\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 05, pp. 1--17.\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-2015/05\hfil Comparison results for ellipti inequalities] {Comparison results for elliptic variational inequalities related to Gauss measure} \author[Y. Tian, C. Ma \hfil EJDE-2015/05\hfilneg] {Yujuan Tian, Chao Ma} % in alphabetical order \address{Yujuan Tian (corresponding author) \newline School of Mathematical Sciences, Shandong Normal University, Jinan, Shandong 250014, China} \email{tianyujuan0302@126.com} \address{Chao Ma \newline School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China} \email{chaos\_ma@163.com} \thanks{Submitted November 19, 2014. Published Janaury 5, 2015.} \subjclass[2000]{35J86, 35J70, 35B45} \keywords{Comparison results; rearrangements; gauss measure; \hfill\break\indent Elliptic variational inequality} \begin{abstract} In this article, we study linear elliptic variational inequalities that are defined on a possibly unbounded domain and whose ellipticity condition is given in terms of the density of Gauss measure. Using the notion of rearrangement with respect to the Gauss measure, we prove a comparison result with a problem of the same type defined in a half space, with data depending only on the first variable. \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} This article concerns the problem \begin{equation} \begin{gathered} a(u,\psi-u)\geq\int_\Omega f(\psi-u)\varphi\,dx, \quad \forall\psi\in H_0^1(\varphi,\Omega),\; \psi\geq0,\\ u\in H_0^1(\varphi,\Omega),\quad u\geq0, \end{gathered}\label{P1} \end{equation} where \begin{align*} a(u,\psi-u)&=\int_\Omega\sum _{i,j=1}^{n}a_{ij}D_{i}uD_j(\psi-u)\,dx -\int_\Omega\sum _{i=1}^{n}b_iuD_i(\psi-u)\,dx\\ &\quad +\int_\Omega\sum _{i=1}^{n}d_iD_{i}u(\psi-u)\,dx +\int_\Omega cu(\psi-u)\,dx, \end{align*} $\varphi(x)=(2\pi)^{-\frac{n}{2}}e^{-\frac{|x|^{2}}{2}}$ is the density of Gauss measure, $\Omega$ is an open subset of $\mathbb{R}^{n}(n\geq2)$ with Gauss measure less than one, $a_{ij}$, $b_i$, $d_i$, $c$ and $f$ are measurable functions on $\Omega$ that satisfy the following assumptions: \begin{itemize} \item[(A1)] $a_{ij}/\varphi, c/\varphi\in L^{\infty}(\Omega)$, $f\in L^{2}(\varphi,\Omega)$; \item[(A2)] $\sum _{i,j=1}^{n}a_{ij}(x)\xi_{i}\xi_{j}\geq\varphi(x)|\xi|^{2}$, a.e. $x\in\Omega,\forall\xi\in \mathbb{R}^{n}$; \item[(A3)] $\big(\sum _{i=1}^{n}|b_i(x)+d_i(x)|^2\Big)^{1/2}\leq B\varphi(x)$, a.e. $x\in\Omega$, $B>0$; \item[(A4)] $\sum _{i=1}^{n}D_ib_i(x)+c(x)\geq c_0(x)\varphi(x)$ in $\mathcal{D}'(\Omega)$, $c_0 \in L^{\infty}(\Omega)$; \end{itemize} We obtain a priori estimates for the solutions of \eqref{P1} using rearrangement techniques. As $\Omega$ is bounded, the operator is uniformly elliptic. This issue has been studied by many authors, firstly by Weinberger \cite{Weinberger} and Talenti \cite{Talenti1}. Actually it is well known that, one can use Schwarz symmetrization to estimate the solutions of elliptic and parabolic equations in terms of the solutions of (one dimentional) radially symmetric problems (for a comprehensive bibliography see the paper \cite{Diaz 1} and \cite{Trombetti}). For variational inequalities, similar comparison results in terms of linear elliptic variational inequalities can be found, for example, in \cite{Alvino2, Alvino3, Maderna, Posteraro2}; while nonlinear elliptic variational inequalities are discussed, for example, in \cite{Bandle2, Posteraro1}. In the same case, comparison results for parabolic variational inequalities can be found in \cite{Diaz, Ferone}. In the elliptic variational inequalities \eqref{P1}, since $\Omega$ maybe unbounded, the degeneracy of the operator does not allow to use the classical approach via Schwarz symmetrization. Based on the structure of the problem, it is more appropriate to use the Gauss symmetrization as has been done for elliptic and parabolic equations in \cite{Betta, Chiacchio, Di Blasio1, Di Blasio2}. Our aim is to compare the solutions of problem \eqref{P1} with the symmetric solutions of a problem in which the data depend only on the first variable and the domain is a half-space, i.e. the following ``symmetrized" problem \begin{equation} \begin{gathered} a^\sharp(v,\psi-v)\geq\int_{\Omega^\sharp} f^\sharp\varphi(\psi-v)dx, \quad \forall\psi\in H_0^1(\varphi,\Omega^\sharp),\; \psi\geq0,\\ v\in H_0^1(\varphi,\Omega^\sharp),\quad