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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 171, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2015/171\hfil Existence and asymptotic behavior of solutions]
{Existence and asymptotic behavior of solutions to nonlinear radial 
$p$-Laplacian equations}

\author[S. Masmoudi, S. Zermani \hfil EJDE-2015/171\hfilneg]
{Syrine Masmoudi, Samia Zermani}

\address{Syrine Masmoudi \newline
 D\'epartement de Math\'ematiques, Facult\'e
des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia}
\email{syrine.sassi@fst.rnu.tn}

\address{Samia Zermani \newline
 D\'epartement de Math\'ematiques, Facult\'e
des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia}
\email{zermani.samia@yahoo.fr}


\thanks{Submitted February 2, 2015. Published June 22, 2015.}
\subjclass[2010]{34B18, 35J66}
\keywords{$p$-Laplacian problem; positive solution; boundary behavior;
\hfil\break\indent Schauder fixed point theorem}

\begin{abstract}
 This article concerns the existence, uniqueness and boundary behavior of
 positive solutions to the nonlinear problem
 \begin{gather*}
 \frac{1}{A}(A\Phi _p(u'))'+a_1(x)u^{\alpha_1}+a_2(x)u^{\alpha_2}=0, \quad \text{in } (0,1), \\
 \lim_{x\to 0}A\Phi _p(u')(x)=0,\quad u(1)=0,
 \end{gather*}
 where $p>1$, $\alpha _1,\alpha _2\in (1-p,p-1)$,
 $\Phi_p(t)=t|t|^{p-2}$, $t\in \mathbb{R}$, $A$ is a positive differentiable
 function and $a_1,a_2$ are two positive measurable functions in $(0,1)$
 satisfying some  assumptions related to Karamata regular
 variation theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}

 In recent years, the existence of positive solutions for elliptic
problems involving the $p$-Laplacian has found considerable interest and
different approaches have been developed. This is due to their
signification in various areas of pure and applied mathematics including
geological sciences, fluid dynamics, electrostatics, cosmology (see
\cite{AM,ADT,CN,FOP,KYY}), as well as in relation to inequalities of
Poincar\'e, Writinger, Sobolev type and isoperimetric inequalities
(see \cite{BFK,DM,DT,K,MGS}). Motivations for studying radial solutions
can be found in \cite{BM,BD,G,K,RW} and references therein.

 M\^{a}agli et al \cite{MGS} considered the  quasilinear
elliptic problem
\begin{equation}
\begin{gathered}
-\Delta _pu:=-\operatorname{div}( |\nabla u|^{p-2}\nabla u) =q( x) u^{\alpha }
\quad \text{in }\Omega \\
u=0\quad \text{on }\partial \Omega ,
\end{gathered} \label{1.1}
\end{equation}
where $\Omega $ is a $C^{2}$ bounded domain of $\mathbb{R}^{n}$
 $( n\geq 2) $, $p>1$, the exponent $\alpha \in (-1,p-1) $ and $q\in C( \Omega ) $
is a positive function having  singular behavior near the boundary
$\partial \Omega $. More
precisely, let $d(x)$ be the Euclidean distance of $x\in \Omega $ to
$\partial \Omega $ then $q( x) =d( x) ^{-\beta }L(d(x)) $, with
$0<\beta <p$ and $L$ belongs to a functional class
$\mathcal{K}$ called Karamata class and defined on
$(0,\eta ],(\eta >\operatorname{diam}( \Omega ) )$ by
\[
\mathcal{K}:=\big\{ t\to L(t):=c\exp \Big(\int_{t}^{\eta }\frac{z(s)
}{s}ds\Big): c>0 \text{ and }z\in C([0,\eta ]),\;\;z(0)=0\big\} .
\]
For the convenience of the readers, we briefly describe the result proved in
\cite{MGS}.

\begin{theorem}[\cite{MGS}] \label{thm1}
 Let $L\in \mathcal{K\cap C}^{2}( ( 0,\eta ]) $ such that
$\int_{0}^{\eta }t^{\frac{1-\beta }{p-1}}L(t)^{\frac{1}{
p-1}}dt<\infty $. Then problem \eqref{1.1} has a unique positive and
continuous solution $u$ satisfying, for $x\in (0,1)$,
\begin{equation}
u(x)\approx \begin{cases}
\Big( \int_{0}^{d(x)}s^{-1}L(s)^{\frac{1}{p-1}}ds\Big) ^{\frac{p-1}{
p-1-\alpha }} & \text{if }\beta =p\text{,} \\[4pt]
d(x)^{\frac{p-\beta}{p-1-\alpha }}L( d(x)) ^{\frac{1}{p-1-\alpha }} &
\text{if }1+\alpha \leq \beta <p, \\[4pt]
d(x)\Big( \int_{d(x)}^{\eta }s^{-1}L(s)ds\Big) ^{\frac{1}{p-1-\alpha }} &
\text{if }\beta =1+\alpha , \\[4pt]
d(x) & \text{if }\beta <1+\alpha .
\end{cases}
  \label{1.3}
\end{equation}
\end{theorem}

Here and throughout this paper, the notation $f(x)\approx g(x)$, $x\in S$
for $f$ and $g$ nonnegative functions defined on a set $S$, means that there
exists $c>0$ such that $\frac{1}{c}g(x)\leq f(x)\leq cg(x)$, for each $x\in
S $.

 If $\Omega $ is the unit ball, similar result was shown in \cite{BM}
for radial solution of problem \eqref{1.1} which becomes in the radial form
\begin{equation}
\begin{gathered}
\frac{1}{A}(A\Phi _p(u'))'+q(x)u^{\alpha}=0,\quad \text{in } (0,1),  \\
\lim_{x\to 0}A\Phi _p(u')(x)=0,\quad u(1)=0,
\end{gathered}  \label{1.4}
\end{equation}
where $\Phi _p(t)=t|t|^{p-2}$, $t\in \mathbb{R}$ and $A(t)=t^{n-1}$.
Indeed, by using Karamata variation theory, the authors in \cite{BM}
established existence and asymptotic behavior of a unique positive
continuous solution to \eqref{1.4} for $\alpha <p-1$ and for a large class
of functions $A$ including the example $A(t)=t^{n-1}$. More precisely, they
proved Theorem \ref{thm2} below under the following assumptions:
\begin{itemize}
\item[(H0)] $A$ is a continuous function in $[0,1)$, positive and
differentiable in $(0,1)$ such that
\[
A(x)\approx x^{\lambda }(1-x)^{\mu },
\]
where $\lambda \geq 0$ and $\mu <p-1$.

