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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 63, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/63\hfil Existence of solutions]
{Existence of solutions to singular fourth-order elliptic equations}

\author[M. Benalili, K. Tahri \hfil EJDE-2013/63\hfilneg]
{Mohammed Benalili, Kamel Tahri}  % in alphabetical order

\address{Mohammed Benalili
Faculty of Sciences, Mathematics Dept.,
 University Aboubakr BelKa\"{\i}d,
Tlemcen, Algeria}
\email{m\_benalili@mail.univ-tlemcen.dz}

\address{Kamel Tahri
Faculty of Sciences, Mathematics Dept.,
University Aboubakr Belka\"{\i}d,
Tlemcen, Algeria}
\email{tahri\_kamel@yahoo.fr}

\thanks{Submitted October 3, 2012. Published March 1, 2013.}
\subjclass[2000]{58J05}
\keywords{Fourth-order elliptic equation;
Hardy-Sobolev inequality; \hfill\break\indent critical Sobolev exponent}

\begin{abstract}
 Using a method developed by Ambrosetti et al \cite{1,2} we prove the
 existence  of weak non trivial solutions to fourth-order elliptic equations
 with singularities and with critical Sobolev growth.
\end{abstract}

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

\section{Introduction}

Fourth-order elliptic equations have been widely studied,
because of their importance in the analysis on manifolds particularly those
involving the Paneitz-Branson operators; see for example
\cite{1,2,3,4,5,6,7,8,9,10,13,16}.
Different techniques have been used for solving fourth-order equations,
as example the variational method which was developed by Yamabe to solve
the problem of the prescribed scalar curvature. Let $(M,g)$ a compact smooth
Riemannian manifold of dimension $n\geq 5$ with a metric $g$. We denote by
$H_2^2(M)$ the standard Sobolev space which is the completed of the space
$C^{\infty }( M) $ with respect to the norm
\[
\| \varphi \| _{2,2}=\sum_{k=0}^{k=2}\|\nabla ^{k}\varphi \| _2.
\]
$H_2^2(M)$ will be endowed with the suitable equivalent norm
\[
\| u\| _{H_2^2(M)}=\Big(\int_{M}( ( \Delta
_{g}u) ^2+| \nabla _{g}u| ^2+u^2)dv_{g}\Big)^{1/2}.
\]
In 1979, Vaugon \cite{17} proved the existence of a positive value $\lambda$
and a non trivial solution $u\in C^{4}( M) $ to the equation
\[
\Delta _{g}^2u-\operatorname{div}_{g}( a(x)\nabla _{g}u) +b(x)u=\lambda f(t,x)
\]
where $a$, $b$ are smooth functions on $M$ and $f(t,x)$ is odd and
increasing function in $t$ fulfilling the inequality
\[
| f(t,x)| <a+b| t| ^{\frac{n+4}{n-4}}.
\]
Edminds,  Fortunato and Jannelli \cite{14}  showed that all the
solutions in $\mathbb{R}^n$ to the equation
\[
\Delta ^2u=u^{\frac{n+4}{n-4}}
\]
are positive, symmetric, radial and decreasing functions of the form
\[
u_{\epsilon }(x)=\frac{( (n-4)n(n^2-4)\epsilon ^{4}) ^{\frac{n-4
}{8}}}{(r^2+\epsilon ^2)^{\frac{n-4}{2}}}.
\]
In 1995, Van Der Vorst \cite{15} obtained the same results
for the problem
\begin{gather*}
\Delta ^2u-\lambda u=u| u| ^{\frac{8}{n-4}}\quad \text{in } \Omega , \\
\Delta u=u=0\quad \text{on }\partial \Omega ,
\end{gather*}
where $\Omega $ is a bounded domain of $\mathbb{R}^n$.

In 1996,  Bernis,  Garcia-Azorero and Peral \cite{9} obtained the
existence at least of two positive solutions to the  problem
\begin{gather*}
\Delta ^2u-\lambda u| u| ^{q-2}=u|
u| ^{\frac{8}{n-4}}\text{ in }\Omega , \\
\Delta u=u=0\text{ on }\partial \Omega ,
\end{gather*}
where $\Omega $ is bounded domain of $\mathbb{R}^n$, $1<q<2$ and $\lambda >0$
in some interval. In 2001,  Caraffa \cite{12} obtained the existence of a
non trivial solution of class $C^{4,\alpha }$,
$\alpha \in (0,1) $ for the equation
\[
\Delta _{g}^2u-\nabla ^{\alpha }( a(x)\nabla _{\alpha }u)
+b(x)u=\lambda f(x)| u| ^{N-2}u
\]
with $\lambda >0$, first for $f$ a constant and next for a positive function
$f$ on $M$.

Recently the first author \cite{4} showed the existence of at least two
distinct non trivial solutions in the subcritical case and a non trivial
solution in the critical case for the equation
\[
\Delta _{g}^2u-\nabla ^{\alpha }( a(x)\nabla _{\alpha }u)
+b(x)u=f(x)| u| ^{N-2}u
\]
where $f$ is a changing sign smooth function and $a$ and $b$ are smooth
functions. In \cite{6} the same author proved the existence of at least two
non trivial solutions to
\[
\Delta _{g}^2u-\nabla ^{\alpha }( a(x)\nabla _{\alpha }u)
+b(x)u=f(x)| u| ^{N-2}u+| u|
^{q-2}u+\varepsilon g(x)
\]
where $a$, $b$, $f$, $g$ are smooth functions on $M$ with $f>0$, $2<q<N$,
$\lambda >0$ and $\epsilon >0$ small enough. Let $S_{g}$ denote the scalar
curvature of $M$. In 2011,  the authors proved the following result

\begin{theorem}[\cite{8}]\label{thm1}
 Let $(M,g)$ be a compact Riemannian manifold of
dimension $n\geq 6$ and $a$, $b$, $f$  smooth functions on $M$,
 $\lambda \in ( 0,\lambda _{\ast }) $  for some specified
$\lambda _{\ast}>0$, $1<q<2$  such that
\begin{itemize}
\item[(1)] $f(x)>0$ on $M$.
\item[(2)] At the point $x_0$ where $f$ attains its maximum,
 we suppose that for $n=6$,
$S_{g}(x_0)+3a(x_0)>0$,
and for $n>6$
\[
\Big( \frac{( n^2+4n-20) }{2(n+2)(n-6)}S_{g}(x_0)+\frac{
(n-1)}{(n+2)(n-6)}a(x_0)-\frac{1}{8}\frac{\Delta f(x_0)}{
f(x_0)}\Big) >0.
\]
\end{itemize}
Then the equation
\[
\Delta _{g}^2u+\operatorname{div}_{g}( a(x)\nabla _{g}u) +b(x)u=\lambda
| u| ^{q-2}u+f(x)| u| ^{N-2}u
\]
admits a non trivial solution of class $C^{4,\alpha }( M) $,
$\alpha \in (0,1)$.
\end{theorem}

Recently  Madani \cite{14} studied  the Yamabe problem with
singularities when the metric $g$ admits a finite number of points with
singularities and is smooth outside these points. More precisely,
 let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$,
we denote by $T^{\ast }M$ the cotangent space of $M$.
The space $H_2^{p}(M,T^{\ast}M\otimes T^{\ast }M)$ is the set of sections
 $s$ ($2$-covariant tensors)
such that in normal coordinates the components $s_{ij}$ of $s$ are in
$H_2^{p}$ the complement of the space $C_0^{\infty }(\mathbb{R}^n)$
with respect to the norm
$\| \varphi \|_{2,p}=\sum_{k=0}^{k=2}\| \nabla ^{k}\varphi \| _p$.

Solving the singular Yamabe problem is equivalent to finding a positive
solution $u\in H_2^{p}(M)$ of the equation
\begin{equation}
\Delta _{g}u+\frac{n-2}{4(n-1)}S_{g}u=k|u|^{N-2}u,  \label{*}
\end{equation}
where $S_{g}$ is the scalar curvature of the $g$ and $k$ is a real
constant. 
The Christoffels symbols belong to $\ H_{1}^{p}( M) $,
the Riemannian curvature tensor, the Ricci tensor $Ric_{g}$ and scalar
curvature $S_{g}$ are in $L^{p}(M)$, hence equation \ref{*} is the singular
Yamabe equation. 

Under the  assumptions that
 $g$ is a metric in the Sobolev space
$H_2^{p}(M,T^{\ast}M\otimes T^{\ast }M)$ with $p>n/2$
and that there exist a point $P\in M$ and $\delta >0$ such that $g$
is smooth in the ball $B_p(\delta )$,
Madani \cite{14} proved 
the existence of a metric $\overline{g}=u^{N-2}g$ conformal to $g$
such that $u\in H_2^{p}(M)$, $u>0$ and the scalar curvature
$S_{\overline{g}}$ of $\overline{g}$ is constant if $(M,g)$ is not
conformal to the round sphere.

The author in \cite{7}  considered fourth-order  elliptic
equations, with singularities, of the form
\begin{equation}
\Delta ^2u-\nabla ^{i}( a(x)\nabla _{i}u) +b(x)u=f|u| ^{N-2}u    \label{1}
\end{equation}
where the functions $a$ and $b$ are in $L^{s}(M)$, $s>\frac{n}{2}$ and in
$L^{p}(M)$, $p>\frac{n}{4}$ respectively, $N=\frac{2n}{n-4}$ is the Sobolev
critical exponent in the embedding
$H_2^2( M) \hookrightarrow L^{N}( M) $. He established the following results.
Let $(M,g) $ be a compact $n$-dimensional Riemannian manifold, $n\geq 6$,
$a\in L^{s}(M)$, $b\in L^{p}(M)$, with $s>\frac{n}{2}$, $p>\frac{n}{4}$,
$f\in C^{\infty }(M)$ a positive function and $x_0\in M$ such that
$f(x_0)=\max_{x\in M}f(x)$.

\begin{theorem}
For $n\geq 10$, or $n=8,9$ and $2<p<5$, $\frac{9}{4}<s<11$ or $n=7$,
$\frac{7}{2}<s<9$ and $\frac{7}{4}<p<9$ we suppose that
\[
\frac{n^2+4n-20}{6( n-6) (n^2-4)}S_{g}( x_0) -
\frac{n-4}{2n( n-2) }\frac{\Delta f(x_0)}{f(x_0)}>0.
\]
For $n=6$ and $\frac{3}{2}<p<2$, $3<s<4$, we assume that
\[
S_{g}(x_0)>0.
\]
Then  \eqref{1} has a non trivial weak solution $u$ in
$H_2^2( M) $. Moreover if $a\in H_{1}^{s}( M) $,
then $u\in $ $C^{0,\beta }( M) $, for some $\beta \in ( 0,1-
\frac{n}{4p}) $.
\end{theorem}

In this article, we extend results obtained in Theorem \ref{thm1}
to the case of singular elliptic fourth order, more precisely we are
concerned with the following problem:
Let $(M,g)$ be a Riemannian compact manifold of dimension
$n\geq 5$.
Let $a\in L^{r}(M)$, $b\in L^{s}(M)$ where $r>\frac{n}{2}$,
$s>\frac{n}{4}$ and $f$ a positive $C^{\infty }$-function on $M$;
we look for non trivial solution of the equation
\begin{equation}
\Delta _{g}^2u+\operatorname{div}_{g}( a(x)\nabla _{g}u) +b(x)u=\lambda
| u| ^{q-2}u+f(x)| u| ^{N-2}u  \label{2}
\end{equation}
where $1<q<2$ and $N=\frac{2n}{n-4}$ is the critical Sobolev exponent and
$\lambda >0$ a real number.