v\geq0, \end{gathered} \label{P2} \end{equation} where $$ a^\sharp(v,\psi-v)=\int_{\Omega^\sharp} \varphi D_{1}vD_1(\psi-v)\,dx -\int_{\Omega^\sharp} B\varphi D_1v(\psi-v)\,dx +\int_{\Omega^\sharp} c_{0\sharp}\varphi v(\psi-v)\,dx , $$ where $\Omega^\sharp$ is a half space with the same Gauss measure as $\Omega$, $f^{\sharp}$ is the Gauss symmetrization of $f$ and $c_{0\sharp}$ is the decreasing Gauss symmetrization of $c_0$. To this end, by following arguments in \cite{Alvino2, Posteraro2}, we first discuss the existence of symmetric solutions to the ``symmetrized" problem \eqref{P2}, which is a key step for the comparison results. However, in the equation case, the papers \cite{Di Blasio1, Di Blasio2} always assume that the ``symmetrized" problem has a symmetric solution instead of studying the existence conditions for such solutions. Our results (Theorem \ref{thm3.1}) make up for that in large extent. In addition, as an application of the comparison results, we prove an estimates of the Lorentz-Zygmund norm of $u$ in terms of the norm of the symmetric solutions $v$. The main tools we use are Gauss symmetrization and the properties of the weighted rearrangement. It is worth noting that the method used in equation case for obtaining the comparison results can not be applied to the variational inequalities \eqref{P1}. In this paper, we combine the property of the first eigenvalue (Lemma \ref{lem4.3}) with the maximum principle to overcome the difficulties and get the desired results. This article is organized as follows: Section 2 is devoted to give some notation and preliminary results; in Section 3, the main results of this paper are stated; in Section 4, we finish the proof of the main results. \section{Notation and preliminary results} In this section, we recall some definitions and results which will be useful in what follows. First, we recall that the wieghted Sobolev space $W_0^{1,p}(\varphi,\Omega)$ is the closure of $C_0^{\infty}(\Omega)$ under the norm $$ \|u\|_{W^{1,p}(\varphi,\Omega)}=\Big(\int_{\Omega}|\nabla u(x)|^p\varphi\,dx +\int_{\Omega}|u(x)|^p\varphi\,dx\Big)^{\frac{1}{p}}. $$ When $p=2$, the space $W_0^{1,2}(\varphi,\Omega)$ is also denoted by $H_0^1(\varphi,\Omega)$. Let $\gamma_{n}$ be the n-dimensional normalized Gauss measure on $\mathbb{R}^{n}$ defined as $$ d\gamma_{n}=\varphi(x)dx=(2\pi)^{-\frac{n}{2}}\exp \big(-\frac{|x|^2}{2}\big)\,dx,\quad x\in \mathbb{R}^{n}. $$ Set \begin{align*} \Phi(\tau) &=\gamma_{n}(\{x\in\mathbb{R}^{n}:x_{1}>\tau\})\\ &=(2\pi)^{-1/2}\int_{\tau}^{+\infty} \exp(-\frac{t^2}{2})dt,\quad \forall\tau\in \mathbb{R}\cup\{-\infty,+\infty\}. \end{align*} In \cite{Ledoux} we observe that \begin{equation} \label{2.1} \lim _{t\to 0^{+},1^{-}}(2\pi)^{-1/2} \frac{\exp(-\frac{\Phi^{-1}(t)^{2}}{2})}{t(2\log \frac{1}{t})^{1/2}}=1. \end{equation} \begin{remark}[\cite{Tian}] \label{rmk2.1} \rm By $\lim _{t\to 0^{+}}\frac{t(2\log\frac{1}{t})^{1/2}}{t(1-\log t)^{1/2}}=\sqrt{2}$ and $\lim _{t\to 1^{-}}\frac{t(2\log\frac{1}{t})^{1/2}}{t(1-\log t)^{1/2}}=0$ and note that \eqref{2.1} and the fact $\gamma_n(\Omega)<1$, we have \begin{gather}\label{2.2} \exp\Big(-\frac{\Phi^{-1}(t)^{2}}{2}\Big)\leq \alpha t(1-\log t)^{1/2},\quad t\in(0,\gamma_n(\Omega)),\\ \label{2.3} \exp\Big(-\frac{\Phi^{-1}(t)^{2}}{2}\Big)\geq \beta t(1-\log t)^{1/2},\quad t\in(0,\gamma_n(\Omega)), \end{gather} where $\alpha$ and $\beta$ are two positive constants depending on $\gamma_n(\Omega)$. \end{remark} Now we give the notion of rearrangement. \begin{definition} \label{def2.1}\rm If $u$ is a measurable function in $\Omega$ and $\mu(t)=\gamma_{n}(\{x\in\Omega:|u|>t\})$ is the distribution function of $u$, then we define the decreasing rearrangement of $u$ with respect to Gauss measure as \[ u^{\star}(s)=\inf\{t\geq0:\mu(t)\leq s\},\quad s\in[0,\gamma_n(\Omega)]. \] Let $\Omega^{\sharp}=\{x=(x_1,x_2,\dots,x_n)\in\mathbb{R}^{n}:x_1>\lambda\}$ be the half-space such that $\gamma_n(\Omega)=\gamma_n(\Omega^{\sharp})$. Then \[ u^\sharp(x)=u^\star(\Phi(x_1)),\quad x\in\Omega^{\sharp} \] denote the increasing Gauss symmetrization of $u$ (or Gauss symmetrization of $u$). Similarly, the decreasing Gauss symmetrization of $u$ will be $$ u_{\sharp}(x)=u_{\star}(\Phi(x_1)),\quad x\in\Omega^{\sharp}, $$ with $$ u_{\star}(s)=u^{\star}(\gamma_n(\Omega)-s),\quad s\in(0,\gamma_n(\Omega)). $$ \end{definition} Properties of