\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 136, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/136\hfil Boundary behavior of large solutions ] {Boundary behavior of large solutions for semilinear elliptic equations in borderline cases} \author[Z. Zhang \hfil EJDE-2012/136\hfilneg] {Zhijun Zhang} % in alphabetical order \address{Zhijun Zhang \newline School of Mathematics and Information Science, Yantai University, Yantai, Shandong, 264005, China} \email{zhangzj@ytu.edu.cn, chinazjzhang@yahoo.com.cn} \thanks{Submitted June 30, 2012. Published August 19, 2012.} \thanks{Supported by grant 10671169 from NNSF of China and 2009ZRB01795 from NNSF of \hfill\break\indent Shandong Province} \subjclass[2000]{35J55, 35J60, 35J65} \keywords{Semilinear elliptic equations; boundary blow-up; boundary behavior;\hfill\break\indent borderline cases} \begin{abstract} In this article, we analyze the boundary behavior of solutions to the boundary blow-up elliptic problem $$\Delta u =b(x)f(u), \quad u\geq 0,\; x\in\Omega,\; u|_{\partial \Omega}=\infty,$$ where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $f(u)$ grows slower than any $u^p$ ($p > 1$) at infinity, and $b \in C^{\alpha}(\bar{\Omega})$ which is non-negative in $\Omega$ and positive near $\partial\Omega$, may be vanishing on 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 boundary blow-up elliptic problem $$\label{e1.1} \Delta u =b(x)f(u), \ u\geq 0, \quad x\in \Omega,\quad u|_{\partial \Omega}=\infty,$$ where the last condition means that $u(x) \to \infty$ as $d(x)=\operatorname{dist}(x, \partial \Omega) \to 0$, $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $f$ satisfies \begin{itemize} \item[(F1)] $f\in C[0, \infty) \cap C^1(0, \infty)$, $f(0)=0$ and $f(s)$ is increasing on $(0, \infty)$; \item[(F2)] the Keller-Osserman (\cite{KE}, \cite{OS}) condition $$\Theta (r):= \int_r^\infty \frac{ds}{\sqrt{2F(s)}}<\infty, \quad \forall r>0,\quad F(s)=\int_0^s f(\tau) d\tau;$$ \end{itemize} the function $b$ satisfies \begin{itemize} \item[(B1)] $b \in C^{\alpha}(\bar{\Omega})$, is non-negative in $\Omega$ and positive near $\partial\Omega$. \end{itemize} The model problem \eqref{e1.1} arises from many branches of mathematics and has generated a good deal of research, see, for instance, \cite{AGQ}-\cite{BM}, \cite{CD}-\cite{DDGR}, \cite{KE}-\cite{LM}, \cite{OS}-\cite{Z} and the references therein. When $b\equiv 1$ in $\Omega$ and $f$ satisfies (F1), it is well-known that \eqref{e1.1} has one solution $u\in C^2(\Omega)$ if and only if (F2) holds. Moreover, the blow-up rate of $u(x)$ near $\partial \Omega$ can be described by (see, e.g., \cite{BM} and \cite[Theorem 6.8]{DU}) $$\label{e1.2} \frac{\Theta (u(x))}{d(x)} \to 1 \quad\text{as } d(x) \to 0.$$ Moreover, if one assumes that $$\label{e1.3} \liminf_{r\to\infty} \frac {\Theta (\lambda r)}{\Theta (r)}> 1,\quad \forall \lambda\in (0, 1),$$ then it holds (see \cite{BM}) $$\label{e1.4} \frac{u(x)}{\phi(d(x))} \to 1 \quad\text{as } d(x) \to 0,$$ where $\phi$ is the inverse of $\Theta$; i.e., $\phi$ satisfies $$\label{e1.5} \int_{\phi(t)}^\infty \frac{ds}{\sqrt{2F(s)}}=t, \quad \forall t>0.