\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 80, pp. 1--22.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/80\hfil Self similar solutions] {Self similar solutions of generalized Burgers equation} \author[A. Gmira, A. Hamydy, S. Ouailal\hfil EJDE-2005/80\hfilneg] {Abdelilah Gmira, Ahmed Hamydy, Salek Ouailal} % in alphabetical order \address{Universite Abdelmalek Essaadi, Faculte des Sciences Departement \\ de Mathematiques et informatique B. P. 2121 Tetouan - Maroc} \email[A. Gmira]{gmira@fst.ac.ma} \email[A. Hamydy]{hamydy@caramail.com} \email[S. Ouailal]{salekouailal@yahoo.com} \date{} \thanks{Submitted December 6, 2004. Published July 15, 2005.} \subjclass[2000]{35k55, 35k65} \keywords{Burgers equation; self similar; classification; asymptotic behavior} \begin{abstract} In this paper, we study the initial-value problem \begin{gather*} (|u'|^{p-2}u')'+\beta r u'+\alpha u-\gamma |u|^{q-1}u|u'|^{p-2}u'=0, \quad r>0, \\ u(0)=A,\quad u'(0)=0, \end{gather*} where $A>0$, $p>2$, $q>1$, $\alpha>0$, $\beta>0 $ and $ \gamma \in{\mathbb{R}}$. Existence and complete classification of solutions are established. Asymptotic behavior for nonnegative solutions is also presented. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \section{Introduction} This paper concerns the nonlinear parabolic equation \begin{equation} U_t-( |U_x |^{p-2}U_x)_x=-\frac kt U+|U|^{q-1}U|U_x|^{p-2}U_x \quad \text{in }{\mathbb{R}\times\mathbb{R}}^{+},\label{E11} \end{equation} where $p>2$, $q>1$ and $k>0$. As is often the case in nonlinear PDE's of parabolic type the characteristic properties of an equation, are displayed by means of the existence of so-called self similar solutions; this is our main interest. It is worth mentioning that if $p=2$, $q=1$ and $k=0$, we get the classical one dimensional Burgers equation \begin{equation} U_t=U_{xx}+U_xU, \label{E12} \end{equation} which is originally proposed as a simplified model of Navier-Stokes Turbulence (see \cite{B1} and \cite {B2}) . By design, Burgers equation is the simplest model of hydrodynamic flow that captures the interaction of nonlinear wave propagation and viscosity. Burgers turbulence is often viewed as a pared-down model of acoustic turbulence (see \cite{GM} and \cite{SZ}). The importance and popularity of equation (\ref{E12}) lie in its simplicity and in the fact that the well known Hopf-cole substitution $w=\frac{U_{x}}{U}$ reduces it to the linear heat equation. This nonlinear change of variables permits an explicit description of solutions of (\ref{E12}) and explains their essentially nonlinear first order asymptotic as $t$ goes to infinity. If $p=2$ and $k\neq 0$, equation (\ref{E11}) becomes \begin{equation} U_t=U_{xx}+|U|^{q-1}UU_x-\frac ktU \label{E13} \end{equation} which is studied by \cite{SR}. Note that, if $q=1$, equation (\ref{E13}) describes the propagation of weakly nonlinear longitudinal waves in gases or liquids from a non planar source (see \cite {LS} and \cite {L}). If $p>2$, equations (\ref{E11}) appears in the description of ice sheet dynamics (see \cite{F} ) where the reaction term $-\frac{k}{t}U$ can be considered as a turbulent term. In this case the selfsimilar solutions of problem (\ref{E11}) take the form \[ U(x,t)=t^\sigma f(y),\quad \text{where}\quad y= xt^\eta, \] with \[ \sigma=\frac{-1}{pq+p-2} \quad \text{and}\quad \eta=\frac{- q}{pq+p-2}. \] Then the profile $f$ is determined as a solution in of the ODE \[ ( |f'|^{p-2}f') '+\frac{q}{pq+p-2}yf' -(k-\frac{1}{pq+p-2})f+|f|^{q-1}f|f'|^{p-2}f'=0,\quad y\in {\mathbb{R}}. \] Where the prime denotes the differentiation with respect to $y$. If we set \[ \ g(y)= \left\{ \begin{array}{l} f(y)\quad \forall $ y $\in \mathbb {R}^{+},\\ f(-y)\quad \forall $ y $\in \mathbb {R}^{-}, \end{array} \right . \] then, $g$ satisfies \[ ( |g'|^{p-2}g') '+\alpha g +\beta yg'+ \gamma|g|^{q-1}g|g'|^{p-2}g'=0, \] in ${\mathbb{R}}$; with $\alpha=-k+\frac{1}{pq+p-2}$, $\beta=\frac{q}{pq+p-2}$ \[ \alpha=-k+\frac{1}{pq+p-2}, \beta=\frac{q}{pq+p-2} \] and $\gamma=-1$, if $y>0$; $\gamma=1$, if $y<0$. Consequently, we have just to focus on the study of the initial-value problem \begin{equation} \begin{gathered} \label{Q} ( |u'|^{p-2}u') '+\beta ru'+\alpha u-\gamma |u|^{q-1}u|u'|^{p-2}u'=0, \quad r>0 \\ u(0)=A,\quad u'(0)=0, \end{gathered} \end{equation} when $\alpha >0$, $\beta >0$ and $\gamma \in \mathbb{R}$. We will mainly discuss: (i) The existence and uniqueness of solutions for \eqref{Q}; (ii) the asymptotic behavior of positive solutions, and (iii) a classification of solutions. The main results of this paper are the following. \begin{theorem} \label{thm1} Assume $p>2$, $q>1$, $\alpha >0$, $\beta >0$, and $\gamma \in \mathbb{R}$. Then for each $A>0$, there exists a real $R_{\rm max}>0$ such that \eqref{Q} has a unique solution $u\equiv u(.,A)$ defined in the right open interval $[0,R_{\rm max}[$, meaning that $u$ and $|u'|^{p-2}u'$ are a $C^1$ functions in $[0,R_{\rm max}[$, satisfying \eqref{Q}. \end{theorem} The following result gives the monotonicity of solutions of problem \eqref{Q} with respect to initial data. \begin{theorem} \label{thm2} Assume $\alpha>0$, $\beta>0$ and $\gamma<0$. Let $u(.,A)$ and $u(.,B)$ be two solutions of problem \eqref{Q} with $u(0,A)=A$, $u(0,B)=B$ and $A\neq B$. Then $u(.,A)$ and $u(.,B)$ can not intersect each other before their first zero. \end{theorem} Concerning the asymptotic behavior, we have the following results. \begin{theorem} \label{thm3} Let $u$ be a strictly positive solution of \eqref{Q}. Then \[ \lim_{r\to +\infty }u(r)=\lim_{r\to+\infty }u'(r)=0. \] Furthermore, if $\alpha>0$, $\beta>0$ and $\gamma\leq0$, then \[ \lim_{r\to+\infty }r^{\frac \alpha \beta }u(r)=L \] exists and lies in $[ 0,+\infty [$. Moreover this limit $L$ is strictly positive for $0<\alpha\leq\beta$ and $\gamma<0$. \end{theorem} Finally, the structure of solutions of problem \eqref{Q} consists of three families: The set of strictly positive solutions, the set of changing sign solutions and finally solutions with compact support. This classification depends strongly on the sign of $\gamma(\alpha-\beta)$. \begin{theorem} \label{thm4} Assume $p>2, q>1$, and $\gamma<0$. Then we have \begin{itemize} \item[(i)] For $\alpha>\beta$ there exist two constants $A_1$ and $A_2$ such that for any $A>A_1$, the solution $u(.,A)$ is strictly positive and for $A0$ is quite the opposite. \begin{theorem} \label{thm5} Assume $p>2$, $q>1$ and $\gamma>0$. Then \begin{itemize} \item[(i)] If $\alpha\geq\beta$ any solution of \eqref{Q} change sign. \item[(ii)] If $\alpha<\beta$, there exist two constants $A_1$ and $A_2$ such that for any $AA_1, u(.,A)$ changes sign. \end{itemize} \end{theorem} The organization of this paper is as follows. Theorems \ref{thm1}, \ref{thm2} and \ref{thm3} are proved in section 2. In section 3 a classification of solutions is investigated and then Theorems \ref{thm4} and \ref{thm5} are established. \section{Existence and asymptotic behavior of solutions} In this section, we investigate existence, uniqueness and asymptotic behavior of solutions of the problem \eqref{Q}. We start with a local existence and uniqueness result. \begin{proposition} \label{prop2.1} Assume $p>2$, $q>1$, $\alpha >0$, $\beta >0$ and $\gamma \in {\mathbb{R}}$. Then for each $A>0$, there exists a right open interval $I=[0,R_{\rm max}[$ and a unique function $u$ such that, $u$ and $|u'|^{p-2}u'$ lie in $C^1(I)$ and satisfy \eqref{Q}. \end{proposition} First of all, we note that, for a fixed $\alpha, \beta$ and $\gamma$, it easy to see that $u(.,\gamma,A)=-u(.,\gamma,-A)$. Therefore, in the sequel we restrict ourselves to the case of $A>0$. \begin{remark} \label{rmk2.1} \rm The first equation in \eqref{Q} can be reduced to the first order system \begin{equation} \begin{gathered} X'=|Y|^{-\frac{p-2}{p-1}}Y \\ Y'=-\alpha X-\beta|Y|^{-\frac{p-2}{p-1}}Y+\gamma|X|^{q-1}XY. \end{gathered} \label{E21} \end{equation} Since the mapping \[ (X,Y)\mapsto \Big( |Y|^{-\frac{p-2}{p-1}}Y , -\alpha X-\beta|Y|^{-\frac{p-2}{p-1}}Y+\gamma|X|^{q-1}XY \Big) \] is a locally Lipschitz continuous function in the set $\{ (X,Y)\in \mathbb{R}\times \mathbb{R}^{*}\}$, we deduce that, for any $ r_0>0, A\geq0$ and $B\neq0$, there exists a unique solution of \eqref{Q} in a neighborhood of $r_0$ such that $u(r_0)=A$ and $u'(r_0)=B$. \end{remark} Because of the presence of the term $|Y|^{-\frac{p-2}{p-1}}Y$, the above function is not locally Lipschitz continuous near $r_0$ whenever $u'(r_0)=0$. Consequently, for our problem \eqref{Q} the above argument does not work. To avoid this difficulty, we make use an idea from \cite{GV}. Then the proof becomes similar to that of \cite[proposition 1.1]{GB1} and \cite[proposition 2.1]{GB2}. We present it here for the convenience of the reader. \begin{proof}[Proof proposition \ref{prop2.1}] The idea of the proof is to convert our initial value problem \eqref{Q} to a fixed point problem of some operator. This will be done in two steps. \noindent\textbf{Step 1.} Local existence and uniqueness. It is clear that to solve problem \eqref{Q} is equivalent to find a function $u \in C^1(I)$ defined in some interval $I=[0,R[$ with $R>0$ such that $|u'|^{p-2}u'\in C^1(I)$ and satisfies the integral equation \begin{equation} u(r)=A-\int_0^rG(F_u)(s)ds, \label{E22} \end{equation} where $G(s)=|s|^{(2-p)/(p-1} s$, for all $s\in \mathbb{R}$, and \begin{equation} F_u(s)=\beta su(s) +( \alpha -\beta ) \int_0^s u(\tau)d\tau-\gamma \int_0^s|u|^{q-1}u(\tau ) |u'|^{p-2}u'(\tau)d\tau. \label{E23} \end{equation} Now, let us define on $[0,A]$ the following two functions \begin{equation} f_1(X)=\begin{cases} \alpha (A-X)-|\gamma |X^{p-1}(A+X)^q & \text{if }\alpha \geq \beta , \\ \alpha (A+X)-2\beta X-|\gamma |X^{p-1}(A+X)^q & \text{if }\alpha <\beta, \end{cases} \label{E24} \end{equation} and \begin{equation} f_2(X)=\begin{cases} (A+X)\left\{ \alpha +|\gamma |X^{p-1}(A+X)^{q-1}\right\} & \text{if }\alpha \geq \beta , \\ \alpha (A-X)+|\gamma |X^{p-1}(A+X)^q &\text{if }\alpha <\beta . \end{cases} \label{E25} \end{equation} Since $f_1$ is continuous and $f_1(0)=\alpha A>0$, then there exists some interval $[0,A_0]\subset[0,A]$ such that \[ f_1(X)>0 \quad \forall X\in[0,A_0]. \] Let us introduce some useful notation for the proof: \begin{equation} f_1(A_0)=K_1, \quad f_2(A_0)=K_2,\quad R_0=\inf \{ 1, \frac{A_0^{p-1}}{2\Gamma A}, \frac{K_1^{p-2}}{% (2\Gamma )^{p-1}}\}, \label{E26} \end{equation} where \begin{equation} \Gamma =\beta +|\alpha -\beta |+(2p-2+q)2^{q-1}|\gamma |A^{q+p-2}. \label{E27} \end{equation} It is easy to see that the function $f_2$ satisfies the estimate \begin{equation} f_2(X)\leq2A\Gamma, \quad \forall X\in [0,A]. \label{E28} \end{equation} Now, we consider the complete metric space \begin{equation} X=\{ \varphi \in C^1([ 0,R_0] ):\| \varphi-A\| _X\leq A_0\} \label{E29} \end{equation} where \begin{equation} \|\varphi\|_X=\max(\|\varphi\|_0,\|\varphi'\|_0). \label{E210} \end{equation} and $\|.\|_0$ denotes the sup norm. Next we define the mapping $\mathcal{T}$ on $X$, by \begin{equation} T(\varphi)=A-\int_0^rG(F_\varphi)(s)ds,\quad \forall r\in[0,R_0] \label{E211} \end{equation} \noindent \textbf{Claim 1.} $\mathcal{T}$ maps $X$ into itself. In fact, take $\varphi\in X$. First, it is easy to see that $\mathcal{T}(\varphi )\in C^1( [ 0,R_0] )$. Also by a simple calculation we get \begin{equation} 0From the choice of $R_0$, these last two equations imply that $\mathcal{T}$ is a contraction. The use of the Banach's Contraction theorem leads to the existence of a unique function $u$ solving problem \eqref{Q} in $(0,R_0)$. \noindent\textbf{Step 2.