\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 147, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2004/147\hfil Generalized scalar curvature type equation]
{Generalized scalar curvature type equation on complete Riemannian manifolds}

\author[M. Benalili, Y. Maliki\hfil EJDE-2004/147\hfilneg]
{Mohammed Benalili, Youssef Maliki} % in alphabetical order

\address{Mohammed Benalili \hfill\break 
Facult\'{e} des Sciences, Dept. de
Math\'{e}matiques, B. P. 119, Universit\'{e} Aboubekr Belka\"{\i}d, Tlemcen,
Algerie}
\email{m\_benalili@mail.univ-tlemcen.dz}

\address{Youssef Maliki \hfill\break 
Facult\'{e} des Sciences, Dept. de
Math\'{e}matiques, B. P. 119, Universit\'{e} Aboubekr Belka\"{\i}d, Tlemcen,
Algerie}
\email{m-youssef@mail.univ-tlemcen.dz}

\date{}
\thanks{Submitted November 26, 2004. Published December 13, 2004.}
\subjclass[2000]{58J05}
\keywords{Complete manifolds; p-Laplacian; critical Sobolev exponent}

\begin{abstract}
  In this work, we investigate positive solutions for a quasilinear 
  elliptic equation on complete manifold $M$. This equation extends
  to the $p$-Laplacian the equation of the prescribed scalar curvature. 
  A minimizing sequence is constructed which converges to a non trivial 
  solution belonging to $C^{1,\alpha }(K)$ for any compact set 
  $K\subset M$ and some $\alpha \in (0,1)$.
\end{abstract}

\maketitle

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


\section{Introduction}

Let $(M,g)$ be a complete Riemannian manifold of dimension $n\geq 3$, with
bounded geometry, $R(x)$ its scalar curvature and $p\in (1,n)$. Let 
$H_{1}^{p}(M)$ be the standard Sobolev space endowed with the norm 
\begin{equation*}
\Vert u\Vert _{H_{1}^{p}(M)}=\Vert \nabla u\Vert _{L^{p}(M)}+\Vert u\Vert
_{L^{p}(M)}.
\end{equation*}
In this paper, we seek for a positive solution 
$u\in H_{1,\mathrm{loc}}^{p}(M)$ to the equation 
\begin{equation}
\Delta _{p}u+a(x)u^{p-1}=f(x)u^{p\ast -1}.  \label{e1}
\end{equation}
where $\Delta _{p}u=-div(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian of 
$u$ on $M$ and $p^{\ast }=\frac{pn}{n-p}$.

Our results extend those of Druet \cite{d1} obtained in the case of compact
manifolds. On complete Riemannian manifold conditions at infinity on $f$ 
must be added.

In the case $p=2$ and the function $a(x)=\frac{n-2}{n(n-1)}R(x)$, where 
$R(x) $ is the scalar curvature of the manifold $M$, the problem of the
existence of a positive solution of the equation \eqref{e1} is originated
from the study of pointwise conformal deformation of Riemannian metric with
prescribed scalar curvature. If in case $p=2$, $u$ is a positive solution of 
\eqref{e1} on $(M,g)$, then the scalar curvature of the pointwise conformal
metric $g'=u^{\frac{4}{n-2}}g$ \ is $\frac{4(n-1)}{n-2}f$ (cf. \cite{d1}). 
The equation \eqref{e1} is referred as the generalized scalar
curvature type equation.

Our main result in this paper is as follows.

\begin{theorem} \label{thm1}
Let $( M,g) $ be a complete non-compact Riemannian $n$-manifold
with $n\geq 3$ , $1<p<n$ such that $p^{2}<n$. Let $a,\;f\in C^{\infty }(M)$
be real valued function on $M$. Suppose that operator
$L_{p}u=\Delta_{p}u+a(x)u^{p-1}$ is coercive.
Under the following assumptions:
\begin{enumerate}
\item At a point $x_{o}$ where $f$ is maximal, we are in one of the following
cases
\begin{itemize}
\item[(i)]  $p<2$,  $n>3p-2$ and $a(x_{o})<0$

\item[(ii)] $p=2$ and $\frac{8( n-1) }{( n-2) (
n-4) }a(x_{o})<\frac{-\Delta f(x_{o})}{f(x_{o})}+\frac{2R(x_{o})}{n-4}$

\item[(iii)] $p>2$ and $( \frac{n-3p+2}{p}) \frac{\Delta f(x_{o})}{
f(x_{o})}<R(x_{o})$.
\end{itemize}
\item  $( M,g) $ is of bounded geometry, that is: $Ricci >-c$,  where
$c\geq 0$ is a constant, and the injectivity radius is strictly positive.

\item There exists a constant $C>0$ such that
\[
|\nabla f| \leq Cf,|\nabla ^{2}f| \leq
Cf,\int_{M}|a| ^{\frac{n}{p}}dv_{g}\leq C\int_{M}fdv_{g}<\infty
\text{ and }\int_{M}f^{p/p\ast}dv_{g}<\infty .
\]
\item The functions $a$ and $f$ are bounded and $f$ is strictly positive.
\end{enumerate}
Then, there exists a positive solution $u\in H_{1,loc}^{p}(M)$ of
\eqref{e1} such that $u\in C^{^{1,\alpha }}(K)$ on any compact set  $K$
of $M$ for some $\alpha \in ( 0,1)$.
\end{theorem}

This article is organized as follows: in the second section we construct a
sequence of minimizing weak solutions, in the third section we give
sufficient geometric conditions to guarantee the strong convergence of the
minimizing sequence. Using the Aubin's test functions, we show in the last
section that these geometric conditions are satisfied.

\section{Convergence of the minimizing sequence}

In this section, we construct a sequence of weak solutions for \eqref{e1}.
The following theorem has been  proved in \cite{d1}.

\begin{theorem} \label{thm2}
Let $( M,g) $ be a Riemannian compact manifold $1<p<n$ such that
$p^{2}<n$,  and let $a,\;f\in C^{\infty }(M)$ be real functions on $M$. We
assume that the operator $L_{p}u=\Delta_{p}u+a(x)u^{p-1}$ is coercive.
If at a point $x_{o}$ where $f$ is maximal, we have one of the following
cases
\begin{itemize}
\item[(i)] $p<2$,  $n>3p-2$ and $a(x_{o})<0$

\item[(ii)] $p=2$ and $\frac{8( n-1) }{( n-2) (
n-4) }a(x_{o})<\frac{-\Delta f(p)}{f(p)}+\frac{2R(x_{o})}{n-4}$

\item[(iii)] $p>2$ and $( \frac{n-3p+2}{p}) \frac{\Delta f(x_{o})}{
f(x_{o})}<R(x_{o}).$
\end{itemize}
Then, there exists a positive solution $u\in H_{1}^{p}(M)$ of
\eqref{e1} such that $u\in C^{^{1,\alpha }}(M)$ for some
$\alpha \in (0,1)$.
\end{theorem}

Let $\Omega _{j}$ be an exhaustion of the complete manifold $M$ \ by compact
manifolds with smooth boundary such that 
$\Omega _{j}\subset \overset{o}{\Omega }_{j+1}$. Let $u_{j}$ be the 
minimizer given by Theorem \ref{thm2} for 
\begin{equation}
\begin{gathered} \Delta_{p}u_{j}+a(x)u_{j}^{p-1}=\mu
(\Omega_{j})fu_{j}^{p\ast -1} \quad \text{in }\Omega_{j} \\ u_{j}>0 \quad
\text{in }\Omega_{j} \\ u_{j}=0 \quad \text{on }\partial \Omega_{j}\,.
\end{gathered}  \label{e2}
\end{equation}
By the monotone decreasness of $\mu (\Omega _{j})$ and the coercivity of the
operator $L_{p}u=\Delta _{p}u+a(x)u^{p-1}$, we have 
\begin{equation}
\Vert u\Vert _{H_{1}^{p}(\Omega _{j})}\leq \frac{1}{c}\mu (\Omega _{1})
\label{e3}
\end{equation}
where $c>0$ is a constant. Since \eqref{e3} implies the boundedness of 
$\{u_{i}\}$ in $H_{1}^{p}(M)$, we can choose a subsequence of $\{u_{i}\}$
still denoted $\{u_{i}\}$ such that $u_{i}\rightarrow u$ weakly in 
$H_{1}^{p}(M)$

\begin{proposition} \label{prop1}
The sequence $\{ u_{i}\} $  converges weakly on every compact
set $K$ of $M$ to a solution $u\in C^{1,\alpha }(K)$ of
\begin{equation}
\begin{gathered}
\Delta_{p}u+a(x)u^{p-1}=fu^{p\ast -1} \quad \text{in }K \\
u>0 \quad \text{in }K \\
u=0 \quad \text{on }\partial K
\end{gathered}  \label{e4}
\end{equation}
for some $\alpha \in (0,1)$.
\end{proposition}

To prove the boundedness of $\{ u_{i}\} $in $C^{1,\alpha }(K)$, we use
propositions from the paper of Druet \cite{d1} which have their origin in
Tolksdorf \cite{t2} Guedda and Veron \cite{g1} and Vazquez \cite{v1}.

