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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 155, pp. 1--9.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/155\hfil Solving $p$-Laplacian equations]
{Solving $p$-Laplacian equations on complete manifolds}
\author[M. Benalili, Y. Maliki\hfil EJDE-2006/155\hfilneg]
{Mohammed Benalili, Youssef Maliki}

\address{Mohammed Benalili \\
Universit\'{e} Abou -Bekr Belka\"{\i}d, 
Facult\'{e} des sciences, \newline
D\'{e}part. Mth\'{e}matiques, 
B.P. 119, Tlemcen, Algerie}
\email{m\_benalili@mail.univ-tlemcen.dz}

\address{Youssef Maliki \\
Universit\'{e} Abou -Bekr Belka\"{\i}d,
Facult\'{e} des sciences, \newline
D\'{e}part. Mth\'{e}matiques, 
B.P. 119, Tlemcen, Algerie}
\email{malyouc@yahoo.fr}

\thanks{Submitted June 28, 2005. Published December 14, 2006.}
\subjclass[2000]{31C45, 53C21}
\keywords{Differential geometry; nonlinear partial differential equations}

\begin{abstract}
 Using a reduced version of the sub and super-solutions method, 
 we prove that the equation $\Delta _{p}u+ku^{p-1}-Ku^{p^{\ast }-1}=0$
 has a positive solution on a complete Riemannian manifold for 
 appropriate functions $k,K:M\to \mathbb{R}$.
\end{abstract}

\maketitle


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

\section{Introduction}

Let $(M,g)$ be an $n$-dimensional complete and connected Riemannian manifold 
$(n\geq 3)$ and let $p\in (1,n)$. We are interested in the existence of
positive solutions $u\in H_{1,\mathrm{loc}}^{p}(M)$ (the standard Sobolev
space of order $p$) of the equation 
\begin{equation}
\Delta _{p}u+ku^{p-1}-Ku^{p^{\ast }-1}=0  \label{1}
\end{equation}
with $p^{\ast }=\frac{pn}{n-p}$ and $\Delta _{p}u=\mathop{\rm div}(| \nabla
u| ^{p-2}\nabla u)$ is the $p$-Laplacian of $u$ .

As usual $u\in H_{1,\mathrm{loc}}^{p}(M)$ is defined to be a weak solution
of \eqref{1} if 
\begin{equation}
\int_{M}-| \nabla u| ^{p-2}\nabla u \nabla v+(ku^{p-1}-Ku^{p^{\ast }-1})v=0
\label{2}
\end{equation}
for each $v\in C_{0}^{\infty }(M)$. A supersolution (respectively a
subsolution) $u\in H_{1,\mathrm{loc}}^{p}(M)$ is defined in the same way by
changing $=$ by $\leq $ (respect $\geq $) in equation(\ref{2}) and requiring
that the test function $v\in C_{0}^{\infty }(M)$ to be non negative.
Throughout this paper, we will assume that $k$ and $K$ are smooth real
valued functions on $M $. Following the terminology in \cite{3}, this
equation is referred to as the generalized scalar curvature type equation,
it's an extension of the equation of prescribed scalar curvature. In the
case of a compact manifold, the problem was considered in \cite{3}. One of
the results obtained in this latter paper is the following theorem

\begin{theorem} \label{thm1}
Let $(M,g)$ be a compact Riemannian manifold with $n\geq 2$ and
let $p\in (1,n)$. Let $k$ and $K$ be smooth real functions on $M$. If we
assume that $k$ and $K$ are both positive, then  \eqref{1}
possesses a positive solution $u\in C^{1,\alpha }(M)$.
\end{theorem}

In this paper, we look for positive solutions of \eqref{1} on complete
Riemannian manifolds. To achieve this task, we use a recent result obtained
by the authors in \cite{2}. Before quoting this result we recall some
definitions. A nonnegative and smooth function $K$ on a complete manifold is
said \textit{essentially positive} if there exists an exhaustion by compact
domains $\{ \Omega _{i}\} _{i\geq 0}$ such that $M= \cup_{i\geq 0}\Omega
_{i} $ and $K|_{\partial \Omega _{j}}>0$ for any $j\geq 0$. Moreover, if
there is a positive supersolution $u\in H_{1}^{p}(\Omega _{i})\cap
C^{0}(\Omega _{i})$ on each $\Omega _{i}$ of \eqref{1} the essentially
positive function $K$ is said to be \textit{permissible}. With this
terminology the following theorem has been established in \cite{2}