} $|u'|^{p-2}u'\in C^1([0,R_0))$. We have just to prove the regularity at $r = 0$. For this purpose, note that the first equation in \eqref{Q} gives \begin{equation} \lim_{r\to 0}(|u'|^{p-2}u')'(r)=-\alpha A. \label{E224} \end{equation} Integrating equation \eqref{Q} from $0$ to $r$, and letting r go to 0, we obtain \begin{equation} \lim_{r\to 0}\frac{|u'|^{p-2}u'(r)}r=-\alpha A. \label{E225} \end{equation} Hence $|u'|^{p-2}u'\in C^1(([0,R_0[)$. This completes the proof of Proposition \ref{prop2.1}. \end{proof} \begin{remark} \label{rmk2.2} \rm It is not difficult to see that the solution $u$ of \eqref{Q} is a $C^\infty $ function at any $r>0$ whenever $u'(r)\neq 0$. \end{remark} The remaining of this section is devoted to the proof of the Theorem \ref{thm2}. For this purpose we start with the following lemma. \begin{lemma} \label{lem2.1} Let $A>0$ and $u$ be the corresponding solution of \eqref{Q}. Then as long as $u$ is strictly positive we have $00$. \end{proof} The next result gives the monotonicity of solutions of the problem \eqref{Q} with respect to initial data. More exactly, we have \begin{proposition} \label{prop2.2} Assume $\alpha>0$, $\beta>0$ and $\gamma<0$. Let $00$, we set \begin{equation} u_k(r)=k^{-p/(p-2)}u(kr),\quad r\in [0,\frac{R_1}k]. \label{E230} \end{equation} Then $u_k$ satisfy the equation \begin{equation} (|u_k'|^{p-2}u_k'(r))'+\beta ru_k'(r)+\alpha u_k-\gamma k^\mu u_k^q(s)|u_k'|^{p-2}u_k'(r)=0, \label{E231} \end{equation} with $\mu =1+\frac{pq}{p-2}$. %\label{E234} Since $u $ is strictly positive and decreasing on $[0,R_1[$, the function $k\mapsto u_k$ is strictly increasing. Moreover for any $r\in [0,R_0]$ $\lim_{k\to 0 }u_k(r)=+\infty$. Then there exists a small $k_0>$0 such that \[ u_k(r)>v(r)\quad \text{for } r\in [ 0,R_0]\quad\text{and} k\in [ 0,k_0] \] Therefore, the set \begin{equation} \Omega\equiv \left\{ k\in ]0,k_0[; u_k(r)>v(r) \text{for \ }r\in [0,R_0]\right\} \label{E232} \end{equation} is not empty and open. In particular if we denote by $K $ the supremum of $\Omega $, the real $K\notin\Omega$ and thereby, necessarily there exists $r_0\in [ 0,R_0]$ such that $u_K(r_0) = v(r_0)$. As $k_0 $ is small without loss of generality we assume \begin{equation} k_0=(\frac{A_1}{2A_2})^{(p-2)/p}. \label{E233} \end{equation} If $r_0=R_0$, then \begin{equation} u_K( R_0) =K^{- p/(p-2)}u(KR_0)=v( R_0) . \label{E234} \end{equation} But $u( R_0) =v( R_0)$, then using again the strictly increasing of the function $k\mapsto u_k$ is strictly increasing we deduce necessarily $K=1$. This is a contradiction with the choice of the real $k_0$. If $r_0=0$ we get \[ u_K(0)=K^{-p/p-2)}u(0)=K^{-p/(p-2)}A_1=A_2 \] which contradicts (\ref{E233}). Consequently we deduce that there exists some point $r_0\in [0,R_0[$ such that \begin{equation} u_K > v\quad \mbox{on }] 0,R_0[ \mbox{ and }u_K( r_0) = v(r_0). \label{E235} \end{equation} So $u_K-v$ has a local minimum at the point $r_0$, where the graphs of $u_K$ and $v$are tangent. Moreover, as $v'$ and $u'$ are strictly negative, the equation satisfied by $v$ (respectively by $u_K$) can be written in the form \begin{equation} ( p-1) |v'|^{p-2}v''+\beta rv'+\alpha v-\gamma v^q|v'|^{p-2}v'=0, \label{E236} \end{equation} and respectively \begin{equation} ( p-1) |u_K'|^{p-2}u_K'' +\beta ru_k'+\alpha u_K-\gamma K^\mu u_K^q|u_K'| ^{p-2}u_K'=0. \label{E237} \end{equation} Subtract (\ref{E236}) from (\ref{E237}), we obtain at point $r_0$, \begin{equation} ( p-1)|v'|^{p-2}( u_K-v)''(x)=\gamma ( K^\mu -1) v^q|v'|^{p-2}v'(r_0). \label{E238} \end{equation} Since $\gamma <0$, $v' <0,K<1$ and $\mu >0$, we get \begin{equation} ( p-1) |v'|^{p-2}( u_K-v)''(r_0)=\gamma ( k^\mu -1) v^q| v'|^{p-2}v'( r_0) <0. \label{E239} \end{equation} This is impossible because $( u_K-v)$ has a local minimum at $x$ and then the proposition is proved. \end{proof} In the next result, we investigate the asymptotic behavior of positive solutions. \begin{proposition} \label{prop2.3} Let $u$ be a positive solution of \eqref{Q} defined on $[ 0,+\infty[$. Then \[ \lim_{r\to +\infty }u(r)=\lim_{r\to+\infty }u'(r)=0. \] \end{proposition} The proof of this result depends strongly on the sign of $\gamma$. In fact, if $\gamma \leq 0$, the result follows from the energy function. However, for $\gamma >0$ we need some information about the monotonicity of $u'$ this is given in the following lemma. \begin{lemma} \label{lem2.2} Assume $\gamma >0$. Let a real $A>0$ and $u(.,A)$ be a strictly positive solution of \eqref{Q} defined in $[0,+\infty [$. Then there exists a unique real number $R(A)>0$ such that \[ u''<0\quad \mbox{on }[ 0,R(A)[\quad\mbox{and}\quad u''\geq 0\quad \mbox{on} [R(A),+\infty [ . \] \end{lemma} \begin{proof} First, note that from Lemma \ref{lem2.1}, the solution $u=u(.,A)$ is decreasing and converges to some nonnegative constant. On the other hand (\ref{E228}), implies that $( |u'|^{p-2}u') '$ must change sign. Let $R(A)>0$ its first zero. For simplicity we set $ R=R(A)$. Then $( |u'|^{p-2}u') '( r) <0$ for any $r$ in $[0,R[$. Furthermore, \begin{equation} \alpha u( R) =-[ \beta R-\gamma u^q|u'| ^{p-2}( R) ] u'(R). \label{E240} \end{equation} As $u$ is strictly positive we deduce that $u'(R) \neq 0$, and then $u$ is a $C^\infty$ function at the point $R$. So, the first equation in \eqref{Q} can be written in some neighborhood of $R$, say $] R-\varepsilon ,R+\varepsilon [$ ($\varepsilon >0$), in the form \begin{equation}\label{E241} ( p-1) |u'|^{p-2}u''+\beta ru'+\alpha u-\gamma u^q|u'|^{p-2}u'=0. \end{equation} Differentiating this last equality and taking $r=R$, we obtain \begin{equation} ( p-1) |u'|^{p-2}u^{( 3) }(R) =-( \alpha +\beta ) u'( R) +\gamma qu^{q-1}|u'|^p( R). \label{E242} \end{equation} But, since $\gamma >0$, $\alpha +\beta \geq 0$ and $u'( R) <0$, then the left hand side of the last equation is strictly positive. By continuity of $u^{( 3) }$ we get \[ u^{( 3) }( r) >0\quad \text{ for any }r\in [R,R+\varepsilon [. \] Hence $u''$ is non-negative on\ $[R,R+\varepsilon [ $. Finally, using \eqref{Q} we deduce \[ u''( r) \geq 0\quad \text{ for any $r$ in }[ R,+\infty [, \] which completes the proof. \end{proof} \begin{remark} \label{rmk2.3} \rm Note that the right hand side of (\ref{E242}) satisfies \[ -( \alpha +\beta ) u'( R) + \gamma qu^{q-1}|u'|^p( R) =\frac{u'(R)}{ u(R)}\{ \gamma qu^q|u'|^{p-2}u'(R)-(\alpha +\beta ) u(R)\}. \] Using \eqref{E240}, the relation (\ref{E242}) becomes \[ ( p-1) |u'|^{p-2}u^{( 3) }( R) =\beta (1+\frac \beta \alpha )R\frac{|u'(R)|^2 }{u(R)} - \gamma (1+\frac \beta \alpha -q)u^{q-1}|u'| ^p(R). \] Hence, if $\gamma <0$ and $q\leq 1+\frac \beta \alpha $, we get $u^{( 3) }( R) >0$ and thereby the Lemma \ref{lem2.2} also holds in this case. \end{remark} \begin{proof}[Proof of Proposition \ref{prop2.3}] By Lemma \ref{lem2.1} $ \lim_{r\to+\infty }u(r)=l$ exists and lies in $[ 0,A[$. We start by