%This is a template LaTeX file for the unified format of the Electronic Journal of Qualitative Theory of Differential Equations. Please read and follow very carefully the instructions included in the comments (the lines starting with %). This file should be compiled by pdflatex, but after slight modifications Latex->dvi->ps->pdf and Latex->dvi->pdf compilation is also possible. You only have to rewrite all the .png extensions to .eps in the template and one option of the hyperref package has to be modified where it is indicated. Please make sure that the tink.png file (or the tink.eps file respectively) is included in the same folder with this template file.

\documentclass[11pt, twoside, a4paper]{article}

\usepackage{amsmath}     %obligatory package
\usepackage{amssymb}     %obligatory package
\usepackage{amsthm}      %obligatory package
\usepackage{graphicx}    %obligatory package

\usepackage{textcomp}    %obligatory package
\usepackage[T1]{fontenc} %obligatory package
\usepackage{marvosym}    %obligatory package(for the \Letter symbol for indicating corresponding author)
\usepackage[sc]{mathpazo}  	 %obligatory package(font package)

%In case you miss any of the packages below and you do not want to install them, then please download packeges.zip from our website available at http://www.math.u-szeged.hu/EditorEJQTDE/packages.zip
%After unpacking the zip file you will find the files you need. Please copy the missing files in the folder where you have your LaTeX file to be compiled. Note that some the hyperref package uses a number of files so we put them into a subfolder. If you miss one of these packages it is recommended to copy all files from its subfolder to your folder.

\usepackage{authblk} 			%obligatory package
\usepackage{enumerate} 			%recommended package
\usepackage[usenames]{xcolor}   %obligatory package
\usepackage{lastpage} 			%obligatory package

%\usepackage[dvipdfm]{hyperref} %Use this option in dvi->pdf mode (e.g. in case you prefer eps figures) 						
								%obligatory package
%\usepackage[dvips]{hyperref}   %Use this option in dvi->ps->pdf mode
								%obligatory package	
								
\usepackage{hyperref} 			%Use this in pdflatex mode  %obligatory package
%\usepackage[all]{hypcap}		%Recommended if the paper containes figures or tables

\definecolor{ForestGreen}{rgb}{0.15,0.416,0.18}
\definecolor{EgyptBlue}{rgb}{0.063,0.2,0.65}
\hypersetup{
	colorlinks=true,
	linkcolor=EgyptBlue,         
   citecolor=EgyptBlue,          
   urlcolor=ForestGreen          
}

\linespread{1.05}
\hoffset -1in
\voffset -1in
\oddsidemargin 25mm
\textwidth 160mm
\topmargin 10mm
\headheight 10mm
\headsep 10mm
\textheight 237mm


\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}

\numberwithin{equation}{section}
\numberwithin{table}{section}
\numberwithin{figure}{section}


\title{Asymptotic behaviour of positive large solutions of quasilinear logistic problems}
%Please write the title of your paper here. Please use capital letters only for the first letter of the title and for proper names

%Please write the full names of all authors here, using the order first name - middle names (optionally initials only) - last (family) name.
\author[1]{\textbf{Ramzi Alsaedi}}
\author[1]{\textbf{Habib M\^{a}agli}}
\author[1, 2]{\textbf{Vicen\c{t}iu D. R\u{a}dulescu}\footnote{Corresponding author. Email: vicentiu.radulescu@imar.ro; vicentiu.radulescu@math.cnrs.fr}}%Please write the email address of the corresponding author here
\author[1]{\textbf{ Noureddine Zeddini}}
%\author[2, 3]{\textbf{Fifth Author}}


%Please enter the affiliations and addresses of the authors here.

\affil[1]{Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia}
\affil[2]{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania}
%\affil[3]{Institute of Mathematics, Third University, 3 University Road, City Name, H--9888, Country Name}

%In case you have no co-authors, use the form below for your name and affiliation(s) instead the above one.

%\author{\textbf{First Author}\footnote{Email: firstauthor@firstuniversity.com}}
%\affil{First University, 1 University Street, City Name, H--1234, Country Name\\
%Second University, 2 University Square, City Name H--9876, Country Name}

%Please write here the short title of your paper not exceeding 65 characters including spaces
\newcommand\shorttitle{Asymptotic behaviour of positive large solutions of quasilinear logistic problems}


%Please fill here the names of all authors, first name and middle name should be abbreviated to single letter
\newcommand\authorsshort{R. Alsaedi,  H. M\^{a}agli, V. D. R\u{a}dulescu, N. Zeddini}

%Please rewrite .png to .eps if you use dvi mode
\newcommand{\ejqtdelogo}{tink.png}
%\newcommand{\ejqtdelogo}{tink.eps}  



% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% % % % % % % % % Please do not change this paragraph.% % % % % % % % % % 
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\makeatletter
\renewcommand{\maketitle}{\bgroup\setlength{\parindent}{0pt}


\begin{picture}(20,20)
     \put(-50,-5){\includegraphics[width=2.9truecm]{\ejqtdelogo}}
  \end{picture}


\vspace{1truecm}
\begin{center}{\vbox{\titlefont\@title}}\end{center}
\vspace{0.5truecm}
\begin{center}{\@author} \end{center}

\egroup
}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\renewcommand{\@fnsymbol}[1]{%
    \ifcase#1 \or {\,\Letter\!} \or\textasteriskcentered\or \textasteriskcentered\textasteriskcentered 
    \else\@ctrerr\fi}
\makeatother

\makeatletter
\newcommand{\hbibitem}[4]{\bibitem{#1}{#2}
\def\@tempa{#3}%
\def\@tempb{#4}%
\ifx\@tempa\@empty\ifx\@tempb\@empty{}{}\else{}{\href{http://dx.doi.org/#4}{url}}\fi\else
{\href{http://www.ams.org/mathscinet-getitem?mr=#3}{MR#3}}\ifx\@tempb\@empty{}\else{; \href{http://dx.doi.org/#4}{url}}\fi\fi}
\makeatother

\renewcommand{\Affilfont}{\small\normalfont}
\renewcommand{\Authfont}{\Large}
\renewcommand\Authands{ and }
\newcommand*{\titlefont}{\fontsize{18}{21.6}\selectfont\textbf}

\makeatletter
\renewcommand\@author{\ifx\AB@affillist\AB@empty\AB@author\else
      \ifnum\value{affil}>\value{Maxaffil}\def\rlap##1{##1}%
    \AB@authlist\\[\affilsep]\vbox{\AB@affillist}
    \else  \AB@authors\fi\fi}
\makeatother

\makeatletter
\def\ps@plain{
\def\@oddhead{\ifnum\thepage=1\hss\baselineskip8pt
\vtop to 0 pt{\vskip-0.4truecm\noindent\hbox{\hspace{1.3truecm}\Large
Electronic Journal of Qualitative Theory of Differential Equations\hss\linebreak}%
\vspace{0.1truecm} 
\noindent\hbox{\hspace{1.3truecm}\Large 2015, No.\ {\bf XX}, {1--\begin{NoHyper}\pageref{LastPage}\end{NoHyper}};\quad {\small {http:/\!/www.math.u-szeged.hu/ejqtde/}}\hss}\vss}
\else
\hss\textit{\shorttitle}\hss\hbox to 0pt{\hss\thepage}\fi}\def\@oddfoot{}
\def\@evenhead{\hbox to 0pt{\thepage\hss} \hss\textit{\authorsshort}\hss}
\def\@evenfoot{}
}
\makeatother

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %


%You may define your own macros here e.g
\def\R{\mathbb{R}} 	
\newcommand\dom{\operatorname{dom}}
\newcommand{\eps}[1]{{#1}_{\varepsilon}}




\begin{document}

\maketitle


\pagestyle{plain}



\begin{center}
\noindent
\begin{minipage}{0.85\textwidth}

\medskip
\begin{center}
\noindent Received XX XXXXXX 201X, appeared XX XXXXXX 2015 % Please do not change this line, it will be edited by the technical editors.
\smallskip

\noindent Communicated by Handling Editor
\end{center}
\bigskip

{\small{
\noindent {\bf Abstract.} We are interested in the asymptotic analysis of singular solutions with blow-up boundary for a class of quasilinear logistic equations with indefinite potential. Under natural assumptions, we study the competition between the growth of the variable weight and the behaviour of the nonlinear term, in order to establish the blow-up rate of the positive solution. The proofs combine  the Karamata regular variation theory with a related comparison principle. The abstract result is illustrated with an application to the logistic problem with convection.}
\smallskip

% Please enter at most 6 keywords here with lowercase letters separated by commas.
\noindent {\bf{Keywords:}} Asymptotic behaviour; positive solution; boundary blow-up; maximum principle.
\smallskip

% Please enter at most 5 Mathematics Subject Classification codes here. Please use 2010 classification codes, which can be found on the following link: http://www.ams.org/msc//msc2010.html.
\noindent{\bf{2010 Mathematics Subject Classification:}} 35B44; 35B50; 35J62.
}

\end{minipage}
\end{center}

\section{Introduction} % Please enter the title of your first section (only the first letter of the title should be capital).

