varepsilon^{4}(\Delta
U_{\alpha,\beta}^{\varepsilon})^{2}+\varepsilon^{2}\beta|\nabla
U_{\alpha,\beta}^{\varepsilon}|^{2}+\alpha(U_{\alpha,\beta}^{\varepsilon})^{2}\big{)}\,dx\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad=(1+2\delta)\int_{\mathbb{R}^{n}}\big{(}(\Delta
U_{\alpha,\beta})^{2}+\beta|\nabla
U_{\alpha,\beta}|^{2}+\alpha(U_{\alpha,\beta})^{2}\big{)}\,dx.\end{array}$

In the same way we can assume that for the same values of
$\delta,\varepsilon_{0}$, and any $\varepsilon\in(0,\varepsilon_{0})$ to get

(4.8)
$\begin{array}[t]{ll}\displaystyle\varepsilon^{-n}\int_{\mathbb{R}^{n}}\big{(}\varepsilon^{4}(\Delta
U_{\alpha,\beta}^{\varepsilon,s})^{2}+\varepsilon^{2}\beta|\nabla
U_{\alpha,\beta}^{\varepsilon,s}|^{2}+\alpha(U_{\alpha,\beta}^{\varepsilon,s})^{2}\big{)}\,dx\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad\geq(1-2\delta)\int_{\mathbb{R}^{n}}\big{(}\Delta(U_{\alpha,\beta})^{2}+\beta|\nabla
U_{\alpha,\beta}|^{2}+\alpha(U_{\alpha,\beta})^{2}\big{)}\,dx.\end{array}$

We can also see from (4.3) and (4.6), that

(4.9) $\lim_{\varepsilon\rightarrow 0}\
\varepsilon^{-n}\int_{\mathbb{R}^{n}}(U_{\alpha,\beta}^{\varepsilon,s})^{q+1}dx=\int_{\mathbb{R}^{n}}(U_{\alpha,\beta})^{q+1}.$

Since $U_{\alpha,\beta}\in\mathcal{N}({\alpha,\beta,q})$ we have that
$\lim_{\varepsilon\rightarrow
0}\lambda^{+,\varepsilon}(U_{\alpha,\beta}^{\varepsilon,s})=1$.

Therefore we are only left to prove that $\lim_{\varepsilon\rightarrow
0}E_{\alpha,\beta,q}^{\varepsilon,+}(U_{\alpha,\beta}^{\varepsilon,s})=m_{\alpha,\beta}$.
But this also follows directly from (4.7), (4.8), (4.9). ∎

##  5\. The variational setting on a general Riemannian manifold

Consider a closed $n$-dimensional Riemannian manifold $(M,g)$. Let
$\alpha,\beta$ be positive constants, $\beta^{2}\geq 4\alpha$. As defined in
the previous section, for Euclidean space, we consider
$J^{g,\varepsilon}_{\alpha,\beta}:H^{2}(M)\rightarrow\mathbb{R}$ where

$J^{g,\varepsilon}_{\alpha,\beta}(u)=\frac{\varepsilon^{-n}}{2}\int_{M}\left(\varepsilon^{4}(\Delta_{g}u)^{2}+\beta\varepsilon^{2}|\nabla
u|^{2}+\alpha u^{2}\right)dv_{g}.$

For $q>1$ such that if $n\geq 5$, $q<\frac{n+4}{n-4}$, let

$E^{g,\varepsilon,+}_{\alpha,\beta,q}(u)=\frac{\varepsilon^{-n}}{2}\int_{M}\left(\varepsilon^{4}(\Delta_{g}u)^{2}+\varepsilon^{2}\beta|\nabla
u|^{2}+\alpha
u^{2}\right)dv_{g}-\frac{\varepsilon^{-n}}{q+1}\int_{M}(u^{+})^{q+1}dv_{g}.$

Let

$\mathcal{N}^{+}_{g}(\alpha,\beta,q,\varepsilon)=\bigg{\\{}u\in
H^{2}(M)\setminus\\{0\\}:2J^{\varepsilon}_{\alpha,\beta}(u)=\varepsilon^{-n}\int_{M}(u^{+})^{q+1}dv_{g}\bigg{\\}},$
$m^{g,+}_{\alpha,\beta,\varepsilon}=\inf_{u\in\mathcal{N}^{+}_{g}(\alpha,\beta,q,\varepsilon)}E^{\varepsilon,+}_{\alpha,\beta,q}(u)=\inf_{u\in\mathcal{N}^{+}_{g}(\alpha,\beta,q,\varepsilon)}\varepsilon^{-n}\frac{q-1}{2(q+1)}\|u^{+}\|^{q+1}_{L^{q+1}(M)}\geq
0.$

Note as before that for any $u\in H^{2}(M)$ such that $u^{+}\neq 0$ there is a
unique positive number $\lambda^{g,\varepsilon,+}(u)$ such that
$\lambda^{g,\varepsilon,+}(u).u\in\mathcal{N}^{+}_{g}(\alpha,\beta,q,\varepsilon)$.

Also define $Y^{g,\varepsilon,+}_{\alpha,\beta,q}:\\{u\in H^{2}(M):u^{+}\neq
0\\}\rightarrow\mathbb{R}$,

$Y^{g,\varepsilon,+}_{\alpha,\beta,q}(u)\begin{array}[t]{ll}\displaystyle=\frac{\frac{\varepsilon^{-n}}{2}\int_{M}\left(\varepsilon^{4}(\Delta_{g}u)^{2}+\varepsilon^{2}\beta|\nabla
u|^{2}+\alpha
u^{2}\right)dv_{g}}{\left(\varepsilon^{-n}\int_{M}(u^{+})^{q+1}dv_{g}\right)^{\frac{2}{q+1}}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\varepsilon^{-n\frac{q-1}{q+1}}\frac{\frac{1}{2}\int_{M}\left(\varepsilon^{4}(\Delta_{g}u)^{2}+\varepsilon^{2}\beta|\nabla
u|^{2}+\alpha
u^{2}\right)dv_{g}}{\left(\int_{M}(u^{+})^{q+1}dv_{g}\right)^{\frac{2}{q+1}}}.\end{array}$

Note that if $u\in\mathcal{N}^{+}_{g}(\alpha,\beta,q,\varepsilon)$ then
$Y^{g,\varepsilon,+}_{\alpha,\beta,q}(u)=(1/2)\varepsilon^{-n\frac{q-1}{q+1}}\|u^{+}\|^{q-1}_{L^{q+1}(M)}$.
Then