\item[(H1)] $q$ is a positive measurable function on $(0,1)$ such that
\[
q(x)\approx (1-x)^{-\beta }L(1-x)
\]
with $\beta \leq p$ and $L\in \mathcal{K}$ defined on
$(0,\eta ]$ $(\eta >1) $ such that
\[
\int_{0}^{\eta }t^{\frac{1-\beta }{p-1}}L(t)^{\frac{1}{p-1}}dt<\infty .
\]
\end{itemize}

\begin{remark} \label{rmk1} \rm
We need to verify condition $\int_{0}^{\eta }t^{\frac{1-\beta }{p-1}}L(t)^{
\frac{1}{p-1}}dt<\infty $ in hypothesis (H1), only
for $\beta =p$. This is due to Karamata's theorem which we recall in Lemma
 \ref{kar} below.
\end{remark}

\begin{theorem}[\cite{BM}] \label{thm2}
 Assume {\rm (H0)--(H1)} hold. Then  problem \eqref{1.4} has a unique positive
and continuous solution $u$ satisfying, for $x\in (0,1)$,
\begin{equation}
u(x)\approx (1-x)^{\min {(\frac{p-\beta }{p-1-\alpha },\frac{p-1-\mu }{p-1})}
}\Psi _{L,\beta ,\alpha }(x),  \label{1.5}
\end{equation}
where $\Psi _{L,\beta ,\alpha }$ is the function defined on $(0,1)$ by
\begin{equation}
\Psi _{L,\beta ,\alpha }( x) :=\begin{cases}
\Big( \int_{0}^{1-x}s^{-1}L(s)^{\frac{1}{p-1}}ds\Big) ^{\frac{p-1}{
p-1-\alpha }} & \text{if }\beta =p,  \\[4pt]
L(1-x)^{\frac{1}{p-1-\alpha }} & \text{if }\frac{(\mu +1)(p-1-\alpha
)+\alpha p}{p-1}<\beta <p, \\[4pt]
\Big( \int_{1-x}^{\eta }s^{-1}L(s)ds\Big) ^{\frac{1}{p-1-\alpha }} &
\text{if }\beta =\frac{(\mu +1)(p-1-\alpha )+\alpha p}{p-1},  \\[4pt]
1 & \text{if }\beta <\frac{(\mu +1)(p-1-\alpha )+\alpha p}{p-1}.
\end{cases}  \label{1.6}
\end{equation}
\end{theorem}

For the special case when $A(t)=t^{n-1}$ the estimates \eqref{1.3} and
\eqref{1.5} are the same.

In this article, we study the  boundary-value problem
\begin{equation}
\begin{gathered}
\frac{1}{A}(A\Phi _p(u'))'+a_1(x)u^{\alpha_1}+a_2(x)u^{\alpha _2}=0,\quad
\text{in }(0,1),\\
\lim_{x\to 0}A\Phi _p(u')(x)=0,\quad u(1)=0,
\end{gathered} \label{1.7}
\end{equation}
where $\alpha _1,\alpha _2\in (1-p,p-1)$ and $A$ satisfies (H0).
 Our purpose is to establish an existence and a
uniqueness  of a continuous solution to \eqref{1.7} and to give
estimates on such solution, where appear the combined effects of singular
and sublinear terms in the nonlinearity.

 The pure elliptic semilinear problem corresponding to \eqref{1.7} is
\begin{equation}
\begin{gathered}
\Delta u+a_1(x)u^{\alpha _1}+a_2(x)u^{\alpha _2}=0,\quad
\text{in }\Omega , \\
 u=0,\quad \text{on } \partial \Omega ,
\end{gathered}  \label{1.8}
\end{equation}
which has been studied by several authors on smooth domains, see for example
\cite{CMMZ,Mag,R,RR,SWL} and references therein.

Let us introduce our conditions on the functions $a_i$:
\begin{itemize}
\item[(H2)] For $i\in \{1,2\}$, $a_i$ is a positive measurable
function and satisfies for each $x\in (0,1)$
\[
a_i(x)\approx (1-x)^{-\beta _i}L_i(1-x),
\]
where $\beta _i\leq p$ and $L_i\in \mathcal{K}$ defined on
$(0,\eta]$, $(\eta >1)$ such that
\[
\int_{0}^{\eta }t^{\frac{1-\beta _i}{p-1}}L_i(t)^{\frac{1}{p-1}}dt<\infty .
\]
\end{itemize}

As it turns out,  estimates \eqref{1.5} depend closely on
$\min {(\frac{ p-\beta }{p-1-\alpha },\frac{p-1-\mu }{p-1})}$.
 Also, as it will be seen, the numbers
\begin{gather*}
\delta _1=\min{(\frac{p-\beta _1}{p-1-\alpha _1},\frac{p-1-\mu }{p-1})}, \\
\delta _2=\min{(\frac{p-\beta _2}{p-1-\alpha _2},\frac{p-1-\mu }{p-1})}
\end{gather*}
play a crucial role in the combined effects of singular and sublinear
nonlinearities in problem \eqref{1.7} and lead to a competition. However,
without loss of generality, we can suppose that
$\frac{p-\beta _1}{p-1-\alpha _1}\leq \frac{p-\beta _2}{p-1-\alpha _2}$
and we introduce the function $\theta $ defined on $(0,1)$ by
\begin{equation}
\theta ( x) :=\begin{cases}
(1-x)^{\delta _1}\Psi _{L_1,\beta _1,\alpha _1}(x) &
\text{if }\delta _1<\delta _2, \\
(1-x)^{\delta _1}( \Psi _{L_1,\beta _1,\alpha
_1}(x)+\Psi _{L_2,\beta _2,\alpha _2}(x)) & \text{if }\delta
_1=\delta _2,
\end{cases}  \label{1.9}
\end{equation}
where for $i\in \{1,2\}$, $\Psi _{L_i,\beta _i,\alpha _i}$ is the
function defined by \eqref{1.6}.