Let $\Omega$ be a $C^2$-bounded domain of ${\mathbb{R}}^n$, $(n\geq 2)$.
Let $\Delta_p$ denote the $p$-Laplace operator ($p>1$), that is,
$$\Delta_pu:=\mbox{div}\,(|\nabla u|^{p-2}\nabla u)\,.$$

We first recall some notations used in this paper. For some $\alpha\in (0,1)$, we denote by
$ C^{0,\alpha}_{\rm loc}(\Omega)$ the Banach space of locally H\"older continuous functions, that is, real-valued functions defined on $\Omega$
which are uniformly H\"older continuous with exponent $\alpha$ on any compact subset of $\Omega$.
The local H\"older space $ C^{1,\alpha}_{\rm loc}(\Omega)$ consists of functions whose first order derivatives are locally H\"older continuous with exponent $\alpha$ in $\Omega$. Similarly, for $p>1$, we denote by $W_{\rm loc}^{1,p}(\Omega)$ the Banach space of locally $L^p$-integrable functions with locally $L^p$-integrable weak derivatives, that is,
$$W_{\rm loc}^{1,p}(\Omega):=\{u:\Omega\rightarrow {\mathbb{R}}\ \mbox{measurable};\ u|_K\in W^{1,p}(K)\ \mbox{for all compact set $K\subset\Omega$}\}.$$

In this paper, we study the existence and the boundary behaviour
of solutions for the following quasilinear elliptic problem
\begin{align} \label{main problem}
\left\{
\begin{array}{ll}
\Delta_p u = a(x)\,f(u)\,,\;\; x \in \Omega\,,\;\; \\
u>0\;\; \mbox{ in } \Omega\\
\displaystyle\lim_{\delta(x) \rightarrow 0}u(x)=\infty\,,
\end{array}
\right.
\end{align}
where $p>1$,  $f:(0,\infty)\rightarrow (0,\infty)$ is a $C^1$ function, $a$ is a positive function, locally H\"{o}lder continuous in $\Omega$
and satisfies some conditions related to Karamata regular variation theory and $\delta(x)$ denotes the Euclidean distance from $x$ to the boundary
$\partial \Omega$.

 By a weak solution of (\eqref{main problem}), we mean a positive function $u\in W_{\rm loc}^{1,p}(\Omega)\cap C_{\rm loc}^{1,\beta}(\Omega)$ for some $0<\beta<1$, and satisfies in the distributional sense
$$-\int_{\Omega}|\nabla u|^{p-2}\, \nabla u \nabla \varphi\, dx= \int_{\Omega} a(x)f(u)\varphi\,dx,\mbox{ for any test function } \varphi \in C_{c}^{\infty}(\Omega).$$ A solution of (\eqref{main problem}) is called a large solution (or boundary blow-up or explosive solution).


\smallskip
Problems such as (\eqref{main problem}) arise in the study of the subsonic motion of a gas \cite{Poh}, the electric potential in some bodies \cite{LMcK},
and Riemann geometry \cite{CNi}.

 When $p=2$, problem (\eqref{main problem}) becomes
\begin{align} \label{problem p=2}
\Delta u = a(x)\,f(u)\,,\ x \in \Omega\,,\
u>0\;\; \mbox{ in } \Omega,\
\displaystyle\lim_{\delta(x) \rightarrow 0}u(x)=+\infty.
\end{align}

The subject of large solutions to (\eqref{problem p=2}) has received much attention starting with the pioneering works of Bieberbach in 1916 with $a(x)=1$,
$f(u)=e^u$ , $n=2$ and with $a(x)=1$, $f(u)=e^u$ and $n=3$ in Rademacher's work in 1943 (see \cite{Bi1} and \cite{Rade}). In 1957, Keller \cite{Kel} and Osserman \cite{Osser} gave a necessary and sufficient condition for the existence of a solution to (\eqref{problem p=2}) when $a(x)=1$ and $\Omega$ is bounded, namely  $\displaystyle\int^{\infty}\frac{1}{\sqrt{F(s)}}ds<\infty$, where $F'(s)=f(s)$  is increasing nonlinearity. Later, many authors have considered
questions such as existence, uniqueness and boundary behaviour of the solution and its normal derivative in different domains and for bounded positive weights
$a(x)$.

Problem (\eqref{problem p=2}) arises from many branches of mathematics and applied mathematics, and have been discussed by many
authors in many contexts. For $p=2$, $f=0$ and $u\in C^2$ it is
obtained the classical Laplace equation which was extensively studied
in the literature (see, for example, \cite{Radu}, \cite{stevic1}, \cite{stevic2}, \cite{stevic3}).

 In a significant development, Cirstea and R\u adulescu \cite{CR1} use Karamata's regular variation theory to study the blow-up rate and uniqueness, near the boundary to problem (\eqref{problem p=2}), in the case where $a(x)$ decays to zero on $\partial \Omega$  at a fixed rate along the entire boundary $\partial \Omega$ and $f'$ varies regularly at infinity.

  More recently, some results of existence and nonexistence of solutions to problem (\eqref{problem p=2}) are established when the weight $a(x)$  is unbounded near the boundary $\partial \Omega$ (see \cite{BM1}, \cite{CCEG}-\cite{CR2}, \cite{GLS}-\cite{GR}, \cite{LMcK}, \cite{MBC}, \cite{Radu}, \cite{Zh1}-\cite{ZMML} and the references therein).

 In the general case (not necessarily $p=2$), the problem (\eqref{main problem}) seems to have been first considered in \cite{DL}
when $a(x)=1$. The question of existence, uniqueness and boundary behaviour of solutions were dealt there. Since then, there have been  some others papers which
included similar results for different types of nonlinearities we mention for instance \cite{DuGu}, \cite{GM2}, \cite{GM3},  \cite{Mat}-\cite{Moh}. We also point our the important contributions of Guo and Webb \cite{GuW1}, \cite{GuW2} in the understanding of the structure of boundary blow-up solutions for quasi-linear elliptic problems.

 When the weight $a(x)$ is bounded, problem (\eqref{main problem}) has been considered by several authors. But when the weight $a(x)$ is not necessarily bounded very little is known about the global behaviour of the solution except for the case $p=2$, see for example \cite{DuGu}, \cite{Mat}, \cite{Moh}.

  Our aim in this paper is to establish existence and asymptotic behaviour of solutions of (\eqref{main problem})
with more general nonlinearities $f(u)$ and weights $a(x)$. In particular, we give global estimates of solutions (\eqref{main problem}) in the case where
$a(x)$ may be unbounded and satisfies some hypotheses related to the Karamata class of regularly varying functions at zero.

 In order to use the
method of sub- and super-solutions for (\eqref{main problem}), we begin by giving an auxiliary result in the case $f(u)=u^{\alpha}\,,\,\alpha>p-1$. More precisely, we
prove the existence and asymptotic behaviour of a positive solution for the following problem
\begin{align} \label{p-alpha problem}
\left\{
\begin{array}{ll}
\Delta_p u = b(x)\,u^{\alpha}\,,\;\; x \in \Omega\,,\;\, \alpha>p-1>0\; \\
u>0\;\; \mbox{ in } \Omega\\
\displaystyle\lim_{\delta(x) \rightarrow 0}u(x)=\infty\,,
\end{array}
\right.
\end{align}
where $b(x)$ satisfies the following hypothesis
\begin{enumerate}
\item[${\bf{(B_1)}}$] $\; b:\Omega \rightarrow (0,\infty)$  belongs to  $ C^{0,\gamma}_{\rm loc}(\Omega)$ , $0<\gamma<1$ and there exists $C>1$
such that  for each $x\in \Omega$,
$$\frac{1}{C}\, \delta(x)^{-\lambda}\, L(\delta(x))\leq b(x)\leq C\,\delta(x)^{-\lambda}\, L(\delta(x))\,,$$
where $\lambda \leq p$, $L$ is defined on $(0,\eta]$ for some $\eta> \mbox{diam}\, (\Omega)$ with $\displaystyle\int_0^{\eta}\frac{L(t)^{1/(p-1)}}{t^{\frac{\lambda-1}{p-1}}} \,dt<\infty$ and $L$
belongs to  the set of Karamata functions ${\mathcal{K}}$ defined on $(0,\eta]$ by
\begin{equation*}
L(t):=c\,\exp\left(\int_t^{\eta}\frac{z(s)}{s}\,ds\right)\,
\end{equation*}
with $c>0$ and $z\in C([0,\eta])$ such that $z(0)=0$.
\end{enumerate}
Under this hypothesis, we state our first main result.