\begin{theorem} \label{thm2}
Let $(M,g)$ be a complete non compact Riemannian manifold of dimension
$n\geq 3$ and $k,K$ be smooth real valued functions on $M$. Suppose that $K$
is permissible and $k\leq K$.
If there exists a positive subsolution
$u_{-}\in H_{1,\rm loc}^{p}(M)\cap L^{\infty }(M)\cap C^{0}(M)$ of
\eqref{1} on $M$, then \eqref{1} has a positive and maximal weak
solution $u\in H_{1}^{p}(M)$. Moreover
$u\in C^{1,\alpha }(\Omega _{i})$ on each compact $\Omega _{i}$ for some
$\alpha \in (0,1)$.
\end{theorem}

The Riemannian manifold $M$ will be said of bounded geometry if the Ricci
curvature of $M$ is bounded from below and the injectivity radius is
strictly positive everywhere.

We formulate our main result as follows:

\begin{theorem} \label{thm3}
Let $(M,g)$ be a complete non compact Riemannian manifold of dimension
$n\geq 3$ and $k,K$ be smooth real valued functions on $M$. Suppose that
\begin{itemize}
\item[(a)] the function $K$ is permissible and $K\geq c_{o}>0$ where $c_{o}$ 
is a
real constant, $k$ is bounded and satisfies $k\leq K$, and 
$\int_{\Omega_{i}}k=0$, on each compact domain $\Omega _{i}$ of 
the exhaustion of $M$.

\item[(b)] $M$ is of bounded geometry.
\end{itemize}
Then  \eqref{1} has a weak positive maximal solution
$u\in H_{1}^{p}(M)$. Moreover $u\in C_{loc}^{1,\alpha }(M)$
for some $\alpha \in (0,1)$.
\end{theorem}

Our paper is organized as follows: In the first section we construct a
supersolution of \eqref{1} on each compact subset of $M$. In the second
section, we show the existence of a positive eigenfunction of the nonlinear
operator $L_{p}u=-\Delta _{p}u-ku^{p-1}$ on $M$ which we will use next to
construct a global subsolution of our equation.

First, we establish the following result.

\begin{lemma} \label{lem1}
Let $\Omega $ be a compact domain of $M$ and $f$ be a
$C^{\infty }$ function on $\Omega $. The equation
\begin{equation}
\begin{gathered}
-\Delta _{p}\phi =f\quad \text{in  }\Omega -\partial \Omega \\
\phi =0\quad \text{on }\partial \Omega
\end{gathered} \label{3}
\end{equation}
admits a solution $\phi \in C^{1,\alpha }(\Omega )$.
\end{lemma}

\begin{proof}
Letting $A=\{\phi \in H_{1,0}^{p}(\Omega ):\int_{\Omega }f\phi =1\}$, we put 
\begin{equation*}
\mu =\inf_{\phi \in K}\int_{\Omega }|\nabla \phi |^{p}.
\end{equation*}%
The set $A$ is non empty since it contains the function 
$\phi =\frac{\mathop{\rm sgn}(f)|f|^{p-1}}{\int_{\Omega }|f|^{p}}$.

Let $(\phi_{i})_{i\in \mathbb{N}}$ be a minimizing sequence in $A$, that is, 
\begin{equation*}
\lim_{i\to \infty }\int_{\Omega }| \nabla \phi_{i}| ^{p}=\mu .
\end{equation*}
Then, if $\lambda _{1,p}$ denotes the first nonvanishing eigenvalue of the 
$p $-Laplacian operator, we have 
\begin{equation*}
\lambda _{1,p}\leq \frac{\int_{\Omega }| \nabla \phi _{i}| ^{p}}{%
\int_{\Omega }| \phi _{i}| ^{p}}
\end{equation*}
so 
\begin{equation*}
\int_{\Omega }| \phi _{i}| ^{p}\leq \lambda _{1,p}^{-1}\int_{\Omega }|
\nabla \phi _{i}| ^{p}<\frac{ \mu }{\lambda _{1,p}}+1.
\end{equation*}
The sequence $(\phi _{i})_{i\in \mathbb{N}}$ is bounded in 
$H_{1}^{p}(\Omega)$, hence by the reflexivity of the space 
$H_{1}^{p}(\Omega)$ and the Rellich-Kondrakov theorem, 
there exists a subsequence of $(\phi_{i})_{i\in \mathbb{N}}$ still 
denoted $(\phi _{i})$ such that

\begin{itemize}
\item[(a)] $(\phi _{i})_{i\in \mathbb{N}}$ converges weakly to $\phi \in
H_{1}^{p}(\Omega )$