Now, we are ready to state our main result.

\begin{theorem} \label{thm3}
Assume {\rm (H0)} and {\rm (H2)} hold and suppose that
$\alpha _1,\alpha _2\in (1-p,p-1)$. Then problem \eqref{1.7} has a unique
positive and continuous solution $u$ satisfying, for each $x\in (0,1)$,
\begin{equation}
u(x)\approx \theta (x),  \label{1.10}
\end{equation}
where $\theta $ is the function defined by \eqref{1.9}.
\end{theorem}

 The outline of this article is as follows.
In Section 2, we give some already known results on functions in
$\mathcal{K}$ useful for our study and
we give estimates of some potential functions.
In Section 3, we prove our main result.

\section{Preliminary results}

 Our arguments combine a method of fixed point theorem with Karamata
regular variation theory. So, we are quoting some properties of functions in
$\mathcal{K}$ useful for our study.

\subsection{The Karamata class $\mathcal{K}$}

It is obvious to see that a function $L$ is in $\mathcal{K}$ if and only if
 $L$ is a positive function in $C^{1}( (0,\eta ]) $ such that
\[
\lim_{t\to 0}\frac{tL'(t)}{L(t)}=0.
\]
A standard function belonging to the class $\mathcal{K}$ is given by
\[
L(t):=\prod_{k=1}^{p}(\log _{k}(\frac{\omega }{t}))^{\lambda _{k}},
\]
where $p\in {\mathbb{N}}^{\ast }$,
$(\lambda _1,\lambda _2,\dots ,\lambda_p)\in {\mathbb{R}}^{p}$,
$\omega $ is a positive real number
sufficiently large and $\log _{k}(x)=\log o\log o\dots .o\log (x)$ ($k$ times).

\begin{lemma}[\cite{S}] \label{k1} \quad
\begin{itemize}
\item[(i)] Let $L\in \mathcal{K}$ and $\varepsilon >0$. Then
$\lim_{t\to 0}t^{\varepsilon }L(t)=0$.

\item[(ii)] Let $L_1,L_2\in \mathcal{K},p\in \mathbb{R}$. Then
$L_1+L_2\in \mathcal{K},L_1L_2\in \mathcal{K}$ and $L_1^{p}\in
\mathcal{K}$.
\end{itemize}
\end{lemma}

\begin{lemma}[Karamata's Theorem]\label{kar}
 Let $L\in \mathcal{K}$ be defined on $(0,\eta ]$ and $\sigma \in \mathbb{R}$.
\begin{itemize}
\item[(i)] If $\sigma >-1$, then $\int_{0}^{\eta }t^{\sigma }L(t)dt$ converges and
\[
\int_{0}^{t}s^{\sigma }L(s)ds\sim \frac{t^{1+\sigma }L(t)}{\sigma +1}
\quad\text{as }t\to 0^{+}.
\]

\item[(ii)] If $\sigma <-1$, then $\int_{0}^{\eta }t^{\sigma }L(t)dt$ diverges and
\[
\int_{t}^{\eta }s^{\sigma }L(s)ds\sim -\frac{t^{1+\sigma }L(t)}{\sigma +1}
\quad\text{as }t\to 0^{+}.
\]
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{CMMZ}] \label{k2}
 Let $L\in \mathcal{K}$ be defined on $(0,\eta ]$. Then
\[
t\to \int_{t}^{\eta }\frac{L(s)}{s}ds\in \mathcal{K}.
\]
If further $\int_{0}^{\eta }\frac{L(t)}{t}dt$ converges, then
\[
t\to \int_{0}^{t}\frac{L(s)}{s}ds\in \mathcal{K}.
\]
\end{lemma}

\begin{lemma}[\cite{CMMZ}] \label{estim2}
For $i\in \{1,2\}$, let $\eta _i<1$ and $L_i\in \mathcal{K}$.
For $t\in (0,\eta )$, put
\[
J(t)=\Big( \int_{t}^{\eta }\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{1-\eta_1}}
+\Big( \int_{t}^{\eta }\frac{L_2(s)}{s}ds\Big) ^{\frac{1}{1-\eta _2}}.
\]
Then, for $t\in (0,\eta )$, we have
\[
\int_{t}^{\eta }\frac{(J^{\eta _1}L_1+J^{\eta _2}L_2)(s)}{s}
ds\approx J(t).
\]
\end{lemma}

\begin{lemma}[\cite{CMMZ}] \label{estim1}
 For $i\in \{1,2\}$, let
$\eta _i<1$ and $L_i\in \mathcal{K}$ such that
$\int_{0}^{\eta }\frac{ L_i(s)}{s}ds<\infty $. For $t\in (0,\eta )$, put
\[
I(t)=\Big( \int_{0}^{t}\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{1-\eta _1}
}+\Big( \int_{0}^{t}\frac{L_2(s)}{s}ds\Big) ^{\frac{1}{1-\eta _2}}.
\]
Then, for $t\in (0,\eta )$, we have
\[
\int_{0}^{t}\frac{(I^{\eta _1}L_1+I^{\eta _2}L_2)(s)}{s}ds\approx
I(t).
\]
\end{lemma}