\begin{proposition} \label{prop2}
Let $( M,g) $ be a compact Riemannian $n$-manifold. Assume that
$u\in H_{1}^{p}( M) $ is a solution of $\Delta_{p}u+a(x)u^{p-1}=f$,
 where $n\geq 2$,  $1<p<n,\;a(x)\in L^{\frac{n}{p}}(M)$
and $f\in L^{\frac{n}{p}}(M)$,  then $u\in L^{t}(M)$ for $t\in [1,\infty )$.
\end{proposition}

\begin{proposition} \label{prop3}
Let $( M,g) $ be a compact Riemannian $n$-manifold. Assume
$n\geq 2$, $1<p<n$, $f\in L^{s}(M)$ for some $s>\frac{n}{p}$ and
$u\in H_{1}^{p}(M) $ is a solution of $\Delta_{p}u=f$ on $M$.
Then $u\in L^{\infty}(M)$.
\end{proposition}

\begin{proposition} \label{prop4}
Let $( M,g) $ be a compact Riemannian $n$-manifold and
$h(x,r) \in C^{o}( M\times R) $. Assume $n\geq 2$,  $1<p<n$
and $\forall ( x,r) \in M\times R$,   $|h( x,r)| \leq C|r| ^{p\ast -1}+D$.
If $u\in H_{1}^{p}( M) $ is a solution of $\Delta_{p}u+h(
x,u) =0$,  then $u\in C^{^{1,\alpha }}(M)$. Moreover
$\| u\|_{C^{^{1,\alpha }}(M)}\leq \tilde{c}$, where $\tilde{c}$ is a
constant depending only on $\| u\|_{L^{\infty }(M)}$ and $\| h(
x,r) \|_{_{L^{\infty }(M)}}$.
\end{proposition}

\begin{proof}[Proof of Proposition \protect\ref{prop1}]
First we show that the sequence $\{u_{j}\} $ is bounded in $L^{t}(K)$ for
any $t\in [1,+\infty) $. Involving Proposition \ref{prop2}, we have only to
check that the sequence $\{ a(x)-fu_{j}^{p\ast -p}\} $ is bounded in 
$L^{\frac{n}{p}}(K)$. We have 
\begin{align*}
\int_{K}|a(x)-fu_{j}^{p\ast -p}| ^{\frac{n}{p}}dv_{g} &\leq 2^{\frac{n}{p}
-1}\int_{K}( |a(x)| ^{\frac{n}{p}}+| f| ^{\frac{n}{p}}u_{j}^{p\ast }) dv_{g}
\\
&= 2^{\frac{n}{p}-1}\big[( \| a\|_{\frac{n}{p} }^{K}) ^{\frac{n}{p}}+( \|
f\|_{\infty }^{K}) ^{ \frac{n}{p}}( \| u_{j}\|_{p\ast }^{K}) ^{p\ast }\big]
\end{align*}
where $\| u\|_{p}^{K}=\big( \int_{K}|u|^{p}dv_{g}\big) ^{1/p}$. Since by the
relation(3) the sequence$\{ u_{j}\} $ is bounded in $L^{p}(K)$, so is in 
$L^{p\ast }(K)$, we have the desired conclusion.

Next, we show that $\{ u_{j}\} $ is bounded in $L^{\infty }(K)$. According
to Proposition \ref{prop3}, we have to show that the sequence $\{ g_{j}\} $
given by $g_{j}(x)=-a(x)u_{j}(x)^{p-1}+f(x)u_{j}(x)^{p\ast-1}$, is bounded
in $L^{s}(K)$ for some $s>\frac{n}{p}$. But this fact is a consequence of
proposition \ref{prop2}.

Finally, we take $h(x,u_{j})=a(x)u_{j}^{p-1}-f(x)u_{j}^{p\ast -1}$, and
since by assumption the functions $a$ and $f$ are bounded on the manifold $M$,
 one has the boundness of the sequence $\{h(x,u_{j}(x)\}$ in the compact
set $K$.

By proposition \ref{prop4}, $u_{j}\in C^{1,\alpha }(K)$ and 
$\Vert u_{j}\Vert _{K}^{1,\alpha }\leq c(p,n,K,\Vert g_{j}
\Vert _{L^{\infty }(K)})$.
 The boundedness of $\{u_{j}\}$ in $L^{\infty }(K)$ implies that 
$\{g_{j}\} $ and  $C(p,n,K,\Vert g_{j}\Vert _{L^{\infty }(K)})$ are
bounded. Consequently, $\{u_{j}\}$ is bounded in $C^{1,\alpha}(K)$. 
So by Arzela-Ascoli theorem $\{u_{j}\}$ converges uniformly towards a
weak solution $u$ of \eqref{e4} on each compact set.
\end{proof}

\section{Strong convergence}

In this section, we have to show that the solution $u$ is not trivial. To
achieve this task, we give sufficient conditions that guarantee the strong
convergence of minimizers constructed in the previous section. Let $K$ be
any compact set of the complete manifold $M$, $2K$ a compact set containing 
$K$ and $\eta \in C^{\infty }(M)$ be the function 
\begin{equation*}
\eta (x)= 
\begin{cases}
0 & \text{on }K \\ 
1 & \text{on }M-2K\,.
\end{cases}
\end{equation*}
Let $k>1$ and $\{ u_{q}\} $ be the sequence of minimizers given by
Proposition \ref{prop1} and $\| .\|_{p}$ be the $L^{p}(M)$-norm. we are
going to estimate the ratio 
$\| \nabla ( \eta f^{\frac{1}{ p\ast }}u_{q}^{\frac{k+p-1}{p}}) \|_{p}$.

Letting $\{ \Omega_{k}\} $ be the exhaustion, of the complete manifold $M$,
considered in the previous section. Denote by $\Lambda_{k}=\{ u\in
H_{1}^{p}(\Omega_{k}):\;\int_{\Omega_{k}}f|u| ^{p\ast }dv_{g} =1\} $ and 
$I_{k}(u)$ the functional $I_{k}(u)=\int_{\Omega_{k}}( |\nabla u| ^{p}+|u|
^{p}) dv_{g}$.

\begin{proposition} \label{prop5}
Under the conditions (2), (3), (4), of Theorem \ref{thm1} and
\[
( \sup_{M-K}f(x)) ^{p/p\ast} \inf_{u\in \Lambda_{k}}I_{k}(u)<K(n,p)^{-p},
\]
the ratio $\| \nabla ( \eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}}) \|_{p}$
is bounded.
\end{proposition}