\item[(b)] $(\phi _{i})_{i\in \mathbb{N}}$ converges strongly to $\phi \in
L^{p}(\Omega )$.
\end{itemize}

From (b) we deduce that $\phi _{i}\longrightarrow \phi $ in $L^{1}(\Omega )$
then $\phi \in A$ and from (a) we get 
\begin{equation*}
\| \phi \| _{H_{1}^{p}(\Omega )}\leq \lim_{i\to +\infty }\inf \| \phi _{i}\|
_{H_{1}^{p}(\Omega )}.
\end{equation*}
Taking into account of (b) again, we obtain 
\begin{equation*}
\int_{\Omega }|\nabla \phi |^{p}\leq \liminf_{i\to +\infty } \int_{\Omega
}|\nabla \phi _{i}|^{p}=\mu\,.
\end{equation*}
Since $\phi \in A$, we get 
\begin{equation*}
\int_{\Omega }| \nabla \phi | ^{p}=\mu =\inf_{\psi \in K}\int_{\Omega }|
\nabla \psi | ^{p}.
\end{equation*}
The Lagrange multiplier theorem allows us to say that $\phi $ is a weak
solution of (\ref{3}).
\end{proof}

The regularity of $\phi $ follows from the next proposition, with the
following notation 
\begin{equation*}
W^{1,p}(\Omega )=
\begin{cases}
H_{1}^{p}(\Omega ) & \text{if }\partial \Omega =\phi \\ 
H_{1,0}^{p}(\Omega ) & \text{if }\partial \Omega \neq \phi \, .
\end{cases}
\end{equation*}

\begin{proposition} \label{p1}
Let $h\in C^{o}(\Omega \times R)$ be such that, for any
$(x,r)\in \Omega \times R$,  $| h(x,r)| \leq C|r| ^{p^{\ast }-1}+D$.

If $u\in W^{1,p}(\Omega )$ is a solution of $-\Delta _{p}u+h(x,u)=0$,
then $u\in C^{1,\alpha }(\Omega )$.
\end{proposition}

The above proposition was proved in (\cite{3}), in the context of compact
Riemannian manifolds without boundary. The proof is in its essence based on
the Sobolev inequality and since this latter is also valid in 
$\mathaccent"7017 {H}_{1}^{p}(\Omega )$ as in $H_{1}^{p}(\Omega )$, 
it follows that
proposition (\ref{p1}) remains true in the case of compact Riemannian
manifolds with boundary.

\section{Existence of a supersolution}

In this section we construct a positive supersolution of \eqref{1} on each
compact domain of $M$.

\begin{theorem} \label{thm4}
Let $\Omega $ be a compact domain of $M$. If $K$ is a smooth function
such that $K\geq c_{0}>0$ and $k$ is a smooth function with $k\leq K$ ,
then there exists a positive supersolution of  \eqref{1} in
$\Omega $.
\end{theorem}

\begin{proof}
Letting $u=e^{v}$ where $v\in H_{1}^{p}(\Omega )$ is a function which will
be precise later and $\ q=p^{\ast }-1$, then we get for every $\phi \in
H_{1}^{p}(\Omega )$ with $\phi \geq 0$ 
\begin{equation*}
\int_{\Omega }\Delta _{p}u\phi =\int_{\Omega }e^{(p-1)v}(\Delta
_{p}v+(p-1)|\nabla v|^{p})\phi
\end{equation*}%
and 
\begin{equation*}
\int_{\Omega }(\Delta _{p}u+ku^{p-1}-Ku^{q})\phi =\int_{\Omega
}e^{(p-1)v}(\Delta _{p}v+(p-1)|\nabla v|^{p}+k-Ke^{(q-p+1)v})\phi \text{.}
\end{equation*}%
So it suffices to show the existence of $v$ such that 
\begin{equation}
\int_{\Omega }e^{(p-1)v}(\Delta _{p}v+(p-1)|\nabla
v|^{p}+k-Ke^{(q-p+1)v}).\phi \leq 0  \label{4}
\end{equation}%
Let $b>0$ be a constant and consider the solution of $\Delta _{p}h=-b^{1-p}k$
\ which is guaranteed by Lemma \ref{lem1}.