\begin{theorem}\label{p-alpha theorem}
Let $p>1$, $\alpha>\max\{p-1,1\}$ and  assume that $b$ satisfies $(B_1)$. Then  problem
(\eqref{p-alpha problem}) has a positive weak solution $u\in C_{\rm loc}^{1,\beta}(\Omega)$, for some $0<\beta<1$, satisfying for each $x\in \Omega$,
\begin{align}\label{estimates solution p-alpha problem}
\frac{1}{C}\, \delta(x)^{\frac{p-\lambda}{p-1-\alpha}}\; \theta_{L,\lambda,p,\alpha}(\delta(x))\leq u(x)\leq C\,
\delta(x)^{\frac{p-\lambda}{p-1-\alpha}}\; \theta_{L,\lambda,p,\alpha}(\delta(x))\,,
\end{align}
where $C>1$ is a constant and $\theta_{L,\lambda,p,\alpha}$ is the function defined on $(0,\eta]$ by
\begin{align} \label{definition theta}
\theta_{L,\lambda,p,\alpha}(t):=\left\{
\begin{array}{ll}
 L(t)^{1/(p-1-\alpha)}\,,\; \; \;  \quad \quad \mbox{ if } \lambda<p,\\
\left(\displaystyle\int_0^t\frac{L(s)^{1/(p-1)}}{s}\,ds\right)^{\frac{p-1}{p-1-\alpha}}\,,\; \mbox{ if } \lambda=p.
\end{array}
\right.
\end{align}
\end{theorem}
In order to establish our main result for problem (\eqref{main problem}), we assume that the functions $f$ and $a$ satisfy the following conditions:
\begin{enumerate}
\item[${\bf{(H_1)}}$] The function $a$ is positive, belongs to $C^{0,\gamma}_{\rm loc}(\Omega)$, $0<\gamma<1$ and there exist two $\gamma$-H\"{o}lder continuous
functions $a_1$ and $a_2$ such that for each $x\in \Omega$, $$a_1(\delta(x))\leq a(x)\leq a_2(\delta(x))\,,$$
where $a_i(t)=t^{-\lambda_i}\,L_i(t)$, with $\lambda_i\leq p$ and $L_i\in {\mathcal{K}}$ defined on $(0,\eta]$, $(\eta> \mbox{diam}\,(\Omega))$ such that $\displaystyle\int_0^{\eta}\frac{\left(L_i(t)\right)^{\frac{1}{p-1}}}{t^{\frac{\lambda_i-1}{p-1}}}\,dt<\infty$.
\item[${\bf{(H_2)}}$] The function $f:(0,\infty)\rightarrow (0,\infty)$ is of class $C^1$ and there exist constants $k_1$, $k_2$, $\alpha_1\,,\;\alpha_2$
with $0<k_1\leq k_2$, $0<p-1<\alpha_1\leq \alpha_2$ such that $$f(t)\leq k_2\,t^{\alpha_2}\;
\mbox{ for }t>0\;\mbox{ and } f(t)\geq k_1\,t^{\alpha_1}\; \mbox{ for } t\geq 1.$$
\end{enumerate}

We now give an example of weight $a(x)$ that satisfies hypothesis ${\bf{(H_1)}}$. Consider the simplest case corresponding to $\Omega =B(0,1)\subset {\mathbb R}^n$ and assume that $p>1$, $\lambda \leq p$, and $q>p-\lambda$. Then the function $$a(x)=(1-|x|)^{q-\lambda}\qquad x\in\Omega$$
satisfies hypothesis ${\bf{(H_1)}}$. We refer to \cite{CR1} for more examples of functions
belonging to the Karamata class.

We now state the second main result in this paper.
\begin{theorem}\label{main result}
Under hypotheses ${(H_1)}$-${(H_2)}$, problem (\eqref{main problem}) has a weak solution $u\in C^{1,\beta}(\Omega)$, for some $0<\beta<1$, satisfying
\begin{align}\label{estimates  solution main problem}
\frac{1}{C}\, \delta(x)^{\frac{p-\lambda_2}{p-1-\alpha_2}}\,  \theta_{L_2,\lambda_2,p,\alpha_2^*}(\delta(x))\leq u(x)\leq C\, \delta(x)^{\frac{p-\lambda_1}{p-1-\alpha_1}}\,  \theta_{L_1,\lambda_1,p,\alpha_1^*}(\delta(x))\,,
\end{align}
where
\begin{align} \label{definition theta lambda i p-alpha i}
\theta_{L_i,\lambda_i,p,\alpha_i}(t):=\left\{
\begin{array}{ll}
 (L_i(t))^{\frac{1}{p-1-\alpha_i}}\,,\; \; \;  \quad \quad \mbox{ if } \lambda_i<p,\\
\left(\displaystyle\int_0^t\frac{\left(L_i(s)\right)^{\frac{1}{p-1}}}{s}\,ds\right)^{\frac{p-1}{p-1-\alpha_i}}\,,\; \mbox{ if } \lambda_i=p
\end{array}
\right.
\end{align}
for $i\in \{1,2\}$, $C>1$ and
$\alpha_i^*=[\alpha_i-\beta (p-1)]/(1-\beta)$ ($i=1,2$).
\end{theorem}
Throughout this paper, we need the following notations.

For two nonnegative functions $f$ and $g$ defined on a set $S$,  the notation $f (x) \approx g (x)$, $x \in S$, means that there exists $c > 0$ such that
$$ \frac{1}{c} \,f(x) \leq g(x) \leq c\,f(x)\,,\; \mbox{ for all } x \in S.$$
We denote by $\varphi_1$ the positive normalized (i.e., $\displaystyle\max_{x\in \Omega} \varphi_1(x)=1$) eigenfunction corresponding
to the first positive eigenvalue $\lambda_1$ of the $p$-Laplace operator $(-\Delta_p)$
in $W_0^{1,p}(\Omega)$.  By definition $\varphi_1$ is the unique normalized
function satisfying the following eigenvalue problem
\begin{align*}
\left\{
\begin{array}{ll}
-\Delta_p \varphi_1 = \lambda_1\,\varphi_1^{p-1}\,,\;\; x \in \Omega\,,\;\, \\
\varphi_1 \geq 0\;\; \mbox{ in } \Omega \mbox{ and }
\displaystyle\lim_{\delta(x) \rightarrow 0}\varphi_1(x)=0.
\end{array}
\right.
\end{align*}
We recall that, from Moser iterations \cite{Mos} and \cite[Theorem 1]{Lieb}, $\varphi_1\in C^{1,\beta}(\overline{\Omega})$, for some $0<\beta<1$, and
from strong maximum principle for quasilinear operators (see \cite[Theorem 10]{Vaz}), $\varphi_1$ satisfies
\begin{align} \label{phi 1 comparable distance}
\varphi_1(x) \approx \delta (x)\; \mbox{ in } \Omega
\end{align}
and
\begin{align} \label{phi1 plus grad phi1 comparable 1}
\varphi_1^p(x) +|\nabla \varphi_1(x)|^p \;\approx 1\; \mbox{ in } \Omega.
\end{align}
Our paper is organized as follows. In Section 2, we collect some useful properties of Karamata functions. Section 3 deals with the proof of our main results.
The last section is reserved to some applications.

\section{The Karamata Class ${\mathcal{K}}$}
To let the paper self-contained, we begin this section by recapitulating some properties of Karamata regular
variation theory established by Karamata in 1930. This theory has been applied to study the asymptotic behaviour
of solutions to differential equations. We refer to  \cite{CR1}, \cite{CR2}, \cite{Maag}, \cite{Radu}, \cite{ZMA} and \cite{ZMML} for more details.
\begin{lemma}\label{lemma 2.1}
The following hold.
\\
$(i)$ Let $L\in \mathcal{K}\ $and $\varepsilon >0,$ then we have
\begin{equation*}
\displaystyle\lim_{t\rightarrow 0^{+}}t^{\varepsilon }L(t)=0.
\end{equation*}
$(ii)$ Let $L_{1},L_{2}\in \mathcal{K}$ and $p\in \mathbb{R}.$ Then we have $
L_{1}+L_{2}\in \mathcal{K}$, $L_{1}L_{2}\in \mathcal{K}$ and $L_{1}^{p}\in
\mathcal{K}.$
\end{lemma}
\begin{example} \label{example 2.2}
Let $m$ be a positive integer.  Let $c>0$, $\left( \mu _{1},\mu _{2},...,\mu _{m}\right) \in
{\mathbb{R}}^{m}$ and $d$ be a sufficiently large positive real number
such that the function
\begin{equation*}
L(t)=c\displaystyle\prod_{k=1}^{m}\left( \log _{k}\left( \frac{d }{t}\right)
\right) ^{\mu _{k}}
\end{equation*}
is defined and positive on $\left( 0,\eta \right] ,$ for some $\eta >1,$
where $\log _{k}x=\log \circ \log \circ ...\circ \log x$ $($k times$).$ Then
$L\in \mathcal{K}.$
\end{example}
\begin{lemma}\label{lemma 2.2}
A function $L$ is in ${\mathcal{K}}$ if and only if $L$ is a positive function in $C^1((0,\eta])$ satisfying
\begin{align}\label{caracterization of K}
\lim_{t \rightarrow 0^+}\frac{t\,L'(t)}{L(t)}=0.
\end{align}
\end{lemma}
{\bf{Proof.}} Let $L \in {\mathcal{K}}$. Since $L(t):=c\exp\left(\int_t^{\eta}\frac{z(s)}{s}\,ds\right)$, then for $t\in (0,\eta]$, we have
$$\frac{t\,L'(t)}{L(t)}=-z(t)\,.$$
So, using the fact that $z(0)=0$, we deduce (\eqref{caracterization of K}).