establishing the proposition when $\gamma \leq 0$, in this case the energy function given by (\ref{E226}) is positive and decreasing. It then converges, and $\lim_{r\to +\infty }u'(r) = 0$. Moreover integrating equation \eqref{Q} between $0$ and $r$, we get \begin{equation} |u'|^{p-2}u'( r) +\beta ru( r) +\int_0^r\left\{ ( \alpha -\beta ) u( s) -\gamma u^q( s) |u'|^{p-2}u'( s) \right\} ds=0. \label{E243} \end{equation} Therefore, \begin{equation} \lim_{r\to +\infty }\frac 1r\int_0^r\{ ( \alpha -\beta ) u( s) -\gamma u^q( s) | u'|^{p-2}u'( s)\} ds=- \beta l. \label{E244} \end{equation} On the other hand, if $l\neq 0$ the L'H\^opital rule implies that \[ \lim_{r\to +\infty }\frac 1r\int_0^r\{ ( \alpha -\beta ) u( s) -\gamma u^q( s) | u'|^{p-2}u'( s) \} ds=(\alpha -\beta )l. \] This contradicts (\ref{E244}) and therefore $l=0$. To handle the case $\gamma >0$, we use the above Lemma \ref{lem2.2}. Assume that $l\neq 0$ and\ integrate equation \eqref{Q} on $( r,2r)$ for some $r>0$. We obtain \begin{equation} \begin{aligned} |u'|^{p-2}u'( 2r) =& |u'|^{p-2}u'( r) +\beta ru( r) -2\beta ru( 2r) \\ & +( \beta -\alpha ) \int_r^{2r}u( s) ds+\gamma \int_r^{2r}u^q( s) |u'|^{p-2}u'( s) ds. \end{aligned}\label{E245} \end{equation} Since $\gamma >0$ and $u'\leq 0$, for i$\beta\geq\alpha$, we obtain \begin{equation} \frac{|u'|^{p-2}u'( 2r) }{2r}\leq ( \beta -\frac \alpha 2) ( u( r) -u(2r)) -\frac \alpha 2u(2r)\,. \label{E246} \end{equation} On the other hand if $\beta<\alpha$, \begin{equation} \frac{|u'|^{p-2}u'( 2r) }{r}\leq \beta ( u( r) -u(2r))-\alpha u(2r). \label{E247} \end{equation} Now, observe that \[ \lim_{r\to +\infty }[ u(r)-u(2r)] =0 \quad \text{and}\quad \lim_{r\to +\infty }u(2r)=l\neq 0; \] therefore, for any $\alpha, \beta \geq 0$, we get \begin{equation} \frac{|u'|^{p-2}u'( 2r) }{2r}\leq -\frac \alpha 2u( 2r) \quad \text{for large }r. \label{E248} \end{equation} This gives \begin{equation} ( u^{(p-2)/(p-1)}) '( r) \leq -\frac{p-2}{p-1}(\frac {\alpha}{2})^{1/(p-1)}r^{1/(p-1)}, \label{E249} \end{equation} which contradicts that $u$is strictly positive. Consequently $l=0$ and the proof is complete. \end{proof} Now, we pass to the asymptotic behavior of positive solutions. \begin {proposition} \label{prop2.4} Assume $\alpha >0$, $\beta >0$ and $\gamma \leq 0$. Let $u$ be a strictly positive solution of \eqref{Q}. Then $ \lim_{r\to+\infty }r^{\alpha/ \beta }u(r)=L$ exists and lies in $[ 0,+\infty [$. \end{proposition} Some preliminary results are needed for the proof of this proposition. \begin {lemma} \label{lem2.3} Assume $\alpha >0$, $\beta >0$ and $\gamma \leq 0$. Let $u$ be a strictly positive solution of \eqref{Q} such that \begin{equation} u(r)\leq K (1+r)^{-\sigma }\quad \text{for } r\geq 0. \label{E250} \end{equation} Then, there exists a constant $M$ depending on $K$ and $\sigma$ such that \begin{equation} |u'(r)|\leq M (1+r)^{-\sigma -1}\quad \text{for } r\geq 0. \label{E251} \end{equation} \end{lemma} \begin {proof} Without loss of generality we have just to prove (\ref{E251}) for $r>2$. In fact, as $u'$ is a continuous function, it is bounded in $[ 0,2]$. So there exists some constant $C>0$ such that \begin{equation} |u'(r)|\leq C,\quad \text{for }r \text{ in }[ 0,2]. \label{E252} \end{equation} Hence, if we take $M \geq C3^{\sigma +1}$, then (\ref{E251}) holds for $r$ in $[ 0,2]$. For any $r>2$, we set \begin{equation} F(r)=\exp \big[ \frac{-\gamma }{p-1}\int_0^ru^q(s)ds\big] \label{E253} \end{equation} and consider the function \begin{equation} G(r)=\exp \big[ \frac \beta {p-1}\int_0^rs|u'(s)| ^{2-p}ds\big] ,\quad r > 2. \label{E254} \end{equation} In view of (\ref{E228}) we have \begin{equation} u'(r)\sim -(\alpha A)^{1/(p-1)}r^{1/(p-1)} \quad \text{as } r \to 0. \label{E255} \end{equation} Recalling that $u'$ is strictly negative, we deduce that the function $G$ is well posed. Now, we write equation \eqref{Q} in the form \begin{equation} (FGu')'(r)+\frac \alpha {\beta }\frac{F(r)}% ru(r)G'(r)=0. \label{E256} \end{equation} Integrating the above equation, we obtain \begin{equation} |u'(r)|= \frac \alpha {\beta F(r)G(r)}\int_0^r% \frac{F(s)}su(s)G'(s)ds. \label{E257} \end{equation} Since $\gamma \leq 0 $ the function $F$ is increasing and \begin{equation} |u'(r)|\leq \frac \alpha {\beta G(r)}\int_0^r% \frac{u(s)}sG'(s)ds. \label{E258} \end{equation} Next, we find a bound the right-hand side of the above inequality. We set \begin{equation} I_1=\int_0^1\frac{u(s)}sG'(s)ds,\quad I_2=\int_1^{r/2}\frac{u(s)}sG'(s)ds,\quad I_3=\int_{r/2}^r\frac{u(s)}sG'(s)ds, \label{E259} \end{equation} so that \begin{equation} \int_0^r\frac{u(s)}sG'(s)ds=I_1+I_2+I_3 \,.\label{E260} \end{equation} First, note that (\ref{E255}) implies easily that $I_{1}$ is bounded. On the other hand, in view of Proposition \ref{prop2.3}, there exists a constant $K>0$ such that \begin{equation} |u'(r)|^{2-p}\geq K\quad \text{for }r\geq 0. \label{E261} \end{equation} Then \begin{equation} G(r)\geq \exp (Kr^2)\quad \text{for }r > 2. \label{E262} \end{equation} To estimate $I_2$, we use (\ref{E253}) to obtain \begin{equation} I_2\leq C\int_1^{r/2}\frac{(s+1)^{-\sigma }}sG'(s)ds\leq C\int_1^{r/2}G'(s)ds\leq CG(\frac r2). \label{E263} \end{equation} Or \begin{equation} \frac 1{G(r)}I_2\leq C\frac{G(r/2)}{G(r)}\leq C\exp [ \frac{% -\beta }{p-1}\int_{\frac r2}^rs|u'(s)| ^{2-p}ds] . \label{E264} \end{equation} Now recalling (\ref{E261}), we get \begin{equation} \frac 1{G(r)}I_2\leq C\exp (-K_1r^2),\label{E265} \end{equation} with $K_1=\frac{3\beta }{8(p-1)}K$. But as the solution $u$ is decreasing, then \[ \frac 1{G(r)}I_3=\frac 1{G(r)}\int_{r/2}^r\frac{u(s)}sG'(s)ds\leq \frac 2ru(r/2). \] Using again the estimate (\ref{E250}), we obtain \begin{equation} \frac 1{G(r)}I_3\leq C(r+1)^{-\sigma -1}\quad \text{for }r>2. \label{E266} \end{equation} Finally, putting together (\ref{E258}), (\ref{E265}) and (\ref{E266}) the desired estimate (\ref{E254}) follows. This completes the proof of the lemma. \end{proof} \begin {lemma} \label{lem2.4} Assume $\alpha >0$, $\beta >0$ and $\gamma \leq 0$. Let $u$ be a strictly positive solution of \eqref{Q}. Then \begin{equation} u(r)\leq C r^{-\frac \alpha \beta },\quad \text{ for large }r . \label{E267} \end{equation} \end{lemma} \begin{proof} Using the first equation in \eqref{Q}, the function $u(r)$ satisfies \begin{equation} \begin{aligned} \frac \alpha 2\frac{u^2(r)^{ }}r=&\frac{|u'|^p}{2r} -\frac \beta 2uu'-\frac 1{2r^2}u|u'|^{p-2}u'\\ & +\frac \gamma {2r}\ u^{q+1}|u'| ^{p-2}u'(r)-\frac 12[ \frac{u|u'|^{p-2}u'}r] '. \end{aligned}\label{E268} \end{equation} Recalling the expression of the energy function given by (\ref{E226}) we deduce \begin{equation} \begin{aligned} \frac{E(r)}r=&\frac{3p-2}{2p}\frac{|u'|^p}r-\frac{\beta}4(u^2)' -\frac 1{2r^2}u|u'|^{p-2}u'\\ &-\frac{\ 1}2[ \frac{u|u'|^{p-2}u'}r] '+\frac \gamma {2r}\ u^{q+1}|u'| ^{p-2}u'(r). \end{aligned}\label{E269} \end{equation} Integrating the above inequality on the interval $(r,R)$ we obtain \begin{align*} \int_r^R\frac{E(s)}sds =& \frac{3p-2}{2p}\int_r^R\frac{|u'(s)|^p}sds+\frac{u(r)|u'|^{p-2}u'(r)}{2r}\\ & -\frac{u(R)|u'|^{p-2}u'(R)}{2R}+\frac{\beta \ }4u^2(r)-\frac{\beta}4u^2(R) \\ & -\frac 12\int_r^R\frac{u(s)|u'(s)|^{p-2}u'(s)}{s^2}ds +\frac \gamma 2\int_r^R\frac{u^{q+1}(s)|u'(s)|^{p-2}u'(s)}sds. \end{align*}% \label{E273} Since $u'$ is negative, $\beta \geq 0$ and $\gamma \leq 0$ we get \begin{equation} \begin{aligned} \int_r^R\frac{E(s)}s\,ds \leq& \frac{3p-2}{2p}\int_r^R\frac{|u'(s)|^p}sds+\frac{u(R)|u'(R)|^{p-1}}{2R} +\frac{\beta \ }4u^2(r) \\ &+\frac 12\int_r^R\frac{u(s)|u'(s)|^{p-1}}{s^2}ds + \frac{|\gamma |}2\int_r^R\frac{u^{q+1}(s)|u'(s)|^{p-1}}sds. \end{aligned}\label{E270} \end{equation} Since $E$ is strictly decreasing and converges to zero when $r$ approaches to infinity, we deduce that $E' \in L^1(] 0,\infty [)$. In particular $r|u'|^2$ and $u^q|u'|^p$ lie in $L^1(] 0,\infty [)$. Letting $R\to$ $\infty $, \begin{equation} \begin{aligned} \int_r^\infty \frac{E(s)}sds \leq & \frac \beta 4u^2(r)+\frac{ 3p-2}{2p}\int_r^\infty \frac{|u'(s)|^p}s\,ds \\ & +\frac 12\int_r^\infty \frac{u(s)|u'(s)|^{p-1}}{s^2} ds+\frac{|\gamma |}2\int_r^\infty \frac{u^{q+1}(s)|u'(s)|^{p-1}}s\,ds. \end{aligned}\label{E271} \end{equation} Now, we set \begin{equation} H(r)=\int_r^\infty \frac{E(s)}sds. \label{E272} \end{equation} First, using the fact that $u^2(r)\leq \frac 2\alpha E(r)$, we obtain \begin{equation} H(r)\geq \int_r^{2r}\frac{E(s)}sds\geq \frac{E(2r)}2\geq \frac \alpha 4u^2(2r). \label{E273} \end{equation} On the other hand, inequality (\ref{E271}) gives \begin{equation} \begin{aligned} H(r)+\frac \beta {2\alpha }rH'(r) \leq& \frac{3p-2}{2p}\int_r^\infty \frac{|u'(s)|^p}sds +\frac{|\gamma|}2\int_r^\infty\frac{u^{q+1}(s)|u'(s)|^{p-1}}s\,ds\\ &+\frac 12\int_r^\infty\frac{u(s)}{s^2}|u'(s)|^{p-1}ds. \end{aligned}\label{E274} \end{equation} Assume that the function $u$ satisfies \begin{equation} u(r)\leq Cr^{-\sigma }\quad \text{for }r\geq 1, \label{E275)} \end{equation} for some fixed $\sigma \geq 0$ and some constant $C$ (this is possible because $u(r)\leq A$ for all $r\geq 0$). If $\sigma\geq (2\alpha)\beta $ we have obviously (\ref{E267}). Assume now that $\sigma< (2\alpha)\beta$. Lemma \ref{lem2.3} implies $|u'(r)|\leq Cr^{-\sigma -1}$ for any $r\geq 1$. Consequently \begin{equation} [ r^{2\alpha /\beta }H(r)] '\leq Cr^{2\alpha /\beta -1-p(\sigma +1)} [ 1+r^{1-q\sigma }] \quad \text{for } r\geq 1. \label{E276)} \end{equation} By a simple integration, we obtain \begin{equation} H(r)\leq Cr^{-2\alpha /\beta }+Cr^{-p(1+\sigma )}+Cr^{1-p-(p+q)\sigma } \label{E277} \end{equation} when $[p(1+\sigma )-2\alpha/\beta][2\alpha /\beta -p+1-(p+q)\sigma]\neq 0$. Otherwise if $[p(1+\sigma )-2\alpha/\beta][2\alpha /\beta -p+1-(p+q)\sigma]= 0$, we have \begin{equation} H(r)\leq Cr^{-2\alpha /\beta }+Cr^{-2\alpha /\beta }\ln r +Cr^{-(1-q\sigma +2\alpha /\beta )}. \label{E278} \end{equation} Combining (\ref{E273}), (\ref{E277}), (\ref{E278}) and using the fact that $\sigma< (2\alpha)\beta $, we deduce that there exists $m>\sigma $ such that \begin{equation} u(r)\leq Cr^{-m}{}^{\quad }\text{for all }r\geq 1. \label{E279} \end{equation} If $m=\alpha /\beta $ we have exactly the estimate (\ref{E267}). Otherwise if $m\neq \alpha /\beta $, the desired estimate (\ref{E267}) follows by induction starting with $\sigma =m$. This completes the proof. \end{proof} \begin{proof}[Proof of Proposition \ref{prop2.4}] Set \begin{equation} I(r)=r^{\alpha /\beta }\big[ u+\frac 1{\beta r}|u'|^{p-2}u'\big]\,. \label{E280} \end{equation} Then we have \begin{equation} I'(r)=\frac 1\beta r^{ \alpha /\beta -1}\big[ (\frac \alpha \beta -1)\frac{|u'|^{p-2}u'(r)}r+\gamma u^q|u'|^{p-2}u'\big]. \label{E281} \end{equation} In view of Lemma \ref{lem2.3} and Lemma \ref{lem2.4}, the functions $r\mapsto r^{\alpha /\beta -1}u^q|u'(r)|^{p-1}$ and $r\mapsto r^{\alpha /\beta -2}|u'(r)|^{p-1}$ are in $L^1(] 0,\infty[)$. Consequently$, I'(r)\in{L^1(] 0,\infty[ )}$. Moreover (\ref{E255}) implies I(0)=0, and therefore \[ \lim_{r\to +\infty }I(r)=\int_0^\infty I'(s)ds\quad \] exists. Since $ \lim_{r\to +\infty }r^{\alpha /\beta -1}|u'|^{p-2}u'=0$, we deduce that \[\lim_{r\to +\infty }r^{\alpha /\beta }u(r)=L \in [ 0,\infty [. \] This completes the proof. \end{proof} \begin{proposition} \label{prop2.5} Assume $\alpha >0$, $\beta >0$, $\gamma <0$ and $L=0$ in Proposition \ref{prop2.4}. Then $r^mu(r)\to 0$ and $ r^mu'(r)\to 0$\ as $r\to +\infty$ for all positive integers $m$. \end{proposition} \begin{proof} >From the proof of the previous proposition, $\lim_{r\to +\infty }I(r)=0$. Thus, $I(r)=-\int_r^\infty I'(s)ds$. Therefore ,(\ref{E280}) gives \begin{equation} u(r)=\frac{-1}{\beta r}|u'|^{p-2}u'(r)-r^{-\alpha /\beta }\int_r^\infty I'(s)ds. \label{E282} \end{equation} Since $\gamma <0$ and $u'<0$, we deduce from (\ref{E285}) that \begin{equation} u(r)\leq \frac 1{\beta r}|u'|^{p-1}+\frac 1\beta |\frac \alpha \beta -1|r^{- \alpha /\beta }\int_r^\infty s^{ \alpha /\beta -2}|u'|^{p-1}ds. \label{E283} \end{equation} Then in view of Lemma \ref{lem2.3}, we get \begin{equation} u(r)\leq Cr^{-(p+(p-1)\alpha )/\beta} . \label{E284} \end{equation} Define the sequence $(m_k)_{k\in {\mathbb N}} $ by \begin{equation} \begin{cases} m_0=\alpha /\beta, \\ m_k=p+(p-1)m_{k-1}. \end{cases}\label{E285} \end{equation} Then $\lim_{k\to +\infty }m_k=+ \infty$. Consequently, the proposition follows by induction starting with $m_0= \alpha /\beta$. This completes the proof. \end{proof} \begin{proposition} \label{prop2.6} Assume $\alpha \leq \beta $ and $\gamma <0$. Let $u$ be a strictly positive solution of \eqref{Q}. Then $\lim_{r\to +\infty}r^{ \alpha /\beta }u(r)>0$. \end{proposition} \begin{proof} By Proposition \ref{prop2.4}, \[ \lim_{r\to +\infty }r^{ \alpha /\beta }u(r)\in [ 0,\infty[. \] Suppose that $\lim_{r\to +\infty }r^{\alpha /\beta }u(r)=0$. Then Proposition \ref{prop2.5} implies \[ \lim_{r\to +\infty } r^{ \alpha /\beta -1}|u'|^{p-2}u'=0, \] and therefore, $\lim_{r\to +\infty }I(r)=0$. On the other hand, (\ref{E281}) implies that the function $I$ given by (\ref{E280}) is strictly increasing; this is a contradiction which completes the proof. \end{proof} \section{ Classification of solutions} In this section we give a classification of solutions of \eqref{Q}. For this purpose we Set \begin{gather*} P=\{ A>0: u(r,A)>0, \forall r>0\},\\ N=\{ A>0: \exists\, r_0>0; u(r,A)> 0 \text{for }r\in [ 0,r_0[, u(r_0,A)=0 \text{ and }u'(r_0,A)<0\},\\ C=\{ A>0;\exists\, r_0>0 ;u(r_0,A)=u'(r_0,A)=0\}. \end{gather*} This classification depends strongly on the sign of$ \gamma $ and $ \alpha -\beta$. First, we start with the following result. \begin{proposition} \label{prop3.1} Assume $\gamma >0$ and $ \alpha \geq\beta$. Then any solution of \eqref{Q} changes sign. \end{proposition} \begin{proof} Let $u$ be a solution of \eqref{Q}. Then \begin{equation} |u'|^{p-2}u'( r) =-\beta ru( r) -( \alpha -\beta ) \int_0^ru(s)ds+\gamma \int_0^ru^q( s) |u'|^{p-2}u'(s) ds. \label{E31} \end{equation} for any $r\in [ 0,R_{\max }$. If the set $C$ is not empty, there exists a finite $r_0>0 $ such that $u(r_0)= u'(r_0)=0 $ and $u(r)>0$ on $] 0,r_0[ $. Taking $r=r_{0}$ in (\ref{E31}) and using again Lemma \ref{lem2.1}, we get \begin{equation} ( \alpha -\beta ) \int_0^{r_0}u(s)ds+\gamma \int_0^{r_0}u^q| u'|^{p-1}ds=0. \label{E32} \end{equation} This contradicts $\gamma >0$ and $\alpha -\beta \geq 0$. Hence $C=\emptyset$. If the set $P $is not empty, without loss of generality we can assume that equation \eqref{E31} holds with $u$ strictly positive and $u'$ negative. Then since $\alpha -\beta \geq 0$ and $\gamma >0$ we deduce \begin{equation} |u'|^{p-2}u'( r) \leq - \beta ru( r).\label{E33} \end{equation} By integrating this last inequality we get a contradiction. \end{proof} \begin{proposition} \label{prop3.2} Assume $\gamma >0$ and $\alpha< \beta $. Then \begin{itemize} \item[(i)] $ u(.,A)$ is strictly positive for any $A\in ] 0,A_0[$ with \[ A_0=[ \frac {q-1}\beta( \frac {\beta-\alpha }\gamma) ^{ p/(p-1)}] ^{(p-1)/(p(q+1) -2)} \] \item[(ii)] $u(.,A)$ changes sign for large $A$. \end{itemize} \end{proposition} For the proof we need some preliminary results. Let $u$ be a solution of problem \eqref{Q} defined in $[ 0,R_{\rm max}[ $. Set \begin{equation} h(r)=( \beta -\alpha ) u(r)+\gamma |u|^{q-1}u| u'|^{p-2}u'(r)\quad \forall r\in [0,R_{\rm max }[ .\label{E34} \end{equation} Then the following result holds. \begin{lemma} \label{lem3.1} Assume $\alpha <\beta $. Let $u$ be a solution of \eqref{Q}. Then $u$ cannot vanish before the first zero of $h$. \end{lemma} \begin{proof} On the contrary, suppose that $u$ vanishes before$ h $and let $r_0$ be the first zero of $u$. As $h(0)=( \beta -\alpha ) u(0)>0$, then $h(r_0)\geq 0$ and $h(r)>0$ for $r\in [0,r_0[$. %\label{E35} Integrating \eqref{Q}, we obtain \begin{equation} |u'|^{p-2}u'(r_0)=\int_0^{r_0}h(s)ds>0. \label{E35} \end{equation} This contradicts$ u'(r_0)\leq 0$ and then, the Lemma is proved. \end{proof} Now, assume that there exists some initial data $A>0$ such that $u(.,A) (=u) $ is a strictly positive solution of \eqref{Q} and set \begin{equation} g(r)=\beta r-\gamma u^q|u'|^{p-2}(r),\quad r>0. \label{E36} \end{equation} \begin{lemma} \label{lem3.2} There exists $\rho (=\rho (A))>0$ such that \begin{equation} g(\rho )=0\quad\mbox{and}\quad g(r)<0\mbox{ for $r$ in $] 0,\rho[$}. \label{E37} \end{equation} Furthermore, \begin{equation} \lim_{A\to +\infty }u(\rho ,A)=0\quad \mbox{and} \quad \lim_{A\to +\infty }u'(\rho ,A)=-\infty. \label{E38} \end{equation} \end{lemma} \begin{proof} First, we observe that the function $g$ satisfies \begin{equation} ( |u'|^{p-2}u'(r)) '=-\alpha u(r)-u'(r)g(r)\quad \text{for all }r>0. \label{E39} \end{equation} The proof is divided in 3 steps. \noindent{\bf Step 1.} $g(\rho )=0$ and $g(r)<0$ for $r\in [ 0,\rho [ $. Recalling (\ref{E225}) we get \begin{equation} |u'|^{p-2}u'(r)\sim - \alpha Ar, \quad \text{as } r\to 0.\label{E310)} \end{equation} Hence, \begin{equation} g(r)\sim - \gamma A^q( \alpha A r)^{(p-2)/(p-1)}, \quad\text{as } r\to 0. \label{E311} \end{equation} and therefore $g $ starts with a negative value. If $g$ has a constant sign for all $r>0$, equation (\ref{E39}) gives \[ ( |u'|^{p-2}u') '(r)<- \alpha u(r)<0,\quad \text{for }r>0. \] and then the solution $u(.,A)$ must change sign; this is a contradiction and then (\ref{E37}) follows. \noindent{\bf Step 2.} We claim that $\lim_{A\to +\infty }u'(\rho,A)=- \infty $. In fact, equation \eqref{E39} implies that the solution $u $ satisfies \begin{equation} \big[ \frac{p-1}p|u'|^p+\frac \alpha 2u^2\ \big] '(r)=-( u') ^2g(r),\quad \text{for }r>0. \label{E312} \end{equation} Integrating this last equality on $[0,R]\subset[0,\rho [$ and using the fact that $g$ is negative on $[0,\rho [$, we get \begin{equation} |u'|^p( r) \geq \frac{p\alpha }{2( p-1) }[ A^2-u^2(r)] ,\quad \forall r\in [0,\rho [. \label{E313} \end{equation} Hence, if $u(\rho ,A)$ is bounded, by letting $A$ approach $ \infty $, we deduce \[ \lim_{A\to +\infty }u'(\rho(A) ,A)=- \infty. \] Otherwise if $u(\rho ,A)$ is not bounded, then there exists a subsequence, denoted also $\rho (A)$ such that \[ \lim_{A\to +\infty }u( \rho (A),A) =+ \infty . \] Now, recalling (\ref{E37}) and (\ref{E39}) we get \begin{equation} ( |u'|^{p-2}u'(r)) '\leq - \alpha u(r)<0,\quad \text{for any }r\in [ 0,\rho [ . \label{E314} \end{equation} In particularly, we deduce that $u$ is concave in $[ 0,\rho [ $ and therefore \begin{equation} u(r)\geq A+\frac{u(\rho )-A}\rho r,\quad \text{for any }r\in [ 0,\rho[ .