\begin{proof}
For $p\geq 2$, using Simon's inequality \cite{s1}, that is to say: for any
vector fields $X$ and $Y$ on the manifold $M$, 
\begin{equation*}
|X+Y|^{p}\leq C_{p}\left\langle |X|^{p-2}X+|Y|^{p-2}Y,X+Y\right\rangle
\end{equation*}
where $C_{p}$ is a constant depending on $p$ and $\left\langle
.,.\right\rangle $ denoting the metric. We get 
\begin{align*}
& \Vert \nabla (\eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}})\Vert
_{p}^{p} \\
& =\int_{M}|\nabla (\eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}})|^{p}dv_{g} \\
& =\int_{M}|(u_{q}^{\frac{k+p-1}{p}}\nabla (\eta f^{\frac{1}{p\ast }})
+\frac{k+p-1}{p}(\eta f^{\frac{1}{p\ast }})u_{q}^{\frac{k-1}{P}}\nabla
u_{q})|^{p}dv_{g} \\
& \leq C_{p}\int_{M}\Big[u_{q}^{(\frac{k+p-1}{p})(p-1)}|\nabla 
(\eta f^{\frac{1}{p\ast }})|^{p-2}\nabla (\eta f^{\frac{1}{p\ast }}) \\
& \quad +(\tfrac{k+p-1}{p})^{p-1}(\eta f^{\frac{1}{p\ast }})^{p-1}u_{q}
^{(\frac{k-1}{p})(p-1)}|\nabla u_{q}|^{p-2}\nabla u_{q}\Big] \\
& \quad \times \big[u_{q}^{\frac{k+p-1}{p}}\nabla (\eta f^{\frac{1}{p\ast }
})+\tfrac{k+p-1}{p}(\eta f^{\frac{1}{p\ast }})^{p-1}u_{q}^{\frac{k-1}{p}
}\nabla u_{q}\big]dv_{g} \\
& =C_{p}\Big[\int_{M}u_{q}^{k+p-1}|\nabla (\eta f^{\frac{1}{p\ast }
})|^{p}dv_{g}+(\tfrac{k+p-1}{p})^{p}\int_{M}(\eta f^{\frac{1}{p\ast }
})^{p}u_{q}^{k-1}|\nabla u_{q}|^{p}dv_{g} \\
& \quad \times \tfrac{k+p-1}{p}\int_{M}\eta f^{\frac{1}{p\ast }
}u_{q}^{k+p-2}|\nabla (\eta f^{\frac{1}{p\ast }})|^{p-2}\langle \nabla (\eta
f^{\frac{1}{p\ast }}),\nabla u_{q}\rangle dv_{g} \\
& \quad +(\tfrac{k+p-1}{p})^{p-1}\int_{M}(\eta f^{\frac{1}{p\ast }
})^{p-1}u_{q}^{k}|\nabla u_{q}|^{p-2}\langle \nabla (\eta f^{\frac{1}{p\ast }
}),\nabla u_{q}\rangle dv_{g}].
\end{align*}
On the other hand, 
\begin{align*}
\int_{M}\eta ^{p}f^{p/p\ast }u_{q}^{k}\Delta _{p}u_{q}dv_{g}& =k\int_{M}\eta
^{p}f^{p/p\ast }u_{q}^{k-1}|\nabla u_{q}|^{p}dv_{g} \\
& \quad +p\int_{M}(\eta f^{\frac{1}{p\ast }})^{p-1}u_{q}^{k}|\nabla
u_{q}|^{p-2}\langle \nabla u_{q},\nabla (\eta f^{\frac{1}{p\ast }})\rangle
dv_{g}
\end{align*}
and 
\begin{align*}
& \int_{M}\eta f^{\frac{1}{p\ast }}u_{q}^{k+p-1}\Delta _{p}(\eta 
f^{\frac{1}{p\ast }})dv_{g} \\
& =\int_{M}u_{q}^{k+p-1}|\nabla (\eta f^{\frac{1}{p\ast }})|^{p}dv_{g} \\
& \quad +(k+p-1)\int_{M}(\eta f^{\frac{1}{p\ast }})u_{q}^{k+p-2}|\nabla
(\eta f^{\frac{1}{p\ast }})|^{p-2}\langle \nabla u_{q},\nabla \eta 
f^{\frac{1}{p\ast }}\rangle dv_{g}
\end{align*}
so 
\begin{align*}
& \Vert \nabla (\eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}})\Vert
_{p}^{p} \\
& \leq C_{p}\Big[\int_{M}\eta f^{\frac{1}{p\ast }}u_{q}^{k+p-1}\Delta
_{p}(\eta f^{\frac{1}{p\ast }})dv_{g}+\frac{1}{k}(\frac{k+p-1}{p}
)^{p}\int_{M}\eta ^{p}f^{p/p\ast }u_{q}^{k}\Delta _{p}u_{q}dv_{g} \\
& \quad -\tfrac{p-1}{p}(k+p-1)\int_{M}\eta f^{\frac{1}{p\ast }
}u_{q}^{k+p-2}|\nabla (\eta f^{\frac{1}{p\ast }})|^{p-2}\langle \nabla
u_{q},\nabla (\eta f^{\frac{1}{p\ast }})\rangle dv_{g} \\
& \quad -(\tfrac{k+p-1}{p})^{p-1}\tfrac{p-1}{k}\int_{M}(\eta 
f^{\frac{1}{p\ast }})^{p-1}u_{q}^{k}|\nabla u_{q}|^{p-2}\langle \nabla u_{q},\nabla
(\eta f^{\frac{1}{p\ast }})\rangle dv_{g}\Big].
\end{align*}
Multiplying \eqref{e1} by $(\eta f^{\frac{1}{p\ast }})^{p}u_{q}^{k}$ and
integrating over $M$, we get 
\begin{equation}
\begin{aligned} &\int_{M}( \eta f^{\frac{1}{p\ast }}) ^{p}u_{q}^{k}\Delta
_{p}u_{q}dv_{g}\\ &=-\int_{M}a(x)( \eta f^{\frac{1}{p\ast }})
^{p}u_{q}^{k+p-1}dv_{g}+\mu (_{\Omega_{q}})\int_{M}( \eta f^{\frac{1}{ p\ast
}}) ^{p}fu_{q}^{k+p\ast -1}dv_{g}. \end{aligned}  \label{e5}
\end{equation}
Using H\"{o}lder inequality, we obtain 
\begin{equation}
\begin{aligned} &\int_{M}( \eta f^{\frac{1}{p\ast }}) ^{p}fu_{q}^{k+p\ast
-1}dv_{g}\\ &\leq ( \sup_{M-K}f) ^{p/p\ast} \Big(\int_{M-K}fu_{q}^{p\ast
}dv_{g}\Big) ^{1-\frac{p}{p\ast }} \big(\int_{M}( \eta f^{\frac{1}{p\ast
}}u_{q}^{\frac{k+p-1}{p}}) ^{p\ast }dv_{g}\Big) ^{p/p\ast}. \end{aligned}
\label{e6}
\end{equation}
The first term of the right-hand side of \eqref{e5} is estimated as 
\begin{align*}
& \int_{M}a(x)(\eta f^{\frac{1}{p\ast }})^{p}u_{q}^{k+p-1}dv_{g} \\
& \leq \Big(\int_{M-K}|a(x)|^{\frac{n}{p}}dv_{g}\Big)^{\frac{p}{n}}
\Big(\int_{M-K}(\eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}})^{p\ast }
dv_{g}\Big)^{p/p\ast }.
\end{align*}
Since by assumption 
\begin{equation*}
\big(\int_{M}|a(x)|^{\frac{n}{p}}dv_{g}\big)^{p/n}<C\int_{M}fdv_{g}<\infty ,
\end{equation*}
we choose the compact set $K$ so that 
\begin{equation*}
\int_{M-K}fdv_{g}<\frac{\varepsilon }{C}.
\end{equation*}
Then 
\begin{equation}
\begin{aligned} &\int_{M}( \eta f^{\frac{1}{p\ast }}) ^{p}u_{q}^{k}\Delta
_{p}u_{q}dv_{g} \\ &\leq \Big( \big( (\sup_{M-K}f)
^{p/p\ast}\int_{M}fu_{q}^{p\ast }) ^{1-\frac{p}{p\ast }}+\varepsilon \Big)
\Big( \int_{M}\big( \eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p} }\big)
^{p\ast }dv_{g}\Big) ^{p/p\ast}. \end{aligned}  \label{e7}
\end{equation}
On the other hand we have 
\begin{equation*}
\nabla (\eta f^{\frac{1}{p\ast }})=f^{\frac{1}{p\ast }}\nabla \eta 
+\frac{1}{p\ast }\eta f^{\frac{1}{p\ast }-1}\nabla f
\end{equation*}
and since by assumption $|\nabla f|\leq C$, we obtain 
\begin{equation*}
|\nabla \eta f^{\frac{1}{p\ast }}|\leq f^{\frac{1}{p\ast }}|\nabla \eta |+
\frac{\eta c}{p\ast }f^{\frac{1}{p\ast }}\leq Cf^{\frac{1}{p\ast }}
\end{equation*}
where $C$ is a universal constant. So 
\begin{equation}
\begin{aligned} &\int_{M}\eta f^{\frac{1}{p\ast }}u_{q}^{k+p-2}|\nabla (
\eta f^{ \frac{1}{p\ast }}) | ^{p-2}\langle \nabla ( \eta f^{ \frac{1}{p\ast
}}) ,\nabla u_{q}\rangle dv_{g}\\ &\leq C\int_{M-K}f^{ \frac{p}{p\ast
}}u_{q}^{k+p-2}|\nabla u_{q}| dv_{g}. \end{aligned}  \label{e8}
\end{equation}
Using H\"{o}lder inequality we obtain that the right-hand side of this
inequality is bounded above by 
\begin{equation}
\begin{aligned} &\int_{M-K}f^{p/p\ast}u_{q}^{k+p-2}|\nabla u_{q}| dv_{g}\\
&\leq \Big( \int_{M-K}|\nabla u_{q}| ^{p}dv_{g}\Big) ^{ \frac{1}{p}}\Big(
\int_{M-K}( f^{p/p\ast}u_{q}^{k+p-2}\Big) ^{\frac{p}{p-1}}dv_{g})
^{1-\frac{1}{p}}. \end{aligned}  \label{e9}
\end{equation}
Applying H\"{o}lder's inequality again, we get 
\begin{equation}
\begin{aligned} &\int_{M-K}( f^{p/p\ast}u_{q}^{k+p-2}) ^{\frac{p}{p-1}
}dv_{g}\\ &\leq \Big( \int_{M-K}u_{q}^{p\ast }dv_{g}\Big) ^{\frac{p(k+p-2)
}{p\ast ( p-1) }} \Big( \int_{M-K}f^{\frac{p( p-1) }{p\ast ( p-1) -p( k+p-2)
}}dv_{g}\Big) ^{1-\frac{p( k+p-2) }{p\ast ( p-1) }}\\ &\leq ( \sup_{M-K}f)
^{\frac{p( k+p-2) }{p\ast ( p-1) }}\Big( \int_{M-K}u_{q}^{p\ast }dv_{g}\Big)
^{\frac{p( k+p-2) }{p\ast ( p-1) }} \Big( \int_{M-K}f^{p/p\ast}dv_{g}\Big)