(5.1)
$m^{g,+}_{\alpha,\beta,\varepsilon}\begin{array}[t]{lll}\displaystyle=\inf_{u\in\mathcal{N}_{g}^{+}(\alpha,\beta,q,\varepsilon)}\varepsilon^{-n}\frac{q-1}{2(q+1)}\|u^{+}\|_{L^{q+1}(M)}^{q+1}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\frac{q-1}{2(q+1)}\bigg{(}\inf_{u\in\mathcal{N}_{g}^{+}(\alpha,\beta,q,\varepsilon)}\varepsilon^{-n\frac{q-1}{q+1}}\|u^{+}\|_{L^{q+1}(M)}^{q-1}\bigg{)}^{\frac{q+1}{q-1}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\frac{q-1}{q+1}2^{\frac{q+1}{q-1}}\left(\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)}Y^{g,\varepsilon,+}_{\alpha,\beta,q}(u)\right)^{\frac{q+1}{q-1}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\frac{q-1}{q+1}2^{\frac{2}{q-1}}\left(\inf_{u\in\\{u\in
H^{2}(M):u^{+}\neq
0\\}}Y^{g,\varepsilon,+}_{\alpha,\beta,q}(u)\right)^{\frac{q+1}{q-1}}.\end{array}$

######  Remark 5.1.

For any $x\in M$ identify the geodesic ball of some small radius $r$ with the
ball of radius $r$ centered at 0 in $\mathbb{R}^{n}$ using the exponential map
$exp_{x}$. Let $\overline{U}_{\alpha,\beta,x}^{\varepsilon,s}=exp_{x}\circ
U_{\alpha,\beta}^{\varepsilon,s}$. Note that for any $\delta>0$ we can find
$r>0$ such that on the ball $B_{g}(0,r)$ is $\delta$-close to the Euclidean
metric in the $C^{1}$-topology. It follows that the volume elements,
gradients, and Laplacian corresponding to $g$ and to the Euclidean metric are
$\delta$-close.

Since for $\varepsilon$ small enough we can get that
$U_{\alpha,\beta}^{\varepsilon,s}$ is concentrated in any small ball it
follows directly from the previous remark and Lemma 4.3 that:

######  Proposition 5.2.

We have

$\lim_{\varepsilon\rightarrow
0}\lambda^{g,\varepsilon,+}(\overline{U}_{\alpha,\beta,x}^{\varepsilon,s})=1\quad\mbox{and}\quad\lim_{\varepsilon\rightarrow
0}E^{\varepsilon,+}_{\alpha,\beta,q}(\lambda^{g,\varepsilon,+}(\overline{U}_{\alpha,\beta,x}^{\varepsilon,s})\overline{U}_{\alpha,\beta,x}^{\varepsilon,s})=m_{\alpha,\beta}.$

######  Definition 5.3.

With the notations from the previous remark define ${\bf
i}:M\rightarrow\mathcal{N}(\alpha,\beta,q,\varepsilon)$ by

${\bf
i}(x):=\lambda^{g,\varepsilon,+}(\overline{U}_{\alpha,\beta,x}^{\varepsilon,s})\overline{U}_{\alpha,\beta,x}^{\varepsilon,s}.$

We prove

######  Theorem 5.4.

For any $n$-dimensional closed Riemannian manifold $(M,g)$ we have that

$\lim_{\varepsilon\rightarrow
0}m^{g,+}_{\alpha,\beta,\varepsilon}=m_{\alpha,\beta}.$

###### Proof.

Since for any $x\in M$, $m^{g,+}_{\alpha,\beta,\varepsilon}\leq
E^{\varepsilon,+}_{\alpha,\beta,q}({\bf i}(x))$ it follows from the previous
proposition that $\limsup_{\varepsilon\rightarrow
0}m^{g,+}_{\alpha,\beta,\varepsilon}\leq m_{\alpha,\beta}$.

To complete the proof of the theorem we prove that
$\liminf_{\varepsilon\rightarrow 0}m^{q,+}_{\alpha,\beta,\varepsilon}\geq
m_{\alpha,\beta}$.

For any $\delta\in(0,1)$ we can pick normal neighborhoods $U_{1},...,U_{L}$
such that in normal coordinates $g$ is $C^{1}$-close to the Euclidean metric.
Namely

$(1+\delta)^{-1}\leq g_{ii}\leq(1+\delta)$ $|g_{ij}|\leq\delta,\ \ \ i\neq j$
$|g_{ij,k}|\leq\delta$

and

$(1+\delta)^{-1}dx\leq dv_{g}\leq(1+\delta)dx$

Let $\varphi_{k}^{2}$, $k=1,\dots,L$ be a partition of unity subordinate to
$U_{1},\dots,U_{L}$. Let $D$ be an upper bound for
$|\nabla\varphi_{k}|^{2},(\Delta\varphi_{k})^{2}$, $k=1,\dots,L$.

For any $u\in H^{2}(M)$ we have

$\|\varepsilon^{\frac{-n}{q+1}}u^{+}\|^{2}_{L^{q+1}(M)}\begin{array}[t]{ll}\displaystyle\leq\|\varepsilon^{\frac{-n}{q+1}}u\|^{2}_{L^{q+1}(M)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\|\varepsilon^{\frac{-2n}{q+1}}u^{2}\|_{L^{\frac{q+1}{2}}(M)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\Big{\|}\sum_{k}\varepsilon^{\frac{-2n}{q+1}}\varphi_{k}^{2}u^{2}\Big{\|}_{L^{\frac{q+1}{2}}(M)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\leq\sum_{k}\left(\varepsilon^{-n}\int_{U_{k}}|\varphi_{k}u|^{q+1}dv_{g}\right)^{\frac{2}{q+1}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\leq(1+\delta)^{\frac{2}{q+1}}\sum_{k}\left(\varepsilon^{-n}\int_{U_{k}}|\varphi_{k}u|^{q+1}dx\right)^{\frac{2}{q+1}}.\end{array}$

Using (4.5) we get

$\left(\varepsilon^{-n}\int_{U_{k}}|\varphi_{k}u|^{q+1}dx\right)^{\frac{2}{q+1}}\leq\left({\bf
Y}^{\varepsilon}_{\alpha,\beta,q}\right)^{-1}J^{\varepsilon}_{\alpha,\beta}(\varphi_{k}u).$

But

$J^{\varepsilon}_{\alpha,\beta}(\varphi_{k}u)\begin{array}[t]{ll}\displaystyle=\frac{\varepsilon^{-n}}{2}\int_{U_{k}}\big{(}\varepsilon^{4}(\Delta(\varphi_{k}u))^{2}+\varepsilon^{2}\alpha|\nabla(\varphi_{k}u)|^{2}+\beta(\varphi_{k}u)^{2}\big{)}\,dx\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\leq(1+\delta)^{2}\frac{\varepsilon^{-n}}{2}\int_{U_{k}}\big{(}\varepsilon^{4}(\Delta_{g}(\varphi_{k}u))^{2}+\varepsilon^{2}\alpha|\nabla(\varphi_{k}u)|^{2}+\beta(\varphi_{k}u)^{2}\big{)}\,dv_{g}.\end{array}$

Arguing as in the proof of Lemma 4.3 we can now see that for $\varepsilon$
small enough

$J^{\varepsilon}_{\alpha,\beta}(\varphi_{k}u)\leq(1+\delta)(1+\delta)^{2}\frac{\varepsilon^{-n}}{2}\int_{U_{k}}\varepsilon^{4}\varphi_{k}^{2}(\Delta_{g}u)^{2}+\varepsilon^{2}\alpha\varphi_{k}^{2}|\nabla
u|^{2}+\beta\varphi_{k}^{2}u^{2}dv_{g}.$

