\documentclass[twoside]{article} \usepackage{amsfonts,amsmath} \pagestyle{myheadings} \markboth{\hfil Nonlinear elliptic systems with indefinite terms \hfil EJDE--2002/83} {EJDE--2002/83\hfil Ahmed Bensedik \& Mohammed Bouchekif \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 83, pp. 1--16. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % On certain nonlinear elliptic systems with indefinite terms % \thanks{ {\em Mathematics Subject Classifications:} 35J20, 35J25, 35J60, 35J65, 35J70. \hfil\break\indent {\em Key words:} Elliptic systems, p-Laplacian, variational methods, mountain-pass Lemma, \hfil\break\indent Palais-Smale condition, potential function, Moser iterative method. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Work supported by research project B1301/02/2000. \hfil\break\indent Submitted April 2, 2002. Published October 2, 2002.} } \date{} % \author{Ahmed Bensedik \& Mohammed Bouchekif} \maketitle \begin{abstract} We consider an elliptic quasi linear system with indefinite term on a bounded domain. Under suitable conditions, existence and positivity results for solutions are given. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \numberwithin{equation}{section} \section{Introduction} The purpose of this article is to find positive solutions to the system \begin{equation} \label{Spq} \begin{gathered} -\Delta_pu=m( x) \frac{\partial H}{\partial u}( u,v) \quad \text{in }\Omega \\ -\Delta_qv=m( x) \frac{\partial H}{\partial v}( u,v) \quad \text{in }\Omega \\ u=v=0 \quad \text{on }\partial \Omega \end{gathered} \end{equation} where $\Omega $ is a bounded regular domain of $\mathbb{R}^N$, with a smooth boundary $\partial \Omega $, $\Delta_pu:=\mathop{\rm div}(| \nabla u | ^{p-2}\nabla u) $ is the $p$-Laplacian with $1
0$, for all $x\in \Omega $, for all $ (u,v) \in D_3$ such that $0\leq H( x,u,v) \leq C( |u| ^{p'}+ | v| ^{q'})$ \item There exists $C'>0$, for all $x\in \Omega$, for all $( u,v) \in D_2$ such that $H( x,u,v) \leq C'$ \item There exists a positive function $a$ in $L^\infty ( \Omega )$, such that for each $x\in \Omega$ and $(u,v) \in D_1\cap \mathbb{R}_{+}^2$, $H( x,u,v)=a( x)u^{\alpha +1}v^{\beta +1}$, \end{itemize} where \begin{gather*} D_1=\big\{ ( u,v) \in \mathbb{R}_{}^2 : | u| \geq A\text{ or }| v| \geq A\big\} ,\\ D_2=\big\{ ( u,v) \in \mathbb{R}_{}^2\backslash D_{1} : |u| \geq \delta \text{ or }| v| \geq \delta \big\}, \end{gather*} and $D_3=\mathbb{R}^2\backslash (D_1\cup D_2)$ with $A>\delta >0$, $1
1 \quad\mbox{and}\quad
\frac{\alpha +1}{p^{*}}+\frac{\beta +1}{q^{*}}<1
\]
by using a suitable application of the variational method due to
Ambrosetti-Rabinowitz \cite{AR}.
M. Bouchekif \cite{Bo1} generalized the
work of F. de Th\'elin and J.Velin \cite{TV} for the large class of functions
of the form
\[
H( u,v) =a| u| ^\gamma +c| v| ^\delta+b| u| ^{\alpha +1}| v| ^{\beta +1}
\]
where $\alpha $, $\beta \geq 0$; $\gamma $, $\delta >1$ and $a$,
$b$ and $c$ are real numbers.
The case where the system \eqref{Spq} is governed by a single operator
$\Delta_p$ has been studied by Baghli \cite {Ba}.