Conversely, let $L$ be a positive function in $C^2((0,\eta])$ satisfying (\eqref{caracterization of K}). For $t\in (0,\eta]$, put
\begin{align}\label{def de z}
z(t)=-\frac{t\,L'(t)}{L(t)}\,,
\end{align}
then $z\in C((0,\eta])$ and $\lim_{t\rightarrow 0^+}z(t)=0$. Moreover, we have $$L(t)=L(\eta)\exp\left(\int_t^{\eta}\frac{z(s)}{s}\,ds\right).$$
This proves  that $L\in {\mathcal{K}}$.

Applying Karamata's theorem (see \cite{Mar,Sen}), we get the following.
\begin{lemma} \label{lemma 2.3}
Let $\mu \in{\mathbb{R}}$ and $L$ be a function in $\mathcal{K}$ defined on $(0,\eta ]$. We have
\begin{enumerate}
\item[$(i)$] If $\mu <-1,$ then $\displaystyle\int_{0}^{\eta }s^{\mu }L(s)ds$ diverges and $
\displaystyle\int_{t}^{\eta }s^{\mu }L(s)ds \displaystyle\sim_{t\rightarrow 0^{+}}-\displaystyle\frac{
t^{1+\mu }L(t)}{\mu +1}$.
\item[$(ii)$] If $\mu >-1,$ then $\displaystyle\int_{0}^{\eta }s^{\mu }L(s)ds$ converges and $
\displaystyle\int_{0}^{t}s^{\mu }L(s)ds \displaystyle\sim_{t\rightarrow 0^{+}}\displaystyle\frac{
t^{1+\mu }L(t)}{\mu +1}$.
\end{enumerate}
\end{lemma}
\begin{lemma} \label{lemma 2.4}\cite{MMZ}
Let $L\in \mathcal{K}$ be defined on $(0,\eta ]$. Then we have
\begin{align}\label{lemma 2.4 eq1}
\displaystyle\lim_{t\rightarrow 0^{+}}\frac{L\left( t\right) }{\int_{t}^{\eta }
\frac{L(s)}{s}ds}=0.
\end{align}
If further $\displaystyle\int_{0}^{\eta }\frac{L(s)}{s}ds$ converges, then we have
\begin{align}\label{lemma 2.4 eq2}
\displaystyle\lim_{t\rightarrow 0^{+}}\frac{L\left( t\right) }{\int_{0}^{t}
\frac{L(s)}{s}ds}=0.
\end{align}
\end{lemma}

\begin{remark} \label{remark 2.5}
Let $L\in \mathcal{K}$ defined on $(0,\eta ],$ then using $\left( \eqref{caracterization of K}
\right)$ and $\left( \eqref{lemma 2.4 eq1}\right)$, we deduce that
\begin{equation*}
t\rightarrow \int_{t}^{\eta }\frac{L(s)}{s}ds\in \mathcal{K}.
\end{equation*}
If further $\int_{0}^{\eta }\frac{L(s)}{s}ds$ converges, we have by
$\left( \eqref{lemma 2.4 eq2}\right)$ that
\begin{equation*}
t\rightarrow \int_{0}^{t}\frac{L(s)}{s}ds\in \mathcal{K}.
\end{equation*}
\end{remark}

\begin{lemma} \label{lemma 2.6}
Let $L\in \mathcal{K}$, $0<\varepsilon<\eta$ and $\varphi_1$ be the first eigenfunction of $(-\Delta_p)$ in $\Omega$. Then we have
\begin{align}\label{L phi1 comparable L delta}
L(\varepsilon\,\varphi_1(x)) \approx L(\delta(x))\,,\; x \in \Omega.
\end{align}
\end{lemma}
{\bf{Proof.}} Let $L \in {\mathcal{K}}$. Then there exist $c>0$ and  $z\in C([0,\eta])$ such that $z(0)=0$ and for $t \in (0,\eta]$, $\eta>\mbox{diam}\,(\Omega)$, $$L(t):=c\exp\left(\int_t^{\eta}\frac{z(s)}{s}\,ds\right).$$ Let $M=\displaystyle\max_{s \in [0,\eta]}|z(s)|$. By using (\eqref{phi 1 comparable distance}),
there exists $c_1>0$ such that $$\frac{1}{c_1}\,\delta(x)\leq \varepsilon \,\varphi_1(x)\leq c_1\,\delta(x).$$ Using this fact we deduce that
$$\left|\int_{\delta(x)}^{\varepsilon\,\varphi_1(x)}\frac{z(s)}{s}\,ds\right|\leq M \, \log c_1.$$
Hence, $$c_1^{-M} \, L(\delta(x))\leq L(\varepsilon\,\varphi_1(x))\leq c_1^{M} \, L(\delta(x))\,,\; x\in \Omega.$$
This ends the proof.

\smallskip
We point out that the constants in asymptotic relation (\eqref{L phi1 comparable L delta}) depend on $\varepsilon$ (the first one goes to zero
but the second one goes to infinity as $\varepsilon \to 0+$).

\section{Proof of main results}
First, we recall some classical results about the sub- and super-solution method.
\begin{definition}
A function $v\in W_{\rm loc}^{1,p}(\Omega)\cap C_{\rm loc}^{1,\beta}(\Omega)$, $0<\beta<1$, is called a weak sub-solution of (\eqref{main problem}) if $v=\infty$
on $\partial \Omega$ and
\begin{equation*}
-\int_{\Omega}|\nabla v|^{p-2}\,\nabla v\nabla \varphi \,dx\geq \int_{\Omega}a(x)\,f(v)\varphi \,dx\;;\; \;\forall \varphi \in C_c^{\infty}(\Omega)
\mbox{ with } \varphi \geq 0.
\end{equation*}
If the above inequality is reversed, $v$ is called a weak super-solution of (\eqref{main problem}).
\end{definition}

We point out that this definition agrees with the sub- and super-solutions used in the proofs of Theorem
\ref{p-alpha theorem} and Theorem \eqref{main result}, since the corresponding relations in those proofs are viewed in the weak sense.

The following property is an adaptation of Lemma 2.1 in \cite{GM1}. In the statement of the next result, we can assume without loss of generality that the H\"older exponent is the same for all functions
$\underline{u}$, $\overline{u}$, and $u$. Indeed, if the corresponding exponents are $\beta_1$, $\beta_2$ and $\beta_3$, it is enough to consider $\beta=\min\{\beta_1,\beta_2,\beta_3\}$.

\begin{lemma}\label{lemma sub supersolution}
Let $a(x)$ be a locally $\gamma$-H\"{o}lder continuous function in $\Omega$, $0<\gamma<1$ and  $f$ be continuously differentiable
on $[0,\infty)$. Assume that there exist a weak sub-solution $\underline{u}$ and a weak  super-solution $\overline{u}$ to
the problem (\eqref{main problem}) such that $\underline{u}\leq \overline{u}$. Then there exists at least one weak solution $u\in W_{\rm loc}^{1,p}(\Omega)\cap C_{\rm loc}^{1,\beta}(\Omega)$, for some $0<\beta<1$,
such that $\underline{u}\leq u \leq \overline{u}$.
\end{lemma}

{\bf{Proof.}} For $n \in {\mathbb{N}}$, we set $$\Omega_n:=\left\{x \in \Omega :\  \delta(x)<\frac{1}{n}\right\}\,.$$
Consider the boundary value problem
\begin{align} \label{omega-n-problem}
\left\{
\begin{array}{ll}
\Delta_p u = a(x)\,f(u)\,,\;\; x \in \Omega_n\,,\;\; \\
u|_{\partial \Omega_n}=\underline{u}.
\end{array}
\right.
\end{align}
Since $\underline{u}$ is a sub-solution and $\overline{u}$ is a super-solution, this problem has at least one positive weak solution $u_n$
such that $\underline{u}\leq u_n\leq \overline{u}$, see R\u adulescu \cite{Radu}. This in particular gives bounds on any compact set $K\subset\Omega$ for the sequence ${u_n}$ which in turn leads to
 bounds in $C^{1,\gamma}_{\rm loc}(\Omega)$. Since the embedding of $C^{1,\gamma}(\overline{\Omega'})$ into $C^1(\overline{\Omega'})$ is compact for every
$\overline{\Omega'}\subset \Omega$, then for every $k\in {\mathbb{N}}$, we can select a subsequence of ${u_n}$ which converges in $C^1(\overline{\Omega_k})$.
A diagonal procedure gives a subsequence (denoted again by ${u_n}$) which converges to a function $u\in C_{\rm loc}^1(\Omega)$. Passing to the limit in
(\eqref{omega-n-problem}) we see that $u$ is a weak solution of the equation in (\eqref{main problem}), verifying $\underline{u}\leq u \leq \overline{u}$. In particular, we deduce that $u=\infty$ on $\partial \Omega$. This proves the lemma.