Then

$\begin{array}[t]{ll}\displaystyle\left(\varepsilon^{-n}\int_{M}(u^{+})^{q+1}dv_{g}\right)^{\frac{2}{q+1}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad\leq(1+\delta)^{\frac{2}{q+1}}\left({\bf
Y}^{\varepsilon}_{\alpha,\beta,q}\right)^{-1}(1+\delta)^{3}\sum_{k}\frac{\varepsilon^{-n}}{2}\int_{U_{k}}\big{(}\varepsilon^{4}\varphi_{k}^{2}(\Delta_{g}u)^{2}+\varepsilon^{2}\alpha\varphi_{k}^{2}|\nabla
u|^{2}+\beta\varphi_{k}^{2}u^{2}\big{)}\,dv_{g}\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \displaystyle\qquad=(1+\delta)^{\frac{2}{q+1}}\left({\bf
Y}^{\varepsilon}_{\alpha,\beta,q}\right)^{-1}(1+\delta)^{3}J^{\varepsilon}_{\alpha,\beta}(u).\end{array}$

Therefore

$Y^{g,\varepsilon,+}_{\alpha,\beta,q}(u)\geq\frac{1}{(1+\delta)^{4}}{\bf
Y}^{\varepsilon}_{\alpha,\beta,q}.$

Then by (4.4) and (5.1) it follows that

$m^{g,+}_{\alpha,\beta,\varepsilon}\geq\frac{1}{(1+\delta)^{4}}m_{\alpha,\beta}.$

Since this could be done for any $\delta>0$ it follows that

$\liminf_{\varepsilon\rightarrow 0}m^{g,+}_{\alpha,\beta,\varepsilon}\geq
m_{\alpha,\beta}.$

∎

Now define

$J_{\varepsilon}^{g}(u):=\frac{\varepsilon^{-n}}{2}\int_{M}\varepsilon^{4}u\,P_{g+\varepsilon^{2}h}u\,dv_{g},$

where $P_{g+\varepsilon^{2}h}$ is given in (2.3). Observe that
$\mathcal{N}^{g}_{\varepsilon}$ in (3.2) is given by

$\mathcal{N}^{g}_{\varepsilon}=\bigg{\\{}u\in
H^{2}(M):\int_{M}\big{(}\varepsilon^{4}uP_{g+\varepsilon^{2}h}u\big{)}\,dv_{g}=\int_{M}(u^{+})^{q+1}dv_{g}\bigg{\\}},$

and $J^{g,+}_{\varepsilon,q}(u)$ in (3.1) is given by

$J^{g,+}_{\varepsilon,q}(u):=\frac{\varepsilon^{-n}}{2}\int_{M}\varepsilon^{4}u\,P_{g+\varepsilon^{2}h}u\,dv_{g}-\frac{\varepsilon^{-n}}{q+1}\int_{M}(u^{+})^{q+1}\,dv_{g}.$

We now put

${\bf
m}_{\varepsilon}:=\inf_{u\in\mathcal{N}^{g}_{\varepsilon}}J^{g,+}_{\varepsilon,q}(u).$

It follows from Remark 2.5 and Theorem 5.4 that

######  Theorem 5.5.

It is verified that

$\lim_{\varepsilon\rightarrow 0}{\bf m}_{\varepsilon}=m_{a_{N,m},b_{N,m}}.$

##  6\. Concentration for functions on $\Sigma_{\varepsilon,{\bf
m}_{\varepsilon}+\delta}$

For any $d>0$ we let
$\Sigma_{\varepsilon,d}=\\{u\in\mathcal{N}^{g}_{\varepsilon}:J^{g,+}_{\varepsilon,q}(u)<d\\}$.
In this section we show that for $\varepsilon>0$, $\delta>0$ small enough, the
functions in $\Sigma_{\varepsilon,{\bf m}_{\varepsilon}+\delta}$ concentrate
on a small ball. Namely, we prove:

######  Theorem 6.1.

Fix $\eta<1$ and $r>0$. There exist $\varepsilon_{0},\delta_{0}>0$ such that
for any $\varepsilon\in(0,\varepsilon_{0})$, $\delta\in(0,\delta_{0})$ and any
$u\in\Sigma_{\varepsilon,{\bf m}_{\varepsilon}+\delta}$ there exists $x\in M$
such that

$\int_{B_{g}(x,r)}(u^{+})^{q+1}dv_{g}\geq\eta\int_{M}(u^{+})^{q+1}dv_{g}.$

Let $\alpha=a_{N,m}$, $\beta={b_{N},m}$. Consider as in the previous section

$J^{g,\varepsilon}_{\alpha,\beta}(u)=\frac{\varepsilon^{-n}}{2}\int_{M}\big{(}\varepsilon^{4}(\Delta_{g}u)^{2}+\beta\varepsilon^{2}|\nabla
u|^{2}+\alpha u^{2}\big{)}\,dv_{g},$
$E^{g,\varepsilon,+}_{\alpha,\beta,q}(u)=\frac{\varepsilon^{-n}}{2}\int_{M}\big{(}\varepsilon^{4}(\Delta_{g}u)^{2}+\varepsilon^{2}\beta|\nabla
u|^{2}+\alpha
u^{2}\big{)}\,dv_{g}-\frac{\varepsilon^{-n}}{q+1}\int_{M}(u^{+})^{q+1}dv_{g},$
$\mathcal{N}_{g}^{+}(\alpha,\beta,q,\varepsilon)=\bigg{\\{}u\in
H^{2}(M)\setminus\\{0\\}:2J^{g,\varepsilon}_{\alpha,\beta}(u)=\varepsilon^{-n}\int_{M}(u^{+})^{q+1}dv_{g}\bigg{\\}},$
(6.1)
$m^{g,+}_{\alpha,\beta,\varepsilon}\begin{array}[t]{ll}\displaystyle=\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)}E^{\varepsilon,+}_{\alpha,\beta,q}(u)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)}J^{g,\varepsilon}_{\alpha,\beta}(u)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\inf_{u\in\mathcal{N}^{+}(\alpha,\beta,q,\varepsilon)}\frac{q-1}{2(q+1)}\varepsilon^{-n}\int_{M}(u^{+})^{q+1}dv_{g}.\end{array}$

Let

$\Sigma^{\alpha,\beta}_{\varepsilon,d}=\\{u\in\mathcal{N}_{g}^{+}(\alpha,\beta,q,\varepsilon):E^{g,\varepsilon,+}_{\alpha,\beta,q}(u)\leq
d\\}.$

In the rest of the paper, we will use the following notation: $B_{g}(x,r)$ is
the ball in $M$ centered at $x$ with radius r with respect to the distance
induced by the metric $g$.

Let $r_{0}=r_{0}(M,g)$ be such that for any $0<r<r_{0}$ and any $x\in M$,
$B_{g}(x,r_{0})$ is strongly convex.