Our aim is to extend to the system \eqref{Spq} the
results obtained in the scalar case (see \cite{Bo2}). Our
existence results follow from modified quasilinear system in order
to apply the Palais-Smale
condition (P.S.) and then the Moser's Iterative Scheme as in
T. \^{O}tani \cite{O} or in F. de Th\'elin and J. V\'elin \cite{TV}.
We consider only weak solutions, and assume that $H$ satisfies the following
hypothesis.
\begin{enumerate}
\item[(H1)] $H\in C^1( \mathbb{R}^{+}\times \mathbb{R}^{+}) $
\item[(H2)] $H(u,v)=o(u^p+v^q)$ as $(u,v)\to (0^{+},0^{+})$
\item[(H3)] There exists $R_0>0$ and $\mu$, $1<\mu <\min ( p^{*}/p,q^{*}/q)$,
such that
$$\frac up \frac{\partial H}{\partial u}(u,v)+\frac vq
\frac{\partial H}{\partial v}(u,v)\geq \mu H(u,v)>0
\; \forall (u,v)\in \mathbb{R}_{+}^{*}\times
\mathbb{R}_{+}^{*},\; u^p+v^q\geq R_0.$$
\end{enumerate}
\section{Preliminaries and existence results}
The values of $H( u,v) $ are irrelevant for $u\leq 0$ or
$v\leq 0$. We set
\[
I( u,v) =\frac 1p\int_\Omega | \nabla u|
^pdx+\frac 1q\int_\Omega | \nabla v|
^qdx-\int_\Omega m( x) H( u,v) dx
\]
defined on $E:=W_0^{1,p}( \Omega ) \times W_0^{1,q}( \Omega)$.
The solutions of the system \eqref{Spq} are
critical points of the functional $I$. Note that the functional
$I$ does not satisfy in general the Palais-Smale condition since
\[
B_\mu H( u,v) :=\frac up\frac{\partial H}{\partial u}(
u,v) +\frac vq\frac{\partial H}{\partial v}( u,v) -\mu
H( u,v)
\]
$\,\,$is not always bounded. In order to apply
Ambrosetti-Rabinowitz Theorem \cite{AR}, we modify $H$ so that the
corresponding $B_\mu H( u,v) $ becomes bounded.
Let
\[
A( R) =\max \Big\{ \frac{H( u,v) }{( u^p+v^q)^\mu } :
R\leq u^p+v^q\leq R+1\Big\}
\]
and
\begin{align*}
C_R =\max &\Big\{ \sup_{u^p+v^q\leq R+1} \big|
\frac{\partial H}{\partial u}( u,v) \big| +2p\mu A(
R) ( R+1) ^{\mu +1-\frac 1p}\sup_{R\leq r\leq R+1}| \eta
_R'( r) | ;\\
&\sup_{u^p+v^q\leq R+1}\big| \frac{\partial H}{\partial v}( u,v) \big|
+2q\mu A( R)( R+1) ^{\mu +1-\frac 1q}\sup_{R\leq r\leq R+1}
| \eta_R'( r) | \Big\}
\end{align*}
where $\eta_R\in C^1( \mathbb{R})$ is a cutting function defined by
\[
\eta_R( r) \begin{cases}
=1 & \text{if } r\leq R \\
< 0& \text{if } R From the definition of $I_R'$, we write
\begin{eqnarray*}
\lefteqn{ \int_\Omega ( | \nabla u_n| ^{p-2}\nabla
u_n-| \nabla u_l| ^{p-2}\nabla u_l) \nabla (u_n-u_l) dx}\\
&=&\langle I_R'( u_n,v_n)-I_R'( u_l,v_l) ,( u_n-u_l,0)\rangle \\
&&+\int_\Omega m(x) \Big[ \frac{\partial H_R}{\partial u}( u_n,v_n)
-\frac{\partial H_R}{\partial u}( u_l,v_l) \Big] (u_n-u_l) dx.
\end{eqnarray*}
By assumptions on $I_R'$,
$\langle I_R'( u_n,v_n)-I_R'( u_l,v_l) ,( u_n-u_l,0)\rangle $
converges to 0 as $n$ and $l$ tend to $+\infty $.
In what follows, $C$ denotes a generic positive constant.