\medskip
Next, we give the proof of Theorem \ref{p-alpha theorem}.

{\bf{Proof of Theorem \ref{p-alpha theorem}.}} Let $\varphi_1$ be the positive normalized eigenfunction associated to the first eigenvalue $\lambda_1$ of
$-\Delta_p$ in $W_0^{1,p}(\Omega)$ and let $0<\varepsilon<\eta$.  In order to construct a sub-solution $\underline{u}$ and a super-solution $\overline{u}$ of
(\eqref{main problem}), we  define the function $$v(x)=\left(\displaystyle\int_0^{\varepsilon \, \varphi_1(x)}t^{\frac{1-\lambda}{p-1}}\, \left(L(t)\right)^{\frac{1}{p-1}}dt\right)^{\frac{p-1}{p-1-\alpha}}\quad \mbox{if $x \in \Omega$}$$ and we will prove that $\Delta_p v(x)\approx b(x)\, v(x)^{\alpha}$.\\
A straightforward computation shows that
\begin{equation*}
\nabla v=\varepsilon^{\frac{p-\lambda}{p-1}}\,(v(x))^{\frac{\alpha}{p-1}}\left(\varphi_1(x)\right)^{\frac{1-\lambda}{p-1}}\,\left(L(\varepsilon \varphi_1(x))\right)^{\frac{1}{p-1}}\,\nabla \varphi_1
\end{equation*}
and
\begin{align*}
\Delta_pv=\varepsilon^{p-\lambda}\, \varphi_1^{-\lambda}(x)L(\varepsilon \varphi_1(x))(v(x))^{\alpha}\Bigg[\lambda_1\varphi_1^p+|\nabla \varphi_1|^{^{p}}\Bigg(\alpha \frac{\left(\varepsilon \varphi_1(x)\right)^{\frac{p-\lambda}{p-1}}(L(\varepsilon \varphi_1(x)))^{\frac{1}{p-1}}}{\left(v(x)\right)^{\frac{p-1-\alpha}{p-1}}}\big.\Big.\qquad  \qquad \\ \qquad \qquad \big.\big.  +(\lambda-1)-\varepsilon \varphi_1(x)\frac{L'(\varepsilon \varphi_1(x))}{L(\varepsilon \varphi_1(x))}\Bigg)\Bigg].
\end{align*}
Since $L\in {\mathcal{K}}$ and $\displaystyle\int_0^{\eta}\displaystyle\frac{(L(t))^{\frac{1}{p-1}}}{t^{\frac{\lambda-1}{p-1}}}\,dt<\infty$, then we deduce
from Lemmas \ref{lemma 2.2}, \ref{lemma 2.3} (ii) and \ref{lemma 2.4} that
$$\lim_{\varepsilon \rightarrow 0}\, \sup_{x\in \Omega}\frac{\left(\varepsilon \varphi_1(x)\right)^{\frac{p-\lambda}{p-1}}(L(\varepsilon \varphi_1(x)))^{\frac{1}{p-1}}}{\displaystyle\int_0^{\varepsilon \varphi_1(x)}t^{\frac{1-\lambda}{p-1}}\,(L(t))^{\frac{1}{p-1}}\,dt}=\frac{p-\lambda}{p-1}$$
and $$\lim_{\varepsilon \rightarrow 0}\, \sup_{x\in \Omega}\frac{(\varepsilon \varphi_1(x))L'(\varepsilon \varphi_1(x))}{L(\varepsilon \varphi_1(x))}=0.$$
Hence, there exists $\varepsilon>0$ such that for each $ x \in \Omega$,
\begin{equation*}
-\frac{(p-\lambda)(\alpha-p+1)+(p-1)^2}{4(p-1)}\leq-\varepsilon \varphi_1(x)\frac{L'(\varepsilon \varphi_1(x))}{L(\varepsilon \varphi_1(x))}\leq
\frac{(p-\lambda)(\alpha-p+1)+(p-1)^2}{4(p-1)}
\end{equation*}
and
\begin{align*}
\frac{(p-\lambda)(3\alpha+p-1)-(p-1)^2}{4(p-1)}\leq \alpha \frac{\left(\varepsilon \varphi_1(x)\right)^{\frac{p-\lambda}{p-1}}(L(\varepsilon \varphi_1(x)))^{\frac{1}{p-1}}}{\displaystyle\int_0^{\varepsilon \varphi_1(x)}t^{\frac{1-\lambda}{p-1}}\,(L(t))^{\frac{1}{p-1}}\,dt}  \qquad \qquad \qquad \qquad \qquad \qquad \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \leq \frac{(p-\lambda)(5\alpha-p+1)+(p-1)^2}{4(p-1)}.
\end{align*}
This gives
\begin{align*}
\frac{(p-\lambda)(\alpha-p+1)+(p-1)^2}{2(p-1)}\leq \alpha \frac{\left(\varepsilon \varphi_1(x)\right)^{\frac{p-\lambda}{p-1}}(L(\varepsilon \varphi_1(x)))^{\frac{1}{p-1}}}{\displaystyle\int_0^{\varepsilon \varphi_1(x)}t^{\frac{1-\lambda}{p-1}}\,(L(t))^{\frac{1}{p-1}}\,dt}-\varepsilon \varphi_1(x)\frac{L'(\varepsilon \varphi_1(x))}{L(\varepsilon \varphi_1(x))}+(\lambda-1)   \\  \qquad \qquad \qquad   \leq \frac{(p-\lambda)(3\alpha-p+1)+3(p-1)^2}{2(p-1)}.
\end{align*}
Therefore using these inequalities and (\eqref{phi1 plus grad phi1 comparable 1}) we obtain $\Delta_pv(x)\approx (\varphi_1(x))^{-\lambda}\,L(\varepsilon \varphi_1(x))(v(x))^{\alpha}$.\\ Now, using (\eqref{phi 1 comparable distance}),  Lemma \ref{lemma 2.6} and hypothesis ${\bf{(B_1)}}$ we obtain
\begin{align*}
\Delta_p v(x)&\approx &(\varphi_1(x))^{-\lambda}\,L(\varepsilon \varphi_1(x))\,(v(x))^{\alpha}\\ &\approx& \delta(x)^{-\lambda}\,L(\delta(x))(v(x))^{\alpha}\\ &\approx& b(x) (v(x))^{\alpha}.
\end{align*}
This proves that for every $\lambda\leq p$, there exists $M>0$ such that for every $x\in \Omega$, we have
\begin{align}\label{equation-sub super solution}
\frac{1}{M}\,b(x)v^{\alpha}(x)\leq \Delta_p v(x)\leq M b(x)v^{\alpha}(x).
\end{align}
By putting $c=M^{\frac{1}{\alpha-p+1}}$, it follows from (\eqref{equation-sub super solution}) that $\underline{u}=\frac{1}{c}\,v$
and $\overline{u}=c\,v$ are respectively sub-solution and super-solution of problem (\eqref{p-alpha problem}). Thus, we conclude
by Lemma \eqref{lemma sub supersolution} that problem (\eqref{p-alpha problem}) has a positive solution $u$ such that $\underline{u}\leq u\leq \overline{u}$.
Applying Lemma \ref{lemma 2.6}, Remark \ref{remark 2.5}, and Lemma \ref{lemma 2.3},  we deduce that
\begin{equation*}
u(x) \approx \delta(x)^{\frac{p-\lambda}{p-1-\alpha}}\,\theta_{L,\lambda,p,\alpha}(\delta(x)).
\end{equation*}

