\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 19, pp. 1--18.\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-2015/19\hfil Boundary behavior of solutions] {Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection} \author[B. Li, Z. Zhang \hfil EJDE-2015/19\hfilneg] {Bo Li, Zhijun Zhang} \address{Bo Li \newline School of mathematics and statistics, Lanzhou University, Lanzhou 730000, Gansu, China. \newline School of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, China} \email{libo\_yt@163.com} \address{Zhijun Zhang \newline School of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, China} \email{chinazjzhang2002@163.com, zhangzj@ytu.edu.cn} \thanks{Submitted December 3, 2014. Published January 20, 2015.} \subjclass[2000]{35J65, 35B05, 35J25, 60J50} \keywords{Semilinear elliptic equation; singular Dirichlet problem; \hfill\break\indent nonlinear convection term; classical solution; boundary behavior} \begin{abstract} In this article we analyze the exact boundary behavior of solutions to the singular nonlinear Dirichlet problem \begin{gather*} -\Delta u=b(x)g(u)+\lambda|\nabla u|^q+\sigma, \quad u>0, \; x \in \Omega,\\ u\big|_{\partial \Omega}=0, \end{gather*} where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $q\in (0, 2]$, $\sigma>0$, $\lambda> 0$, $g\in C^1((0,\infty), (0,\infty))$, $\lim_{s \to 0^+}g(s)=\infty$, $g$ is decreasing on $(0, s_0)$ for some $s_0>0$, $b \in C_{\rm loc}^{\alpha}({\Omega})$ for some $\alpha\in (0, 1)$, is positive in $\Omega$, but may be vanishing or singular on the boundary. We show that $\lambda |\nabla u|^q$ does not affect the first expansion of classical solutions near the boundary. \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} In this article, we consider the boundary behavior of solutions to the singular nonlinear Dirichlet problem \begin{equation}\label{e1.1} -\Delta u=b(x)g(u)+\lambda|\nabla u|^q+\sigma, \quad u>0,\; x\in \Omega, \quad u\big|_{\partial\Omega}=0, \end{equation} where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $q\in (0, 2]$, $\lambda> 0$, $\sigma> 0$, $ b$ satisfies \begin{itemize} \item[(B1)] $b \in C_{\rm loc}^{\alpha}({\Omega})$ for some $\alpha \in (0,1)$, is positive in $\Omega$, \end{itemize} and $g$ satisfies \begin{itemize} \item[(G1)] $g\in C^1((0,\infty), (0,\infty))$ and $\lim_{s\to 0}g(s)=\infty$; \item[(G2)] there exists $s_0>0$ such that $g'(s)<0$, for all $s\in (0, s_0)$; \item[(G3)] there exists $C_g\geq 0$ such that $$ \lim_{s\to 0^+}g'(s)\int_0^s \frac {d\tau}{g(\tau)}=-C_g. $$ \end{itemize} A typical example of functions which satisfy (G1)-(G3) is $$ g(s)=s^{-\gamma}+\mu s^p, \ s>0, $$ where $\gamma, p, \mu>0$. In this case, $C_g=\gamma/(1+\gamma)$. A complete characterization of $g$ in (G1)-(G3) is provided in Lemma \ref{lem2.3}. For convenience, we denote by $\psi$ the solution to the problem \begin{equation}\label{e1.2} \int_0^{\psi(t)}\frac {ds}{g(s)}=t,\quad \forall t>0. \end{equation} When $\lambda=0$, \eqref{e1.1} 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 (see, for instance, \cite{CRT,GR4,FM,NC,RA,STU}) and has been discussed by many authors and in many contexts. With regard to the existence, nonexistence, uniqueness, multiplicity, regularity, local (near the boundary) and global estimates of (classical or weak) solutions, see, for instance, \cite{AMM}-\cite{BGP}, \cite{CMMZ,CP,CRT,DP,GR3,GR4,FM}, \cite{GO}-\cite{HM}, \cite{LS}-\cite{LM}, \cite{MO2}, \cite{SAN}-\cite{ZAM}, \cite{Z5} and the references therein. When $\lambda>0$, $b\equiv 1$ in $\Omega$ and $g(u)=u^{-\gamma}$ with $\gamma>0$, the authors \cite{Z1} considered the existence and regularities of the unique solution to \eqref{e1.1}. Cui \cite{CUI} established a sub-supersolution method to more general problem than \eqref{e1.1}. When $\lambda= 1$, $\sigma=0$, $01$ such that $G(s)0$, \end{itemize} then the unique solution $u$ to \eqref{e1.1} has the following properties: \begin{itemize} \item[(I1)] $|u(x)-\phi (d(x))|0. \end{equation} For further works, see \cite{CGP}, \cite{DGR}-\cite{GR2}, \cite{GH,GP2,HS,PV}, \cite{Z2}-\cite{Z4} and the references therein. We introduce two types of functions. First, we denote by $K$ the set of all Karamata functions $\hat{L}$ which are \textbf{normalized} slowly varying at zero (see, Bingham-Goldie-Teugels's book \cite{BGT} and Maric's book \cite{MAR}) defined on $(0, \eta]$ for some $\eta>0$ by \begin{equation}\label{e1.4} \hat{L}(s)=c_0 {\rm exp} \Big( \int_s^{\eta} \frac {y(\tau)}{\tau} d\tau \Big), \quad s \in (0, \eta], \end{equation} where $c_0>0$ and the function $y\in C([0, \eta])$ with $y(0)=0$. Next let $\Lambda$ denote the set of all positive monotonic functions $\theta$ in $C^1(0,\delta_0)\cap L^1(0, \delta_0)$ ($\delta_0>0$) which satisfy \begin{equation}\label{e1.5} \lim_{t \to 0} \frac {d}{dt}\big(\frac{\Theta(t)}{\theta(t)}\big):= C_\theta\in [0, \infty),\quad \Theta(t): =\int_0^t \theta(s)ds. \end{equation} The set $\Lambda$ was first introduced by C\^{i}rstea