$$ However, there are less results for the boundary behavior of the solution to problem \eqref{e1.1} under the condition that $$\label{e1.6} \lim_{r\to\infty} \frac {\Theta (\lambda r)}{\Theta (r)} =1,\quad \forall \lambda\in (0, 1).$$ When $f$ satisfies \begin{itemize} \item[(A)] $f$ is locally Lipschitz continuous and non-negative on $[0, \infty)$, and $f(s)/s$ is increasing on $(0,\ \infty)$; \item[(B)] $f(s) = C_1^2s(\ln s)^{2\alpha}+ C_2 s(\ln s)^{2\alpha-1}(1 + o(1))$ as $s \to \infty$ with $C_1>0$, $\alpha>1$ and $C_2\in \mathbb{R}$, \end{itemize} C\^{i}rstea and Du \cite{CD} first showed that problem \eqref{e1.1} has a unique solution $u$ satisfying $$\label{e1.7} \lim_{d(x)\to\infty} \frac {u(x)} {\exp \big((C_1(\alpha-1) K(d(x)))^{-1/(\alpha-1)}\big)} =\exp (\xi_0),$$ where $$\label{e1.8} \xi_0=\frac {1}{2}-\frac {C_2}{2\alpha C_1^2}.$$ Then they extended the above result to weight $b$ which can be vanishing on the boundary. It is worthwhile to point out that \eqref{e1.7} depends not only on $C_1^2s(\ln s)^{2\alpha}$ but also on the lower term $C_2 s(\ln s)^{2\alpha-1}$ in (B). This is completely different from the case $f(s) = s^p[C_1 + o(1)]$ as $s \to \infty$ for some $p > 1$, since problem \eqref{e1.1} has a unique positive solution $u$ which satisfies $$\lim_{d(x)\to 0}u(x)(d(x))^{2/(p-1)}= \Big(\frac {2(p+1)}{C_1(p-1)^2}\Big)^{1/(p-1)}$$ in such a situation and $b\equiv 1$ in $\Omega$ (see \cite{BM}). On the other hand, when $b\equiv 1$ in $\Omega$, $f$ satisfies (F1), (F2) and the conditions that \begin{itemize} \item[(F03)] there exists $\alpha>1$ such that $$\frac {2F(s) f'(s)}{f^2(s)}=1-(\alpha +o(1))(\ln s)^{-1}\quad\text{as } s\to \infty;$$ \item[(F04)] there exist $\theta_0\in (0, 1)$ and $S_0>1$ such that $$\theta f(s)\geq f( \theta s),\quad \forall \theta \in (\theta_0, 1),\; \forall s>S_0;$$ \item[(F05)] there exist $C_0>0$ and $S_1\geq S_0$ such that $$\frac {s^2 |f''(\theta s)|}{f''(s)}\leq C_0 (\ln s)^{-1},\quad \forall s>S_1,\ \forall\, \theta \in (1/2, 2),$$ \end{itemize} Anedda and Porru \cite{AP} showed that for any $\varepsilon>0$ there is $C_\varepsilon > 0$ such that the solution of problem \eqref{e1.1} satisfying \begin{align*} & 1+ \frac {(\alpha- 1)(N-1)}{ 2(2\alpha-1)} K(x)d(x)-\varepsilon d(x)- C_\varepsilon d^2(x) \\ &< \frac {u(x)}{ \phi(d(x))} \\ &< 1+ \frac {(\alpha- 1)(N-1)}{ 2(2\alpha-1)} K(x)d(x)+\varepsilon d(x)+ C_\varepsilon d^2(x), \end{align*} where $K(x)$ is the mean curvature of the surface $\{x\in \Omega: d(x) = {\rm constant} \}$. We also note that an example which satisfies the above requirements is the following $$f(s) = 0,\quad s\in [0, 1],\quad f (s) = s (\ln s)^{2\alpha}, \quad s>1, \quad \alpha>1.