\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 57, pp. 1--33.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2015/57\hfil Existence and asymptotic behavior] {Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term} \author[H. Wan \hfil EJDE-2015/57\hfilneg] {Haitao Wan} \address{Haitao Wan \newline School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China} \email{wht200805@163.com, Phone +8618954556896} \thanks{Submitted December 9, 2014. Published March 6, 2015.} \subjclass[2000]{35A01, 35B40, 35J25} \keywords{Singular Dirichlet problem; Karamata regular variation theory; \hfill\break\indent convection term; boundary asymptotic behavior; global asymptotic behavior} \begin{abstract} In this article, we consider the problem $$ -\Delta u =b(x)g(u)+ \lambda a(x)|\nabla u|^{q}+\sigma(x),\; u > 0,\; x\in \Omega,\quad u|_{\partial \Omega }= 0 $$ with $\lambda\in\mathbb{R}$, $q\in [0, 2]$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^{N}$. The weight functions $b, a,\sigma$ belong to $C^{\alpha}_{\rm loc}(\Omega)$ satisfying $b(x),a(x)>0$, $\sigma(x)\geq0$, $x\in \Omega$, which may vanish or be singular on the boundary. $g\in C^1((0,\infty),(0,\infty))$ satisfies $\lim_{t\to 0^{+}}g(t)=\infty$. Our results include the existence, uniqueness and the exact boundary asymptotic behavior and global asymptotic behavior of the solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and main results} In this article we study the existence and asymptotic behavior of the unique classical solution to the problem \begin{equation}\label{M} -\Delta u =b(x)g(u)+ \lambda a(x)|\nabla u|^{q}+\sigma(x),\, u > 0,\, x\in \Omega,\quad u|_{\partial \Omega }= 0, \end{equation} where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^{N}$, $\lambda\in\mathbb{R}$, $q\in[0, 2]$, $b,a,\sigma$ satisfy \begin{itemize} \item[(H1)] $b,a,\sigma \in C^{\alpha}_{\rm loc}(\Omega) $ for some $ \alpha \in (0, 1)$, and $b(x),a(x)>0,\sigma(x)\geq0, x\in\Omega$, \end{itemize} and $g$ satisfies the following hypotheses, not necessary simultaneously: \begin{itemize} \item[(G1)] $g \in C^1((0, \infty), (0, \infty))$, $\lim_{t\to 0^{+}}g(t) =\infty$; \item[(G2)] there exists $t_0>0$ such that $g'(t)<0$, for all $ t\in(0, t_0)$; \item[(G3)] $g$ is decreasing on $(0, \infty)$; \item[(G4)] there exists $D_{g}\geq 0$ such that \[ \lim_{t\to0^{+}}g'(t)\int_0^{t}\frac{ds}{g(s)}=-D_{g}. \] \end{itemize} When $\lambda=0$ and $\sigma\equiv0$ in $\Omega$, problem \eqref{M} becomes \begin{equation}\label{MI} -\Delta u =b(x)g(u),\; u > 0,\; x\in \Omega,\quad u|_{\partial \Omega}= 0. \end{equation} This problem arises in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrical materials, and has been studied and extended by many authors, for instance \cite{CA}-\cite{CMV}, \cite{CRT}, \cite{FM,GR1}, \cite{GR}, \cite{GMMT,GuiLin}, \cite{Lair}-\cite{Maagli}, \cite{MR,MI}, \cite{NC}, \cite{JSMY}-\cite{ZM}, \cite{Z.2}, \cite{Zhang1}-\cite{ZhangL} and the references therein. Next, we review works about the existence, uniqueness and asymptotic behavior of classical solutions to \eqref{M}, which are summarized as the following two parts. \subsection*{Part I: Existence and boundary behavior} For $b\equiv1$ in $\Omega$, when $g$ satisfies (G1) and (G3), Crandall, Rabinowitz and Tartar \cite{CRT}, Fulks and Maybee \cite{FM}, Stuart \cite{CAS} showed that \eqref{MI} has a unique solution $u \in C^{2,\alpha}_{\rm loc}(\Omega) \cap C(\bar{\Omega})$, and the authors in \cite{CRT} established the asymptotic behavior of the unique solution. Moreover, Anedda \cite{CA}, Berhanu, Gladiali and Porru \cite{BGP}, Berhanu, Cuccu and Porru \cite{BCP}, Ghergu and R\u{a}dulescu \cite{GR1}, Ghergu and R\u{a}dulescu \cite{GR}, McKenna and Reichel \cite{MR}, Mi and Liu \cite{MI}, Zhang \cite{Zhang5} analyzed the first or second estimate of the solution near the boundary to \eqref{MI}. In particular, when $b\in C^{\alpha}(\bar{\Omega})$ satisfies the following assumptions: there exist a constant $\delta>0$ and a positive non-decreasing function $k_1\in C((0, \delta))$ such that \begin{itemize} \item[(B01)] $\lim_{d(x)\to0}\frac{b(x)}{k_1(d(x))}=b_0\in(0, \infty)$, where $d(x):=\operatorname{dist}(x, \partial\Omega)$; \item[(B02)] $\lim_{t\to0^{+}}k_1(t)g(t)=\infty$; \end{itemize} and $g$ satisfies (G1), (G3) and the conditions \begin{itemize} \item[(G01)] there exist positive $c_0, \eta_0$ and $\gamma\in (0, 1)$ such that $g(t)\leq c_0t^{-\gamma}$, for all $t\in (0, \eta_0)$; \item[(G02)] there exist $\theta>0$ and $t_0\geq1$ such that $g(\xi t)\geq \xi^{-\theta}g(t)$ for all $\xi\in(0, 1)$ and $00,\;t\in(0,c),\;\phi(0)=0. \] Zhang \cite{Z.2} extended the above result to the case where $g$ is normalized regularly varying at zero with index $-\gamma$ $(\gamma>0)$ and $k_1$ in (B01) is normalized regularly varying at zero with index $-\beta$ $(\beta\in(0, 2))$. Later, Ben Othman et al \cite{OMMZ}, Gontara et al \cite{GMMT} extended the results in \cite{GR1,Z.2} to a large class of functions $b$ which belongs to the Kato class $K(\Omega)$ and $g$ is normalized regularly varying at zero with index $-\gamma$ $(\gamma\geq0)$. In particular, they established an exact boundary behavior of the unique solution to the problem \begin{equation}\label{B} -\Delta v=b(x),\; v>0,\; x\in\Omega,\; v|_{\partial\Omega}=0, \end{equation} when $b$ satisfies (H1) and the condition \begin{itemize} \item[(B03)] \[ 0<\tilde{b}_2:=\liminf_{d(x)\to0}\frac{b(x)}{k_1(d(x))}\leq \tilde{b}_1:=\limsup_{d(x)\to0}\frac{b(x)}{k_1{d(x)}}<\infty \] with $k_1(t)=t^{-2}\prod_{i=1}^{m}(\ln_i(t^{-1}))^{-\mu_i}$, $t\in(0,\delta)$, for some $\delta>0, $ \end{itemize} where $\ln_i(t^{-1})=\ln\circ\ln\circ\ln\circ\dots\circ\ln (t^{-1})$ ($i$ times) and $\mu_1=\mu_2=\dots=\mu_{j-1}=1,\,\mu_{j}>1$ and $\mu_i\in\mathbb{R}$ for $j+1\leq i\leq1$. For the convenience of discussions, we introduce two classes of Karamata functions as follows. (i) Denote by $\Lambda$ the set of all positive functions in $C^1((0, \delta_0])\cap L^1((0, \delta_0])$ which satisfy \begin{equation}\label{K} \lim_{t\to 0^{+}}\frac{d}{dt}\Big(\frac{K(t)}{ k(t)}\Big)\in(0, \infty),\quad K(t) = \int^{t}_0 k(s)ds \end{equation} and for each $k\in \Lambda$ there exists $\delta_{k}\in(0, \delta_0]$ such that $k$ is monotonic on $(0, \delta_{k}]$. (ii) Denote by $\mathcal {K}$ the set of all positive functions $k$ defined on $(0, \delta_0]$ by \begin{equation}\label{qx} \begin{gathered} k(t):=c\exp\Big(\int_{t}^{\delta_0}\frac{y(s)}{s}ds\Big),\quad c>0 \mbox { and }\\ y\in C((0, \delta_0]) \text{ with } \lim_{t\to0^{+}}y(t)=0. \end{gathered} \end{equation} Define \[ D_{k}:=\lim_{t\to 0^{+}}\frac{d}{dt}\Big(\frac{K(t)}{ k(t)}\Big) \quad \text{for each } k\in\Lambda\cup\mathcal {K}. \] Indeed, if $k\in\mathcal {K}$, then it follows by Proposition \ref{P5}(i) and a direct calculation that $D_{k}=1$. The set $\Lambda$ was first introduced by C\^{\i}rstea and R\u{a}dulescu \cite{C.1}-\cite{C.4} for non-decreasing functions and by Mohammed \cite{Mo} for non-increasing functions to study the boundary behavior and uniqueness of solutions for boundary blow-up elliptic problems, which enables us to obtain significant information about the qualitative behavior of the large solution in a general framework. Later, Based on their ideas, Huang et al. \cite{Huang1}-\cite{Huang2}, Mi and Liu \cite{Mi2}, Zhang \cite{ZhangB} and Repov\v{s} \cite{Repovs} further studied the asymptotic behavior of boundary blow-up solutions. \smallskip Recently, Zhang and Li \cite{ZhangL} obtained the following results. (i) Let $b$ satisfy (H1), (B03), $g$ satisfy (G1), (G3)-(G4) with $D_{g}>0$, then for the unique classical solution $u$ of \eqref{MI}, \[ \Big(\frac{\tilde{b}_2}{\mu_{j}-1}\Big)^{1-D_{g}} \leq\liminf_{d(x)\to0}\frac{u(x)}{\psi(h(d(x)))} \leq\limsup_{d(x)\to0} \frac{u(x)}{\psi(h(d(x)))} \leq\Big(\frac{\tilde{b}_1}{\mu_{j}-1}\Big)^{1-D_{g}}, \] where \[ h(t)=(\ln_{j}(t^{-1}))^{1-\mu_{j}}\prod_{i=j+1}^{m}(\ln_i(t^{-1}))^{-\mu_i},\, t\in(0, \delta), \] and the function $\psi$ is uniquely determined by \begin{equation}\label{PS} \int_0^{\psi(t)}\frac{ds}{g(s)}=t,\quad t>0. \end{equation} (ii) Let $b$ satisfy (H1) and the condition that there exists $k\in\Lambda$ such that \begin{equation}\label{CM} 00$. If \begin{equation}\label{DD} D_{k}+2D_{g}>2, \end{equation} then for the unique classical solution $u$ of \eqref{MI}, \[ \xi_2^{1-D_{g}}\leq\liminf_{d(x)\to0} \frac{u(x)}{\psi(K^2(d(x)))}\leq\limsup_{d(x)\to0} \frac{u(x)}{\psi(K^2(d(x)))}\leq\xi_1^{1-D_{g}}, \] where \begin{equation}\label{ZL} \xi_i=\frac{b_i}{2(D_{k}+2D_{g}-2)},\,i=1,2. \end{equation} Later, Zeddini, Alsaedi and M\^{a}agli \cite{ZM} extended the above results so that they cover the case $b(t)= t^{-2}k(t)$, where $k$ belongs to $\mathcal {K}$ and satisfies \begin{equation}\label{youxian} \int_0^{\delta_0}\frac{k(s)}{s}ds<\infty. \end{equation} They obtained the following theorem. \begin{theorem} \label{thmA} Let $b$ satisfy {\rm (H1)} and there exist $k\in\mathcal {K}$ such that \[ 0<\tilde{b}_2:=\liminf_{d(x)\to0}\frac{b(x)}{(d(x))^{\gamma-1}k(d(x))} \leq\tilde{b}_1:=\limsup_{d(x)\to0}\frac{b(x)}{(d(x))^{\gamma-1}k(d(x))}<\infty, \] where $\gamma\geq0$ and \[ \int_0^{\delta_0}\frac{k(s)}{s}ds=\infty, \] then the unique classical solution $u$ of \eqref{MI} in the case of $g(u)=u^{-\gamma}$ satisfies \begin{align*} a_2^{1/(1+\gamma)} &\leq\liminf_{d(x)\to0}\frac{u(x)}{d(x) \Big(\int_{d(x)}^{\delta_0}\frac{k(s)}{s}ds\Big)^{1/(1+\gamma)}}\\ &\leq\limsup_{d(x)\to0}\frac{u(x)}{d(x) \Big(\int_{d(x)}^{\delta_0}\frac{k(s)}{s}ds\Big)^{1/(1+\gamma)}} \leq a_1^{1/(1+\gamma)}, \end{align*} where $a_i=\tilde{b}_i(1+\gamma),\,i=1,2$. \end{theorem} This improves the result of Lazer and Mckenna \cite{LM}. Recently, Alsaedi, M\^{a}agli and Zeddini \cite{Alsaedi} extended the