Now putting $v=bh+t$ where $t$ is a real constant to be chosen later. The
inequality (\ref{4}) becomes 
\begin{equation*}
\int_{\Omega }e^{(p-1)(bh+t)}(b^{p-1}\Delta _{p}h+(p-1)b^{p}| \nabla h|
^{p}+k-Ke^{(q-p+1)(bh+t)})\phi \leq 0
\end{equation*}
If we choose $t$ such that $e^{(q-p+1)t}=b^{p-1}$, we will find that 
\begin{align*}
&\int_{\Omega }e^{(p-1)(bh+t)}((p-1)b| \nabla h| ^{p}-Ke^{(q-p+1)bh})\phi \\
&\leq \int_{\Omega }e^{(p-1)(bh+t)}((p-1)b| \nabla h| ^{p}-Km_{o})\phi \leq 0
\end{align*}
where $m_{o}=\min_{x\in \Omega }e^{(q-p+1)bh(x)}$ and since the function $K$ 
$\geq c_{o}>0$, we choose $b$ small enough so that 
\begin{equation*}
| \nabla h| ^{p}\leq \frac{c_{o}m_{o}}{b(p-1)}
\end{equation*}
we get the desired result.
\end{proof}

\section{Existence of a subsolution}

The operator $L_{p}u=-\Delta _{p}u-ku^{p-1}$ under Dirichlet conditions has
a first eigenvalue $\lambda _{1,p}^{\Omega }$ on each open and bounded
domain $\Omega \subset M$ which is variationally defined as 
\begin{equation}
\lambda _{1,p}^{\Omega }=\inf (\int_{\Omega }|\nabla \phi |^{p}-k|\phi |^{p})
\label{5}
\end{equation}%
where the infimum is extended to the set 
\begin{equation*}
A=\{\phi \in H_{1,0}^{p}(\Omega ):\int_{\Omega }|\phi |^{p}=1\}.
\end{equation*}%
Since $|\nabla \phi |=|\nabla |\phi ||$, we can assume that $\phi \geq 0$.
The corresponding positive eigenfunction is solution of the Dirichlet
problem 
\begin{equation}
\begin{gathered} \Delta _{p}\phi +k\phi ^{p-1}=-\lambda _{1,p}^{\Omega }\phi
^{p-1} \quad \text{in }\Omega \\ \phi >0 \quad \text{in }\Omega \\ \phi =0
\quad \text{on }\partial \Omega \end{gathered}  \label{6}
\end{equation}%
Let $\{\Omega _{i}\}_{i\geq 0}$ be an exhaustion of $M$ by compact domains
with smooth boundary such that 
$\Omega _{i}\subset \mathaccent"7017{\Omega }_{i+1}$

\begin{lemma} \label{lem2}
If $k$ is bounded function, then the sequence
$\lambda_{1,p}^{\Omega _{i}}$ defined by \eqref{5} converges.
\end{lemma}

\begin{proof}
By definition, $\lambda _{1,p}^{\Omega _{i}}$ is a decreasing sequence. Let 
$\lambda _{1,p}$ its limit, since the function $k$ is bounded, there exists a
constant $c>0$ such that $-k+c\geq 1$, then 
\begin{align*}
\int_{\Omega }| \nabla \phi | ^{p}+(c-k)\phi ^{p} &\geq \int_{\Omega }|
\nabla \phi | ^{p}+\phi ^{p} \\
&\geq 2^{1-p}((\int_{\Omega }| \nabla \phi | ^{p})^{1/p}+(\int_{\Omega }\phi
^{p})^{1/p})^{p} \\
&=2^{1-p}\Vert \phi \Vert _{H_{1}^{p}(\Omega )}^{p}
\end{align*}
so the operator $L_{p}u=-\Delta _{p}u+(c-k)u^{p-1}$ is coercive and we have,
for $\phi _{i}$ any eigenfunction corresponding to $\lambda
_{1,p}^{\Omega_{i}}$, 
\begin{align*}
\lambda _{1,p}^{\Omega _{i}} &=\int_{\Omega _{i}}| \nabla \phi _{i}|
^{p}-k\phi _{i}^{p} \\
&\geq -c+2^{1-p}\Vert \phi _{i}\Vert _{H_{1}^{p}(\Omega )}^{p} \\
&\geq -c+2^{1-p}\geq -c+2^{1-n}\,.
\end{align*}
Then $\lambda _{1,p}>-\infty$.
\end{proof}