^{1-\frac{p( k+p-2) }{p\ast ( p-1) }}. \end{aligned}  \label{e10}
\end{equation}
As above, we get 
\begin{align*}
& \int_{M}(\eta f^{\frac{1}{p\ast }})^{p-1}u_{q}^{k}|\nabla
u_{q}|^{p-2}\langle \nabla u_{q},\nabla (\eta f^{\frac{1}{p\ast }})\rangle
dv_{g} \\
& \leq C\int_{M-K}f^{p/p\ast }u_{q}^{k}|\nabla u_{q}|^{p-1}dv_{g} \\
& \leq C\Big(\int_{M-K}f^{p/p\ast }u_{q}^{p\ast }\Big)^{\frac{k}{p\ast }}
\Big(\int_{M-K}f^{p/p\ast }|\nabla u_{q}|^{\frac{p\ast (p-1)}{p\ast -k}}\Big)
^{1-\frac{k}{p\ast }}.
\end{align*}
Since $\alpha =(p-1)\frac{p\ast }{p\ast -k}$, we have 
$p-\alpha =\frac{p(p\ast -k)-p\ast (p-1)}{p\ast -k}
=\frac{p\ast -pk}{p\ast -k}>0$ and 
\begin{equation}
\begin{aligned} &\int_{M-K}f^{p/p\ast}|\nabla u_{q}| ^{\alpha }dv_{g}\\
&\leq \Big( \int_{M-K}|\nabla u_{q}| ^{p}dv_{g}\Big) ^{ \frac{\alpha
}{p}}\Big( \int_{M-K}f^{\frac{p^{2}}{p\ast ( p-\alpha ) }}dv_{g}\Big)
^{1-\frac{\alpha }{p}}\\ &\leq ( \sup_{M-K}f) ^{\frac{p( p-1) }{p\ast
-pk}}\Big( \int_{M-K}|\nabla u_{q}| ^{p}dv_{g}\Big) ^{\frac{\alpha }{p}
}\Big( \int_{M-K}f^{p/p\ast}dv_{g}\Big) ^{1-\frac{\alpha }{p}}\,.
\end{aligned}  \label{e11}
\end{equation}
On the other hand, 
\begin{align*}
\Delta _{p}(\eta f^{\frac{1}{p\ast }})& =-\mathop{\rm div}(|\nabla (\eta f^{
\frac{1}{p\ast }})|^{p-2}\nabla (\eta f^{\frac{1}{p\ast }})) \\
& =|\nabla \eta f^{\frac{1}{p\ast }}|^{p-2}\Delta (\eta f^{\frac{1}{p\ast }
})-\mathop{\rm trace}\big(\nabla |\nabla (\eta f^{\frac{1}{p\ast }
})|^{p-2}\otimes \nabla (\eta f^{\frac{1}{p\ast }})\big)
\end{align*}
and 
\begin{align*}
\Delta (\eta f^{\frac{1}{p\ast }})& =f^{\frac{1}{p\ast }}\Delta \eta +\eta
\Delta f^{\frac{1}{p\ast }}-trace(\nabla \eta \otimes \nabla f^{\frac{1}{
p\ast }}) \\
& \leq f^{\frac{1}{p\ast }}\Delta \eta +\frac{1}{p\ast }(1-\frac{1}{p\ast }
)\eta f^{\frac{1}{p\ast }-2}|\nabla f|^{2}+\frac{1}{p\ast }\eta f^{\frac{1}{
p\ast }-1}\Delta f+\frac{1}{p\ast }f^{\frac{1}{p\ast }-1}|\nabla f|
\end{align*}
then 
\begin{equation*}
|\Delta (\eta f^{\frac{1}{p\ast }})|\leq Cf^{\frac{1}{p\ast }}
\end{equation*}
and 
\begin{equation*}
|\nabla (\eta f^{\frac{1}{p\ast }})|^{p-2}|\Delta (\eta f^{\frac{1}{p\ast }
})|\leq Cf^{\frac{p-1}{p\ast }}.
\end{equation*}
From 
\begin{equation*}
|\nabla |\nabla (\eta f^{\frac{1}{p\ast }})|^{p-2}|=(p-2)|\nabla (\eta f^{
\frac{1}{p\ast }})|^{p-3}|\nabla |\nabla (\eta f^{\frac{1}{p\ast }})||
\end{equation*}
and Kato's inequality, we deduce that 
\begin{equation*}
\big|\nabla |\nabla (\eta f^{\frac{1}{p\ast }})|^{p-2}\big|\leq (p-2)|\nabla
(\eta f^{\frac{1}{p\ast }})|^{p-3}|\nabla ^{2}(\eta f^{\frac{1}{p\ast }})|.
\end{equation*}
Now, since 
\begin{align*}
& \nabla ^{2}(\eta f^{\frac{1}{p\ast }}) \\
& =f^{\frac{1}{p\ast }}\nabla ^{2}\eta +\frac{2}{p\ast }f^{\frac{1}{p\ast }
-1}\nabla \eta \otimes \nabla f+\frac{1}{p\ast }(1-\frac{1}{p\ast })\eta f^{
\frac{1}{p\ast }-2}\nabla f\otimes \nabla f+\frac{1}{p\ast }\nabla ^{2}f\,,
\end{align*}
we obtain 
\begin{equation*}
\big|\nabla |\nabla (\eta f^{\frac{1}{p\ast }})|^{p-2}\big|\leq Cf^{\frac{p-1}{p\ast }}.
\end{equation*}
Finally, we get 
\begin{equation*}
|\Delta _{p}(\eta f^{\frac{1}{p\ast }})|\leq Cf^{\frac{p-1}{p\ast }}
\end{equation*}
and 
\begin{equation}
\begin{aligned} &\int_{M}\eta f^{\frac{1}{p\ast }}u_{q}^{k+p-1}\Delta_{p}(
\eta f^{ \frac{1}{p\ast }}) dv_{g}\\ &\leq
C\int_{M-K}f^{p/p\ast}u_{q}^{k+p-1}\\ &\leq C\Big( \int_{M-K}u_{q}^{p\ast
}dv_{g}\Big) ^{\frac{k+p-1}{p\ast } }\Big( \int_{M-K}f^{\frac{p\ast }{p\ast
k-p+1}}dv_{g}\Big) ^{1-\frac{k+p-1 }{p\ast }}. \end{aligned}  \label{e12}
\end{equation}
Sobolev's inequality leads to 
\begin{align*}
& \Big(\int_{M-K}u_{q}^{p\ast }dv_{g}\Big)^{p/p\ast } \\
& \leq (K(n,p)^{p}+\varepsilon )\int_{M-K}|\nabla
u_{q}|^{p}dv_{g}+A\int_{M-K}u_{q}^{p}dv_{g} \\
& \leq (K(n,p)^{p}+\varepsilon )\Big(\int_{M-K}|\nabla u_{q}|^{p}dv_{g}+
\frac{A}{K(n,p)^{p}+\varepsilon }\int_{M-K}u_{q}^{p}dv_{g}\Big).
\end{align*}
 From the coercivity of the operator $L_{p}u=-\Delta _{p}u-a(x)|u|^{p-2}u$,
we get 
\begin{align*}
& \Big(\int_{M-K}u_{q}^{p\ast }dv_{g}\Big)^{p/p\ast } \\
& \leq \frac{1}{c}(K(n,p)^{p}+\varepsilon )\max \big(1,\frac{A}{
K(n,p)^{p}+\varepsilon }\big)\int_{M-K}(|\nabla u_{q}|^{p}+u_{q}^{p})dv_{g}
\\
& \leq \tilde{C}\int_{M}(|\nabla u_{q}|^{p}+u_{q}^{p})dv_{g},
\end{align*}
where $\tilde{C}=\frac{1}{c}(K(n,p)^{p}+\varepsilon )\max \big(1,\frac{A}{
K(n,p)^{p}+\varepsilon }\big)$ and by construction of the sequence 
$\{u_{q}\} $, which has a compact support in $\Omega _{q}$, we have 
\begin{equation*}
\int_{M}(|\nabla u_{q}|^{p}+a(x)u_{q}^{p})dv_{g}=\lambda _{q}
\end{equation*}
hence 
\begin{equation*}
\Big(\int_{M-K}u_{q}^{p\ast }dv_{g}\Big)^{p/p\ast }\leq \tilde{C}\lambda
_{q}.
\end{equation*}
Since by assumption the Lagrange multipliers satisfy 
\begin{equation*}
\lambda _{q}<\frac{1}{K(n,p)^{p}(\sup_{M-K}f)^{p/p\ast }},
\end{equation*}
we have 
\begin{equation}
\Big(\int_{M-K}u_{q}^{p\ast }dv_{g}\Big)^{p/p\ast }\leq C.  \label{e13}
\end{equation}
Combining inequalities (5) to (13) we obtain 
\begin{align*}
\Vert \nabla (\eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}})\Vert
_{p}^{p}& \leq \lambda _{q}\Big(\big(\sup_{M-K}f\big)^{p/p\ast }\big(
\int_{M-K}fu_{q}^{p\ast }dv_{g}\big)^{1-\frac{p}{p\ast }}+\varepsilon \Big)
\\
& \quad \times \Big(\int_{M-K}\big(\eta f^{\frac{1}{p\ast }}u_{q}^{\frac{
k+p-1}{p}}\big)^{p\ast }dv_{g}\Big)^{p/p\ast }+C\,.
\end{align*}
Using Sobolev's inequality, this expression is bounded by 
\begin{equation}
\begin{aligned} & \lambda_{q}\Big( \big( \sup_{M-K}f\big) ^{p/p\ast}
\big(\int_{M-K}fu_{q}^{p\ast }dv_{g}\big) ^{1-\frac{p}{p\ast
}}+\varepsilon\Big)\\ &\times \Big( ( K( n,p) ^{p}+\varepsilon ) \| \nabla (
\eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}}) \|_{p}^{p}+A\| \eta
f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p} }\|_{p}^{p}\Big) + C\,,
\end{aligned}  \label{e14}
\end{equation}
where $K(n,p)$ is the best constant in the Sobolev's inequality. For the
last term in \eqref{e14}, we write 
\begin{align*}
& \int_{M-K}f^{p/p\ast }u_{q}^{k+p-1}dv_{g} \\
& \leq \Big(\int_{M-K}u_{q}^{p\ast }dv_{g}\Big)^{\frac{\kappa +p-1}{p\ast }}
\Big(\int_{M-K}f^{\frac{p}{p\ast -k-p+1}}dv_{g}\Big)^{1-\frac{\kappa +p-1}{
p\ast }} \\
& \leq \big(\sup_{M-K}f\big)^{\frac{p(1-p-k)}{p\ast ^{2}}}\Big(\int
u_{q}^{p\ast }dv_{g}\Big)^{\frac{\kappa +p-1}{p\ast }}\Big(\int f^{p/p\ast
}dv_{g}\Big)^{1-\frac{\kappa +p-1}{p\ast }}<\infty .
\end{align*}
From the assumption on the Lagrange multipliers, $\Vert \nabla (\eta f^{
\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}})\Vert _{p}^{p}$ is bounded.