\subsection{Potential estimates}

For a nonnegative measurable function $f$ in $(0,1)$, let
\[
G_pf(x):=\int_{x}^{1}( \frac{1}{A(t)}
\int_{0}^{t}A(s)f(s)ds) ^{\frac{1}{p-1}}dt.
\]
We point out that if $f$ is a nonnegative measurable function such that the
mapping $x\to A(x)f(x)$ is integrable in $\left[ 0,1\right] $, then $
G_pf$ is the solution of the problem
\begin{equation}
\begin{gathered}
\frac{1}{A}(A\Phi _p(u'))'+f=0,\quad \text{in } (0,1), \\
\lim_{x\to 0}A\Phi _p(u')(x)=0,\quad u(1)=0.
\end{gathered}  \label{potentiel}
\end{equation}
In what follows, we aim to prove Proposition \ref{compasymp}. To this end,
we need the following two lemmas which are proved in \cite{BM} and
\cite{CMMZ}.

\begin{lemma}[\cite{BM}] \label{estimarad}
 Assume {\rm (H0)} and {\rm (H1)} hold. Then for $x\in (0,1)$, we have
\[
G_p(q)(x)\approx (1-x)^{\min {(\frac{p-\beta }{p-1},\frac{p-1-\mu }{p-1})}}
\begin{cases}
\int_{0}^{1-x}\frac{L(s)^{\frac{1}{p-1}}}{s}ds & \text{if }\beta =p, \\[4pt]
 L(1-x)^{\frac{1}{p-1}} & \text{if }\mu +1<\beta <p,\\[4pt]
( \int_{1-x}^{\eta }\frac{L(s)}{s}ds) ^{\frac{1}{p-1}} & \text{if }\beta =\mu +1, \\[4pt]
1 & \text{if }\beta <\mu +1.
\end{cases}
\]
\end{lemma}

\begin{lemma}[\cite{CMMZ}] \label{estim}
For $s,t>0$,  $\eta _1<1$ and $\eta _2<1$, we have
\[
2^{-max(1-\eta _1,1-\eta _2)}(t+s)\leq t^{1-\eta _1}(t+s)^{\eta
_1}+s^{1-\eta _2}(t+s)^{\eta _2}\leq 2(t+s).
\]
\end{lemma}

\begin{proposition}\label{compasymp}
Assume {\rm (H0)} and {\rm (H2)} hold. Let $\theta $ be the function given
by \eqref{1.9}. Then for $x\in (0,1)$,
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
\theta (x).
\]
\end{proposition}

\begin{proof}
For $t\in(0,1)$, let
\begin{gather*}
K(t):=\Big(L_1^{\frac{1}{p-1-\alpha_1}}+L_2^{\frac{1}{p-1-\alpha_2}}\Big)(t),\\
N(t):=\Big(\int_{t}^{\eta}\frac{L_1(s)}{s}ds\Big)^{\frac{1 }{p-1-\alpha_1}}
+\Big(\int_{t}^{\eta}\frac{L_2(s)}{s}ds\Big)^{\frac{1}{p-1-\alpha_2}},\\
M(t):=\Big( \int_{0}^{t}\frac{(L_1(s))^{\frac{1}{p-1}}}{s}
ds\Big) ^{\frac{p-1}{p-1-\alpha _1}}+\Big( \int_{0}^{t}
\frac{(L_2(s))^{\frac{1}{p-1}}}{s}ds\Big) ^{\frac{p-1}{p-1-\alpha _2}}.
\end{gather*}
Since $\delta _1<\delta _2$ is equivalent to
$\frac{p-\beta _1}{p-1-\alpha _1}<\frac{p-\beta _2}{p-1-\alpha _2}$ and
$\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}<\beta _1<p$,  we can write
\[
\theta ( x) =\begin{cases}
\Big( \int_{0}^{1-x}\frac{L_1(s)^{\frac{1}{p-1}}}{s}
ds\Big) ^{\frac{p-1}{p-1-\alpha _1}} \quad \text{if }\beta _1=p,\; \beta_2<p,
\\[4pt]
(1-x)^{\frac{p-\beta _1}{p-1-\alpha _1}}L_1(1-x)^{\frac{1
}{p-1-\alpha _1}} \\
\quad \text{if }\frac{p-\beta _1}{p-1-\alpha _1}<\frac{
p-\beta _2}{p-1-\alpha _2},\;\frac{(\mu +1)(p-1-\alpha _1)+p\alpha
_1}{p-1}<\beta _1<p,
\\[4pt]
(1-x)^{\frac{p-\beta _1}{p-1-\alpha _1}}K(1-x) \\
\quad \text{if }
\frac{p-\beta _1}{p-1-\alpha _1}=\frac{p-\beta _2}{p-1-\alpha _2}
,\; \frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}<\beta_1<p,
\\[4pt]
(1-x)^{\frac{p-1-\mu }{p-1}}N(1-x) \\
\quad \text{if }\beta _1=\frac{
(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1},\; \beta _2=\frac{(\mu
+1)(p-1-\alpha _2)+p\alpha _2}{p-1},
\\[4pt]
(1-x)^{\frac{p-1-\mu }{p-1}}( \int_{1-x}^{\eta }\frac{
L_1(s)}{s}ds) ^{\frac{1}{p-1-\alpha _1}} \\
\quad \text{if }\beta _1=
\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1},\; \beta _2<\frac{(\mu
+1)(p-1-\alpha _2)+p\alpha _2}{p-1}, \\
(1-x)^{\frac{p-1-\mu }{p-1}} \quad \text{if }\beta _1<\frac{(\mu
+1)(p-1-\alpha _1)+p\alpha _1}{p-1},
 \\
 M(1-x) \quad \text{if }\beta _1=\beta _2=p.
\end{cases}
\]

The main idea is to prove that the function
$a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2}$ satisfies
(H1) and then to apply Lemma \ref{estimarad}.