In case the $\lambda =0$ and
\[
a=\frac{4}{n-2}R_{ic_{g}}-\frac{( n-2) ^2+4}{2( n-1)( n-2) }S_{g}.g,
\quad
b=\frac{n-4}{2}Q_{g}^{n},
\]
where
\[
Q_{g}^{n}=\frac{1}{2( n-1) }\Delta S_{g}+\frac{n^{3}-4n^2+16n-16
}{8( n-1) ^2( n-2) ^2}S_{g}^2-\frac{2}{(
n-2) ^2}| Ric_{g}| ^2
\]
suppose that $g$\ is a metric in the Sobolev space
 $H_{4}^{p}(M,T^{\ast}M\otimes T^{\ast }M)$\ with $\ p>\frac{n}{4}$,
 then the Ricci $Ric_{g}$
curvature and the scalar curvature $S_{g}$ are in the Sobolev spaces
$H_2^{p}(M,T^{\ast }M\otimes T^{\ast }M)$ and $H_2^{p}(M)$ respectively,
hence $b\in L^{s}( M) $ with $s>\frac{n}{4}$ and by Sobolev
embedding $a\in L^{r}( M) $ with $r>\frac{n}{2}$. In this latter
case the equation
\begin{equation}
\Delta _{g}^2u+\operatorname{div}_{g}( a(x)\nabla _{g}u) +b(x)u=f(x)|
u| ^{N-2}u  \label{**}
\end{equation}
is called singular $Q$-curvature equation$.$For more general coefficients
$a\in L^{r}( M) $ with $r>\frac{n}{2}$ and $b\in L^{s}(M) $ with $s>\frac{n}{4}$,
the equation \eqref{**} is called singular $Q$-curvature type equation.
To solve equation \eqref{2}, we use a method
developed in \cite{1} and \cite{2} which resumes to study the variations of
functional associated to equation \ref{2} on the manifold $M_{\lambda }$
defined in section 2. Serious difficulties appear compared with the smooth
case: considering the equation \eqref{12} in section 4, we need a
Hardy-Sobolev inequality and Releich-Kondrakov embedding on a manifolds. In
the case of the singular Yamabe equation theses latters were established in
\cite{14} and in the case of singular $Q$-curvature type equations by the
first author in \cite{7}. In the sharp cases (see section 5) the Hardy
Sobolev inequality and the Releich-Kondrakov embedding are no more valid so
we need an additional assumption with some tricks combined with the Lebesgue
dominated convergence theorem.

Denote by $P_{g}$ the operator defined in the weak sense on
$H_2^2(M) $ by $P_{g}(u)=\Delta ^2u+\operatorname{div}(a\nabla u)+bu$.  $P_{g}$ is called
coercive if there exits $\Lambda >0$ such that for any $u\in H_2^2(M)$
\[
\int_{M}uP_{g}(u)dv_{g}\geq \Lambda \| u\|_{H_2^2(M)}^2.
\]
Our main result reads as follows.

\begin{theorem}
Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 6$ and $f$
a positive function. Suppose that $P_{g}$ is coercive and at a point $x_0$
where $f$ attains its maximum the following two conditions hold:
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac{\Delta f(x_0)}{f( x_0) }
&<\Big(\frac{n(n^2+4n-20) }{3( n+2) ( n-4) ( n-6) }
\frac{1}{( 1+\| a\| _r+\| b\|_s) ^{n/4}}\\
&\quad -\frac{n-2}{3( n-1) }\Big)
S_{g}( x_0) \quad \text{when }n>6,
\end{aligned} \\
S_{g}(x_0)>0\quad \text{when }n=6.
\end{gathered}
\label{C}
\end{equation}
Then there is $\lambda ^{\ast }>0$ such that for any
$\lambda \in (0, \lambda ^{\ast })$, the equation \eqref{2} has a non
trivial weak solution.
\end{theorem}

For fixed $R\in M$, we define the function $\rho $ on $M$ by
\begin{equation}
\rho (Q)=\begin{cases}
d(R,Q)&\text{if } d(R,Q)<\delta (M) \\
\delta (M)&\text{if }d(R,Q)\geq \delta (M)
\end{cases} \label{3}
\end{equation}
where $\delta (M)$ denotes the injectivity radius of $M$.

For real numbers $\sigma $ and $\mu $, consider the following equation, in the
distribution sense,
\begin{equation}
\Delta ^2u-\nabla ^{i}(\frac{a}{\rho ^{\sigma }}\nabla _{i}u)+\frac{bu}{
\rho ^{\mu }}=\lambda | u| ^{q-2}u+f(x)|
u| ^{N-2}u    \label{4}
\end{equation}
where the functions $a$ and $b$ are smooth on $M$.

\begin{corollary} \label{coro1}
Let $0<\sigma <\frac{n}{r}<2$ and $0<\mu <\frac{n}{s}<4$. Suppose that
\begin{gather*}
\frac{\Delta f(x_0)}{f( x_0) }<\frac{1}{3}\Big( \frac{
(n-1)n( n^2+4n-20) }{( n^2-4) ( n-4)
( n-6) }\frac{1}{( 1+\| a\| _r+\|
b\| _s) ^{n/4}}-1\Big) S_{g}( x_0) \\
\text{when } n>6 , \\
S_{g}(x_0)>0 \quad \text{when }n=6.
\end{gather*}
Then there is $\lambda _{\ast }>0$ such that if
 $\lambda \in (0,\lambda _{\ast })$, the \eqref{4} possesses a weak
non trivial solution $ u_{\sigma ,\mu }\in M_{\lambda }$.
\end{corollary}

In the sharp case $\sigma =2$ and $\mu =4$, letting $K(n,2,\gamma )$ be the
best constant in the Hardy-Sobolev inequality given by Theorem \ref{thm7} we
obtain the following result.

\begin{theorem}
Let $(M,g)$ be a Riemannian compact manifold of dimension $n\geq 5$.
 Let $( u_{_{\sigma _{m},\mu _{m}}}) _{m}$ be a sequence in
$M_{\lambda}$ such that
\begin{gather*}
J_{\lambda ,\sigma ,\mu }(u_{_{\sigma _{m},\mu _{m}}})\leq c_{\sigma ,\mu }
\\
\nabla J_{\lambda }(u_{_{\sigma ,\mu }})-\mu _{_{\sigma ,\mu }}\nabla \Phi
_{\lambda }(u_{_{\sigma ,\mu }})\to 0
\end{gather*}
Suppose that
\[
c_{\sigma ,\mu }<\frac{2}{nK_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}
\]
and
\[
1+a^{-}\max ( K(n,2,\sigma ),A( \varepsilon ,\sigma )
) +b^{-}\max ( K(n,2,\mu ),A( \varepsilon ,\mu )) >0
\]
then the equation
\[
\Delta ^2u-\nabla ^{\mu }(\frac{a}{\rho ^2}\nabla _{\mu }u)+\frac{bu}{
\rho ^{4}}=f| u| ^{N-2}u+\lambda | u|
^{q-2}u
\]
in the distribution has a weak non trivial solution.
\end{theorem}

Our paper is organized as follows: in a first section we show that the
manifold of constraints is non empty, in the second one we establish a
generic existence result to equation \ref{2}. The third section deals with
applications to particular equations which could arise from conformal
geometry. In the fourth section and under supplementary assumption we obtain
non trivial solution in the critical case. The last section is devoted to
tests functions which verify geometric assumptions and by the same way
complete the proofs of our claimed theorems in the introduction.

\section{The manifold $M_{\protect\lambda }$ of constraints is non empty}

In this section, we consider on $H_2^2(M)$ the functional
\[
J_{\lambda }(u)=\frac{1}{2}\int_{M}( | \Delta _{g}u|
^2-a(x)| \nabla _{g}u| ^2+b(x)u^2) dv_{g}-
\frac{\lambda }{q}\int_{M}| u| ^qdv_{g}-\frac{1}{N}
\int_{M}f(x)| u| ^{N}dv_{g}
\]
associated to Equation \ref{2}. First, we put
\[
\Phi _{\lambda }(u)=\langle \nabla J_{\lambda }(u),\text{ }
u\rangle
\]
hence
\[
\Phi _{\lambda }(u)=\int_{M}( ( \Delta _{g}u)
^2-a(x)| \nabla _{g}u| ^2+b(x)u^2)
dv_{g}-\lambda \int_{M}| u|
^qdv_{g}-\int_{M}f(x)| u| ^{N}dv_{g}.
\]
We let
\[
M_{\lambda }=\{ u\in H_2^2(M):\text{ }\Phi _{\lambda }(u)=0\text{
and }\| u\| \geq \tau >0\} .
\]

\begin{proposition} \label{prop1}
The norm
$$
\| u\| =(\int_{M}| \Delta _{g}u|
^2-a(x)| \nabla _{g}u| ^2+b(x)u^2dv_{g})^{1/2}
$$
is  equivalent to the usual norm on $H_2^2(M)$ if and only if
$P_{g}$ is coercive.
\end{proposition}

\begin{proof}
If $P_{g}$ is coercive there is $\Lambda >0$ such that for any
$u\in H_2^2(M)$,
\[
\int_{M}P_{g}(u)udv_{g}\geq \Lambda \| u\|_{H_2^2(M)}^2
\]
and since $a\in L^{r}(M)$ and $b\in L^{s}(M)$ where
 $r>\frac{n}{2}$ and $s>\frac{n}{4}$, by H\"{o}lder's inequality we obtain
\[
\int_{M}uP_{g}(u)dv_{g}\leq \| \Delta _{g}u\|
_2^2+\| a\| _{\frac{n}{2}}\| \nabla
_{g}u\| _{2^{\ast }}^2+\| b\| _{\frac{n}{4}
}\| u\| _{N}^2
\]
where $2^{\ast }=2n/(n-2)$.

The Sobolev's inequalities lead to: for any $\eta >0$,
\[
\| \nabla _{g}u\| _{2^{\ast }}^2\leq \max ((1+\eta
)K(n,1)^2,A_{\eta })\int_{M}( | \nabla _{g}^2u|
^2+| \nabla _{g}u| ^2) dv_{g}
\]
where $K(n,1)$ denotes the best Sobolev's constant in the embedding
$H_{1}^2( \mathbb{R}^n) \hookrightarrow L^{\frac{2n}{n-2}}(\mathbb{R}^n) $,
 and for any $\epsilon >0$,
\[
\| u\| _{N}^2\leq \max ((1+\varepsilon)K_0,B_{\varepsilon })
\| u\| _{H_2^2(M)}^2
\]
where in this latter inequality $K_0$ is the best Sobolev's constant in
the embedding $H_{1}^2( M) \hookrightarrow L^{\frac{2n}{n-2}
}( M) $ and $B_{\epsilon }$ the corresponding (see \cite{3}).
 Now by the well known formula (see \cite[page 115]{3})
\[
\int_{M}| \nabla _{g}^2u| ^2dv_{g}=\int_{M}(| \Delta _{g}u| ^2-R_{ij}
\nabla ^{i}u\nabla ^{j}u) dv_{g}
\]
where $R_{ij}$ denote the components of the Ricci curvature, there is a
constant $\beta >0$ such that
\[
\int_{M}| \nabla _{g}^2u| ^2dv_{g}\leq
\int_{M}| \Delta _{g}u| ^2+\beta | \nabla_{g}u| ^2dv_{g}
\]
so we obtain
\[
\| \nabla _{g}u\| _{2^{\ast }}^2\leq (\beta +1)\max
((1+\eta )K(n,1)^2,A_{\eta })\int_{M}( | \Delta
_{g}u| ^2+| \nabla _{g}u| ^2+u^2)
dv_{g}
\]
and we infer that
\begin{align*}
\int_{M}P_{g}(u)udv_{g}
&\leq \| u\|_{H_2^2(M)}^2+(\beta +1)\| a\| _{\frac{n}{2}}
\max((1+\eta )K(n,1)^2,A_{\eta })\| u\| _{H_2^2(M)}^2\\
&\quad +\| b\| _{\frac{n}{4}}\max ((1+\varepsilon
)K_0,B_{\varepsilon })\| u\| _{H_2^2(M)}^2.
\end{align*}
Hence
\begin{align*}
&\int_{M}uP_{g}(u)dv_{g}\\
&\leq \underbrace{\max ( 1,\|
b\| _{\frac{n}{4}}\max ((1+\varepsilon )K_0,B_{\varepsilon
}),(\beta +1)\| a\| _{\frac{n}{2}}\max ((1+\varepsilon
)K(n,1)^2,A_{\varepsilon })) }_{>0}\\
&\quad\times \| u\| _{H_2^2(M)}^2.
\end{align*}
\end{proof}