In the case $1<p<2$ , the Simon's inequality writes 
\begin{equation*}
|X+Y|^{p}\leq C_{p}\left\langle |X|^{p-2}X+|Y|^{p-2}Y,X+Y\right\rangle ^{
\frac{p}{2}}(|X|^{p}+|Y|^{p})^{1-\frac{p}{2}}
\end{equation*}
where $X,Y$ are any vector fields on the manifold $M.$

Putting 
\begin{gather*}
X=u_{q}^{\frac{k+p-1}{p}}\nabla (\eta f^{\frac{1}{p\ast }}), \\
Y=\frac{k+p-1}{p}(\eta f^{\frac{1}{p\ast }})u_{q}^{\frac{k-1}{P}}\frac{k+p-1
}{p}(\eta f^{\frac{1}{p\ast }})u_{q}^{\frac{k-1}{P}}\nabla u_{q}
\end{gather*}
we get 
\begin{align*}
& \Big|u_{q}^{\frac{k+p-1}{p}}\nabla (\eta f^{\frac{1}{p\ast }})+\frac{k+p-1
}{p}(\eta f^{\frac{1}{p\ast }})u_{q}^{\frac{k-1}{P}}\nabla u_{q}\Big|
^{p}dv_{g} \\
& \leq C_{p}\Big[\Big(u_{q}^{(\frac{k+p-1}{p})(p-1)}|\nabla (\eta f^{\frac{1
}{p\ast }})|^{p-2}\nabla (\eta f^{\frac{1}{p\ast }}) \\
& \quad +(\tfrac{k+p-1}{p})^{p-1}(\eta f^{\frac{1}{p\ast }})^{p-1}u_{q}^{(
\frac{k-1}{p})(p-1)}|\nabla u_{q}|^{p-2}\nabla u_{q}\Big) \\
& \quad \times \Big(u_{q}^{\frac{k+p-1}{p}}\nabla (\eta f^{\frac{1}{p\ast }
})+\frac{k+p-1}{p}(\eta f^{\frac{1}{p\ast }})u_{q}^{\frac{k-1}{P}}\nabla
u_{q}\Big)\Big]^{p/2} \\
& \quad \times \Big[u_{q}^{k+p-1}|\nabla (\eta f^{\frac{1}{p\ast }})|^{p}+(
\frac{k+p-1}{p})^{p}(\eta f^{\frac{1}{p\ast }})^{p}u_{q}^{k-1}|\nabla
u_{q}|^{p}\Big]^{1-\frac{p}{2}}.
\end{align*}
Then 
\begin{align*}
& \big\|\nabla \big(\eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}}\big)
\big\|_{p}^{p} \\
& \leq C_{p}\int_{M}\Big[u_{q}^{k+p-1}|\nabla (\eta f^{\frac{1}{p\ast }
})|^{p}+(\frac{k+p-1}{p})^{p}(\eta f^{\frac{1}{p\ast }})^{p}u_{q}^{k-1}|
\nabla u_{q}|^{p} \\
& \quad +\frac{k+p-1}{p}(\eta f^{\frac{1}{p\ast }})u_{q}^{k+p-2}|\nabla
(\eta f^{\frac{1}{p\ast }})|^{p-2}\langle \nabla u_{q},\nabla (\eta f^{\frac{
1}{p\ast }})\rangle \\
& \quad +(\frac{k+p-1}{p})^{p-1}(\eta f^{\frac{1}{p\ast }})^{p-1}u_{q}^{k}|
\nabla u_{q}|^{p-2}\langle \nabla u_{q},\nabla (\eta f^{\frac{1}{p\ast }
})\rangle \Big]^{p/2} \\
& \quad \times \Big[u_{q}^{k+p-1}|\nabla (\eta f^{\frac{1}{p\ast }})|^{p}+(
\frac{k+p-1}{p})^{p}(\eta f^{\frac{1}{p\ast }})^{p}u_{q}^{k-1}|\nabla
u_{q}|^{p}\Big]^{1-\frac{p}{2}}dv_{g}.
\end{align*}
And by H\"{o}lder's inequality, the above expression is less than or equal
to 
\begin{align*}
& C_{p}\Big(\int_{M}\Big[u_{q}^{k+p-1}|\nabla (\eta f^{\frac{1}{p\ast }
})|^{p}+(\frac{k+p-1}{p})^{p}(\eta f^{\frac{1}{p\ast }})^{p}u_{q}^{k-1}|
\nabla u_{q}|^{p} \\
& +\frac{k+p-1}{p}(\eta f^{\frac{1}{p\ast }})u_{q}^{k+p-2}|\nabla (\eta f^{
\frac{1}{p\ast }})|^{p-2}\langle \nabla u_{q},\nabla (\eta f^{\frac{1}{p\ast 
}})\rangle \\
& +(\frac{k+p-1}{p})^{p-1}(\eta f^{\frac{1}{p\ast }})^{p-1}u_{q}^{k}|\nabla
u_{q}|^{p-2}\langle \nabla u_{q},\nabla (\eta f^{\frac{1}{p\ast }})\rangle 
\Big]dv_{g}\Big)^{2/p} \\
& \times \Big(\int_{M}(u_{q}^{k+p-1}|\nabla (\eta f^{\frac{1}{p\ast }
})|^{p}+(\frac{k+p-1}{p})^{p}(\eta f^{\frac{1}{p\ast }})^{p}u_{q}^{k-1}|
\nabla u_{q}|^{p})dv_{g}\Big)^{1-\frac{p}{2}}.
\end{align*}
Arguing as in the case $p\geq 2$, we obtain that $\Vert \nabla (\eta f^{
\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}})\Vert _{p}^{p}$ is bounded.
\end{proof}

