\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 106, pp. 1--10.\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 2003 Texas State University-San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/106\hfil A reduction method for elliptic equations] {A reduction method for proving the existence of solutions to elliptic equations involving the $p$-laplacian} \author[Mohamed Benalili \& Youssef Maliki\hfil EJDE--2003/106\hfilneg] {Mohamed Benalili \& Youssef Maliki} \address{Mohamed Benalili \hfill\break Universit\'{e} Abou-Bakr BelKa\"{i}d\\ Facult\'{e} des Sciences\\ Depart. Math\'{e}matiques\\ B. P. 119, Tlemcen, Algerie} \email{m\_benalili@mail.univ-tlemcen.dz} \address{Youssef Maliki\hfill\break Universit\'{e} Abou-Bakr BelKa\"{i}d\\ Facult\'{e} des Sciences\\ Depart. Math\'{e}matiques\\ B. P. 119, Tlemcen, Algerie} \email{malyouc@yahoo.fr} \date{} \thanks{Submitted March 10, 2003. Published October 21, 2003.} \subjclass{58J05, 53C21} \keywords{Analysis on manifolds, semi-linear elliptic PDE} \begin{abstract} We introduce a reduction method for proving the existence of solutions to elliptic equations involving the $p$-Laplacian operator. The existence of solutions is implied by the existence of a positive essentially weak subsolution on a manifold and the existence of a positive supersolution on each compact domain of this manifold. The existence and nonexistence of positive supersolutions is given by the sign of the first eigenvalue of a nonlinear operator. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \section{introduction} Let $(M,g)$ be a complete non-compact Riemannian manifold of dimension $n\geq 3$. On this manifold, we consider the elliptic quasilinear equation \begin{equation} \label{e0.1} \Delta_p u+ku^{p-1}-Ku^{q}=0, \end{equation} with $q>p-1$, where $K\geq 0$ and $k\leq K$ are smooth functions on the manifold $M$ and $\Delta_p u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-laplacian operator of $u$. Under some positivity assumption on the function $K$, we reduce the existence of a weak positive solution to \eqref{e0.1} on $M$ to the existence of a positive essentially weak subsolution on $M$ together with the existence of a positive supersolution on each compact subdomain of $M$. The difficulty we face using the method of sub and supersolutions resides in seeking a positive subsolution $\underline{u}$ and a positive supersolution $\overline{u}$ that at the same time satisfy the condition $\underline{u}\leq \overline{u}$. Our reduction method makes easier the analysis of \eqref{e0.1} on general complete non-compact manifolds. This result extends the case studied by Peter Li et al \cite{t1} for the Laplace-Beltrami operator (i.e. $p=2$). In the third section, we show that the existence and the nonexistence of positive supersolutions to \eqref{e0.1} on arbitrary bounded subdomains of $M$ is completely determined by the sign of the first eigenvalue of the non-linear operator $L_p u=-\Delta_p u-k|u| ^{p-2}u$ on the zero set $Z_{o}=\{ x\in M : K(x)=0\} $ of the function $K$. This property was also obtained in \cite{t1} for the Laplace-Beltrami operator. \section{Reduction Result} \begin{definition} \label{def1}\rm A positive and smooth function $K$ is said to be essentially positive if there exists an exhaustion by compact domains $\{ \Omega _{i}\}_{i\geq 0}$ such that \begin{equation*} M=\cup_{i\geq 0} \Omega _{i}\quad \text{and}\quad K\big| _{\partial \Omega_{i}}>0\quad \forall i\geq 0. \end{equation*} Furthermore, If there is a positive weak supersolution $u_{i}\in H_{1}^p (\Omega _{i})\cap C^{o}(\Omega _{i})$ on each $\Omega _{i}$, then $K$ is called permissible. \end{definition} \begin{definition} \label{def2} \rm A positive solution $u$ of the equation \eqref{e0.1} is said to be maximal if for