\begin{lemma} \label{lem3}
 If $k$ is bounded, then the eigenfunction problem
\begin{equation}
\begin{gathered}
\Delta _{p}\phi +k\phi ^{p-1}=-\lambda _{1,p}\phi ^{p-1}\quad
\text{in }M \\
\phi >0\quad \text{in }M
\end{gathered}  \label{7}
\end{equation}
has a positive solution $\phi \in C_{loc}^{1,\alpha }(M)$.
\end{lemma}

\begin{proof}
Letting $(\Omega _{i})_{i\geq 1}$ be an exhaustive covering of the complete
manifold $M$ by compact subsets and $(\phi _{i})$ be the sequence of the
first nonvanishing eigenfonctions (positive) of the operator $L_{p}u=-\Delta
_{p}u-ku^{p-1}$ on each $\Omega _{i}$. Multiplying \eqref{7} by $\phi _{i}$
and integrating over $\Omega _{i}$, we get 
\begin{equation*}
\int_{\Omega _{i}}|\nabla \phi _{i}|^{p}-k\phi _{i}^{p}=\lambda
_{1,p}^{\Omega _{i}}\int_{\Omega _{i}}\phi _{i}^{p}=\lambda _{1,p}^{\Omega
_{i}}\leq \lambda _{1,p}^{\Omega _{1}}
\end{equation*}%
so that 
\begin{equation*}
\int_{\Omega _{i}}|\nabla \phi _{i}|^{p}\leq \max_{x\in M}|k|+\lambda
_{1,p}^{\Omega _{1}}<\infty .
\end{equation*}%
On the other hand, 
\begin{equation}
\begin{aligned} \Big(\Big(\int_{\Omega _{i}}| \nabla \phi _{i}|
^{p}\Big)^{1/p}+\Big(\int_{\Omega _{i}}\phi _{i}^{p}\Big)^{1/p}\Big)^{p}
&\leq 2^{p-1}(\int_{\Omega _{i}}|\nabla \phi _{i}| ^{p}+\phi _{i}^{p})\\
&\leq 2^{p-1}\Big(1+\max_{x\in M}| k| +\lambda _{1,p}^{\Omega _{1}}\Big)
<\infty \end{aligned}  \label{8}
\end{equation}%
and by the reflexivity of the space $H_{1}^{p}(M)$, we deduce that 
\begin{equation*}
\phi _{i}\rightarrow \phi \text{ weakly in }H_{1}^{p}(M)
\end{equation*}%
and 
\begin{equation}
\Vert \phi \Vert _{H_{1}^{p}(M)}^{p}\leq \liminf \Vert \phi _{i}\Vert
_{H_{1}^{p}(M)}^{p}.  \label{9}
\end{equation}%
Now since $\int_{M}\phi _{i}^{p}=1$, for every $\varepsilon >0$ there exists
a compact domain $K_{i}\subset M$ such that $\int_{M\backslash K_{i}}\phi
_{i}^{p}<\frac{\varepsilon }{2^{i}}$, let $K=\cap _{i=1}^{\infty }K_{i}$ and 
\begin{equation*}
\int_{M\backslash K}\phi _{i}^{p}=\int_{\cup _{i=1}^{\infty }(M\backslash
K_{i})}\phi _{i}^{p}\leq \sum_{i=1}^{\infty }\int_{M\backslash K_{i}}\phi
_{i}^{p}<\epsilon .
\end{equation*}%
From (\ref{8}) we obtain by Rellich-Kondrakov theorem that 
\begin{equation*}
\phi _{i}\rightarrow \phi \text{ strongly in }L^{p}(K).
\end{equation*}%
We claim that 
\begin{equation}
\int_{M}\phi ^{p}=1;  \label{10}
\end{equation}%
since, if it is not the case we have by (\ref{9}) 
\begin{equation*}
1-\int_{M}\phi ^{p}>0,
\end{equation*}%
consequently 
\begin{equation*}
1=\lim_{i\rightarrow \infty }\int_{M}\phi _{i}^{p}\leq \varepsilon
+\lim_{i\rightarrow \infty }\int_{K}\phi _{i}^{p}=\varepsilon +\int_{K}\phi
^{p}
\end{equation*}%
and hence $\varepsilon \geq 1-\int_{M}\phi ^{p}$. A contradiction with the
fact that $\varepsilon $ is arbitrary fixed.