$$ Inspired by the above works, in this article, we analyze the boundary behavior of solutions to problem \eqref{e1.1} for more general $f$ which satisfies the condition \eqref{e1.6}. In particular, we consider functions $f$ which satisfy (F1), (F2) and the following conditions that \begin{itemize} \item[(F3)] there exist two functions $f_1\in C^1[S_0,\ \infty)$ for some large $S_0>0$ and $f_2$ such that $$f(s): =f_1(s)+f_2(s),\quad s\geq S_0;$$ \item[(F4)] $$\label{e1.9} \frac {f_1'(s)s}{f_1(s)}:=1+g(s), \quad s\geq S_0,$$ with $g\in C^1[S_0,\ \infty)$ satisfying \begin{gather}\label{e1.10} g(s)>0,\quad s\geq S_0,\quad \lim_{s\to \infty}g(s)=0,\\ \label{e1.11} \lim_{s\to \infty}\frac {sg'(s)}{g(s)}=0, \quad \lim_{s\to \infty}\frac {sg'(s)}{g^2(s)}=C_g\in \mathbb{R}, \quad \lim_{s\to \infty} \frac {\sqrt{\frac {s}{f_1(s)}}}{g(s)}=0; \end{gather} \item[(F5)] either there exists a constant $E_1\neq 0$ such that $$\label{e1.12} \lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)}=E_1$$ or $$\label{e1.13} \lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)}=0$$ and there exists a constant $\mu\leq 1$ such that $$\label{e1.14} \lim_{s\to \infty}\frac{f_2(\xi s)}{f_2(s)}=\xi^\mu,\quad \forall \xi>0.$$ \end{itemize} Our main result is stated using the assumption \begin{itemize} \item[(B2)] There exist $k\in \Lambda$ and a positive constant $b_0$ such that $$\lim_{d(x) \to 0 } \frac{b(x)}{(k(d(x)))^{2}}=b_0^2,$$ where $\Lambda$ denotes the set of all positive non-decreasing functions in $C^1(0,\delta_0)$ ($\delta_0>0$) which satisfy $$\label{e1.15} \lim_{t \to 0^+} \frac {d}{dt}\left(\frac{K(t)}{k(t)}\right): = D_k\in [0, \infty),\ \ K(t)=\int_0^t k(s)ds,$$ \end{itemize} \begin{theorem}\label{thm1.1} Let $f$ satisfy {\rm (F1)--(F5)}. If $b$ satisfies {\rm (B1)--(B2)}, then for any solution $u$ of problem \eqref{e1.1}, $$\label{e1.16} \lim_{d(x)\to 0}\frac {u(x)}{\psi(b_0K(d(x)))}=\exp (\xi_0),$$ where $$\label{e1.17} \begin{gathered} \xi_0=\frac {1}{2}-E_2-(1-D_k)\big(\frac {1}{2}+C_g\big),\\ E_2= \begin{cases} E_1 &\text{if \eqref{e1.12} holds};\\ 0, &\text{if \eqref{e1.13} and \eqref{e1.14} hold}, \end{cases} \end{gathered}$$ and $\psi$ is the unique solution of the problem $$\label{e1.18} \int_{\psi(t)}^\infty \frac {ds} {\sqrt{sf_1(s)}}=t, \quad \forall t>0.$$ \end{theorem} \begin{remark}\label{rmk1.1} \rm (F3), \eqref{e1.10}, and \eqref{e1.12} or \eqref{e1.13} imply $$\lim_{s\to \infty} \frac {f_2(s)}{f(s)}=0,\quad \lim_{s\to\infty}\frac {f_1(s)}{f(s)}=1.$$ \end{remark} \begin{remark}\label{rmk1.2} \rm Some basic examples which satisfy all our requirements are the following: \begin{itemize} \item[(1)] $f_1(s)=C_1^2s(\ln s)^{2\alpha}$ in (F3), where $\alpha>1$, \begin{gather*} g(s)=2\alpha (\ln s)^{-1}; \quad \lim_{s\to \infty} \frac {\sqrt{\frac {s}{f_1(s)}}}{g(s)}= \frac {1}{2\alpha C_1}\lim_{s\to \infty}(\ln s)^{-(\alpha-1)}=0; \\ \frac {sg'(s)} {g^2(s)}\equiv C_g=-\frac {1}{2\alpha};\quad \lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)}=\frac {1}{2\alpha C_1^2}\lim_{s\to \infty}\frac {f_2(s)}{s(\ln s)^{2\alpha-1}}=E_2; \\ \psi(t)=\exp\big(C_1(\alpha-1) t\big)^{-1/(\alpha-1)}. \end{gather*} In particular, when $f_2(s)=C_2 s^\mu (\ln s)^\beta$ with $\beta\leq 2\alpha-1$, $E_1=0$ for $\mu<1$ or $\mu=1$ and $\beta< 2\alpha-1$, and $E_1=\frac {C_2}{2\alpha C_1^2}$ for $\mu=1$ and $\beta= 2\alpha-1$. \item[(2)] $f_1(s)=C_1^2 s e^{(\ln s)^{q}}$ in (F3), where $q\in (0, 1)$, \begin{gather*} g(s)=q(\ln s)^{-(1-q)};\quad \lim_{s\to \infty} \frac {\sqrt{\frac {s}{f_1(s)}}}{g(s)}= \frac {1}{q C_1}\lim_{s\to \infty} \frac { \exp (-\frac {1}{2}(\ln s)^q)}{(\ln s)^{-(1-q)}}=0; \\ \lim_{s\to \infty}\frac {sg'(s)}{g^2(s)}=-\frac {1-q}{q} \lim_{s\to \infty} (\ln s)^{-q}=C_g=0; \\ \lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)}=\frac {1}{q C_1^2}\lim_{s\to \infty}\frac {f_2(s)}{s(\ln s)^{-(1-q)}\exp ((\ln s)^q)}=E_2; \\ \int_{\ln (\psi(t))}^\infty\exp(-s^q/2)ds=C_1 t. \end{gather*} \item[(3)] $f_1(s)=C_1^2 s(\ln s)^{2}(\ln (\ln s))^{2\alpha}$ in (F3), where $\alpha>1$, \begin{gather*} g(s)=2(\ln s)^{-1}\big(1+\alpha (\ln (\ln s))^{-1}\big); \\ \lim_{s\to \infty} \frac {\sqrt{\frac {s}{f_1(s)}}}{g(s)}= \frac {1}{2 C_1}\lim_{s\to \infty}\frac {(\ln (\ln s))^{-\alpha}}{1+\alpha(\ln (\ln s))^{-1}}=0; \\ \lim_{s\to \infty} \frac {sg'(s)}{g^2(s)} =-\lim_{s\to \infty} \frac {1+\alpha(\ln (\ln s))^{-1} +\alpha(\ln (\ln s ))^{-2}} {2(1+\alpha(\ln (\ln s ))^{-1})^2}=C_g=-\frac {1}{2}; \\ \lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)} =\frac {1}{2 C_1^2}\lim_{s\to \infty}\frac {f_2(s)}{s \ln s (\ln (\ln s))^{2\alpha}(1+\alpha(\ln (\ln s ))^{-1})}=E_2; \\ \psi(t)=\exp\big(\exp\big(C_1(\alpha-1)t\big)^{-1/(\alpha-1)}\big). \end{gather*} \end{itemize} \end{remark} \begin{remark}\label{rmk1.3} \rm When $f$ further satisfies the condition $f(s)/s$ being increasing on $(0, \infty)$, in a similar proof in \cite{CD}, problem \eqref{e1.1} has a unique solution. \end{remark} \begin{remark}\label{rmk1.4} \rm For the existence of the minimal solution to problem \eqref{e1.1}, see \cite{LA}. \end{remark} \begin{remark}\label{rmk1.5} \rm For each $k\in \Lambda$, $D_k\in [0, 1]$ and $$\label{e1.19} \lim_{t \to 0^+} \frac{K(t)}{k(t)}=0,\quad \lim_{t \to 0^+} \frac{K(t)k'(t)}{k^2(t)}=1-\lim_{t \to 0^+} \frac {d}{dt}\Big(\frac{K(t)}{k(t)}\Big)=1-D_k.$$ \end{remark} \section{Preliminaries} Our approach relies on Karamata regular variation theory established by Karamata in 1930 which is a basic tool in stochastic process (see, for instance, Bingham, Goldie and Teugels \cite{BGT}, Maric \cite{MA} and the references therein.), and has been applied to study the asymptotic behavior of solutions to differential equations and problem \eqref{e1.1} (see Maric \cite{MA}, C\^{i}rstea and R\v{a}dulescu \cite{CR}, R\v{a}dulescu \cite{RAD}, C\^{i}rstea and Du \cite {CD}, the authors \cite{Z} and the references therein.). In this section, we present some bases of Karamata regular variation theory. \begin{definition}\label{def2.1}\rm A positive measurable function $f$ defined on $[a,\infty)$, for some $a>0$, is called \emph{regularly varying at infinity} with index $\rho$, written $f \in RV_\rho$, if for each $\xi>0$ and some $\rho \in \mathbb{R}$, $$\label{e2.1} \lim_{t \to \infty} \frac{f(\xi t)}{f(t)}= \xi^\rho.