results in \cite{ZM} to the case where $\Omega$ is an exterior domain in $\mathbb{R}^{N}$ with $N\geq3$. When $\lambda>0$, $q=2$, $b,a\equiv1$, $\sigma\equiv0$ in $\Omega$ and $g(u)=u^{-\gamma}$, $\gamma>0$, by using the change of variable $v=e^{\lambda u}-1$, Zhang and Yu \cite{Z.1} proved that \eqref{M} possesses a unique classical solution for each $\lambda\in(0, \infty)$. This was then used to deduce the existence and nonexistence of classical solutions to \eqref{M} in the case $q\in(0, 2)$. When $\lambda=\pm1$, $q\in(0, 2)$, $b,a\equiv 1$, $\sigma\equiv0$ in $\Omega$ and the function $g:(0, \infty)\to(0, \infty)$ is locally Lipschitz continuous and decreasing, Giarrusso and Porru \cite{EGP} showed that if $g$ satisfies the following conditions: \begin{itemize} \item[(i)] $\int_0^1g(s)ds=\infty$, $\int_1^{\infty}g(s)ds<\infty$; \item[(ii)] there exist positive constants $\delta$ and $M$ with $M>1$ such that \[ G_1(t)0,\] \end{itemize} then the unique solution $u$ to \eqref{M} has the properties: \begin{itemize} \item[(i)] $|u(x)-\Psi(d(x))|0. \end{equation} \end{itemize} This implies \[ \lim_{d(x)\to0}\frac{u(x)}{\Psi(d(x))}=1. \] When $\lambda\in\mathbb{R}$ and $g$ satisfies (G1) with $\lim_{t\to\infty}g(t)=0$, (G3), Zhang \cite{Z.4} showed that \begin{itemize} \item[(i)] if $q=2$, $b$ satisfies (H1) and \eqref{B} possesses a unique solution which belongs to $C^{2,\alpha}_{\rm loc}(\Omega)\cap C(\bar{\Omega})$, then \eqref{M} has a unique solution $u_{\lambda}\in C^{2, \alpha}_{\rm loc}(\Omega) \cap C(\bar{\Omega})$ for every $\lambda\geq0$; \item[(iii)] if $b\equiv1$ in $\Omega$, then \eqref{M} has a unique solution $u_{\lambda}\in C^{2,\alpha}_{\rm loc}(\Omega)\cap C(\bar{\Omega})$ in one of the following three cases: (i) $q\in[0, 2]$, $\lambda\leq0$; (ii) $q\in[0, 1)$, $\lambda\geq0$; (iii) $q=1$, $0\leq\lambda<\lambda_1^{1/2}$, where $\lambda_1$ is the first eigenvalue of Laplace operator $(-\Delta)$ with the Dirichlet boundary condition. \end{itemize} When $\lambda>0$, $a\equiv1$, $\sigma\equiv0$ in $\Omega$, and $g$ satisfies (G1), (G3), (G4), Zhang et al \cite{zhangz} studied the boundary asymptotic behavior of the unique solution to \eqref{M} in the following two cases: (i) $q=2$ and $b\in C^{\alpha}_{\rm loc}(\Omega)$; (ii) $q\in(0, 2)$ and $b\equiv1$ in $\Omega$. For other works, we refer the reader to \cite{CGP}-\cite{LDMR}, \cite{GR2}-\cite{Gr2}, \cite{GP}, \cite{PV}, \cite{LiZhang2} and the references therein. \subsection*{Part II: Existence and global behavior} In this part, we review these works about the existence and global asymptotic behavior of classical solutions to \eqref{MI} in the case that $g\in C^1((0, \infty))$ is a nonnegative function. For the convenience , we introduce the notation below. For two nonnegative functions $f$ and $g$ defined on a set $\Omega$, \[ f(x)\approx g(x),\quad x\in \Omega, \] means that there exists some constant $c>0$ such that \[ \frac{f(x)}{c} \leq g(x)\leq cf(x), \quad \text{for all } x\in \Omega. \] Let $\varphi_1$ denote the positive normalized (i.e, $\max_{x\in\Omega}\varphi_1(x)=1$) eigenfunction corresponding to the first positive eigenvalue $\lambda_1$ of the Laplace operator $(-\Delta)$. It is well known (please refer to \cite{Redulescu}) that $\varphi_1\in C^2(\bar{\Omega})$ is a positive function, and we have for $x\in\Omega$, \begin{equation}\label{f1} \varphi_1(x)\approx d(x). \end{equation} When $g(u)=u^{-\gamma}$, $\gamma>1$ and $b$ satisfies the condition that $b(x)\approx (d(x))^{-\mu},\,x\in \Omega$, where $\mu\in(0,2)$. Lazer and Mckenna \cite{LM} showed that \eqref{MI} has a unique solution $u$ satisfying \[ c_2(d(x))^{2/(1+\gamma)}\leq u(x)\leq c_1(d(x))^{(2-\mu)/(1+\gamma)}, \quad \text{for } x\in\Omega, \] where $c_1,\,c_2$ are two positive constants. When $g$ satisfies (G1), (G3) and the conditions \begin{itemize} \item[(G04)] there exist $\gamma>1$ and $c>0$ such that $\lim_{t\to0^{+}}t^{\gamma}g(t)=c$; \item [(G05)] $\int_1^{\infty}g(t)dt<\infty$, \end{itemize} and the weight function $b$ satisfies \begin{itemize} \item[(B04)] there exists $\beta\in(0, 2)$ such that $b(x)\approx (\varphi_1(x))^{-\beta}$, $x\in\Omega$, \end{itemize} Zhang and Cheng \cite{ZhangCheng} obtained the following results: \begin{itemize} \item[(i)] problem \eqref{MI} has a unique solution $u\in C_{\rm loc}^{2,\alpha}(\Omega)\cap C(\bar{\Omega})$ satisfying \[ u(x)\approx\Psi((\varphi_1(x))^{\eta}),\quad x\in\Omega, \] where $\Psi$ is uniquely determined by \eqref{jixx} and $\eta=(2-\beta)/2$; \item[(ii)] $u\in H_0^1(\Omega)$ if and only if \[ \int_{\Omega}\varphi_1^{\beta}g(\Psi(\varphi_1^{\eta})) \Psi(\varphi_1^{\eta})dx<\infty .\] \end{itemize} Moreover, when $g(u)=u^{-\gamma}$, $\gamma>0$, they also obtained some more precise results. specifically, for $b\equiv1$ in $\Omega$, i.e., $\beta=0$, they proved that the above results still hold as the condition (G04) is omitted. Later, applying Karamata regular variation theory, many authors further studied the global estimate of solutions to \eqref{MI} (please refer to \cite{Othman}, \cite{Chemmam}, \cite{GMMT}, \cite{Maagli}). In particular, M\^{a}agli \cite{Maagli} proved the following theorem. \begin{theorem} \label{thmB} If $b$ satisfies {\rm (H1)} and for all $x\in\Omega$, \[ b(x)\approx (d(x))^{-\mu}\tilde{k}(d(x)),\quad \mu\leq2 \] and $\int_0^{l}\frac{\tilde{k}(s)}{s}ds<\infty$, where $\tilde{k}\in C^1((0, l))$ ($l>\max\{\delta_0,\,\operatorname{diam}(\Omega)\}$) is a positive extension of $k\in\mathcal {K}$, i.e., \[ \tilde{k}:=\begin{cases} k(t), &0-1$ has a unique classical solution $u$ satisfying, for $x\in\Omega$, \[ u(x)\approx (d(x))^{\min\{1,(2-\mu)/(1+\gamma)\}}\Psi_{\tilde{k}, \mu,\gamma}(d(x)), \] where \begin{equation}\label{kdv} \Psi_{\tilde{k}, \mu, \gamma}(t):= \begin{cases} \big(\int_0^{t}\frac{\tilde{k}(s)}{s}ds\big)^{1/(1+\gamma)}, &\text{if }\mu=2, \quad {\rm (i)}\\ (\tilde{k}(t))^{1/(1+\gamma)}, &\text{if }1-\gamma<\mu<2, \quad {\rm (ii)}\\ \big(\int_{t}^{l}\frac{\tilde{k}(s)}{s}ds\big)^{1/(1+\gamma)}, &\text{if }\mu=1-\gamma, \quad {\rm (iii)}\\ 1, &\text{if }\mu<1-\gamma.\quad {\rm (iv)} \end{cases} \end{equation} \end{theorem} Recently, Ben Othman and Khamessi \cite{Othman} improved and generalized the above result as follows. Let $\tilde{k}_1, \tilde{k}_2\in C^1((0, l))$ $(l>\max\{\delta_0,\operatorname{diam}(\Omega)\})$ be, respectively, the extensions of $k_1,\, k_2\in\mathcal {K}$ and \[ \int_0^{\delta_0}\frac{k_i(s)}{s}ds<\infty,\quad i=1,2. \] Assume that $b$ satisfies (H1) and the condition \[ (d(x))^{-\mu_2}\tilde{k}_2(d(x))\leq b(x)\leq (d(x))^{-\mu_1}\tilde{k}_1(d(x)),\quad x\in\Omega \] with $\mu_2\leq\mu_1\leq2$, and $g\in C^1((0, \infty))$ is a nonnegative function satisfying \[ c_2u^{-\gamma_2}\leq g(u)\text{ for }00, \] where $\gamma_1\geq \gamma_2>-1$ and $c_1>c_2>0$. Then \eqref{MI} has a unique classical solution $u$ satisfying for each $x\in\Omega$, \begin{align*} &c^{-1}(d(x))^{\min\{1,\,(2-\mu_2)/(1-\gamma_2)\}}\Psi_{\tilde{k}_2,\, \mu_2,\, \gamma_2}(d(x))\\ &\leq u(x) \leq c (d(x))^{\min\{1,\,(2-\mu_1)/(1-\gamma_1)\}}\Psi_{\tilde{k}_1,\, \mu_1,\, \gamma_1}(d(x)), \end{align*} for some constant $c>0$. Inspired by the above works, in this paper we continue to study the existence and asymptotic behavior of the unique classical solution to \eqref{M}. For $q\in[0, 1]$, we first establish a local comparison principle of the unique solution to \eqref{M}, where we omit the usual condition that $g$ is decreasing on $(0, \infty)$ as in \cite{zhangz}. Then we consider the exact asymptotic behavior of the unique solution near the boundary to \eqref{M} and reveal that the nonlinear term $\lambda a(x)|\nabla u|^{q}+\sigma(x)$ does not affect the asymptotic behavior for several kinds of functions $b,\,a$ and $\sigma$. For $q\in [0, 2]$, in view of the ideas of boundary estimate we investigate the existence and global asymptotic behavior of the unique solution to \eqref{M}, and our approach is very different from that one in \cite{Maagli}. In particular, when $\lambda=0$ and $\sigma\equiv0$ in $\Omega$, we improve and extend the results in \cite{Maagli} and \cite{ZM} as follows: \begin{itemize} \item[(I1)] By Theorem \ref{theorem1.2}, we extend the result of Theorem \ref{thmA} from the nonlinearity $g(u)=u^{-\gamma}$ with $\gamma\geq0$ to the case where $g$ is normalized regularly varying at zero with index $-\gamma$, $\gamma=D_{g}/(1-D_{g})\geq0$, $D_{g}<1$; \item[(I2)] By Theorems \ref{theorem121}\,-\ref{theorem1.6}, we extend partial results of Theorem \ref{thmB}, i.e., the nonlinearity $g(u)=u^{-\gamma}$, $\gamma\geq0$ is extended to the case where $g$ is normalized regularly varying at zero with index $-\gamma$, $\gamma=D_{g}/(1-D_{g})\geq0$, $D_{g}<1$. Exactly, we extend expressions (i), (ii) and (iii)-(iv) in \eqref{kdv} by Theorems \ref{theorem1.6}, \ref{theorem121} and \ref{theorem1.5}, respectively. It is worthwhile to point out that in our results, if $b$ satisfies (B3) and \eqref{DD} holds or $b$ satisfies (B4), then $g$ is admitted to be rapidly varying at zero. \end{itemize} Moreover, when $\lambda>0$, $a\equiv1,\,\sigma\equiv0$ in $\Omega$ and $q\in[0, 1)$, we extend the result of \cite[Theorem 1.4]{zhangz} from the case where $b\equiv1$ in $\Omega$ to $b\in C^{\alpha}_{\rm loc}(\Omega)$ for some $\alpha\in(0, 1)$, i.e., we extend the range of $D_{g}$ in (G4) from $D_{g}>1/2$ to $D_{g}\geq0$. To our aim, we assume that $b$ satisfies one of the following conditions: \begin{itemize} \item[(B1)] there exists $k\in\Lambda\cup\mathcal {K}$ such that \begin{equation}\label{CM2} 00,\, i=1,2$ for each $00$, $i=1,2$ for each $00$ such that the unique solution $u_{\lambda}$ of \eqref{M} for each $\lambda\in (-\lambda_0, \lambda_0)$ satisfies \eqref{T1}. \end{itemize} \end{theorem} \begin{theorem}\label{theorem1.2} Let $b,a,\sigma$ satisfy {\rm (H1), (H3), (B1)}, and $g$ satisfy {\rm(G1)--(G2)} with $\liminf_{t\to0^{+}}t^{\gamma}g(t)>0$. Also let {\rm (G4)} and $D_{k}+2D_{g}=2 $ hold, where $\gamma=D_{g}/(1-D_{g}),\, D_{g}<1$. Further assume that \begin{itemize} \item[(H5)] $\int_0^{\delta_0}k^2(s)s^{-\gamma}ds=\infty$; \item[(H6)] \[ \lim_{t\to0^{+}}\Big(g'(t)\int_0^{t}\frac{1}{g(s)}ds+D_{g}\Big) \frac{\int_{t}^{\delta_0}k^2(s)s^{-\gamma}ds}{k^2(t) t^{1-\gamma}}=E\in(-\infty,\,(1-D_{g})^2). \] \end{itemize} Then the following hold: \begin{itemize} \item[(i)] when $q\in[0, 1)$, the unique solution $u_{\lambda}$ of \eqref{M} for each $\lambda\in\mathbb{R}$ satisfies \begin{equation}\label{T2} \begin{split} \xi_2^{1-D_{g}} &\leq\liminf_{d(x)\to0}\frac{u_{\lambda}(x)} {\psi\Big((d(x))^{1+\gamma}\int_{d(x)}^{\delta_1}k^2(s)s^{-\gamma}ds\Big)}\\ &\leq\limsup_{d(x)\to0}\frac{u_{\lambda}(x)} {\psi\Big((d(x))^{1+\gamma}\int_{d(x)}^{\delta_1}k^2(s)s^{-\gamma}ds\Big)} \leq\xi_1^{1-D_{g}}, \end{split} \end{equation} for some $\delta_1>0$, where \[ \xi_i=\frac{b_i}{1-(1-D_{g})^{-2}E},\quad i=1,2; \] \item[(ii)] when $q=1$, there exists $\lambda_0>0$ such that the unique solution $u_{\lambda}$ of \eqref{M} for each $\lambda\in(-\lambda_0, \lambda_0)$ satisfies \eqref{T2}. \end{itemize} \end{theorem} \begin{theorem}\label{theorem1.3} Let $b,a,\sigma$ satisfy{\rm (H1), (H4), (B2)}, $g$ satisfy {\rm (G1)--(G2), (G4)}. Also, $D_{g}<1$ in {\rm (G4)} if $\rho_1=2-q$ in {\rm (H4)}. Then the following hold: \begin{itemize} \item[(i)] when $q\in[0, 1)$, the unique solution $u_{\lambda}$ of \ref{M} for each $\lambda\in\mathbb{R}$ satisfies \begin{equation}\label{T3} b_4^{1-D_{g}}\leq\liminf_{d(x)\to0} \frac{u_{\lambda}(x)}{\psi\Big(\int_0^{d(x)}\frac{k(s)}{s}ds\Big)} \leq\limsup_{d(x)\to0}\frac{u_{\lambda}(x)} {\psi\Big(\int_0^{d(x)}\frac{k(s)}{s}ds\Big)} \leq b_3^{1-D_{g}}; \end{equation} \item[(ii)] when $q=1$, there exists $\lambda_0>0$ such that the unique solution $u_{\lambda}$ of \eqref{M} for each $\lambda\in(-\lambda_0, \lambda_0)$ satisfies \eqref{T3}. \end{itemize} \end{theorem} \begin{remark} \rm Let $k\in \Lambda\cup\mathcal {K}$ and $D_{k}+2D_{g}=2$ hold. Combining with Lemma \ref{Lemma1} (iv) and Proposition \ref{p4}, we know that there exists $k_1\in \mathcal {K}$ such that $k^2(t)=t^{2(1-D_{k})/D_{k}}k_1,\,t\in(0, \delta_0]$. Hence, it follows by Lemma \ref{lemma3} that \[ \lim_{t\to 0^{+}}\frac{\int_{t}^{\delta_0} k^2(s)s^{-\gamma}ds}{k^2(t)t^{1-\gamma}}=\lim_{t\to0^{+}} \frac{\int_{t}^{\delta_0}\frac{k_1(s)}{s}ds}{k_1(t)}=\infty, \] where $\gamma=D_{g}/(1-D_{g}),\, D_{g}<1$. \end{remark} \begin{remark} \rm In Theorem \ref{theorem1.2}, let $\delta_0=1$, $k^2(t)=t^{\gamma-1}(-\ln t)^{\beta}$ and \begin{align*} g(t)&=ct^{-\gamma}\exp\Big(\int_{t}^1\frac{-E(1+\gamma)^2(1+\beta)}{s(-\ln s)}ds\Big)\\ &=c t^{-\gamma}(-\ln t)^{-E(1+\gamma)^2(1+\beta)}, \end{align*} $c>0$, $E\leq 0$, $\beta\geq0$, $t\in(0, 1)$. By \cite[Lemma 3 (iii)]{MI}, we know that (H6) holds. \end{remark} \subsection*{Part 2: Existence and global asymptotic behavior} \begin{theorem}\label{theorem121} Let $b,a,\sigma$ satisfy {\rm (H1)--(H2), (B3)}, $g$ satisfy {\rm (G1), (G3)--(G4)} with $D_{g}+q<3$ and \eqref{DD} hold. \begin{itemize} \item[(i)] If $q\in(0, 1)$, then for each $\lambda\in\mathbb{R}$ problem \eqref{M} has a unique classical solution $u_{\lambda}$ satisfying \begin{equation}\label{Y} u_{\lambda}(x)\approx\psi(\tilde{K}^2(d(x))),\quad x\in\Omega \end{equation} with $ \tilde{K}(t)=\int_0^{t}\tilde{k}(s)ds$, $t\in(0, \infty)$, where $\tilde{k}\in C^1((0, \infty))$ is a positive extension of $k\in C^1((0, \delta_0])$. \item[(ii)] If $q\in[1, 2]$, then there exists $\lambda_0>0$ such that for each $\lambda\in(-\infty, \lambda_0)$ problem \eqref{M} has a unique classical solution $u_{\lambda}$ satisfying \eqref{Y}. \end{itemize} \end{theorem} \begin{theorem}\label{theorem1.5} Let $b,a,\sigma$ satisfy {\rm (H1), (H3), (B3)}, $g$ satisfy {\rm (G1), (G3)--(G4)} and $2/(2+\gamma)\max\{\delta_0, \operatorname{diam}(\Omega)\}$ and $\tilde{k}\in C^1((0, l))$ is a positive extension of $k\in C^1((0, \delta_0])$; \item[(ii)] when $q\in[1, 2]$, there exists $\lambda_0>0$ such that for each $\lambda\in(-\infty, \lambda_0)$ problem \eqref{M} has a unique classical solution $u_{\lambda}$ satisfying \eqref{c1}. \end{itemize} \end{theorem} \begin{theorem}\label{theorem1.6} Let $b,a,\sigma$ satisfy {\rm (H1), (H4), (B4)}, $g$ satisfy {\rm (G1), (G3)-(G4)}. Moreover, $D_{g}<1$ in {\rm (G4)} if\, $\rho_1=2-p$ in (H4). \begin{itemize} \item[(i)] If $q\in(0, 1)$, then for each $\lambda\in\mathbb{R}$, problem \eqref{M} has a unique classical solution $u_{\lambda}$ satisfying \begin{equation}\label{T4} u_{\lambda}(x)\approx \psi\Big(\int_0^{d(x)}\frac{\tilde{k}(s)}{s}ds\Big),\quad x\in\Omega, \end{equation} where $\tilde{k}\in C^1((0, l))$ is a positive extension of $k\in C^1((0, \delta_0])$ and $l>\max\{\delta_0,\operatorname{diam}(\Omega)\}$. \item[(ii)] If $q\in[1, 2]$, then there exists $\lambda_0>0$ such that for each $\lambda\in(-\infty, \lambda_0)$, problem \eqref{M} has a unique classical solution satisfying \eqref{T4}. \end{itemize} \end{theorem} \begin{remark}\label{remark} \rm Assume that $b(x)\approx(d(x))^{-\alpha}k_1(d(x))$, $x\in \Omega,\,\alpha<2$, where \begin{equation*} k_1=\exp\Big(\int_{t}^{l}\frac{y(s)}{s}\Big)ds, \quad y\in C((0,l]),\quad \lim_{t\to0^{+}}y(t)=0,\quad l>\max\{\operatorname{diam} (\Omega),\; \delta_0\}. \end{equation*} Then we can take $k\in \Lambda\cup\mathcal {K}$ such that \[ k^2(t)=t^{-\alpha}\exp\Big(\int_{t}^{\delta_0}\frac{y(s)}{s}ds\Big),\quad t\in(0, \delta_0] \] such that \eqref{f3} holds for each $c\in(0, \min\{\delta_0,1/c_1\})$ and \[ a_1(c)=c_0c^{\alpha}c_1^{|\alpha|}(c_1/c)^{\beta}M_0,\quad a_2(c)=c_0^{-1}c^{\alpha}c_1^{-|\alpha|}(c/c_1)^{\beta}M_0, \] where $\beta=\max_{t\in(0, l]}|y(t)|$, $M_0=\exp\Big(\int_{\delta_0}^{l}\frac{y(s)}{s}ds\Big)$, and $c_0, c_1$ are two large enough constants. \end{remark} \begin{remark} \label{rmk1.4} \rm As in Remark \ref{remark}, If $b(x)\approx (d(x))^{-2}k_1(d(x))$, then we can choose $k\in\mathcal {K}$ such that $\eqref{f4}$ holds. \end{remark} \begin{remark} \label{rmk1.5} \rm For each $k\in C^1((0, \delta_0])$, there exists a positive function $\tilde{k}\in C^1((0, \infty))$ such that $\tilde{k}\equiv k$ on $(0, \delta_0]$, for instance, define \[ \tilde{k}(t):= \begin{cases} k(t), &0 0$, is called \emph{regularly varying at zero} with index $\rho$, written as $g\in \text{RVZ}_{\rho}$, if for each $\xi> 0$ and some $\rho\in\mathbb{R}$, \begin{equation}\label{D} \lim_{t\to 0^{+}}\frac{g(\xi t)}{g(t)}=\xi^{\rho}. \end{equation} In particular, when $\rho=0$, $g$ is called \emph{slowly varying at zero}. \end{definition} Clearly, if $g\in\text{RVZ}_{\rho}$, then $t\mapsto g(t)t^{-\rho}$ is slowly varying at zero. Some basic examples of slowly varying functions at zero are \begin{itemize} \item[(i)] every measurable function on $(0, a_1)$ which has a positive limit at zero; \item[(ii)] $(-\ln t)^{p},\, (\ln(-\ln t))^{p}$, $t\in(0, 1)$, $p\in\mathbb{R}$; \item[(iii)] $\exp\big((-\ln t)^{p}\big)$, $t>0$, $00$, is called \emph{rapidly varying at zero} if for each $p>1$ \[ \lim_{t\to0^{+}}g(t)t^{p}=\infty. \] \end{definition} \begin{proposition}[Uniform convergence theorem] \label{P1} If $g\in RVZ_{\rho}$, then \eqref{D} holds uniformly for $\xi\in [c_1, c_2]$ with $00$. \end{proposition} We call that \begin{equation}\label{H} k(t)= c\exp\Big(\int_{t}^{a_0}\frac{y(\tau)}{\tau}d\tau\Big),\quad t\in (0, a_0], \end{equation} is \emph{normalized} slowly varying at zero and $ g(t)=t^{\rho}k(t),\, t\in (0, a_0] $ is \emph{normalized} regularly varying at zero with index $\rho$ and written $g\in NRVZ_{\rho}$. By the above definition, we know that $\mathcal {K}\subseteq NRVZ_0$. On the other hand, if $k\in NRVZ_0\cap C^1((0, \delta_0])$, then $k\in\mathcal {K}$. Assume that $g$ belongs to $ C^1((0, a_0])$ for some $a_0>0$ and is positive on $(0, a_0]$. Then, $g\in NRVZ_{\rho}$ if and only if \[ \lim_{t\to 0^{+}}\frac{tg^{'}(t)}{g(t)}=\rho. \] \begin{proposition}\label{p3} If functions $k, k_1$ are slowly varying at zero, then \begin{itemize} \item[(i)] $k^{\rho}$ for every $\rho\in\mathbb{R}$, $c_1k+ c_2k_1$ $(c_1 \geq 0, c_2\geq 0$ with $c_1+c_2 >0)$, $k \circ k_1$ if $k_1(t)\to 0 $ as $t\to 0^{+})$, are also slowly varying at zero; \item[(ii)] for every $\rho > 0$ and $t\to 0^{+}$, $ t^{\rho}k(t)\to{0}$, $t^{-\rho}k(t)\to{\infty}$; \item[(iii)] for $\rho\in\mathbb{R}$ and $t\to 0^{+}$, $\ln k(t)/\ln t \to 0$ and $\ln(t^{\rho}k(t))/\ln t\to \rho$. \end{itemize} \end{proposition} \begin{proposition}\label{p4} If $g_1\in RVZ_{\rho_1}$, $g_2\in RVZ_{\rho_2}$ with $\lim_{t\to 0^{+}}g_2(t)=0$, then $g_1\circ g_2\in RVZ_{\rho_1\rho_2}$. \end{proposition} \begin{proposition}\label{p5.0} If $g_1\in RVZ_{\rho_1},\, g_2\in RVZ_{\rho_2}$, then $g_1\cdot g_2\in RVZ_{\rho_1+\rho_2}$. \end{proposition} \begin{proposition}[Asymptotic Behavior] \label{P5} If a function $k$ is slowly varying at zero, then for $a > 0$ and $t\to 0^{+}$, \begin{itemize} \item[(i)] $\int_0^{t}s^{\rho}k(s)ds\cong (1+\rho)^{-1}t^{1+\rho}k(t)$ for $\rho> -1$; \item[(ii)] $\int_{t}^{a}s^{\rho}k(s)ds\cong -(1+\rho)^{-1}t^{1+\rho}k(t)$ for $ \rho < -1$. \end{itemize} \end{proposition} \section{Auxiliary results} In this section, we collect some useful results. \begin{lemma}\label{Lemma1} Let $k\in\Lambda\cup\mathcal {K}$. Then \begin{itemize} \item[(i)] $\lim_{t\to 0^{+}}\frac{K(t)}{k(t)}=0$; \item[(ii)] $\lim_{t\to 0^{+}}\frac{tk(t)}{K(t)}=D_{k}^{-1}$, i.e., $K\in NRVZ_{D_{k}^{-1}}$; \item[(iii)] $\lim_{t\to 0^{+}}\frac{K(t)k'(t)}{k^2(t)}=1-D_{k}$; \item[(iv)] $\lim_{t\to 0^{+}}\frac{tk'(t)}{k(t)}=\frac{1-D_{k}}{D_{k}}$. \end{itemize} \end{lemma} \begin{proof} Here, we