Now, we prove that
$$ C_{n,l}:=\int_\Omega m( x) [ \frac{\partial H_R}{
\partial u}( u_n,v_n) -\frac{\partial H_R}{\partial u}(
u_l,v_l) ] ( u_n-u_l) dx
$$
converges to 0 as $n$ and $l$ tend to $+\infty $.
We have
\[
| C_{n,l}| \leq | m|_0\int_\Omega
[ | \frac{\partial H_R}{\partial u}( u_n,v_n)
| +| \frac{\partial H_R}{\partial u}( u_l,v_l)
| ] | u_n-u_l| dx
\]
and
\begin{eqnarray*}
\lefteqn{\int_\Omega | \frac{\partial H_R}{\partial u}(
u_n,v_n) | | u_n-u_l| dx }\\
&\leq& \int_\Omega ( C_R+\mu pA( R) | u_n|
^{p-1}( | u_n| ^p+| v_n| ^q) ^{\mu -1}) |u_n-u_l| dx \\
& \leq& 2^{\mu -1}C_R\int_\Omega ( 1+| u_n| ^{\mu
p-1}+| u_n| ^{p-1}| v_n| ^{q\mu -q})| u_n-u_l| dx\\
&\leq& 2^{\mu -1}C_R\Big[ \int_\Omega | u_n-u_l|dx
+\int_\Omega | u_n| ^{\mu p-1}| u_n-u_l|dx\\
&&+\int_\Omega | u_n| ^{p-1}| v_n| ^{q\mu -q}| u_n-u_l| dx\Big] .
\end{eqnarray*}
Using H\"older's inequality and Sobolev's
embeddings, we obtain
\begin{eqnarray*}
\lefteqn{ \int_\Omega \Big| \frac{\partial H_R}{\partial u}(
u_n,v_n) \Big| | u_n-u_l| dx }\\
&\leq &2^{\mu -1}C_R( \mathop{\rm meas}\Omega ) ^{\frac{p-1}p}\Big[
\int_\Omega |u_n-u_l| ^pdx\Big] ^{1/p} \\
&&+2^{\mu -1}C_R\Big[ \int_\Omega | u_n| ^{\mu
p}dx\Big] ^{\frac{\mu p-1}{\mu p}}\Big[ \int_\Omega |
u_n-u_l| ^{\mu p}dx\Big] ^{\frac 1{\mu p}} \\
&&+2^{\mu -1}C_R\Big[ \int_\Omega |
u_n| ^{\mu p}dx\Big] ^{\frac{p-1}{\mu p}}\Big[ \int_\Omega
| v_n| ^{\mu q}dx\Big] ^{\frac{\mu -1}\mu }\Big[
\int_\Omega | u_n-u_l| ^{\mu p}dx\Big] ^{\frac 1{\mu p}},
\end{eqnarray*}
(because $( u_n) \in W_0^{1,p}( \Omega ) $ and $\mu p 0$.
\item There exist $( \phi ,\theta )$ in $E$
such that $I_R( \phi ,\theta ) <0$.
\end{enumerate}
\end{proposition}
\paragraph{Proof.}
From (H2) and taking into account that $H_R( u,v) =H( u,v) $
for $u^p+v^q\leq R$, we can write
\[
\forall \varepsilon >0,\exists \delta_\varepsilon >0:u^p+v^q\leq
\delta_\varepsilon \Longrightarrow H_R( u,v) \leq
\varepsilon ( u^p+v^q) ,
\]
and since $H_R( u,v) /( u^p+v^q) ^\mu $ is uniformly
bounded as $u^p+v^q$ tends to $+\infty $
\[
\exists M( \varepsilon ,R) >0:u^p+v^q\geq \delta
_\varepsilon \Longrightarrow H_R( u,v) \leq M(u^p+v^q) ^\mu .
\]
Then for every $( u,v) $ in $\mathbb{R}^{+}\times \mathbb{R}^{+}$ we have
\[
H_R( u,v) \leq \varepsilon ( u^p+v^q) +M(u^p+v^q) ^\mu .