Now from (\ref{9}) and (\ref{10}) we get 
\begin{equation*}
\int_{M}| \nabla \phi | ^{p}\leq \lim \inf \int_{M}| \nabla \phi _{i}| ^{p}
\end{equation*}
hence 
\begin{equation*}
\int_{M}|\nabla \phi |^{p}-k\phi ^{p}\leq \lim \inf ( \int_{M}|\nabla \phi
_{i}|^{p}-k\phi _{i}^{p})
\end{equation*}
which by lemma\ref{lem2} goes to $\lambda _{1,p}$, and since 
$\int_{M}\phi^{p}=1$, we obtain 
\begin{equation*}
\int_{M}| \nabla \phi | ^{p}-k\phi ^{p}=\lambda _{1,p}.
\end{equation*}
So $\phi $ is a weak solution of the equation 
\begin{equation*}
\Delta _{p}\phi +k\phi ^{p-1}=-\lambda _{1,p}\phi ^{p-1}
\end{equation*}
From proposition \ref{p1}, we deduce that $\phi \in C_{loc}^{1,\alpha }(M)$.

It remains to show that $\phi $ is positive, which is deduced from the next
proposition.

\begin{proposition}[Druet \cite{3}]\label{p2}
 Let $(\Omega ,g)$ be a compact
Riemannian $n$-manifold $n\geq 2$, $1<p<n$. Let $u\in C^{1}(\Omega )$ be
such that $-\Delta _{p}u+h(x,u)\geq 0$  on $\Omega $, $h$ fulfilling
the conditions
\begin{gather*}
h(x,r)<h(x,s),\quad x\in \Omega ,\; 0\leq r<s \\
| h(x,u)| \leq C(K+| r|^{p-2})| r| ,\quad  (x,r)\in M\times R,\; C>0.
\end{gather*}
If $u\geq 0$ on $\Omega $ and $u$ does not vanish identically, then $u>0$ on
$\Omega $.
\end{proposition}
\end{proof}

If $\lambda $ is an eigenvalue of the operator 
\begin{equation*}
L_{p}u=-\Delta _{p}\phi -k|\phi |^{p-2}\phi ,
\end{equation*}%
so is $\lambda +c$ for the operator 
\begin{equation*}
L_{c}u=-\Delta _{p}\phi -(k-c)|\phi |^{p-2}\phi 
\end{equation*}%
where $c$ is a constant and since $k$ is bounded function we choose $c$ such
that $c-k>0$, and then we get 
\begin{equation*}
-\Delta _{p}\phi +h(x,\phi )\geq 0
\end{equation*}%
where 
\begin{equation*}
h(x,\phi )=(c-k(x))\phi ^{p-1}.
\end{equation*}%
Obviously the function $h$ satisfies the assumptions of proposition \ref{p2}
and we have $\phi >0$.

Now we establish the following lemma which will be used later.

\begin{lemma}\label{lem4}
Let $M$ be a Riemannian manifold of bounded geometry. Suppose
that $a(x)$ is a bounded smooth function on $M$ and $u\in H_{1}^{p}(M)$ be a
weak solution of the equation
\begin{equation}
\Delta _{p}u+a(x)u^{p-1}=0   \label{11}
\end{equation}
then $u\in L^{\infty }(M)$.
\end{lemma}