every positive solution $v$, we have $v\leq u$. \end{definition} In this section, we prove the following theorem. \begin{theorem} \label{thm1} Suppose that $K$ is permissible and $k\leq K$. If there exists a positive subsolution $\underline{u}\in H_{1,\rm loc}^p (M)\cap L^{\infty }(M)\cap C^{o}(M)$ of \eqref{e0.1} on $M$, then it has a weak positive and maximal solution $u\in H_{1}^p (M)$. Moreover $u$ is of class $C^{1,\alpha }$ on each compact set for some $\alpha \in (0,1)$. \end{theorem} To prove this theorem, we show the following lemmas. \begin{lemma} \label{lm1} Let $\Omega \subset M$ be a bounded domain. Assume that \eqref{e0.1} has a positive subsolution $\underline{u}\in H_{1,\rm loc}^p (\Omega )\cap C^{o}(\Omega )$ and a positive supersolution $\overline{u}\in H_{1,\rm loc}^p (\Omega )$. If $\ (\overline{u}-\underline{u})\big| _{\partial \Omega } \geq 0$ then $\overline{u}\geq \underline{u}$ on $\Omega$. \end{lemma} \begin{proof} First, we note that multiplying a positive supersolution $\overline{u}$ of \eqref{e0.1} by a constant $a\geq 1$ we get a supersolution. Indeed, \begin{align*} \Delta_p (au) +k(au) ^{p-1}-K(au) ^{q} &=a^{p-1}\left( \Delta_p u+ku^{p-1}\right) u^{q}-K(au) ^{q} \\ &\leq a^{p-1}Ku^{q}\left( 1-a^{q-p+1}\right) \\ &\leq 0. \end{align*} So we can assume without loss of generality that $\overline{u}\geq 1$ on a compact domain. Suppose that the set $S=\{ x\in \Omega :\overline{u} (x)<\underline{u}(x)\}$ is not empty. Let $\phi =\max (\underline{u}-\overline{u},0)$ be the test function which is positive and belongs to $H_{1,0}^p (\Omega )$. We have, \begin{align*} &\int_{S}\left\langle \left| \nabla \underline{u}\right| ^{p-2}\nabla \underline{u}-\left| \nabla \overline{u}\right| ^{p-2}\nabla \overline{u}% ,\nabla (\underline{u}-\overline{u})\right\rangle dv_g \\ &\leq \int_{S}(k(\underline{u}^{p-1}-\overline{u}^{p-1})\underline{(u}- \overline{u})-K(\underline{u}^{q}-\overline{u}^{q}))(\underline{u}-\overline{u})dv_g \\ &\leq \int_{S}K(\underline{u}^{p-1}-\overline{u}^{p-1}-\underline{u}^{q}+ \overline{u}^{q}))(\underline{u}-\overline{u})dv_g \\ &\leq \int_{S}K(\underline{u}^{p-1}(1-\underline{u}^{q-p+1})-\overline{u} ^{p-1}(1-\overline{u}^{q-p+1}))(\underline{u}-\overline{u})dv_g \\ &\leq \int_{S}K(\underline{u}^{p-1}(1-\underline{u}^{q-p+1})-\overline{u} ^{p-1}(1-\underline{u}^{q-p+1}))(\underline{u}-\overline{u})dv_g \\ &\leq \int_{S}K(1-\underline{u}^{q-p+1})(\underline{u}^{p-1}-\overline{u} ^{p-1})(\underline{u}-\overline{u})dv_g \\ &\leq 0\quad (q-p+1>0). \end{align*} If $p\geq 2$, by Simon inequality there exists a positive constant $C_p>0$ such that \begin{equation*} C_p \int_{S}\left| \nabla (\underline{u}-\overline{u})\right| ^p dv_g \leq \int_{S}\big\langle \left| \nabla \underline{u}\right| ^{p-2}\nabla \underline{u}-\left| \nabla \overline{u}\right| ^{p-2}\nabla \overline{u},\nabla (\underline{u}-\overline{u})\big\rangle dv_g \leq 0. \end{equation*} Hence, \[ \left\| (\underline{u}-\overline{u})^{+}\right\| _{H_{1,0}^p (\Omega )} =\int_{\Omega }\left| \nabla (\underline{u}-\overline{u})^{+}\right|^p dv_g = 0 \] i.e. $(\underline{u}-\overline{u})^{+}=0$, or $\underline{u}\leq \overline{u}$ on $\Omega$. For $1
0$ such that
\begin{equation*}
C_p'\int_{S}\frac{\left| \nabla (\underline{u}-\overline{u}
)\right| ^2 }{(\left| \nabla \underline{u}\right| +\left| \nabla
\overline{u}\right|)^{2-p}}dv_g \leq \int_{S}\big\langle \left|
\nabla \underline{u}\right| ^{p-2}\nabla \underline{u}-\left| \nabla
\overline{u}\right|^{p-2}\nabla \overline{u},\nabla (\underline{u}-\overline{u})
\big\rangle dv_g \leq 0
\end{equation*}
that is
\begin{equation} \label{e1.1}
\int_{S}\frac{\left| \nabla (\underline{u}-\overline{u})\right| ^2 }{%
(\left| \nabla \underline{u}\right| +\left| \nabla \overline{u}\right|
)^{2-p}}\,dv=0.