Taking into account Remark 2.5 and Theorem 5.5 we see that Theorem 6.1 follows
from

######  Theorem 6.2.

Fix $r<r_{0}(M,g)$. For any $\eta<1$ there exist
$\varepsilon_{0},\delta_{0}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0})$, $\delta\in(0,\delta_{0})$ and
$u\in\Sigma^{\alpha,\beta}_{\varepsilon,m^{g,+}_{\alpha,\beta,\varepsilon}+\delta}$
there exists $x\in M$ such that
$\int_{B_{g}(x,r)}(u^{+})^{q+1}\geq\eta\int_{M}(u^{+})^{q+1}$.

For any function $u\in H^{2}(M)$ such that $u^{+}\neq 0$ we let as before

$\lambda^{g,\varepsilon,+}(u)={\left(\frac{\int_{M}\big{(}\varepsilon^{4}(\Delta_{g}u)^{2}+\varepsilon^{2}\beta|\nabla
u|^{2}+\alpha
u^{2}\big{)}\,dv_{g}}{\int_{M}(u^{+})^{q+1}dv_{g}}\right)}^{\frac{1}{{q-1}}},$

so that
$\lambda^{g,\varepsilon,+}(u)\,u\in\mathcal{N}^{+}_{g}(\alpha,\beta,q,\varepsilon)$

######  Proposition 6.3.

Let $u=u_{1}+u_{2}\in\mathcal{N}^{+}_{g}(\alpha,\beta,q,\varepsilon)$, where
$u_{1}$ and $u_{2}$ have disjoint supports. For $i=1,2$ let
$\varepsilon^{-n}\int_{M}(u_{i}^{+})^{q+1}dv_{g}=a_{i}>0$ and
$\varepsilon^{-n}\int_{M}\big{(}\varepsilon^{4}(\Delta_{g}u)^{2}+\varepsilon^{2}\beta|\nabla
u|^{2}+\alpha u^{2}\big{)}\,dv_{g}=b_{i}>0$. Note that then
$a_{1}+a_{2}=b_{1}+b_{2}$ (since
$u\in\mathcal{N}_{g}^{+}(\alpha,\beta,q,\varepsilon)$) . Then

$E^{\varepsilon,+}_{\alpha,\beta,q}(u)\geq
m^{g,+}_{\alpha,\beta,\varepsilon}\left({\left(\frac{a_{1}}{b_{1}}\right)}^{\frac{2}{q-1}}+{\left(\frac{a_{2}}{b_{2}}\right)}^{\frac{2}{q-1}}\right).$

###### Proof.

Note that
$\lambda^{g,\varepsilon,+}(u_{i})=\left(\frac{b_{i}}{a_{i}}\right)^{\frac{1}{q-1}}.$
Then

$m^{g,+}_{\alpha,\beta,\varepsilon}\leq
E^{\varepsilon,+}_{\alpha,\beta,q}\left(\left(\frac{b_{i}}{a_{i}}\right)^{\frac{1}{q-1}}u_{i}\right)=\left(\frac{b_{i}}{a_{i}}\right)^{\frac{2}{q-1}}\frac{q-1}{2(q+1)}\
b_{i},$

or equivalently,

$b_{i}\geq\frac{2(q+1)}{q-1}\left(\frac{a_{i}}{b_{i}}\right)^{\frac{2}{q-1}}m^{g,+}_{\alpha,\beta,\varepsilon}.$

Then

$E^{\varepsilon,+}_{\alpha,\beta,q}(u)=\frac{q-1}{2(q+1)}(b_{1}+b_{2})\geq
m^{g,+}_{\alpha,\beta,\varepsilon}\left({\left(\frac{a_{1}}{b_{1}}\right)}^{\frac{2}{q-1}}+{\left(\frac{a_{2}}{b_{2}}\right)}^{\frac{2}{q-1}}\right).$

∎

Now consider the following elementary result:

######  Lemma 6.4.

Let $r>0$ and $x_{1},x_{2},y_{1},y_{2}$ be positive numbers such that
$x_{1}+x_{2}=y_{1}+y_{2}=r$. Then

$\varphi(x_{1},x_{2},y_{1},y_{2})={\left(\frac{x_{1}}{y_{1}}\right)}^{\frac{2}{q-1}}+{\left(\frac{x_{2}}{y_{2}}\right)}^{\frac{2}{q-1}}>1.$

For any $\delta\in(0,1)$ consider the set

$A_{\delta}:=\left\\{(x_{1},x_{2},y_{1},y_{2})\in(\mathbb{R}_{>0})^{4}:x_{1}+x_{2}=y_{1}+y_{2}=r,\
x_{1},x_{2}\geq\delta r\right\\}$

and let $\Psi(\delta):=\inf_{A_{\delta}}\varphi.$ Then
$\Psi:(0,1)\rightarrow(1,\infty)$ is a continuous function, and $\Psi$ is
independent of $r$.

We can deduce

######  Corollary 6.5.

Let $u=u_{1}+u_{2}\in\mathcal{N}_{g}^{+}(\alpha,\beta,q,\varepsilon)$, where
the functions $u_{1}$, $u_{2}$ have disjoint supports. Moreover assume that
$\varepsilon^{-n}\int_{M}(u_{i}^{+})^{q+1}dv_{g}>\delta$ for some $\delta>0$,
$i=1,2$. Then

$E^{\varepsilon,+}_{\alpha,\beta,q}(u)\geq\Psi(\delta)\
m^{g,+}_{\alpha,\beta,\varepsilon}.$

Fix $r<r_{0}(M,g)$.

Fix a positive integer $l$ and another positive integer $i$, $1<i<l$. Consider
bump functions $\varphi_{i,l}$ which is equal to 1 in
$\Big{[}0,\frac{(i-1)}{l}r\Big{]}$ and 0 in
$\Big{[}\frac{i}{l}r,\infty\Big{)}$, and $\sigma_{i,l}$ which is 1 in
$\Big{[}\frac{(i+1)}{l}r,\infty\Big{)}$ and 0 in
$\Big{[}0,\frac{i}{l}r\Big{]}$.

######  Remark 6.6.

We can assume that $|(\varphi_{i,l})^{\prime}|,|(\sigma_{i,l})^{\prime}|\leq
3l/r$, $|(\varphi_{i,l})^{\prime\prime}|,|(\sigma_{i,l})^{\prime\prime}|\leq
9(l/r)^{2}$.

For any $x\in M$, consider the function
$\varphi_{i,l}^{r,x}:M\rightarrow[0,1]$ which is 0 away from
$B_{g}(x,r)\subset M$ and in $B_{g}(x,r)$ (identified with the $r$-ball in
$\mathbb{R}^{n}$) is given by $\varphi_{i,l}(d(0,-))$. Define
$\sigma_{i,l}^{r,x}$ in a similar way. Now for any function
$u\in\mathcal{N}^{g}_{\varepsilon}$ define
$u_{1,i,l}^{r,x}=\varphi_{i,l}^{r,x}\ u$ and
$u_{2,i,l}^{r,x}=\sigma_{i,l}^{r,x}\ u$. Note that since $u^{+}\neq 0$ we have
that $(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})^{+}\neq 0$. Note also that these two
functions have disjoint supports.