only prove the results in the case of $k\in\mathcal {K}$ because the ones have been given by Lemma $2.1$ in \cite{Zhang5} when $k\in\Lambda$. (i)-(iii) By Proposition \ref{P5}(i), we obtain that (i)-(iii) hold. (iv) (iv) follows by (ii)-(iii). \end{proof} \begin{lemma}[{\cite[Lemma 2.2]{ZhangL}}] \label{Lemma2} Let $g$ satisfy {\rm (G1)-(G2)}, \begin{itemize} \item[(i)] if $g$ satisfies (G4), then $\lim_{t\to 0^{+}}\frac{g(t)}{t}\int_0^{t}\frac{ds}{g(s)}=1-D_{g}$ and $D_{g}\leq1$; \item[(ii)] (G4) holds with $D_{g}\in[0, 1)$ if and only if $g\in NRVZ_{-D_{g}/(1-D_{g})}$; \item[(iii)] if (G4) holds with $D_{g}=1$, then $g$ is rapidly varying at zero. \end{itemize} \end{lemma} \begin{lemma}[{\cite[lemma 2.3]{ZM}}] \label{lemma3} Let $k\in\mathcal{K}$, then \[ \lim_{t\to 0^{+}}\frac{k(t)}{\int_{t}^{\delta_0}\frac{k(s)}{s}ds}=0. \] If further $\int_0^{\delta_0}\frac{k(s)}{s}ds$ converges, then we have \[ \lim_{t\to 0^{+}}\frac{k(t)}{\int_0^{t}\frac{k(s)}{s}ds}=0. \] \end{lemma} \begin{lemma}\label{Lemma5} Suppose $g$ satisfies {\rm (G1)--(G2), (G4)} and let $\psi$ be the solution of \eqref{PS}. Then \begin{itemize} \item[(i)] $\psi'(t)=g(\psi(t)),\,\psi(t)>0$, $\psi(0)=0$ and $\psi''(t)=g(\psi(t))g'(\psi(t))$, $t>0$; \item[(ii)] $\lim_{t\to 0^{+}}\frac{t\psi'(t)}{\psi(t)}=1-D_{g}$; \item[(iii)] $\lim_{t\to 0^{+}}\frac{t\psi''(t)}{\psi'(t)}=-D_{g}$; \item[(iv)] if $k\in\Lambda\cup\mathcal {K}$ and $D_{k}(1+\gamma)\leq2$, $\gamma=D_{g}/(1-D_{g}),\,D_{g}<1$, then \[ \lim_{t\to0^{+}}\frac{t^{1-\gamma}k^2(t)}{\int_{t}^{\delta_0 }\frac{k(s)}{s}ds}=0; \] \item[(v)] if $k\in\Lambda\cup\mathcal {K}$ and \eqref{DD} holds, then \begin{gather*} \lim_{t\to 0^{+}}t^{-\rho_1}\hat{k}(t)(\psi'(K^2(t)))^{q-1}K^{q}(t)k^{q-2}(t)=0;\\ \lim_{t\to0^{+}}t^{-\rho_2}\hat{k}(t)\big(\psi'(K^2(t))k^2(t)\big)^{-1}=0, \end{gather*} where $q\in[0, 2]$, $D_{g}+q<3$, $\hat{k}\in\mathcal {K}$ and $\rho_1,\rho_2$ are as defined in {\rm(H2)}; \item[(vi)] if $k\in\Lambda\cup\mathcal {K}$ and $2/(2+\gamma)0,\quad \tau_2=\frac{2D_{k}+2D_{g}-2}{D_{k}}>0. \] Thus, there exist $k_1,k_2\in\mathcal {K}$ such that \begin{gather*} (\psi'\circ K^2(t))^{q-1}\cdot K^{q}(t)\cdot k^{q-2}(t)=t^{\tau_1}k_1(t),\,t\in(0, \delta_0], \\ (\psi'\circ K^2(t))^{-1}k^{-2}(t)=t^{\tau_2}k_2(t),\quad t\in(0, \delta_0]. \end{gather*} It follows by Proposition \ref{p3} (ii) that \begin{gather*}%\label{L1} \lim_{t\to0^{+}}t^{-\rho_1}\hat{k}(t)(\psi'\circ K^2(t))^{q-1}\cdot K^{q}(t)\cdot k^{q-2}(t)=\lim_{t\to0^{+}}t^{\tau_1-\rho_1}\hat{k}(t)k_1(t)=0;\\ \lim_{t\to0^{+}}t^{-\rho_2}\hat{k}(t)(\psi'(K^2(t))k^2(t))^{-1} =\lim_{t\to0^{+}}t^{\tau_2-\rho_2}\hat{k}(t)k_2(t)=0. \end{gather*} (vi) By Lemma \ref{Lemma1} (iv) and Proposition \ref{p4}, we know that \eqref{px} holds. Hence, there exists $k_0\in\mathcal {K}$ such that \eqref{puj} holds here. As before, when $D_{k}=2/(1+\gamma)$, we have $(2(1-D_{k})/D_{k})-\gamma=-1$. So, it follows by Lemma \ref{lemma3} that \begin{equation}\label{zbf} \lim_{t\to0^{+}}\frac{t\big(\int_{t}^{\delta_0}\frac{k_0(s)}{s}ds\big)'} {\int_{t}^{\delta_0}\frac{k_0(s)}{s}ds} =-\lim_{t\to0^{+}}\frac{k_0(t)}{\int_{t}^{\delta_0}\frac{k_0(s)}{s}ds}=0, \end{equation} This implies \[ \int_{t}^{\delta_0}k^2(s)s^{-\gamma}ds\in\mathcal {K}. \] Moreover, when $2/(2+\gamma)0,\quad \tau_2=\frac{q(2-D_{k}(1+\gamma))+D_{k}(\gamma+2)-2}{D_{k}}>0. \] Hence, (vi) follows by Proposition \ref{p3} (ii). (vii) By (iii), we see that there exists $k_1\in\mathcal {K}$ such that \[ k(t)\psi'\Big(\int^{t}_0\frac{k(s)}{s}ds\Big) = \frac{k(t)}{\int_0^{t}\frac{k(s)}{s}ds} \Big(\int_0^{t}\frac{k(s)}{s}ds\Big)^{1-D_{g}}k_1 \Big(\int_0^{t}\frac{k(s)}{s}ds\Big), \] where \begin{equation}\label{kq} \int_0^{t}\frac{k(s)}{s}ds\in\mathcal {K}, \end{equation} which can be obtained by a simple calculation as for \eqref{zbf}. If $\rho_1<2-q$, then by Proposition \ref{p3} (ii) we have \begin{equation}\label{cs} \lim_{t\to0^{+}}t^{-\rho_1}\hat{k}_1(t)t^{2-q}k^{q-1}(t) \Big[\psi'\Big(\int^{t}_0\frac{k(s)}{s}ds\Big)\Big]^{q-1}=0. \end{equation} If $\rho_1=2-q$, then it follows by Lemma \ref{lemma3} and Proposition \ref{p3} (ii) that \eqref{cs} holds. On the other hand, we conclude by Proposition \ref{p3} (ii) that \[ \lim_{t\to0^{+}}t^{2-\rho_2}\hat{k}_2(t)\Big[k(t)\psi' \Big(\int_0^{t}\frac{k(s)}{s}ds\Big)\Big]^{-1}=0. \] \end{proof} \section{Boundary asymptotic behavior} In this section, we prove Theorems \ref{Theorem0}-\ref{theorem1.3}. First, we introduce some notations and two significant lemmas, which are necessary for the proofs. For $\delta>0$, we define $\Omega_{\delta}=\{x\in\Omega: d(x)<\delta\}$. Since $\Omega$ is a $C^2$ - smooth domain, we take $\delta_1\in (0, \delta_0]$ such that \begin{equation}\label{bian} d\in C^2(\Omega_{\delta_1}),\, \,|\nabla d(x)|=1,\quad \Delta d(x)=-(N-1)H(\bar{x})+o(1),\quad x\in\Omega_{\delta_1}, \end{equation} where, for all $x\in\Omega$ near the boundary of $\Omega$, $\bar{x}\in\partial\Omega$ is the nearest point to $x$, and $H(\bar{x})$ denotes the mean curvature of $\partial\Omega$ at $\bar{x}$ (please refer to \cite[Lemmas 14.6 and 14.7]{GT}). For $a$ satisfies (H1), let $Va\in C^{2,\alpha}_{\rm loc}(\Omega)\cap C(\bar{\Omega})$ be the unique solution to the following problem \[ -\Delta v=a(x),\,v>0,\,v|_{\partial\Omega}=0. \] specifically, if $a\equiv1$ in $\Omega$, then $V1\in C^{2,\alpha}_{\rm loc}(\Omega)\cap C^1(\bar{\Omega})$. It follows by H\"{o}pf's maximum principle in \cite{GT} that \begin{equation*}\label{hopf} \nabla V1(x)\neq0,\,\forall x\in\partial\Omega \text{ and } V1(x)\approx d(x), \forall x\in\Omega. \end{equation*} \begin{lemma}\label{lemma3.1} For fixed $\lambda\in\mathbb{R}$, let $g$ satisfy {\rm (G1)-(G2)}, $b,a$ and $\sigma$ satisfy (H1) and $q\in[0, 1]$. Let $u_{\lambda}\in C^2(\Omega_{\delta})\cap C(\bar{\Omega}_{\delta})$ be a unique solution to \eqref{M}, $\overline{u}_{\lambda}\in C^2(\Omega_{\delta})\cap C(\bar{\Omega}_{\delta})$ satisfy \[ -\Delta \overline{u}_{\lambda}\geq b(x)g(\overline{u}_{\lambda})+\lambda a(x)|\nabla \overline{u}_{\lambda}|^{q}+\sigma(x),\, \overline{u}_{\lambda}(x)>0,\,x\in\Omega_{\delta},\, \overline{u}_{\lambda}|_{\partial\Omega}=0, \] and $\underline{u}_{\lambda}\in C^2(\Omega_{\delta})\cap C(\bar{\Omega}_{\delta})$ satisfy \[ -\Delta \underline{u}_{\lambda}\leq b(x)g(\underline{u}_{\lambda})+\lambda a(x)|\nabla \underline{u}_{\lambda}|^{q}+\sigma(x),\, \underline{u}_{\lambda}(x)>0,\,x\in\Omega_{\delta},\, \underline{u}_{\lambda}|_{\partial\Omega}=0, \] where $\delta$ sufficiently small such that $\overline{u}_{\lambda}(x), \underline{u}_{\lambda}(x), u_{\lambda}(x) \in (0, t_1)$, $x\in\Omega_{\delta}$. The constant $t_1\Big(|\lambda|\sup_{x\in\Omega}|\nabla Va(x)|\Big)^{1/(1-q)} \] such that \begin{equation}\label{buchong} \underline{u}_{\lambda}-MVa\leq u_{\lambda}(x)\leq \overline{u}_{\lambda}+MVa \quad \text{on } \{x\in\Omega: d(x)=\delta\}. \end{equation} We assert that for all $x\in\Omega_{\delta}$ \begin{gather}\label{U1} u_{\lambda}(x)\leq \overline{u}_{\lambda}(x)+MVa(x),\\ \label{U2} u_{\lambda}(x)\geq \underline{u}_{\lambda}(x)-MVa(x). \end{gather} Assume the contrary, there exists $x_0\in\Omega_{\delta}$ such that the following hold, \[ u_{\lambda}(x_0)-(\overline{u}_{\lambda}(x_0)+MVa(x_0))>0. \] By the continuity of $u_{\lambda}$ and $\overline{u}_{\lambda}$ on $\Omega_{\delta}$ and $u_{\lambda}(x)=\overline{u}_{\lambda}(x)+MVa(x)=0$, $x\in\partial\Omega$, we see that there exists $x_1\in\Omega_{\delta}$ such that \[ u_{\lambda}(x_1)-(\overline{u}_{\lambda}(x_1)+MVa(x_1)) =\max_{x\in\Omega_{\delta}}u_{\lambda}(x)-(\overline{u}_{\lambda}(x)+MVa(x))>0. \] At the point $x_1$, by using \cite[Theorem 2.2]{GT} we have \begin{equation}\label{Min} \nabla \overline{u}_{\lambda}-\nabla u_{\lambda}=M\nabla Va\,\text{ and } -\Delta (u_{\lambda}-(\overline{u}_{\lambda}+MVa))\geq 0. \end{equation} By using the backward Minkowski inequality, we obtain \begin{equation}\label{Min1} ||\nabla u_{\lambda}|^{q}-|\nabla \overline{u}_{\lambda}|^{q}|\leq||\nabla u_{\lambda}|-|\nabla \overline{u}_{\lambda}||^{q}. \end{equation} Moreover, combining with \eqref{Min}, \eqref{Min1} and the basic fact \begin{equation*}%\label{Min2} ||\nabla u_{\lambda}|-|\nabla \overline{u}_{\lambda}||^{q}\leq |\nabla u_{\lambda}-\nabla\overline{u}_{\lambda}|^{q}, \end{equation*} we have \begin{equation}\label{Fan} ||\nabla u_{\lambda}|^{q}-|\nabla \overline{u}_{\lambda}|^{q}|\leq M^{q}|\nabla Va|^{q}. \end{equation} Thus, it follows by (H1), (G2) and \eqref{Fan} that \begin{equation*}\label{Min3} \begin{split} &-\Delta(u_{\lambda}-(\overline{u}_{\lambda}+MVa))(x_1)\\ &\leq b(x_1)(g(u_{\lambda}(x_1))-g(\overline{u}_{\lambda}(x_1)))-Ma(x) +|\lambda|a(x)\big||\nabla u_{\lambda}|^{q}-|\nabla \overline{u}_{\lambda}|^{q}\big|\\ &\leq b(x_1)(g(u_{\lambda}(x_1))-g(\overline{u}_{\lambda}(x_1)))-Ma(x) +M^{q}|\lambda|a(x)|\nabla Va|^{q} <0, \end{split} \end{equation*} which is a contradiction. Hence, \eqref{U1} holds. In the same way, we can show that \eqref{U2} holds. \smallskip (I2) When $q=1$, we can still choose a large $M>0$ such that \eqref{buchong} holds. By the same proof as the above, we obtain that \eqref{U1} and \eqref{U2} hold in the case of \[ |\lambda|<\lambda_0=\big(\sup_{x\in\Omega}|\nabla Va(x)|\big)^{-1}. \] \end{proof} For the next lemma we assume that $a$ satisfies \begin{equation}\label{leng} a(x)\approx (d(x))^{-\rho}\tilde{k}(d(x)),\quad x\in\Omega, \end{equation} where $\rho\leq2$ and $\tilde{k}\in C^1((0, l])$ ($l>\max\{\delta_0,\,\operatorname{diam}(\Omega)\}$) is a positive extension of $k\in\mathcal {K}$, moreover, if $\rho=2$, then \eqref{youxian} holds. \begin{lemma}[{\cite[Proposition 1]{Maagli}}] \label{lemma3.2} Assume that $a$ satisfies {\rm (H1)} and \eqref{leng}. Then $Va(x)\approx \varphi(d(x))$, $x\in\Omega$, where \begin{equation}\label{fai} \varphi(t):= \begin{cases} \int_0^{t}\frac{\tilde{k}(s)}{s}ds, &\rho=2;\\ t^{2-\rho}\tilde{k}(t), &1<\rho<2;\\ t\int_{t}^{l}\frac{\tilde{k}(s)}{s}ds, &\rho=1;\\ t, &\rho<1. \end{cases} \end{equation} \end{lemma} \begin{proof}[Proof of Theorem \ref{Theorem0}] Let $\varepsilon\in(0, b_2/2)$ and put \[ \tau_1=\xi_1+\varepsilon\xi_1/b_1,\quad \tau_2=\xi_2-\varepsilon\xi_2/b_2. \] We see that \[ \xi_2/{2}<\tau_2<\tau_1<3\xi_1/2. \] Let \[ \overline{u}_{\varepsilon}=\psi(\tau_1K^2(d(x))),\quad \underline{u}_{\varepsilon}=\psi(\tau_2K^2(d(x))). \] A straightforward calculation shows that \begin{align*} &\Delta \overline{u}_{\varepsilon}+b(x)g(\overline{u}_{\varepsilon})+\lambda a(x)|\nabla \overline{u}_{\varepsilon}|^{q}+\sigma(x)\\ &=\psi'(\tau_1K^2(d(x)))k^2(d(x))\Big[4\tau_1 \Big(\tau_1K^2(d(x))g'(\psi(\tau_1K^2(d(x))))+D_{g}\Big)\\ &\quad +2\tau_1\Big(\frac{K (d(x))k'(d(x))}{k^2(d(x))}-(1-D_{k})\Big) +2\tau_1\Big(\frac{K(d(x))}{k(d(x))}\Big)\Delta d(x) \\ &\quad +\Big(\frac{b(x)}{k^2(d(x))}-b_1\Big) -4\tau_1D_{g}+2\tau_1+2\tau_1(1-D_{k})+b_1\\ &\quad +\lambda a(x)(2\tau_1)^{q}(\psi'(\tau_1K^2(d(x))))^{q-1} K^{q}(d(x))k^{q-2}(d(x))\\ &\quad +\sigma(x)\big(\psi'(\tau_1K^2(d(x)))k^2(d(x))\big)^{-1}\Big]. \end{align*} Combining Lemma \ref{Lemma1}, Lemma \ref{Lemma5} (v) with the hypotheses (B1), (H2) and (G4), we obtain that for fixed $\varepsilon>0$, there exists $\delta_{\varepsilon}\in(0, \delta_1)$ such that for $x\in\Omega_{\delta_{\varepsilon}}$, \begin{align*} &\Big|4\tau_1\Big(\tau_1K^2(d(x))g'(\psi(\tau_1K^2(d(x))))+D_{g}\Big)\\ &+2\tau_1\Big(\frac{K(d(x))k'(d(x))}{k^2(d(x))}-(1-D_{k})\Big) +2\tau_1\Big(\frac{K(d(x))}{k(d(x))}\Big)\Delta d(x)\\ &+\Big(\frac{\lambda a(x)}{(d(x))^{-\rho_1}\hat{k}_1(d(x))}\Big) (d(x))^{-\rho_1}\hat{k}_1(d(x)) (2\tau_1)^{q}(\psi'(\tau_1K^2(d(x))))^{q-1}K^{q}(d(x))\\ &\times k^{q-2}(d(x)) +\Big(\frac{\sigma(x)}{(d(x))^{-\rho_2}\hat{k}_2(d(x))}\Big) (d(x))^{-\rho_2}\hat{k}_2(d(x))\\ &\times \left(\psi'(\tau_1K^2(d(x)))k^2(d(x))\right)^{-1}\Big|\\ &<\varepsilon/2 \end{align*} and \[ k^2(d(x))(b_2-\varepsilon/2)0$ such that \begin{equation}\label{soc} \underline{u}_{\varepsilon}(x)-MVa(x)\leq u_{\lambda}(x)\leq \overline{u}_{\varepsilon}(x)+MVa(x),\quad x\in \Omega_{\delta}, \end{equation} i.e., for any $x\in\Omega_{\delta}$ \begin{equation}\label{sls} \begin{gathered} 1+\frac{MVa(x)}{\psi(\tau_1K^2(d(x)))}\geq \frac{u_{\lambda}(x)}{\psi(\tau_1K^2(d(x)))}\\ 1-\frac{MVa(x)}{\psi(\tau_2K^2(d(x)))}\leq \frac{u_{\lambda}(x)}{\psi(\tau_2K^2(d(x)))} . \end{gathered} \end{equation} Subsequently, we prove \begin{equation}\label{yu} \lim_{d(x)\to0}\frac{MVa(x)}{\psi(\tau_iK^2(d(x)))}=0,\quad i=1,2. \end{equation} In fact, by (H2) we can take a constant $c_1>0$ such that \begin{equation}\label{ska} a(x)\frac{q(D_{k}+2D_{g}-2)+2(1-D_{g})}{D_{k}}, \end{equation} where $\tilde{\hat{k}}_1\in C^1((0,l))\,(l>\max\{\delta_0,\, \operatorname{diam}(\Omega)\})$ is a positive extension of $\hat{k}_1$. A basic fact, \cite[Theorem 3.1]{GT}, shows that $Va(x)\leq Vw(x)$, $x\in\Omega$. We conclude by Lemma \ref{lemma3.2} that there exists a constant $c_2>0$ such that \begin{equation}\label{cp} Va(x)\leq c_2\varphi(d(x)),\quad x\in\Omega, \end{equation} where $\varphi$ is defined by \eqref{fai}. Combining Lemma \ref{Lemma1} (ii), Lemma \ref{Lemma5} (ii) with Propositions \ref{p4} and \ref{p5.0}, we obtain \[ \varphi\cdot(\psi\circ \tau_iK^2)^{-1}\in NRVZ_{\rho},\,i=1,2 \text{ with } \rho=\min\{2-\rho_1,\,1\}-2(1-D_{g})/D_{k}. \] It follows by \eqref{DD} and \eqref{nis} that $\rho>0$. Hence, we have \[ \lim_{d(x)\to0}\frac{c_2\varphi(x)}{\psi(\tau_iK^2(d(x)))}=0. \] This fact, combined with \eqref{cp}, shows that \eqref{yu} holds. It follows by \eqref{sls} that \begin{equation}\label{gbx} \begin{split} \limsup_{d(x)\to 0}\frac{u_{\lambda}(x)}{\psi(\tau_1K^2(d(x)))}\leq 1\quad \text{and}\quad \liminf_{d(x)\to 0} \frac{u_{\lambda}(x)}{\psi(\tau_2K^2(d(x)))}\geq1. \end{split} \end{equation} Consequently, by Lemma \ref{Lemma5} (ii), we deduce that \begin{equation}\label{hua} \begin{gathered} \tau_1^{1-D_{g}}=\lim_{d(x)\to0}\frac{\psi(\tau_1(K^2(d(x))))}{\psi((K^2(d(x))))} \geq \limsup_{d(x)\to0}\frac{u_{\lambda}(x)}{\psi(K^2(d(x)))};\\ \tau_2^{1-D_{g}}=\lim_{d(x)\to0}\frac{\psi(\tau_2(K^2(d(x))))}{\psi((K^2(d(x))))} \leq \liminf_{d(x)\to0}\frac{u_{\lambda}(x)}{\psi(K^2(d(x)))}. \end{gathered} \end{equation} \noindent \textbf{Case 2: $p=1$.} By Lemma \ref{lemma3.1} (I2), we know that there exist positive constants $M,\lambda_0$ such that \eqref{soc} holds here if $\lambda\in(-\lambda_0, \lambda_0)$. As in the proof of the above, we obtain \eqref{hua} still holds. The proof is complete when passing to the limit as $\varepsilon\to0$. \end{proof} \begin{proof}[Proof of Theorem \ref{theorem1.2}] Let $\varepsilon\in(0, b_2/2)$ and put \[ \tau_1=\xi_1+\varepsilon\xi_1/b_1,\quad \tau_2=\xi_2-\varepsilon\xi_2/b_2. \] Clearly, $\xi_2/{2}<\tau_2<\tau_1<3\xi_1/2$. Let \[ \overline{u}_{\varepsilon}(x)=\psi\left(\tau_1(d(x))^{1+\gamma}\theta(x)\right),\quad \underline{u}_{\varepsilon}(x)=\psi\left(\tau_2(d(x))^{1+\gamma}\theta(x)\right), \] where $\gamma=D_{g}/(1-D_{g})$ and $\theta(x)=\int_{d(x)}^{\delta}k^2(s)s^{-\gamma}ds$. Denote \begin{align*} &I(x)\\ &=\tau_1(1+\gamma)^2\Big(\frac{\tau_1(d(x))^{1+\gamma} \theta(x)\psi''\big(\tau_1(d(x))^{1+\gamma}\theta(x)\big)} {\psi'\left(\tau_1(d(x))^{1+\gamma}\theta(x)\right)} +D_{g}\Big)\\ &\quad\times \frac{(d(x))^{\gamma-1}\theta(x)}{k^2(d(x))} -\tau_1(1+\gamma)^2E\\ &\quad +\frac{\tau_1(d(x))^{1+\gamma} \theta(x)\psi''\left(\tau_1(d(x))^{1+\gamma}\theta(x)\right)} {\psi'\left(\tau_1(d(x))^{1+\gamma}\theta(x)\right)} \frac{\tau_1(d(x))^{1-\gamma}k^2(d(x))}{\theta(x)}\\ &\quad -\Big(\frac{2(1+\gamma)\tau_1^2(d(x))^{1+\gamma}\theta(x) \psi''\left(\tau_1(d(x))^{1+\gamma}\theta(x)\right)} {\psi'\left(\tau_1(d(x))^{1+\gamma}\theta(x)\right)} +2\tau_1\gamma\Big)\\ &\quad -\Big(\frac{2\tau_1d(x)k'(d(x))}{k(d(x))} -\frac{2\tau_1(1-D_{k})}{D_{k}}\Big) +\Big(\frac{\tau_1(1+\gamma)(d(x))^{\gamma}\theta(x)}{k^2(d(x))} -\tau_1d(x)\Big)\Delta d(x)\\ &\quad +\Big(\frac{\lambda\tau_1^{q} a(x)}{(d(x))^{-\rho_1}\hat{k}_1(d(x))}\Big) (d(x))^{-\rho_1}\hat{k}_1(d(x)) (k^2(d(x)))^{q-1}\\ &\quad\times \left|\psi'\left(\tau_1(d(x))^{1+\gamma}\theta(x)\right)\right|^{q-1} \Big|\frac{(1+\gamma)(d(x))^{\gamma}\theta(x)}{k^2(d(x))}-d(x)\Big|^{q}\\ &\quad +\Big(\frac{\sigma(x)}{(d(x))^{-\rho_2}\hat{k}_2(d(x))}\Big) (d(x))^{-\rho_2}\hat{k}_2(d(x)) \big(\psi'\left(\tau_1(d(x))^{1+\gamma}\theta(x)\right)k^2(d(x))\big)^{-1}. \end{align*} Combining Lemma \ref{Lemma1} (iv) and Lemma \ref{Lemma5} (iii)-(iv) (vi) with the hypotheses (B1), (H3), (H6) and (G4), we obtain that for fixed $\varepsilon>0$, there exists $\delta_{\varepsilon}\in(0, \delta_1)$ such that $|I(x)|<\varepsilon/2$ for $x\in\Omega_{\delta_{\varepsilon}}$, and \[ k^2(d(x))(b_2-\varepsilon/2)0$ such that \eqref{soc} holds here, i.e., for any $x\in\Omega_{\delta}$ \begin{equation}\label{sld} \begin{gathered} 1+\frac{MVa(x)}{\psi(\tau_1(d(x))^{1+\gamma}\theta(x))} \geq\frac{u_{\lambda}(x)}{\psi(\tau_1(d(x))^{1+\gamma}\theta(x))};\\ 1-\frac{MVa(x)}{\psi(\tau_2(d(x))^{1+\gamma}\theta(x))} \leq\frac{u_{\lambda}(x)}{\psi(\tau_2(d(x))^{1+\gamma}\theta(x))}. \end{gathered} \end{equation} Subsequently, we prove \begin{equation}\label{xmg} \lim_{d(x)\to0}\frac{MVa(x)}{\psi\left(\tau_i(d(x))^{1+\gamma}\theta(x)\right)}=0, \quad i=1,2. \end{equation} As before, by (H3), we can take a constant $c_1>0$ such that \eqref{ska} holds here with $\rho_1<1$. On the other hand, we can also take a constant $c_2>0$ such that \eqref{cp} holds here. By Lemma \ref{Lemma2} (ii) and the hypotheses on $g$, we know that there exists $k_1\in\mathcal {K}$ such that \[ g(t)=t^{-\gamma}k_1(t),\,t\in(0, \delta_0] \text{ and }\liminf_{t\to0^{+}}k_1(t)>0. \] Therefore, by \eqref{PS} and Proposition \ref{P5} (i), as $t\to0^{+}$, we obtain \begin{equation*}\label{jini} \psi(t)\cong((1+\gamma)tk_1(\psi(t)))^{1/(1+\gamma)}. \end{equation*} This fact, combined with Lemma \ref{lemma3.2} and \eqref{cp}, shows that \eqref{xmg} holds. Combining with \eqref{sld}, we have \begin{gather*} \limsup_{d(x)\to0}\frac{u_{\lambda}(x)}{\psi(\tau_1(d(x))^{1+\gamma}\theta(x))}\leq 1\quad \text{and}\quad \liminf_{d(x)\to0}\frac{u_{\lambda}(x)}{\psi(\tau_2(d(x))^{1+\gamma}\theta(x))}\geq 1. \end{gather*} Consequently, by Lemma \ref{Lemma5} (ii), we deduce that \begin{equation}\label{wyx} \begin{gathered} \tau_1^{1-D_{g}}=\lim_{d(x)\to0}\frac{\psi(\tau_1(d(x))^{1+\gamma}\theta(x))}{\psi((d(x))^{1+\gamma}\theta(x))}\geq \limsup_{d(x)\to0}\frac{u_{\lambda}(x)}{\psi((d(x))^{1+\gamma}\theta(x))};\\ \tau_2^{1-D_{g}}=\lim_{d(x)\to0}\frac{\psi(\tau_2(d(x))^{1+\gamma}\theta(x))}{\psi((d(x))^{1+\gamma}\theta(x))}\leq \liminf_{d(x)\to0}\frac{u_{\lambda}(x)}{\psi((d(x))^{1+\gamma}\theta(x))}. \end{gathered} \end{equation} \noindent\textbf{Case 2: $p=1$.} By Lemma \ref{lemma3.1} (I2), we know that there exist positive constants $M,\,\lambda_0$ such that \eqref{soc} holds here if $\lambda\in(-\lambda_0, \lambda_0)$. As in the proof of the above, we obtain that \eqref{wyx} still holds. The proof is complete, when passing to the limit $\varepsilon\to0$. \end{proof} \noindent\textbf{Proof of Theorem \ref{theorem1.3}.