\begin{lemma}\label{lem0}
The set $M_{\lambda }$ is non empty provided that
$\lambda \in (0,\lambda _0)$ where
\[
\lambda _0=\frac{( 2^{q-2}-2^{q-N}) \Lambda ^{\frac{N-q}{
N-2}}}{V(M)^{(1-\frac{q}{N})}( \max_{x\in M}f(x)) ^{\frac{2-q}{N-2
}}(\max ((1+\varepsilon )K( n,2) ,A_{\varepsilon }))^{\frac{N-q}{
N-2}}}.
\]
\end{lemma}

\begin{proof}
The proof of this lemma is the same as in \cite{8}, but we give it
here for convenience. Let $t>0$ and $u\in H_2^2(M)-\{ 0\} $.
Evaluating $\Phi _{\lambda }$ at $tu$, we obtain
\[
\Phi _{\lambda }(tu)=t^2\| u\| ^2-\lambda
t^q\| u\| _{q}^q-t^{N}\int_{M}f(x)|
u| ^{N}dv_{g}.
\]
Put
\begin{gather*}
\alpha (t)=\| u\|^2-t^{N-2}\int_{M}f(x)| u| ^{N}dv(g),\\
\beta (t)=\lambda t^{q-2}\| u\| _{q}^q\text{;}
\end{gather*}
by Sobolev's inequality, we obtain
\[
\alpha (t)\geq \| u\| ^2-\max_{x\in M}f(x)(\max
((1+\varepsilon )K_0,A_{\varepsilon }))^{N/2}
\|u\| _{H_2^2(M)}^{N}t^{N-2}.
\]
By the coercivity of the operator
$P_{g}=\Delta _{g}^2-\operatorname{div}_{g}(a\nabla _{g}) +b$ there
is a constant $\Lambda >0$ such that
\[
\alpha (t)\geq \| u\| ^2-\Lambda ^{-N/2}\max_{x\in M}f(x)
(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{
\frac{N}{2}}\| u\| ^{N}t^{N-2}.
\]
Letting
\[
\alpha _{1}(t)=\| u\| ^2-\Lambda ^{-N/2}\max_{x\in M}f(x)
(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{N/2}\| u\| ^{N}t^{N-2}
\]
H\"{o}lder and Sobolev inequalities lead to
\[
\beta (t)\leq \lambda V(M)^{(1-\frac{q}{N})}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\| u\|
_{H_2^2(M)}^qt^{q-2}
\]
and the coercivity of $P_{g}$ assures the existence of a constant
 $\Lambda>0 $ such that
\[
\beta (t)\leq \lambda \Lambda ^{-q/2}V(M)^{(1-\frac{q}{N})}(\max
((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\|
u\| ^qt^{q-2}.
\]
Put
\[
\beta _{1}(t)=\lambda \Lambda ^{-q/2}V(M)^{(1-\frac{q}{N})}(\max
((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\|
u\| ^qt^{q-2}.
\]
Let $t_0$ such $\alpha _{1}(t_0)=0$;  i.e.,
\[
t_0=\frac{\Lambda ^{\frac{N}{2(N-2)}}}{\|
u\| ( \max_{x\in M}f(x)) ^{\frac{1}{N-2}}(\max
((1+\varepsilon )K_0,A_{\varepsilon }))^{\frac{N}{2(N-2)}}}
\]
Now since $\alpha _{1}(t)$ is a decreasing and a concave function and
$\beta_{1}(t)$ is a decreasing and convex function,  then
\begin{gather*}
\min_{t\text{ }\in (0,\text{ }\frac{t_0}{2}]}\alpha _{1}(t)=\alpha
_{1}(\frac{t_0}{2})=\| u\| ^2(1-2^{2-N})>0,\\
\min_{t\text{ }\in (0,\text{ }\frac{t_0}{2}]}\beta _{1}(t)=\beta _{1}(
\frac{t_0}{2})>0,
\end{gather*}
where
\[
\beta _{1}(\frac{t_0}{2})
=\frac{2^{2-q}\lambda V(M)^{(1-\frac{q}{N})}
\Lambda ^{\frac{q-N}{N-2}}\| u\| ^2}{(\max
((1+\varepsilon )K_0,A_{\varepsilon }))^{\frac{q-N}{N-2}}(
\max_{x\in M}f(x)) ^{\frac{q-2}{N-2}}}.
\]
Consequently $\Phi _{\lambda }(tu)=0$ with
 $t\in (0,$ $\frac{t_0}{2}]$ has a solution if
\[
\min_{t\in (0,\frac{t_0}{2}]}\alpha _{1}(t)\geq
\max_{t\in (0,\frac{t_0}{2}]}\beta _{1}(t);
\]
that is to say
\[
0<\lambda <\frac{( 2^{q-2}-2^{q-N}) ( \max_{x\in
M}f(x)) ^{\frac{q-2}{N-2}}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{\frac{q-N}{N-2}}}{\Lambda ^{\frac{N-q}{N-2}}V(M)^{(1-
\frac{q}{N})}}=\lambda _0
\]
Let $t_{1}\in (0,\frac{t_0}{2}]$ such that
$\Phi _{\lambda }(t_{1}u)=0$. If we take $u\in H_2^2( M) $ such that
$ \| u\| \geq \frac{\rho }{t_{1}}$ and $v=t_{1}u$ we obtain
$\Phi _{\lambda }(v)=0$ and $\| v\| =t_{1}\|u\| \geq \rho $; i.e.,
$v\in M_{\lambda }$ provided that $\lambda \in (0,\lambda _0)$.
\end{proof}

\section{Existence of non trivial solutions in $M_{\lambda }$}

The following lemmas whose proofs are similar modulo minor modifications as
in \cite{8} give the geometric conditions to the functional $J_{\lambda }$.

\begin{lemma}\label{lem1}
 Let $(M,g)$ be a Riemannian compact manifold of dimension
$n\geq 5$. For all $u\in M_{\lambda }$ and all
 $\lambda \in ( 0,\min( \lambda _0,\lambda _{1}) ) $ there is $A>0$
such that $J_{\lambda }(u)\geq A>0$ where
\[
\lambda _{1}=\frac{\frac{( N-2) q}{2( N-q) }\Lambda ^{q/2}}
{V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon
)K(n,2),A_{\varepsilon }))^{q/2}\tau ^{q-2}}.
\]
\end{lemma}

\begin{lemma}\label{lem2}
Let $(M,g)$ be a Riemannian compact manifold of dimension $n\geq 5$.
The following assertions are true:
\begin{itemize}
\item[(i)] $\langle \nabla \Phi _{\lambda }(u),u\rangle <0$ for all
$u\in M_{\lambda }$ and for all
$\lambda \in (0,\min (\lambda _0,\lambda _{1}))$.

\item[(ii)] The critical points of $J_{\lambda }$ are points of $M_{\lambda }$.
\end{itemize}
\end{lemma}

Now, we show that $J_{\lambda }$ satisfies the Palais-Smale condition on
$M_{\lambda }$ provided that $\lambda >0$ is sufficiently small. The result
is given by the following lemma whose proof is different from the one in the
case of smooth coefficients.

\begin{lemma}\label{lem3}
Let $(M,g)$ be a compact Riemannian manifold of dimension
$n\geq 5$. Let $( u_{m}) _{m}$ be a sequence in $M_{\lambda }$
such that
\begin{gather*}
J_{\lambda }(u_{m})\leq c \\
\nabla J_{\lambda }(u_{m})-\mu _{m}\nabla \Phi _{\lambda }(u_{m})\to 0.
\end{gather*}
Suppose that
\[
c<\frac{2}{nK_0^{n/4}(f(x_0))^{(n-4)/4}}
\]
then there is a subsequence $( u_{m}) _{m}$ converging strongly
in $H_2^2(M)$.
\end{lemma}

\begin{proof}
Let $( u_{m}) _{m}\subset M_{\lambda }$ and
\[
J_{\lambda }(u_{m})=\frac{N-2}{2N}\| u_{m}\| ^2-\lambda
\frac{N-q}{Nq}\int_{M}| u_{m}| ^qdv_{g}\,.
\]
As in the proof of Lemma \ref{lem2}, we have
\begin{gather*}
J_{\lambda }(u_{m})\geq \frac{N-2}{2N}\| u_{m}\|
^2-\lambda \frac{N-q}{Nq}\Lambda ^{-q/2}V(M)^{1-\frac{q}{N}}(\max
((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\|
u_{m}\| ^q, \\
\begin{aligned}
J_{\lambda }(u_{m})&\geq \| u_{m}\| ^2(\frac{N-2}{2N}
-\lambda \frac{N-q}{Nq}\Lambda ^{-q/2}V(M)^{1-\frac{q}{N}}(\max
((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\tau ^{q-2})\\
&>0\,.
\end{aligned}
\end{gather*}
Since $0<\lambda <\frac{\frac{( N-2) q}{2( N-q) }
\Lambda ^{q/2}}{V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon
)K(n,2),A_{\varepsilon }))^{q/2}\tau ^{q-2}}$ and $J_{\lambda
}(u_{m})\leq c$, we obtain
\begin{align*}
c&\geq J_{\lambda }(u_{m}) \\
&\geq \big[ \frac{N-2}{2N}-\lambda \frac{N-q}{Nq}\Lambda ^{-\frac{q}{2}
}V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{\frac{q
}{2}}\tau ^{q-2}\big]\| u_{m}\| ^2>0
\end{align*}
so
\[
\| u_{m}\| ^2\leq \frac{c}{\frac{N-2}{2N}-\lambda \frac{
N-q}{Nq}\Lambda ^{-q/2}V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon
)K_0,A_{\varepsilon }))^{q/2}\tau ^{q-2}}<+\infty .
\]
Then $( u_{m}) _{m}$ is a bounded in $H_2^2(M)$. By the
compactness of the embedding $H_2^2(M)\subset H_p^{k}(M)$
 ($k\ =0,1$; $p<N$) we obtain a subsequence still denoted
$( u_{m}) _{m}$ such that
\begin{gather*}
u_{m}\to u\quad\text{weakly in } H_2^2(M),\\
u_{m}\to u \quad\text{strongly in $L^{p}(M)$ where $p<N$},\\
\nabla u_{m}\to \nabla u\quad\text{strongly in $L^{p}(M)$ where
$p<2^{\ast}=\frac{2n}{n-2}$}\\
u_{m}\to u\quad\text{a.e. in } M.
\end{gather*}
On the other hand since $\frac{2s}{s-1}<N=\frac{2n}{n-4}$, we obtain
\begin{align*}
| \int_{M}b(x)| u_{m}-u| ^2dv_{g}|
&\leq \| b\| _s\| u_{m}-u\| _{\frac{2s}{s-1}}^2\\
&\leq \| b\| _s( ( K_0+\epsilon )\| \Delta ( u_{m}-u) \| _2^2+A_{\epsilon
}\| u_{m}-u\| _2^2) .
\end{align*}
Now taking into account
\begin{equation}
K_0=\frac{16}{n(n^2-4)( n-4) \omega _{n}^{n/4}}<1  \label{K}
\end{equation}
we obtain
\[
\int_{M}b(x)( u_{m}-u) ^2dv_{g}\leq \| b\|
_s\| \Delta ( u_{m}-u) \| _2^2+o(
1) .
\]
By the same process as above, we obtain
\[
\int_{M}a(x)| \nabla ( u_{m}-u) |
^2dv_{g}\leq \| a\| _r\| \Delta (u_{m}-u) \| _2^2+o( 1) .
\]
By Brezis-Lieb lemma, we write
\[
\int_{M}( \Delta _{g}u_{m}) ^2dv_{g}=\int_{M}( \Delta
_{g}u) ^2dv_{g}+\int_{M}( \Delta _{g}(u_{m}-u))
^2dv_{g}+o(1)
\]
and
\[
\int_{M}f(x)| u_{m}| ^{N}dv_{g}=\int_{M}f(x)|
u| ^{N}dv_{g}+\int_{M}f(x)| u_{m}-u|
^{N}dv_{g}+o(1).
\]
Now we claim that $\mu _{m}\to 0$ as $m \to +\infty $
Testing with $u_{m}$ we obtain 
\[
\langle \nabla J_{\lambda }(u_{m})-\mu _{m}\nabla \Phi _{\lambda
}(u_{m}),u_{m}\rangle =o(1);
\]
then
\begin{align*}
\langle \nabla J_{\lambda }(u_{m})-\mu _{m}\nabla \Phi _{\lambda
}(u_{m}),u_{m}\rangle 
 =\underset{=0}{\underbrace{\langle \nabla J_{\lambda
}(u_{m}),u_{m}\rangle }}-\mu _{m}\langle \nabla \Phi _{\lambda
}(u_{m}),u_{m}\rangle =o(1);
\end{align*}
hence
\[
\mu _{m}\langle \nabla \Phi _{\lambda }(u_{m}),u_{m}\rangle =o(1).
\]
By Lemma \ref{lem2}, we obtain
$\lim \sup_{m}\langle \nabla \Phi _{\lambda }(u_{m}),u_{m}\rangle <0$ so
$\mu _{m}\to 0$  as $m\to +\infty$.