rearrangement with respect to Gauss measure or a positive measure have been widely considered in \cite{Chong, Rakotoson1, Rakotoson2, Talenti3}, for instance. Here we just recall the following: Hardy-Little inequality: \begin{align*} \int_{0}^{\gamma_n(\Omega)}u_\star(s)v^\star(s)ds &=\int_{\Omega^{\sharp}}u_\sharp(x)v^\sharp(x)d\gamma_n \leq\int_{\Omega}|u(x)v(x)|d\gamma_{n}\\ &\leq\int_{\Omega^{\sharp}}u^\sharp(x)v^\sharp(x)d\gamma_n= \int_{0}^{\gamma_n(\Omega)}u^\star(s)v^\star(s)ds, \end{align*} where $u$ and $v$ are measurable functions. Polya-Sz\"{e}go principle: Let $u\in W_0^{1,p}(\varphi,\Omega)$ with $1s \text{ and } \alpha+\frac{1}{q}>\beta+\frac{1}{s}. $$ \end{remark} \begin{remark} \label{rmk2.5}\rm The space $L^{p,q}(\log L)^{\alpha}(\varphi,\Omega)$ is nontrivial if and only if one of the following conditions holds \begin{gather*} p<+\infty,\\ p=+\infty \text{ and } \alpha+\frac{1}{q}<0,\\ p=+\infty,q=+\infty \text{ and }\alpha=0. \end{gather*} \end{remark} The following imbedding theorem in Lorentz-Zygmund space is a straight consequence of the Sobolev logarithmic inequalities. It has been proved in \cite{Di Blasio2} by using the properties of rearrangement. \begin{proposition} \label{prop2.1} Let $\Omega$ be an open subset of $\mathbb{R}^{n}$ with $\gamma_{n}(\Omega)<1$. If $f\in W_0^{1,p}(\varphi,\Omega)$ with $1\leq p<+\infty$, then $ f\in L^p(\log L)^{1/2}(\varphi,\Omega)$ and $$ \|f\|_{L^{p}(\log L)^{1/2}(\varphi,\Omega)}\leq C\|\nabla f\|_{L^p(\varphi,\Omega)}. $$ \end{proposition} The following Hardy inequalities are also needed in this article \cite{Bennett}. \begin{proposition} \label{prop2.2} Suppose that $r>0$, $1\leq q \leq+\infty$ and $-\infty<\alpha<+\infty$. Let $\Psi$ be a nonegative measurable function on $(0, 1)$. If $1\leq q <+\infty$, then \begin{equation}\label{2.5} \Big(\int_{0}^{1}\Big(t^{-r}(1-\log t)^\alpha\int_{0}^{t}\Psi(s)ds \Big)^q\frac{dt}{t}\Big)^{1/q} \leq C\Big(\int_{0}^{1}\big(t^{1-r}(1-\log t)^\alpha\Psi(t)\big)^q \frac{dt}{t}\Big)^{1/q} \end{equation} holds. Moreover if $q=+\infty$, then \begin{equation}\label{2.6} \sup_{00$ such that \begin{equation}\label{a3.1} \begin{aligned} &\int_{\Omega^\sharp}\varphi |D_1Z|^2dx -\int_{\Omega^\sharp}B\varphi ZD_1Zdx +\int_{\Omega^\sharp}c_{0\sharp}\varphi Z^2dx\\ &\geq\xi\int_{\Omega^\sharp}\varphi |D_1Z|^2dx,\quad \forall Z\in H_0^1(\varphi,\Omega^\sharp). \end{aligned} \end{equation} \end{itemize} Then the maximum principle holds for $A^\sharp$; i.e., \begin{equation} %\label{a3.2} A^\sharp v\geq0\text{ in }\Omega^\sharp\text{ and } v\geq0 \text{ on }\partial\Omega^\sharp \text{ imply } v\geq0 \text{ in }\Omega^\sharp, \label{P4} \end{equation} and problem \eqref{P2} has unique symmetric solution. \end{theorem} Now, the comparison result between problems \eqref{P1} and \eqref{P2} can be stated in the following theorem. \begin{theorem} \label{thm3.2} Let {\rm (A1)--(A4)} and one of the conditions in Theorem \ref{thm3.1} hold. Suppose that $u$ is the solution of problem \eqref{P1} and $v=v^{\sharp}$ is the solution of problem \eqref{P2}. Then \begin{itemize} \item[(1)] As $c_0(x)\leq 0$, we have \begin{equation}\label{3.1}u^{\star}(s)\leq v^\star(s), \quad s\in[0,\gamma_n(\Omega)]. \end{equation} \item[(2)] As $c_0^+(x)\not\equiv0$, it follows that \begin{gather}\label{3.2} u^{\star}(s)\leq v^\star(s),~s\in[0,s_1],\\ \label{3.3} \int_{s_1}^se^{B\Phi^{-1}(\sigma)}u^\star(\sigma) d\sigma \leq \int_{s_1}^se^{B\Phi^{-1}(\sigma)}v^\star(\sigma) d\sigma,\quad s\in [s_1,\gamma_n(\Omega)], \end{gather} where $s_1=\inf\{s\in[0,\gamma_n(\Omega)]:c_{0\star}(s)>0\}$. \end{itemize} \end{theorem} Using the above comparison results, it is possible to obtain estimates of the solutions to problem \eqref{P1} in terms of the solutions to ``symmetrized" problem \eqref{P2}. \begin{theorem} \label{thm3.3} Suppose that Lorentz-Zygmund spaces $L^{p,q}(\log L)^\alpha(\varphi,\Omega)$ are nontrivial. Under the same assumptions of Theorem \ref{thm3.2}, we have (1) If $c_0(x)\leq 0$, then \begin{equation}\label{3.4} \|u\|_{L^{p,q}(\log L)^\alpha(\varphi,\Omega)} \leq \|v\|_{L^{p,q}(\log L)^\alpha(\varphi,\Omega^\sharp)} \end{equation} with $0 0$, then for all $0<\varepsilon<\frac{1}{p}$, \begin{equation}\label{3.5} \|u\|_{L^{p,q}(\log L)^\alpha(\varphi,\Omega)}\leq Ce^{[\frac{B^2}{2\varepsilon}-B\Phi^{-1}(\gamma_n(\Omega))]} \|v\|_{L^{\frac{p}{1-p\varepsilon},q} (\log