and R\v{a}dulescu \cite{CR} for non-decreasing functions and by Mohammed \cite{MO1} for non-increasing functions to study the boundary behavior of solutions to boundary blow-up elliptic problems. We assume that $b$ satisfies \begin{itemize} \item[(B2)] there exists $\theta\in \Lambda$ such that $$ 01/2$; \item[(ii)] $q=2$, $b$ satisfies (B1) and (B2), $g$ satisfies (G3) with \begin{equation}\label{e1.6} C_\theta+2C_g>2, \end{equation} and one of the following two conditions holds \begin{itemize} \item[(S01)] $C_g>0$; \item[(S02)] $C_g=0$ and $\lambda \limsup_{s\to 0^+}\frac {g(s)}{|g'(s)|}<1$ \end{itemize} \end{itemize} and obtained the boundary behavior of the unique solutions to \eqref{e1.1}. In this article, we extend \cite{Z4} for more general $g$ and $b$. We first establish a local comparison principle for $q\in (0, 1)$ under (G2). More precisely, we show the first exact asymptotic behaviour of any classical solution near the boundary to \eqref{e1.1} and reveal that the nonlinear gradient term $\lambda |\nabla u|^q$ does not affect the behaviour. For $q\in [1, 2]$, by using a nonlinear change, the local comparison principle and the results in \cite{Z5} and \cite{LZ}, we show the same results as $q\in (0, 1)$. Our main results are summarized as follows. \begin{theorem}\label{thm1.1} For fixed $\lambda >0$, let $g$ satisfy {\rm (G1)--(G3)}, $b$ satisfy {\rm (B1)--(B2)}. If both \eqref{e1.6} and one of the following conditions hold \begin{itemize} \item[(S1)] $q\in (0, 1)$; \item[(S2)] $q\in [1, 2]$ and $C_g>0$; \item[(S3)] $q\in [1, 2]$, $C_g=0$ and $$ \lambda \limsup_{s\to 0^+}\frac {g(s)}{|g'(s)|}<1, $$ \end{itemize} then for any classical solution $u_\lambda$ to \eqref{e1.1}, it holds \begin{equation}\label{e1.7} \xi_1^{1-C_g}\leq \lim_{d(x) \to 0 } \inf \frac{u_\lambda(x)}{\psi(\Theta^2(d(x)))}\leq \lim_{d(x) \to 0 } \sup \frac{u_\lambda(x)}{\psi(\Theta^2(d(x)))} \leq \xi_2^{1-C_g}, \end{equation} where \begin{equation}\label{e1.8} \xi_1=\frac { b_1}{2(C_\theta+2C_g-2)}, \quad \xi_2=\frac { b_2}{2(C_\theta+2C_g-2)}. \end{equation} In particular, \begin{itemize} \item[(i)] when $C_g=1$, $u_\lambda$ satisfies $$ \lim_{d(x) \to 0}\frac{u_\lambda(x)}{\psi(\Theta^2(d(x)))}=1; $$ \item[(ii)] when $C_g<1$ and $b_1=b_2=b_0$ in {\rm (B2)}, $u_\lambda$ satisfies $$ \lim_{d(x) \to 0}\frac{u_\lambda(x)}{\psi(d^2(x)\theta^2(d(x)))} =(\xi_{01}C_\theta^2)^{1-C_g}, $$ where $$ \xi_{01}=\frac {b_0}{2(C_\theta+2C_g-2)}. $$ \end{itemize} \end{theorem} \begin{theorem}\label{thm1.2} For fixed $\lambda >0$, let $q\in (0, 2]$, $g$ satisfy {\rm (G1)--(G3)}, and let $b$ satisfy {\rm (B1)} and \begin{itemize} \item[(B3)] there exists $\hat{L}\in K$ with $\int_0^{\eta}\frac {\hat{L}(s)} {s}ds<\infty$ such that $$ 00$, let $q\in (0, 2]$, $b$ satisfy {\rm (B1)}, $g$ satisfy {\rm (G1)} and $g(s)=s^{-\gamma}+\mu s^p$, $s\in (0, s_0)$, for some $s_0>0$, where $\gamma,\ p, \mu>0$. If $b$ satisfies \begin{itemize} \item[(B4)] there exists $\hat{L}\in K$ with $\int_0^{\eta}\frac {\hat{L}(s)} {s}ds=\infty$ such that $$ 00$, $s\in (0, s_0)$; \item[(ii)] $g(s)=(\ln (-\ln s))^\gamma$, $\gamma>0$, $s\in (0, s_0)$; \item[(iii)] $ g(s)=e^{(-\ln s)^\gamma}$, $0<\gamma<1$, $s\in (0, s_0)$, where $s_0>0$ sufficiently small. \end{itemize} \end{remark} \begin{remark}\label{rmk1.2}\rm When $\gamma>0$, we note that $C_g=\frac {\gamma}{1+\gamma}$ and $C_\theta=\frac {2}{\gamma+1}$ in Theorem \ref{thm1.3}, i.e., $C_\theta+2C_g=2$. \end{remark} The outline of this paper is as follows. In section 2, we present some basic facts from Karamata regular variation theory and some preliminaries. Some comparison principles are given in section 3. In section 4, we prove Theorems \ref{thm1.1}--\ref{thm1.3}. \section{Preliminaries} Our approach relies on Karamata regular variation theory established by Karamata in 1930 which is a basic tool in stochastic processes (see Bingham, Goldie and Teugels' book \cite{BGT}, Maric's book \cite{MAR} and the references therein). In this section, we present some basic facts from Karamata regular variation theory. \begin{definition}\label{def2.1} \rm A positive continuous function $g$ defined on $(0, \eta]$, for some $\eta>0$, is called \textbf{regularly varying at zero} with index $\rho$, denoted by $g \in RVZ_\rho$, if for each $\xi>0$ and some $\rho \in \mathbb{R}$, \begin{equation}\label{e2.1} \lim_{s \to 0^+} \frac{g(\xi s)}{g(s)}= \xi^\rho. \end{equation} In particular, when $\rho=0$, $g$ is called \textbf{slowly varying at zero}. \end{definition} Clearly, if $g\in RVZ_\rho$, then $L(s):=g(s)/{s^\rho}$ is slowly varying at zero. \begin{definition}\label{def2.2} \rm A positive continuous function $g$ defined on $(0, \eta]$, for some $\eta>0$, is called \textbf{rapidly varying to infinity at zero} if for each $\xi\in (0,1)$ \begin{equation}\label{e2.2} \lim_{s \to 0^+} \frac{g(\xi s)}{g(s)}=\infty. \end{equation} \end{definition} \begin{definition}\label{def2.3} \rm A positive function $g\in C(0, \eta]$ with $\lim_{s\to 