$$ \end{definition} In particular, when $\rho=0$, $f$ is called \emph{slowly varying at infinity}. Clearly, if $f\in RV_\rho$, then $L(t):\ =f(t)/{t^\rho}$ is slowly varying at infinity. Some basic examples of slowly varying functions at infinity are \begin{itemize} \item[(i)] every measurable function on $[a, \infty)$ which has a positive limit at infinity; \item[(ii)] $(\ln t)^q$ and $\big(\ln (\ln t)\big)^q$, $q\in \mathbb{R}$; \item[(iii)] $e^{(\ln t)^q}$, $00$, is \emph{regularly varying at zero} with index $\rho$ (and denoted by $g \in RVZ_\rho$) if $t\to g(1/t)$ belongs to $RV_{-\rho}$. \begin{proposition}[Uniform convergence theorem]\label{pro2.1} If $f\in RV_\rho$, then \eqref{e2.1} holds uniformly for $\xi \in [c_1, c_2]$ with $00$; if $\rho>0$, then uniform convergence holds on intervals $(0, a_1]$ provided $f$ is bounded on $(0, a_1]$ for all $a_1>0$. \end{proposition} \begin{proposition}[Representation theorem]\label{prop2.2} A function $L$ is slowly varying at infinity if and only if it may be written in the form $$\label{e2.2} L(t)=\varphi(t) \exp \Big( \int_{a_1}^t \frac {y(\tau)}{\tau} d\tau \Big), \quad t \geq a_1,$$ for some $a_1\geq a$, where the functions $\varphi$ and $y$ are measurable and for $t \to \infty$, $y(t)\to 0$ and $\varphi(t)\to c_0$, with $c_0>0$. \end{proposition} We say that $$\label{e2.3} \hat{L}(t)=c_0 \exp \Big( \int_{a_1}^t\frac {y(\tau)}{\tau} d\tau \Big), \quad t \geq a_1,$$ is \emph{normalized} slowly varying at infinity and $$\label{e2.4} f(t)=t^\rho\hat{L}(t), \quad t \geq a_1,$$ is \emph{normalized} regularly varying at infinity with index $\rho$ (and written $f\in NRV_\rho$). A function $f\in RV_\rho$ belongs to $NRV_\rho$ if and only if $$\label{e2.5} f\in C^1[a_1, \infty),\text{ for some a_1>0 and } \lim_{t \to \infty} \frac{tf'(t)}{f(t)}=\rho.$$ Then, we see that $f_1\in NRV_1$, $f_2\in RV_\mu$, $f\in RV_1$ and $g$ is normalized slowly varying at infinity in (F3)-(F5). Similarly, $g$ is called \emph{normalized} regularly varying at zero with index $\rho$, and denoted by $g \in NRVZ_\rho$, if $t\to g(1/t)$ belongs to $NRV_{-\rho}$. \begin{proposition}\label{prop2.3} If functions $L, L_1$ are slowly varying at infinity, then \begin{itemize} \item[(i)] $L^\rho$ (for every $\rho\in \mathbb{R}$), $L\circ L_1$ (if $L_1(t)\to \infty$ as $t\to \infty$) , are also slowly varying at infinity. \item[(ii)] For every $\rho >0$ and $t\to \infty$, $$t^{\rho} L(t)\to \infty, \quad t^{-\rho} L(t)\to 0.