\end{equation}
It follows from the H\"{o}lder inequality that,
\begin{align*}
\int_{S}\left| \nabla (\underline{u}-\overline{u})\right| ^p dv_g
&=\int_{S}\frac{\left| \nabla (\underline{u}-\overline{u})\right| ^p }{%
(\left| \nabla \underline{u}\right| +\left| \nabla \overline{u}\right|
)^{p(1-\frac{p}{2})}}((\left| \nabla \underline{u}\right| +\left| \nabla
\overline{u}\right| )^{p(1-\frac{p}{2})}dv_g \\
&\leq \Big(\int_{S}\frac{\left| \nabla (\underline{u}-\overline{u})\right| ^2
}{(\left| \nabla \underline{u}\right| +\left| \nabla \overline{u}\right|
)^{2-p}}dv_g \Big)^{p/2}\Big(\int_{S}(\left| \nabla \underline{u}\right|
+\left| \nabla \overline{u}\right| )^p )^{1-\frac{p}{2}}dv_g \Big).
\end{align*}
By \eqref{e1.1}, we get
\[
\left\| (\underline{u}-\overline{u})^{+}\right\| _{H_{1,0}^p (\Omega )}
=\int_{\Omega }\left| \nabla (\underline{u}-\overline{u})^{+}\right|
^p dv_g = 0\,.
\]
Hence $\underline{u}\leq \overline{u}$ on $\Omega $.
\end{proof}
Let $H^{n}(-1)$ be the $n$-dimensional simply connected hyperbolic space of
sectional curvature equals to $-1$.
\begin{lemma} \label{lm2}
Let $\varepsilon >0$, $\beta >0$ and $\lambda $ constants, then there
exists a positive and increasing function $\phi _{\varepsilon }$ such that
the function $V_{\varepsilon }(x)=\phi _{\epsilon }(r(x))$, defined on the
geodesic ball $B(\varepsilon )\subset H^{n}(-1)$ satisfies
\begin{gather*}
\Delta_p V_{\varepsilon }+\lambda V_{\varepsilon }^{p-1}-\beta
V_{\varepsilon }^{q}\leq 0,\\
V_{\varepsilon }\big|_{\partial B(\varepsilon )}=\infty .
\end{gather*}
Here $r(x)$ is the distance function on the ball $B(\varepsilon )$
\end{lemma}
\begin{proof}
In polar coordinates, the metric of $H^{n}(-1)$ is
\begin{equation*}
ds^2 =dr^2 +\sinh ^2 (r)W^2
\end{equation*}
where $W^2 $ is the metric on the sphere $S^{n-1}$.
We get easily
\begin{equation*}
\Delta _{H^{n}(-1)}=\frac{\partial ^2 }{\partial r^2 }+( n-1)
\coth (r) \frac{\partial }{\partial r}+\frac{1}{\sinh ^2 (r)}\Delta _{S^{n-1}}
\end{equation*}
where $\Delta _{M}$ is the Laplace-Beltrami operator on the manifold $M$.
and
\begin{equation*}
\Delta_p u=|\nabla u|^{p-2}\Delta _{M}u+\big\langle \nabla
u,\nabla |\nabla u|^{p-2}\big\rangle .
\end{equation*}
For $p\in \left( 1,n\right)$, let $\Delta_p ^{M}u=\mathop{\rm div}
\left(|\nabla u| ^{p-2}\nabla u\right) $ be the $p$-Laplacian operator of $u$
on the manifold $M$.
For $q>p-1$ we consider the function $\phi :\left( 0,\varepsilon
\right) \rightarrow R$,
\begin{equation*}
\phi (r) =\left( \sinh ^2 ( \frac{\varepsilon }{2})
-\sinh ^2 ( \frac{r}{2}) \right) ^{-\alpha },
\end{equation*}
with $\alpha =\frac{p}{q-p+1}$. Setting
\begin{equation*}
a(r) =\sinh ^2 ( \frac{\varepsilon }{2}) -\sinh
^2 ( \frac{r}{2}), \quad
V( x) =\phi \left( r(x) \right),
\end{equation*}
we obtain
\begin{equation} \label{e1.2}
\Delta_p ^{H^{n}(-1)}V=\phi '{}^{{p-2}}\Delta _{H^{n}(-1)}V
+(p-2) \phi '{}^{p-2}\phi ''.