Let

$\overline{u}:=\overline{u}_{i,l}^{r,x}=\lambda^{q,\varepsilon,+}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})\,(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x}).$

######  Lemma 6.7.

For the closed Riemannian manifold $(M,g)$ fix $0<r<r_{0}(M,g)$ and
$\delta_{0}>0$. There exists $\varepsilon_{0}>0$ and $l\in\mathbb{N}$ such
that for any $\varepsilon\in(0,\varepsilon_{0})$ we have that for any
$u\in\Sigma^{\alpha,\beta}_{\varepsilon,m^{g,+}_{\alpha,\beta,\varepsilon}+\delta_{0}}$
and any $x\in M$ there exist $i$, $1<i<l$ such that
$\overline{u}\in\Sigma^{\alpha,\beta}_{\varepsilon,m^{g,+}_{\alpha,\beta,\varepsilon}+2\delta_{0}}$

###### Proof.

Assume from the beginning that $\varepsilon$ is small enough so that
$m^{g,+}_{\alpha,\beta,\varepsilon}$ is close to $m_{\alpha,\beta}$ (using
Theorem 5.4). Pick $\eta\in(0,1)$, close to 1, such that (for any such
$\varepsilon$)

(6.2)
$(m^{g,+}_{\alpha,\beta,\varepsilon}+\delta_{0}){\left(\frac{9-8\eta}{\eta}\right)}^{\frac{2}{q-1}}(9-8\eta)<m^{g,+}_{\alpha,\beta,\varepsilon}+2\delta_{0}.$

We assume that the positive integer $l$ is large enough so that $8/l<1-\eta.$

Consider any $\varepsilon$ small and
$u\in\Sigma_{\varepsilon,m^{+}_{\alpha,\beta,\varepsilon}+\delta_{0}}$. Pick
any $x\in M$. Assume $l$ is even and divide $B_{g}(x,r)$ into $l/2$ annular
regions of radius $(2/l)r$. Call $A_{j}$, $j=1,\ldots,l/2$ each of these
regions.

For any nonegative integrable function $f$ on $B(x,r)$ we have

$\int_{B_{g}(x,r)}fdv_{g}=\sum_{j=1}^{l/2}\int_{A_{j}}fdv_{g}.$

Then $\\#\\{j:\int_{A_{j}}fdv_{g}\geq(8/l)\int_{B_{g}(x,r)}fdv_{g}\\}\leq
l/8$. Applying this observation to $(u^{+})^{q+1},(\Delta_{g}u)^{2}$ and
$|\nabla u|^{2}$ we see that there exists $j>1$ such that

(6.3)
$\int_{A_{j}}(\Delta_{g}u)^{2}dv_{g}\leq\frac{8}{l}\int_{M}(\Delta_{g}u)^{2}dv_{g}<(1-\eta)\int_{M}(\Delta_{g}u)^{2}dv_{g},$

(6.4) $\int_{A_{j}}|\nabla u|^{2}dv_{g}\leq\frac{8}{l}\int_{M}|\nabla
u|^{2}dv_{g}<(1-\eta)\int_{M}|\nabla u|^{2}dv_{g}$

and

(6.5)
$\int_{A_{j}}(u^{+})^{q+1}dv_{g}\leq\frac{8}{l}\int_{M}(u^{+})^{q+1}dv_{g}<(1-\eta)\int_{M}(u^{+})^{q+1}dv_{g}.$

Note that if $i=2j-1$ then outside of $A_{j}$ we have that
$u^{+}=(u_{1,i,l}^{r,x})^{+}+(u_{2,i,l}^{r,x})^{+}$. Therefore from (6.5) we
get

(6.6)
$\big{\|}(u_{1,i,l}^{r,x})^{+}+(u_{2,i,l}^{r,x})^{+}\big{\|}_{q+1}^{q+1}>\eta\|u^{+}\|_{q+1}^{q+1}.$

Note that the gradients and Laplacians of $u_{1,i,l}^{r,x}$ and
$u_{2,i,l}^{r,x}$ also have disjoint supports. Also note that
$|u_{1,i,l}^{r,x}|+|u_{2,i,l}^{r,x}|\leq|u|$ and away from $A_{j}$,
$u=u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x}$.

We have

$\int_{A_{j}}(\Delta_{g}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x}))^{2}dv_{g}=\int_{A_{j}}(\Delta_{g}u_{1,i,l}^{r,x})^{2}dv_{g}+\int_{A_{j}}(\Delta_{g}u_{2,i,l}^{r,x})^{2}dv_{g}$

and

$\int_{A_{j}}(\Delta_{g}(u_{1,i,l}^{r,x}))^{2}dv_{g}\begin{array}[t]{lll}\displaystyle=\int_{A_{j}}\left(\varphi_{i,l}^{x,r}\Delta_{g}u+u\Delta_{g}\varphi_{i,l}^{x,r}+2\langle\nabla\varphi_{i,l}^{x,r},\nabla
u\rangle\right)^{2}dv_{g}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\leq
3\int_{A_{j}}\big{(}(\varphi_{i,l}^{x,r})^{2}(\Delta_{g}u)^{2}+u^{2}(\Delta_{g}\varphi_{i,l}^{x,r})^{2}+4|\nabla
u|^{2}|\nabla\varphi_{i,l}^{x,r}|^{2}\big{)}\,dv_{g}\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ \displaystyle\leq
3\int_{A_{j}}(\Delta_{g}u)^{2}dv_{g}+3\Big{(}9\frac{l^{2}}{r^{2}}\Big{)}^{2}\int_{A_{j}}u^{2}dv_{g}+12\Big{(}3\frac{l}{r}\Big{)}^{2}\int_{A_{j}}|\nabla
u|^{2}dv_{g}\end{array}$

A similar inequality is obtained for $u_{2,i,l}^{r,x}$ and we have:

$\int_{A_{j}}(\Delta_{g}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x}))^{2}dv_{g}\leq
6\int_{A_{j}}(\Delta_{g}u)^{2}dv_{g}+6\Big{(}9\frac{l^{2}}{r^{2}}\Big{)}^{2}\int_{A_{j}}u^{2}dv_{g}+24\Big{(}3\frac{l}{r}\Big{)}^{2}\int_{A_{j}}|\nabla
u|^{2}dv_{g}$

Using (6.3) we obtain

$\begin{array}[t]{ll}\displaystyle\int_{M}(\Delta_{g}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x}))^{2}dv_{g}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\int_{A_{j}}(\Delta_{g}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x}))^{2}dv_{g}+\int_{M\setminus
A_{j}}(\Delta_{g}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x}))^{2}dv_{g}\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \displaystyle\leq(6-6\eta)\int_{M}(\Delta
u)^{2}dv_{g}+\int_{M}(\Delta
u)^{2}dv_{g}+6\Big{(}9\frac{l^{2}}{r^{2}}\Big{)}^{2}\int_{M}u^{2}dv_{g}+24\Big{(}3\frac{l}{r}\Big{)}^{2}\int_{M}|\nabla
u|^{2}dv_{g}.\end{array}$