\]
Hence
\begin{eqnarray*}
\lefteqn{\int_\Omega m( x) H_R( u,v) dx }\\
&\leq& |m|_0\Big[ \varepsilon \int_\Omega ( u^p+v^q)
dx+M\int_\Omega ( u^p+v^q) ^\mu dx\Big] \\
&\leq& | m|_0\Big[ \int_\Omega ( \varepsilon
u^p+2^{\mu -1}Mu^{p\mu }) dx+\int_\Omega (
\varepsilon v^q+2^{\mu -1}Mv^{q\mu }) dx\Big]\\
&\leq& C| m|_0\big[ \varepsilon ( \| u\|
_{1,p}^p+\| v\|_{1,q}^q) +M( \| u\|
_{1,p}^{\mu p}+\| v\|_{1,q}^{\mu q}) \big] .
\end{eqnarray*}
For $I_R(u,v)$, we obtain
\begin{eqnarray*}
I_R( u,v) &\geq &\| u\|_{1,p}^p\big[ \frac 1p-C| m|_0( \varepsilon +M\| u\|
_{1,p}^{\mu p-p}) \big] \\
&&+\| v\|_{1,q}^q\big[ \frac 1q-C| m|_0( \varepsilon +M\|
v\|_{1,q}^{\mu q-q})\big]
\geq \sigma >0,
\end{eqnarray*}
for every $( u,v) $ in the sphere $S( 0,\rho ) $ of
$E$ where $\rho $ is such that $0<\rho <\min ( \rho_1,\rho
_2) $ with
\[
\rho_1=\big[ \frac 1{pMC| m|_0}-\frac \varepsilon M\big]
^{\frac 1{\mu p-p}}\quad\text{and}\quad
\rho_2=\big[ \frac 1{qMC|m|_0}-\frac \varepsilon M\big] ^{\frac 1{\mu q-q}}
\]
with $\varepsilon $ sufficiently small.
\noindent 2.\quad Choose $( \phi ,\theta )\in E$ such that:
$\phi >0$, $\theta >0$,
\[
\mathop{\rm supp}\phi \subset \Omega ^{+},
\quad \mathop{\rm supp}\theta \subset \Omega ^{+},
\]
where $\Omega ^{+}=\{ x\in \Omega ; m( x) >0\}$.
Hence, for $t$ sufficiently large,
\begin{eqnarray*}
I_R( t^{1/p}\phi ,t^{1/q}\theta ) &=&
\frac tp\| \phi\|_{1,p}^p+\frac tq\| \theta \|
_{1,q}^q-\int_\Omega m( x) H_R( t^{1/p}\phi,t^{1/q}\theta ) dx\\
&\leq& t\big[ \frac{\| \phi \|_{1,p}^p}p+\frac{\|
\theta \|_{1,q}^q}q\big] -t^\mu \frac{m_{R_0}}{R_0^\mu
}\int_\Omega m( x) ( \phi ^p+\theta ^q) ^\mu dx
\end{eqnarray*}
and so $\lim_{t\to +\infty }I_R( t^{1/p}\phi ,t^{1/q}\theta ) =-\infty$,
(because $\mu>1$). By continuity of $I_R$ on $E$, there exists $( \phi
,\theta ) $ in $E\setminus B( 0,\rho ) $ such that
$I_R( \phi ,\theta ) <0$. By the usual Mountain-Pass
Theorem, we know that there exists a critical point of $I_R$ which
we denote by $( u_R,v_R) $, and corresponding to a critical
value $c_R\geq \sigma $. Since $( u_R^{+},v_R^{+}) $, where
$u_R^{+}:=\max ( u_R,0) $, is
also solution for the system $( S_{p,q}^{H_R}) $, we assume $u_R\geq 0$
and $v_R\geq 0$. Positivity of $u_R$ and $v_R$ follows from
Harnack's inequality (see J. Serrin \cite{Se}). We prove now that, under
some conditions, $( u_R,v_R) $ is also solution of the system
\eqref{SpqH}.