\end{equation}
A direct computation shows that
\[
\Delta _{H^{n}(-1)}V=\frac{1}{4}\alpha \left( \alpha +1\right) a(r) ^{-\left( \alpha +2\right) }\sinh ^2 (r) +\frac{1}{2}%
n\alpha a(r) ^{-\left( \alpha +1\right) }\cosh (r)\,
\]
Therefore,
\begin{align*}
\Delta_p ^{H^{n}(-1)}V+\lambda V^{p-1}
&=\left( \frac{\alpha }{2}\right)^{p-1}a(r)^{-\alpha p+\alpha -p}
\big[ \frac{1}{2}\left( p-1\right) \left(\alpha +1\right) \sinh ^p (r) \\
&\quad+ (n+p-2)a(r)\sinh ^{p-2}(r) \cosh (r)+\lambda a(r)^p\big] .
\end{align*}
Taking
\begin{align*}
C\left( \varepsilon ,\lambda ,p,q\right)
&=\frac{1}{2}(p-1)\left( \alpha +1\right) \left( \frac{\alpha }{2}\right) ^{p-1}\sinh ^p \left( \varepsilon
\right) \\
&\quad + \left( n+p-2\right) \left( \frac{\alpha }{2}\right) ^{p-1}a
\left( 0\right) \cosh \left( \varepsilon \right) +\lambda \left( a(0)\right) ^p ,
\end{align*}
we obtain
$\Delta_p ^{H^{n}(-1)}V+\lambda V^{p-1}\leq CV^{q}$
and putting
\begin{equation} \label{e1.3}
\psi =\big( \frac{C}{\beta }\big) ^{1/(q-p+1)}\phi \,,
\end{equation}
we obtain the desired function.
\end{proof}
\begin{lemma} \label{lm3}
Let $\Omega $ be a bounded domain. Suppose that there exists a compact
domain $X\subset \Omega $ such that $K\big|_{\partial X}>0$, then there
exists a constant $C>0$ such that for any positive regular solution $u$ of
\eqref{e0.1} on $\Omega $, we have
$u\big| _{\partial X}\leq C$,
where $\partial X$ is the boundary of $X$.
\end{lemma}
\begin{proof}
Since $X\subset \Omega $ is compact, it follows that there exist a positive
constant $\varepsilon >0$ less than the injectivity radius of $X$ and a
positive constant $\beta >0$ such that the $\varepsilon $-neighborhood of
$\partial X$, $U_{\varepsilon }(\partial X) $ is contained in
$\Omega $ and
\begin{equation} \label{e1.4}
K\big|_{_{U_{\varepsilon }(\partial X) }}\geq \beta >0.
\end{equation}
Let $x_{0}\in \partial X$ and let $r_{o}(x) =\mathop{\rm dist}(x_{0},x)$ be
the distance function on the geodesic ball $B(x_{0},\varepsilon )$.
Let $\Delta_p ^{M}$ be the $p$-lapalcian operator on the manifold $M$.
Let $\lambda =\sup_{x\in \Omega }k(x)$.
By Lemma \ref{lm2}, there exists a positive and increasing function
$V(x)=\phi _{\varepsilon}(r_{o}(x))$ defined on the geodesic ball
$B(\varepsilon )\subset H^{n}(-1)$
satisfying
\begin{equation} \label{e1.5}
\Delta_p ^{H^{n}(-1)}V_{\varepsilon }+\lambda V_{\varepsilon }^{p-1}\leq
\beta V_{\varepsilon }^{q}.
\end{equation}
Since $\Omega $ is bounded, by rescaling the metric if
necessary, we can assume that
\begin{equation*}
\mathop{\rm Ricci}\big|_{\Omega }\geq -(n-1) .
\end{equation*}
Knowing that the gradient of the distance function satisfies $| \nabla r|=1$,
we have
\begin{equation*}
\Delta_p ^{M}r=\Delta ^{M}r\,.
\end{equation*}
By a geometric comparison argument, we have
\begin{equation} \label{e1.6}
\Delta_p ^{M}r\leq \Delta_p ^{_{^{H^{n}(-1)}}}r.
\end{equation}
On the other hand,
\[
\Delta _{M}V_{\varepsilon } =\mathop{\rm div}\left( \nabla \phi _{\varepsilon
}(r(x))\right)
=\phi _{\varepsilon }'\Delta _{M}r+\phi _{\varepsilon }''.