0^+}g(s)=0$, for some $\eta>0$, is called \textbf{rapidly varying to zero at zero} if for each $\xi\in (0, 1)$ \begin{equation}\label{e2.3} \lim_{s \to 0^+} \frac{g(\xi s)}{g(s)}=0. \end{equation} \end{definition} \begin{proposition}[Uniform convergence theorem] \label{prop2.1} If $g\in RVZ_\rho$, then \eqref{e2.1} holds uniformly for $\xi \in [c_1, c_2]$ with $00$. \end{proposition} Note that \begin{equation}\label{e2.5} \hat{L}(s)=c_0 {\rm exp} \Big( \int_s^{\eta} \frac {y(\tau)}{\tau} d\tau \Big), \quad s \in (0, \eta], \end{equation} is \textbf{normalized} slowly varying at zero, and \begin{equation}\label{e2.6} g(s)=s^\rho\hat{L}(s), \quad s \in (0, \eta], \end{equation} is \textbf{normalized} regularly varying at zero with index $\rho$ (and denoted by $g\in NRVZ_\rho$). A function $g\in NRVZ_\rho$ if and only if \begin{equation}\label{e2.7} g\in C^1(0, \eta],\text{ for some } \eta>0 \text{ and } \lim_{s \to 0^+} \frac{sg'(s)}{g(s)}=\rho. \end{equation} \begin{proposition}\label{prop2.3} If functions $L, L_1$ are slowly varying at zero, then \begin{itemize} \item[(i)] $L^\rho$ for every $\rho\in \mathbb{R}$, $c_1L+c_2L_1$ {\rm ($c_1\geq0,\ c_2\geq0$ with $ c_1+c_2>0$)}, $L\cdot L_1$, $L\circ L_1$ {\rm (}if $L_1(s)\to 0$ as $s\to 0^+${\rm)} , are also slowly varying at zero; \item[(ii)] For every $\rho >0$ and $s\to 0^+$, $s^{\rho} L(s)\to 0$, $s^{-\rho} L(s)\to \infty$; \item[(iii)] For $\rho\in\mathbb{R}$ and $s\to 0^+$, $\ln (L(s))/{\ln s}\to 0$ and $\ln (s^\rho L(s))/{\ln s}\to \rho$. \end{itemize} \end{proposition} \begin{proposition}\label{prop2.4} If $g_1\in {R}VZ_{\rho_1}$, $g_2\in {R}VZ_{\rho_2} $ with $\lim_{s\to 0} g_2 (s)=0$, then $g_1\circ g_2\in {R}VZ_{\rho_1 \rho_2}$. \end{proposition} \begin{proposition}[Asymptotic behavior]\label{prop2.5} If a function $L$ is slowly varying at zero, then for $\eta>0$ and $t\to 0^+$, \begin{itemize} \item[(i)] $\int_0^t s^{\rho}L(s)ds\cong (1+\rho)^{-1} t^{1+\rho}L(t)$, for $\rho>-1$; \item[(ii)] $\int_t^\eta s^{\rho}L(s)ds\cong (-\rho-1)^{-1}t^{1+\rho}L(t)$, for $\rho<-1$. \end{itemize} \end{proposition} \begin{proposition}\label{prop2.6} Let $g\in C^1(0, \eta]$ be positive and $$ \lim_{s \to 0^+} \frac{sg'(s)}{g(s)}=+\infty. $$ Then $g$ is rapidly varying to zero at zero. \end{proposition} \begin{proposition}\label{prop2.7} Let $g\in C^1(0, \eta)$ be positive and $$ \lim_{s \to 0^+} \frac{sg'(s)}{g(s)}=-\infty. $$ Then $g$ is rapidly varying to infinity at zero. \end{proposition} \begin{proposition}[{\cite[Lemma 2.3]{ZAM}}]\label{prop2.8} Let $\hat{L}$ be defined on $(0, \eta]$ and be normalized slowly varying at zero. Then $$ \lim_{t\to 0^+}\frac {L(t)}{\int_t^{\eta} \frac {L(\tau)}{\tau} d\tau}=0. $$ If further $\int_0^{\eta} \frac {L(\tau)}{\tau} d\tau$ converges, then $$ \lim_{t\to 0^+}\frac {L(t)}{\int_0^t \frac {L(\tau)}{\tau} d\tau}=0. $$ \end{proposition} Our results in the section are summarized in the following lemmas. \begin{lemma}\label{lem2.1} Let $\theta\in \Lambda$. \begin{itemize} \item[(i)] $\lim_{t \to 0^+} \frac{\Theta(t)}{\theta(t)}=0$; \item[(ii)] $\lim_{t\to 0^+} \frac{\Theta(t)\theta'(t)}{\theta^2(t)} =1-\lim_{t\to 0^+} \frac{d}{dt}\left(\frac{\Theta(t)}{\theta(t)}\right) =1-C_\theta$, and $C_\theta\in [0, 1]$ when $\theta$ is non-decreasing, $C_\theta\geq 1$ provided $\theta$ is non-increasing; \item[(iii)] when $C_\theta>0$, $\theta\in NRVZ_{(1-C_\theta)/{C_\theta}}$ and $\Theta\in NRVZ_{1/{C_\theta}}$. \end{itemize} \end{lemma} \begin{proof} For an arbitrary $\theta\in \Lambda$, we have: (i) When $\theta$ is non-decreasing, we have that $0<\Theta(t)\leq t \theta(t)$, for all $t\in (0, \delta_0)$ and (i) holds; when $\theta$ is non-increasing, it follows by $\theta\in L^1(0, \delta_0)$ that $$ \lim_{t \to 0^+} \frac{\Theta(t)}{\theta(t)}=\lim_{t \to 0^+} \frac{1}{\theta(t)}\lim_{t \to 0^+} \Theta(t)=0. $$ (ii) Since \begin{equation}\label{e2.8} \lim_{t \to 0^+} \frac{\Theta(t)\theta'(t)}{\theta^2(t)} =1-\lim_{t\to 0^+} \frac {d}{dt}\big(\frac{\Theta(t)}{\theta(t)}\big)=1-C_\theta, \end{equation} it follows that $C_\theta\in [0, 1]$ when $\theta$ is non-decreasing, and $C_\theta\geq 1$ provided $\theta$ is non-increasing; (iii) \eqref{e1.5} and the l'Hospital's rule imply \begin{equation}\label{e2.9} \lim_{t \to 0^+} \frac {\Theta(t)}{t \theta(t) }=\lim_{t \to 0^+} \frac {\frac {\Theta(t)}{\theta(t)}}{t}=\lim_{t \to 0^+}\frac {d}{dt}\big(\frac{\Theta(t)}{\theta(t)}\big)= C_\theta. \end{equation} So, when $C_\theta>0$, $\Theta\in NRVZ_{C_\theta^{-1}}$ and it follows by \eqref{e2.8} and \eqref{e2.9} that \begin{equation}\label{e2.10} \lim_{t\to 0}\frac {t\theta'(t)}{\theta(t)} =\lim_{t\to 0}\frac {\Theta(t)\theta'(t)}{\theta^2(t)}\lim_{t\to 0} \frac {t\theta(t)}{\Theta(t)}=\frac {1-C_\theta}{C_\theta}, \end{equation} i.e., $\theta\in NRVZ_{(1-C_\theta)/{C_\theta}}$. \end{proof} \begin{lemma}[{\cite[Lemma 2.2]{Z5}}] \label{lem2.2} Let $g$ satisfy {\rm (G1), (G2)}. \begin{itemize} \item[(i)] If $g$ satisfies {\rm (G3)}, then $C_g\leq 1;$ \item[(ii)] {\rm (G3)} holds with $C_g \in (0, 1)$ if and only if $g\in NRVZ_{-{C_g}/(1-C_g)}$; \item[(iii)] {\rm (G3)} holds with $C_g=0$ if and only if $g$ is normalized slowly varying at zero; \item[(iv)] if {\rm (G3)} holds with $C_g=1$, then $g$ is rapidly varying to infinity at zero. \end{itemize} \end{lemma} \begin{lemma}[{\cite[Lemma 2.3]{Z5}}] \label{lem2.3} Let $g$ satisfy {\rm (G1), (G2)} and let $\psi$ be uniquely determined by $$ \int_0^{\psi(t)}\frac {d\tau}{g(\tau)}=t,\ t\in [0, \infty). $$ Then \begin{itemize} \item[(i)] $\psi'(t)=g(\psi(t))$, $\psi(t)>0$, $t>0$, $\psi(0)=0$ and $\psi''(t)=g(\psi(t))g'(\psi(t))$, $t>0$; \item[(ii)] $ \lim_{t\to 0^+}t g(\psi(t))=0$ and $ \lim_{t\to 0^+}t g'(\psi(t))=-C_g$; \item[(iii)] $\psi\in NRVZ_{1-C_g}$ and $\psi'\in NRVZ_{-C_g}$; \item[(iv)] when $C_\theta+2C_g>2$ and $\theta\in \Lambda$, $\lim_{t\to 0^+}\frac {t}{\psi(\xi \Theta^2(t))}=0$ uniformly for $\xi \in [c_1, c_2]$ with $02$, then $$ \lim_{s\to 0^+}(g(\psi(\Theta^2(t))))^{q-1}\frac {(\Theta(t))^{q}}{(\theta(t))^{2-q}}=0,\quad \lim_{s\to 0^+} g(\psi(\Theta^2(t))) \theta^2(t)=\infty. $$ \end{lemma} \begin{proof} Using Proposition \ref{prop2.4}, Lemma \ref{lem2.2} (iii) and Lemma \ref{lem2.4} (iii), we see that $ g(\psi(\Theta^2(t))) \theta^2(t)$ belongs to $NRVZ_{\rho_1}$ with $$ \rho_1=\frac {-2C_g}{C_\theta}+\frac {2(1-C_\theta)}{C_\theta}=-\frac {C_\theta+2C_g-2+C_\theta}{C_\theta}<0, $$ and $(g(\psi(\Theta^2(t))))^{q-1}\frac {(\Theta(t))^{q}}{(\theta(t))^{2-q}}$ belongs to $NRVZ_{\rho_2}$ with \begin{align*} \rho_2&=\frac {q}{C_\theta}-\frac {2C_g(q-1)}{C_\theta}-\frac {(2-q)(1-C_\theta)}{C_\theta}\\ &=\frac {C_\theta+2C_g-2+C_\theta(1-q)+2q(1-C_g)}{C_\theta}>0. \end{align*} Thus the results follow by Proposition \ref{prop2.3} (ii). \end{proof} \section{Local comparison principles} In this section we give some comparison principles near the boundary. For any $\delta>0$, we define $$ \Omega_{\delta}: =\{x\in\Omega: d(x)<\delta\},\quad \Gamma_{\delta}:=\{x\in \Omega: d(x)=\delta\}. $$ Since $\partial\Omega\in C^2$, there exists a constant $\delta\in (0, \min\{s_0, \delta_0\})$ which only depends on $\Omega$ such that (see, \cite[Lemmas 14.16 and 14.17]{GT}) \begin{equation}\label{e3.1} d\in C^2(\Omega_{\delta}), \quad |\nabla d(x)|= 1, \quad \Delta d(x) =-(N-1)H(\bar{x})+o(1), \quad \forall x\in \Omega_{{\delta}}, \end{equation} where $\bar{x}$ is the nearest point to $x$ on $\partial\Omega$, and $H(\bar{x})$ denotes the mean curvature of $\partial\Omega$ at $\bar{x}$. Next let $v_0 \in C^{2+\alpha} (\Omega)\cap C^1 (\bar{\Omega}) $ be the unique solution of the problem \begin{equation}\label{e3.2} -\Delta v=1,\quad v>0, \quad x\in\Omega,\quad v|_{\partial\Omega}=0. \end{equation} By the H\"{o}pf maximum principle in \cite{GT}, we see that \begin{equation}\label{e3.3} \nabla v_0 (x) \neq 0, \quad \forall x \in \partial \Omega \quad\text{and}\quad c_1 d(x)\leq v_0(x) \leq c_2 d(x),\quad \forall x \in \Omega, \end{equation} where $c_1$, $c_2$ are positive constants. We have the lower bound estimations near the boundary of solutions to \eqref{e1.1}. \begin{lemma}[A local comparison principle]\label{lem3.1} For fixed $\lambda>0$, let $q\in (0, 2]$, $g$ satisfy {\rm (G1), (G2)}, $b$ satisfy {\rm (B1)}, and let $u_\lambda\in C^2(\Omega)\cap C(\bar{\Omega})$ be an arbitrary solution to \eqref{e1.1} and $u_0\in C^2(\Omega)\cap C(\bar{\Omega})$ be an arbitrary solution to the problem \begin{equation}\label{e3.4} -\Delta u=b(x)g(u), \quad u>0,\quad x\in \Omega, \quad u|_{\partial\Omega}=0. \end{equation} Then there exists a positive constant $M_0$ such that \begin{equation}\label{e3.5} u_0(x)\leq u_\lambda(x)+ M_0 v_0(x),\quad x\in \Omega_\delta, \end{equation} where $\delta>0$ sufficiently small such that $$ u_0(x),\quad u_\lambda(x)\in (0, s_0),\quad x\in \Omega_\delta, $$ where $s_0$ is given as in {\rm (G2)}. \end{lemma} \begin{proof} First, by $u_\lambda(x)=v_0(x)=u_0 (x)=0$, for all $x\in \partial\Omega$, and \begin{equation}\label{e3.6} u_0, v_0, u_{\lambda}\in C^2(\Omega)\cap C(\bar{\Omega}), \end{equation} we can choose a large $M_0$ such that \begin{equation}\label{e3.7} u_0(x)\leq u_\lambda(x)+ M_0 v_0(x), \quad x\in \Gamma_{\delta}. \end{equation} Now we prove \eqref{e3.5}. Assume the contrary, there exists $x_0\in \Omega_{\delta}$ such that $$ {u_0}(x_0) - (u_\lambda(x_0)+ M_0v_0 (x_0))>0. $$ It follows that there exists $x_1\in \Omega_{\delta}$ such that $$ 0<{u_0}(x_1) -(u_{\lambda}(x_1)+ M_0 v_0(x_1)) =\max_{x\in \bar{\Omega}_{\delta}} ({u_0}(x)-(u_{\lambda}(x)+ M_0 v_0 (x))). $$ Then (\cite[Theorem 2.2]{GT}) $$ \Delta ({u_0}-(u_{\lambda}+ M_0 v_0))(x_1)\leq 0. $$ On the other hand, we see by (B1), (G1) and (G2) that \begin{align*} &\Delta (u_0-(u_{\lambda}+ M_0 v_0))(x_1)\\ &=-\Delta u_{\lambda} (x_1)+ M_0 +\Delta u_0(x_1)\\ &= b(x_1)(g(u_{\lambda}(x_1))-g({u_0}(x_1)) )+ M_0+\lambda |\nabla u_{\lambda}(x_1)|^q +\sigma >0, \end{align*} which is a contradiction. Hence \eqref{e3.5} holds. \end{proof} Next