\section{Existence results}
We adapt the Moser iteration used in \cite{O,TV} to
construct two strictly unbounded sequences $( \lambda_k)
_{k\in \mathbb{N}}$ and $( \mu_k)_{k\in \mathbb{N}}$ such that
$( u_R,v_R) $ satisfies
\[
\text{if }\left\{\begin{array}{c}
u_R\in L^{\lambda_k}( \Omega ) \\
v_R\in L^{\mu_k}( \Omega )\end{array}\right\}\quad\text{then}\quad
\left\{\begin{array}{c}
u_R\in L^{\lambda_{k+1}}( \Omega ) \\
v_R\in L^{\mu_{k+1}}( \Omega ) .
\end{array} \right\}
\]
\subsection*{Bootstrap argument}
\begin{proposition} \label{prop5}
Under the assumptions of Theorem \ref{thm1}, there exist two sequences
$( \lambda_k)_k$ and $( \mu_k)_k$ such that
\begin{enumerate}
\item For each $k$, $u_R$ and $v_R$
belong to $L^{\lambda_k}( \Omega ) $ and $L^{\mu_k}( \Omega ) $ respectively
\item There exist two positive constants $C_p$ and $C_q$ such that
\[
\| u_R\|_\infty \leq \limsup_{k\to +\infty }\| u_R\|_{L^{\lambda_k}}
\leq C_p,\quad\text{and}\quad
\|v_R\|_\infty \leq \limsup_{k\to +\infty }\| v_R\|_{L^{\mu_k}}\leq C_q.
\]
\end{enumerate}
\end{proposition}
\begin{lemma} \label{lm6}
Let $( a_k)_{k\in \mathbb{N}}$ and $( b_k)_{k\in \mathbb{N}}$
be two positive sequences satisfying, for each integer $k$,
the relations
\begin{equation}
\frac{p+a_k}{\lambda_k}+\frac{q( \mu -1) }{\mu_k}=1,\quad\text{and}
\quad \frac{q+b_k}{\mu_k}+\frac{p( \mu -1) }{\lambda_k}=1. \label{3.1}
\end{equation}
If $u_R$ and $v_R$ are in $L^{\lambda_k}( \Omega ) $ and
$L^{\mu_k}( \Omega )$ respectively,
$\lambda_{k+1}\leq (1+\frac{a_k}p) \pi_p$,
$\mu_{k+1}\leq ( 1+\frac{b_k}q) \pi_q$ with
$1<\pi_p 1$, $1<\mu p\widehat{C} 0$. The sequences $%
( \lambda_k)_k$ and $( \mu_k)_k$ are defined by
$\lambda_k=pf_k$ and $\mu_k=qf_k$,
where
\[
f_k=\frac C{C-1}[ \delta C^{k-1}+( \mu -1) ] .
\]
We remark that the three last sequences are strictly increasing and
unbounded. Furthermore $( f_k) $ satisfies the relation
$f_{k+1}=C[ f_k-( \mu -1) ]$.
\paragraph{Proof of Proposition 2.}
1. We show by induction that for all integer $k$,
$u_R\in L^{\lambda_k}( \Omega )$ and $v_R\in L^{\mu_k}( \Omega )$.
For $k=0$,
\[
\lambda_0=pf_0=\frac{pC}{C-1}\big[ \frac \delta C+( \mu-1)\big]
=p\frac Np\big[ \frac pN\mu \widehat{C}^{k_0}\big]
=\widehat{\lambda }_{k_0},
\]
and similarly, $\mu_0=\widehat{\mu }_{k_0}$.
By Lemma 4, $u_R\in L^{\lambda_0}( \Omega )$ and $v_R\in L^{\mu_0}( \Omega )$.