\]
Then
\begin{equation*}
\Delta_p ^{M}V_{\varepsilon }=\phi _{\varepsilon }{'^{p-2}}
\Delta _{M}V_{\varepsilon }+\left( p-2\right) \phi _{\varepsilon
}^{\prime ^{^{p-2}}}\phi _{\varepsilon }''
\end{equation*}
and
\begin{equation*}
\Delta_p ^{M}V_{\varepsilon }=\phi _{\varepsilon }^{'^{p-1}}
\Delta _{M}r+\left( p-1\right) \phi _{\varepsilon }^{'^{p-2}}\phi _{\varepsilon }'' .
\end{equation*}
By the inequality \eqref{e1.6}, we have
\begin{equation*}
\Delta_p ^{M}V_{\varepsilon }\leq \Delta_p ^{H^{n}(-1)}V_{\varepsilon }
\end{equation*}
and from the inequalities \eqref{e1.4} and \eqref{e1.5},
we deduce that
\[
\Delta_p ^{M}V_{\varepsilon }+kV_{\varepsilon }^{p-1}-KV_{\varepsilon
}^{q} \leq \Delta_p ^{H^{n}(-1)}V_{\varepsilon }+\lambda V_{\varepsilon
}^{p-1}-\beta V_{\varepsilon }^{q} \leq 0.
\]
which implies that $V_{\varepsilon }$ is a positive supersolution of the
equation\eqref{e0.1} on $B(x_{0},\varepsilon )$.
Since $V_{\varepsilon }\big| _{\partial B(x_{0},\varepsilon )}=\infty $,
Lemma \ref{lm1} shows that for any solution $u$ of the equation \eqref{e0.1},
we have
\begin{equation*}
u(x)\leq V_{\varepsilon }(x)\text{ }\forall x\in B(x_{0},\varepsilon )
\end{equation*}
hence
\begin{equation*}
u(x_{0})\leq V_{\varepsilon }(x_{0})=\phi _{\varepsilon }(0) =C,
\end{equation*}
where $C$ is a positive constant independent of $x_0$ and $u$.
\end{proof}
\begin{lemma} \label{lm4}
Let $\Omega \subset M$ be a bounded domain. Suppose that
$K\big| _{\partial\Omega}>0$
and there is a positive and bounded solution
$v\in H_{1}^p (\Omega) \cap L^{\infty}(\Omega)$ of
the equation \eqref{e0.1} such that $v$ is bounded from below by a positive
constant. Then there exists a positive weak\ solution $u$ of the boundary-value
problem
\begin{gather*}
\Delta_p u+ku^{p-1}-Ku^{q}=0 \quad \mbox{on }\Omega \\
u=\infty \quad \mbox{on }\partial \Omega
\end{gather*}
and $u\geq v$ on $\Omega $. Moreover $u\in C^{1,\alpha }(X)$ on each compact
$X\subset \Omega $, and some $\alpha \in (0,1)$.
\end{lemma}
\begin{proof}
Let $C=\inf_{\Omega }v$ (which is positive by hypothesis). Since $v$ is
bounded from above on $\Omega $ then there exists $n_{0}\in N^{\ast }$ such
that $\sup_{\Omega }v\leq n_{0}C$.
Consider the boundary-value problem
\begin{equation}
\begin{gathered}
\Delta_p u+ku^{p-1}-Ku^{q}=0\quad \mbox{on }\Omega \\
u=nC\,,\quad n\geq n_{0} \quad \mbox{on }\partial \Omega\,.
\end{gathered}
\end{equation}
Obviously, $v\in H_{1}^p (\Omega) \cap L^{\infty }(\Omega) $ and
$nv\in H_{1}^p (\Omega) \cap L^{\infty}(\Omega)$ are respectively positive
sub and supersolutions of problem (2.7), and hence by the sub
and supersolutions method, the problem (2.7) has for each $n\geq n_{0}$
a positive solution $v_{n}\in H_{1}^p (\Omega) \cap L^{\infty }(\Omega)$ such
that $v\leq v_{n}\leq nv$.
Since $( v_{n+1}-v_{n}) \big|_{\partial \Omega }=C>0$, it follows
from Lemma \ref{lm1} that $\{ v_{n}\} _{n\geq n_{0}}$ in an increasing
sequence of positive solutions of the equation \eqref{e0.1} on $\Omega $.
Consider the set
\begin{equation*}
\Omega _{\varepsilon }=\left\{ x\in \Omega :\mathop{\rm dist}(x,\partial \Omega)
>\varepsilon \right\}
\end{equation*}
and setting $X=\overline{\Omega }_{\varepsilon }\subset \Omega $, which is
compact, then by Lemma \ref{lm3} there exists for each $\varepsilon >0$ (small
enough) a constant $C\left( \varepsilon \right) >0$ such that
\begin{equation}
\underset{\partial \Omega _{\varepsilon }}{\sup }v_{n}\leq C\left(
\varepsilon \right) \;\forall n\geq n_{0}.