\section{Generic Theorem}

Letting $K$ be any compact set of the complete manifold $M$, we formulate in
this section a generic theorem. First, we establish

\begin{lemma} \label{lem1}
Assume that every subsequence of  $\{ u_{q}\} $ which
converges in $L^{p}(M)$ with $p>1$,  converges to $0$.
Also assume there exists a constant $C>0$, independent of $q$ such
that $\| \nabla (\eta f^{\frac{1}{p\ast }}u_{q}^{\frac{k+p-1}{p}})
\|_{p}^{p}\leq C$,  $k>1$.
Then 
$$
\lim_{q\to \infty }\sup \int_{M}( \eta f^{\frac{1}{p\ast }}u_{q}) ^{p\ast }dv_{g}=0.
$$
\end{lemma}

\begin{proof}
Suppose that $\lim_{q\to \infty }\sup \int_{M}( \eta f^{ \frac{1}{p\ast }
}u_{q}) ^{p\ast }>0$. Using H\"{o}lder's inequality we obtain 
\begin{align*}
&\int_{M}( \eta f^{\frac{1}{p\ast }}u_{q}) ^{p\ast }dv_{g} \\
&\leq \sup_{M-K}f\Big( \int_{M}\big( \eta f^{\frac{1}{p\ast }}u_{q}^{\frac{
k+p-1 }{p}}\big) ^{p\ast }\Big) ^{\frac{n( p-1) +p}{n( k+p-1) }}\Big( 
\int_{M}u_{q}^{\frac{n( k+p-1) }{nk-p} }dv_{g}\Big) ^{\frac{nk-p}{n( k+p-1) }
}
\end{align*}
then 
\begin{equation*}
\lim_{q\to \infty }\sup \int_{M}u_{q}^{\frac{n( k+p-1) }{nk-p}}dv_{g}>0.
\end{equation*}
A contradiction with the fact that every subsequence of $u_{q}$ converging
in $L^{p}(M)$, $p>1$, converges to $0.$
\end{proof}

As a consequence of the above lemma, we obtain the following generic
theorem. Denote by $\Lambda =\{ u\in H_{1}^{p}(M):\;\int_{M}f|u| ^{p\ast
}dv_{g}=1\} $ and $I(u)$ is the functional given by $I(u)=\int_{M}( |\nabla
u| ^{p}+|u| ^{p}) dv_{g}$ where $M$ is a complete Riemannian manifold.

\begin{theorem} \label{thm3}
Let $( M,g) $ be a complete Riemannian manifold of bounded
geometry, $1<p<n$,  and let $a,\;f\in C^{\infty }(M)$ be real functions on $M$
with $f>0$. We assume that:
\begin{itemize}
\item[(i)] The operator $L_{p}u=\Delta_{p}u+a(x)u^{p-1}$ is coercive

\item[(ii)] Conditions (3) and (4) of Theorem \ref{thm1} at infinity on $f$ are satisfied

\item[(iii)] $(\sup_{M}f)^{\frac{p}{p^{\ast }}}
\inf_{u\in \Lambda }I(u)<K(n,p)^{-p}$.
\end{itemize}
Then  \eqref{e1} possesses a positive solution $u\in C^{1,\alpha }(K)$
for any compact set $K\subset M$ and some $\alpha \in (0,1).$
\end{theorem}

\begin{proof}
Suppose that 
\begin{equation*}
\mu f(x))^{p/p\ast }K(n,p)^{p}\lim_{q\rightarrow \infty }\sup
\int_{B(x_{o},\delta )}fu_{q}^{p\ast }dvg<1
\end{equation*}
then by Lemma8, we get that 
\begin{equation*}
\lim_{q\rightarrow \infty }\sup \int_{B(x_{o},\delta )}fu_{q}^{p\ast
}dv_{tg}=0
\end{equation*}
which contradicts the fact that 
\begin{equation}
\int_{M}fu_{q}^{p\ast }dv_{g}=1.  \label{e15}
\end{equation}
In fact 
\begin{equation}
\int_{M}fu_{q}^{p\ast }dv_{g}=\int_{\cup _{i=1}^{\infty }B(x_{i},\delta
)}fu_{q}^{p\ast }dv_{g}\leq \sum_{i=1}^{\infty }\int_{B(x_{i},\delta
)}fu_{q}^{p\ast }dv_{g}  \label{e16}
\end{equation}
where $M=\cup _{i=1}^{\infty }B(x_{i},\delta ).$

So for sufficient large $q$ the last term in \eqref{e16} is strictly smaller
that 1. Consequently 
\begin{equation*}
\mu f(x))^{p/p\ast }K(n,p)^{p}\lim_{q\rightarrow \infty }\sup
\int_{B(x_{o},\delta )}fu_{q}^{p\ast }dv_{g}\geq 1
\end{equation*}
and since by assumption $\mu f(x))^{p/p\ast }K(n,p)^{p}<1$, we obtain 
\begin{equation*}
\lim_{q\rightarrow \infty }\sup \int_{B(x_{o},\delta )}fu_{q}^{p\ast
}dv_{g}>1.
\end{equation*}
which is a contradiction with \eqref{e15}.

Then the condition that every subsequence of the sequence of minimizers $
\{u_{q}\}$ which converges has $0$ as a limit is false and the theorem is
proved.
\end{proof}

\subsection*{Examples of functions satisfying the conditions of Theorem \ref
{thm3}}

The conditions at infinity in Theorem \ref{thm3} are satisfied, for example
by functions decreasing like power functions: $f\sim r^{-q}$, $\nabla f\sim
\rho ^{-q-1}$ and $\nabla ^{2}f\sim r^{-q-2}$ with $q>n\frac{p\ast }{p}$.
Since $\int_{M}f^{p/p\ast }dv_{g}<+\infty $ implies that $\frac{1}{r^{(1-
\frac{1}{n})q+1-n}}$ is integrable.

If the function $a$ decays at infinity as $r^{-q}$ , then the condition that 
$\int_{M}f^{\frac{n}{p}}dv_{g}\leq C\int_{M}fdv_{g}<+\infty $ implies that
the decay rate $q$ satisfies $q>p$.

\section{Test functions}

In this section we give the proof of our main result (Theorem \ref{thm1}).
For this task we check that the condition (iii) of the generic theorem
proved in section 4 is satisfied.

Let $K$ be any compact set of the manifold $M$ and $x_{o}\in M-K$ be the
maximum on of the function $f$ as given in Theorem \ref{thm1}. Let $
r=d(x_{o},x)$ the distance function from $x_{o}$ to any point $x$ in the
manifold $M-K$.

Let $\delta >0$ be smaller than the injectivity radius; for $\epsilon >0$,
we consider the test function 
\begin{equation*}
u_{\varepsilon }(x)= 
\begin{cases}
( \varepsilon +r^{\frac{p}{p-1}}) ^{1-\frac{n}{p}}- ( \varepsilon +\delta ^{
\frac{p}{p-1}}) ^{1-\frac{n}{p}} & \mbox{if }r<\delta \\ 
0 & \mbox{if } r\geq \delta\,.
\end{cases}
\end{equation*}
% where $\nu =( \varepsilon +\delta ^{\frac{p}{p-1}}) ^{1-\frac{n}{p}}$.
Note that the function $u_{\varepsilon }$ was introduced by Aubin in \cite
{a1}. We have 
\begin{equation*}
|\nabla u_{\varepsilon }(x)| ^{p}= 
\begin{cases}
( \frac{n-p}{p-1})^{p}( \varepsilon +r^{\frac{p}{p-1}}) ^{-\frac{n}{p}}r^{( 
\frac{p}{p-1}) } & \mbox{if } r<\delta \\ 
0 & \mbox{if } r\geq \delta
\end{cases}
\end{equation*}
so 
\begin{equation}
\int_{B(x_{o},\delta )}|\nabla u_{\varepsilon }(x)| ^{p}dv_{g}=( \frac{n-p}{
p-1}) ^{p}\int_{0}^{\delta }( \varepsilon +r^{\frac{p}{p-1}}) ^{-n}r^{n+
\frac{1}{p-1} }dr\int_{S^{n-1}(r)}d\Omega .  \label{e17}
\end{equation}
where $d\Omega $ denotes the element volume on the sphere $S^{n-1}(r)$.