Throughout the proof, we use Lemma \ref{k1} and Lemma \ref{k2} to verify
that some functions are in $\mathcal{K}$.
We distinguish the following cases.
\smallskip

\noindent\textbf{Case 1.} $\beta _1=p$ and $\beta _2<p$.
Using (H2) we have
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{-p}L_1(1-x)\theta ^{\alpha _1}(x)+(1-x)^{-\beta
_2}L_2(1-x)\theta ^{\alpha _2}(x).
\]
Since $\theta (x)=( \int_{0}^{1-x}s^{-1}L_1(s)^{\frac{1}{p-1}
}ds) ^{\frac{p-1}{p-1-\alpha _1}}<\infty $, by Lemmas \ref{k1}
and \ref{k2}, the function $x\to L_i(1-x)\theta ^{\alpha
_i}(x)\in \mathcal{K}$, for $i\in \{1,2\}$.

Now, using $\beta _2<p$ we deduce by Lemma \ref{k1}(i) that
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{-p}L_1(1-x)\theta ^{\alpha _1}(x).
\]
Moreover,  by calculus we have
\begin{align*}
\int_{0}^{\eta }\frac{(L_1(t)\theta ^{\alpha _1}(1-t))^{\frac{1}{p-1}}}{t
}dt
&= \int_{0}^{\eta }\frac{L_1(t)^{\frac{1}{p-1}}}{t}( \int_{0}^{t}
\frac{L_1(s)^{\frac{1}{p-1}}}{s}ds) ^{\frac{\alpha _1}{p-1-\alpha
_1}}dt \\
&= \frac{p-1-\alpha _1}{p-1}( \int_{0}^{\eta }\frac{L_1(t)^{\frac{1
}{p-1}}}{t}dt) ^{\frac{p-1}{p-1-\alpha _1}}<\infty \,.
\end{align*}
So applying Lemma \ref{estimarad}, for $\beta =p$ and $L(t)=L_1(t)\theta
^{\alpha _1}(1-t)$, we obtain
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
\Big( \int_{0}^{1-x}\frac{L_1(s)^{\frac{1}{p-1}}}{s}ds\Big) ^{\frac{p-1
}{p-1-\alpha _1}}=\theta (x).
\]
\smallskip

\noindent\textbf{Case 2.} $\frac{p-\beta _1}{p-1-\alpha _1}
<\frac{p-\beta _2}{p-1-\alpha _2}$ and $\frac{(\mu +1)(p-1-\alpha
_1)+p\alpha _1}{p-1}<\beta _1<p$.
Using (H2) we have
\[
a_1(x)\theta ^{\alpha _1}(x)\approx (1-x)^{\frac{p\alpha _1+\beta
_1(1-p)}{p-1-\alpha _1}}L_1(1-x)^{\frac{p-1}{p-1-\alpha _1}}
\]
and
\[
a_2(x)\theta ^{\alpha _2}(x)\approx (1-x)^{\alpha _2\frac{p-\beta _1
}{p-1-\alpha _1}-\beta _2}(L_2L_1^{\frac{\alpha _2}{p-1-\alpha _1
}})(1-x).
\]
Now, since $\frac{p-\beta _1}{p-1-\alpha _1}<\frac{p-\beta _2}{
p-1-\alpha _2}$, we have
\[
(p-\beta _1)(1+\frac{\alpha _1-\alpha _2}{p-1-\alpha _1})<p-\beta
_2,
\]
and so
\[
\frac{p\alpha _1+\beta _1(1-p)}{p-1-\alpha _1}<\alpha _2\frac{
p-\beta _1}{p-1-\alpha _1}-\beta _2.
\]
We deduce by Lemma \ref{k1}(i) that
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{\frac{p\alpha _1+\beta _1(1-p)}{p-1-\alpha _1}}L_1(1-x)^{
\frac{p-1}{p-1-\alpha _1}}.
\]
Moreover, since $\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}<\beta
_1<p$, we deduce by a simple calculus that
\[
\frac{-p\alpha _1+\beta _1(p-1)}{p-1-\alpha _1}\in (\mu +1,p).
\]
So applying Lemma \ref{estimarad}, for $\beta =\frac{-p\alpha _1+\beta
_1(p-1)}{p-1-\alpha _1}$ and $L(t)=(L_1(t))^{\frac{p-1}{p-1-\alpha _1
}}$ we obtain that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-\beta _1}{p-1-\alpha _1}}L_1(1-x)^{\frac{1}{p-1-\alpha
_1}}=\theta (x).
\]
\smallskip

\noindent\textbf{Case 3.}
 $\frac{p-\beta _1}{p-1-\alpha _1}
=\frac{p-\beta _2}{p-1-\alpha _2}$ and $\frac{(\mu +1)(p-1-\alpha
_1)+p\alpha _1}{p-1}<\beta _1<p$.
Using (H2) we have
\[
a_1(x)\theta ^{\alpha _1}(x)\approx (1-x)^{\frac{p\alpha _1+\beta
_1(1-p)}{p-1-\alpha _1}}(L_1K^{\alpha _1})(1-x)
\]
and
\[
a_2(x)\theta ^{\alpha _2}(x)\approx (1-x)^{\alpha _2\frac{p-\beta _1
}{p-1-\alpha _1}-\beta _2}(L_2K^{\alpha _2})(1-x).
\]
Now, $\frac{p-\beta _1}{p-1-\alpha _1}=\frac{p-\beta _2}{
p-1-\alpha _2}$ is equivalent to
\[
(p-\beta _1)(1+\frac{\alpha _1-\alpha _2}{p-1-\alpha _1})=p-\beta
_2;
\]
that is,
\[
\frac{p\alpha _1+\beta _1(1-p)}{p-1-\alpha _1}=\alpha _2\frac{
p-\beta _1}{p-1-\alpha _1}-\beta _2.
\]
Then
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{\frac{p\alpha _1+\beta _1(1-p)}{p-1-\alpha _1}}(L_1K^{\alpha
_1}+L_2K^{\alpha _2})(1-x).
\]
Furthermore, the function $x\to (L_1K^{\alpha _1}+L_2K^{\alpha
_2})(1-x)\in \mathcal{K}$.
Now using that $\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}<\beta
_1<p$, we deduce that
\[
\frac{-p\alpha _1+\beta _1(p-1)}{p-1-\alpha _1}\in (\mu +1,p).
\]
So applying Lemma \ref{estimarad}, for $\beta =\frac{-p\alpha _1+\beta
_1(p-1)}{p-1-\alpha _1}$ and $L=L_1K^{\alpha _1}+L_2K^{\alpha
_2}$, we obtain that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-\beta _1}{p-1-\alpha _1}}(L_1K^{\alpha
_1}+L_2K^{\alpha _2})^{\frac{1}{p-1}}(1-x).
\]
Hence, since $\frac{\alpha _i}{p-1}<1$ for $i\in \{1,2\}$ and
\begin{equation}
(s+t)^{\frac{1}{p-1}}\approx s^{\frac{1}{p-1}}+t^{\frac{1}{p-1}},\quad
\text{for }s,t>0,  \label{st}
\end{equation}
applying Lemma \ref{estim} for
$t=L_1(x)^{\frac{1}{p-1-\alpha _1}}$,
$s=L_2(x)^{\frac{1}{p-1-\alpha _2}}$ and
$\eta _i=\frac{\alpha _i}{p-1}$, $(i\in \{1,2\})$, we obtain that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)
\approx (1-x)^{\frac{p-\beta _1}{p-1-\alpha _1}}K(1-x)
=\theta (x).
\]
\smallskip