Our last claim is that $u_{m}\to u$ strongly in $H_2^2(M)$,
indeed
\[
J_{\lambda }(u_{m})-J_{\lambda }(u)
=\frac{1}{2}\int_{M}( \Delta _{g}(u_{m}-u)) ^2dv_g-\frac{1}{N}
\int_{M}f(x)| u_{m}-u| ^{N}dv_{g}+o(1).
\]
Since $u_{m}-u\to 0$ weakly in $H_2^2(M)$,  testing with
$\nabla J_{\lambda }(u_{m})-\nabla J_{\lambda }(u)$, we have
$\langle \nabla J_{\lambda }(u_{m})-\nabla J_{\lambda}(u),u_{m}-u\rangle =o(1)$ 
and
\begin{equation}
\begin{aligned}
&\langle \nabla J_{\lambda }(u_{m})-\nabla J_{\lambda}(u),u_{m}-u\rangle  \\
&=\int_{M}( \Delta _{g}(u_{m}-u))^2dv_{g}-\int_{M}f(x)| u_{m}-u| ^{N}dv_{g}=o(1);
\end{aligned}  \label{5}
\end{equation}
then
\[
\int_{M}( \Delta _{g}(u_{m}-u)) ^2dv_{g}=\int_{M}f(x)|
u_{m}-u| ^{N}dv_{g}+o(1),
\]
and taking account of \eqref{5} we obtain
\[
J_{\lambda }(u_{m})-J_{\lambda }(u)=\frac{1}{2}\int_{M}( \Delta
_{g}(u_{m}-u)) ^2dv_{g}-\frac{1}{N}\int_{M}( \Delta
_{g}(u_{m}-u)) ^2dv_{g}+o(1);
\]
i. e.,
\[
J_{\lambda }(u_{m})-J_{\lambda }(u)=\frac{2}{N}\int_{M}( \Delta
_{g}(u_{m}-u)) ^2dv_{g}+o(1).
\]
Independently, by the Sobolev's inequality, we have
\begin{equation}
\| u_{m}-u\| _{N}^2\leq (1+\varepsilon
)K_0\int_{M}( \Delta _{g}(u_{m}-u)) ^2dv_{g}+o(1).  \label{6}
\end{equation}
Since
\[
\int_{M}f(x)| u_{m}-u| ^{N}dv_{g}\leq \max_{x\in
M}f(x)\| u_{m}-u\| _{N}^{N}
\]
we infer by \eqref{6} that
\[
\int_{M}f(x)| u_{m}-u| ^{N}dv_{g}\leq (1+\varepsilon )^{
\frac{n}{n-4}}\max_{x\in M}f(x)K_0^{\frac{n}{n-4}}\| \Delta
_{g}(u_{m}-u)\| _2^{N}+o(1)
\]
and using equality \eqref{5},
\[
o(1)
\geq \| \Delta _{g}(u_{m}-u)\| _2^2-(1+\varepsilon
)^{\frac{n}{n-4}}\max_{x\in M}f(x)K_0^{\frac{n}{n-4}}\| \Delta
_{g}(u_{m}-u)\| _2^{N}
\]
and
\begin{align*}
&\| \Delta _{g}(u_{m}-u)\| _2^2-(1+\varepsilon
)^{\frac{n}{n-4}}\max_{x\in M}f(x)K_0^{\frac{n}{n-4}}\| \Delta
_{g}(u_{m}-u)\| _2^{N}
\\
&= \| \Delta _{g}(u_{m}-u)\| _2^2(1-(1+\varepsilon )^{
\frac{n}{n-4}}\max_{x\in M}f(x)K_0^{\frac{n}{n-4}}\| \Delta
_{g}(u_{m}-u)\| _2^{N-2})
\end{align*}
so if
\begin{equation}
\limsup_{m\to +\infty }\| \Delta _{g}(u_{m}-u)\| _2^2<
\frac{1}{K_0^{n/4}( \max_{x\in M}f(x)) ^{\frac{n}{4}-1}}   \label{8}
\end{equation}
then $u_{m}\to u$ strongly in $H_2^2( M) $. The
condition \eqref{8} is fulfilled since by Lemma \ref{lem1}
$J_{\lambda}(u)>0 $ on $M_{\lambda }$ with $\lambda $ is as
 in Lemma \ref{lem1} and by hypothesis,
\[
c\geq J_{\lambda }( u_{m}) >( J_{\lambda }(
u_{m}) -J_{\lambda }( u) ) =\frac{2}{n}\int_{M}(
\Delta _{g}(u_{m}-u)) ^2dv_{g}
\]
and
\[
c<\ \frac{2}{n K_0^{n/4}(\max_{x\in M}f(x))^{\frac{n}{4}-1}}.
\]
It is obvious that
\[
\Phi _{\lambda }(u)=0\text{ \ and }\| u\| \geq \tau
\]
i.e. $u\in M_{\lambda }$.
\end{proof}

Now we show the existence of a sequence in $M_{\lambda }$ satisfying the
conditions of Palais-Smale.

\begin{lemma}\label{lem4}
Let $(M,g)$ be a compact Riemannian manifold of dimension
$n\geq 5$, then there is a couple $( u_{m},\mu _{m}) \in
M_{\lambda }\times R$ such that
$\nabla J_{\lambda }(u_{m})-\mu _{m}\nabla \Phi _{\lambda }(u_{m})\to 0$
strongly in $(H_2^2(M))^{\ast }$ and $J_{\lambda }( u_{m}) $ is bounded
provide that $\lambda \in( 0,\lambda _{\ast }) $ with
$\lambda _{\ast }=\{ \min (\lambda _0,\lambda _{1}),0\} $.
\end{lemma}

\begin{proof}
Since $J_{\lambda }$ is Gateau differentiable and by Lemma \ref{lem1}
bounded below on $M_{\lambda }$ it follows from Ekeland's principle \ that
there is a couple $(u_{m},$ $\mu _{m})$ $\in M_{\lambda }\times R$ such that
$\nabla J_{\lambda }(u_{m})-\mu _{m}\nabla \Phi _{\lambda
}(u_{m})\to 0$ strongly in $(H_2^2(M))^{^{\prime }}$ and $
J_{\lambda }( u_{m}) $ is bounded i.e. $(u_{m},$ $\mu _{m})_{m}$
is a Palais-Smale sequence on $M_{\lambda }$.
\end{proof}

Now we are in position to establish the following generic existence result.

\begin{theorem}\label{thm3}
Let $(M,g)$ be a compact Riemannian manifold of dimension
$n\geq 5$ and $f$ a positive function. Suppose that $P_{g}$ is coercive and
\begin{equation}
c<\frac{2}{nK_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}.  \label{C1}
\end{equation}
Then there is $\lambda ^{\ast }>0$ such that for any
$\lambda \in (0,\lambda ^{\ast })$, the equation \eqref{2} has a non
trivial weak solution.
\end{theorem}

\begin{proof}
By Lemma \ref{lem3} and \ref{lem4} there is $u\in H_2^2( M) $
such that
\[
J_{\lambda }(u)=\min_{\varphi \in M_{\lambda }}J_{\lambda }(\varphi ).
\]
By Lagrange multiplicative theorem there is a real number $\mu $ such that
for any $\varphi \in H_2^2( M) $,
\begin{equation}
\langle \nabla J_{\lambda }(u),\varphi \rangle =\mu \langle
\nabla \Phi _{\lambda }(u),\varphi \rangle    \label{9}
\end{equation}
and letting $\varphi =u$ in the equation \eqref{9}, we obtain
\[
\Phi _{\lambda }(u)=\langle \nabla J_{\lambda }(u),u\rangle =\mu
\langle \nabla \Phi _{\lambda }(u),u\rangle .
\]
By Lemma \ref{lem2} we obtain that $\mu =0$ and by equation \eqref{9}, we infer
that for any $\varphi \in H_2^2( M) $
\[
\langle \nabla J_{\lambda }(u),\varphi \rangle =0
\]
hence $u$ is weak non trivial solution to equation \eqref{2} and since by
Lemma \ref{lem2}, $u$ is a  critical points of $J_{\lambda }$.
We conclude that $u\in M_{\lambda }$.
\end{proof}

\section{Applications}

Let $P\in M$, we define a function on $M$ by

\begin{equation}
\rho _{_{P}}(Q)=\begin{cases}
d(P,Q)&\text{if }d(P,Q)<\delta (M) \\
\delta (M)&\text{if }d(P,Q)\geq \delta (M)
\end{cases}  \label{10}
\end{equation}
where $\delta (M)$ is the injectivity radius of $M$. For brevity we denote
this function by $\rho $. The weighted
$L^{p}( M,\rho ^{\gamma })$ space will be the set of measurable functions
$u$ on $M$ such that $\rho ^{\gamma }| u| ^{p}$ are integrable where $p\geq 1$.
We endow $L^{p}( M,\rho ^{\gamma }) $ with the norm
\[
\| u\| _{p,\rho }=( \int_{M}\rho ^{\gamma }|
u| ^{p}dv_{g}) ^{1/p}.
\]
In this section we need  the  Hardy-Sobolev inequality
and the Releich-Kondrakov embedding  whose proofs are given in
\cite{7}.

\begin{theorem} \label{thm7}
Let $(M,g)$ be a Riemannian compact manifold of dimension
$n\geq 5$ and $p$, $q$ , $\gamma $ are real numbers such that
$\frac{\gamma }{p}=\frac{n}{q}-\frac{n}{p}-2$ $\ $and
$2\leq p\leq \frac{2n}{n-4}$.
For any $\epsilon >0$, there is $A(\epsilon ,q,\gamma )$ such that for
any $u\in H_2^2(M)$,
\begin{equation}
\| u\| _{p,\rho ^{\gamma }}^2\leq (1+\epsilon
)K(n,2,\gamma )^2\| \Delta _{g}u\| _2^2+A(\epsilon
,q,\gamma )\| u\| _2^2   \label{11}
\end{equation}
where $K(n,2,\gamma )$ is the optimal constant.
\end{theorem}

In the case $\gamma =0$, $K(n,2,0)=K(n,2)=K_0^{1/2}$ is the best
constant in the Sobolev's embedding of $H_2^2(M)$ in $L^{N}(M)$ where
$N=\frac{2n}{n-4}$.

\begin{theorem}
Let $(M,g)$ be a compact Riemannian manifold of dimension
$n\geq 5$ and $p$, $q$, $\gamma $ are real numbers satisfying
$1\leq q\leq p\leq \frac{nq}{n-2q}$, $\gamma <0$ and $l=1, 2$.