\end{equation}
Consider the function\ $u=C\left( \varepsilon \right) C^{-1}v$ and take
$C\left( \varepsilon \right) $ such that $C\left( \varepsilon \right)
C^{-1}>1$, so that $u$ is a positive supersolution of the equation \eqref{e0.1}.
Since $\left( u-v_{n}\right) |_{\partial \Omega _{\varepsilon }}\geq 0$, it
follows from Lemma \ref{lm1} that $v_{n}\leq C\left( \varepsilon \right) C^{-1}v$ on
$\Omega _{\varepsilon }$ for all $n\geq n_{0}$, and then
$\left\{v_{n}\right\} _{n\geq n_{0}}$ is uniformly bounded on compact subsets of
$\Omega $. Hence$\left\{ v_{n}\right\} _{n\geq n_{0}}$, converges in the
distribution sense to a weak positive solution $u$ of the equation \eqref{e0.1} on
$\Omega $. By the regularity theorem $u\in C^{1,\alpha }(\Omega _{\epsilon})$
for some $\alpha \in \left( 0,1\right) $. It obvious that
$u|_{\partial\Omega }=\infty $.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm1}]
Let $\underline{u}\in H_{1,loc}^p ( M) \cap L^{\infty }(M) \cap C^{o}(M)$ a
positive subsolution of the equation \eqref{e0.1} on $M$. Since $K$ is
permissible then there exists an increasing sequence of
compact domains $\{ \Omega _{i}\} _{i\geq 0}$ such that
$M =\cup_{i}\Omega _{i}$ and $K\big|_{\partial \Omega _{i}}>0$ for all $i\geq 0$
and a positive supersolution
$\overline{u}_{i}\in H_{1}^p ( \Omega _i) \cap C^{o}( \Omega _i) $ on each
$\Omega _{i}$.
Since $\alpha \overline{u}$ (where $\alpha $ is a constant greater than
1) is again a positive supersolution of the equation \eqref{e0.1} on $\Omega _{i},$
we can assume that $\overline{u}_{i}\geq \underline{u}$ on $\Omega _{i}$.
Hence by the method of sub and supersolutions there exists a positive
solution $u_{i}\in C^{1,\alpha }(\Omega _{i})$ of the equation \eqref{e0.1} such
that $\underline{u}\leq u_{i}\leq \overline{u}_{i}$.
Since $u_{i}$ is bounded from below by $\underline{u}$ and $\Omega _{i}$ is
compact, then $u_{i}$ is bounded from below by a positive constant, thus it
follows from Lemma \ref{lm4} that there exists a positive
$C^{1,\alpha }(\Omega _{i})$-solution still denoted by $u_{i}$ of the
boundary-value problem
\begin{gather*}
\Delta_p u_{i}+ku_{i}^{p-1}-Ku_{i}^{q}=0\quad \mbox{in }\Omega _{i} \\
u_{i}=\infty \quad \mbox{on } \partial \Omega _{i}\,.
\end{gather*}
Since for each $i_{0}\geq 1$ we have
$( u_{i+1}-u_{i}) \big|_{\partial \Omega _{i_{0}}}\leq 0$, Lemma \ref{lm1}
implies that $\left\{u_{i}\right\} _{i\geq i_{0}}$ is a decreasing sequence of
positive solutions of the equation \eqref{e0.1} on $\Omega _{i_{0}}$.
Moreover, all $u_{i}$ are bounded
from below by $\underline{u}$, thus the sequence
$\left\{ u_{i}\right\}_{i\geq i_{0}}$ converges in distribution sense to a weak
solution of \eqref{e0.1}. By regularity theorem $u\in C^{1,\alpha }(\Omega _{i})$ for
some $\alpha \in \left( 0,1\right) $.
Now, if $v$ is an other solution of the equation \eqref{e0.1} on
$M =\underset{i}{\cup }\Omega _{i}$, then for $x_{0}\in M$ there exist
$i_{0}\geq 1$ such that $x_{0}\in \Omega _{i}$ for all $i\geq i_{0}$ , as
$u_{i}|_{\partial\Omega _{i}}=\infty $, Lemma \ref{lm1} implies that
$v\leq u_{i}$ for all $i\geq i_{0}$. In particular
$v\leq \underset{i\rightarrow \infty }{\lim }u_{i}=u$. Thus $u$ is maximal.