\noindent\textbf{Case 4.} $\beta _1=\frac{(\mu
+1)(p-1-\alpha _1)+p\alpha _1}{p-1}$ and $\beta _2=\frac{(\mu
+1)(p-1-\alpha _2)+p\alpha _2}{p-1}$.
Using (H2) we have
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{-(1+\mu )}(L_1N^{\alpha _1}+L_2N^{\alpha _2})(1-x).
\]
Furthermore, the function $x\to (L_1N^{\alpha _1}+L_2N^{\alpha
_2})(1-x)\in \mathcal{K}$. So applying Lemma \ref{estimarad} for $\beta
=1+\mu $ and $L=L_1N^{\alpha _1}+L_2N^{\alpha _2}$, we obtain that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-1-\mu }{p-1}}\Big( \int_{1-x}^{\eta }\frac{
(L_1N^{\alpha _1}+L_2N^{\alpha _2})(t)}{t}dt\Big) ^{\frac{1}{p-1}}.
\]
Since
\[
N^{p-1}(t)\approx \Big( \int_{t}^{\eta }\frac{L_1(s)}{s}
ds\Big) ^{\frac{p-1}{p-1-\alpha _1}}+\Big( \int_{t}^{\eta }
\frac{L_2(s)}{s}ds\Big) ^{\frac{p-1}{p-1-\alpha _2}},
\]
it follows that $N^{p-1}\approx J$, where $J$ is the function given in Lemma
\ref{estim2}, for $\eta _i=\frac{\alpha _i}{p-1},i\in \{1,2\}$. So, we have
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-1-\mu }{p-1}}\Big( \int_{1-x}^{\eta }\frac{
(L_1J^{\eta _1}+L_2J^{\eta _2})(t)}{t}dt\Big) ^{\frac{1}{p-1}}.
\]
Hence, since $\eta _i<1$ for $i\in \{1,2\}$,  by Lemma
\ref{estim2}, it follows that
\begin{align*}
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)
&\approx (1-x)^{\frac{p-1-\mu }{p-1}}J^{\frac{1}{p-1}}(1-x) \\
&\approx (1-x)^{\frac{p-1-\mu }{p-1}}N(1-x)=\theta (x).
\end{align*}
\smallskip

\noindent\textbf{Case\thinspace \thinspace 5.}
 $\beta _1=\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}$ and
$\beta _2<\frac{(\mu +1)(p-1-\alpha _2)+p\alpha _2}{p-1}$.
Let $b(1-x)=\Big( \int_{1-x}^{\eta }\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{
p-1-\alpha _1}}$. Using (H2), we have
\begin{align*}
&a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\\
&\approx (1-x)^{-(1+\mu )}(L_1b^{\alpha _1})(1-x)+(1-x)^{-\beta _2+\alpha _2
\frac{p-1-\mu }{p-1}}(L_2b^{\alpha _2})(1-x).
\end{align*}
Now, since for $i\in \{1,2\}$, the function
$x\to (L_ib^{\alpha _i})(1-x)\in \mathcal{K}$ and
$-(1+\mu )<-\beta _2+\alpha _2\frac{p-1-\mu }{p-1}$, it follows from
 Lemma \ref{k1}(i) that
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{-(1+\mu )}(L_1b^{\alpha _1})(1-x).
\]
Applying again Lemma \ref{estimarad}, for $\beta =1+\mu $ and
 $L=L_1b^{\alpha _1}$, we obtain
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-1-\mu }{p-1}}\Big(\int_{1-x}^{\eta }\frac{(L_1b^{\alpha
_1})(s)}{s}ds\Big) ^{\frac{1}{p-1-\alpha _1}}\approx \theta (x).
\]
\smallskip

\noindent\textbf{Case 6.}
 $\beta _1<\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}$.
Using (H2), we have
\[
a_1(x)\theta ^{\alpha _1}(x)\approx (1-x)^{-\beta _1+\alpha _1\frac{
p-1-\mu }{p-1}}L_1(1-x).
\]
Applying Lemma \ref{estimarad} for $\beta =\beta _1-\alpha _1\frac{
p-1-\mu }{p-1}<1+\mu $ and $L=L_1$, we deduce that
\[
G_p(a_1\theta ^{\alpha _1})(x)\approx (1-x)^{\frac{p-1-\mu }{p-1}}.
\]
Moreover, since $\frac{p-1-\mu }{p-1}<\frac{p-\beta _1}{p-1-\alpha _1}
\leq \frac{p-\beta _2}{p-1-\alpha _2}$, it follows that $\beta _2<\frac{(\mu
+1)(p-1-\alpha _2)+p\alpha _2}{p-1}$.