} Let $\varepsilon\in (0, b_4/2)$ and put \[ \tau_1=b_3+\varepsilon,\,\,\tau_2=b_4-\varepsilon. \] We see that \[ b_4/2<\tau_2<\tau_1<3b_3/2. \] Let \[ \overline{u}_{\varepsilon}(x)=\psi\Big(\tau_1\int_0^{d(x)}\frac{k(s)}{s}ds\Big), \quad \underline{u}_{\varepsilon}(x) =\psi\Big(\tau_2\int_0^{d(x)}\frac{k(s)}{s}ds\Big). \] By using Lemma \ref{Lemma1} (iv), Lemma \ref{lemma3} and Lemma \ref{Lemma5} (vii), combining with the hypotheses (B2), (H4) and (G4) we obtain that for fixed $\varepsilon>0$, there exists $\delta_{\varepsilon}\in (0, \delta)$ such that for $x\in\Omega_{\delta_{\varepsilon}}$ \begin{align*} &\Big|\tau_1^2k(d(x))g'(\overline{u}_{\varepsilon}) +\tau_1\Big(\frac{d(x)k'(d(x))}{k(d(x))}+d(x)\Delta d(x)\Big) +\Big(\frac{\lambda a(x)}{(d(x))^{-\rho_1}\hat{k}_1(d(x))}\Big) \\ &\times (d(x))^{-\rho_1}\hat{k}_1(d(x))\tau_1^{q}(d(x))^{2-q}(k(d(x)))^{q-1} \Big[\psi'\Big(\tau_1\int_0^{d(x)}\frac{k(s)}{s}ds\Big)\Big]^{q-1} |\nabla d(x)|^{q}\\ &+\Big(\frac{\sigma(x)}{(d(x))^{-\rho_2}\hat{k}_2(d(x))} \Big)(d(x))^{2-\rho_2}\hat{k}_2(d(x))\Big[k(d(x)) \psi'\Big(\tau_1\int_0^{d(x)}\frac{k(s)}{s}\Big)\Big]^{-1}\Big|\\ &<\varepsilon/2 \end{align*} and \[ (d(x))^{-2}k(d(x))(b_4-\varepsilon/2)0$ such that \eqref{soc} holds here, i.e., for any $x\in\Omega_{\delta}$, \begin{gather*} 1+\frac{MVa(x)}{\psi\big(\tau_1\int_0^{d(x)}\frac{k(s)}{s}ds\big)}\geq \frac{u_{\lambda}(x)}{\psi\big(\tau_1\int_0^{d(x)}\frac{k(s)}{s}ds\big)}, \\ 1-\frac{MVa(x)}{\psi\big(\tau_2\int_0^{d(x)}\frac{k(s)}{s}ds\big)}\leq \frac{u_{\lambda}(x)}{\psi\big(\tau_2\int_0^{d(x)}\frac{k(s)}{s}ds\big)}. \end{gather*} Subsequently, we prove \begin{equation}\label{xxi} \lim_{d(x)\to0}\frac{MVa(x)}{\psi\big(\tau_i\int_0^{d(x)}\frac{k(s)}{s}ds\big)}=0, \quad i=1,2. \end{equation} By \eqref{kq}, Lemma \ref{Lemma5} (ii) and Proposition \ref{p5.0}, we can see that \[ \psi\Big(\tau_i\int_0^{d(x)}\frac{k(s)}{s}ds\Big)\in\mathcal{K},\quad i=1,2. \] It follows by (H4) and Lemma \ref{lemma3.2} that \eqref{xxi} holds. Hence, we have \[ \limsup_{d(x)\to0}\frac{u_{\lambda}(x)} {\psi\big(\tau_1\int_0^{d(x)}\frac{k(s)}{s}ds\big)}\leq1\,\quad \liminf_{d(x)\to0}\frac{u_{\lambda}(x)}{\psi \big(\tau_2\int_0^{d(x)}\frac{k(s)}{s}ds\big)}\geq1. \] Consequently, by Lemma \ref{Lemma5} (ii), we deduce that \begin{equation}\label{tsl} \begin{gathered} \tau_1^{1-D_{g}} =\lim_{d(x)\to0}\frac{\psi\big(\tau_1\int_0^{d(x)} \frac{k(s)}{s}ds\big)} {\psi\big(\int_0^{d(x)}\frac{k(s)}{s}ds\big)} \geq\limsup_{d(x)\to0} \frac{u_{\lambda}(x)} {\psi\big(\int_0^{d(x)}\frac{k(s)}{s}ds\big)}; \\ \tau_2^{1-D_{g}}=\lim_{d(x)\to0} \frac{\psi\big(\tau_2\int_0^{d(x)}\frac{k(s)}{s}ds\big)} {\psi\big(\int_0^{d(x)}\frac{k(s)}{s}ds\big)} \leq\liminf_{d(x)\to0}\frac{u_{\lambda}(x)} {\psi\big(\int_0^{d(x)}\frac{k(s)}{s}ds\big)}. \end{gathered} \end{equation} \noindent\textbf{Case 2: $p=1$.} As in the proofs of Theorems \ref{Theorem0}-\ref{theorem1.2}, there exists a positive constant $\lambda_0$ such that if $\lambda\in(-\lambda_0, \lambda_0)$, then \eqref{tsl} still holds. The proof is complete when passing to the limit $\varepsilon\to0$. \section{Existence and global asymptotic behavior} In this section, we prove Theorems \ref{theorem121}-\ref{theorem1.6}. \begin{proof}[Proof of Theorem \ref{theorem121}] Our proof is done in the following two steps. \noindent\textbf{Step 1 (Existence and global behavior)} For $00$ such that \[ |\nabla\varphi_1|^2\geq\delta_1, \quad \text{in } \Omega \setminus \omega. \] Put \[ M\geq\max\big\{2/(c_0^2\delta_1(D_{k}+2D_{g}-2)),\,1/a_1(c_0)\big\}. \] Combining with (B3) and (G3), we derive that for $x\in\Omega\setminus\omega$, \begin{equation}\label{g1} MI(x)/2\geq b(x)g(Ma_1(c_0)\psi(K^2(c_0\varphi_1(x)))). \end{equation} On the other hand, by (H2), \eqref{f1}, Proposition \ref{P1} and Lemma \ref{Lemma5} (v) we see that \begin{equation}\label{ga} \begin{split} &\lim_{d(x)\to0}\Big[\Big(\frac{a(x)}{(d(x))^{-\rho_1}\hat{k}_1(d(x))}\Big) (d(x))^{-\rho_1}\hat{k}_1(d(x))(\psi'(K^2(c_0\varphi_1(x))))^{q-1}\\ &\quad \times K^{q}(c_0\varphi_1(x))k^{q-2}(c_0\varphi_1(x)) +\Big(\frac{\sigma(x)}{(d(x))^{-\rho_2}\hat{k}_2(d(x))}\Big)(d(x))^{-\rho_2}\\ &\quad\times \hat{k}_2(d(x)) \big(\psi'(K^2(c_0\varphi_1(x)))k^2(c_0\varphi_1(x))\big)^{-1}\Big]=0. \end{split} \end{equation} Hence, there exists $\omega'\Subset\Omega$ satisfying $\omega\Subset\omega'$ and $\operatorname{dist}(\omega', \partial\Omega)<\delta_0$ such that for $x\in\Omega\setminus\omega'$, \begin{equation}\label{g2} \begin{split} &MI(x)/2\\ &=k^2(c_0\varphi_1(x))\psi'(K^2(c_0\varphi_1(x))) \big(a_1(c_0)c_0^2(D_{k}+2D_{g}-2)|\nabla\varphi_1|^2/2\big)\\ &\geq k^2(c_0\varphi_1(x))\psi'(K^2(c_0\varphi_1(x))) \Big[\lambda(2Ma_1(c_0)c_0)^{q}|\nabla\varphi_1(x)|^{q} \Big(\frac{a(x)}{(d(x))^{-\rho_1}\hat{k}_1(d(x))}\Big)\\ &\quad \times(d(x))^{-\rho_1}\hat{k}_1(d(x))(\psi'(K^2(c_0\varphi_1(x))))^{q-1} K^{q}(c_0\varphi_1(x))k^{q-2}(c_0\varphi_1(x))\\ &\quad +\Big(\frac{\sigma(x)}{(d(x))^{-\rho_2}\hat{k}_2(d(x))}\Big) (d(x))^{-\rho_2}\hat{k}_2(d(x))\big(k^2(c_0\varphi_1(x)) \psi'(K^2(c_0\varphi_1(x)))\big)^{-1}\Big]\\ &=\lambda a(x)(2Ma_1(c_0))^{q}Q_2^{q}(c_0,x)|\nabla\varphi_1(x)|^{q}+\sigma(x). \end{split} \end{equation} This together with \eqref{g1} implies that $\overline{u}_{\lambda}$ is a supersolution of \eqref{M} in $\Omega\setminus\omega'$. Now, by taking a suitable constant $M>0$, we prove $\overline{u}_{\lambda}$ is a supersolution of Eq. \eqref{M} in $\omega'$. Define \begin{gather*} m_1:=\sup_{x\in\omega'}b(x)g(\psi(K^2(c_0\varphi_1(x))));\quad m_2:=\inf_{x\in\omega'}a_1(c_0)\varphi_1(x)Q_2(c_0, x);\\ m_3:=\sup_{x\in\omega'}a(x)(a_1(c_0))^{q}Q_2^{q}(c_0, x)|\nabla\varphi_1(x)|^{q};\quad m_4:=\sup_{x\in\omega'}\sigma(x). \end{gather*} Let \[ M\geq\max\big\{m_1/m_2\lambda_1,\,\,1/a_1(c_0)\big\}. \] It follows from the monotonicity of $g$ that for any $ x\in \omega'$ \begin{equation}\label{g3} M\lambda_1a_1(c_0)\varphi_1(x)Q_2(c_0, x)\geq M\lambda_1m_2\geq m_1\geq b(x)g(Ma_1(c_0)\psi(K^2(c_0\varphi_1(x)))). \end{equation} Here, we distinguish the following two cases. \smallskip \noindent\textbf{Case 1: $q\in[0, 1))$.} Let \[ M>\max\big\{\big(2^{q+1}m_3\max\{0, \lambda\}/m_2\lambda_1\big)^{1/(1-q)},\,2m_4/\lambda_1m_2\big\}. \] By a direct calculation, we have for any $x\in \omega'$, \begin{equation}\label{g4y} \begin{split} &M\lambda_1a_1(c_0)\varphi_1(x)Q_2(c_0, x)/2\\ &\geq M\lambda_1m_2/2 \geq m_3(2M)^{q}\max\{0, \lambda\}\\ &\geq \lambda a(x)(2Ma_1(c_0))^{q}Q_2^{q}(c_0, x)|\nabla\varphi_1(x)|^{q} \end{split} \end{equation} and \begin{equation}\label{g4} M\lambda_1a_1(c_0)\varphi_1(x)Q_2(c_0, x)/2\geq M\lambda_1m_2/2\geq m_4\geq\sigma(x). \end{equation} So, we see that $\overline{u}_{\lambda}$ is a supersolution of \eqref{M} in $\omega'$. Finally, combining with \eqref{g1}, \eqref{g2}-\eqref{g4}, we conclude by choosing \begin{align*} M\geq \max\big\{&2/(c_0^2\delta_1(D_{k}+2D_{g}-2)),\, 1/a_1(c_0),\,m_1/(m_2\lambda_1),\\ &\big(2^{q+1}m_3\max\{0, \lambda\})/m_2\lambda_1\big)^{1/(1-q)},\,2m_4/\lambda_1m_2\big\} \end{align*} that $\overline{u}_{\lambda}$ is a supersolution of \eqref{M}. \smallskip \noindent\textbf{Case\ 2: $q\in[1, 2]$.} In this case, let \begin{gather} M\geq\max\big\{2/(c_0^2\delta_1(D_{k}+2D_{g}-2)),\,1/a_1(c_0), \,m_1/(m_2\lambda_1),\,2m_4/\lambda_1m_2\big\},\nonumber \\ \label{fm} \lambda0$ such that $\overline{u}_{\lambda}$ is a supersolution of \eqref{M}. On the other hand, let \[ \underline{u}_{\lambda}:=ma_2(c_0)\psi(K^2(c_0\varphi_1))\quad \text{in }\Omega, \] where $m$ is a positive constant to be determined. Next, by choosing a suitable $m>0$, we prove $\underline{u}_{\lambda}$ is a subsolution of \eqref{M}. Indeed, by \eqref{ga}, we arrive at \[ \sup_{x\in\Omega}a(x)(\psi'(K^2(c_0\varphi_1(x))))^{q-1}K^{q}(c_0\varphi_1(x))k^{q-2}(c_0\varphi_1(x))<\infty. \] Hence, we can take sufficiently small $00$. By Lemma \ref{Lemma1} (ii), Lemma \ref{Lemma5} (ii) and Proposition \ref{p4}, we have \[ \psi\circ \tilde{K}^2\in NRVZ_{2(1-D_{g})/D_{k}}. \] Since $\psi\circ \tilde{K}^2\in C^1((0, \infty))$, we can take a positive constant $\delta<\min\{\delta_0, \operatorname{diam}(\Omega)\}$ and a function $y\in C((0, \delta])$ with $\lim_{t\to0^{+}}y(t)=0$ such that \[ \psi\circ \tilde{K}^2(t)=\psi\circ K^2(t)=\bar{c}\,t^{2(1-D_{g})/D_{k}} \exp\Big(\int_{t}^{\delta}\frac{y(s)}{s}ds\Big),\quad t\in (0, \delta],\;\bar{c}>0. \] On the other hand, by \eqref{f1}, we obtain that there exists $c_1>1$ such that \[ d(x)/c_1\leq\varphi_1(x)\leq c_1d(x),\,x\in\Omega. \] In fact, we can adjust $c_0>0$ such that $c_0<\min\{\delta, 1/c_1\}$. Let $\beta=\max_{t\in(0,\delta]}|y(t)|$. Then we deduce that \[ \big|\exp\Big(\int_{d(x)}^{c_0\varphi_1(x)}\frac{y(s)}{s}ds\Big)\big| \leq\left(c_1/c_0\right)^{\beta},\,x\in\Omega_{\delta}. \] Hence \[ \left(c_0/c_1\right)^{(2(1-D_{g})/D_{k})+\beta}\leq\psi\circ K^2(c_0\varphi_1(x))/\psi\circ \tilde{K}^2(d(x))\leq (c_1c_0)^{2(1-D_{g})/D_{k}}\left(c_1/c_0\right)^{\beta}, \] for $x\in \Omega_{\delta}$. Let \begin{gather*} \begin{aligned} M_1=\max\Big\{&(c_1c_0)^{2(1-D_{g})/D_{k}}\left(c_1/c_0\right)^{\beta},\\ &\sup_{x\in\Omega\setminus\Omega_{\delta}}\psi\circ K^2(c_0\varphi_1(x))/\inf_{x\in\Omega\setminus \Omega_{\delta}}\psi\circ \tilde{K}^2(d(x))\Big\}, \end{aligned}\\ M_2=\min\big\{\left(c_0/c_1\right)^{(2(1-D_{g})/D_{k})+\beta}, \inf_{x\in\Omega\setminus\Omega_{\delta}}\psi\circ K^2(c_0\varphi_1(x))/\sup_{x\in\Omega\setminus \Omega_{\delta}}\psi\circ \tilde{K}^2(d(x))\big\}. \end{gather*} Then we obtain that for $x\in\Omega$, \[ M_2\psi\circ \tilde{K}^2(d(x))\leq\psi\circ K^2(c_0\varphi_1(x))\leq M_1\psi\circ \tilde{K}^2(d(x)), \] i.e., \eqref{Y} holds. \smallskip \noindent\textbf{Step 2 (Uniqueness)} Since uniqueness is an easy consequence of the relationship $v\leq w$ whenever $v\leq w$ on $\partial\Omega$, we prove only this relationship, where $v$ and $w$ are two solutions of \eqref{M} in $\Omega$. Suppose $\min_{x\in\Omega}(w(x)-v(x))<0$, then there exists $x_0\in\Omega$ such that $w(x_0)-v(x_0)=\min_{x\in\Omega}(w(x)-v(x))$. At the point, we have by the basic fact \[ \nabla (w-v)=0 \text{ and } -\Delta (w-v)\leq0. \] On the other hand, we see by (H1) and (G3) that \[ -\Delta (w -u)=b(x_0)(g(w(x_0))-g(v(x_0))))>0, \] which is a contradiction. Hence, $w\geq v$ in $\Omega$. The proof is complete \end{proof} \noindent \textbf{Proof of Theorem \ref{theorem1.5}.