\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 305, pp. 1--16.\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.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/305\hfil Nonlinear elliptic equations]
{Nonlinear elliptic equations with general growth in the
gradient related to Gauss measure}

\author[Y. Tian, C. Ma, F. Li \hfil EJDE-2015/305\hfilneg]
{Yujuan Tian, Chao Ma, Fengquan Li}

\address{Yujuan Tian (corresponding author) \newline
School of Mathematical Sciences, 
Shandong Normal University,
Jinan 250014, China}
\email{tianyujuan0302@126.com}

\address{Chao Ma \newline
School of Mathematical Sciences,
University of Jinan, 
Jinan  250022, China}
\email{chaos\_ma@163.com}

\address{Fengquan Li \newline
School of Mathematical Sciences, 
Dalian University of Technology, 
Dalian 116024, China}
\email{fqli@dlut.edu.cn}

\thanks{Submitted July 22, 2015. Published December 16, 2015.}
\subjclass[2010]{35J60, 35J70, 35J15}
\keywords{Comparison results; symmetrization; gauss measure; 
\hfill\break\indent nonlinear elliptic equation}

\begin{abstract}
 In this article, we establish a comparison result through symmetrization
 for solutions to some problems with general growth in the gradient.
 This allows to get sharp estimates for the solutions, obtained by
 comparing them with solutions of simpler problems whose data depend
 only on the first variable. Furthermore, we use such result to prove
 the existence of bounded solutions. All the above results are based
 on the study of a class of nonlinear integral operator of Volterra type.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega$ be an open subset of $\mathbb{R}^n$. We consider the Dirichlet
 problem whose prototype is
\begin{equation}
\begin{gathered}
-\operatorname{div}(\varphi(x)|\nabla u|^{p-2}\nabla u)=H(x,u,\nabla u)
\quad \text{in }\Omega,\\
u=0\quad \text{on }\partial \Omega,
\end{gathered}\label{P1}
\end{equation}
where $|H(x,s,\xi)|\leq \varphi(x)\left(f(x)+\theta|\xi|^q\right)$
 with $p-1<q\leq p$, $1<p<2$, $\theta>0$ and
$\varphi(x)=(2\pi)^{-n/2}\exp\big(-\frac{|x|^{2}}{2}\big)$
is the density of Gauss measure.
Problem \eqref{P1} is related to the generator of Ornstein-Uhlenbeck semigroup.

 As $\Omega$ is bounded, the operator in \eqref{P1} is uniformly elliptic.
 In this case, it is well known that one can use Schwarz symmetrization
to estimate the solutions of elliptic equations in terms of the solutions
of radially symmetric problems.
 This kind of issue has been faced in \cite{Diaz2,Talenti1,Mossino1,Alvino1}
for linear equations.
 As regards nonlinear equations, for the case of $p-1$ growth in the gradient,
comparison results are obtained in \cite{Betta1,Messano1}.
Optimal summability of solutions are discussed in \cite{Cianchi}.
 For the case of $p$ growth in the gradient, using Schwarz symmetrization,
  the existence of bounded solutions are obtained in
\cite{Diaz1,Grenon1,Grenon2,Messano1}.
  For the case of $q  (p-1<q\leq p)$ growth in the gradient,
  similar results  can be found in \cite{Ferone1,Tian1}.
  Recently, symmetrization techniques have also been applied to equations
involving fractional Laplacian operators (see \cite{Vazquaz1,Vazquaz2,Blasio1}).



  In our case, since $\Omega$ maybe unbounded,
the degeneracy of the operator does not allow to use the classical approach
via Schwarz symmetrization.
   This leads us to consider Gauss symmetrization based on the structure of
   the problem. By Gauss symmetrization, comparison results for
 linear equations have been obtained, with a simpler problem which is defined
in a half space and has data depending only on the first variable
(see \cite{Chiacchio1, Blasio2, Blasio4}).
   Nonlinear equations with $p-1$ growth in the gradient have also been discussed
  in \cite{Blasio3}. However,
  the case with other growth in the gradient has not been studied until now.
   In this paper, we deal with such problem (with $q  (p-1<q\leq p)$
    growth in the gradient).

 Our aim is to prove a sharp comparison result which allows us to estimate
the solutions of  \eqref{P1}  in terms of the symmetric solutions of the
following ``symmetrized" problem
\begin{equation}
\begin{gathered}
-D_1(\varphi|D_1v|^{p-2}D_1v)=\varphi f^\sharp+\theta\varphi|D_1v|^{q}
\quad  \text{in } \Omega^\sharp,\\
 v=0 \quad \text{on }\partial \Omega^\sharp,
\end{gathered}\label{P2}
\end{equation}
where $\Omega^\sharp$ is a half space with the same Gauss
measure as $\Omega$ and $f^\sharp$ is Gauss symmetrization of $f$.
To this end, we first discuss the existence of symmetric solutions to
 \eqref{P2} and give the regularity results of such solutions,
which is a key step for the comparison results.
Moreover, by the comparison results, we are able to prove the existence of
bounded solutions to  \eqref{P1}   in weighted Sobolev
space $W_0^{1,p}(\varphi,\Omega)$. Note that the assumption $1<p<2$
is necessary to the existence of bounded solutions. We will give an example
in the Appendix to show that there may be no solution to  \eqref{P1} if $p\geq2$.