So, in the same manner we obtain
\[
G_p(a_2\theta ^{\alpha _2})(x)\approx (1-x)^{\frac{p-1-\mu }{p-1}}.
\]
Then, we conclude that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-1-\mu }{p-1}}=\theta (x).
\]
\smallskip

\noindent\textbf{Case 7.} $\beta _1=\beta _2=p$.
Using (H2), we have
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{-p}(L_1M^{\alpha _1}+L_2M^{\alpha _2})(1-x).
\]
Furthermore, the function $x\to (L_1M^{\alpha _1}+L_2M^{\alpha
_2})(1-x)\in \mathcal{K}$.
Now, since $\frac{\alpha _i}{p-1}<1$ and
$\int_{0}^{\eta }\frac{ (L_i(s))^{\frac{1}{p-1}}}{s}ds<\infty $, for
$i\in \{1,2\}$,  
 applying \eqref{st} and Lemma \ref{estim1} for
$\eta _i=\frac{\alpha _i}{p-1}$, we obtain 
\begin{align*}
\int_{0}^{\eta }\frac{(L_1M^{\alpha _1}+L_2M^{\alpha _2})^{\frac{1}{
p-1}}(t)}{t}dt
&\approx \int_{0}^{\eta }\frac{(L_1^{\frac{1}{p-1}}M^{
\frac{\alpha _1}{p-1}}+L_2^{\frac{1}{p-1}}M^{\frac{\alpha _2}{p-1}})(t)
}{t}dt \\
&\approx M(\eta )<\infty .
\end{align*}
So applying Lemma \ref{estimarad} for $\beta =p$ and $L=L_1M^{\alpha
_1}+L_2M^{\alpha _2}$, by \eqref{st} we deduce that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
\int_{0}^{1-x}\frac{((L_1M^{\alpha _1})^{\frac{1}{p-1}
}+(L_2M^{\alpha _2})^{\frac{1}{p-1}})(t)}{t}dt.
\]
Then, using again Lemma \ref{estim1} we conclude that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
M(1-x)=\theta (x).
\]
\end{proof}

\begin{proposition}\label{equicont}
Assume {\rm (H0)--(H1)} hold. Then
the family of functions
\[
\mathcal{F}_{q}=\{x\to G_pf(x);f\in B((0,1)),|f|\leq q\}
\]
is uniformly bounded and equicontinuous in $[0,1]$. Consequently
$\mathcal{F} _{q}$ is relatively compact in $C([0,1])$.
\end{proposition}

\begin{proof}
Let $f$ be a measurable function such that $|f(x)|\leq q(x),x\in (0,1)$. By
Proposition \ref{estimarad}, we have
\[
|G_pf(x)|\leq G_pq(x)\approx (1-x)^{\min {(\frac{p-\beta }{p-1},
\frac{p-1-\mu }{p-1})}}\Psi _{L,\beta ,0}(x).
\]
From Lemma \ref{k1} and Lemma \ref{k2}, the function
$x\to (1-x)^{\min {(\frac{p-\beta }{p-1},\frac{p-1-\mu }{p-1})}}\Psi _{L,\beta
,0}(x)$ is continuous on $[0,1)$ and tends to zero as $x\to 1$.
Then, we prove that $\mathcal{F}_{q}$ is uniformly bounded and
$\lim_{x\to 1}G_p(f)(x)=0$, uniformly on $f$.

Moreover, let $x,x'\in \lbrack 0,1]$ such that $x<x'$.
Then
\[
|G_pf(x)-G_pf(x')|\leq \int_{x}^{x'}\Big( \frac{1}{
A(t)}\int_{0}^{t}A(s)q(s)ds\Big) ^{\frac{1}{p-1}}dt.
\]
Since $G_pq(0)<\infty $, it follows by the dominated convergence theorem,
the equicontinuity of $\mathcal{F}_{q}$ in $[0,1]$.
Hence, by Ascoli's theorem, we deduce that $\mathcal{F}_{q}$ is relatively
compact in $C([0,1])$.
\end{proof}

\section{Proof of main result}

\subsection{Existence and boundary behavior}

Assume (H0) and (H2) hold. Let $\theta $ be the
function given by \eqref{1.9}.
By Proposition \ref{compasymp}, there
exists a constant $m\geq 1$ such that for each $x\in \lbrack 0,1)$,
\begin{equation}
\frac{1}{m}\theta (x)\leq G_p(a_1\theta ^{\alpha _1}+a_2\theta
^{\alpha _2})(x)\leq m\theta (x).  \label{3.1}
\end{equation}
Now, we look at the existence of positive solution of problem \eqref{1.7}
satisfying $u(x)\approx \theta (x)$.
We will use a fixed-point argument. We consider the set
\[
\Gamma :=\{u\in C([0,1]):\frac{1}{c_{0}}\theta \leq u\leq c_{0}\theta \},
\]
where $c_{0}=m^{\frac{p-1}{p-1-\max(|\alpha _1|,|\alpha _2|)}}$.
Obviously, the set $\Gamma $ is not empty. We consider the integral operator
$T$ on $\Gamma $ defined by
\[
Tu(x):=G_p(a_1u^{\alpha _1}+a_2u^{\alpha _2})(x),\quad x\in [0,1].
\]
First, we observe that $T\Gamma \subset \Gamma $. Indeed, let $u\in \Gamma $,
then for $i\in \{1,2\}$, we have
\[
c_{0}^{-|\alpha _i|}a_i(x)\theta ^{\alpha _i}(x)\leq a_i(x)u^{\alpha
_i}(x)\leq c_{0}^{|\alpha _i|}a_i(x)\theta ^{\alpha _i}(x),\quad
x\in [ 0,1).
\]
This and  \eqref{3.1} imply
\[
\frac{1}{mc_{o}^{\frac{max(|\alpha _1|,|\alpha _2|)}{p-1}}}\theta
(x)\leq Tu(x)\leq mc_{o}^{\frac{max(|\alpha _1|,|\alpha _2|)}{p-1}
}\theta (x).
\]
Since $mc_{o}^{\frac{max(|\alpha _1|,|\alpha _2|)}{p-1}}=c_{0}$ and
$T\Gamma \subset C([0,1])$, it follows that
 $T$ leaves invariant the convex $\Gamma $.