Suppose that $( u_R,v_R) \in L^{\lambda_k}( \Omega )
\times L^{\mu_k}( \Omega ) $. First we establish that
$\lambda_k=a_k+p\mu$. By condition \eqref{3.1},
\[
1=\frac{p+a_k}{\lambda_k}+q\frac{\mu -1}{\mu_k}=\frac p{\lambda_k}-\frac
q{\mu_k}+\frac{a_k}{\lambda_k}+\mu \frac q{\mu_k},
\]
thus
\[
\frac{a_k}{pf_k}+\frac \mu {f_k}=1
\]
which implies
$a_k=p( f_k-\mu ) =\lambda_k-p\mu$, and similarly
$\mu_k=b_k+q\mu =q( f_k-\mu )$.
Now when we take $\pi_p=Cp$ and $\pi_q=Cq$, we then have
\[
\big[ 1+\frac{a_k}p\big] \pi_p=( 1+f_k-\mu )Cp=pf_{k+1}=\lambda_{k+1}.
\]
and similarly
$[ 1+\frac{b_k}q] \pi_q=\mu_{k+1}$.
Since $( u_R,v_R) \in L^{\lambda_k}( \Omega ) \times
L^{\mu_k}( \Omega ) $, we conclude, according to Lemma 3,
that
\[
( u_R,v_R) \in L^{\lambda_{k+1}}( \Omega ) \times
L^{\mu_{k+1}}( \Omega ) .
\]
So $u_R\in L^{\lambda_k}( \Omega ) $, and $v_R\in L^{\mu_k}( \Omega ) $,
for all integer $k$.
\noindent 2. Now we prove that $u_R$ and $v_R$ are bounded. By Lemma 3,
we have
\begin{gather*}
\| u_R\|_{\lambda_{k+1}}^{\lambda_{k+1}}\leq
K_p\Big\{ \theta_p\big[ 1+\frac{a_k}p\big] \big\{ C_R|
m|_0( \| u_R\|_{\lambda_k}^{\lambda_k}+\| v_R\|_{\mu_k}^{\mu_k})
\big\} ^{1/p}\Big\} ^{\frac{\lambda_{k+1}}{1+\frac{a_k}p}},
\\
\| v_R\|_{\mu_{k+1}}^{\mu_{k+1}}\leq K_q\Big\{
\theta_q\big[ 1+\frac{b_k}q\big] \big\{ C_R| m|
_0( \| u_R\|_{\lambda_k}^{\lambda_k}+\|
v_R\|_{\mu_k}^{\mu_k}) \big\} ^{\frac 1q}\Big\}
^{\frac{\mu_{k+1}}{1+\frac{b_k}q}}.
\end{gather*}
We remark that
\[
\frac{\lambda_{k+1}}{1+\frac{a_k}p}=pC\quad\text{and}\quad
\frac{\mu_{k+1}}{1+\frac{b_k}q}=qC.
\]
Consequently,
\begin{gather*}
\| u_R\|_{\lambda_{k+1}}^{\lambda_{k+1}}\leq
2^CK_p\theta_p^{pC}\big[ 1+\frac{a_k}p\big]_\infty ^{pC}(
| m|_0^{}C_R^{}) ^C\max \big( \| u_R\|
_{\lambda_k}^{\lambda_kC},\| v_R\|_{\mu_k}^{\mu_kC}\big),
\\
\| v_R\|_{\mu_{k+1}}^{\mu_{k+1}}\leq 2^CK_q\theta
_q^{qC}\big[ 1+\frac{b_k}q\big] ^{qC}( | m|
_0^{}C_R^{}) ^C\max \big( \| u_R\|_{\lambda
_k}^{\lambda_kC},\| v_R\|_{\mu_k}^{\mu_kC}\big) .
\end{gather*}
We have
\[
1+\frac{a_k}p=1+\frac{b_k}q=1+f_k-\mu <\frac C{C-1}
\big[ \frac \delta C+\mu-1\big] C^k.