\end{proof}
\section{Existence of supersolution}
Let $K\geq 0$ and $k$ be smooth functions on the manifold $M$. In this
section we show that the existence or the nonexistence of a positive
supersolution on a bounded domain $\Omega \subset M$ is completely
determined by the sign of the first eigenvalue of the non linear operator $%
L_p u=-\Delta_p u-k|u| ^{p-2}u$ on the zero set $Z=\left\{x\in \Omega
:K(x)=0\right\} $ of the function $K$. Let us recall some definitions first.
\begin{definition} \label{def3} \rm
Let $\Omega \subset M$ be a bounded and smooth open set. The first eigenvalue
of the non linear operator $L_p u=-\Delta_p u-k|u| ^{p-2}u$
on $\Omega $ is
\begin{equation}
\lambda _{1,p}^{\Omega }=\inf \Big( \int_{\Omega }\left( \left| \nabla
u\right| ^p -k|u| ^p \right) dv_g \Big)
\end{equation}
where the infimum is taken over all functions $u\in H_{1,0}^p \left( \Omega
\right) $ such that $\int_{\Omega }|u| ^p dv_g =1$.
\end{definition}
\begin{definition} \label{def4} \rm
Let $S\subset M$ be a bounded subset. The first eigenvalue of the non linear
operator $L_p u=-\Delta_p u-k|u| ^{p-2}u$ on $\Omega $ is
\begin{equation}
\lambda _{1,p}^{S}=\sup \lambda _{1,p}^{\Omega }
\end{equation}
where the $sup$ is taken over all smooth open sets $\Omega $ containing $S$.
In particular $\lambda _{1,p}^{\phi }=+\infty $ .
\end{definition}
\begin{definition} \label{def5} \rm
Let $S\subset M$ be an unbounded subset. The first eigenvalue of the non-linear
operator $L_p u=-\Delta_p u-k|u| ^{p-2}u$ on $\Omega $ is
\begin{equation}
\lambda _{1,p}^{S}=\underset{r\rightarrow +\infty }{\lim }\lambda
_{1,p}^{\Omega _{r}}
\end{equation}
where $\Omega _{r}=S\cap \overline{B}\left( o,r\right) $ for all $r>0$
and $o\in M$ a fixed point.
\end{definition}
Let $\Omega $ be a bounded domain. It is known that there exists a unique $%
C^{1,\alpha }(\Omega )$-eigenfunction satisfying
\begin{gather*}
\Delta_p \phi +k\phi ^{p-1}+\lambda _{1,p}^{\Omega _{0}}\phi ^{p-1}=0 \quad%
\mbox{in }\Omega \\
\phi >0 \quad \mbox{in }\Omega \\
\phi =0 \quad \mbox{on }\partial \Omega \\
\frac{\partial \phi }{\partial \nu }<0 \quad\mbox{on }\partial \Omega\,.
\end{gather*}
Let $Z=\{ x\in M:K(x)=0\} $ the zero set of the smooth function $K $ and $%
\lambda _{1,p}^{Z\cap \Omega }$ be the first eigenvalue of the non-linear
operator $L_p u=-\Delta_p u-k|u| ^{p-2}u$ on $\Omega\cap Z$.
\begin{theorem} \label{thm2}
Let $K\geq 0$ be a smooth function on a bounded domain $\Omega$.
If $\lambda_{1,p}^{Z\cap \Omega }>0$, then there exists a positive supersolution
$\overline{u}\in H_{1}^p (\Omega )\cap L^{\infty }\left( \Omega \right) $ of
the equation \eqref{e0.1} on $\Omega $. Conversely if there exists a positive
supersolution $\overline{u}\in H_{1}^p (\Omega )\cap L^{\infty }(\Omega)$
of the equation \eqref{e0.1} then $\lambda _{1,p}^{Z\cap \Omega}\geq 0$.
\end{theorem}
\begin{proof}
Let $\Omega \subset M$ be a bounded domain. Suppose that
$\lambda_{1,p}^{Z\cap \Omega }>0$, it follows from the continuity of the first
eigenvalue with respect to $C^{0}$ deformation of the domain that there
exists a bounded domain $\Omega _{0}$ such that $Z\cap \Omega \subset \Omega
_{0}\subset \Omega $ and $\lambda _{1,p}^{\Omega _{0}}>0$.
On $\Omega _{0}$ there exists a unique positive eigenfunction $\phi \in
C^{1,\alpha }\left( \overline{\Omega }_{0}\right) $ such that
\begin{gather*}
\Delta_p \phi +k\phi ^{p-1}+\lambda _{1,p}^{\Omega _{0}}\phi ^{p-1}=0
\quad\mbox{in } \Omega _{0} \\
\phi >0 \quad \mbox{in } \Omega _{0} \\
\phi =0 \quad \mbox{on }\partial \Omega _{0} \\
\frac{\partial \phi }{\partial \nu }<0 \quad\mbox{on }\partial \Omega _{0}\,.