If $\frac{\gamma }{p}=n$ $(\frac{1}{q}-\frac{1}{p})-l$ then the inclusion
$H_{l}^q(M)\subset $ $L^{p}(M,\rho ^{\gamma })$ is continuous.
If $\frac{\gamma }{p}>n$ $(\frac{1}{q}-\frac{1}{p})-l$ then inclusion
$H_{l}^q(M)\subset $ $L^{p}(M,\rho ^{\gamma })$ is compact.
\end{theorem}

We consider the  equation
\begin{equation}
\Delta _{g}^2u+\operatorname{div}_{g}\Big( \frac{a(x)}{\rho ^{\sigma }}\nabla
_{g}u\Big) +\frac{b(x)}{\rho ^{\mu }}u=\lambda | u|
^{q-2}u+f(x)| u| ^{N-2}u    \label{12}
\end{equation}
where $a$ and $b$ are smooth functions and $\rho $ denotes the distance
function defined by \eqref{10}, $\lambda >0$ in some interval
$(0,\lambda _{\ast }) $, $1<q<2$, $\sigma $, $\mu $ will be precise
later and we associate to \eqref{12} on $H_2^2(M)$ the functional
\begin{align*}
J_{\lambda }(u)
&=\frac{1}{2}\int_{M}( ( \Delta _{g}u) ^2-
\frac{a(x)}{\rho ^{\sigma }}| \nabla _{g}u| ^2+\frac{
b(x)}{\rho ^{\mu }}u^2) dv_{g}\\
&\quad -\frac{\lambda }{q}\int_{M}|
u| ^qdv_{g}-\frac{1}{N}\int_{M}f(x)| u|^{N}dv_{g}.
\end{align*}
If we put
\[
\Phi _{\lambda }(u)=\langle \nabla J_{\lambda }(u),u\rangle
\]
we obtain
\[
\Phi _{\lambda }(u)=\int_{M}( \Delta _{g}u) ^2-\frac{a(x)}{\rho
^{\sigma }}| \nabla _{g}u| ^2+\frac{b(x)}{\rho ^{\mu }}
u^2dv_{g}-\lambda \int_{M}| u|
^qdv_{g}-\int_{M}f(x)| u| ^{N}dv_{g}.
\]

\begin{theorem}\label{thm4}
Let $0<\sigma <\frac{n}{s}<2$ and $0<\mu <\frac{n}{p}<4$.
Suppose that
\[
\sup_{u\in H_2^2( M) }J_{\lambda ,\sigma ,\mu }(u)<\frac{2}{n
\text{ }K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}
\]
then there is $\lambda _{\ast }>0$ such that if
 $\lambda \in (0,\lambda_{\ast })$, equation \eqref{12} possesses
 a weak non trivial solution $u_{\sigma ,\mu }\in M_{\lambda }$.
\end{theorem}

\begin{proof}
Let $\tilde{a}=\frac{a(x)}{\rho ^{\sigma }}$ and
$\tilde{b}=\frac{b(x)}{\rho ^{\mu }}$, so if
$\sigma \in (0,\min ( 2,\frac{n}{s}) )$ and
$\mu \in (0,\min (4,\frac{n}{p}))$, obliviously $\tilde{a}\in L^{s}(M)$,
 $\tilde{b}\in L^{p}(M)$, where $s>\frac{n}{2}$ and $p>\frac{n}{4}$.
Theorem \ref{thm4} is a consequence of Theorem \ref{thm3}.
\end{proof}

\section{The critical cases $\sigma =2$ and $\mu =4$}

In the  cases $\sigma =2$ and $\mu =4$ the Hardy-Sobolev inequality
proved in case of manifolds by the first author in \cite{7} and is
formulated in Theorem \ref{thm7} is no longer valid,
 so we consider the subcritical cases
$0<\sigma <2$ and $0<\mu <4$ and we tend $\sigma $ to $2$and $\mu $ to
$4$. This can be done successfully by adding an appropriate assumption and by
using the Lebesgue dominated converging theorem.

By section four, for any $\sigma \in (0,\min ( 2,\frac{n}{s}) )$
and $\mu \in (0,\min (4,\frac{n}{p}))$, there is a solution
$u_{\sigma ,\mu}\in M_{\lambda }$ of equation \eqref{2}.
Now we are going to show that the sequence $( u_{\sigma ,\mu }) _{\sigma ,\mu }$
is bounded in $H_2^2( M) $. Evaluating $J_{\lambda ,\sigma ,\mu }$ at
$u_{\sigma ,\mu }$
\[
J_{\lambda ,\sigma ,\mu }(u_{\sigma ,\mu })=\frac{1}{2}\| u_{\sigma
,\mu }\| ^2-\frac{1}{N}\int_{M}f(x)| u_{\sigma ,\mu
}| ^{N}dv_{g}-\frac{1}{q}\lambda \int_{M}| u_{\sigma ,\mu
}| ^qdv_{g}
\]
and taking account of $u_{\sigma ,\mu }\in M_{\lambda }$, we infer that
\[
J_{\lambda ,\sigma ,\mu }(u_{\sigma ,\mu })=\frac{N-2}{2N}\|
u_{\sigma ,\mu }\| ^2-\lambda \frac{N-q}{Nq}\int_{M}|
u_{\sigma ,\mu }| ^qdv_{g}.
\]

For a smooth function $a$ on $M$, denotes by $a^{-}=\min ( 0,\min_{x\in
M}(a(x)) $. Let $K(n,2,\sigma )$ the best constant and $A(
\varepsilon ,\sigma ) $ the corresponding constant in the Hardy-
Sobolev inequality given in Theorem \ref{thm7}.

\begin{theorem}
Let $(M,g)$ be a Riemannian compact manifold of dimension $n\geq 5$.
Let $( u_{m}) _{m}=( u_{\sigma _{m},\mu _{m}}) _{m}$ be a
sequence in $M_{\lambda }$ such that
\begin{gather*}
J_{\lambda ,\sigma ,\mu }(u_{m})\leq c_{\sigma ,\mu } \\
\nabla J_{\lambda }(u_{m})-\mu _{_{\sigma ,\mu }}\nabla \Phi _{\lambda
}(u_{m})\to 0.
\end{gather*}
Suppose that
\[
c_{\sigma ,\mu }<\frac{2}{n\text{ }K( n,2) ^{n/4}
(\max_{x\in M}f(x))^{(n-4)/4}}
\]
and
\[
1+a^{-}\max ( K(n,2,\sigma ),A( \varepsilon ,\sigma )
) +b^{-}\max ( K(n,2,\mu ),A( \varepsilon ,\mu )) >0.
\]
Then the equation
\[
\Delta ^2u-\nabla ^{\mu }(\frac{a}{\rho ^2}\nabla _{\mu }u)+\frac{bu}{
\rho ^{4}}=f| u| ^{N-2}u+\lambda | u|
^{q-2}u
\]
has a non trivial solution in the sense of distributions.
\end{theorem}