We have

$\int_{A_{j}}|\nabla(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})|^{2}\
dv_{g}=\int_{A_{j}}|\nabla u_{1,i,l}^{r,x}|^{2}\ dv_{g}\ +\
\int_{A_{j}}|\nabla u_{2,i,l}^{r,x}|^{2}\ dv_{g}$

and

$\int_{A_{j}}|\nabla u_{1,i,l}^{r,x}|^{2}\
dv_{g}\begin{array}[t]{lll}\displaystyle=\int_{A_{j}}|\varphi_{i,l}^{x,r}\nabla
u+u\nabla\varphi_{i,l}^{x,r}|^{2}\ dv_{g}\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \displaystyle\leq\
2\int_{A_{j}}\left((\varphi_{i,l}^{x,r})^{2}|\nabla
u|^{2}+u^{2}|\nabla\varphi_{i,l}^{x,r}|^{2}\right)dv_{g}\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \displaystyle\leq 2\int_{A_{j}}\left(|\nabla
u|^{2}+9\frac{l^{2}}{r^{2}}u^{2}\right)dv_{g}.\end{array}$

The same estimate can be carried out for $\int_{A_{j}}|\nabla
u_{2,i,l}^{r,x}|^{2}\ dv_{g}$. Hence

$\int_{A_{j}}|\nabla(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})|^{2}\ dv_{g}\leq
4\int_{A_{j}}\left(|\nabla u|^{2}+9\frac{l^{2}}{r^{2}}u^{2}\right)\ dv_{g}.$

Hence, we obtain from (6.4)

$\int_{M}|\nabla(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})|^{2}\
dv_{g}\begin{array}[t]{lll}\displaystyle=\int_{A_{j}}|\nabla(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})|^{2}\
dv_{g}+\int_{M\setminus A_{j}}|\nabla(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})|^{2}\
dv_{g}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\leq(4-4\eta)\int_{M}|\nabla u|^{2}dv_{g}+\int_{M\setminus
A_{j}}|\nabla u|^{2}dv_{g}+36\frac{i^{2}}{r^{2}}\int_{M}u^{2}dv_{g}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle\leq(5-4\eta)\int_{M}|\nabla
u|^{2}dv_{g}+36\frac{i^{2}}{r^{2}}\int_{M}u^{2}dv_{g}.\end{array}$

We have:

$\begin{array}[t]{lll}\displaystyle\int_{M}\big{(}\varepsilon^{4}\big{(}\Delta_{g}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})\big{)}^{2}+\beta\varepsilon^{2}|\nabla(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})|^{2}+\alpha(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})^{2}\big{)}\,dv_{g}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\qquad\displaystyle\leq(7-6\eta)\int_{M}\varepsilon^{4}(\Delta_{g}u)^{2}dv_{g}+O(\varepsilon^{4})\int_{M}\big{(}|\nabla
u|^{2}+u^{2}\big{)}\,dv_{g}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad\qquad+(5-4\eta)\varepsilon^{2}\int_{M}|\nabla
u|^{2}dv_{g}+O(\varepsilon^{2})\int_{M}u^{2}dv_{g}+\alpha\int_{M}u^{2}dv_{g}.\end{array}$

Therefore, for the values of $\eta$ and $l$ we have already fixed, there is
$\varepsilon_{0}>0$ such that for any $\varepsilon\in(0,\varepsilon_{0})$

$\begin{array}[t]{lll}\displaystyle\int_{M}\big{(}\varepsilon^{4}\big{(}\Delta_{g}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})\big{)}^{2}+\beta\varepsilon^{2}|\nabla(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})|^{2}+\alpha(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})^{2}\big{)}\,dv_{g}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\qquad\displaystyle\leq(9-8\eta)\int_{M}\big{(}\varepsilon^{4}(\Delta_{g}u)^{2}+\beta\varepsilon^{2}|\nabla
u|^{2}+\alpha u^{2}\big{)}\,dv_{g}.\end{array}$

It follows from the previous inequality, the formula for
$\lambda^{g,\varepsilon,+}$ and (6.6) that

$\lambda^{g,\varepsilon,+}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})\leq{\left(\frac{9-8\eta}{\eta}\right)}^{\frac{1}{q-1}}.$

Then, for
$\overline{u}=\lambda^{g,\varepsilon,+}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})\in\mathcal{N}_{g}^{+}(\alpha,\beta,q,\varepsilon)$,
we have

$J^{g,\varepsilon}_{\alpha,\beta}(\overline{u})=$
$\begin{array}[t]{ll}\displaystyle\frac{(\lambda^{g,\varepsilon,+})^{2}(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})}{2\varepsilon^{n}}\int_{M}\big{(}\varepsilon^{4}(\Delta(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x}))^{2}+\beta\varepsilon^{2}|\nabla(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})|^{2}+\alpha(u_{1,i,l}^{r,x}+u_{2,i,l}^{r,x})^{2}\big{)}\,dv_{g}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\leq{\left(\frac{9-8\eta}{\eta}\right)}^{\frac{2}{q-1}}(9-8\eta)J^{g,\varepsilon}_{\alpha,\beta}(u).\end{array}$

So if
$u\in\Sigma^{\alpha,\beta}_{\varepsilon,m^{+}_{\alpha,\beta,\varepsilon}+\delta_{0}}$
we have that

$E^{\varepsilon,+}_{\alpha,\beta,q}(\overline{u})\leq{\left(\frac{9-8\eta}{\eta}\right)}^{\frac{2}{q-1}}(9-8\eta)E^{\varepsilon,+}_{\alpha,\beta,q}(u),$

and by (6.2) we obtain

$E^{\varepsilon,+}_{\alpha,\beta,q}(\overline{u})<m^{g,+}_{\alpha,\beta,\varepsilon}+2\delta_{0}.$

∎

######  Remark 6.8.

On any closed Riemannian manifold for any $\varepsilon>0$ there is a set of
points $x_{j}$, $j=1,...,K_{\varepsilon}$ such that the closed balls
$\overline{B_{g}(x_{j},\varepsilon)}$ are disjoint, and the set is maximal
under this condition. It follows that the balls
$\overline{B_{g}(x_{j},2\varepsilon)}$ cover $M$. It is easy to construct
closed sets $A_{j}$, $\overline{B_{g}(x_{j},\varepsilon)}\subset
A_{j}\subset\overline{B_{g}(x_{j},2\varepsilon)}$ which cover $M$ and only
intersect in their boundaries. Moreover one can see by a volume comparison
argument that if $\varepsilon$ is small enough there is a constant $K$,
independent of $\varepsilon$, such that for any point in $M$ belongs to at
most $K$ of the balls $B_{g}(x_{j},3\varepsilon)$. ($K$ would be approximately
the maximal number of disjoint balls of radius $1/4$ in the Euclidean space
contained in the ball of radius $1$).

The next result shows that for $\varepsilon$ small there is a small ball which
contains a meaningful part of $u^{+}$. It is similar to the results in [2,
Lemma 5.3] and [25, Proposition 3.6] for a similar situation for (second
order) Yamabe-type equations.