\]
Take
\[
A:=\frac C{C-1}\big[ \frac \delta C+\mu -1\big] [ K_p+K_q]
\]
and $\theta :=2| m|_0\max ( \theta_p^p,\theta_q^q)$,
then we can write
\[
\max \Big( \| u_R\|_{\lambda_{k+1}}^{\lambda
_{k+1}},\| v_R\|_{\mu_{k+1}}^{\mu_{k+1}}\Big)
\leq ( A^q\theta ) ^CC^{kqC}C_R^C\max \Big( \| u_R\|
_{\lambda_k}^{C\lambda_k},\| v_R\|_{\mu_k}^{C\mu_k}\Big) .
\]
We construct an iterative relation
\[
E_{k+1}\leq r_k+CE_k
\]
where
$E_k=\ln \max ( \| u_R\|_{\lambda_k}^{\lambda_k},
\|v_R\|_{\mu_k}^{\mu_k}) $, and $r_k=ak+b$, with
$a=\ln C^{qC}$ and $b=\ln [ A^q\theta C_R] ^C$.
Proceeding step by step, we find
\begin{eqnarray*}
E_{k+1} &\leq &r_k+Cr_{k-1}+C^2r_{k-2}+\cdots +C^kr_0+C^{k+1}E_{0,} \\
E_{k+1} &\leq &C^{k+1}E_0+\sum_{i=0}^kC^ir_{k-i}.
\end{eqnarray*}
Let us evaluate
\[
\sigma_k:=\sum_{i=0}^kC^ir_{k-i}.
\]
We have
$r_{k-i}=a( k-i) +b=ak+b-ai$,
then
\begin{eqnarray*}
\sigma_k&=&( ak+b) \sum_{i=0}^kC^i-a\sum_{i=0}^kiC^i\\
&=& \frac{bC^{k+2}+( a-b) C^{k+1}+( 1-C) ak-[ C(
a+b) -b] }{( C-1) ^2}.
\end{eqnarray*}
Since $C>1$, and $a$, $b$ are positive, we have
\[
\sigma_k\leq \frac{bC^{k+2}+( a-b) C^{k+1}}{( C-1)^2}
\]
then
\[
E_{k+1}\leq \frac{bC^{k+2}}{( C-1) ^2}+C^{k+1}
\big[ \frac{a-b}{( C-1) ^2}+E_0\big] .
\]
By an appropriate choice for the constants $K_p$ and $K_q$, we ensure that
\[
\frac{b-a}{( C-1) ^2}\geq E_0.
\]
Recall that
\[
b-a=C\ln \frac{A^q\theta C_R}{C^q}\quad\text{with}\quad
A=\frac C{C-1}\big[ \frac \delta C+\mu -1\big] [K_p+K_q] ;
\]
hence $E_{k+1}\leq bC^{k+2}/ ( C-1) ^2$.
By the definition of $E_{k+1}$ and the last inequality, we obtain
\[
\lambda_{k+1}\ln \| u_R\|_{\lambda_{k+1}}\leq
E_{k+1}\leq \frac{bC^{k+2}}{( C-1) ^2},
\]
thus
\[
\ln \| u_R\|_{\lambda_{k+1}}\leq
\frac{bC^{k+2}}{\lambda_{k+1}( C-1) ^2}.
\]
Letting $k\to +\infty $, we find
\[
\ln \| u_R\|_\infty \leq \frac{bC}{p\delta (C-1) },
\quad\text{or}\quad
\ln \| u_R\|_\infty \leq \frac N{\delta p^2}b.
\]
Similarly
\[
\ln \| v_R\|_\infty \leq \frac N{\delta q^2}b.
\]
We deduce the existence of constants $C_p$ and $C_q$ such that:
\[
\| u_R\|_\infty \leq C_p\quad \text{and}\quad
\| v_R\|_\infty \leq C_q.
\]
Take
\[
C_p=\exp \frac N{\delta p^2}b,\quad\text{and}\quad
C_q=\exp \frac N{\delta q^2}b.
\]
Then $C_p$ and $C_q$, are greater than $1$, which is compatible with the
remark noted at the beginning of the proof of Lemma 3.
This completes the proof of proposition 1.
\paragraph{Proof of Theorem \ref{thm1}.}
If $\| u_R\|_\infty ^p+\| v_R\|_\infty^q