\begin{proof}
Let $( u_{m}) _{m}\subset M_{\lambda ,\sigma ,\mu }$,
\[
J_{\lambda ,\sigma ,\mu }(u_{m})=\frac{N-2}{2N}\| u_{m}\|
^2-\lambda \frac{N-q}{Nq}\int_{M}| u_{m}| ^qdv_{g}
\]
As in proof of Theorem \ref{thm3}, we obtain
\begin{align*}
J_{\lambda ,\sigma ,\mu }(u_{m})
&\geq \| u_{m}\| ^2\Big(\frac{N-2}{2N}\\
&\quad -\lambda \frac{N-q}{Nq}\Lambda _{\sigma ,\mu }^{-q/2}V(M)^{1-
\frac{q}{N}}(\max ((1+\varepsilon )K( n,2) ,
A_{\varepsilon }))^{q/2}\tau ^{q-2}\Big)>0
\end{align*}
where
\[
0<\lambda <\frac{\frac{( N-2) q}{2( N-q) }
\Lambda _{\sigma ,\mu }^{q/2}}{V(M)^{1-\frac{q}{N}}(\max
((1+\varepsilon )K( n,2) ,A_{\varepsilon }))^{q/2}\tau
^{q-2}}.
\]
First we claim that
\[
\lim_{( \sigma ,\mu ) \to ( 2^{-},4^{-}) }\inf \Lambda _{\sigma ,\mu }>0.
\]
Indeed, if $\nu _{1,\sigma ,\mu }$ denotes the first nonzero eigenvalue of
the operator
$$
P_{g}=\Delta _{g}^2-\operatorname{div}( \frac{a}{\rho ^{\sigma }}
\nabla _{g}) +\frac{b}{\rho ^{\mu }},
$$
 then clearly
$\Lambda _{\sigma ,\mu }\geq \nu _{1,\sigma ,\mu }$. Suppose on the contrary
that $\lim_{(\sigma ,\mu ) \to ( 2^{-},4^{-}) }\inf \Lambda
_{\sigma ,\mu }=0$, then
$\lim \inf_{( \sigma ,\mu ) \to ( 2^{-},4^{-}) }\nu _{1,\sigma ,\mu }=0$.
Independently, if $ u_{\sigma ,\mu }$ is the corresponding eigenfunction to
$\nu _{1,\sigma ,\mu}$ we have
\begin{equation}
\begin{aligned}
\nu _{1,\sigma ,\mu }
&=\| \Delta u_{\sigma,\mu}\| _2^2+\int_{M}
\frac{ a| \nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }}dv_{g}
+\int_{M}\frac{bu_{\sigma,\mu}^2}{\rho ^{\mu }}dv_{g} \\
&\geq \| \Delta u_{\sigma,\mu}\| _2^2+a^{-}\int \frac{|
\nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }}dv_{g}+b^{-}\int_{M}
\frac{u_{\sigma,\mu}^2}{\rho ^{\mu }}dv_{g}
\end{aligned}  \label{13}
\end{equation}
where $a^{-}=\min ( 0,\min_{x\in M}a(x)) $ and
$b^{-}=\min (0,\min_{x\in M}b(x)) $. The Hardy- Sobolev's inequality given by
Theorem \ref{thm7} leads to
\[
\int_{M}\frac{| \nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }}
dv_{g}\leq C(\| \nabla | \nabla u_{\sigma,\mu}| \|
^2+\| \nabla u_{\sigma,\mu}\| ^2),
\]
and since
\[
\| \nabla | \nabla u_{\sigma,\mu}| \| ^2\leq
\| \nabla ^2u_{\sigma,\mu}\| ^2\leq \| \Delta u_{\sigma,\mu}\|
^2+\beta \| \nabla u_{\sigma,\mu}\| ^2
\]
where $\beta >0$ is a constant and it is well known that for any
$\varepsilon >0$ there is a constant $c( \varepsilon ) >0$ such
that
\[
\| \nabla u_{\sigma,\mu}\| ^2\leq \varepsilon \| \Delta u_{\sigma,\mu}\| ^2
+c\| u_{\sigma,\mu}\| ^2.
\]
Hence
\begin{equation}
\int_{M}\frac{| \nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }}
dv_{g}\leq C( 1+\varepsilon ) \| \Delta u_{\sigma,\mu}\|
^2+A( \varepsilon ) \| u_{\sigma,\mu}\| ^2  \label{14}
\end{equation}
Now if $K(n,2,\sigma )$ denotes the best constant in inequality \eqref{14}
we obtain that for any $\varepsilon >0$,
\begin{equation}
\int_{M}\frac{| \nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }}
dv_{g}\leq ( K(n,2,\sigma )^2+\varepsilon ) \| \Delta
u_{\sigma,\mu}\| ^2+A( \varepsilon ,\sigma ) \| u_{\sigma,\mu}\| ^2. 
   \label{15}
\end{equation}
By  inequalities \eqref{11}, \eqref{13} and \eqref{15}, we have
\begin{align*}
\nu _{1,\sigma ,\mu }
&\geq ( 1+a^{-}\max ( K(n,2,\sigma ),A(\varepsilon ,\sigma ) ) \\
&\quad +b^{-}\max ( K(n,2,\mu ),A(
\varepsilon ,\mu ) ) ) ( \| \Delta u_{\sigma
,\mu }\| ^2+\| u_{\sigma ,\mu }\| ^2)
\end{align*}
So if
\[
1+a^{-}\max ( K(n,2,\sigma ),A( \varepsilon ,\sigma )
) +b^{-}\max ( K(n,2,\mu ),A( \varepsilon ,\mu )) >0
\]
then we obtain $\lim_{\sigma ,\mu }( u_{\sigma ,\mu }) =0$ and
$\| u_{\sigma ,\mu }\| =1$ a contradiction.
The reflexivity of $H_2^2(M)$ and the compactness of the embedding
$H_2^2(M)\subset H_p^{k}(M)$ ($k=0,1$; $p<N$), imply that up to
a subsequence, we have
\begin{gather*}
u_{m}\to u\quad\text{weakly in }H_2^2(M),\\
u_{m}\to u\quad\text{strongly in }L^{p}(M),\; p<N,\\
\nabla u_{m}\to \nabla u\quad\text{strongly in }L^{p}(M),\;
p<2^{\ast }=\frac{2n}{n-2},\\
u_{m}\to u\quad\text{a. e. in }M.
\end{gather*}
The Br\'{e}zis-Lieb lemma allows us to write
\[
\int_{M}( \Delta _{g}u_{m}) ^2dv_{g}=\int_{M}( \Delta
_{g}u) ^2dv_{g}+\int_{M}( \Delta _{g}(u_{m}-u))
^2dv_{g}+o(1)
\]
and
\[
\int_{M}f(x)| u_{m}| ^{N}dv_{g}=\int_{M}f(x)|
u| ^{N}dv_{g}+\int_{M}f(x)| u_{m}-u|
^{N}dv_{g}+o(1).
\]
Now by the boundedness of the sequence ($u_{m}$)$_{m}$, we have that
$u_{m}\to u$ weakly in $H_2^2(M)$,
$\nabla u_{m}\to \nabla u$ weakly in $L^2(M,\rho ^{-2})$ and
 $ u_{m}\to u$ weakly in $L^2(M,\rho ^{-4})$; i.e., for any
$\varphi \in L^2(M)$,
\[
\int_{M}\frac{a(x)}{\rho ^2}\nabla u_{m}\nabla \varphi dv_{g}=\int_{M}
\frac{a(x)}{\rho ^2}\nabla u\nabla \varphi dv_{g}+o(1)
\]
and
\[
\int_{M}\frac{b(x)}{\rho ^{4}}u_{m}\varphi dv_{g}=\int_{M}\frac{b(x)}{\rho
^{4}}u\varphi dv_{g}+o(1).
\]
For every $\phi \in H_2^2(M)$ we have
\begin{equation}
\begin{aligned}
&\int_{M}\Big( \Delta _{g}^2u_{m}+\operatorname{div}_{g}
\big( \frac{a(x)}{\rho ^{\sigma
_{m}}}\nabla _{g}u_{m}\big) +\frac{b(x)}{\rho ^{\delta _{m}}}u_{m}\Big)
\phi dv_{g}\\
&=\int_{M}( \lambda | u_{m}|^{q-2}u_{m}+f(x)| u_{m}| ^{N-2}u_{m}) \phi dv_{g}.
\end{aligned}  \label{16}
\end{equation}
By the weak convergence in $H_2^2(M)$, we have immediately that
\[
\int_{M}\phi \Delta _{g}^2u_{m}dv_{g}=\int_{M}\phi \Delta
_{g}^2udv_{g}+o(1)
\]
and
\begin{align*}
&\int_{M}( \frac{a(x)}{\rho ^{\sigma _{m}}}\nabla _{g}u_{m}-\frac{a(x)}{
\rho ^2}\nabla _{g}u) \phi dv_{g}\\
&=\int_{M}( \frac{a(x)}{\rho
^{\sigma _{m}}}\nabla _{g}u_{m}+\frac{a(x)}{\rho ^2}( \nabla
_{g}u_{m}-\nabla _{g}u_{m}) -\frac{a(x)}{\rho ^2}\nabla _{g}u)
\phi dv_{g}
\end{align*}
Then
\begin{equation}
\begin{aligned}
&\big| \int_{M}\Big( \frac{a(x)}{\rho ^{\sigma _{m}}}\nabla _{g}u_{m}-
\frac{a(x)}{\rho ^2}\nabla _{g}u\Big) \phi dv_{g}\big|
\\
&\leq \big| \int_{M}\Big( \frac{a(x)}{\rho ^{\sigma _{m}}}\nabla _{g}u_{m}-
\frac{a(x)}{\rho ^2}\nabla _{g}u_{m}\Big) \phi dv_{g}\big|
+\big| \int_{M}( \frac{a(x)}{\rho ^2}\nabla _{g}u_{m}-\frac{a(x)
}{\rho ^2}\nabla _{g}u) \phi dv_{g}\big|
\\
&\leq \int_{M}| a(x)\phi \nabla _{g}u_{m}| |
\frac{1}{\rho ^{\sigma _{m}}}-\frac{1}{\rho ^2}|
dv_{g}+| \int_{M}\frac{a(x)}{\rho ^2}\nabla _{g}(
u_{m}-u) \phi dv_{g}| .
\end{aligned} \label{17}
\end{equation}
The weak convergence in $L^2(M,\rho ^{-2})$ and the Lebesgue's dominated
convergence theorem imply that the second right hand side of \eqref{17} goes
to $0$. For the third term of the left hand side of \eqref{15}, we write
\[
\int_{M}\Big( \frac{b(x)}{\rho ^{\delta _{m}}}u_{m}-\frac{b(x)}{\rho ^{4}}
u\Big) \phi dv_{g}
=\int_{M}\Big( \frac{b(x)}{\rho ^{\delta _{m}}}u_{m}-
\frac{b(x)}{\rho ^{4}}u_{m}+\frac{b(x)}{\rho ^{4}}u_{m}-\frac{b(x)}{\rho ^{4}
}u\Big) \phi dv_{g}
\]
and
\begin{equation}
\begin{aligned}
&\big| \int_{M}( \frac{b(x)}{\rho ^{\delta _{m}}}u_{m}-\frac{b(x)}{
\rho ^{4}}u) \phi dv_{g}\big|\\
&\leq \int_{M}| b(x)\phi u_{m}| | \frac{1}{\rho
^{\delta _{m}}}-\frac{1}{\rho ^{4}}| dv_{g}+| \int_{M}
\frac{b(x)}{\rho ^{4}}( u_{m}-u) \phi dv_{g}| .
\end{aligned}  \label{18}
\end{equation}
Here also the weak convergence in $L^2(M,\rho ^{-4})$ and the Lebesgue's
dominated convergence allows us to affirm that the left hand side of
\eqref{18} converges to $0$.

It remains to show that $\mu _{m}\to 0$ as $m$ $\to +\infty $
and $u_{m}\to u$ strongly in $H_2^2( M) $ but this is
the same as in the proof of Theorem \ref{thm3} which implies also $u\in
M_{\lambda }$.
\end{proof}

\section{Test Functions}

In this section, we give the proof of the main theorem to do so, we consider
a normal geodesic coordinate system centered at $x_0$. Denote by
$S_{x_0}(\rho )$ the geodesic sphere centered at $x_0$ and of radius
$\rho$ ($\rho <d$ which is the injectivity radius).
Let $d\Omega $ be the volume element
of the $n-1$-dimensional Euclidean unit sphere $S^{n-1}$ and put
\[
G(\rho )=\frac{1}{\omega _{n-1}}\int_{S(\rho )}\sqrt{|g(x)| }d\Omega
\]
where $\omega _{n-1}$ is the volume of $S^{n-1}$ and $|
g(x)| $ the determinant of the Riemannian metric $g$. The Taylor's
expansion of $G(\rho )$ in a neighborhood of $x_0$ is given by
\[
G(\rho )=1-\frac{S_{g}(x_0)}{6n}\rho ^2+o(\rho ^2)
\]
where $S_{g}(x_0)$ denotes the scalar curvature of $M$ at $x_0$.
Let $B(x_0,\delta )$ be the geodesic ball centered at $x_0$
and of radius $\delta $ such that $0<2\delta <d$ and denote by $\eta $ a
smooth function on $M$ such that
\[
\eta (x)=\begin{cases}
1&\text{on }B(x_0,\delta ) \\
0&\text{on }M-B(x_0,2\delta ).
\end{cases}
\]

Consider the  radial function
\[
u_{\epsilon }(x)=(\frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)})^{\frac{
n-4}{8}}\frac{\eta (\rho )}{(( \rho \theta ) ^2+\epsilon ^2)^{
\frac{n-4}{2}}}
\]
with
\[
\theta =( 1+\| a\| _r+\| b\|_s) ^{1/n}
\]
where $\rho =d(x_0,x)$ is the distance from $x_0$ to $x$ and
$f(x_0)=\max_{x\in M}f(x)$. For further computations we need the following
integrals: for any real positive numbers $p$, $g$ such that $p-q>1$ we put
\[
I_p^q=\int_{0}^{+\infty }\frac{t^q}{(1+t)^{p}}dt\,.
\]
The following relations are immediate
\[
I_{p+1}^q=\frac{p-q-1}{p}I_p^q, \quad
I_{p+1}^{q+1}= \frac{q+1}{p-q-1}I_{p+1}^q.
\]

\subsection{Application to compact Riemannian manifolds of dimension $n>6$}

\begin{theorem}\label{thm5}
 Let $( M,g) $ be a compact Riemannian manifold of
dimension $n>6$. Suppose that at a point $x_0$ where $f$ attains its
maximum the following condition
\[
\frac{\Delta f(x_0)}{f( x_0) }<\frac{1}{3}\Big( \frac{
(n-1)n( n^2+4n-20) }{( n^2-4) ( n-4)
( n-6) }\frac{1}{( 1+\| a\| _r+\|
b\| _s) ^{n/4}}-1\Big) S_{g}( x_0)
\]
holds. Then  \eqref{1} has a non trivial solution with energy
\[
J_{\lambda }(u)<\frac{1}{K_0^{n/4}( \max_{x\in M}f(x)) ^{\frac{n}{4}-1}}.
\]
\end{theorem}

\begin{proof}
The proof of Theorem \ref{thm5} reduces to show that the condition \eqref{C1}
 of Theorem \ref{thm3} is satisfied and since by Lemma \ref{lem0} there is a
$t_0>0$ such that $t_0u_{\epsilon }\in M_{\lambda }$ for sufficiently
small $\lambda $, so it suffices to show that
\[
\sup_{t>0}J_{\lambda }( tu_{\epsilon }) <\frac{1}{K_0^{
\frac{n}{4}}( \max_{x\in M}f(x)) ^{\frac{n}{4}-1}}.
\]
To compute the term $\int_{M}f(x)| u_{\epsilon }(x)|^{N}dv_{g}$,
 we need the following Taylor's expansion of $f$ at the point  $x_0$
\[
f(x)=f(x_0)+\frac{\partial ^2f(x_0)}{2\partial y^{i}\partial
y^{j}}y^{i}y^{j}+o(\rho ^2)
\]
and also that of the Riemannian measure
\[
dv_{g}=1-\frac{1}{6}R_{ij}(x_0)y^{i}y^{j}+o(\rho ^2)
\]
where $R_{ij}(x_0)$ denotes the Ricci tensor at $x_0$. The
expression of $\int_{M}f(x)| u_{\epsilon }(x)| ^{N}dv_{g}$
is well known (see for example \cite{11} ) and is given in case $n>6$ by
\[
\int_{M}f(x)| u_{\epsilon }(x)| ^{N}dv_{g}=\frac{\theta
^{-n}}{K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\Big( 1-(
\frac{\Delta f(x_0)}{2(n-2)f(x_0)}+\frac{S_{g}(x_0)}{
6(n-2)})\epsilon ^2+o(\epsilon ^2)\Big)
\]
where $K_0$ is given by \eqref{K} and
$\omega _{n}=2^{n-1}I_{n}^{\frac{n}{2}-1}\omega _{n-1}$ and $\omega _{n}$
is the volume of $S^{n}$, the standard
unit sphere of $R^{n+1}$ endowed with its round metric.