\end{gather*}
Writting
$\Omega =\left( \Omega \backslash \Omega _{0}\right) \cup \left( \Omega \cap
\Omega _{0}\right)$
and setting
\begin{equation*}
\overline{u}=\chi _{\Omega _{0}}\phi +C\left( 1-\chi _{\Omega _{0}}\right)
\end{equation*}
where $\chi _{\Omega }$ is the characteristic function,
\begin{equation*}
\chi _{\Omega }=\begin{cases}
1 & \mbox{if } x\in \Omega \\
0 & \mbox{if } x \notin \Omega
\end{cases}
\end{equation*}
and $C$ is a positive constant large enough so that $\overline{u}=C$, on
$\Omega -\Omega _{o}$, is a positive supersolution of \eqref{e0.1}.
On $\Omega \cap \Omega _{0}$, $\overline{u}=\phi $, but
\[
\Delta_p \overline{u}+k\overline{u}^{p-1}-K\overline{u}^{q} =-\lambda
_{1,p}^{\Omega _{0}}\overline{u}^{p-1}-K\overline{u}^{q}
\leq 0
\]
because $\lambda _{1,p}^{\Omega _{0}}>0$.
Therefore, $\overline{u}\in H_{1}^p (\Omega )\cap L^{\infty }\left( \Omega
\right)$ is positive supersolution of the equation \eqref{e0.1} on $\Omega $.
Conversely, suppose that there exists a positive supersolution
$\overline{u}\in H_{1}^p (\Omega )\cap L^{\infty }\left( \Omega \right)$ of
\eqref{e0.1} and $\lambda _{1,p}^{Z\cap \Omega }<0$.
It follows again from the continuity of the first eigenvalue with respect to
$C^{0}$-deformation of the domain that there exists a bounded domain
$\Omega_{1}$ such that $Z\cap \Omega \subset \Omega _{1}\subset \Omega $ and
$\lambda _{1,p}^{\Omega _{1}}<0$. By the same way as above, we can find a
decreasing sequence $\left\{ \Omega _{i}\right\} _{i\geq 0}$ of bounded
domains such that $\Omega _{i}\subset \Omega $,
$Z\cap \Omega =\cap_{i}\Omega _{i}$ and $\lambda _{1,p}^{\Omega _{i}}<0$.
On $\Omega _{i}$ there exists a positive eigenfunction
$\phi _{i}\in C^{1,\alpha }(\overline{\Omega }_{i})$ and
$\frac{\partial \phi _{i}}{\partial \nu }<0$ on $\partial \Omega _{i}$ satisfying
\begin{gather*}
\Delta_p \phi _{i}+k\phi _{i}^{p-1}+\lambda _{1,p}^{\Omega _{i}}
\phi_{i}^{p-1}=0 \quad\mbox{in } \Omega _{i} \\
\phi _{i}=0 \quad\mbox{on }\partial \Omega _{i}\,.
\end{gather*}
Consider the boundary-value problem, with $q>p-1$,
\begin{equation} \label{e2.4}
\begin{gathered}
\Delta_p u_{i}+ku_{i}^{p-1}-Ku_{i}^{q-1}=0 \quad \mbox{in }\Omega_{i} \\
u_{i}=0 \quad\mbox{on }\partial \Omega _{i}\,.
\end{gathered}
\end{equation}
One can check that for $\varepsilon >0$ small and $C>0$ large,
$\varepsilon \phi _{i}$ and $C\overline{u}$ are respectively positive sub
and supersolutions of the boundary-value problem \eqref{e2.4}
and $\varepsilon \phi _{i}\leq C\overline{u}$.
Therefore, by the sub and supersolutions method there exists a positive
$C^{1,\alpha }$ solution $u_{i}$ of the problem \eqref{e2.4} such that
$\varepsilon \phi _{i}\leq u_{i}\leq C\overline{u}$, we have also
$\frac{\partial u_{i}}{\partial \nu }<0$ on $\partial \Omega _{i}$.
Thus $\frac{\phi _{i}}{u_{i}}$ and
$\frac{u_{i}}{\phi _{i}}\in L^{\infty }( \Omega _i)$.
Consider now the set $\Omega _{i,C}=\left\{ x\in \Omega _{i}:C\phi
_{i}(x)