we consider the upper bound estimations near the boundary to $u_\lambda$. For $q\in (0, 1)$, we have the following lemma. \begin{lemma}[A local comparison principle]\label{lem3.2} For fixed $\lambda>0$, let $g$ satisfy {\rm (G1), (G2)}, $b$ satisfy {\rm (B1)}, and let $u_\lambda\in C^2(\Omega)\cap C(\bar{\Omega})$ be an arbitrary solution to \eqref{e1.1}, $\bar{u}_\lambda\in C^2(\Omega_\delta)\cap C(\bar{\Omega}{_\delta})$ satisfy \begin{equation}\label{e3.8} -\Delta \bar{u}_\lambda\geq b(x)g(\bar{u}_\lambda)+ \lambda|\nabla \bar{u}_\lambda|^q+\sigma, \quad \bar{u}_\lambda>0,\quad x\in \Omega_\delta, \quad \bar{u}_\lambda|_{\partial\Omega}=0, \end{equation} where $\delta>0$ sufficiently small such that $$ \bar{u}_\lambda (x),\quad u_\lambda(x)\in (0, s_0),\quad x\in \Omega_\delta, $$ where $s_0$ is given as in {\rm (G2)}. Then there exists a positive constant $M_0$ such that \begin{equation}\label{e3.9} u_\lambda(x)\leq \bar{u}_\lambda(x)+\lambda M_0 v_0(x),\quad x\in \Omega_\delta. \end{equation} \end{lemma} \begin{proof} From $u_\lambda(x)=\bar{u}_\lambda(x)=v_0 (x)=0$, for all $x\in \partial\Omega$, and \begin{equation}\label{e3.10} u_{\lambda}\in C^2(\Omega)\cap C(\bar{\Omega}), \ \ v_0\in C^2(\Omega)\cap C^1(\bar{\Omega}),\ \ \bar{u}_{\lambda}\in C^2(\Omega_\delta)\cap C(\bar{\Omega}{_\delta}), \end{equation} we can choose a large $M_0$ such that \begin{gather}\label{e3.11} u_\lambda(x)\leq \bar{u}_\lambda(x)+\lambda M_0 v_0(x), \quad x\in \Gamma_{\delta}, \\ \label{e3.12} M_0^{1-q}\geq \lambda^q\max_{x\in \bar{\Omega}}|\nabla v_0(x)|^q. \end{gather} Now we prove \eqref{e3.9}. Assume the contrary, there exists $x_0\in \Omega_{\delta}$ such that $$ {u}_\lambda(x_0)- (\bar{u}_\lambda(x_0)+\lambda M_0v_0 (x_0))>0. $$ It follows that there exists $x_1\in \Omega_{\delta}$ such that $$ 0<{u}_ {\lambda}(x_1) -(\bar{u}_{\lambda}(x_1)+\lambda M_0 v_0(x_1))=\max_{x\in \bar{\Omega}_{\delta}} ({u}_ {\lambda}(x)-(\bar{u}_{\lambda}(x)+\lambda M_0 v_0 (x))). $$ Then (\cite[Theorem 2.2]{GT}) $$ \nabla ({u}_ {\lambda}-(\bar{u}_{\lambda}+\lambda M_0 v_0))(x_1)=0\quad\text{and} \quad \Delta ({u}_ {\lambda}-(\bar{u}_{\lambda}+\lambda M_0 v_0))(x_1)\leq 0. $$ On the other hand, using the basic inequality for $q\in (0, 1)$ $$ |s_2^q-s_1^q|\leq |s_2-s_1|^q,\quad \forall s_2, s_1\geq 0, $$ it follows by (B1), (G1) and (G2) that \begin{align*} &\Delta \big({u}_ {\lambda}-(\bar{u}_{\lambda}+\lambda M_0 v_0)\big)(x_1)\\ &=-\Delta \bar{u}_{\lambda} (x_1)+\lambda M_0 +\Delta {u}_{\lambda}(x_1)\\ &\geq b(x_1)(g(\bar{u}_{\lambda}(x_1))-g({u}_{\lambda}(x_1)))+ \lambda (M_0+|\nabla \bar{u}_{\lambda}(x_1)|^q-|\nabla u_{\lambda}(x_1)|^q)\\ &>\lambda (M_0-\lambda^q M_0^q|\nabla v_0(x_1)|^q) >0, \end{align*} which is a contradiction. Hence \eqref{e3.9} holds. \end{proof} For $q\in [1, 2]$ and an arbitrary positive constant $C$, by using the following inequality \cite[(3.10)]{Z1} $$ s^q\leq \frac {s^2}{C^{1-{q/2}}}+C^{q/2},\quad \forall s\geq 0, $$ we see that \begin{equation}\label{e3.13} -\Delta u_\lambda\leq b(x)g(u_\lambda)+\lambda C^{{q/2}-1}|\nabla u_\lambda|^2+\lambda C^{q/2}+\sigma, \ u_\lambda>0, \ x \in \Omega,\ u_\lambda|_{\partial \Omega}=0. \end{equation} We can choose $C$ such that the problem \begin{equation}\label{e3.14} \begin{gathered} -\Delta \bar{u}_\lambda=b(x)g(\bar{u}_\lambda)+\lambda C^{{q/2}-1}|\nabla \bar{u}_\lambda|^2+\lambda C^{q/2}+\sigma, \ \bar{u}_\lambda>0, \ x \in \Omega,\\ \bar{u}_\lambda|_{\partial \Omega}=0, \end{gathered} \end{equation} has one classical solution $\bar{u}_\lambda$ (\cite[Theorem 4.1]{Z2}). For a fixed $\lambda$, let ${u}_\lambda$ and $\bar{u}_\lambda$ be arbitrary solutions to \eqref{e3.13} and \eqref{e3.14}, we see that the nonlinear changes of variable $$ {w}_\lambda=\exp(\eta {u}_\lambda)-1\ \ {\rm and }\ \ \bar{w}_\lambda=\exp(\eta \bar{u}_\lambda)-1 $$ transform problems \eqref{e3.13} and \eqref{e3.14} into the equivalent problems \begin{equation}\label{e3.15} -\Delta w_\lambda\leq b(x)\tilde{g}(w_\lambda)+\eta f(w_\lambda), \ w_\lambda>0,\quad x\in \Omega, \quad w_\lambda|_{\partial\Omega}=0, \end{equation} and \begin{equation}\label{e3.16} -\Delta \bar{w}_\lambda= b(x)\tilde{g}(\bar{w}_\lambda)+\eta f(\bar{w}_\lambda), \quad \bar{w}_\lambda>0,\quad x\in \Omega, \quad \bar{w}_\lambda|_{\partial\Omega}=0, \end{equation} respectively. Where \begin{gather}\label{e3.17} \tilde{g}(s)=\eta (1+s)g(\eta ^{-1}\ln (1+s)),\quad \eta=\lambda C^{{q/2}-1}, \\ \label{e3.18} f(s)=(\eta C+\sigma )(1+s). \end{gather} \begin{lemma}\label{lem3.3} For fixed $\lambda>0$. Let $g$ satisfy {\rm (G1)--(G3)}. Then \begin{itemize} \item[(i)] $\tilde{g}\in C^1((0,\infty), (0,\infty))$ and $\lim_{s\to 0}\tilde{g}(s)=\infty$; \item[(ii)] when one of the following conditions holds \begin{itemize} \item[(S01)] $C_g>0$; \item[(S02)] $C_g=0$ and $\lambda \limsup_{s\to 0^+}\frac {g(s)}{|g'(s)|}<1$, \end{itemize} there exists $s_1>0$ such that $\tilde{g}'(s)<0,\ \forall s\in (0, s_1)$; \item[(iii)] $$ \lim_{s\to 