L)^{\alpha-\frac{\varepsilon}{2}}(\varphi,\Omega^\sharp)}, \end{equation} where $C$ is a positive constant depending on $p,q$ and $\gamma_{n}(\Omega)$, $1\lambda\}$, we have $x_1>\lambda\geq0$. Therefore, \begin{equation}\label{4.1} |x_1|^{\star}(s)=x_1^{\star}(s)=\Phi^{-1}(s). \end{equation} Moreover, from Remark \ref{rmk2.1} it follows that \begin{align*} \lim _{s\to 0^+}\frac{\Phi^{-1}(s)}{(1-\log s)^{1/2} } &=\lim _{s\to 0^+}\frac{-\sqrt{2\pi} e^{\frac{\Phi^{-1}(s)^{2}}{2}}}{\frac{1}{2}(1-\log s)^{-1/2} (-\frac{1}{s})}\\ &=\lim _{s\to 0^+}2\sqrt{2\pi}\frac{s(1-\log s)^{1/2}} {e^{-\frac{\Phi^{-1}(s)^{2}}{2}}}\\ &=\lim _{s\to 0^+}2\sqrt{2\pi}\,\frac{s(1-\log s)^{1/2}}{s(2\log\frac{1}{s})^{1/2}}\, \frac{s(2\log \frac{1}{s})^{1/2}}{e^{-\frac{\Phi^{-1}(s)^{2}}{2}}}\\ &=2\sqrt{2\pi}\,\frac{1}{\sqrt{2}}\,\frac{1}{\sqrt{2\pi}}=\sqrt{2}. \end{align*} Then there exist $M_1>0$ and $\delta_0\in(0,\gamma_n(\Omega))$ such that $$ \Phi^{-1}(s)(1-\log s)^{-1/2}\leq M_1,\quad s\in(0,\delta_0). $$ Since $\Phi^{-1}(s)(1-\log s)^{-1/2}$ is continuous on $[\delta_0,\gamma_n(\Omega)]$, there exists a constant $M_2>0$ such that $$ \Phi^{-1}(s)(1-\log s)^{-1/2}\leq M_2,\quad s\in[\delta_0,\gamma_n(\Omega)). $$ Thus, $$ \Phi^{-1}(s)(1-\log s)^{-1/2}\leq \max\{M_1,M_2\},\quad s\in(0,\gamma_n(\Omega)). $$ Recalling \eqref{4.1}, we have \[ \| |x_1| \|_{L^\infty(\log L)^{-1/2}(\varphi,\Omega^\sharp)} =\sup _{0t\}),\\ \nu(t)=\gamma_n(\{x\in\Omega^\sharp: x_1>t\}), \end{gather*} we have \[ \mu(t)=\begin{cases} \nu(t) &t\geq-\lambda,\\ 2\nu(t)-(1-\gamma_n(\Omega)) &0\leq t<-\lambda,\\ \gamma_n(\Omega) &t<0 \end{cases} \] and \begin{align*} |x_1|^\star(s) &=\inf\{t\geq0:\mu(t)\leq s\} \\ &=\begin{cases} \inf\{t\geq0:\nu(t)\leq s\} & s\in [0,1-\gamma_n(\Omega)],\\ \inf\{t\geq0:\nu(t)\leq\frac{s+1-\gamma_n(\Omega)}{2} \} & s\in (1-\gamma_n(\Omega),\gamma_n(\Omega) ] \end{cases} \\ &=\begin{cases} x_1^\star(s) &s\in[0,1-\gamma_n(\Omega)],\\ x_1^\star\big(\frac{s+1-\gamma_n(\Omega)}{2}\big) &s\in\big(1-\gamma_n(\Omega),\gamma_n(\Omega)\big] \end{cases}\\ &=\begin{cases} \Phi^{-1}(s) &s\in [0,1-\gamma_n(\Omega)],\\ \Phi^{-1}\big(\frac{s+1-\gamma_n(\Omega)}{2}\big) &s\in\big(1-\gamma_n(\Omega),\gamma_n(\Omega)\big]. \end{cases} \end{align*} Using the same method as in the case $\lambda>0$, we obtain that there exists $M>0$ such that $$ |x_1|^\star(s)(1-\log s)^{-1/2}\leq M, $$ which implies $|x_1|\in L^\infty(\log L)^{-1/2}(\varphi,\Omega^\sharp)$. Now we give the proof of the equivalence of (a) and (b). (a)$\Rightarrow$ (b) The first eigenvalue of \eqref{P3} can be characterized by the Rayleigh principle as \begin{equation} \label{4.2} \lambda_1=\min _{Q\in H_0^1(\varphi,\Omega^\sharp),Q\neq 0}\frac{\int_{\Omega^\sharp}\varphi |D_1Q|^2 \,dx-B\int_{\Omega^\sharp}\varphi QD_1Q \,dx+\int_{\Omega^\sharp}c_{0\sharp}\varphi Q^2 \,dx}{\int_{\Omega^\sharp}\left(\frac{B|x_1|}{2}+c_0^{-\sharp}(x)\right)\varphi Q^2\,dx}. \end{equation} In view of \eqref{4.2} and the fact that $c_{0\sharp}=c_{0\sharp}^+-c_0^{-\sharp}$, integrating by parts, we have \begin{align*} &\int_{\Omega^\sharp} \varphi|D_1Z|^2 \,dx -B\int_{\Omega^\sharp}\varphi Z D_1Z \,dx +\int_{\Omega^\sharp}c_{0\sharp}\varphi Z^2\,dx\\ &\geq (1-\xi)\lambda_1\int_{\Omega^\sharp} \Big(\frac{B|x_1|}{2}+c_0^{-\sharp}\Big)\varphi Z^2\,dx +\xi\int_{\Omega^\sharp}\varphi|D_1Z|^2\,dx\\ &\quad -\xi B\int_{\Omega^\sharp}\varphi Z D_1Z \,dx +\xi\int_{\Omega^\sharp}(c_{0\sharp}^+-c_0^{-\sharp})\varphi Z^2\\ & =(1-\xi)\lambda_1\int_{\Omega^\sharp} \big(\frac{B|x_1|}{2}+c_0^{-\sharp}\big)\varphi Z^2\,dx -\xi \int_{\Omega^\sharp}\frac{B}{2}x_1\varphi Z^2\,dx\\ &\quad +\xi\int_{\Omega^\sharp}(c_{0\sharp}^+ -c_0^{-\sharp})\varphi Z^2\,dx +\xi\int_{\Omega^\sharp}\varphi|D_1Z|^2 \,dx\\ &\geq((1-\xi)\lambda_1-\xi)\int_{\Omega^\sharp} \big(\frac{B|x_1|}{2}+c_0^{-\sharp}\big)\varphi Z^2 \,dx +\xi\int_{\Omega^\sharp}\varphi|D_1Z|^2\,dx, \end{align*} for all $Z\in H_0^1(\varphi,\Omega^\sharp)$. Then we can choose $0<\xi<\frac{\lambda_1}{\lambda_1+1}$ such that \eqref{a3.1} holds. (b) $\Rightarrow$ (a) Assume that $\Psi$ is the eigenfunction corresponding to $\lambda_1$. Then \begin{equation*}\begin{split} \lambda_1=\frac{\int_{\Omega^\sharp} |D_1\Psi|^2\varphi \,dx-B\int_{\Omega^\sharp}\varphi \Psi D_1\Psi \,dx+\int_{\Omega^\sharp}c_{0\sharp} \Psi^2\varphi \,dx}{\int_{\Omega^\sharp}\big(\frac{B|x_1|}{2}+c_0^{-\sharp}(x)\big)\Psi^2\varphi \,dx}. \end{split}\end{equation*} Recalling (b) and Proposition \ref{prop2.1} and noting that $\Psi\not\equiv 0$, we have \[ \lambda_1\geq\frac{\xi\int_{\Omega^\sharp} |D_1\Psi|^2\varphi \,dx}{\int_{\Omega^\sharp}\big(\frac{B|x_1|}{2}+c_0^{-\sharp}\big)\Psi^2\varphi \,dx}\geq\frac{C\xi\int_{\Omega^\sharp} |\Psi|^2\varphi \,dx}{\int_{\Omega^\sharp}\big(\frac{B|x_1|}{2}+c_0^{-\sharp}\big)\Psi^2\varphi \,dx}>0, \] where $C$ is a positive constant depending on $\gamma_n(\Omega)$. Thus (a) holds. Now, we prove that the maximum principle holds for $A^\sharp$. As $c_0(x)\geq0$, it is obvious. As $c_0^-(x)\not\equiv 0$, taking $v^-$ as a test function of \eqref{P4} and using (b), we obtain \[ 0\leq\xi\int_{\Omega^\sharp}\varphi|D_1v^-|^2dx\leq\int_{\Omega^\sharp} v^-A^\sharp v^-dx\leq0, \] which implies $v^-=0$. Thus $v\geq0$ on $\partial\Omega^\sharp$. On the other hand, from \cite{Lions} we see that \eqref{P2} has unique solution $v$. Thus the solution depends only on the first variable. To prove $v=v^\sharp$, for the sake of simplicity, we suppose that $v$ is sufficiently smooth. Set $E=\{x\in\Omega^\sharp:v>0\}$. Then $v$ satisfies \begin{equation*} A^\sharp v=f^\sharp\quad \text{on } E, \end{equation*} which gives \begin{equation} \label{4.3} -\varphi D_{11}v+(x_1-B)\varphi D_1v+c_{0\sharp}\varphi v=f^\sharp\varphi \quad \text{on } E. \end{equation} Differentiating \eqref{4.3} with respect to $x_1$ and taking $K=D_1v$, we obtain \begin{align*} (A^\sharp+\varphi)K &=-\varphi D_{11}K+(x_1-B)\varphi D_1K+c_{0\sharp}\varphi K+\varphi K\\ &=\varphi D_1f^\sharp-v\varphi D_1c_{0\sharp}\geq0 \quad \text{on }E. \end{align*} Observing $E=\{x\in\Omega^\sharp:x_1>\xi_0\}$, where $\xi_0=\Phi^{-1}(\gamma_n(\{x\in\Omega^\sharp:v>0\}))$, it follows $v(\xi_0)=0$. Thus $K\geq 0$ on $\partial E$. Since $A^\sharp+\varphi$ satisfies the property (b), we obtain $K=D_1v\geq 0$ on $E$ by applying the maximum principle to the operator $A^\sharp+\varphi$. Thus $v=v^\sharp$ on $\Omega^\sharp$. The proof is complete. \end{proof} Before proving Theorem \ref{thm3.2}, we need the following lemmas. \begin{lemma} \label{lem4.1} Assume that {\rm (A1)--(A4)} hold. Let $u$ be the solution to problem \eqref{P1} and $v=v^\sharp$ be the solution to problem \eqref{P2}. Then \begin{equation} \label{4.4} -u^{\star'}(s)\leq 2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]} \int_{0}^{s}e^{B\Phi^{-1}(\sigma)} [f^\star(\sigma)-c_{0\star}(\sigma)u^\star(\sigma)]d\sigma, \end{equation} for $ s\in[0,\gamma_{n}(\{u>0\})]$, and \begin{equation} \label{4.5} -v^{\star'}(s)= 2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]} \int_{0}^{s}e^{B\Phi^{-1}(\sigma)} [f^\star(\sigma)-c_{0\star}(\sigma)v^\star(\sigma)]d\sigma, \end{equation} for $s\in[0,\gamma_{n}(\{v>0\})]$. \end{lemma} The proof of the above lemma follows the same lines as in \cite{Di Blasio1,Di Blasio2}; we omit it. \begin{lemma} \label{lem 4.2} Assume that {\rm (A1)--(A4)} hold. Let $u$ and $v=v^\sharp$ be the solutions to problem \eqref{P1} and \eqref{P2} respectively. If $c_{0\star}$ is continuous at $s_1$, then $w=u^\star-v^\star$ satisfies \begin{equation} \label{4.6} -w'(s)\leq- 2\pi e^{\left[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)\right]} \int_{0}^{s}e^{B\Phi^{-1}(\sigma)} c_{0\star}(\sigma)w(\sigma)d\sigma, \end{equation} for $s\in[0,\gamma_{n}(\{u>0\})]$. \end{lemma} \begin{proof} As $\gamma_{n}(\{u>0\})\leq\gamma_{n}(\{v>0\})$, it is obvious that \eqref{4.6} holds. As $\gamma_{n}(\{u>0\})>\gamma_{n}(\{v>0\})$, \eqref{4.6} holds on $[0,\gamma_{n}(\{v>0\})]$. Moreover, using the regularity theory (see \cite{Gilbarg}), $v$ belongs to $H^2(\varphi,\Omega^\sharp)$ and then $v^\star\in \mathcal{C}^1(0,\gamma_{n}(\Omega)]$. Thus $v^{\star'}(\gamma_{n}(\{v>0\}))=0$. It follows from \eqref{4.5} that \begin{equation} \int_{0}^{\gamma_{n}(\{v>0\})}e^{B\Phi^{-1}(\sigma)} [f^\star(\sigma)-c_{0\star}(\sigma)v^\star(\sigma) ]d\sigma=0.