$$ \item[(iii)] For $\rho\in\mathbb{R}$ and $t\to \infty$, $\ln (L(t))/{\ln t}\to 0$ and $\ln (t^\rho L(t))/{\ln t}\to \rho$. \end{itemize} \end{proposition} Our results in the section are summarized as follows. \begin{lemma}[{\cite[Lemma 2.1]{Z}}] \label{lem2.1} Let $k\in \Lambda$. \begin{itemize} \item[(i)] When $D_k\in (0, 1)$, $k$ is \emph{normalized} regularly varying at zero with index $(1-D_k)/{D_k}$; \item[(ii)] when $D_k=1$, $k$ is normalised slowly varying at zero; \item[(iii)] when $D_k=0$, $k$ grows faster than any $t^p$ ($p > 1$) near zero. \end{itemize} \end{lemma} Denote $$\label{e2.6} \Theta(r)=\int_r^\infty\frac {ds}{\sqrt{2F(s)}},\quad \Theta_1(r)=\int_r^\infty\frac {ds}{\sqrt{s f_1(s)}}, \quad r>0.$$ Then $$\label{e2.7} \Theta'(r)=-\frac {1}{\sqrt{2F(r)}},\quad \ \Theta_1'(r)=-\frac {1}{\sqrt{r f_1(r)}},\quad r>0.$$ \begin{lemma}\label{lem2.2} Under the hypotheses in Theorem \ref{thm1.1}: \begin{itemize} \item[(i)] $$\lim_{r\to\infty} \frac {\Theta (\lambda r)}{\Theta ( r)} = \lim_{r\to\infty} \frac {\Theta_1 (\lambda r)}{\Theta_1 ( r)}=1,\quad \forall \lambda\in (0, 1);$$ \item[(ii)] $$\lim_{r\to \infty} \frac {(r/ f_1(r))^{1/2} } {\Theta_1(r)g(r)}=\frac {1}{2}+C_g;$$ \item[(iii)] $$\lim_{r\to \infty} \frac {\frac {f_1(\xi r)} {\xi f_1(r)}-1}{g(r)}= \ln \xi$$ uniformly for $\xi \in [c_1, c_2]$ with $00. $$Then \begin{itemize} \item[(i)] -\psi'(t)=\sqrt{\psi(t)f_1(\psi(t))}, \psi(t)>0, t>0, \psi(0):=\lim_{t\to 0^+}\psi(t)=\infty, \psi''(t)=\frac {1}{2}\big(f_1(\psi(t))+\psi(t)f_1'(\psi(t))\big), t>0; \item[(ii)]$$\lim _{t\to 0}\big(g(\psi(t))\big)^{-1}\Big(\frac {1}{2}\big( 1+\frac {\psi(t)f_1'(\psi(t))}{f_1(\psi(t))}\big)-\frac {f_1(\xi \psi(t))}{\xi f_1(\psi(t))}\Big)=\frac {1}{2}-\ln \xi; $$\item[(iii)]$$\lim _{t\to 0} \frac {\sqrt{\psi(t)f_1(\psi(t))}} {t g(\psi(t))f_1(\psi(t))}=\frac {1}{2}+C_g; $$\item[(iv)]$$ \lim _{t\to 0} \frac {f_2(\xi\psi(t))}{\xi g(\psi(t))f_1(\psi(t))}=E_2 $$uniformly for \xi \in [c_1, c_2] with 00, \quad x\in\Omega,\quad v|_{\partial\Omega}=0. By the H\"{o}pf maximum principle \cite[Lemma 3.4]{GT}, we see that $$\label{e3.2} \nabla v_0 (x) \neq 0, \; \forall x \in \partial \Omega \text{ and } c_1 d(x)\leq v_0(x) \leq c_2 d(x),\; \forall x \in \Omega,$$ where c_1, c_2 are positive constants. Denote \varsigma_0=\exp (\xi_0), where \xi_0 is given in \eqref{e1.17},$$ \varsigma_2=\varsigma_0+\varepsilon,\quad \varsigma_1=\varsigma_0-\varepsilon,\quad \varepsilon\in (0,\min\{\varsigma_0,\ b_0^2\}/2). $$It follows that$$\varsigma_0/2<\varsigma_1<\varsigma_2<2\varsigma_0, \quad \lim_{\varepsilon\to 0}\varsigma_1 =\lim_{\varepsilon\to 0}\varsigma_2=\varsigma_0.$$Since$\ln(1+s)\cong s$as$s\to 0^+$, we can choose$\varepsilon$sufficiently small such that \begin{gather}\label{e3.3} \ln (\varsigma_0)-\ln (\varsigma_2)=\ln \big(1- \frac {\varepsilon}{\varsigma_0+\varepsilon}\big) < -\frac {1}{4\varsigma_0}\varepsilon; \\ \label{e3.4} \ln (\varsigma_0)-\ln (\varsigma_1)=\ln \big(1+ \frac {\varepsilon}{\varsigma_0-\varepsilon}\big)> \frac {1}{4 \varsigma_0}\varepsilon. \end{gather} Fix the above$\varepsilon$. For any$\delta>0$, we define$\Omega_\delta =\{x\in\Omega: 0