Next, let $q=c_{0}^{|\alpha _1|}a_1\theta ^{\alpha _1}+c_{0}^{|\alpha
_2|}a_2\theta ^{\alpha _2}$. By the proof of Proposition
\ref{compasymp}, the function $q$ satisfies (H1). Since $T\Gamma $
is a closed set of $\mathcal{F}_{q}$, it follows by Proposition
\ref{equicont}, that $T\Gamma $ is relatively compact in $C([0,1])$.

Now, we shall prove the continuity of operator $T$ in $\Gamma $.
Let $(u_{n}) $ be a sequence in $\Gamma $ converging uniformly to $u$ in
 $\Gamma$. Then, by applying the dominated convergence theorem, we conclude that
for each $x\in \lbrack 0,1],Tu_{n}(x)\to Tu(x)$ as
$n\to +\infty $. Consequently, as $T\Gamma $ is relatively compact in $C([0,1])$,
we deduce that the pointwise convergence implies the uniform convergence.
Namely, $\| Tu_{n}-Tu\| _{\infty }\to 0$ as $n\to +\infty $.

Therefore, $T$ is a compact operator from $\Gamma $ into itself. So the
Schauder fixed-point theorem implies the existence of $u\in \Gamma $ such
that
\[
u=G_p(a_1u^{\alpha _1}+a_2u^{\alpha _2})(x),\,\,x\in \lbrack 0,1].
\]
We conclude that $u$ is a positive continuous solution of problem \eqref{1.7}
 which satisfies \eqref{1.10}.

\subsection{Uniqueness}

Let $u$ and $v$ be two solutions of \eqref{1.7} in $\Gamma $. Then, there
exists a constant $M>1$ such that
\[
\frac{1}{M}\leq \frac{u}{v}\leq M.
\]
This implies that the set
\[
J=\{t\in (1,\infty ):\frac{1}{t}u\leq v\leq tu\}
\]
is not empty. Now, put $c:=\inf J$. We aim to show that $c=1$.
Suppose that $c>1$ and let $\alpha =max(|\alpha _1|,|\alpha _2|)$, then
we have
\begin{gather*}
\begin{aligned}
&-\frac{1}{A}(A\Phi _p(v'))'+\frac{1}{A}(A\Phi _p(c^{
\frac{-\alpha }{p-1}}u'))'\\
&=a_1(x)(v^{\alpha _1}-c^{-\alpha }u^{\alpha _1})+a_2(x)(v^{\alpha _2}-c^{-\alpha
}u^{\alpha _2})\geq 0,\quad \text{in }(0,1),
\end{aligned}\\
\lim_{x\to 0}(A\Phi _p(v')-A\Phi _p(c^{\frac{-\alpha }{p-1}}u'))(x)=0, \\
u(1)=v(1)=0.
\end{gather*}
This implies that the function
\[
\psi (x):=A\Phi _p(c^{\frac{-\alpha }{p-1}}u')-A\Phi _p(v')(x)
\]
is nondecreasing on $(0,1)$ with $\lim_{x\to 0}\psi (x)=0$.
Hence from the monotonicity of $\Phi _p$, we obtain that the
function $c^{\frac{-\alpha }{p-1}}u-v$ is nondecreasing on $(0,1)$
satisfying $(c^{\frac{-\alpha }{p-1}}u-v)(1)=0$. This implies that
 $c^{\frac{ -\alpha }{p-1}}u\leq v$. On the other hand, we deduce by
symmetry that $v\leq c^{\frac{\alpha }{p-1}}u$. Then, we have
$c^{\frac{\alpha }{p-1}}\in J$. Now, since $\alpha \in ( 0,p-1) $,
it follows that $c^{\frac{\alpha }{p-1}}<c$ and this yields to a contradiction
with the definition of $c$. Hence, $c=1$ and so $u=v$.

\begin{example} \rm
Let $1-p<\alpha _1<0<\alpha _2<p-1$ and $\beta _1,\beta _2<p$ such
that $\frac{p-\beta _1}{p-1-\alpha _1}\leq \frac{p-\beta _2}{
p-1-\alpha _2}$. Let $a_1,a_2$ be continuous functions on $(0,1)$ such
that $a_i(x)\approx (1-x)^{-\beta _i}$, for $i\in \{1,2\}$. Then problem
\eqref{1.7} has a unique continuous solution $u$ satisfying for
$x\in (0,1)$,
\[
u( x) \approx \begin{cases}
(1-x)^{\frac{p-\beta _1}{p-1-\alpha _1}}
 \quad  \text{if }\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}<\beta _1<p,
 \\[4pt]
(1-x)^{\frac{p-1-\mu }{p-1}}(\log (\frac{\eta }{1-x}))^{\frac{1
}{p-1-\alpha _2}}\\
\quad \text{if }\beta _1=\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1},\;
\beta _2=\frac{(\mu +1)(p-1-\alpha _2)+p\alpha _2}{p-1},
 \\[4pt]
(1-x)^{\frac{p-1-\mu }{p-1}}(\log (\frac{\eta }{1-x}))^{\frac{1}{p-1-\alpha _1}}\\
\quad  \text{if }\beta _1=\frac{(\mu +1)(p-1-\alpha
_1)+p\alpha _1}{p-1},\; \beta _2<\frac{(\mu +1)(p-1-\alpha_2)+p\alpha _2}{p-1},
\\[4pt]
(1-x)^{\frac{p-1-\mu }{p-1}} \quad \text{if }\beta _1<\frac{(\mu +1)(p-1-\alpha _1
)+p\alpha _1}{p-1}.
\end{cases}
\]
\end{example}

\subsection*{Acknowledgements}
The authors gratefully acknowledge the referees for valuable comments.

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\end{document}