\label{E315} \end{equation} Thus integrating (\ref{E314}) on $(0,\rho )$, we obtain \begin{equation} |u'|^{p-2}u'(\rho )\leq -\alpha \int_0^\rho \big( A+\frac{u(\rho )-A}\rho r\big) dr. \label{E316} \end{equation} Hence, \begin{equation} |u'|^{p-2}u'( \rho ) \leq -\frac \alpha 2\rho [ A+u(\rho )] . \label{E317} \end{equation} But $g(\rho )=0$, then $u^q|u'|^{p-2}(\rho )=\frac \beta \gamma \rho$. %\label{319} Inserting this last equality in (\ref{E317}) the following estimate holds \begin{equation} |u'( \rho ) |\geq \frac{\gamma \alpha }{ 2\beta }u^q( \rho ) [ A+u( \rho ) ] . \label{E318} \end{equation} Consequently, $\lim_{A\to +\infty }u'(\rho,A)=- \infty $. \noindent{\bf Step3.} We assert that $\lim_{A\to +\infty }u(\rho ,A)=0$. In fact, integrating \eqref{E314} on an interval $] 0,r[ \subset ] 0,\rho[ $ and using (\ref{E315}) we obtain \begin{equation} |u'|^{p-2}u'( r) \leq -\alpha r[ A+\frac{u(\rho )-A}{2\rho }r] \text{ for any }00; h(s)>0\text{\ on\ }[ 0,r[ \}. \label{328} \end{equation} Since $h(0)=( \beta -\alpha ) u(0)>0$, the set $ \{ r>0: h(r)>0\text{\ on\ }[ 0,r[ \} $ is not empty. We claim that $R_0$ is infinite. To the contrary, assume that $R_0$ is a real number. Then $h(R_0)=0$ and $h'(R_0)\leq 0$, so from Lemma \ref{lem3.1}, $u(R_0)>0 $. Moreover, by continuity, $u(r)\neq 0$ for r $\in ] R_0-\varepsilon ,R_0+\varepsilon [ $ (with some $\varepsilon >0)$. Thus, we can write $h(r) $ in the form \begin{equation} h(r)=u^q(r)\widetilde{h}(r), \label{329} \end{equation} for any $r\in ] R_0-\varepsilon ,R_0+\varepsilon [$, with \begin{equation} \widetilde{h}(r)=( \beta -\alpha ) u^{1-q}(r)+\gamma | u'|^{p-2}u'(r). \label{330} \end{equation} We clearly have \begin{eqnarray} \begin{aligned} \widetilde{h}'(R_0)=&u(R_0) [-\gamma \beta +\gamma \beta R_0( \frac{\beta -\alpha }\gamma ) ^{1/(p-1)}u^{-(q-1)(p-1)-1}(R_0) \\ &+ (q-1)(\beta -\alpha )( \frac{\beta -\alpha }\gamma ) ^{ 1/(p-1)} u^{(2-p(q+1))/(p-1)}(R_0)]. \end{aligned} \label{E331} \end{eqnarray} Since $u$ is decreasing, $\gamma >0$ and $\beta -\alpha \geq 0$, \begin{equation} \widetilde{h}'(R_0)>u(R_0)[ -\gamma \beta +(q-1)(\beta -\alpha )( \frac{\beta -\alpha }\gamma ) ^{ 1/(p-1)}A^{(2-p(q+1))/(p-1)}]. \label{E332} \end{equation} Consequently, for any $A$ such that \begin{equation} A^{(p(q+1)-2)/(p-1)}<\frac {q-1}\beta ( \frac{\beta -\alpha } \gamma ) ^{ p/(p-1)}, \label{333} \end{equation} we obtain $\widetilde{h}'(R_0)>0$. This contradicts $ \widetilde{h}(r)>0 $ for r$\in[ 0,R_0[ $ and $ \widetilde{h}(R_0)=0$. Hence $ R_0$ is infinite; meaning that the function $h$ is strictly positive and therefore by Lemma \ref{lem3.1}, $u$ is also strictly positive. \noindent{\bf Step 2.} The proof of part (ii). Assume for contradiction that $u$ is positive for all $A$. Since $u'(R(A),A)<0$ and $u''(R(A),A)=0;$ then by putting $r=R(A)$ in \eqref{Q} we get \begin{equation} \alpha \frac{u(R(A))}{|u'(R(A)|^{p-2}}=u'(R(A)) \big\{ -\beta \frac{R(A)}{|u'(R(A)|^{p-2}} +\gamma u^q(R(A))\big\}. \label{E334} \end{equation} Invoking Lemma \ref{lem3.3}, we deduce \begin{equation} \lim_{A\to +\infty }\frac{R(A)}{|u'(R(A)|^{p-2}}=0. \label{E335} \end{equation} Integrating equation \eqref{Q} on $] R(A),r[ $, we obtain \begin{equation} \begin{aligned} &|u'(r)|^{p-2}u'(r)-|u'(R(A)|^{p-2}u'(R(A))+ \beta ru(r)\\ &-\beta R(A)u(R(A))+(\alpha -\beta )\int_{R(A)}^ru(s)ds -\gamma \int_{R(A)}^ru^q|u'|^{p-2}u'(s)ds=0.\label{E336} \end{aligned} \end{equation} Since $u'$ is negative and strictly increasing in $[ R(A),\infty[ $ we get \begin{equation} |u'(r)|^{p-2}u'(r)>|u'(R(A)|^{p-2}u'(r), \label{E337} \end{equation} for any $r>R(A)$. Hence, equation \eqref{E336} gives \begin{equation} \begin{aligned} |u'(R(A))|^{p-2}u'(r) <&|u'(R(A))|^{p-2}u'(R(A))-\beta ru(r)+\beta R(A)u(R(A))\\ &+(\beta-\alpha )\int_{R(A)}^ru(s)ds.\label{E338} \end{aligned} \end{equation} Now using the fact that $u $ is decreasing and that $\beta >\alpha >0$, we obtain \begin{equation} u'(r)0 $such that $u(r_0)=0$ and $u(r)>0$ in $[ 0,r_0[ $. Then from Lemma \ref{lem2.1}, $u'(r)<0$ in $] 0,r_0[$ and $u'(r_0)\leq 0$. Now integrating equation \eqref{Q} on $] 0,r_0[ $ we get \begin{equation} |u'|^{p-2}u'( r_0) =( \beta -\alpha ) \int_0^{r_0}u(s)ds+\gamma \int_0^{r_0}u^q( s) |u'|^{p-2}u'( s) ds. \label{E341} \end{equation} This is a contradiction with $\gamma <0$ {\it and\ }$\alpha \leq \beta $. \end{proof} In the case of $\gamma <0 $ and $\alpha >\beta >0 $ we will prove that the three sets $P$, $N$ and $C $ are not empty. More precisely we have the following statement. \begin{proposition} \label{prop3.4} Assume $\gamma <0$ and $\beta <\alpha$. Then there exist two constants $A_{-}$ and $A_{+}$ such that $u(.,A) $ is strictly positive for any $A\geq A_{+}$ and $u(.,A)$ changes sign for any $00; \label{E351} \end{equation} This is a contradiction. As if there exists some $x_0>0$ such that\\ {$W(x_0)=W'(x_0)=0$} and $W(x)>0$ in $] 0,x_0[ $ we obtain \[ (\beta -\alpha )\int_0^{x_0}W(s)ds=0, \] this is also a contradiction because $\alpha \neq \beta . $Thereby $W$ is non positive and consequently $u(.,A)$ changes sign for small $ A$. This completes the proof. \end{proof} We have also the following result. \begin{proposition} \label{prop3.5} Assume $\gamma <0$ and $ 0<\beta <\alpha $. Then $N$ and $P$ are non-empty open sets. \end{proposition} Before to start the proof we introduce the function \begin{equation} \Gamma(r)=u(r)+|u'|^{p-2}u'(r). \label{352} \end{equation} \begin{lemma} \label{lem3.4} Assume $\gamma <0$, $\alpha >0 $, and $\beta >0$. Let $u$ be a strictly positive solution of \eqref{Q}. Then the function $\Gamma(r)$ is strictly positive for large $r$. \end{lemma} \begin{proof} Since $u(r)>0$, Proposition \ref{prop2.4} implies \[ \lim_{r\to +\infty }r^{\alpha /\beta }u(r)=L\in [ 0,+\infty[ . \] If $L>0$, $u(r)\approx Lr^{-\alpha /\beta }$ for large r and then Lemma \ref{lem2.3} implies \[ \lim_{r\to +\infty }r^{\alpha /\beta }| u'|^{p-2}u'=0. \] Thus, the function $\Gamma(r)$ behaves like $Lr^{-\alpha /\beta}$, as $r\to\infty$ and therefore, $\Gamma$ is strictly positive. For the case $L=0$, the proof will be done into two steps. \noindent{\bf Step 1.