\begin{proof}
We are going to use Moser's iteration scheme. Let $k\geq 1$ be any real and 
$t=k+p-1$. Multiplying (\ref{11}) by $u^{k}$ ($k>1$) and integrating over 
$M$, we get 
\begin{equation}
-k\int_{M}|\nabla u|^{p}u^{k-1}+\int_{M}a(x)u^{p+k-1}=0.  \label{12}
\end{equation}%
Using Sobolev's inequality, we get for any fixed $\varepsilon >0$ 
\begin{equation}
\begin{aligned} \| u^{\frac{t}{p}}\| _{p^{\ast }}^{p} &=\|u\|
_{t\frac{p^{\ast }}{p}}^{t}\\ &\leq (K(n,p)^{p}+\varepsilon)\| \nabla
u^{\frac{t}{p}}\| _{p}^{p}+B\| u\| _{t}^{t} \\ &=(K(n,p)^{p}+\varepsilon
)(\frac{t}{p})^{p}\| u^{\frac{t }{p}-1}\nabla u\| _{p}^{p}+B\| u\| _{t}^{t}
\end{aligned}  \label{13}
\end{equation}%
where $K(n,p)$ is the best constant in the Sobolev's embedding 
$H_{1}^{p}(R^{n})\subset L^{p\ast }(R^{n})$ (see Aubin \cite{1} or Talenti 
\cite{4}) and $B$ a positive constant depending on $\epsilon $; since 
\begin{equation*}
\Vert u^{\frac{t}{p}-1}\nabla u\Vert _{p}^{p}=\int u^{t-p}|\nabla u|^{p}
\end{equation*}%
and taking account of (\ref{12}) we get 
\begin{equation*}
\int u^{k}\Delta _{p}u=-k\int u^{k-1}|\nabla u|^{p}\leq \Vert a\Vert
_{\infty }\Vert u\Vert _{t}^{t}\,.
\end{equation*}%
Then (\ref{13}) becomes 
\begin{equation*}
\Vert u\Vert _{t\frac{p^{\ast }}{p}}^{t}\leq (K(n,p)^{p}+\varepsilon )
(\frac{t}{p})^{p}\frac{1}{k}(\Vert a\Vert _{\infty }+B)\Vert u\Vert _{t}^{t}
\end{equation*}%
so that 
\begin{equation}
\Vert u\Vert _{t\frac{p^{\ast }}{p}}\leq \Big((K(n,p)^{p}+\varepsilon )
\big(\frac{t}{p}\big)^{p}\frac{1}{k}(\Vert a\Vert _{\infty }+B)\Big)
^{\frac{1}{t}}\Vert u\Vert _{t}\,.  \label{14}
\end{equation}
Putting 
\begin{equation*}
\frac{t}{p}=\beta ^{i}
\end{equation*}
where $i$ is a positive integer and $\beta =\frac{p^{\ast }}{p}=\frac{n}{n-p}
$, (\ref{14}) becomes 
\begin{equation}
\Vert u\Vert _{p\beta ^{i+1}}\leq ((K(n,p)^{p}+\varepsilon )\beta
^{pi}(\Vert a\Vert _{\infty }+B))^{\frac{1}{_{_{p\beta ^{i}}}}}\Vert u\Vert
_{_{p\beta ^{i}}}\,.  \label{15}
\end{equation}
Recurrently, we obtain 
\begin{equation}
\ \Vert u\Vert _{p\beta ^{i+1}}\leq (K(n,p)^{p}+\varepsilon )^{\frac{1}{p}
(\sum_{j=0}^{i}\frac{1}{\beta ^{j}})}\beta ^{\sum_{j=0}^{i}\frac{j}{\beta
^{j}}}(\Vert a\Vert _{\infty }+B)^{\frac{1}{p}(\sum_{j=0}^{i}\frac{1}{\beta
^{j}})}\Vert u\Vert _{p}.  \label{16}
\end{equation}
Now, since 
\begin{equation*}
\sum_{j=0}^{\infty }\frac{1}{\beta ^{j}}=\frac{\beta }{\beta -1}=\frac{n}{p}
\end{equation*}%
and 
\begin{align*}
\sum_{j=0}^{\infty }\frac{j}{\beta ^{j}}& =\sum_{j=1}^{\infty }\frac{j}{%
(1+\pi )^{j}} \\
& \leq \sum_{j=1}^{\infty }\frac{j}{\sum_{p=0}^{j}C_{j}^{p}\pi ^{p}}%
=\sum_{j=1}^{\infty }\frac{1}{\pi \sum_{p=0}^{j-1}C_{j}^{p}\pi ^{p}} \\
& =\frac{1}{\pi }\sum_{j=1}^{\infty }\frac{1}{(1+\pi )^{j-1}}=\frac{1}{\pi }
\sum_{j=0}^{\infty }\frac{1}{(1+\pi )^{j}} \\
& =\frac{n-p}{p}\sum_{j=0}^{\infty }\frac{1}{\beta ^{j}}=\frac{n(n-p)}{p^{2}}
\,,
\end{align*}
it follows by letting $j\rightarrow \infty $ in (\ref{16}) that $u\in
L^{\infty }(M)$.
\end{proof}

\begin{theorem} \label{thm5}
Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension
$n\geq 3$ with bounded geometry. Suppose that
$k\in C^{\infty }(M)\cap L^{\infty}(M)$; then there exists a
positive subsolution of the equation
$\Delta_{p}u+ku^{p-1}-Ku^{p^{\ast }-1}=0$ on $M$.
\end{theorem}

\begin{proof}
Since $k\in L^{\infty }(M)$, there exists a positive constant $c>0$ such
that the operator $L_{c}u=-\Delta _{p}\phi +(c-k)\phi ^{p-1}$ is coercive,
so by lemma \ref{lem3} its first non vanishing eigenvalue $\lambda
_{1,p}+c>0 $. If $\phi $ denotes the corresponding positive eigenfunction to 
$\lambda_{1,p},$by lemma \ref{lem4} we may assume that $\phi <1$.