######  Proposition 6.9.

There exist $\gamma>0$ and $\varepsilon_{0}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0})$ if
$u\in\mathcal{N}_{g}^{+}(\alpha,\beta,q,\varepsilon)$ then there exists $x\in
M$ such that

$\varepsilon^{-n}\int_{B_{g}(x,2\varepsilon)}(u^{+})^{q+1}dv_{g}\geq\gamma.$

###### Proof.

Let us consider $\varepsilon_{0}>0$ such that for
$\varepsilon<\varepsilon_{0}$ one can construct sets like in the previous
remark.

Consider $u\in\mathcal{N}_{g}^{+}(\alpha,\beta,q,\varepsilon)$. Let
$u_{j}=u^{+}\chi_{A_{j}}$ be the restriction of $u^{+}$ to $A_{j}$ (extended
by 0 away from $A_{j}$ in order to consider it as a function on $M$). Then

(6.7)
$\varepsilon^{-n}\int_{M}\big{(}\varepsilon^{4}(\Delta_{g}u)^{2}+\beta\varepsilon^{2}|\nabla
u|^{2}+\alpha
u^{2}\big{)}\,dv_{g}\begin{array}[t]{ll}\displaystyle=\varepsilon^{-n}\int_{M}(u^{+})^{q+1}dv_{g}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle=\sum_{j=1}^{K_{\varepsilon}}\
(\varepsilon^{-\frac{n(q-1)}{q+1}}\|u_{j}\|_{q+1}^{q-1})\
(\varepsilon^{-\frac{2n}{q+1}}\|u_{j}\|^{2}_{q+1})\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\displaystyle\leq\left(\max_{j}{\varepsilon^{-\frac{n(q-1)}{q+1}}}{\|u_{j}\|_{{q+1}}^{q-1}}\right)\sum_{j}\
{\varepsilon^{-\frac{2n}{q+1}}}{\|u_{j}\|^{2}_{q+1}}.\end{array}$

Let

$\Gamma=\max_{j}{\varepsilon^{-\frac{n(q-1)}{q+1}}}{\|u_{j}\|_{q+1}^{q-1}}.$

Note that we have to prove that there exists a positive constant $\gamma$,
independent of $\varepsilon$ and $u$ such that

$\Gamma^{\frac{q+1}{q-1}}\geq\gamma.$

To obtain such a lower bound for $\Gamma$ we have to bound from above each of
the terms $\varepsilon^{-\frac{2n}{q+1}}\|u_{j}\|_{{q+1}}^{2}$.

We first point out that from the local, second order, Sobolev inequalities,
there is a constant $C$ independent of $\varepsilon$ and $x$ such that for any
$x\in M$, $v\in H_{0}^{2}(B_{g}(x,\varepsilon))$,

$\varepsilon^{\frac{-2n}{q+1}}\left(\int_{B_{g}(x,\varepsilon)}|v|^{q+1}dv_{g}\right)^{\frac{2}{q+1}}\leq
C\varepsilon^{-n}\int_{B_{g}(x,\varepsilon)}\big{(}\varepsilon^{4}(\Delta_{g}v)^{2}+\beta\varepsilon^{2}|\nabla
v|^{2}+\alpha v^{2}\big{)}\,dv_{g}.$

Now, let $\varphi_{\varepsilon}$ be a cut-off function on $\mathbb{R}^{n}$
which is 1 in $B_{g}(0,2\varepsilon)$, vanishes away from
$B_{g}(0,3\varepsilon)$, $|\nabla\varphi_{\varepsilon}|\leq 3/\varepsilon$ and
$|\Delta\varphi_{\varepsilon}|\leq 9/\varepsilon^{2}$ in the intermediate
annulus.

Define, for $j=1\dots K_{\varepsilon}$,

$u_{j,\varepsilon}(x)=u(x)\,\varphi_{\varepsilon}(d(x,x_{j})).$

Since $u_{j}\leq|u_{j,\varepsilon}|$ we have that

(6.8)
$\varepsilon^{-\frac{2n}{q+1}}\|u_{j}\|_{{q+1}}^{2}\leq\varepsilon^{-\frac{2n}{q+1}}\|u_{j,\varepsilon}\|_{{q+1}}^{2}$

and by using the Sobolev inequalities we obtain that, for some constant $C$,

(6.9) $\varepsilon^{-\frac{2n}{q+1}}\|u_{j,\varepsilon}\|_{{q+1}}^{2}\leq
C\varepsilon^{-n}\int_{B_{g}(x_{j},3\varepsilon)}\big{(}\varepsilon^{4}(\Delta_{g}u_{j,\varepsilon})^{2}+\beta\varepsilon^{2}|\nabla
u_{j,\varepsilon}|^{2}+\alpha u_{j,\varepsilon}^{2}\big{)}\,dv_{g}.$

Since $|u_{j,\varepsilon}|\leq|u|$ we have that

$\varepsilon^{2}|\nabla u_{j,\varepsilon}|^{2}\leq 2\varepsilon^{2}|\nabla
u|^{2}+18(u)^{2}$

and

$\varepsilon^{4}(\Delta_{g}u_{j,\varepsilon})^{2}\leq
3\varepsilon^{4}(\Delta_{g}u)^{2}+108\varepsilon^{2}|\nabla u|^{2}+243u^{2}.$

Since $\alpha$, $\beta$ are fixed positive constants and any point $x\in M$
could be in at most $K$ of the balls $B_{g}(x_{j},3\varepsilon)$ (and
therefore, it has an open neighbourhood which can only intersect the support
of the corresponding $\nabla u_{j,\varepsilon}$’s), it follows from the
previous inequalities that there is a constant $E$ (independent of
$\varepsilon$) such that

(6.10)
$\sum_{j}\int_{B_{g}(x_{j},3\varepsilon)}\big{(}\varepsilon^{4}(\Delta_{g}u_{j,\varepsilon})^{2}+\varepsilon^{2}\beta|\nabla
u_{j,\varepsilon}|^{2}+\alpha u_{j,\varepsilon}^{2}\big{)}\,dv_{g}\leq
E\int_{M}\big{(}\varepsilon^{4}(\Delta_{g}u)^{2}+\varepsilon^{2}\beta|\nabla
u|^{2}+\alpha u^{2}\big{)}\,dv_{g}.$

Then, from (6.7), (6.8), (6.9) and (6.10), we get

$\varepsilon^{-n}\int_{M}\big{(}\varepsilon^{4}(\Delta_{g}u)^{2}+\varepsilon^{2}\beta|\nabla
u|^{2}+\alpha u^{2}\big{)}\,dv_{g}\leq\Gamma
CE\varepsilon^{-n}\int_{M}\big{(}\varepsilon^{4}(\Delta_{g}u)^{2}+\varepsilon^{2}\beta|\nabla
u|^{2}+\alpha u^{2}\big{)}\,dv_{g}$