} For $00$. Let \[ \overline{u}_{\lambda}:=Ma_1(c_0)\psi((c_0\varphi_1)^{1+\gamma}\theta(c_0, \cdot)) \quad \text{in } \Omega, \] where $M$ is a positive constant to be determined. As before, by choosing a suitable constant $M>0$, we prove $\overline{u}_{\lambda}$ is a supersolution of \eqref{M}. By a straightforward calculation, \[ \begin{split} -\Delta \overline{u}_{\lambda}&=Ma_1(c_0)k^2(c_0\varphi_1)\psi'((c_0\varphi_1)^{1+\gamma}\theta(c_0, \cdot))Q_1(c_0, \cdot)|\nabla\varphi_1|^2+Ma_1(c_0)Q_2(c_0, \cdot)\\ &\geq MI+Ma_1(c_0)Q_2(c_0, \cdot), \end{split} \] where \[ I=(1/2D_{k})(2-D_{k}\gamma-(1+\gamma)^2ED_{k})c_0^2a_1(c_0)k^2(c_0\varphi_1)\psi'((c_0\varphi_1)^{1+\gamma}\theta(c_0, \cdot))|\nabla\varphi_1|^2. \] By Hopf's maximum principle, there exist $\omega\Subset\Omega$ and a constant $\delta_1>0$ such that \[ |\nabla \varphi_1|\geq\delta_1,\text{ in }\Omega\setminus\omega. \] Let \[ M>\max\big\{4D_{k}/((2-D_{k}\gamma-(1+\gamma)^2ED_{k})\delta_1c_0^2),\,1/a_1(c_0)\big\}. \] Combining with (B3) and (G3), for $x\in\Omega\setminus\omega$, we obtain \begin{equation}\label{w1} MI(x)/2\geq b(x)g(Ma_1(c_0)\psi((c_0\varphi_1(x))^{1+\gamma}\theta(c_0,x))). \end{equation} On the other hand, by (H3), \eqref{f1}, Proposition \ref{P1} and Lemma \ref{Lemma5} (vi) we see that \begin{equation}\label{oba} \begin{split} &\lim_{d(x)\to0}\Big[\Big(\frac{a(x)}{(d(x))^{-\rho_1}\hat{k}_1(d(x))}\Big) (d(x))^{-\rho_1}\hat{k}_1(d(x))\Big((k^2(c_0\varphi_1(x)))^{-1}(c_0\varphi_1(x))^{q\gamma}\\ &\times(\psi'((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x)))^{q-1}\theta^{q}(c_0, x)+(k^2(c_0\varphi_1(x)))^{q-1}(c_0\varphi_1(x))^{q}\\ &\times(\psi'((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x)))^{q-1}\Big)+\Big(\frac{\sigma(x)}{(d(x))^{-\rho_2}\hat{k}_2(d(x))}\Big) (d(x))^{-\rho_2}\hat{k}_2(d(x))\\ &\times\big(\psi'((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x))k^2(c_0\varphi_1(x))\big)^{-1}\Big]=0. \end{split} \end{equation} Hence, there exists $\omega'\Subset\Omega$ satisfying $\omega\Subset\omega'$ and $\operatorname{dist}(\omega', \partial\Omega)<\delta_0$ such that for $x\in\Omega\setminus\omega'$, \begin{equation}\label{w2} \begin{split} MI(x)/2 &\geq k^2(c_0\varphi_1(x))\psi'((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x))\Big[\lambda M^{q}(a_1(c_0))^{q}|\nabla\varphi_1(x)|^{q}\\ &\times \Big(\frac{a(x)}{(d(x))^{-\rho_1}\hat{k}_1(d(x))}\Big) (d(x))^{-\rho_1}\hat{k}_1(d(x))\Big((2c_0(1+\gamma))^{q}(k^2(c_0\varphi_1(x)))^{-1}\\ &\times(c_0\varphi_1(x))^{q\gamma} (\psi'((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x)))^{q-1}\theta^{q}(c_0, x) \\ &\quad +(2c_0)^{q}(k^2(c_0\varphi_1(x)))^{q-1}(c_0\varphi_1(x))^{q}\\ &\times(\psi'((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x)))^{q-1}\Big)\\ &+\Big(\frac{\sigma(x)}{(d(x))^{-\rho_2}\hat{k}_2(d(x))} \Big)(d(x))^{-\rho_2}\hat{k}_2(d(x)) \big(k^2(c_0\varphi_1(x))\\ &\quad\times \psi'((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x))\big)^{-1}\Big]\\ &\geq\lambda a(x) M^{q}Q^{q}_3(c_0, x)+\sigma(x). \end{split} \end{equation} This fact, combined with \eqref{w1}, shows that $\overline{u}_{\lambda}$ is a supersolution of Eq. \eqref{M} in $\Omega\setminus\omega'$. Now, by taking a suitable constant $M>0$, we prove $\overline{u}_{\lambda}$ is a supersolution of \eqref{M} in $\omega'$. As before, we define \begin{gather*} m_1:=\sup_{x\in\omega'}b(x)g(\psi((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x)));\quad m_2:=\inf_{x\in\omega'}(a_1(c_0)/2)Q_2(c_0, x);\\ m_3:=\sup_{x\in\omega'}a(x) Q_3^{q}(c_0, x);\quad m_4:=\sup_{x\in\omega'}\sigma(x). \end{gather*} Let $M>\max\big\{m_1/m_2,\,1/a_1(c_0)\big\}$. Using the monotonicity of $g$, we obtain that \begin{equation}\label{w3} \begin{aligned} M(a_1(c_0)/2)Q_2(c_0, x) &\geq Mm_2\geq m_1\\ &\geq b(x)g(Ma_1(c_0)\psi((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x))),\quad x\in\omega'. \end{aligned} \end{equation} Here, we distinguish the following cases. \smallskip \noindent\textbf{Case 1: $q\in[0, 1)$.} Put \[ M>\max\big\{\big(2m_3\max\{0, \lambda\}/m_2\big)^{1/(1-q)},\,2m_4/m_2\big\}. \] We obtain \begin{equation}\label{w4y} M(a_1(c_0)/4)Q_2(c_0, x)\geq Mm_2/2\geq \max\{0, \lambda\} M^{q}m_3\geq \lambda M^{q}a(x)Q_3^{q}(c_0, x), \end{equation} for $x\in \omega'$, and \begin{equation}\label{w4} M(a_1(c_0)/4)Q_2(c_0, x)\geq Mm_2/2\geq m_4\geq\sigma(x),\quad x\in\omega'. \end{equation} Thus, $\overline{u}_{\lambda}$ is a supersolution of \eqref{M} in $\omega'$. Finally, combining with \eqref{w1}, \eqref{w2}-\eqref{w4}, we conclude by choosing \[ \begin{split} M>\max\big\{&4D_{k}/((2-D_{k}\gamma-(1+\gamma)^2ED_{k})\delta_1c_0^2),\,1/a_1(c_0),\,m_1/m_2,\\ &\big(2m_3\max\{0, \lambda\}/m_2\big)^{1/(1-q)},\,2m_4/m_2\big\} \end{split} \] that $\overline{u}_{\lambda}$ is a supersolution of \eqref{M}. \smallskip \noindent\textbf{Case 2: $q\in[1, 2]$.} Put \begin{gather} M>\max\big\{4D_{k}/((2-D_{k}\gamma-(1+\gamma)^2ED_{k})\delta_1c_0^2), \,1/a_1(c_0),\,m_1/m_2,\,2m_4/m_2\big\},\nonumber \\ \label{kee} \lambda<(M^{1-q}m_2)/2m_3. \end{gather} It follows by a direct calculation that \eqref{w4y}-\eqref{w4} still hold. So, for every $\lambda$ satisfying \eqref{kee}, we can take $M>0$ such that $\overline{u}_{\lambda}$ is a supersolution of \eqref{M}. On the other hand, let \[ \underline{u}_{\lambda}:=ma_2(c_0)\psi((c_0\varphi_1)^{1+\gamma}\theta(c_0, \cdot)) \quad \text{in }\Omega, \] where $m$ is a positive constant to be determined. Next, by choosing a suitable $m>0$, we prove that $\underline{u}_{\lambda}$ is a subsolution of \eqref{M}. By \eqref{oba}, we arrive at \begin{align*} &\sup_{x\in\Omega}\big[a(x)(\psi'((c_0\varphi_1(x))^{1+\gamma}\theta(c_0, x)))^{q-1}(k^2(c_0\varphi_1(x)))^{-1}\\ &\times\big|c_0(1+\gamma)(c_0\varphi_1(x))^{\gamma}\theta(c_0, x)-c_0(c_0\varphi_1(x))k^2(c_0\varphi_1(x))\big|^{q}\big]<\infty. \end{align*} Moreover, by Lemma \ref{Lemma5} (vi), we obtain \[ \sup_{x\in\Omega}\big[(c_0\varphi_1(x))^{\gamma}(k^2(c_0\varphi_1(x)))^{-1} \theta(c_0, x)\big]<\infty. \] Using a similar proof as for Theorem \ref{theorem121}, we can take a small enough $00$ satisfying \[ c_0<\delta<\min\big\{\delta_0,\text{ diam}(\Omega)\big\} \] and a function $y\in C((0, \delta])$ with $\lim_{t\to0^{+}}y(t)=0$, such that \[ \psi\Big((c_0\varphi_1(x))^{1+\gamma} \int_{c_0\varphi_1(x)}^{\delta}k^2(s)s^{-\gamma}ds\Big) =\bar{c}\,c_0\varphi_1(x)\int_{c_0\varphi_1(x)}^{\delta} \frac{y(s)}{s}ds,\,x\in\Omega,\,\bar{c}>0. \] Let $\widetilde{k}\in C^1((0, l))$ be a positive extension of $k\in C^1((0, \delta_0])$. As in the proof of Theorem \ref{theorem121}, we can take $M_1>M_2>0$ such that \begin{align*} M_2\psi\Big((d(x))^{1+\gamma}\int_{d(x)}^{l}\tilde{k}(s)s^{-\gamma}ds\Big) &\leq \psi\Big((c_0\varphi_1(x))^{1+\gamma}\theta(c_0,x)\Big)\\ &\leq M_1\psi\Big((d(x))^{1+\gamma}\int_{d(x)}^{l}\tilde{k}(s)s^{-\gamma}ds\Big). \end{align*} The proof is complete. \begin{proof}[Proof of Theorem \ref{theorem1.6}] We note that this proof is essentially the same as the proofs of Theorems \ref{theorem121} and \ref{theorem1.5}, so we only provide an outline. For $00$ such that \[ 1/20$ such that \[ |\nabla\varphi_1|\geq \delta_1,\text{ in }\Omega\setminus\omega. \] Let \[ M>\big\{4/(c_0^2\delta_1),\,1/a_1(c_0)\big\}. \] Combining with (B4) and (G3), we have for any $x\in\Omega\setminus \omega$ \begin{equation}\label{zh} \begin{split} MI(x)/2 &\geq a_1(c_0)(c_0\varphi_1)^{-2}k(c_0\varphi_1(x)) g\Big(\psi\Big(\int_0^{c_0\varphi_1(x)}\frac{k(s)}{s}ds\Big)\Big)\\ &\geq b(x)g\Big(Ma_1(c_0)\psi\Big(\int_0^{c_0\varphi_1(x)} \frac{k(s)}{s}ds\Big)\Big). \end{split} \end{equation} On the other hand, by (H4), \eqref{f1}, Proposition \ref{P1} and Lemma \ref{Lemma5} (vii) we see that \begin{align*} &\lim_{d(x)\to0}\Big[\Big(\frac{a(x)}{(d(x))^{-\rho_1}\hat{k}_1(d(x))}\Big) (d(x))^{-\rho_1}\hat{k}_1(d(x))(c_0\varphi_1(x))^{2-p}\Big(k(c_0\varphi_1(x))\\ &\times\psi'\Big(\int_0^{c_0\varphi_1(x)}\frac{k(s)}{s}ds\Big)\Big)^{q-1} +\Big(\frac{\sigma(x)}{(d(x))^{-\rho_2}\hat{k}_2(d(x))}\Big) (d(x))^{-\rho_2}\hat{k}_2(d(x))(c_0\varphi_1(x))^2\\ &\times\Big(k(c_0\varphi_1(x))\psi' \Big(\int_0^{c_0\varphi_1(x)}\frac{k(s)}{s}ds\Big)\Big)^{-1}\Big]=0. \end{align*} By the same arguments as for Theorems \ref{theorem1.5} and \ref{theorem1.6}, we know that there exists $\omega'\Subset\Omega$ satisfying $\omega\Subset\omega'$ and $\operatorname{dist}(\omega', \partial\Omega)<\delta_0$ such that for $x\in\Omega\setminus\omega'$ \begin{equation}\label{hz} MI(x)/2\geq \lambda a(x) M^{q}Q_3^{q}(c_0, x)+\sigma(x). \end{equation} It follows by \eqref{zh} and \eqref{hz} that $\overline{u}_{\lambda}$ is a supersolution of \eqref{M} in $\Omega\setminus\omega'$. Define \begin{gather*} m_1:=\sup_{x\in\omega'}b(x)g\Big(\psi\Big(\int_0^{c_0\varphi_1(x)} \frac{k(s)}{s}ds\Big)\Big);\quad m_2:=\inf_{x\in\omega'}(a_1(c_0)/2)Q_2(c_0, x);\\ m_3:=\sup_{x\in\omega'}a(x)Q_3^{q}(c_0, x);\quad m_4:=\sup_{x\in\omega'}\sigma(x). \end{gather*} As in the proof of Theorem \ref{theorem1.6}, when $q\in[0, 1)$, we can take \[ M\geq\max\big\{4/(c_0^2\delta_1),\,1/a_1(c_0),\,m_1/m_2,\,\big(2m_3\max\{0, \lambda\}/m_2\big)^{1/(1-q)},\,2m_4/m_2\big\} \] such that $\overline{u}_{\lambda}$ is a supersolution of \eqref{M}. When $q\in[1, 2]$, let \[ M\geq\max\big\{4/(c_0^2\delta_1),\,1/a_1(c_0),\,m_1/m_2,\,2m_4/m_2\big\} \text{ and }\lambda0$, we show that \[ \underline{u}_{\lambda}:=ma_2(c_0)\psi\Big(\int_0^{c_0\varphi_1(x)}\frac{k(s)}{s}ds\Big) \] is a subsolution of \eqref{M} with $q\in(0, 2]$. Hence, by \cite[Lemma 3]{Cui}, problem \eqref{M} possesses a classical solution $u_{\lambda}$ satisfying \[ \underline{u}_{\lambda}\leq u_{\lambda}\leq \overline{u}_{\lambda} \quad \text{in }\Omega, \] i.e., \[ u_{\lambda}(x)\thickapprox \psi\Big(\int_0^{c_0\varphi_1(x)}\frac{k(s)}{s}\Big),\quad x\in\Omega. \] As in