The following proposition plays a key role in the proof of Theorem \eqref{main result}.
\begin{proposition}\label{prop u2 inf c0 u1}
Let $a_1\,,\,a_2$ be the functions defined in hypothesis ${(H_1)}$ and let $\alpha_1\,,\;\alpha_2$ be such that $0<p-1<\alpha_1\leq \alpha_2$.
Let $u_i$ be the solution,  given in theorem \ref{p-alpha theorem}, of
the following problem
\begin{align} \label{alpha i-problem}
\left\{
\begin{array}{ll}
\Delta_p u_i = a_i(\delta(x))\,u^{\alpha_i}\,,\;\; x \in \Omega\,,\;\; \\
u_i>0\;\; \mbox{ in } \Omega\\
\displaystyle\lim_{\delta(x) \rightarrow 0}u_i(x)=\infty.
\end{array}
\right.
\end{align}
Then there exists a  constant $c_0>0$ such that
\begin{align}\label{u2 inf c0 u1}
u_2\leq c_0\,u_1\; \mbox{ in } \Omega.
\end{align}
\end{proposition}
{\bf{Proof.}} By Theorem \ref{p-alpha theorem}, problem (\eqref{alpha i-problem}) has a solution $u_i$ and there exist two constants $c_1>0$,
$c_2>0$ such that for each $x\in \Omega$, we have,
\begin{align}\label{estimation of u1 and u2}
\frac{1}{c_i}\,\psi_{L_i,\lambda_i,p, \alpha_i}(\delta(x))\leq u_i(x)\leq c_i\,\psi_{L_i,\lambda_i,p, \alpha_i}(\delta(x))\,,
\end{align}
where for $i\in \{1,2\}$, $\psi_{L_i,\lambda_i,p, \alpha_i}$ is the function defined on $(0,\eta]$, by
\begin{align}
\psi_{L_i,\lambda_i,p, \alpha_i}(t)=t^{\frac{p-\lambda_i}{p-1-\alpha_i}}\,\,\theta_{L_i,\lambda_i,p, \alpha_i}(t)
\end{align}
and $\theta_{L_i,\lambda_i,p, \alpha_i}$ is given by (\eqref{definition theta lambda i p-alpha i}). To prove Proposition \eqref{prop u2 inf c0 u1}, it is enough to show that $\displaystyle\frac{\psi_{L_2,\lambda_2,p, \alpha_2}}{\psi_{L_1,\lambda_1,p, \alpha_1}}$
is bounded in $(0,\eta]$. Now, using Lemma \ref{lemma 2.1} (i) and hypothesis ${(H_1)}$, we deduce that $\lambda_1\leq \lambda_2\leq p$.
On the other hand, since $p-1<\alpha_1\leq \alpha_2$, then we deduce that $$ 0\leq \frac{p-\lambda_2}{\alpha_2-(p-1)}\leq \frac{p-\lambda_1}{\alpha_1-(p-1)}.$$
Put $\sigma=\frac{(\alpha_2-\alpha_1)(p-\lambda_1)+(\lambda_2-\lambda_1)(\alpha_1-(p-1))}{(\alpha_1-(p-1))(\alpha_2-(p-1))}$. Then $\sigma\geq 0$ and  for each $t\in (0,\eta]$ we have $$\frac{\psi_{L_2,\lambda_2,p, \alpha_2}(t)}{\psi_{L_1,\lambda_1,p,\alpha_1}(t)}=t^{\sigma}\,\,
\frac{\theta_{L_2,\lambda_2,p, \alpha_2}(t)}{\theta_{L_1,\lambda_1,p, \alpha_1}(t)}.$$ Now,  using Lemma \ref{lemma 2.1} and the definition of $\theta_{L_i,\lambda_i,p, \alpha_i}$, we deduce
that $$\frac{\theta_{L_2,\lambda_2,p, \alpha_2}}{\theta_{L_1,\lambda_1,p, \alpha_1}} \in {\mathcal{K}}.$$ So, we distinguish the following two cases.

 {Case 1.} $\sigma>0$.
In this case, we conclude by Lemma \ref{lemma 2.1} that  $$\displaystyle\lim_{t\rightarrow 0}\frac{\psi_{L_2,\lambda_2,p, \alpha_2}(t)}{\psi_{L_1,\lambda_1,p, \alpha_1}(t)}=0.$$ Hence $\frac{\psi_{L_2,\lambda_2,p, \alpha_2}}{\psi_{L_1,\lambda_1,p, \alpha_1}}$ is bounded in $(0,\eta]$.

 {Case 2.} $\sigma=0$. This is equivalent to $\lambda_1=\lambda_2=p$ or $\lambda_1=\lambda_2<p \; \mbox{ and } \alpha_1=\alpha_2$. In this case, we have $L_1\leq L_2$
in $(0,\eta]$. So we will discuss two subcases:\\ $\bullet$ If $\lambda_1=\lambda_2=p$, then for each $t\in (0,\eta]$ we have
\begin{align*}
\frac{\psi_{L_2,\lambda_2,p, \alpha_2}(t)}{\psi_{L_1,\lambda_1,p, \alpha_1}(t)}
&=& \left(\displaystyle\int_0^t\frac{\left(L_1(s)\right)^{\frac{1}{p-1}}}{s}\,ds\right)^{\frac{p-1}{\alpha_1-(p-1)}}\;\left(\displaystyle\int_0^t
\frac{\left(L_2(s)\right)^{\frac{1}{p-1}}}{s}\,ds\right)^{\frac{1-p}{\alpha_2-(p-1)}}
\qquad \qquad \\&\leq& \left(\displaystyle\int_0^t\frac{\left(L_2(s)\right)^{\frac{1}{p-1}}}{s}\,ds\right)^{\frac{p-1}{\alpha_1-(p-1)}}
\left(\displaystyle\int_0^t\frac{\left(L_2(s)\right)^{\frac{1}{p-1}}}{s}\,ds\right)^{\frac{1-p}{\alpha_2-(p-1)}} \qquad \qquad \\ &\leq&
\left(\displaystyle\int_0^t\frac{\left(L_2(s)\right)^{\frac{1}{p-1}}}{s}\,ds\right)^{\frac{(p-1)(\alpha_2-\alpha_1)}{(\alpha_1-(p-1))(\alpha_2-(p-1))}}.
\end{align*}
Since $p-1<\alpha_1\leq \alpha_2$ and $0<\displaystyle\int_0^{\eta}\frac{\left(L_2(s)\right)^{\frac{1}{p-1}}}{s}\,ds<\infty$, then we deduce that $\displaystyle\frac{\psi_{L_2,\lambda_2,p, \alpha_2}}{\psi_{L_1,\lambda_1,p, \alpha_1}}$ is bounded in $(0,\eta]$.\\ $\bullet$ If $\lambda_1=\lambda_2<p$ and $\alpha_1=\alpha_2$, then for each $t\in (0,\eta]$ we have
\begin{align*}
\frac{\psi_{L_2,\lambda_2,p, \alpha_2}(t)}{\psi_{L_1,\lambda_1,p, \alpha_1}(t)}=\frac{\theta_{L_2,\lambda_2,p, \alpha_2}(t)}{\theta_{L_1,\lambda_1,p, \alpha_1}(t)} =\frac{\left(L_2(t)\right)^{\frac{1}{p-1-\alpha_2}}}{\left(L_1(t)\right)^{\frac{1}{p-1-\alpha_1}}}&=&
\left(\frac{L_1(t)}{L_2(t)}\right)^{\frac{1}{\alpha_1-(p-1)}}\leq 1.
\end{align*}
This completes the proof of Proposition \eqref{prop u2 inf c0 u1}.\\ {\bf{Proof of Theorem \eqref{main result}.}}
Let $u_i$ be a solution of the problem (\eqref{alpha i-problem}) and let $c_0$ be a positive constant
such that $ u_2\leq c_0\,u_1$. Since $\lim_{\delta(x)\rightarrow 0} u_1(x)=\infty$, then $\displaystyle\inf_{x \in \Omega} u_1(x)>0$.
Let $\mu_1\,,\,\mu_2$ be two positive constants chosen so that $$\mu_1\geq \max \left(\frac{1}{k_1^{\frac{1}{\alpha_1-(p-1)}}},\frac{1}{\displaystyle\inf_{x\in \Omega}u_1(x)}\right)\quad \mbox{and}\quad
\mu_2\leq \min\left(\frac{\mu_1}{c_0},\frac{1}{k_2^{\frac{1}{\alpha_2-(p-1)}}}\right),$$ where $k_1$, $k_2$ are given in hypothesis ${(H_2)}$. Put
\begin{align}\label{def subsolution u and supersolution}
\overline{u}=\mu_1\,u_1\;\;  \mbox{ and }
\underline{u}=\mu_2\,u_2.
\end{align}
Then using hypotheses ${(H_1)}$  and ${(H_2)}$, we obtain
\begin{align*}
\left\{
\begin{array}{ll}
\Delta_p \overline{u} =\frac{1}{\mu_1^{\alpha_1-(p-1)}} \, a_1(\delta(x))\,{\overline{u}}^{\alpha_1}\leq a(x)f(\overline{u})\,,\;\; x \in \Omega\,,\;\; \\
\overline{u}>0\;\; \mbox{ in } \Omega\; ;\;
\displaystyle\lim_{\delta(x) \rightarrow 0}\overline{u}(x)=\infty
\end{array}
\right.
\end{align*}
and
\begin{align*}
\left\{
\begin{array}{ll}
\Delta_p \,\underline{u} =\displaystyle\frac{1}{k_2\,\mu_2^{\alpha_2-(p-1)}} \, a_2(\delta(x))\,k_2\,{\underline{u}}^{\alpha_2}\geq a(x)f(\underline{u})\,,
\;\; x \in \Omega\,,\;\; \\
\underline{u}>0\;\; \mbox{ in } \Omega\; ;\;
\displaystyle\lim_{\delta(x) \rightarrow 0}\underline{u}(x)=\infty.
\end{array}
\right.
\end{align*}
So $\underline{u}$ and $\overline{u}$ are respectively a sub-solution and a super-solution of problem (\eqref{main problem}). Moreover, for each
$x\in \Omega$ we have
\begin{equation*}
\underline{u}(x)=\mu_2\,u_2(x)\leq \mu_2\,c_0\,u_1(x)\leq \mu_1\,u_1(x)=\overline{u}.
\end{equation*}
Since $a\in C_{\rm loc}^{0,\gamma}(\Omega)$ and $f\in C^1([0,\infty))$, we deduce from Lemma \eqref{lemma sub supersolution} that (\eqref{main problem})
has a  weak solution $u\in W_{\rm loc}^{1,p}(\Omega)\cap C_{\rm loc}^{1,\beta}(\Omega)$, for some $0<\beta<1$, satisfying
\begin{equation*}
\underline{u}\leq \,u\leq \,\overline{u}.
\end{equation*}
This together with (\eqref{estimation of u1 and u2}) and (\eqref{def subsolution u and supersolution}) implies that $u$ satisfies (\eqref{estimates  solution main problem}).