Therefore

$\Gamma\geq\frac{1}{CE}$

and we can take

$\gamma={\left(\frac{1}{CE}\right)}^{\frac{q+1}{q-1}}.$

∎

Finally we can prove Theorem 6.2

###### Proof of Theorem 6.2.

Assume the theorem is not true. Then there exist $\eta<1$ and sequences of
positive numbers $\varepsilon_{j}\rightarrow 0$, $\delta_{j}\rightarrow 0$,
and
$u_{j}\in\Sigma^{\alpha,\beta}_{\varepsilon_{j},{m}^{g,+}_{\alpha,\beta,\varepsilon_{j}}+\delta_{j}}$
such that for any $x\in M$ one has

$\int_{B_{g}(x,r)}(u_{j}^{+})^{q+1}<\eta\int_{M}(u_{j}^{+})^{q+1}.$

For each $j$ large enough the previous Proposition provides $x_{j}\in M$ such
that, for some fixed $\gamma>0$,

$\varepsilon_{j}^{-n}\int_{B_{g}(x_{j},2\varepsilon_{j})}(u_{j}^{+})^{q+1}\
dv_{g}\geq\gamma.$

Lemma 6.7 then gives a function $\overline{u_{j}}=u_{j,1}+u_{j,2}$ such that
$\overline{u_{j}}\in\Sigma^{\alpha,\beta}_{\varepsilon_{j},{m}^{g,+}_{\alpha,\beta,\varepsilon_{j}}+2\delta_{j}}$,
$u_{j,1}$ is supported inside the ball of radius $r$ centered at $x_{j}$,
$u_{j,1}$ and $u_{j,2}$ have disjoint support, $u_{j}=u_{j,1}$ in
$B_{g}(x_{j},\varepsilon_{j})$ and $u_{j}=u_{j,2}$ outside $B_{g}(x_{j},r)$.

We have that

$\varepsilon_{j}^{-n}\int_{M}(u_{j,1}^{+})^{q+1}\ dv_{g}\geq\gamma$

and, using (6.1),

$\varepsilon_{j}^{-n}\int_{M}(u_{j,2}^{+})^{q+1}\
dv_{g}>\varepsilon_{j}^{-n}(1-\eta)\int_{M}(u_{j}^{+})^{q+1}\
dv_{g}\geq(1-\eta)\frac{2(q+1)}{q-1}{m}^{g,+}_{\alpha,\beta,\varepsilon_{j}}.$

Then it follows from Corollary 6.5 that there exists $\delta>0$, independent
of $j$, such that

$E^{\varepsilon_{j},+}_{\alpha,\beta,q}(\overline{u_{j}})\geq\Psi(\delta){m}^{g,+}_{\alpha,\beta,\varepsilon}.$

But for $j$ large enough we have that

$E^{\varepsilon_{j},+}_{\alpha,\beta,q}(\overline{u_{j}})<{m}^{g,+}_{\alpha,\beta,\varepsilon_{j}}+2\delta_{j}<\Psi(\delta){m}^{g,+}_{\alpha,\beta,\varepsilon_{j}},$

reaching a contradiction. ∎

##  7\. The Lusternik-Schnirelmann category of $\Sigma_{\varepsilon,{\bf
m}_{\varepsilon}+\delta}$

In this section we prove:

######  Theorem 7.1.

There exist $\varepsilon_{0}>0$, $\delta_{0}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0})$, we have

$\mathrm{Cat}(\Sigma_{\varepsilon,{\bf
m}_{\varepsilon}+\delta_{0}})\geq\mathrm{Cat}(M).$

Note that then Theorem 1.1 follows from Corollary 3.5 .

In the proof we make use of the center of mass function defined in [25,
Section 5]. We briefly recall the definition. For $r>0$, $\eta\in(0,1)$ let

$L^{1}_{r,\eta}=\bigg{\\{}u\in L^{1}(M)\setminus\\{0\\}:\sup_{x\in
M}\int_{B_{g}(x,r)}|u|\,dv_{g}>\eta\int_{M}|u|\,dv_{g}\bigg{\\}}.$

If $r_{0}>0$ is such that for any $x\in M$ the ball $B_{g}(x,r_{0})$ is
strongly convex then it is proved in [25, Theorem 5.2] for any $r<r_{0}/2$ and
any $\eta>1/2$ that there exist a continuous function ${\bf
CM}(r,\mu):L^{1}_{r,\mu}\rightarrow M$ such that for any $x\in M$ such that
$\int_{B_{g}(x,r)}|u|dv_{g}>\eta\int_{M}|u|dv_{g}$ we have ${\bf
CM}(r,\eta)(u)\in B_{g}(x,2r)$.

###### Proof of Theorem 7.1.

Fix $r<r_{0}/2$. Let $\eta=0.9$. Fix $\varepsilon_{0}>0$, $\delta_{0}>0$ as in
Theorem 6.1 and pick any $\delta\in(0,\delta_{0})$ . Then for any
$\varepsilon\in(0,\varepsilon_{0})$ we have that if
$u\in\Sigma_{\varepsilon,{\bf m}_{\varepsilon}+\delta}$ then $(u^{+})^{q+1}\in
L^{1}_{r,\eta}$.

Define ${\bf b}:\Sigma_{\varepsilon,{\bf m}_{\varepsilon}+\delta}\rightarrow
M$ by ${\bf b}(u)={\bf CM}(r,0.9)\left((u^{+})^{q+1}\right)$. Note that ${\bf
b}$ is a continuous function.

Next we consider for any $x\in M$ the function
${\overline{U}}^{\varepsilon,r}_{\alpha,\beta,x}$ as in Remark 5.1, which is
supported in $B_{g}(x,r)$. Consider as in Definition 5.3 the function ${\bf
i}(x)\in\mathcal{N}(\alpha,\beta,q,\varepsilon)$.

Note that there exists $\lambda>0$ such that $\lambda{\bf
i}(x)\in\mathcal{N}^{g}_{\varepsilon}$, with $\mathcal{N}^{g}_{\varepsilon}$
as in (3.2), and $\lambda$ is close to 1 by Remark 2.5. We can also see from
Remark 2.5, Proposition 5.2, Theorem 5.4, Theorem 5.5 that with $\delta$ fixed
we can choose $\varepsilon_{0}$ smaller if necessary so that for any
$\varepsilon\in(0,\varepsilon_{0})$ we have that
$J^{g}_{\varepsilon}(\lambda{\bf i}(x))<{\bf m}_{\varepsilon}+\delta$.

Define ${\bf p}:M\rightarrow\Sigma_{\varepsilon,{\bf m}_{\varepsilon}+\delta}$
by ${\bf p}(x)=\lambda{\bf i}(x)$.

It follows from Theorem 5.2 in [25] that for any $x$, ${\bf b}({\bf p}(x))\in
B_{g}(x,2r)$. Then if follows from the election of $r$ that for any $x\in M$,
${\bf b}\circ{\bf p}(x)$ is joined to $x$ by a unique minimizing geodesic
contained in $B_{g}(x,2r)$. Then ${\bf b}\circ{\bf p}$ is homotopic to the
identity of $M$ and the theorem follows. ∎

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