0^+}\tilde{g}'(s)\int_0^s \frac {d\tau}{\tilde{g}(\tau)}=-C_g. $$ \end{itemize} \end{lemma} \begin{proof} By (G1), (i) is obvious. (ii) follows by \cite[Lemma 3.1]{Z4}. (iii) Since $g$ satisfies (G1) and is decreasing on $(0, s_0)$, we see that $$ 0<\int_0^s \frac {d\tau}{g(\tau)}<\frac {s}{g(s)},\quad \forall s\in (0, s_0), $$ i.e., \begin{gather}\label{e3.19} 00$, let $f\in C([0, \infty), [0, \infty))$, $g$ satisfy {\rm (G1), (G2)}, $b$ satisfy {\rm (B1)}. Then there exists a positive constant $M_0$ such that \begin{equation}\label{e3.21} {w}_\lambda(x)\leq \bar{w}_\lambda(x)+M_0(\eta C +\sigma ) v_0(x),\quad x\in \Omega_\delta, \end{equation} where $\delta>0$ sufficiently small such that $$ {w}_\lambda(x),\; \bar{w}_\lambda (x) \in (0, s_1),\quad x\in \Omega_\delta, $$ where $s_1$ is as in Lemma \ref{lem3.3}. \end{lemma} \section{Boundary behavior} In this section we prove Theorems \ref{thm1.1}--\ref{thm1.3}. First we have the statement in \cite[Theorem 1.1]{LZ} with $a\equiv 1$ in $\Omega$. \begin{lemma}\label{lem4.1} For a fixed $\lambda>0$, let $f\in C([0, \infty), [0, \infty))$, $g$ satisfy {\rm (G1)--(G3)}, and let $b$ satisfy {\rm (B1), (B2)}. If \begin{equation}\label{e4.1} C_\theta+2C_g>2, \end{equation} then for any classical solution $V_\lambda$ to the problem \begin{equation}\label{e4.2} -\Delta V=b(x)g(V)+\lambda a(x) f(V), \quad V>0,\quad x\in \Omega, \quad V|_{\partial\Omega}=0, \end{equation} it holds that \begin{equation}\label{e4.3} \xi_1^{1-C_g}\leq \lim_{d(x) \to 0 } \inf \frac{V_\lambda(x)}{\psi(\Theta^2(d(x)))}\leq \lim_{d(x) \to 0 } \sup \frac{V_\lambda(x)}{\psi(\Theta^2(d(x)))} \leq \xi_2^{1-C_g}, \end{equation} where $\psi$ is the solution to \eqref{e1.2}, $\xi_1$ and $\xi_2$ are given as in \eqref{e1.8}. In particular, {\rm (i)} and {\rm (ii)} in Theorem \ref{thm1.1} hold. \end{lemma} Next we have the statement in \cite[Theorems 1.2]{Z5} with $a\equiv 1$ in $\Omega$. \begin{lemma}\label{lem4.2} For a fixed $\lambda>0$, let $f\in C([0, \infty), [0, \infty))$, $g$ satisfy {\rm (G1)--(G3)}, and let $b$ satisfy {\rm (B1)}. If $b$ satisfies {\rm (B3)}, then any classical solution $V_\lambda$ to \eqref{e4.2} satisfies \eqref{e1.10}. \end{lemma} Next we have the statement in \cite[Theorems 1.3]{Z5} with $a\equiv b$ in $\Omega$. \begin{lemma}\label{lem4.3} For a fixed $\lambda>0$, let $f\in C([0, \infty), [0, \infty))$, $g$ satisfy {\rm (G1)} and $g(s)=s^{-\gamma}+\mu s^p$, $s\in (0, s_0)$ for some $s_0>0$ and $\gamma, p, \mu> 0$, and let $b$ satisfy {\rm (B1)}. If $b$ satisfies {\rm (B4)}, then any classical solution $V_\lambda$ to \eqref{e4.2} satisfies \eqref{e1.12}. \end{lemma} \begin{remark}\label{rmk3.1} \rm Obviously, when $f\equiv 0$ on $[0, \infty)$, a solution $V_\lambda$ to \eqref{e4.2} is a solution to \eqref{e3.4}. \end{remark} \begin{proof}[Proof of Theorem \ref{thm1.1}] Let $u_\lambda\in C^2(\Omega)\cap C(\bar{\Omega})$ be an arbitrary solution to \eqref{e1.1}. Using \eqref{e3.3}, Lemmas \ref{lem3.1}, \ref{lem3.3}, \ref{lem3.4}, \ref{lem4.1} and \ref{lem2.3} (iv), we obtain that for $q\in (0, 2]$, \begin{equation}\label{e4.4} \xi_1^{1-C_g}\leq \lim_{d(x) \to 0 } \inf \frac{u_0(x)}{\psi(\Theta^2(d(x)))}\leq \lim_{d(x) \to 0 } \inf \frac{u_\lambda(x)}{\psi(\Theta^2(d(x)))}, \end{equation} and for $q\in [1, 2]$, \begin{equation}\label{e4.5} \lim_{d(x) \to 0 } \sup \frac{w_\lambda(x)}{\psi_1(\Theta^2(d(x)))}\leq \lim_{d(x) \to 0 } \sup \frac{\bar{w}_\lambda(x)}{\psi_1(\Theta^2(d(x)))}\leq \xi_2^{1-C_g}. \end{equation} where $w_\lambda(x)=\exp(\eta u_\lambda(x))-1$, $\bar{w}_\lambda(x)=\exp(\eta \bar{u}_\lambda(x))-1$, $\ \eta=\lambda C^{{q/2}-1}$, $\psi_1$ is the solution to the problem \begin{equation}\label{e4.6} \int_0^{\psi_1(t)}\frac {ds}{\tilde{g}(s)}=t,\quad \forall t>0, \end{equation} and $\tilde{g}$ is given in \eqref{e3.17}. From \begin{gather*} \psi(t)=\eta^{-1}\ln (1+\psi_1(t)),\quad \forall t>0,\\ \exp(\eta s)-1\cong \eta s\quad \text{as } s\to 0, \end{gather*} it follows that $q\in [1, 2]$, \begin{equation}\label{e4.7} \lim_{d(x) \to 0 } \sup \frac{u_\lambda(x)}{\psi(\Theta^2(d(x)))}\leq \xi_2^{1-C_g}. \end{equation} Thus \eqref{e1.7} holds for $q\in [1, 2]$. Next we structure an appropriate supersolution near the boundary to \eqref{e1.1} in the case $q\in (0, 1)$. Let $\varepsilon\in(0, b_1/4)$ and let $$ \tau_1=\xi_2+2\varepsilon\xi_2/{b_2}, $$ where $\xi_2$ is given in \eqref{e1.8}. It follows that $\xi_2<\tau_1<2\xi_2$, $\lim_{\varepsilon\to 0} \tau_1=\xi_2$, and \begin{equation}\label{e4.8} -4\tau_1C_g+2\tau_1(2-C_\theta)+b_2=-2\varepsilon. \end{equation} By (B2), \eqref{e3.1}, Lemmas \ref{lem2.1}, \ref{lem2.3} and \ref{lem2.4}, we see that \begin{gather*} \lim _{d(x)\to 0}\tau_1\Theta^2(d(x)) g'(\psi(\tau_1\Theta^2(d(x))))=-C_g, \\ \lim _{d(x)\to 0}\Big(\frac {\theta'(d(x))\Theta(d(x))} {\theta^2(d(x))} +1+\frac {\Theta(d(x))} {\theta(d(x))} \Delta d(x)\Big)=2-C_\theta , \\ \begin{aligned} &\lim _{d(x)\to 0}\Big(\lambda \tau_1^q 2^q \frac{(\Theta(d(x)))^q}{(\theta(d(x)))^{2-q}(g(\psi(\tau_1\Theta^2(d(x)))))^{1-q}}\\ &+ \frac{\sigma} {\theta^2(d(x)) g(\psi(\tau_1\Theta^2(d(x))))}\Big)=0, \end{aligned} \\ \limsup_{d(x)\to 0}\frac{b(x)} {\theta^2(d(x))}\leq b_2. \end{gather*} Thus, corresponding to $\varepsilon,s_0$ and $\delta$, where $s_0$ is given in (G2) and $\delta$ in Lemma \ref{lem3.1}, respectively, there is $\delta_\varepsilon\in (0,\delta)$ sufficiently small such that for $x\in \Omega_{\delta_\varepsilon}$, $$ \bar{u}_\varepsilon=\psi(\tau_1\Theta^2(d(x))) $$ satisfies \begin{equation}\label{e4.9} \bar{u}_\varepsilon (x)\in (0, s_0),\quad x\in \Omega_{\delta_\varepsilon}, \end{equation} and \begin{align*} &\Delta \bar{u}_\varepsilon(x)+ b(x) g(\bar{u}_\varepsilon(x))+\lambda |\bar{u}_\varepsilon(x)|^q+\sigma\\ &=\psi''(\tau_1\Theta^2(d(x))) (2\tau_1 \Theta(d(x)) \theta(d(x)))^2+2\tau_1\psi'(\tau_1\Theta^2(d(x))) \\ &\quad\times \big(\theta^2(d(x)) +\Theta(d(x))\theta'(d(x))+\Theta(d(x))\theta(d(x))\Delta d(x)\big)\\ &\quad + b(x)g(\psi(\tau_1\Theta^2(d(x))))+\lambda (2\tau_1)^q (\theta(d(x)) \Theta(d(x)))^q (g(\psi(\tau_1\Theta^2(d(x)))))^q+\sigma \\ &= g(\psi(\tau_1\Theta^2(d(x))))\theta^2(d(x)) \Big(4\tau_1\tau_1\Theta^2(d(x)) g'(\psi(\tau_1\Theta^2(d(x))))\\ &\quad+2\tau_1 \big(\frac {\theta'(d(x))\Theta(d(x))} {\theta^2(d(x))} +1+\frac {\Theta(d(x))} {\theta(d(x))} \Delta d(x)\big)+\frac{b(x)}{\theta^2(d(x))}\\ &\quad+\lambda \tau_1^q 2^q \frac{(\Theta(d(x)))^q}{(\theta(d(x)))^{2-q}(g(\psi(\tau_1\Theta^2(d(x)))))^{1-q}} + \frac{\sigma} {\theta^2(d(x)) g(\psi(\tau_1\Theta^2(d(x))))}\Big)\\ & \leq 0; \end{align*} i.e., $\bar{u}_\varepsilon$ is a supersolution of equation \eqref{e1.1} in $\Omega_{\delta_\varepsilon}$. Let $u_\lambda\in C(\bar{\Omega})\cap C^{2+\alpha}(\Omega)$ be an arbitrary classical solution to \eqref{e1.1}. By Lemma \ref{lem3.2}, we see that there exists $M_0>0$ such that for $x\in \Omega_{\delta_\varepsilon}$, $$ u_\lambda(x)\leq \bar{u}_\varepsilon(x)+\lambda M_0 v_0(x), $$ i.e., $$ \frac{u_\lambda(x)}{\psi(\tau_1\Theta^2(d(x)))}\leq 1+\lambda M_0 \frac{v_0(x)}{\psi(\tau_1\Theta^2(d(x)))},\quad x\in \Omega_{\delta_\varepsilon}. $$ It follows by \eqref{e3.3} and Lemma \ref{lem2.3} (iv) that $$ \lim_{d(x) \to 0 } \sup \frac{u_\lambda(x)}{\psi(\tau_2\Theta^2(d(x)))} \leq 1. $$ Using Lemma \ref{lem2.3} again, we have $$ \lim_{d(x) \to 0 }\frac{\psi(\tau_1\Theta^2(d(x)))} {\psi(\Theta^2(d(x)))}={\tau_1}^{1-C_g}. $$ Moreover, since $C_\theta>0$, by \eqref{e2.9} and Lemma \ref{lem2.3}, we obtain that $$ \lim_{d(x) \to 0 }\frac{\psi(\Theta^2(d(x)))} {\psi(d^2(x)\theta^2(d(x)))}=C_\theta^{2(1-C_g)}. $$ Thus letting $\varepsilon\to 0$, we have \begin{equation}\label{e4.10} \lim_{d(x) \to 0 } \sup \frac{u_\lambda(x)}{\psi(\Theta^2(d(x)))} \leq \xi_2^{1-C_g}. \end{equation} Combining \eqref{e4.10} with \eqref{e4.4}, we obtain \eqref{e1.7}. In particular, when $C_g=1$, $u_\lambda$ satisfies $$ \lim_{d(x) \to 0}\frac{u_\lambda(x)}{\psi(\Theta^2(d(x)))}=1; $$ and, when $C_g<1$ and $b_1=b_2=b_0$ in (b1), $u_\lambda$ satisfies $$ \lim_{d(x) \to 0}\frac{u_\lambda(x)}{\psi(d^2(x)\theta^2(d(x)))}= (\xi_{01}C_\theta^2)^{1-C_g}. $$ This completes the proof of Theorem \ref{thm1.1}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.2}] Let $u_\lambda\in C^2(\Omega)\cap C(\bar{\Omega})$ be an arbitrary solution to \eqref{e1.1}. For $q\in [1, 2]$, in a similar way as that of Theorem \ref{thm1.1}, by using \eqref{e3.1}, \eqref{e3.3}, Lemmas \ref{lem3.1}, \ref{lem3.3}, \ref{lem3.4}, \ref{lem4.2} and \ref{lem2.3} (v), we can show that Theorem \ref{thm1.2} holds. Next we construct an appropriate supersolution near the boundary to \eqref{e1.1} in the case of $q\in (0, 1)$. Let $\varepsilon\in(0, b_1/4)$ and let $\tau_2=b_2+2\varepsilon$. It follows that $$ b_1/2<\tau_2<2b_2. $$ By (B3), (G1), \eqref{e3.1}, \eqref{e3.3}, Propositions \ref{prop2.3} (iii) and \ref{prop2.8}, and Lemma \ref{lem2.3}, we derive that \begin{gather*} \sigma \lim_{d(x)\to 0} \frac{ d^{2}(x)} {\hat{L}(d(x)) g(\psi(\tau_2h_1(d(x))))}=0, \\ \lambda \tau_2^q \lim_{d(x)\to 0}\big( (d(x))^{2-q}(\hat{L}(d(x)))^{q-1}(g(\psi(\tau_2h_1(d(x)))))^{1-q}\big)=0, \\ \lim_{d(x)\to 0}\tau_2 h_1(d(x)) g'(\psi(\tau_2 h_1(d(x))))=-C_g, \\ \lim_{d(x)\to 0}\frac {\hat{L}(d(x))}{h_1(d(x))}=0, \quad \limsup_{d(x)\to 0}\frac{b(x)}{d^{-2}(x)\hat{L}(d(x))}\leq b_2, \\ \lim_{d(x)\to 0}\tau_2\big(1-\frac {d(x)\hat{L}'(d(x))} {\hat{L}(d(x))}\big)=\tau_2, \quad \lim_{d(x)\to 0}\tau_2d(x) \Delta d(x)=0. \end{gather*} Thus, corresponding to $\varepsilon, s_0$ and $\delta$, where $s_0$ is given as in (G2) and $\delta$ in Lemma \ref{lem3.1}, respectively, there is $\delta_\varepsilon\in (0, \delta)$ sufficiently small such that for $x\in \Omega_{\delta_\varepsilon}$ $$ \bar{u}_\varepsilon= \psi(\tau_2h_1(d(x))) $$ satisfies \eqref{e4.9} and \begin{align*} &\Delta \bar{u}_\varepsilon(x)+b(x) g(\bar{u}_\varepsilon(x))+\lambda |\bar{u}_\varepsilon(x)|^q+\sigma\\ &= \psi''(\tau_2h_1(d(x)))\tau_2^2 h_1^2(d(x))+\psi'(\tau_2 h_1(d(x)))\big(\tau_2 