\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 $10$ 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 $\Omega _{i,C}=\left\{ x\in \Omega _{i}:C\phi _{i}(x) 0$, then there exists a positive supersolution $\overline{u}\in C^{1,\alpha }(\Omega)$ of the equation \eqref{e0.1} on $\Omega $ for some $\alpha \in (0,1)$. \end{theorem} \begin{proof} Let $\Omega _{o}$, $\Omega _{1}$ be bounded domains such that $Z\cap \Omega \subset \Omega _{o}\subset \Omega _{1}$ such that $\lambda _{1,p}^{Z\cap \Omega }>0$. Let $v\in C^{1,\alpha }\left( \Omega _{1}\right) $ be the first eigenfunction on $\Omega _{1}$ and $0\leq \phi \leq 1$ a smooth function such that $\phi =1$ on $\Omega _{o}$, $0$ outside $\Omega _{1}$. We can check easily as in \cite[Theorem 2.1]{t1} that the function $u=\phi v+(1-\phi )C$, where $C$ is a suitably chosen constant, is a positive $C^{1,\alpha }(\Omega) $ supersolution of the equation \eqref{e0.1}. \end{proof} \begin{corollary} \label{coro1} Let $Z$ be the zero set of the function $K$. Suppose that the first eigenvalue $\lambda _{1,p}^{Z}$ of the operator $L_p u=-\Delta_p u-k(x)|u| ^{p-2}u$ is strictly positive. Then the function $K$ is permissible. In particular if $K>0$ on $M$, $K$ is permissible \end{corollary} \section{Example} Consider the cylinder $M=R^{+}\times N$ where $(N,h)$ is a compact manifold with Riemannian metric $h$ and of scalar curvature $S_{h}\geq 0$. We endows $% M$ with the metric \begin{equation*} g=dr^2 +f(r)^2 h \end{equation*} where $f$ is smooth positive function. Denote by $\Gamma _{i,j}^{l},S_{g,}% \overline{R}_{ijl}^{k}$ and $R_{ijl}^{k}$ $1\leq i,j,k,l\leq n$ respectively the Christofell symbols, the scalar curvature, the curvature tensor on $M$ and the curvature tensor on $N$. >From the local expression of $\Gamma _{ij}^{\alpha }$, \begin{equation*} \Gamma _{ij}^{\alpha }=\frac{1}{2}g^{\alpha l}\big(\frac{\partial g_{jl}}{% \partial x_{i}}+\frac{\partial g_{il}}{\partial x_{j}}-\frac{\partial g_{ij}% }{\partial x_{l}}\big), \end{equation*} we have \begin{gather*} \Gamma _{ij}^{1} =-f(r)f^{\prime}(r)h_{ij},\quad 2\leq i,j\leq n \\ \Gamma _{i1}^{1} = 0, \quad 1\leq i\leq n \\ \Gamma _{11}^{\alpha } =0, \quad 1\leq \alpha \leq n \\ \Gamma _{1j}^{\alpha } =-f(r)/f^{\prime}(r)\delta _{j}^{\alpha } , \quad 2\leq \alpha ,\; j\leq n \\ \Gamma _{ij}^{\alpha } =\frac{1}{2}g^{\alpha l}(\frac{\partial g_{jl}}{% \partial x_{i}} +\frac{\partial g_{il}}{\partial x_{j}}-\frac{\partial g_{ij}% }{\partial x_{l}}), \quad 2\leq i,j,\alpha \leq n\,. \end{gather*} A direct computation gives \begin{gather*} \overline{R}_{1\alpha 1}^{\alpha } =-f^{\prime\prime}(r)/f(r), \quad 2\leq \alpha \leq n \\ \overline{R}_{i1j}^{1} = -f(r)f^{\prime\prime}(r)h_{ij},\quad 2\leq i,j\leq n \\ \overline{R}_{i\alpha \alpha }^{\alpha } = 0,\quad 1\leq i,\alpha \leq n \\ \overline{R}_{i\alpha j}^{\alpha } = R_{i\alpha j}^{\alpha }-f^{\prime}(r)^2 h_{ij},\quad 2\leq i,j,\alpha \leq n,\;j\neq \alpha \end{gather*} so \begin{equation*} S_g =-2(n-1)f^{\prime\prime}(r)/f(r)-(n-1)(n-2)f^{\prime}(r)^2 /f(r)^2 +% \frac{S_{h}}{f(r)^2 }\,. \end{equation*} When we take $f(r)=\exp r^2 $, \begin{gather} f^{\prime}(r)>0 \label{e2.5} \\ \lim_{r\rightarrow \infty } f(r)=\lim_{r\rightarrow \infty }\lim f^{\prime}(r)/f(r) =\lim_{r\rightarrow \infty } f^{\prime\prime}(r)/f(r) =\infty \label{e2.6} \end{gather} For $r>0$ large enough, by inequalities \eqref{e2.5} and \eqref{e2.6} we obtain $S_g \leq -\varepsilon$. By re-parametrizing, we can assume that \begin{equation} \label{e2.7} S_g \leq -\varepsilon \quad \text{for any }r>0\,. \end{equation} Let $K=\varepsilon +4(n-1)(1+nr^2 )$ then $k=-S_g \leq K$. Now consider on $M $ the equation \begin{equation} \label{e2.8} \Delta_p u-S_g u^{p-1}-Ku^{p^{\ast }-1}=0 \end{equation} with $2