\section{Application to the singular logistic problem with convection}
Let $a$ be a function satisfying ${(H_1)}$ and let $f$ be a function satisfying ${(H_2)}$ and $\beta \in {\mathbb{R}}$ with $\beta< 1$. In this paragraph, we are interested in the following problem
\begin{align}\label{application 1}
\left\{
\begin{array}{ll}
\displaystyle\Delta_p u -\frac{\beta(p-1)}{u}\, |\nabla u|^p= a(x)\,f(u)\,,\;\; x \in \Omega\,,\;\; \\
u>0\;\; \mbox{ in } \Omega\\
\displaystyle\lim_{\delta(x) \rightarrow 0}u(x)=\infty\,.
\end{array}
\right.
\end{align}
By putting $v=u^{1-\beta}$, we obtain by a simple calculus that $v$ satisfies
\begin{align}\label{eq 4-2}
\left\{
\begin{array}{ll}
\displaystyle\Delta_p v = (1-\beta)^{p-1} a(x) v^{-\frac{\beta(p-1)}{1-\beta}}\,f(v^{\frac{1}{1-\beta}})\,,\;\; x \in \Omega\,,\;\; \\
v>0\;\; \mbox{ in } \Omega\\
\displaystyle\lim_{\delta(x) \rightarrow 0}v(x)=\infty\,.
\end{array}
\right.
\end{align}
Let $g$ be the function defined on $(0,\infty)$ by $g(v)=(1-\beta)v^{-\frac{\beta(p-1)}{1-\beta}}\,f(v^{\frac{1}{1-\beta}})$ and put
$\alpha_1^{\ast}=\frac{\alpha_1-\beta(p-1)}{1-\beta}$ and $\alpha_2^{\ast}=\frac{\alpha_2-\beta(p-1)}{1-\beta}$. Clearly
$0<p-1<\alpha_1^{\ast}\leq \alpha_2^{\ast}$ and the function $g$ satisfies
$$(1-\beta)k_1\,r^{\alpha_1^{\ast}}\leq g(r)\; \mbox{ for } r\geq 1\; \mbox{ and } g(r)\leq (1-\beta)k_2\,r^{\alpha_2^{\ast}}\, \mbox{ for } r>0.$$
Therefore, it follows from Theorem \eqref{main result}  that problem (\eqref{eq 4-2}) has a positive weak solution $v\in W_{\rm loc}^{1,p}(\Omega)
\cap C_{\rm loc}^{1, \nu}(\Omega)$, for some $0<\nu<1$, such that
\begin{equation*}
\frac{1}{C}\left(\delta(x)\right)^{\frac{(p-\lambda_2)(1-\beta)}{p-1-\alpha_2}}\,\theta_{L_2,\lambda_2,p,\alpha_2^{\ast}}(\delta(x))\leq v(x)\leq C\,\left(\delta(x)\right)^{\frac{(p-\lambda_1)(1-\beta)}{p-1-\alpha_1}}\, \theta_{L_1,\lambda_1,p,\alpha_1^{\ast}}(\delta(x))\,
\end{equation*}
for some constant $C>1$. Consequently, we deduce that problem (\eqref{application 1}) has a solution $u\in W_{\rm loc}^{1,p}(\Omega)
\cap C_{\rm loc}^{1, \nu}(\Omega)$ satisfying
\begin{equation*}
\frac{1}{C}\left(\delta(x)\right)^{\frac{p-\lambda_2}{p-1-\alpha_2}}\,\theta_{L_2,\lambda_2,p,\alpha_2^\ast}(\delta(x))\leq u(x)\leq C\,\left(\delta(x)\right)^{\frac{p-\lambda_1}{p-1-\alpha_1}}\, \theta_{L_1,\lambda_1,p,\alpha_1^\ast}(\delta(x))\,
\end{equation*}
for some constant $C>1$.

\medskip
{\bf{Authors' Contribution.}}
All authors contributed equally and significantly in the elaboration of this article and their names are written in alphabetical order.
All authors read and approved the final document.

% For the acknowledgements section, please do not number the section, please begin it with \section*{Acknowledgements}
\section*{Acknowledgements}
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. 39-130-35-HiCi. The authors acknowledge with thanks DSR technical and financial support. The authors are indebted to the anonymous referee for useful comments.


% Please your references as follows in your main tex file.
% Using BibTex is not recommended but can be handled.
% The order of the references should follow alphabetical order of the first author's last (family) name. If there are more papers by the same author, these should be in chronological order.
% Abbreviations of names of journals should follow the Mathematical Reviews (see http://www.ams.org/msnhtml/serials.pdf). Page numbers should be written with an en dash, e.g. 15--23.

%Please make sure that you list only those entries that are cited in your paper.

%Please use our above defined macro \hbibitem for defining your reference entries. It has four arguments. The first is the label by which you wish to refer to it. The second argument should contain all usual information of the entry typeset as it is shown in the examples below. The third and fourth arguments are optional but strongly recommended. The third one should contain the Math Reviews number and the fourth one the DOI number of the entry. Both data can be looked up e.g. on http://www.ams.org/mrlookup using the BibTeX format searching.

\begin{thebibliography}{99}

\hbibitem{BM1}
{\textsc{C. Bandle, M. Marcus}, Large solutions of semilinear elliptic equations: existence, uniqueness and
asymptotic behaviour, \textit{J. Anal. Math.} {\bf 58} (1992), 9--24.}{}{}

\hbibitem{Bi1}
 {\textsc{L. Bieberbach}, $\Delta u = e^u$ und die automorphen Funktionen, \textit{Math. Ann.} {\bf 77} (1916), 173--212.}{}{}

\hbibitem{BGT}
 {\textsc{N. H. Bingham, C. M. Goldie, J. L. Teugels}, Regular Variation, \textit{Cambridge University Press}, Cambridge, (1987).}{}{}

\hbibitem{CNi}
{\textsc{K. S. Cheng, W. M. Ni}, On the structure of the conformal scalar curvative equation on ${\mathbb{R}}^n$, \textit{Indiana Univ. Math. J.} {\bf 41} (1992), 261--278.}{}{}

\hbibitem{CCEG}
{\textsc{M. Chuaqui, C. Cort\'azar, M. Elgueta, J. Garcia-Melian}, Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights, \textit{Comm. Pure Appl. Anal.} {\bf 3} (2004), 653--662.}{}{}

\hbibitem{CR1}
 {\textsc{F. C\^irstea, V. R\u{a}dulescu}, Uniqueness of the blow-up boundary solution of logistic equations with absorption, \textit{C. R. Acad. Sci. Paris}, Ser. I {\bf 335} (2002), 447--452.}{}{}

\hbibitem{CR2}
 {\textsc{F. C\^irstea, V. R\u{a}dulescu}, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, \textit{Asympt. Anal.} {\bf 46} (2006), 275--298.}{}{}

\hbibitem{DL}
{\textsc{G. Diaz, R. Letelier}, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, \textit{Nonlinear Anal.} {\bf 20} (1993), 97--125.}{}{}

\hbibitem{Du}
{\textsc{Y. Du}, Asymptotic behaviour and uniqueness results for boundary blow-up solutions, \textit{Diff. Int. Eqns.} {\bf 17} (2004), 819--834.}{}{}