\label{4.7} \end{equation} Noting that $w(s)=u^\star(s)$ on $[\gamma_{n}(\{v>0\}),\gamma_{n}(\{u>0\})]$, we combine \eqref{4.7} and \eqref{4.4} to discover that \begin{align*} -w'(s) &\leq 2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]} \Big\{\int_{0}^{s}e^{B\Phi^{-1}(\sigma)} [f^\star(\sigma)-c_{0\star}(\sigma)w(\sigma)]d\sigma\\ &\quad -\int_{0}^{\gamma_{n}(\{v>0\})}e^{B\Phi^{-1}(\sigma)} c_{0\star}(\sigma)v^\star(\sigma)d\sigma\Big\}\\ &=2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]} \Big\{-\int_{0}^{s}e^{B\Phi^{-1}(\sigma)} c_{0\star}(\sigma)w(\sigma) d\sigma\\ &\quad +\int_{\gamma_{n}(\{v>0\})}^se^{B\Phi^{-1}(\sigma)} f^\star(\sigma)d\sigma\Big\}, \quad s\in[\gamma_{n}(\{v>0\}),\gamma_{n}(\{u>0\})]. \end{align*} If we show \begin{equation}\label{4.8} f^\star(s)<0 \quad \text{on }[\gamma_{n}(\{v>0\}),\gamma_{n}(\Omega)], \end{equation} then \eqref{4.6} is proved. Now it remains to prove \eqref{4.8}. In fact, as $\gamma_{n}(\{v>0\})\leq s_1$, \eqref{4.7} yields \[ \int_{0}^{\gamma_{n}(\{v>0\})}e^{B\Phi^{-1}(\sigma)} f^\star(\sigma)= \int_{0}^{\gamma_{n}(\{v>0\})}e^{B\Phi^{-1}(\sigma)} c_{0\star}(\sigma)v^\star(\sigma)d\sigma\leq0. \] Therefore $f^\star$ can not be nonnegative on $[0,\gamma_{n}(\{v>0\})]$. As $\gamma_{n}(\{v>0\})> s_1$, taking $V(s)=\int_{s_1}^{s}e^{B\Phi^{-1}(\sigma)} c_{0\star}(\sigma)v^\star(\sigma)d\sigma$, from \eqref{4.5} we obtain \begin{align*} &-\Big(e^{-B\Phi^{-1}(s)}(c_{0\star}(s))^{-1}V'(s)\Big)' +2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]}V\\ &= 2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]} \Big[\int_{0}^{s}e^{B\Phi^{-1}(\sigma)} f^\star(\sigma)d\sigma-\int_{0}^{s_1}e^{B\Phi^{-1} (\sigma)}c_{0\star}(\sigma)v^\star(\sigma)d\sigma \Big]\\ & \geq 2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]} \int_{0}^{s}e^{B\Phi^{-1}(\sigma)} f^\star(\sigma)d\sigma, \quad s\in(s_1,\gamma_{n}(\{v>0\})). \end{align*} If $f^\star\geq0$ on $[0,\gamma_{n}(\{v>0\})]$, observing $c_{0\star}(s_1)=0$, $V$ satisfies \begin{equation} \label{P5} \begin{gathered} -\Big(e^{-B\Phi^{-1}(s)}(c_{0\star}(s))^{-1}V'(s)\Big)' +2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]}V \geq 0 \quad \text{on }(s_1,\gamma_{n}(\{v>0\})),\\ V(s_1)=0,\quad V'(s_1)=0. \end{gathered} \end{equation} By the maximum principle (see \cite{Protter}), we obtain $V\leq0$ in $(s_1,\gamma_{n}(\{v>0\}))$ which contradict with the fact that $V>0$ in $(s_1,\gamma_{n}(\{v>0\}))$. Thus $f^\star$ can not be nonnegative in $[0,\gamma_{n}(\{v>0\})]$ and we obtain the desired result. \end{proof} \begin{lemma}[P.163 in \cite{Bandle1}] \label{lem4.3} Let \[ L=-\sum _{i,j=1}^{n}D_i(a_{ij}(x)D_j)+c(x). \] Consider the eigenvalue problem \begin{equation} \label{P6} \begin{gathered} L\Psi=\lambda_1 P\Psi \quad \text{in }D,\\ \Psi=0\quad \text{on }\Gamma_0,\\ \frac{\partial\Psi}{\partial \nu}+\eta\Psi=0\quad \text{on }\Gamma_1, \end{gathered} \end{equation} where $D\subseteq \mathbb{R}^n$, $\Gamma_0$ is a subset of $\partial D$, $\Gamma_1=\partial D-\Gamma_0$ and $P$ is a positive function in $D$. If $\lambda_1$ is the first eigenvalue of \eqref{P6}, then $$ \lambda_1\leq\sup _{x\in D}\frac{Lh}{Ph}, $$ where $h$ is any positive function in $D$ satisfying the same boundary conditions as $\Psi$. $$ \lambda_1\geq\inf _{x\in D}\frac{Lh}{Ph}, $$ where $h$ is positive in $D$, $h\geq0$ on $\Gamma_0$ and $\frac{\partial h}{\partial \nu}+\eta h\geq0$ on $\Gamma_1$. \end{lemma} \begin{lemma}[\cite{Alvino1}] \label{lem4.4} Let $f,g$ be measurable positive functions such that $$ \int_0^rf(\sigma)d\sigma\leq\int_0^rg(\sigma)d\sigma,~~~~~r\in[0,\rho]. $$ If $h\geq0$ is a decreasing function in $[0,\rho]$, then $$ \int_0^rf(\sigma)h(\sigma)d\sigma\leq\int_0^rg(\sigma)h(\sigma)d\sigma,\quad r\in[0,\rho]. $$ \end{lemma} \begin{proof}[Proof of Theorem \ref{thm3.2}] Suppose that $c_0(x)$ is smooth in $\Omega$. (1) As $c_0(x)\leq 0$, we assume $c_0(x)<0$. In this case, $c_0^{-\star}=-c_{0\star}$. Set $W(s)=\int_0^se^{B\Phi^{-1}(\sigma)}c_0^{-\star}(\sigma)w(\sigma)d\sigma$. By \eqref{4.6}, we have \begin{equation} \label{P7} \begin{gathered} -\big[e^{-B\Phi^{-1}(s)}(c_0^{-\star}(s))^{-1}W'\big]'\leq 2\pi e^{\left[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)\right]}W \quad \text{in } (0,\gamma_{n}(\{u>0\})),\\ W(0)=0,\quad W'(\gamma_{n}(\{u>0\}))\leq 0. \end{gathered} \end{equation} Thus, $w\leq0$ on $[0,\gamma_{n}(\{u>0\})]$. In fact, let $E$ be a half space whose Gauss measure is $\gamma_{n}(\{u>0\})$. Consider the eigenvalue problem \begin{equation} \label{P8} \begin{gathered} A^\sharp \Psi=\widetilde{\lambda} c_0^{-\sharp}\varphi\Psi \quad\text{in }E,\\ \Psi\in H_0^1(\varphi, E). \end{gathered} \end{equation} The first eigenvalue of \eqref{P8} can be characterized as \begin{equation}\label{4.9} \widetilde{\lambda}=\min _{Q\in