Now the restriction of $| \frac{\partial u_{\epsilon }}{\partial
\rho }| $ to the geodesic ball $B(x_0,\delta )$ is computed
as follows
\[
| \frac{\partial u_{\epsilon }}{\partial \rho }|
_{B(x_0,\delta )}=| \nabla u_{\epsilon }| =\theta
^{-2}(n-4)( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}) ^{
\frac{n-4}{8}}\frac{\rho }{(( \frac{\rho }{\theta })
^2+\epsilon ^2)^{\frac{n-2}{2}}}
\]
and Since $a\in L^{r}(M)$ with $r>n/2$ we have
\begin{align*}
\int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }|
^2dv_{g}
&\leq \theta ^{-4}(n-4)^2\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}
}{f(x_0)}\Big) ^{\frac{n-4}{4}}\| a\| _r\omega
_{n-1}^{1-\frac{1}{r}} \\
&\quad \times \Big( \int_{0}^{\delta }\frac{\rho ^{\frac{2r}{r-1}+n-1}}{((
\frac{\rho }{\theta }) ^2+\epsilon ^2)^{\frac{( n-2) r}{
r-1}}}\Big( \int_{S(\rho )}\sqrt{| g(x)| }d\Omega
\Big) d\rho \Big) ^{\frac{r-1}{r}}
\end{align*}
Since
\[
\int_{S(\rho )}\sqrt{| g(x)| }d\Omega =\omega
_{n-1}\Big( 1-\frac{S_{g}(x_0)}{6n}\rho ^2+o(\rho ^2)\Big)
\]
we obtain
\begin{align*}
\int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }|^2dv_{g}
&\leq \theta ^{-4}(n-4)^2( \frac{(n-4)n(n^2-4)\epsilon ^{4}
}{f(x_0)}) ^{\frac{n-4}{4}}\| a\| _r\omega
_{n-1}^{1-\frac{1}{r}} \\
&\quad \times \Big( \int_{0}^{\delta }\frac{\rho ^{\frac{2r}{r-1}+n-1}}{((
\rho \theta ) ^2+\epsilon ^2)^{\frac{( n-2) r}{r-1}}}
d\rho \Big( 1-\frac{S_{g}(x_0)}{6n}\rho ^2+o(\rho ^2)\Big)
\Big) ^{\frac{r-1}{r}}
\end{align*}
and by the following change of variable
\[
t=(\frac{\rho \theta }{\epsilon })^2\quad\text{ i.e. }
\rho =\frac{\epsilon }{\theta }\sqrt{t}
\]
we obtain
\begin{align*}
&\int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }|^2dv_{g}\\
&\leq \theta ^{-n\frac{r}{r-1}}(n-4)^2\Big( \frac{
(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4}
}\| a\| _r\omega _{n-1}^{1-\frac{1}{r}}\epsilon ^{-(
n-4) +2-\frac{n}{r}} \\
&\quad \times \Big( \int_{0}^{(\frac{\delta \theta }{\epsilon })^2}\frac{t^{
\frac{n-2}{2}+\frac{r}{r-1}}}{(t+1)^{\frac{( n-2) r}{r-1}}}dt-
\frac{S_{g}(x_0)}{6n}\theta ^{-2}\epsilon ^2\int_{0}^{(\frac{\delta
\theta }{\epsilon })^2}\frac{t^{\frac{n}{2}+\frac{r}{r-1}}}{(t+1)^{\frac{
( n-2) r}{r-1}}}dt+o(\epsilon ^2)\Big) ^{\frac{r-1}{r}}.
\end{align*}
Letting $\epsilon \to 0$ we obtain
\begin{align*}
&\int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }|^2dv_{g} \\
&\leq 2^{-1+\frac{1}{r}}\theta ^{-n( 1-\frac{1}{r})
}(n-4)^2( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}) ^{
\frac{n-4}{4}}\| a\| _r\omega _{n-1}^{1-\frac{1}{r}
}\epsilon ^{-( n-4) +2-\frac{n}{r}} \\
&\quad \times ( I_{\frac{( n-2) r}{r-1}}^{\frac{n-2}{2}+\frac{r}{r-1
}}-\theta ^{-2}\frac{S_{g}(x_0)}{6n}I_{\frac{( n-2) r}{r-1}
}^{\frac{n}{2}+\frac{r}{r-1}}\epsilon ^2+o(\epsilon ^2)) ^{\frac{
r-1}{r}}.
\end{align*}
Then
\begin{align*}
&\int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }|^2dv_{g}\\
&\leq 2^{-1+\frac{1}{r}}\theta ^{-n\frac{r}{r-1}}(n-4)^2\Big(
\frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4}
}\| a\| _r\omega _{n-1}^{1-\frac{1}{r}}\epsilon
^{\epsilon ^{-( n-4) +2-\frac{n}{r}}} \\
&\quad \times I_{\frac{( n-2) r}{r-1}}^{1+\frac{n-2}{2}.\frac{r-1}{r}}
\Big[ 1-\frac{r-1}{r}\theta ^2\frac{S_{g}(x_0)}{6n}I_{\frac{(
n-2) r}{r-1}}^{\frac{n}{2}+\frac{r}{r-1}}I_{\frac{( n-2) r}{
r-1}}^{-\frac{n-2}{2}-\frac{r}{r-1}}\epsilon ^2+o(\epsilon ^2)\Big].
\end{align*}
It remains to compute the integral
 $\int_{B(x_0,2\delta )-B(x_0,\delta )}a(x)| \nabla u_{\epsilon }| ^2dv_{g}$.