\hbibitem{DuGu}
{\textsc{Y. Du, Z. Guo}, Boundary blow-up solutions and their applications in quasilinear elliptic equations, \textit{J. Anal. Math.} {\bf 89} (2003), 277--302.}{}{}

\hbibitem{GLS}
{\textsc{J. Garcia-Melian, R. Letelier-Albornoz, J. Sabina de Lis}, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, \textit{Proc. Amer. Math. Soc.} {\bf 129} (2001), 3593--3602.}{}{}

\hbibitem{GM1}
{\textsc{J. Garcia-Melian}, A remark on the existence of large solutions via sub and supersolutions, \textit{Electronic J. Differential Equations} 2003 (2003), 1--4.}{}{}

\hbibitem{GM2}
{\textsc{J. Garcia-Melian}, Large solutions for equations involving the $p$-laplacian and singular weights, \textit{Z. Angew. Math. Phys.} {\bf 60} (2009), 594--607.}{}{}

\hbibitem{GM3}
{\textsc{J. Garcia-Melian}, Quasilinear equations with boundary blow-up and exponential reaction, \textit{Adv. Nonl. Stud.} {\bf 9} (2009), 149--160.}{}{}

\hbibitem{GR}
 {\textsc{M. Ghergu, V.D. R\u{a}dulescu}, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, \textit{Springer Monographs in Mathematics}, Springer Verlag (2012).}{}{}

\hbibitem{GuW1}
{\textsc{Z. Guo, J. R. L. Webb}, Structure of boundary blow-up solutions of quasilinear elliptic problems I. Large and small solutions,\textit{ Proc. Roy. Soc. Edinburgh} {\bf 135A} (2005), 615--642.}{}{}

\hbibitem{GuW2}
{\textsc{Z. Guo, J. R. L. Webb}, Structure of boundary blow-up solutions of quasilinear elliptic problems II. Small and intermediate solutions, \textit{J. Differential Equations} {\bf 211} (2005), 187--217.}{}{}

\hbibitem{Kar}
{\textsc{ J. Karamata}, Sur un mode de croissance r\'eguli\`ere de fonctions. Th\'eor\`emes fondamentaux, \textit{Bull. Soc. Math. France} {\bf 61} (1933), 55--62.}{}{}

\hbibitem{Kel}
{\textsc{J. B. Keller}, On solutions of $\Delta u=f(u)$, \textit{Comm. Pure Appl. Math.} {\bf 10} (1957), 503--510.}{}{}

\hbibitem{La}
 {\textsc{A. V. Lair}, A necessary and sufficient condition for the existence of large solutions to semilinear elliptic equations, \textit{J. Math. Anal. Appl.} {\bf 240} (1999), 205--218.}{}{}

\hbibitem{LW}
{\textsc{A. V. Lair, A. W. Wood}, Large solutions of sublinear elliptic equations, \textit{Nonlinear Anal.} {\bf 39} (2000), 745--753.}{}{}

\hbibitem{LMcK}
{\textsc{A. C. Lazer, P. J. McKenna}, On a problem of Bieberbach and Rademacher, \textit{Nonlinear Anal.} {\bf 21} (1993), 327--335.}{}{}

\hbibitem{Lieb}
{\textsc{G. Lieberman}, Boundary regularity for solutions of degenerate elliptic equations, \textit{Nonlinear Anal.} {\bf 12} (1988), 1203--1219.}{}{}

\hbibitem{LuYa}
{\textsc{Q. Lu, Z. Yang, E. H. Twizell}, Existence of entire explosive positive solutions of quasilinear elliptic equations, \textit{Appl. Math. Comput.} {\bf 148} (2004), 359--372.}{}{}

\hbibitem{Maag}
{\textsc{H. M\^aagli}, Asymptotic behaviour of positive solutions of a semilinear Dirichlet problem, \textit{Nonlinear Anal.} {\bf 74} (2011), 2941--2947.}{}{}

\hbibitem{MBC}
{\textsc{H. M\^aagli, S. Ben Othman, R. Chemmam}, Asymptotic behaviour of positive large solutions of semilinear Dirichlet problems, \textit{Electron. J. Qual. Theory Differ. Equ.}, No. 57 (2013),   1--13.}{}{}

\hbibitem{MMZ}
{\textsc{H. M\^{a}agli, N. Mhadhebi,  N. Zeddini}, Existence and exact asymptotic behaviour of positive solutions for a fractional boundary value problem, \textit{Abstract and Applied Analysis}, Vol 2013, Article ID 420514, 6 pages.}{}{}

\hbibitem{Mar}
{\textsc{V. Maric}, Regular Variation and Differential Equations, \textit{Lecture Notes in Math.}, Vol. 1726, Springer-Verlag, Berlin, 2000.}{}{}

\hbibitem{Mat}
{\textsc{J. Matero}, Quasilinear elliptic equations with boundary blow-up,\textit{ J. Anal. Math. }{\bf 69} (1996), 229--247.}{}{}

\hbibitem{MckRW}
{\textsc{P. J. Mckenna, W. Reichel, W. Walter,} Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up, \textit{Nonlin. Anal.} {\bf 28} (1997), 1213--1225.}{}{}

\hbibitem{Moh}
{\textsc{A. Mohammed}, Existence and asymptotic behaviour of blow-up solutions to weighted quasilinear equations,\textit{ J. Math. Anal. Appl.} {\bf 298} (2004), 621--637.}{}{}

\hbibitem{Mos}
{\textsc{J. Moser}, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equation, \textit{Comm. Pure. Appl. Math.} {\bf 13} (1960), 457--468.}{}{}

\hbibitem{Osser}
{\textsc{R. Osserman}, On the inequality $\Delta u\geq f(u)$, \textit{Pacific J. Math.} {\bf 7} (1957), 1641--1647.}{}{}

\hbibitem{Poh}
{\textsc{S. L. Pohozaev}, The Dirichlet problem for the equation $\Delta u=u^2$, \textit{Soviet Math. Dokl.} {\bf 1} (1960), 1143--1146.}{}{}

\hbibitem{Rade}
{\textsc{H. Rademacher}, Einige besondere Probleme der partiellen Differential gleichungen, Die Differential und Integral gleichungen der Mechanik und Physik I, \textit{2nd. edition, (P. Frank und R. von Mises, eds.)}, Rosenberg, New York, (1943), 838--845.}{}{}

\hbibitem{Radu}
 {\textsc{V. R\u{a}dulescu}, Singular phenomena in nonlinear elliptic problems: from boundary blow-up solutions to equations with singular nonlinearities, \textit{in Handbook of Differential Equations: Stationary Partial Differential Equations}, vol. 4, (2007), 483--591.}{}{}

\hbibitem{Sen}
{\textsc{R. Seneta}, Regular Varying Functions, \textit{Lectures Notes in Math.}, Vol. 508, Springer-Verlag, Berlin, 1976.}{}{}

\hbibitem{stevic1} 
{\textsc{S. Stevi\'c}, On harmonic Hardy and Bergman spaces,  \textit{J. Math. Soc. Japan} {\bf 54} (2002), 983--996.}{}{}

\hbibitem{stevic2} 
{\textsc{S. Stevi\'c}, On harmonic function spaces,  \textit{J. Math. Soc. Japan} {\bf 57} (2005), 781--802.}{}{}

\hbibitem{stevic3} 
{\textsc{S. Stevi\'c}, Area type inequalities and integral means of harmonic functions on the unit ball, \textit{J. Math. Soc. Japan} {\bf 59} (2007), 583--601.}{}{}

\hbibitem{Vaz}
{\textsc{J. L. V\'{a}zquez}, A strong maximum principle for some quasilinear elliptic equations, \textit{Appl. Math. Optimiz.} {\bf 1} (1984), 1992--2002.}{}{}

\hbibitem{ZMA}
{\textsc{N. Zeddini, R. Alsaedi, H. M\^aagli}, Exact boundary behaviour of the unique positive solution to some singular elliptic problems, \textit{Nonlin. Anal. (TMA)} {\bf 89} (2013), 146--156.}{}{}

\hbibitem{Zh1}
 {\textsc{Z. Zhang}, A remark on the existence of explosive solutions for a class of semilinear elliptic equations, \textit{Nonlin. Anal.} {\bf 41} (2000), 143--148.}{}{}

\hbibitem{Zh2}
 {\textsc{Z.Zhang}, On the existence and asymptotic behaviour of explosive solutions for nonlinear elliptic problems with convection terms, \textit{Chinese Annals  Math.} {\bf 23A} (2002), 395--406.}{}{}

\hbibitem{ZMML}
 {\textsc{Z. Zhang, Y. Ma, L. Mi, X. Li}, Blow-up rates of large solutions for elliptic equations, \textit{J. Differential Equations} {\bf 249} (2010), 180--199.}{}{}

\end{thebibliography}
\end{document}