H_0^1(\varphi,E),Q\neq 0}\frac{\int_{E}\varphi |D_1Q|^2\,dx-B\int_{E}\varphi QD_1Q \,dx+\int_{E}c_{0\sharp}\varphi Q^2\,dx}{\int_{E}c_0^{-\sharp}\varphi Q^2\,dx}. \end{equation} By (b) of Theorem \ref{thm3.1}, we have \begin{align*} &\int_{E}\varphi |D_1Q|^2 \,dx-B\int_{E}\varphi QD_1Q\,dx+\int_{E}c_{0\sharp}\varphi Q^2 \,dx\\ &\geq\beta\int_{E}\varphi |D_1Q|^2\,dx>0,\quad Q\in H_0^1(\varphi,E) \text{ and } Q\not\equiv 0 \text{ in }E \end{align*} Hence, $\widetilde{\lambda}>0$. Now if $\Psi$ is an eigenfunction corresponding to $\widetilde{\lambda}$, then both $\Psi$ and $|\Psi|$ minimize \eqref{4.9}. $|\Psi|$ is also an eigenfunction. Then we can take $\Psi\geq0$. Moreover, $\Psi$ satisfies \begin{equation}\label{a4.10} \widetilde{\lambda}=\frac{\int_{E}\varphi |D_1\Psi|^2 \,dx-B\int_{E}\varphi \Psi D_1\Psi\,dx+\int_{E}c_{0\sharp}\varphi \Psi^2\,dx}{\int_{E}c_0^{-\sharp}\varphi \Psi^2\,dx}. \end{equation} Integrating by parts and using Hardy-Littlewood inequality and Polya-Sz\"{e}go principle, we obtain \begin{equation}\label{4.10} \begin{aligned} \widetilde{\lambda} &=\frac{\int_{E} |D_1\Psi|^2\varphi \,dx-B\int_{E}\frac{x_1}{2}\Psi^2\varphi\,dx+\int_{E}c_{0\sharp} \Psi^2\varphi\,dx}{\int_{E}c_0^{-\sharp} \Psi^2 \varphi\,dx}\\ &\geq \frac{\int_{E} |D_1\Psi^\sharp|^2\varphi \,dx-B\int_{E}(\frac{x_1}{2})^\sharp\Psi^{\sharp2}\varphi \,dx+\int_{E}c_{0\sharp}\Psi^{\sharp2}\varphi \,dx}{\int_{E}c_0^{-\sharp} \Psi^{\sharp2 }\varphi\,dx}. \end{aligned} \end{equation} Noting that $x_1^\sharp=x_1$, we conclude from \eqref{4.10} that \[ \widetilde{\lambda}\geq \frac{\int_{E} |D_1\Psi^\sharp|^2\varphi \,dx-B\int_{E}\frac{x_1}{2}\Psi^{\sharp2}\varphi \,dx+\int_{E}c_{0\sharp} \Psi^{\sharp2}\varphi \,dx}{\int_{E}c_0^{-\sharp} \Psi^{\sharp2 }\varphi\,dx}. \] Thus, the above inequality, \eqref{4.9} and \eqref{a4.10} imply $\Psi=\Psi^\sharp$. In addition, \begin{equation} \label{4.12} -\Psi^{\star'}(s)=2\pi(\widetilde{\lambda}+1) e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]}\int_{0}^{s}e^{B\Phi^{-1}(\sigma)} c_0^{-\star}(\sigma)\Psi^\star(\sigma)d\sigma, \end{equation} for $s\in[0,\gamma_{n}(\{u>0\})]$. Let $\Theta(s)=\int_{0}^{s}e^{B\Phi^{-1}(\sigma)} c_0^{-\star}(\sigma)\Psi^\star(\sigma)d\sigma$. Then \eqref{4.12} can be written as \begin{equation} \label{P9} \begin{gathered} -\big[e^{-B\Phi^{-1}(s)}(c_0^{-\star}(s))^{-1}\Theta'\big]' =2\pi(\widetilde{\lambda}+1)e^{\left[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)\right]}\Theta \quad \text{in } (0,\gamma_{n}(\{u>0\})),\\ \Theta(0)=0,\quad \Theta'(\gamma_{n}(\{u>0\}))= 0. \end{gathered} \end{equation} If $\lambda$ is the first eigenvalue of the problem \begin{equation} \label{P10} \begin{gathered} -\big[e^{-B\Phi^{-1}(s)}(c_0^{-\star}(s))^{-1}U'\big]' =\lambda2\pi e^{\left[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)\right]}U \quad \text{in } (0,\gamma_{n}(\{u>0\})),\\ U(0)=0,\quad U'(\gamma_{n}(\{u>0\}))= 0, \end{gathered} \end{equation} it follows from Lemma \ref{lem4.3} that \[ \lambda\geq\inf _{s\in(0,\gamma_{n}(\{u>0\}))} \frac{-[e^{-B\Phi^{-1}(s)}(c_0^{-\star}(s))^{-1}\Theta']'} {2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]}\Theta} =\widetilde{\lambda}+1. \] Therefore, $\lambda\geq\widetilde{\lambda}+1>1$. By \cite[Lemma 4.7]{Bandle1}, we have $W'\leq0$ on $[0,\gamma_{n}(\{u>0\})]$. That is $w\leq0$ on $[0,\gamma_{n}(\{u>0\})]$. The result in the case $c(x)\leq 0$ can be proved by approximation techniques (see \cite{Alvino1}). (2) As $c_0^+(x)\not\equiv0$, we let $$ s_2=\inf\{s\in[0,\gamma_n(\Omega)]:c_{0\star}(s)\geq0\}. $$ Then $s_2\leq s_1$ and $c_{0\star}(s_2)=0$. Moreover, noting that $c_{0\star}(s)<0$ on $(0,s_2)$, we obtain $c_0^{-\star}(s)=-c_{0\star}(s)$ on $(0,s_2)$. Take $$ W_1(s)=\int_0^se^{B\Phi^{-1}(\sigma)}c_0^{-\star}(\sigma)w(\sigma)d\sigma. $$ By \eqref{4.6}, we obtain \begin{gather*} -\big[e^{-B\Phi^{-1}(s)}(c_0^{-\star}(s))^{-1}W_1'(s)\big]'\leq 2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]}W_1(s) \quad\text{in } (0,s_2),\\ W_1(0)=0,\quad W_1'(s_2)= 0. \end{gather*} Proceeding as in case (1), it follows that \begin{equation}\label{4.13} u^{\star}(s)\leq v^\star(s), \quad s\in[0,s_2]. \end{equation} On the other hand, letting $$ W_2(s)=\int_{s_1}^se^{B\Phi^{-1}(\sigma)}c_{0\star}(\sigma)w(\sigma)d\sigma, $$ Inequalities \eqref{4.6} and \eqref{4.13} yield \begin{gather*} -\big[e^{-B\Phi^{-1}(s)}(c_{0\star}(s))^{-1}W_2'(s)\big]' \leq -2\pi e^{[\Phi^{-1}(s)^{2}-B\Phi^{-1}(s)]}W_2(s)\\ \text{in } (s_1,\gamma_{n}(\{u>0\})],\\ W_2(s_1)=0,\quad W_2'(\gamma_{n}(\{u>0\}))\leq 0. \end{gather*} By the maximum principle, we have $W_2(s)\leq0$ on $[s_1,\gamma_{n}(\{u>0\})]$. That is, \[ \int_{s_1}^se^{B\Phi^{-1}(\sigma)}c_{0\star}(\sigma)u^\star(\sigma)d\sigma\leq \int_{s_1}^se^{B\Phi^{-1}(\sigma)}c_{0\star}(\sigma)v^\star(\sigma)d\sigma, \quad s\in[s_1,\gamma_{n}(\{u>0\})]. \] According to Lemma \ref{lem4.4}, we obtain \[ \int_{s_1}^se^{B\Phi^{-1}(\sigma)}u^\star(\sigma)d\sigma\leq \int_{s_1}^se^{B\Phi^{-1}(\sigma)}v^\star(\sigma)d\sigma, \quad s\in[s_1,\gamma_{n}(\{u>0\})], \] which implies $u^\star(s_1)\leq v^\star(s_1)$. Finally applying \eqref{4.13} to \eqref{4.6}, it follows \begin{equation}\label{4.14} -u^{\star'}(s)\leq -v^{\star'}(s),\quad s\in [s_2, s_1] \end{equation} Integrating \eqref{4.14} from $s$ to $s_1$, we obtain \[ u^\star(s)\leq v^\star(s)\quad \text{in }[s_2,s_1], \] which completes the proof. \end{proof} At last, we can remove the smooth assumption on $c_0(x)$ by approximations. \begin{remark} \label{rmk4.1}\rm For the variational problem, since $f^\star$ maybe negative in $[0,\gamma_{n}(\{u>0\})]$, the method in the equation case are failed to obtain \eqref{3.1}--\eqref{3.3}. Here, we use the properties of the first eigenvalue (Lemma \ref{lem4.3}) and maximal principle to obtain the desired results. \end{remark} \begin{proof}[Proof of Theorem \ref{thm3.3}] (1) If $c_0(x)\leq 0$, inequality \eqref{3.4} follows from \eqref{3.1}. (2) If $c_0(x)>0$, we have \begin{equation}\label{4.15} \int_{0}^se^{B\Phi^{-1}(\sigma)}u^\star(\sigma) d\sigma \leq \int_{0}^se^{B\Phi^{-1}(\sigma)}v^\star(\sigma) d\sigma,\quad s\in [0,\gamma_n(\Omega)], \end{equation} Set \[ \|u\|^\star_{L^{p,q}(\log L)^{\alpha}(\varphi,\Omega)} =\begin{cases} \big[\int_{0}^{\gamma_{n}(\Omega)} \big(t^{\frac{1}{p}}(1-\log t)^\alpha u^{\star\star}(t)\big)^q \frac{dt}{t}\big]^{1/q} &\text{if } 01$ and $q\geq1$. As $q<+\infty$, using \eqref{4.15}, \eqref{2.3} and \eqref{2.5}, we obtain \begin{align*} &\|u\|^{\star q}_{L^{p,q}(\log L)^{\alpha}(\varphi,\Omega)} \\ &= \int_{0}^{\gamma_{n}(\Omega)} \Big(t^{\frac{1}{p}}(1-\log t)^\alpha \frac{1}{t}\int_0^tu^{\star}(\sigma)d\sigma\Big)^q \frac{dt}{t}\\ &\leq\int_{0}^{\gamma_{n}(\Omega)} \left(t^{\frac{1}{p}}(1-\log t)^\alpha e^{-B\Phi^{-1}(t)} \frac{1}{t}\int_0^te^{B\Phi^{-1}(\sigma)}u^{\star}(\sigma)d\sigma\right)^q \frac{dt}{t}\\ & \leq \int_{0}^{\gamma_{n}(\Omega)} \Big(t^{\frac{1}{p}}(1-\log t)^\alpha e^{-B\Phi^{-1}(t)} \frac{1}{t}\int_0^te^{B\Phi^{-1}(\sigma)}v^{\star}(\sigma)d\sigma\Big)^q \frac{dt}{t}\\ & = \int_{0}^{\gamma_{n}(\Omega)} \Big(t^{\frac{1}{p}}(1-\log t)^\alpha e^{-B\Phi^{-1}(t)} \\ &\quad\times \frac{1}{t}\int_0^te^{[-(\sqrt{\frac{\varepsilon}{2}}\Phi^{-1}(\sigma) -\frac{B}{\sqrt{2\varepsilon}})^2 +\frac{B^2}{2\varepsilon}]} e^{\frac{\varepsilon\Phi^{-1}(\sigma)^{2}}{2}} v^{\star}(\sigma)d\sigma\Big)^q \frac{dt}{t}\\ &\leq e^{[q\frac{B^2}{2\varepsilon}-Bq\Phi^{-1}(\gamma_{n}(\Omega)]} \int_{0}^{\gamma_{n}(\Omega)} \Big(t^{\frac{1}{p}}(1-\log t)^\alpha \frac{1}{t}\int_0^t e^{\frac{\varepsilon\Phi^{-1}(\sigma)^{2}}{2}} v^{\star}(\sigma)d\sigma\Big)^q \frac{dt}{t}\\& \leq Ce^{[q\frac{B^2}{2\varepsilon}-Bq\Phi^{-1}(\gamma_{n}(\Omega)]} \int_{0}^{\gamma_{n}(\Omega)} \Big(t^{\frac{1}{p}}(1-\log t)^\alpha \frac{1}{t}\int_0^t \frac{1}{\sigma^\varepsilon(1-\log\sigma)^{\frac{\varepsilon}{2 }}} v^{\star}(\sigma)d\sigma\Big)^q \frac{dt}{t}\\ & \leq Ce^{[q\frac{B^2}{2\varepsilon}-Bq\Phi^{-1}(\gamma_{n}(\Omega)]} \int_{0}^{\gamma_{n}(\Omega)} \Big(t^{\frac{1}{p}-\varepsilon}(1-\log t)^{\alpha-\frac{\varepsilon}{2}} v^{\star}(t)\Big)^q \frac{dt}{t}\\ &=Ce^{[q\frac{B^2}{2\varepsilon}-Bq\Phi^{-1}(\gamma_{n}(\Omega)]} \|v\|^q_{L^{\frac{p}{1-p\varepsilon},q}(\log L)^{\alpha-\frac{\varepsilon}{2}}(\varphi,\Omega^\sharp)}, \end{align*} where $C$ is a positive constant depending on $p$, $q$ and $\gamma_{n}(\Omega)$. As $q=\infty$, \eqref{3.5} can be obtained by the same method as before with \eqref{2.5} replaced by \eqref{2.6}. (3) It follows from Theorem \ref{thm3.2} that \begin{gather*} u^{\star}(s)\leq v^\star(s),\quad s\in[0,s_1],\\ \int_{s_1}^se^{B\Phi^{-1}(\sigma)}u^\star(\sigma) d\sigma \leq \int_{s_1}^se^{B\Phi^{-1}(\sigma)}v^\star(\sigma) d\sigma,\quad s\in [s_1,\gamma_n(\Omega)], \end{gather*} where $0