Let $S(r)=\int_{S^{n-1}(r)}d\Omega $. Taking into account the expansion of
the determinant in a system of geodesic coordinates at a point $x_{o}$, we
get 
\begin{equation*}
\sqrt{g}=1-R_{ij}x^{i}x^{j}+o(r^{2})\,.
\end{equation*}
A computation in \cite{a1} gives us 
\begin{equation*}
S(r)=\omega _{n-1}(1-\frac{R}{6n}r^{2}+o(r^{2}))
\end{equation*}
where $w_{n-1}$ is the volume of the standard unit sphere $S^{n-1}$ in $
R^{n} $. The integral \eqref{e17} becomes 
\begin{align*}
& \int_{B(x_{o},\delta )}|\nabla u_{\varepsilon }(x)|^{p}dv_{g} \\
& =(\frac{n-p}{p-1})^{p}\omega _{n-1}\int_{0}^{\delta }(\varepsilon +r^{
\frac{p}{1-p}})^{-n}r^{n+\frac{1}{p-1}}\big(1-\frac{R}{6n}r^{2}+o(r^{2})\big)
dr.
\end{align*}
Letting $s=r\varepsilon ^{\frac{1-p}{p}}$, we get 
\begin{equation}
\begin{aligned} \int_{B(x_{o},\delta )}|\nabla u_{\varepsilon
}(x)|^{p}dv_{g} &=( \frac{n-p}{p-1}) ^{p}\omega_{n-1}\varepsilon ^{1-
\frac{n}{p}}\int_{0}^{\delta \varepsilon ^{\frac{1-p}{p}}} \big(
1+s^{\frac{p}{1-p}}\big) ^{-n}s^{n+\frac{1}{p-1}}\\ &\quad\times \big(
1-\frac{R}{6n}s^{2}\varepsilon ^{\frac{2( p-1) }{p} }+o(s^{2}\varepsilon
^{\frac{2( p-1) }{p}})\big) ds, \end{aligned}  \label{e18}
\end{equation}
set 
\begin{gather*}
I_{p}^{q}=\int_{0}^{\infty }t^{q-1}(1+t)^{-p}dt\quad \text{with }p-q-1>0, \\
B(p,q)=\int_{0}^{1}t^{p-1}(1-t)^{q-1}dt\quad \text{with }p>0,\;q>0\,.
\end{gather*}
Put $t=s^{\frac{p}{p-1}}$, then the integral \eqref{e18} becomes 
\begin{align*}
& \int_{B(x_{o},\delta )}|\nabla u_{\varepsilon }(x)|^{p}dv_{g} \\
& =\frac{p-1}{p}(\frac{n-p}{p-1})^{p}\omega _{n-1}\varepsilon ^{1-\frac{n}{p}
}\Big[\int_{0}^{\delta ^{\frac{p}{p-1}}\varepsilon ^{-1}}(1+t)^{-n}t^{n(1-
\frac{1}{p})}dt \\
& \quad -\varepsilon ^{2(1-\frac{1}{p})}\frac{R(x_{o})}{6n}\int_{0}^{\delta
^{\frac{p}{p-1}}\varepsilon ^{-1}}(1+t)^{-n}t^{(n+2)(1-\frac{1}{p}
)}dt+o(\varepsilon ^{\frac{2(p-1)}{p}})\Big] \\
& =\frac{p-1}{p}(\frac{n-p}{p-1})^{p}\omega _{n-1}\varepsilon ^{1-\frac{n}{p}
}\Big[\int_{0}^{\infty }(1+t)^{-n}t^{n(1-\frac{1}{p})}dt \\
& \quad -\varepsilon ^{2(1-\frac{1}{p})}\frac{R(x_{o})}{6n}\int_{0}^{\infty
}(1+t)^{-n}t^{(n+2)(1-\frac{1}{p})}dt-\int_{\delta ^{\frac{p}{p-1}
}\varepsilon ^{-1}}^{\infty }(1+t)^{-n}t^{n(1-\frac{1}{p})}dt \\
& \quad +\varepsilon ^{2(1-\frac{1}{p})}\frac{R(x_{o})}{6n}\int_{\delta ^{
\frac{p}{p-1}}\varepsilon ^{-1}}^{\infty }(1+t)^{-n}t^{(n+2)(1-\frac{1}{p}
)}dt+o(\varepsilon ^{\frac{2(p-1)}{p}})\Big].
\end{align*}
We have 
\begin{equation*}
\lim_{\varepsilon \rightarrow 0}\int_{\delta ^{\frac{p}{p-1}}\varepsilon
^{-1}}^{\infty }(1+t)^{-n}t^{n(1-\frac{1}{p})}dt=0
\end{equation*}
and if $n+2>3p$, then 
\begin{equation*}
\lim_{\varepsilon \rightarrow 0}\int_{\delta ^{\frac{p}{p-1}}\varepsilon
^{-1}}^{\infty }(1+t)^{-n}t^{(n+2)(1-\frac{1}{p})}dt=0\,.
\end{equation*}
So 
\begin{align*}
& \int_{B(x_{o},\delta )}|\nabla u_{\varepsilon }(x)|^{p}dv_{g} \\
& =\frac{p-1}{p}(\frac{n-p}{p-1})^{p}\omega _{n-1}\varepsilon ^{1-\frac{n}{p}
}\Big[I_{n}^{n(1-\frac{1}{p})}-\varepsilon ^{2(1-\frac{1}{p})}\frac{R(x_{o})
}{6n}I_{n}^{(n+2)(1-\frac{1}{p})}+o(\varepsilon ^{2(1-\frac{1}{p})})\Big].
\end{align*}
On the other hand, a simple computation, for $(p>q+1)$, gives the following
formula 
\begin{equation*}
I_{p}^{q}=B(q+1,p-q-1)=\frac{\Gamma (q+1)\Gamma (p-q-1)}{\Gamma (p)},
\end{equation*}
where $\Gamma $ denotes the Euler function from which we obtain the
following relation 
\begin{equation*}
I_{n}^{(n+2)(1-\frac{1}{p})}=\frac{\Gamma ((n+2)(1-\frac{1}{p})+1)\Gamma (
\frac{n+2}{p}-3)}{\Gamma (n(1-\frac{1}{p})+1)\Gamma (\frac{n}{p}-1)}
I_{n}^{n(1-\frac{1}{p})}=a(n,p)I_{n}^{n(1-\frac{1}{p})}.
\end{equation*}
Finally the equality \eqref{e18} becomes 
\begin{equation}
\begin{aligned} &\int_{B(x_{o},\delta )}|\nabla u_{\varepsilon }(x)|
^{p}dv_{g}\\ &=\frac{p-1}{p}( \frac{n-p}{p-1}) ^{p}\omega_{n-1}\varepsilon
^{1- \frac{n}{p}}I_{n}^{n( 1-\frac{1}{p}) }[1-\varepsilon ^{2(
1-\frac{1}{p}) }a(n,p)\frac{R(x_{o})}{6n} +o(\varepsilon ^{2( 1-\frac{1}{p})
})] . \end{aligned}  \label{e19}
\end{equation}
The expansion of $\int_{B(x_{o},\delta )}au_{\varepsilon }^{p}dv_{g}$ is
computed in the same way as above, 
\begin{align*}
& \int_{B(x_{o},\delta )}au_{\varepsilon }^{p}dv_{g} \\
& =\int_{0}^{\delta }u_{\varepsilon }^{p}r^{n-1}dr\int_{S^{n-1}(r)}a\sqrt{g}
d\Omega \\
& =\int_{0}^{\delta }\big((\varepsilon +r^{\frac{p}{p-1}})^{1-\frac{n}{p}
}-\nu \big)^{p}r^{n-1}dr \\
& \quad \times \int_{S^{n-1}(r)}\big(a(x_{o})+\frac{1}{2}\nabla
_{ij}a(x_{o})x^{i}x^{j}+o(r^{2})\big)\big(1-\frac{1}{6}
R_{ij}(x_{o})x^{i}x^{j}+o(r^{2})\big)d\Omega \\
& =\omega _{n-1}a(x_{o})\int_{0}^{\delta }r^{n-1}(\varepsilon +r^{\frac{p}{
p-1}})^{p-n}\Big[1-p\nu (\varepsilon +r^{\frac{p}{p-1}})^{\frac{n}{p}-1}+o
\big((\varepsilon +r^{\frac{p}{p-1}})^{\frac{n}{p}-1}\big)\Big] \\
& \quad \times \Big[1-\big(\frac{R(x_{o})}{6n}+\frac{\Delta a(x_{o})}{
2nR(x_{o})}\big)r^{2}+o(r^{2})\Big]dr\,.
\end{align*}
Putting $s=r^{\frac{1-p}{p}}$, we get 
\begin{align*}
& \int_{B(x_{o},\delta )}au_{\varepsilon }^{p}dv_{g} \\
& =\omega _{n-1}a(x_{o})\varepsilon ^{p-\frac{n}{p}}\int_{0}^{\delta
\varepsilon ^{\frac{1-p}{p}}}s^{n-1}(1+s^{\frac{p}{p-1}})^{p-n}\Big[1-p\nu
\varepsilon ^{\frac{n}{p}-1}(1+s^{\frac{p}{p-1}}) \\
& \quad +o(1+s^{\frac{p}{p-1}})^{\frac{n}{p}-1}\varepsilon ^{\frac{n}{p}-1}
\Big]\Big[1-\big(\frac{R(x_{o})}{6n}+\frac{\Delta a(x_{o})}{2na(x_{o})}\big)
s^{2}\varepsilon ^{2(1-\frac{1}{p})}+\varepsilon ^{2(1-\frac{1}{p})}o(s^{2})
\Big]ds.
\end{align*}
Letting $t=s^{\frac{p}{p-1}}$, we get 
\begin{align*}
& \int_{B(x_{o},\delta )}au_{\varepsilon }^{p}dv_{g} \\
& =\frac{p-1}{p}\omega _{n-1}a(x_{o})\varepsilon ^{p-\frac{n}{p}
}\int_{0}^{\delta ^{\frac{p}{p-1}}\varepsilon ^{-1}}t^{n(1-\frac{1}{p}
)-1}(1+t)^{p-n} \\
& \quad \times \Big[1-p\nu \varepsilon ^{\frac{n}{p}-1}(1+t)+o(1+t)
\varepsilon ^{\frac{n}{p}-1}\Big] \\
& \quad \times \lbrack 1-\big(\frac{R(x_{o})}{6n}+\frac{\Delta a(x_{o})}{
2na(x_{o})}\big)t^{2(1-\frac{1}{p})}\varepsilon ^{2(1-\frac{1}{p}
)}+\varepsilon ^{2(1-\frac{1}{p})}o(t^{2(1-\frac{1}{p})})\Big]dt \\
& =\frac{p-1}{p}\omega _{n-1}a(x_{o})\varepsilon ^{p-\frac{n}{p}
}\int_{0}^{\delta ^{\frac{p}{p-1}}\varepsilon ^{-1}}t^{n(1-\frac{1}{p}
)-1}(1+t)^{p-n}dt+o(\varepsilon ^{p-\frac{n}{p}})\,.
\end{align*}
Since $p<n^{2}$, we have 
\begin{equation*}
\lim_{\varepsilon \rightarrow 0}\int_{\delta ^{\frac{p}{p-1}}\varepsilon
^{-1}}^{\infty }t^{n(1-\frac{1}{p})-1}(1+t)^{p-n}dt=0\,;
\end{equation*}
therefore, 
\begin{equation*}
\int au_{\varepsilon }^{p}dv_{g}=\frac{p-1}{p}\omega
_{n-1}a(x_{o})\varepsilon ^{p-\frac{n}{p}}I_{n-p}^{n(1-\frac{1}{p}
)-1}+o(\varepsilon ^{p-\frac{n}{p}}).
\end{equation*}
From the formulae 
\begin{gather*}
I_{n-p}^{n(1-\frac{1}{p})-1}=\frac{\Gamma \big(n(1-\frac{1}{p})\big)\Gamma (
\frac{n}{p}-p)}{\Gamma (n-p)}\,, \\
I_{n}^{n(1-\frac{1}{p})}=\frac{\Gamma \big(n(1-\frac{1}{p})+1\big)\Gamma (
\frac{n}{p}-1)}{\Gamma (n)}=\frac{n(1-\frac{1}{p})\Gamma \big(n(1-\frac{1}{p}
)\big)\Gamma (\frac{n}{p}-1)}{\Gamma (n)}
\end{gather*}
we deduce that 
\begin{equation*}
I_{n-p}^{n(1-\frac{1}{p})-1}=\frac{\Gamma (n)\Gamma (\frac{n}{p}-p)}{n(1-
\frac{1}{p})\Gamma (n-p)\Gamma (\frac{n}{p}-1)}I_{n}^{n(1-\frac{1}{p}
)}=b(p,n)I_{n}^{n(1-\frac{1}{p})}.
\end{equation*}
Finally, we get 
\begin{equation}
\int_{B(x_{o},\delta )}au_{\varepsilon }^{p}dv_{g}=\varepsilon ^{p-\frac{n}{p
}}\frac{p-1}{p}\omega _{n-1}a(x_{o})b(p,n)I_{n}^{n(1-\frac{1}{p}
)}+o(\varepsilon ^{p-\frac{n}{p}}).  \label{e20}
\end{equation}
Now, we compute the term 
\begin{align*}
& \int_{B(x_{o},\delta )}fu_{\varepsilon }^{p\ast }dv_{g} \\
& =\int_{0}^{\delta }r^{n-1}\big((\varepsilon +r^{\frac{p}{p-1}})^{1-\frac{n
}{p}}-\nu \big)^{p\ast }dr\int_{S^{n-1}(r)}f\sqrt{g}d\Omega \\
& =\int_{0}^{\delta }r^{n-1}\big((\varepsilon +r^{\frac{p}{p-1}})^{1-\frac{n
}{p}}-\nu \big)^{p\ast }dr \\
& \quad \times \int_{S^{n-1}(r)}\big(f(x_{o})+\frac{1}{2}\nabla
_{ij}f(x_{o})x^{i}x^{j}+o(r^{2})\big)\big(1-\frac{1}{6}
R_{ij}(x_{o})x^{i}x^{j}+o(r^{2})\big)d\Omega \\
& =\int_{0}^{\delta }r^{n-1}\big((\varepsilon +r^{\frac{p}{p-1}})^{1-\frac{n
}{p}}-\nu \big)^{p\ast }dr \\
& \quad \times \int_{S^{n-1}(r)}\Big[f(x_{o})+\big(\frac{1}{2}\nabla
_{ij}f(x_{o})-f(x_{o})\frac{R_{ij}(x_{o})}{6}\big)x^{i}x^{j}+o(r^{2})\Big]
d\Omega \\
& =\omega _{n-1}f(x_{o})\int_{0}^{\delta }\!r^{n-1}\big((\varepsilon +r^{
\frac{p}{p-1}})^{1-\frac{n}{p}}-\nu \big)^{p\ast }\big(1-\big(\frac{R(x_{o})
}{6n}+\frac{\Delta f(x_{o})}{2nf(x_{o})}\big)r^{2}+o(r^{2})\big)dr \\
& =\omega _{n-1}f(x_{o})\int_{0}^{\delta }r^{n-1}(\varepsilon +r^{\frac{p}{
p-1}})^{-n}\big(1-\big(\frac{R(x_{o})}{6n}+\frac{\Delta f(x_{o})}{2nf(x_{o})}
\big)r^{2}+o(r^{2})\big)dr\,.
\end{align*}
Letting $s=r\varepsilon ^{\frac{1-p}{p}}$, we get 
\begin{align*}
\int_{B(x_{o},\delta )}fu_{\varepsilon }^{p\ast }dv_{g}& =\frac{p-1}{p}
\omega _{n-1}f(x_{o})\varepsilon ^{\frac{-n}{p}}\int_{0}^{\delta \varepsilon
^{\frac{1-p}{p}}}s^{n-1}(1+s^{\frac{p}{p-1}})^{-n} \\
& \quad \times \big[1-\big(\frac{R(x_{o})}{6n}+\frac{\Delta f(x_{o})}{