First we remark that
\[
\big| \int_{(\frac{\delta \theta }{\epsilon })^2}^{(\frac{2\delta
\theta }{\epsilon })^2}h(t)\frac{t^q}{(t+1)^{p}}dt\big| \leq
C( \frac{1}{\epsilon }) ^{2(q-p+1)}=C\epsilon ^{2(p-q-1)}
\]
and since $p-q=n-4\geq 3$, we obtain
\[
\int_{(\frac{\delta \theta }{\epsilon })^2}^{(\frac{2\delta \theta }{
\epsilon })^2}h(t)\frac{i^q}{(t+1)^{p}}dt=o(\epsilon ^2)
\]
and then
\begin{equation}
\int_{B(x_0,2\delta )-B(x_0,\delta )}a(x)| \nabla
u_{\epsilon }| ^2dv_{g}=o(\epsilon ^2).
\label{19}
\end{equation}
Finally we obtain
\begin{align*}
&\int_{M}a(x)| \nabla u_{\epsilon }| ^2dv_{g}\\
&\leq 2^{-1+ \frac{1}{r}}\theta ^{-n\frac{r}{r-1}}(n-4)^2\Big( \frac{
(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4}
}\| a\| _r\omega _{n-1}^{1-\frac{1}{r}}\epsilon ^{-(
n-4) +2-\frac{n}{r}} \\
&\quad \times \Big( I_{\frac{( n-2) r}{r-1}}^{1+\frac{n-2}{2}.\frac{r-1
}{r}}+o(\epsilon ^2)\Big) .
\end{align*}
Letting
\begin{equation}
A=K_0^{n/4}\frac{(n-4)^{\frac{n}{4}+1}\times ( \omega
_{n-1}) ^{\frac{r-1}{r}}}{2^{\frac{r-1}{r}}}(n(n^2-4))^{\frac{n-4}{4}
}\Big( I_{\frac{( n-2) r}{r-1}}^{\frac{n-2}{2}+\frac{r}{r-1}
}\Big) ^{\frac{r-1}{r}}    \label{20}
\end{equation}
we obtain
\[
\int_{M}a(x)| \nabla u_{\epsilon }| ^2dv_{g}\leq
\epsilon ^{2-\frac{n}{r}}\theta ^{-n\frac{r}{r-1}}
\frac{A}{K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\| a\|_r( 1+o(\epsilon ^2)) .
\]
Now we compute
\[
\int_{M}b(x)u_{\epsilon }^2dv_{g}=\int_{B(x_0,\delta
)}b(x)u_{\epsilon }^2dv_{g}+\int_{B(x_0,2\delta )-B(x_0,\delta )}b(x)u_{\epsilon }^2dv_{g}
\]
and since $b\in L^{s}(M)$ with $s>\frac{n}{4}$, we have
\[
\int_{M}b(x)u_{\epsilon }^2dv_{g}\leq \| b\|
_s\| u_{\epsilon }\| _{\frac{2s}{s-1}}^2.
\]
Independently,
\begin{align*}
\| u_{\epsilon }\| _{\frac{2s}{s-1},B(x_0,\delta )}^2
&=\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{
\frac{n-4}{4}}\\
&\quad\times \Big( \int_{0}^{\delta }\frac{\rho ^{n-1}}{(( \rho
\theta ) ^2+\epsilon ^2)^{\frac{( n-4) s}{(s-1)}}}
\Big( \int_{S(r)}\sqrt{| g(x)| }d\Omega \Big) dr\Big) ^{\frac{s-1}{s}}
\end{align*}
and
\[
\int_{S(r)}\sqrt{| g(x)| }d\Omega =\omega
_{n-1}\Big( 1-\frac{S_{g}(x_0)}{6n}\rho ^2+o(\rho ^2)\Big).
\]
Consequently,
\begin{align*}
\| u_{\epsilon }\| _{\frac{2s}{s-1},B(x_0,\delta)}^2
&=\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4}}\\
\omega _{n-1}^{\frac{s-1}{s}}
&\quad\times \Big(\int_{0}^{\delta }\frac{\rho ^{n-1}}{(( \rho \theta )
^2+\epsilon ^2)^{\frac{( n-4) s}{(s-1)}}}\Big( 1-\frac{
S_{g}(x_0)}{6n}\rho ^2+o(\rho ^2)\Big) d\rho \Big) ^{\frac{s-1}{s}}.
\end{align*}
And putting $t=(\rho \theta/\epsilon )^2$ , we obtain
\begin{align*}
\| u_{\epsilon }\| _{\frac{2s}{s-1},B(x_0,\delta)}^2
&=\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4}}
( \omega _{n-1}) ^{\frac{s-1}{s}}\epsilon ^{-n+4+4-
\frac{n}{s}}\\
&\quad\times
\Big(\frac{\epsilon ^{n}\theta ^{-n}}{2}\int_{0}^{(\frac{\delta \theta }{
\epsilon })^2}\frac{t^{\frac{n}{2}-1}}{(t+1)^{\frac{( n-4) s}{
(s-1)}}}dt\\
&\quad -\frac{\theta ^{-n-2}S_{g}(x_0)}{12n}\epsilon
^{n+2}\int_{0}^{(\frac{\delta \theta }{\epsilon })^2}\frac{t^{\frac{n}{2}}
}{(t+1)^{\frac{( n-4) s}{(s-1)}}}dt+o(\epsilon ^{n+2})\Big) ^{\frac{s-1}{s}}.
\end{align*}
Letting $\epsilon \to 0$, we obtain
\begin{align*}
\| u_{\epsilon }\| _{\frac{2s}{s-1},B( x_0,\delta ) }^2
&=\Big(\frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}
\Big) ^{\frac{n-4}{4}}( \omega _{n-1}) ^{\frac{s-1}{s}}\epsilon
^{-n+4+4-\frac{n}{s}}\\
&\quad \times \theta ^{-n\frac{s}{s-1}}(\frac{\epsilon ^{n}}{2})^{\frac{s-1}{s}
}\Big( \int_{0}^{+\infty }\frac{t^{\frac{n}{2}}}{(t+1)^{\frac{(
n-4) s}{(s-1)}}}dt\\
&\quad -\frac{S_{g}(x_0)}{12n}\epsilon ^2\theta
^{-2}\int_{0}^{+\infty }\frac{t^{\frac{n}{2}+1}}{(t+1)^{\frac{(
n-4) s}{(s-1)}}}dt+o(\epsilon ^2)\Big) ^{\frac{s-1}{s}}.
\end{align*}
Hence
\begin{align*}
&\| u_{\epsilon }\| _{\frac{2s}{s-1},B(x_0,\delta)}^2\\
&=\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4}}
 ( \omega _{n-1}) ^{\frac{s-1}{s}}\epsilon ^{-n+4+4-
\frac{n}{s}}\theta ^{-n\frac{s}{s-1}}(\frac{\epsilon ^{n}}{2})^{\frac{s-1}{s}
}\\
&\quad\times \Big( \int_{0}^{+\infty }\frac{t^{\frac{n}{2}}}{(t+1)^{\frac{(
n-4) s}{(s-1)}}}dt-\theta ^{-2}\frac{S_{g}(x_0)}{12n}\epsilon
^2\int_{0}^{+\infty }\frac{t^{\frac{n}{2}+1}}{(t+1)^{\frac{(
n-4) s}{(s-1)}}}dt+o(\epsilon ^2)\Big) ^{\frac{s-1}{s}},
\end{align*}
or
\begin{align*}
\| u_{\epsilon }\| _{\frac{2s}{s-1}}^2
&=\Big( \frac{
(n-4)n(n^2-4)}{f(x_0)}\Big) ^{\frac{n-4}{4}}
\Big( \frac{\omega _{n-1}}{2}\Big) ^{\frac{s-1}{s}}
\epsilon ^{4-\frac{n}{s}}\theta ^{-n\frac{s}{s-1}}
\\
&\quad \times \Big[ ( I_{\frac{( n-4) s}{(s-1)}}^{\frac{n}{2}
}) ^{\frac{s-1}{s}}-\frac{\theta ^{-2}(s-1)S_{g}(x_0)}{12n\text{
}s}( I_{\frac{( n-4) s}{(s-1)}}^{\frac{n}{2}}) ^{-
\frac{1}{s}}I_{\frac{( n-4) s}{(s-1)}}^{\frac{n}{2}+1}\epsilon
^2+o(\epsilon ^2)\Big]
\end{align*}
Finally, by the same method as in equality \eqref{19}, we obtain
\begin{align*}
&\int_{M}b(x)u_{\epsilon }^2dv_{g}\\
&\leq \| b\| _s( \frac{(n-4)n(n^2-4)}{f(x_0)}) ^{\frac{n-4}{4}}(\frac{\omega
_{n-1}}{2})^{\frac{s-1}{s}}\epsilon ^{4-\frac{n}{s}}\theta ^{-n\frac{s}{s-1}}
\Big( \Big( I_{\frac{( n-4) s}{(s-1)}}^{\frac{n}{2}
}\Big) ^{\frac{s-1}{s}}+o(\epsilon ^2)\Big).
\end{align*}
Putting
\begin{equation}
B=K_0^{n/4}((n-4)n(n^2-4))^{\frac{n-4}{4}}(\frac{\omega
_{n-1}}{2})^{\frac{s-1}{s}}\Big( I_{\frac{( n-4) s}{(s-1)}}^{
\frac{n}{2}}\Big) ^{\frac{s-1}{s}}    \label{21}
\end{equation}
we obtain
\[
\int_{M}b(x)u_{\epsilon }^2dv_{g}\leq \epsilon ^{4-\frac{n}{s}}\theta ^{-n
\frac{s}{s-1}}\frac{\| b\| _sB}{\text{ }K_0^{\frac{
n}{4}}(f(x_0))^{\frac{n-4}{4}}}( 1+o(\epsilon ^2)) .
\]
The computation of $\int_{M}( \Delta u_{\epsilon }) ^2dv_{g}$
is well known see for example (\cite{11})\ and is given by
\[
\int_{M}( \Delta u_{\epsilon }) ^2dv_{g}=\frac{\theta ^{-n}}{
K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\Big( 1-\frac{
n^2+4n-20}{6(n^2-4)(n-6)}S_{g}(x_0)\epsilon ^2+o(\epsilon
^2)\Big) .
\]
Summarizing,  we obtain
\begin{align*}
&\int_{M}( \Delta u_{\epsilon }) ^2-a(x)| \nabla
u_{\epsilon }| ^2+b(x)u_{\epsilon }^2dv_{g}\\
&\leq \frac{\theta ^{-n}}{K_0^{n/4}f(x_0)^{\frac{n-4}{4}}}
\Big( 1+\epsilon ^{2-\frac{n}{r}}\theta ^{-\frac{n}{r-1}}A\|
a\| _r+\epsilon ^{4-\frac{n}{s}}\theta ^{-\frac{n}{s-1}
}B\| b\| _s\\
&\quad -\frac{n^2+4n-20}{6(n^2-4)(n-6)} S_{g}(x_0)\epsilon ^2
+o(\epsilon ^2)\Big) .
\end{align*}
Now, we have
\begin{align*}
J_{\lambda }( tu_{\epsilon })
&\leq J_0( tu_{\epsilon})
 =\frac{t^2}{2}\| u_{\epsilon }\| ^2-\frac{t^{N}
}{N}\int_{M}f(x)| u_{\epsilon }(x)| ^{N}dv_{g}
\\
&\leq \frac{\theta ^{-n}}{K_0^{n/4}f(x_0)^{\frac{n-4}{4}
}}\Big\{ \frac{1}{2}t^2( 1+\epsilon ^{2-\frac{n}{r}}\theta ^{-\frac{n
}{r-1}}A\| a\| _r+\epsilon ^{4-\frac{n}{s}}\theta ^{-
\frac{n}{s-1}}B\| b\| _s) -\frac{t^{N}}{N}\\
&\quad   +\Big[ \Big( \frac{\Delta f(x_0)}{2( n-2) f(x_0)}+
\frac{S_{g}( x_0) }{6( n-1) }\Big) \frac{t^{N}}{N}-
\frac{1}{2}t^2\frac{n^2+4n-20}{6( n^2-4) ( n-6) }
S_{g}( x_0) \Big] \epsilon ^2\Big\}\\
&\quad +o( \epsilon ^2)
\end{align*}
and letting $\epsilon $ be small enough so that
\[
1+\epsilon ^{2-\frac{n}{r}}\theta ^{-\frac{n}{r-1}}A\| a\|
_r+\epsilon ^{4-\frac{n}{s}}\theta ^{-\frac{n}{s-1}}B\|
b\| _s\leq ( 1+\| a\| _r+\|
b\| _s) ^{\frac{4}{n}}
\]
and since the function $\varphi (t)=\alpha \frac{t^2}{2}-\frac{t^{N}}{N}$,
with $\alpha >0$ and $t>0$, attains its maximum at
$t_0=\alpha ^{\frac{1}{N-2}}$ and
\[
\varphi (t_0)=\frac{2}{n}\alpha ^{n/4}.
\]
Consequently,
\begin{align*}
J_{\lambda }( tu_{\epsilon })
&\leq \frac{2\theta ^{-n}}{nK_0^{n/4}f(x_0)^{\frac{n-4}{4}}}
\Big\{ 1+\|a\| _r+\| b\| _s
 +\Big[ \Big( \frac{\Delta f(x_0)}{2( n-2) f(x_0)}+
\frac{S_{g}( x_0) }{6( n-1) }\Big) \frac{t_0^{N}}{N}\\
&\quad -\frac{1}{2}t_0^2\frac{n^2+4n-20}{6( n^2-4) (
n-6) }S_{g}( x_0) \Big] \epsilon ^2\Big\}
+o( \epsilon ^2) .
\end{align*}
Taking into account the value of $\theta $ and putting
\[
R(t)=\Big( \frac{\Delta f(x_0)}{2( n-2) f(x_0)}+\frac{
S_{g}( x_0) }{6( n-1) }\Big) \frac{t^{N}}{N}-\frac{
1}{2}\frac{n^2+4n-20}{6( n^2-4) ( n-6) }
S_{g}( x_0) t^2
\]
we obtain
\[
\sup_{t\geq 0}J_{\lambda }( tu_{\epsilon })
<\frac{2}{nK_0^{n/4}( \max_{x\in M}f(x)) ^{\frac{n}{4}-1}}
\]
provided that $R(t_0)<0$; i.e.,
\[
\frac{\Delta f(x_0)}{f( x_0) }<
\Big( \frac{n(n^2+4n-20) }{3( n+2) ( n-4) ( n-6) }
\frac{1}{( 1+\| a\| _r+\| b\|_s) ^{n/4}}-\frac{n-2}{3( n-1)}\Big)
S_{g}( x_0) .
\]
Which completes the proof.
\end{proof}

\subsubsection{Application to compact Riemannian manifolds of dimension
 $n=6$}

\begin{theorem}
In case $n=6$, we suppose that at a point $x_0$ where $f$ attains its
maximum $S_{g}( x_0) >0$. Then the equation \eqref{1} has a non
trivial solution.
\end{theorem}

\begin{proof}
The same calculations as in case $n>6$ gives us
\[
\int_{M}f(x)| u_{\epsilon }(x)| ^{N}dv_{g}
=\frac{\theta ^{-n}}{K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}
\Big( 1-\Big(
\frac{\Delta f(x_0)}{2(n-2)f(x_0)}+\frac{S_{g}(x_0)}{
6(n-2)}\Big)\epsilon ^2+o(\epsilon ^2)\Big) .
\]
Also, we have
\[
\int_{M}a(x)| \nabla u_{\epsilon }| ^2dv_{g}\leq \frac{
\| a\| _rA}{\text{ }K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\epsilon ^{2-\frac{n}{r}\theta ^{-\frac{r}{r-1}}}(
1+o( \epsilon ^2) )
\]
and
\[
\int_{M}b(x)u_{\epsilon }^2dv_{g}\leq \frac{\| b\| _sB}{
K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\epsilon ^{4-\frac{n
}{s}}\theta ^{-\frac{s}{s-1}}+( 1+o( \epsilon ^2) ).
\]
where $A$ and $B$ are given by \eqref{20} and \eqref{21} respectively for
$n=6$. The computations of the term
$\int_{M}( \Delta u_{\epsilon}) ^2dv_{g}$ are well known
(see for example \cite{11})
\begin{align*}
&\int_{M}( \Delta u_{\epsilon }) ^2dv(g)\\
&=\theta ^{-n}(n-4)^2\Big(
\frac{(n-4)n(n^2-4)}{f(x_0)}\Big)^{\frac{n-4}{4}}\frac{\omega _{n-1}}{2}\\
&\quad \times \Big( \frac{n(n+2)(n-2)}{(n-4)}I_{n}^{\frac{n}{2}-1}-\frac{2}{n}
\theta ^{-2}S_{g}(x_0)\epsilon ^2\log (\frac{1}{\epsilon ^2}
)+O(\epsilon ^2)\Big) .
\end{align*}
\[
\int_{M}( \Delta u_{\epsilon }) ^2dv_{g}
=\frac{\theta ^{-n}}{K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}
\Big( 1-\frac{
2( n-4) }{n^2(n^2-4)I_{n}^{\frac{n}{2}-1}}S_{g}(x_0)\epsilon ^2
\log ( \frac{1}{\epsilon ^2}) +O(\epsilon^2)\Big) .
\]
Now summarizing and letting $\epsilon $ so that
\[
1+\epsilon ^{2-\frac{n}{r}}\theta ^{-\frac{n}{r-1}}A\| b\|
_s+\epsilon ^{4-\frac{n}{s}}\theta ^{-\frac{n}{s-1}}B\|
a\| _r\leq ( 1+\| a\| _r+\|
b\| _s) ^{\frac{4}{n}}
\]
we obtain
\begin{align*}
J_{\lambda }( u_{\epsilon })
&\leq \frac{1}{2}\|u_{\epsilon }\| ^2-\frac{1}{N}\int_{M}f(x)| u_{\epsilon
}(x)| ^{N}dv_{g} \\
&\leq \frac{\theta ^{-n}}{\text{ }K_0^{n/4}(f(x_0))^{
\frac{n-4}{4}}}\Big[ \frac{t^2}{2}( 1+\| a\|
_r+\| b\| _s) ^{1-\frac{4}{n}}-\frac{t^{N}}{N} \\
&\quad  -\frac{n-4}{n^2( n^2-4) I_{n}^{\frac{n}{2}-1}}\theta
^{-2}S_{g}(x_0)t^2\epsilon ^2\log ( \frac{1}{\epsilon ^2}
) \Big] +O(\epsilon ^2).
\end{align*}
The same arguments as in the case $n>6$ allow us to infer that
\[
\max_{t\geq 0}J_{\lambda }( tu_{\epsilon }) <\frac{2}{n\text{ }
K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}
\]
if
$S_{g}(x_0)>0$.
Which completes the proof.
\end{proof}

\subsection*{Acknowledgment}
The authors want to thank the editor and the anonymous referee 
for their helful comments and suggestions.

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\end{document}