For $r>0$ we consider 
\begin{equation*}
u_{-}=\big(e^{r^{2}}-\phi ^{r^{3}}\big)^{\frac{1}{r}+1}
\end{equation*}
and by a direct computations we obtain in the sense of distribution 
\begin{gather*}
\nabla u_{-}=-r^{2}(r+1)(e^{r^{2}}-\phi ^{r^{3})^{\frac{1}{r}}}\phi
^{r^{3}-1}\nabla \phi, \\
\begin{aligned} \Delta _{p}u_{-}&=\left[ r^{2}(r+1)(e^{r^{2}}-\phi
^{r^{3})^{1/r}} \phi ^{r^{3}-1}\right] ^{p-1}\\ &\quad\times \big[ -\Delta
_{p}\phi +(p-1) \big(\frac{1-r^{3}}{\phi }+\frac{ r^{2}\phi
^{r^{3}-1}}{e^{r^{2}}-\phi ^{r^{3}}}\big)| \nabla \phi| ^{p}\big] .
\end{aligned}
\end{gather*}
Hence 
\begin{align*}
&\Delta _{p}u_{-}+ku_{-}^{p-1}-Ku_{-}^{q} \\
&=\big[ r^{2}(r+1)(e^{r^{2}}-\phi ^{r^{3})^{\frac{1}{r}}}\phi ^{r^{3}}\big] 
^{p-1} \\
&\quad \times \Big[ -\Delta _{p}\phi +(p-1)\big(\frac{1-r^{3}}{\phi }+\frac{
r^{2}\phi ^{r^{3}-1}}{e^{r^{2}}-\phi ^{r^{3}}}\big)| \nabla \phi | ^{p}+k%
\big(\frac{e^{r^{2}}-\phi ^{r^{3}}}{r^{2}(r+1)\phi ^{r^{3}}}\big)^{p-1}\phi
^{p-1} \\
&\quad -K\big(\frac{e^{r^{2}}-\phi ^{r^{3}}}{r^{2}(r+1)\phi ^{r^{3}}} \big)
^{p-1}(e^{r^{2}}-\phi ^{r^{3}})^{(q-p+1) (1+\frac{1}{r})}\phi ^{p-1}\Big] \\
&=\big[ r^{2}(r+1)(e^{r^{2}}-\phi ^{r^{3})^{\frac{1}{r}}}\phi ^{r^{3}-1} 
\big] ^{p-1} \\
&\quad\times \Big[ \lambda _{1,p}+(p-1)\frac{1}{\phi ^{p}} \big(1-r^{3}+
\frac{r^{2}\phi ^{r^{3}}}{e^{r^{2}}-\phi ^{r^{3}}}\big)| \nabla \phi |^{p} +k
\Big(\big(\frac{e^{r^{2}}-\phi ^{r^{3}}}{r^{2}(r+1)\phi ^{r^{3}}} \big)
^{p-1}+1\Big) \\
&\quad -K\big(\frac{e^{r^{2}}-\phi ^{r^{3}}}{r^{2}(r+1)\phi ^{r^{3}}} \big)
^{p-1}(e^{r^{2}}-\phi ^{r^{3}})^{(q-p+1) (1+\frac{1}{r})}\Big] .
\end{align*}
Now since 
\begin{equation*}
\lim_{r\to 0}(e^{r^{2}}-\phi ^{r^{3}})^{1+\frac{1 }{r}}=0
\end{equation*}
and 
\begin{equation*}
\lim_{r\to 0}\frac{r^{2}}{e^{r^{2}}-\phi ^{r^{3}}}=1\,,
\end{equation*}
we deduce that 
\begin{equation*}
u_{-}=(e^{r^{2}}-\phi ^{r^{3}})^{1+\frac{1}{r}}\in H_{1,\mathrm{loc}}^{p}(M)
\end{equation*}
is a subsolution of \eqref{1} and clearly $u_{-}\in C^{o}(M)\cap L^{\infty
}(M)$. The main theorem (Theorem \ref{thm3}) is a consequence of theorem 
\ref{thm4} and theorem \ref{thm5}.
\end{proof}

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\end{document}