} $\Gamma(r)$ is monotone for large $r$. For this purpose we set \begin{equation} J(r)=\beta ru'(r)+\alpha u(r). \label{353} \end{equation} We assert that $J(r) $has a constant sign for large $r$. In fact, assume that there exists a large $r_0$ such that $J(r_0)=0$. According to equation \eqref{Q}, we obtain \begin{equation} (p-1)|u'(r_0)|^{p-2}J'(r_0)=-\beta {(\frac{\alpha}{\beta}) }^{p-1} \frac{u^{p-1}(r_0)}{r_0^{p-1}}\left\{ (p-1)(\alpha/\beta +1) +\gamma r_0u^q(r_0)\right\}. \label{354} \end{equation} Since $\lim_{r\to +\infty }r^{\alpha /\beta }u(r)=0$, we deduce that $J'(r_0)<0$. Consequently $J(r)$ has the same sign for large $r$. Now, note that for any $r>0$, \begin{equation} \Gamma'(r)=u'(r)-\beta ru'-\alpha u(r)+\gamma u^q| u'|^{p-2}u'(r). \label{355} \end{equation} Hence $\Gamma'(r)$ and $J(r)$ have the opposite signs, in particular the function $J$ is monotone for large $r$. \noindent{\bf Step 2.} We claim that $\Gamma$ is not negative for large $r$. In fact if not,using the step1 we deduce that there exists a large $R_1$ such that $\Gamma (r)=u(r) + |u'|^{p-2}u'\leq0$ for any $r\geq R_{1}$. Integrating this last inequality on $(R_{1},r)$ we get \begin{equation} u^{(p-2)/(p-1)}(r)\leq u^{(p-2)/(p-1)}(R_1)-(p-2)/(p-1)r+(p-2)/(p-1)R_1. \label{356} \end{equation} By letting $r\to+\infty $, we obtain a contradiction. Combining step 1 and step 2 we deduce $\Gamma(r)>0$ for large $r$. This completes the proof. \end{proof} Now we use step by step the idea introduced by Brezis et al \cite{BPT}, for studying a very singular solution of the heat equation with absorption. Ever since, this idea was used in many papers, see for example \cite{PW}. In order to do this, we write \eqref{Q} as the system \begin{equation} \label{S} \begin{gathered} u'=|v|^{-(p-2)/(p-1)}v, \\ v'=-\beta r|v|^{-(p-2)/(p-1)}v-\alpha u+\gamma |u|^{q-1}uv. \end{gathered} %\label{359} \end{equation} For each $\lambda >0$, we define the set \begin{equation} { L}_\lambda =\{ (f_{1, }f_2): 00$ there exists $ r_\lambda =[\lambda +\alpha \lambda ^{-1/(p-1)}]/\beta;$ such that $ { L}_\lambda $ is positively invariant for $r\geq r_\lambda $. That is, if $(u_0,v_0)\in { L}_\lambda $ and $(u(r),v(r))$ is the solution of \eqref{S} which satisfies $(u(r_0),v(r_0))=(u_0,v_0) $ for some $r_0>r_\lambda$, then for any $r\geq r_\lambda $ the orbit $(u(r),v(r)) $ lies in ${L}_\lambda $ for all $ r\geq r_0$. \end{lemma} \begin{proof} We shall show that, given $\lambda >0$, there exists $r_\lambda >0$ such that if $r>r_\lambda $, then the vector field determined by \eqref{S} points into ${ L}_\lambda $, except at the critical point $(0,0)$. On the top $(f_2=0)$, \[ v'=-\alpha u<0\quad \text{for all } r>0. \] While on the right side $(u=1)$, \[ v'=-\beta r|v|^{-(p-2)(p-1)}v<0\quad \text{for all }r>0. \] On the line $f_2=- \lambda f_1$ we must prove that $\frac{v'}{u'}<- \lambda $ for large $r$. This is true because \[ \frac{v'}{u'}=\frac{v'|v|^{(p-2)/(p-1)}}v=-\beta r+\alpha \lambda ^{-1/(p-1)}u^{(p-2)/(p-1)}+\gamma \lambda ^{(p-2)/(p-1)}u^{q+(p-2)/(p-1)}. \] Since $\gamma <0$, if $r\geq (\lambda +\alpha \lambda ^{- 1/(p-1)})/\beta=r_\lambda $ we obtain $\frac{v'}{u'}<- \lambda$. \end{proof} \begin{proof}[Proof of Proposition \ref{prop3.5}]. First, note that from Proposition \ref{prop3.4}, the sets $P$ and $N$ are not empty. On the other hand, the continuous dependence of solutions on the initial value implies that $N$ is an open set. To prove that $P$ is open, take $A_0\in P$ and let a large $r_0>0$ be fixed. Then by continuous dependence of solutions on the initial data, there is a neighborhood $O$ of $A_0$ such that $u(r,A)>0$ for any $(r,A)\in[0,r_0] \times O$. In particular $u(r_0,A)>0$ for any $A\in O$ and then from Lemma \ref{lem3.4} \begin{equation} \Gamma(r_0)=u(r_0,A)+|u'|^{p-2}u'(r_0,A)>0. \label{E359} \end{equation} Since $r_0$ is large then $u(r_0,A)<1$ and consequently $(u(r_0,A),|u'|^{p-2}u'(r_0,A))$ is in $L_1$. Recalling Lemma \ref{lem3.5}, we deduce that the trajectory remains in $ L_1$ for any $r\geq r_0$, which implies in particular that $u(r,A)>0 $ for any $r\geq r_0$ and $A\in O$. Therefore, $u(r,A)>0$ for any $r\geq0$ and there by $P$ is open. The proof is complete. \end{proof} The rest of the paper is devoted to the study of solution with compact support \begin{proposition} \label{prop3.6} Assume that $\gamma<0$ and $\beta<\alpha$, then there exists at least one solution with compact support. \end{proposition} \begin{proof} As $P$ and $N$ are open disjoint sets, then there exists $A\in \mathbb {R}^{+}-(P\cup N)$; that is, $u(.,A)$ has a compact support. \end{proof} We conclude this paper with a study of the behavior of solution with a compact support. \begin{lemma} Assume $ \gamma <0. $Let $u $be a solution with compact support $[0,R]$. Then \begin{equation} ( u^{(p-2)/(p-1)}) '(R)=-\frac{p-2}{p-1}\beta ^{^{1/(p-1)}}R^{1/(p-1)}. \label{E360} \end{equation} \end{lemma} \begin{proof} Take $r$ close to $R$. and integrate \eqref{Q} between $r $and $R$; using the fact that $u $is decreasing, we get \[ |u'|^{p-2}u'(r)=\beta ru(r)-( \alpha -\beta ) \int_r^Ru(s)ds+\gamma \int_r^Ru^q(s)|u'| ^{p-2}u'(s)ds. \] Dividing by $u(r) $, we have \[ \frac{|u'|^{p-1}( r) }{u( r) } =\beta r-\frac{\alpha -\beta }{u( r) }\int_r^Ru(s)ds+\frac \gamma {u( r) }\int_r^Ru^q(s)|u'| ^{p-1}u'(s)ds. \] First, note that \[ 0\leq \int_r^Ru(s)ds\leq u(r)( R-r). \] Hence \[ \lim_{r\to R }\frac{\alpha -\beta }{u( r) } \int_r^Ru(s)ds=0. \] On the other hand, since the function $u'(s) $is negative and also $|u'|$ decreasing near $R$, then \begin{align*} \frac \gamma {u( r) }\int_r^Ru^q(s)|u'|^{p-2}u'(s)ds = & \frac{|\gamma |}{u( r) }\int_r^Ru^q(s)|u'|^{p-1}(s)ds\\ \leq & \frac {-|\gamma |}{u( r) }|u'|^{p-2}(r)\int_r^Ru^q(s)u'(s)ds\\ \leq & \frac{|\gamma |}{q+1}u^q(r)|u'(r)|^{p-2}. \end{align*} The last term of this inequality approaches zero as ${r\to R}$ and then we get \[ \lim_{r\to R }\frac{|u'| ^{p-1}( r) }{u( r) }=\beta R. \] This is equivalent to (\ref{E360}); thus the proof of the lemma is complete. \end{proof} \subsection*{Acknowledgments}. 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