2nf(x_{o}p)}\big)s^{2}\varepsilon ^{2(1-\frac{1}{p})}+o(s^{2})\varepsilon
^{2(1-\frac{1}{p})}\big]ds\,.
\end{align*}
Setting $t=s^{\frac{p}{p-1}}$, we obtain 
\begin{align*}
& \int_{B(x_{o},\delta )}fu_{\varepsilon }^{p\ast }dv_{g} \\
& =\frac{p-1}{p}\omega _{n-1}f(x_{o})\varepsilon ^{\frac{-n}{p}
}\int_{0}^{\delta ^{\frac{p}{p-1}}\varepsilon ^{-1}}t^{n(1-\frac{1}{p}
)-1}(1+t)^{-n} \\
& \quad \times \big(1-\big(\frac{R(x_{o})}{6n}+\frac{\Delta f(x_{o})}{
2nf(x_{o})}\big)t^{2(1-\frac{1}{p})}\varepsilon ^{2(1-\frac{1}{p})}+o(t^{2(1-
\frac{1}{p})})\varepsilon ^{2(1-\frac{1}{p})}\big)dt.
\end{align*}
Since 
\begin{equation*}
\lim_{\varepsilon \rightarrow 0}\int_{\delta ^{\frac{p}{p-1}}\varepsilon
^{-1}}^{\infty }t^{n(1-\frac{1}{p})-1}(1+t)^{-n}dt=0
\end{equation*}
and 
\begin{equation*}
\lim_{\varepsilon \rightarrow 0}\int_{\delta ^{\frac{p}{p-1}}\varepsilon
^{-1}}^{\infty }t^{(n+2)(1-\frac{1}{p})-1}(1+t)^{-n}dt=0
\end{equation*}
provided that $p<1+\frac{n}{2}$, which is the case if $p^{2}<n$, we deduce
that 
\begin{align*}
\int_{B(x_{o},\delta )}fu_{\varepsilon }^{p\ast }dv_{g}& =\frac{p-1}{p}
\omega _{n-1}f(x_{o})\varepsilon ^{\frac{-n}{p}}\Big[I_{n}^{n(1-\frac{1}{p}
)-1} \\
& \quad -\big(\frac{R(x_{o})}{6n}+\frac{\Delta f(x_{o})}{2nf(x_{o})}\big)
I_{n}^{(n+2)(1-\frac{1}{p})-1}\varepsilon ^{2(1-\frac{1}{p})}+o(\varepsilon
^{2(1-\frac{1}{p})})\Big].
\end{align*}
Hence, by putting 
\begin{equation*}
c(n,p)=\frac{\Gamma \big((n+2)(1-\frac{1}{p})\big)\Gamma \big(\frac{n-2p+1}{p
}\big)}{\Gamma \big(n(1-\frac{1}{p})\big)\Gamma (\frac{n}{p})}
\end{equation*}
one has 
\begin{align*}
& \int_{B(x_{o},\delta )}fu_{\varepsilon }^{p\ast }dv_{g} \\
& =\varepsilon ^{\frac{-n}{p}}\frac{p-1}{p}\omega _{n-1}f(x_{o})I_{n}^{n(1-
\frac{1}{p})-1}\big[1-\big(\frac{R(x_{o})}{6n}+\frac{\Delta f(x_{o})}{
2nf(x_{o})}\big)c(p,n)I_{n}^{n(1-\frac{1}{p})} \\
& \quad +\varepsilon ^{2(1-\frac{1}{p})}+o\big(\varepsilon ^{2(1-\frac{1}{p}
)}\big)\big]
\end{align*}
and since 
\begin{equation*}
I_{n}^{n(1-\frac{1}{p})-1}=\frac{\Gamma (\frac{n}{p})}{n(n-\frac{1}{p}
)\Gamma (\frac{n}{p}-1)}I_{n}^{n(1-\frac{1}{p})}=d(n,p)I_{n}^{n(1-\frac{1}{p}
)}
\end{equation*}
it follows that 
\begin{align*}
& \int_{B(x_{o},\delta )}fu_{\varepsilon }^{p\ast }dv_{g} \\
& =\varepsilon ^{\frac{-n}{p}}\frac{p-1}{p}\omega
_{n-1}f(x_{o})d(n,p)I_{n}^{n(1-\frac{1}{p})} \\
& \times \big[1-\big(\frac{R(x_{o})}{6n}+\frac{\Delta f(x_{o})}{2nf(x_{o})}
\big)c(n,p)\varepsilon ^{2(1-\frac{1}{p})}+o\big(\varepsilon ^{2(1-\frac{1}{p
})}\big)\big].
\end{align*}
From equalities \eqref{e19} and \eqref{e20}, we get 
\begin{equation}
\begin{aligned} &\int_{B(x_{o},\delta )}( |\nabla u_{\varepsilon }^{p}|
+au_{\varepsilon }^{p}) dv_{g}\\ &=\frac{p-1}{p}\omega_{n-1}(
\frac{n-p}{p-1}) ^{p}\varepsilon ^{1-\frac{n}{p}} I_{n}^{n( 1-\frac{1}{p})
}\big[1- \frac{a(n,p)}{6n}R(x_{o})\varepsilon ^{2( 1-\frac{1}{p})}\\ &\quad
+b(n,p)( \frac{n-p}{p-1}) ^{p}a(x_{o})\varepsilon ^{p-1} +o\big(\varepsilon
^{2( 1-\frac{1}{p}) }\big) +o\big(\varepsilon ^{p-1}\big)\big]\,.
\end{aligned}  \label{e21}
\end{equation}
Since 
\begin{align*}
& \Big(\int_{B(x_{o},\delta )}fu_{\varepsilon }^{p\ast }dv_{g}\Big)^{\frac{p
}{p\ast }} \\
& =\varepsilon ^{1-\frac{n}{p}}\big(\frac{p-1}{p}\omega
_{n-1}f(x_{o})d(n,p)I_{n}^{n(1-\frac{1}{p})}\big)^{p/p\ast } \\
& \quad \times \big[1-\big(\frac{R(x_{o})}{6n}+\frac{\Delta f(x_{o})}{
2nf(x_{o})}\big)c(n,p)\varepsilon ^{2(1-\frac{1}{p})}+o\big(\varepsilon
^{2(1-\frac{1}{p})}\big)]^{p/p\ast }
\end{align*}
we get 
\begin{equation}
\begin{aligned} &\frac{\int_{B(x_{o},\delta )}( |\nabla u_{\varepsilon
}^{p}| +au_{\varepsilon }^{p}) dv_{g}}{\big( \int_{B(x_{o},\delta
)}fu_{\varepsilon }^{p\ast }dv_{g}\big) ^{p/p\ast}}\\ &=\big(
\frac{p-1}{p}\omega_{n-1}f(x_{o})I_{n}^{n( 1-\frac{1}{p}) }\big)
^{\frac{p}{n}}( \frac{n-p}{p-1}) ^{p}\big( d(n,p)f(x_{o})\big)
^{-\frac{p}{p\ast }} \\ &\quad \times \big[1-\big(
\frac{a(n,p)}{6n}a(x_{o})\varepsilon ^{2( 1- \frac{1}{p})
}-b(n,p)a(x_{o})\varepsilon ^{p-1}\big) +o\big(\varepsilon ^{2(
1-\frac{1}{p}) }\big) +o\big(\varepsilon ^{p-1}\big) \big] \\ &\quad \times
\big[1+\frac{p}{p\ast }\big( \frac{R(x_{o})}{6n}+\frac{\Delta
f(x_{o})}{2nf(p)}\big) c(n,p)\varepsilon ^{2( 1-\frac{1}{p})
}+o\big(\varepsilon ^{2( 1-\frac{1}{p}) }\big)\big] . \end{aligned}
\label{e22}
\end{equation}
Now, since the function 
\begin{equation*}
\phi =\big(1+r^{\frac{p}{p-1}}\big)^{1-\frac{n}{p}}
\end{equation*}
realizes the best constant in the Sobolev's imbedding $H_{1}^{p}(R^{n})
\subset L^{p\ast }(R^{n})$, that is 
\begin{equation*}
\Big(\int_{R^{n}}\phi ^{p\ast }dx\Big)^{p/p\ast
}=K(n,p)^{p}\int_{R^{n}}|\phi |^{p}dx\text{ ,}
\end{equation*}
we get 
\begin{align*}
& (\frac{n-p}{p-1})^{p}\omega _{n-1}\int_{0}^{\infty }(1+r^{\frac{p}{p-1}
})^{-n}r^{n+\frac{1}{p-1}}dr \\
& =K(n,p)^{-p}\Big(\omega _{n-1}\int_{0}^{\infty }(1+r^{\frac{p}{p-1}
})^{-n}r^{n-1}dr\Big)^{p/p\ast }.
\end{align*}
By letting $t=r^{\frac{p}{p-1}}$, we have 
\begin{align*}
& \frac{p-1}{p}(\frac{n-p}{p-1})^{p}\omega _{n-1}\int_{0}^{\infty
}(1+t)^{-n}t^{n(1-\frac{1}{p})}dt \\
& =K(n,p)^{-p}\Big(\omega _{n-1}\frac{p-1}{p}\int_{0}^{\infty
}(1+t)^{-n}t^{n(1-\frac{1}{p})-1}dr\Big)^{p/p\ast }\,.
\end{align*}
Therefore, 
\begin{equation*}
\frac{p-1}{p}(\frac{n-p}{p-1})^{p}\omega _{n-1}I_{n}^{n(1-\frac{1}{p}
)}=K(n,p)^{-p}\big(\frac{p-1}{p}\omega _{n-1}d(n,p)I_{n}^{n(1-\frac{1}{p})}
\big)^{p/p\ast }
\end{equation*}
which implies 
\begin{equation*}
K(n,p)^{-p}=(\frac{n-p}{p-1})^{p}\big(\frac{p-1}{p}\omega _{n-1}I_{n}^{n(1-
\frac{1}{p})}\big)^{\frac{p}{n}}d(n,p)^{-\frac{p}{p\ast }}.
\end{equation*}
Then the equality \eqref{e22} becomes 
\begin{equation}
\begin{aligned} &\frac{\int_{B(x_{o},\delta )}( |\nabla u_{\varepsilon
}^{p}| +au_{\varepsilon }^{p}) dv_{g}}{\big( \int_{B(x_{o},\delta
)}fu_{\varepsilon }^{p\ast }dv_{g}\big) ^{\frac{p}{ p\ast }}}\\
&=K(n,p)^{-p}f(x_{o})^{-\frac{p}{p\ast }} \Big[1- \big(
\frac{a(n,p)}{6n}R(x_{o})\varepsilon ^{2( 1-\frac{1}{p}) }\\ &\quad -(
\frac{p-1}{n-p}) ^{p}b(n,p)R(x_{o})\varepsilon^{p-1}\big) +o\big(
\varepsilon ^{2( 1-\frac{1}{p}) }\big) +o(\varepsilon ^{p-1})\Big] \\ &\quad
\times \big[1+\frac{p}{p\ast }\big( \frac{R(x_{o})}{6n}+\frac{\Delta
f(x_{o})}{2nf(x_{o})}\big) c(n,p)\varepsilon ^{2( 1-\frac{1}{p} )
}+o(\varepsilon ^{2( 1-\frac{1}{p}) })\big] \\
&=K(n,p)^{-p}f(x_{o})^{-\frac{p}{p\ast }} \Big[1-\Big\{ \big(
\frac{a(n,p)}{6n}-\frac{p}{p\ast } \frac{c(n,p)}{6n}\varepsilon ^{2(
1-\frac{1}{p} ) }\big) R(x_{o})\\ &\quad -( \frac{p-1}{n-p})
^{p}b(n,p)a(x_{o})\varepsilon ^{p-1}- \frac{p}{p\ast }\frac{\Delta
f(x_{o})}{2nf(x_{o})}c(n,p)\varepsilon ^{2( 1-\frac{1}{p}) }\Big\}\\ &\quad
+o\big(\varepsilon ^{p-1}\big)+o\big(\varepsilon ^{2( 1-\frac{1}{p}) }\big)
\Big] . \end{aligned}  \label{e23}
\end{equation}
If $1<p<2$ and $n+2>3p$, the bracket in the equality \eqref{e23} is
equivalent to 
\begin{equation*}
1+(\frac{p-1}{n-p})^{p}b(n,p)a(x_{o})\varepsilon ^{p-1}\,.
\end{equation*}
Then, if $a(x_{o})<0$, we get 
\begin{equation}
1+(\frac{p-1}{n-p})^{p}b(n,p)a(x_{o})\varepsilon ^{p-1}<1.  \label{e24}
\end{equation}
If $p=2,$ \ the bracket reads 
\begin{equation*}
\big(\frac{a(n,2)}{6n}-\frac{n-2}{n}\frac{c(n,2)}{6n}\big)R(x_{o})-(\frac{1}{
n-2})^{2}b(n,2)a(x_{o})-\frac{n-2}{n}\frac{\Delta f(x_{o})}{2nf(x_{o})}c(n,2)
\end{equation*}
where the quantities $a(n,2)$, $b(n,2)$ and $c(n,2)$ are replaced by their
respective expressions. The condition 
\begin{equation*}
\big(\frac{a(n,2)}{6nc(n,2)}-\frac{n-2}{6n^{2}}\big)R(x_{o})-(\frac{1}{n-2}
)^{2}\frac{b(n,2)}{c(n,2)}a(x_{o})-\frac{n-2}{n}\frac{\Delta f(x_{o})}{
2nf(x_{o})}>0
\end{equation*}
implies 
\begin{align*}
& \big(\frac{(n+2)(n-2)}{6n^{2}(n-4)}-\frac{n-2}{6n^{2}}\big)R(x_{o}) \\
& -(\frac{1}{n-2})^{2}\frac{4(n-1)(n-2)^{2}}{n^{2}(n-4)}a(x_{o})-\frac{n-2}{n
}\frac{\Delta f(x_{o})}{2nf(x_{o})}>0;
\end{align*}
that is, 
\begin{equation}
\frac{\Delta f(x_{o})}{f(x_{o})}-\frac{2}{n-4}a(x_{o})+\frac{8(n-1)}{
(n-2)(n-4)}R(x_{o})<0.  \label{e25}
\end{equation}
Now for, $p>2$, the bracket in question is equivalent to 
\begin{equation*}
\big(\frac{a(n,p)}{6nc(n,p)}-\frac{n-p}{6n^{2}}\big)R(x_{o})-\frac{n-p}{n}
\frac{\Delta f(x_{o})}{2nf(x_{o})}.
\end{equation*}
The condition 
\begin{equation*}
\big(\frac{(n+2)(n-p)}{6n^{2}(n-3p+2)}-\frac{n-p}{6n^{2}}\big)R(x_{o})-\frac{
n-p}{2n^{2}}\frac{\Delta f(x_{o})}{f(x_{o})}>0
\end{equation*}
becomes 
\begin{equation}
\frac{\Delta f(x_{o})}{f(x_{o})}<\frac{p}{n-3p+2}R(x_{o}).  \label{e26}
\end{equation}
Each of the conditions \eqref{e24}, \eqref{e25} and \eqref{e26} assures that 
\begin{equation*}
\frac{\int_{B(x_{o},\delta )}(|\nabla u_{\varepsilon }^{p}|+au_{\varepsilon
}^{p})dv_{g}}{\big(\int_{B(x_{o},\delta )}fu_{\varepsilon }^{p\ast }dv_{g}
\big)^{p/p\ast }}<K(n,p)^{-p}f(x_{o})^{-p/p\ast }
\end{equation*}
and a fortiori the condition (iii) of the generic theorem is satisfied.
Therefore, our main theorem (Theorem \ref{thm1}) is proved.

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\end{thebibliography}

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