h_1''(d(x))+\tau_2^2 h_1'(d(x))\Delta d(x)\big)\\ &\quad + b(x)g(\psi(\tau_2h(d(x))))+\lambda \tau^q \big(\frac {\hat{L}(d(x))}{d(x)}\big)^q (g(\psi(\tau_2h_1(d(x)))))^q+\sigma \\ &= (d(x))^{-2}\hat{L}(d(x))g(\psi(\tau_2h_1(d(x))))\\ &\quad\times \Big(\tau_2 \big(\tau_2 h_1(d(x)) g'(\psi(\tau_2 h_1(d(x))))\big)\frac {\hat{L}(d(x))}{h_1(d(x))}-\tau_2\Big(1-\frac {d(x)\hat{L}'(d(x))} {\hat{L}(d(x))}\Big)\\ &\quad +\tau_2d(x) \Delta d(x)+\frac{b(x)}{d^{-2}(x)\hat{L}(d(x))}+ \sigma\frac{ d^{2}(x)}{\hat{L}(d(x))}\frac {1}{g(\psi(\tau_2h_1(d(x))))}\\ &\quad +\lambda \tau_2^q (d(x))^{2-q}(\hat{L}(d(x)))^{q-1}\frac{1}{(g(\psi(\tau_2h_1(d(x)))))^{1-q}} \Big)\\ &\leq 0, \end{align*} i.e., $\bar{u}_\varepsilon$ is a supersolution to equation \eqref{e1.1} in $\Omega_{\delta_\varepsilon}$. The rest of the proof is the same as that Theorem \ref{thm1.1} and is omitted. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.3}] Let $u_\lambda\in C^2(\Omega)\cap C(\bar{\Omega})$ be an arbitrary solution to \eqref{e1.1}. For $q\in [1, 2]$, in a similar way as that of Theorem \ref{thm1.1}, by using \eqref{e3.1}, \eqref{e3.3}, Lemmas \ref{lem3.1}, \ref{lem3.3}, \ref{lem3.4} and \ref{lem4.3}, we can show that Theorem \ref{thm1.3} holds. Next we construct an appropriate supersolution near the boundary to \eqref{e1.1} in the case of $q\in (0, 1)$. Let $\varepsilon\in(0, 1)$. Let $\tau_3$ be the unique positive solution to the problem $$ b_2t^{-\gamma}-\frac {t}{1+\gamma}=- 2\varepsilon, $$ it follows by the properties of the function $b_it^{-\gamma}-\frac {t}{1+\gamma}$ ($i=1, 2$) that $$ (b_1(1+\gamma))^{1/{(1+\gamma)}}<\tau_3<\zeta_0,\quad \lim_{\varepsilon\to 0}\tau_3=(b_2(1+\gamma))^{1/{(1+\gamma)}}, $$ where $\zeta_0$ is the unique positive solution to the problem $$ b_2t^{-\gamma}-\frac {t}{1+\gamma}=- 2. $$ Since $\hat{L}$ and $\int_{t}^{\eta}\frac{\hat{L}(\tau)}{\tau}d\tau$ are slowly varying at zero, we see that by \eqref{e3.3}, (B4), Propositions \ref{prop2.3} and \ref{prop2.8} that \begin{gather*} \lim_{d(x)\to 0}\frac {d(x)}{\hat{L}(d(x))}\Big(\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)} {\tau}d\tau-\frac {1}{1+\gamma} \hat{L}(d(x))\Big)\Delta d(x)=0, \\ \lim_{d(x)\to 0} \frac {\gamma}{(1+\gamma)^2} \frac {\hat{L}(d(x))}{\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau}=0, \quad \lim_{d(x)\to 0} \frac {1}{1+\gamma} \frac {d(x)\hat{L}'(d(x))}{\hat{L}(d(x))}=0, \\ \lim_{d(x)\to 0}\sup\frac {b(x)} {(d(x))^{\gamma-1}\hat{L}(d(x))} \leq b_2, \end{gather*} and, using (G1) and (B4), there holds \begin{gather*} \sigma \tau_3^{-1}\lim_{d(x)\to 0} \frac {d(x)}{\hat{L}(d(x))} \Big(\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{1/{(1+\gamma)}}=0;\\ \mu \tau_3^{p-1}\lim_{d(x)\to 0}(d(x))^{\gamma+p} \Big(\int_{d(x)}^{\eta}\frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{(\gamma+p)/{(1+\gamma)}}=0;\\ \begin{aligned} &\lambda \tau_3^{q-1}\lim_{d(x)\to 0}\Big( d(x) \Big(\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{(q+\gamma)/{(1+\gamma)}}\\ &\times \Big |1-\frac {1}{1+\gamma}\hat{L}(d(x)) \Big(\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{-1}\Big|^q\Big)=0. \end{aligned} \end{gather*} Thus, corresponding to $\varepsilon,\ s_0$ and $\delta$, where $s_0$ is given as in (G2) and $\delta$ in Lemma \ref{lem3.1}, respectively, there is $\delta_\varepsilon\in (0, \delta)$ sufficiently small such that for $x\in \Omega_{\delta_\varepsilon}$ $$ \bar{u}_\varepsilon= \tau_3 d(x)\Big(\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{1/{(1+\gamma)}}$$ satisfies \eqref{e4.2} and \begin{align*} &\Delta \bar{u}_\varepsilon(x)+ b(x) ((\bar{u}_\varepsilon(x))^{-\gamma}+\mu (\bar{u}_\varepsilon(x))^{p}) +\lambda |\bar{u}_\varepsilon(x)|^q+\sigma\\ &= \tau_3 \frac {\hat{L}(d(x))}{d(x)}\Big(\int_{d(x)}^{\eta}\frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{-\gamma/{(1+\gamma)}}\bigg(-\frac {1}{1+\gamma}- \frac {\gamma}{(1+\gamma)^2}\frac {\hat{L}(d(x))}{\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau}\\ % &\quad - \frac {1}{1+\gamma}\frac {d(x)\hat{L}'(d(x))}{\hat{L}(d(x))} + \frac {d(x)}{\hat{L}(d(x))}\Big(\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau- \frac {1}{1+\gamma} \hat{L}(d(x))\Big)\Delta d(x) \\ &\quad + \frac {b(x)} {(d(x))^{\gamma-1}\hat{L}(d(x))}\Big(\tau_3^{-\gamma-1}+ \mu \tau_3^{p-1}(d(x))^{\gamma+p} \Big(\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{(\gamma+p)/{(1+\gamma)}}\Big)\\ &\quad + \sigma \tau_3^{-1}\frac {d(x)} {\hat{L}(d(x))} \Big(\int_{d(x)}^{\eta}\frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{\gamma/{(1+\gamma)}}\\ &\quad +\lambda \tau_3^{q-1} d(x) \Big(\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{(q+\gamma)/{(1+\gamma)}}\Big|1-\frac {1}{1+\gamma}\hat{L}(d(x)) \Big(\int_{d(x)}^{\eta} \frac {\hat{L}(\tau)}{\tau}d\tau \Big)^{-1}\Big|^q\bigg)\\ & \leq 0, \end{align*} i.e., $\bar{u}_\varepsilon$ is a supersolution of \eqref{e1.1} in $\Omega_{\delta_\varepsilon}$. 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