the proof of Theorem \ref{theorem121}, we obtain that \eqref{T4} holds. \end{proof} \subsection*{Acknowledgments} The author wishes to thank Dr. Shuibo Huang at Northwest University for Nationalities for his very valuable suggestions and discussions. \begin{thebibliography}{00} \bibitem{Alsaedi} R. Alsaedi, H. M\^{a}agli, N. Zeddini; \emph{Exact behavior of the unique positive solution to some singular elliptic problem in exterior domains}, Nonlinear Anal. (2014), http: //dx.doi.org/10.1016/j.na.2014.09.018. \bibitem{CA} C. Anedda; \emph{Second-order boundary estimates for solutions to singular elliptic equations}, Electron. J. Differential Equations 2009 (90) (2009) 1-15. \bibitem{OMMZ} S. Ben Othman, H. M\^{a}agli, S. Masmoudi, M. Zribi; \emph{Exact asymptotic behavior near the boundary to the solution for singular nonlinear Dirichlet problems}, Nonlinear Anal. 71 (2009) 4137-4150. \bibitem{Othman} S. Ben Othman, B. Khamessi; \emph{Asymptotic behavior of positive solutions of a nonlinear Dirichlet problem}, J. Math. Anal. Appl. 409 (2014) 925-933. \bibitem{BGP} S. Berhanu, F. Gladiali, G. Porru; \emph{Qualitative properties of solutions to elliptic singular problems}, J. Inequal. Appl. 3 (1999) 313-330. \bibitem{BCP} S. Berhanu, F. Cuccu, G. Porru; \emph{On the boundary behaviour, including second order effects, of solutions to singular elliptic problems}, Acta Math. Appl. Sin. Engl. Ser. 23 (2007) 479-486. \bibitem{Chemmam} R. Chemmam, H. M\^{a}agli, S. Masmoudi, M. Zribi; \emph{Combined effects in nonlinear singular elliptic problems in a bounded domain}, Adv. Nonlinear Anal. 1 (2012) 301-318. \bibitem{CMV} F. C\^{\i}rstea, M. Ghergu, V. R\u{a}dulescu; \emph{Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type}, J. Math. Pures Appl. 84 (2005) 493-508. \bibitem{C.1} F. C\^{\i}rstea, V. R\u{a}dulescu; \emph{Uniqueness of the blow-up boundary solution of logistic equations with absorption}, C. R. Acad. Sci. Paris, S\'{e}r. I 335 (2002) 447-452. \bibitem{C.2} F. C\^{\i}rstea, V. R\u{a}dulescu; \emph{Asymptotics for the blow-up boundary solution of the logistic equation with absorption}, C. R. Acad. Sci. Paris, S\'{e}r. I 336 (2003) 231-236. \bibitem{C.3} F. C\^{\i}rstea, V. R\u{a}dulescu; \emph{Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach}, Asymptot. Anal. 46 (2006) 275-298. \bibitem{C.4} F. C\^{i}rstea, V. R\u{a}dulescu; \emph{Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type}, Trans. Amer. Math. Soc. 359 (2007) 3275-3286. \bibitem{CRT} M. G. Crandall, P. H. Rabinowitz, L. Tartar; \emph{On a Dirichlet problem with a singular nonlinearity}, Comm. Partial Differential Equations 2 (1977) 193-222. \bibitem{CGP} F. Cuccu, E. Giarrusso, G. Porru; \emph{Boundary behaviour for solutions of elliptic singular equations with a gradient term}, Nonlinear Anal. 69 (2008) 4550-4566. \bibitem{Cui} S. Cui; \emph{Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems}, Nonlinear Anal. 41 (2000) 149-176. \bibitem{LDMR} L. Dupaigne, M. Ghergu, V. R\u{a}dulescu; \emph{Lane-Emden-Fowler equations with convection and singular potential}, J. Math. Pures Appl. 87 (2007) 563-581. \bibitem{FM} W. Fulks, J. S. Maybee; \emph{A singular non-linear equation}, Osaka J. Math. 12 (1960) 1-19. \bibitem{GR1} M. Ghergu, V. R\u{a}dulescu; \emph{Bifurcation and asymptotics for the Lane-Emden-Fowler equation}, C. R. Acad. Sci. Paris, S\'{e}r. I 337 (2003) 259-264. \bibitem{GR2} M. Ghergu, V. R\u{a}dulescu; \emph{Bifurcation for a class of singular elliptic problems with quadratic convection term}, C. R. Acad. Sci. Paris S\'{e}r. I 338 (2004) 831-836. \bibitem{Gr2} M. Ghergu, V. R\u{a}dulescu; \emph{Multi-parameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with a convection term}, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005) 61-83. \bibitem{GR} M. Ghergu, V. R\u{a}dulescu; \emph{Singular Elliptic Problems: Bifurcation and Asymptotic Analysis}, Oxford University Press, 2008. \bibitem{EGP} E. Giarrusso, G. Porru; \emph{Boundary behaviour of solutions to nonlinear elliptic singular problems}, in: J. C. Misra (Ed.), Appl. Math. in the Golden Age, Narosa Publishing House, New Delhi, India, 2003, pp. 163-178. \bibitem{GP} E. Giarrusso, G. Porru; \emph{Problems for elliptic singular equations with a gradient term}, Nonlinear Anal. 65 (2006) 107-128. \bibitem{GT} D. Gilbarg, N. S. Trudinger; \emph{Elliptic Partial Differential Equations of Second Order}, third ed., Springer-Verlag, Berlin, 1998. \bibitem{GMMT} S. Gontara, H. M\^{a}agli, S. Masmoudi, S. Turki; \emph{Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem}, J. Math. Anal. Appl. 369 (2010) 719-729. \bibitem{GuiLin} C. Gui, F. Lin; \emph{Regularity of an elliptic problem with a singular nonlinearity}, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 1021-1029. \bibitem{Huang1} S. Huang, Q. Tian; \emph{Second order estimates for large solutions of elliptic equations}, Nonlinear Anal. 74 (2011) 2031-2044. \bibitem{Huang2} S. Huang, Q. Tian, S. Zhang, J. Xi; \emph{A second-order estimate for blow-up solutions of elliptic equations}, Nonlinear Anal. 74 (2011) 2342-2350. \bibitem{Lair} A. V. Lair, A. W. Shaker; \emph{Classical and weak solutions of a singular semilinear elliptic problem}, J. Math. Anal. Appl. 211 (1997) 371-385. \bibitem{LM} A. C. Lazer, P. J. McKenna; \emph{On a singular nonlinear elliptic boundary-value problem}, Proc. Amer. Math. Soc. 111 (1991) 721-730. \bibitem{LiZhang2} B. Li, Z. Zhang; \emph{Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection}, Electron. J. Differential Equations 2015 (19) (2015) 1-18. \bibitem{Maagli} H. M\^{a}agli; \emph{Asymptotic behavior of positive solutions of a semilinear Dirichlet problem}, Nonlinear Anal. 74 (2011) 2941-2947. \bibitem{Maric} V. Maric; \emph{Regular Variation and Differential Equations}, Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. \bibitem{MR} P. J. McKenna, W. Reichel; \emph{Sign-changing solutions to singular second-order boundary value problems}, Adv. Differential Equations 6 (2001) 441-460. \bibitem{MI} L. Mi, B. Liu; \emph{The second order estimate for the solution to a singular elliptic boundary value problem}, Appl. Anal. Discrete Math. 6 (2012) 194-213. \bibitem{Mi2} L. Mi, B. Liu; \emph{Second order expansion for blowup solutions of semilinear elliptic problems}, Nonlinear Anal. 75 (2012) 2591-2613. \bibitem{Mo} A. Mohammed; \emph{Boundary asymptotic and uniqueness of solutions to the $p$-Laplacian with infinite boundary values}, J. Math. Anal. Appl. 325 (2007) 480-489. \bibitem{NC} A. Nachman, A. Callegari; \emph{A nonlinear singular boundary value problem in the theory of pseudoplastic fluids}, SIAM J. Appl. Math. 38 (1980) 275-281. \bibitem{PV} G. Porru, A. Vitolo; \emph{Problems for elliptic singular equations with a quadratic gradient term}, J. Math. Anal. Appl. 334 (2007) 467-486. \bibitem{Redulescu} V. R\u{a}dulescu; \emph{Singular phenomena in nonlinear elliptic problems: from blow-up boundary solutions to equations with singular nonlinearities}, in Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 4 (2007) 483-591. \bibitem{Repovs} D. Repov\v{s}; \emph{Asymptotics for singular solutions of quasilinear elliptic equations with an absorption term}, J. Math. Anal. Appl. 395 (2012) 78-85. \bibitem {SIR} S. I. Resnick; \emph{Extreme Values, Regular Variation, and Point Processes}, Springer-Verlag, New York, Berlin, 1987. \bibitem{RS} R. Seneta; \emph{Regularly Varying Functions}, Lecture Notes in Math., vol. 508, Springer-Verlag, 1976. \bibitem{JSMY} J. Shi, M. Yao; \emph{Positive solutions for elliptic equations with singular nonlinearity}, Electron. J. Differential Equations 2005 (4) (2005) 1-11. \bibitem{CAS} C. A. Stuart; \emph{Existence and approximation of solutions of non-linear elliptic equations}, Math. Z. 147 (1976) 53-63. \bibitem{Usam} H. Usami; \emph{On a singular elliptic boundary value problem in a ball}, Nonlinear Anal. 13 (1989) 1163-1170. \bibitem{WAN} H. Wan; \emph{The second order expansion of solutions to a singular Dirichlet boundary value problem}, J. Math. Anal. Appl. (2015), http://dx.doi.org/10.1016/j.jmaa.2015.02.031. \bibitem{ZM} N. Zeddini, R. Alsaedi, H. M\^{a}agli; \emph{Exact boundary behavior of the unique positive solution to some singular elliptic problems}, Nonlinear Anal. 89 (2013) 146-156. \bibitem{Z.1} Z. Zhang, J. Yu; \emph{On a singular nonlinear Dirichlet problem with a convection term}, SIAM J. Math. Anal. 32 (2000) 916-927. \bibitem{ZhangCheng} Z. Zhang, J. Cheng; \emph{Existence and optimal estimates of solutions for singular nonlinear Dirichlet problem}, Nonlinear Anal. 57 (2004) 473-484. \bibitem{Z.2} Z. Zhang; \emph{The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation}, J. Math. Anal. Appl. 312 (2005) 33-43. \bibitem{Z.4} Z. Zhang; \emph{The existence and asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem with a convection term}, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 209-222. \bibitem{Zhang1} Z. Zhang; \emph{Boundary behavior of solutions to some singular elliptic boundary value problems}, Nonlinear Anal. 69 (2008) 2293-2302 \bibitem{Zhang5} Z. Zhang; \emph{The second expansion of the solution for a singular elliptic boundary value problem}, J. Math. Anal. Appl. 381 (2011) 922-934. \bibitem{ZhangB} Z. Zhang; \emph{The second expansion of large solutions for semilinear elliptic equations}, Nonlinear Anal. 74 (2011) 3445-3457. \bibitem{ZhangL} Z. Zhang, B. Li; \emph{The boundary behavior of the unique solution to a singular Dirichlet problem}, J. Math. Anal. Appl. 391 (2012) 278-290. \bibitem{zhangz} Z. Zhang, B. Li, X, Li; \emph{The exact boundary behavior of the unique solution to a singular Dirichlet problem with a nonlinear convection term}, Nonlinear Anal. 108 (2014) 14-28. \end{thebibliography} \end{document}