There are two main difficulties in studying  \eqref{P1}.
One is due to the fact that the operator is in general not uniformly elliptic,
for instance when $\Omega$ is an unbounded domain.
  Another is due to the presence of general growth in the gradient.
Therefore, the present approaches   can not be applied to our case.
   To overcome the above difficulties, based on the properties of the weighted
rearrangement, we convert the problems into the study of a class of
Volterra integral operator. This class of Volterra integral operator was
introduced in \cite{pasic1,Korkut}. By discussing the existence of fixed
point to the Volterra integral operator, we obtained the existence and
non-existence of symmetric solution to the ``symmetrized" problem \eqref{P2}.
The sharp comparison results are obtained by proving a new  type of comparison
principle for the Volterra integral operator. The methods developed in this
article can also be used to study the corresponding variational inequalities.


This article is organized as follows:
In Section 2  we give some preliminary results.
In Section 3, the main results of this paper are stated.
In Section 4, three results for a class of Volterra
integral operator are proved.
In Section 5, 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.

Let $\gamma_{n}$ be the $n$-dimensional normalized Gauss measure on
 $\mathbb{R}^n$ defined as
$$
d\gamma_{n}=\varphi(x)dx=(2\pi)^{-n/2}\exp\big(-\frac{|x|^2}{2}\big)dx, \quad
x\in \mathbb{R}^n.
$$
We  denote by $\Phi(\tau)$ the measure of the half space
$\{x\in\mathbb{R}^n:x_{1}>\tau\}$, i.e.
$$
\Phi(\tau)=\gamma_{n}(\{x\in\mathbb{R}^n:x_{1}>\tau\})
=\frac{1}{\sqrt{2\pi}}\int_{\tau}^{+\infty}\exp\big(-\frac{t^2}{2}\big)dt,
\tau\in \mathbb{R}.
$$
Observe that (see \cite{Tian2})
  \begin{gather}\label{2.1}
\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{2.2}\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 positive constants depending on
$\gamma_n(\Omega)$.
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
\[
u^{\star}(s)=\inf\{t\geq0:\mu(t)\leq s\}, \quad s\in[0,\gamma_n(\Omega)].
\]
If $\Omega^{\sharp}=\{x=(x_1,x_2,\cdots,x_n)\in\mathbb{R}^n:x_1>\lambda\}$
 is 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 $u$).
\end{definition}

Similarly, the decreasing Gauss symmetrization of $u$ is
$$
u_{\sharp}(x)=u_{\star}(\Phi(x_1)), \quad x\in\Omega^{\sharp},
$$
with
$$
u_{\star}(s)=u^{\star}(\gamma_n(\Omega)-s),\quad s\in(0,\gamma_n(\Omega)).
$$
 The properties of rearrangement with
respect to Gauss measure or a positive measure have been widely considered,
see \cite{Chong,Carlen,Rakotoson1,Rakotoson2,Talenti2} for instance. Here we just
recall that


(a) (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.


(b) (Polya-Sz\"{e}go principle) Let $u\in W_0^{1,p}(\varphi,\Omega)$
with $1<p<+\infty$. Then
\begin{equation*}
\|\nabla u^{\sharp}\|_{L^{p}(\varphi,\Omega^\sharp)}\leq\|\nabla
u\|_{L^{p}(\varphi,\Omega)},
\end{equation*}
and equality holds if and only if
$\Omega=\Omega^\sharp$ and $|u|=u^\sharp$ modulo a rotation.

Finally, we recall that the weighted Sobolev space $W_0^{1,p}(\varphi,\Omega)$
is the closure of $C_0^{\infty}(\Omega)$ under the norm
$$
\|u\|_{W^{1,p}(\varphi,\Omega)}
=\Big(\int_{\Omega}|\nabla u(x)|^p\varphi\,dx+\int_{\Omega}|u(x)|^p
\varphi\,dx\Big)^{1/p}.
$$

\section{Statement of main results}

In this article, we consider the  problem
\begin{equation}
\begin{gathered}
-\operatorname{div}(a(x,u,\nabla u))=H(x,u,\nabla u) \quad \text{in }\Omega,\\
u\in W^{1,p}_0(\varphi,\Omega)\cap L^{\infty}(\Omega),
\end{gathered}\label{P3}
\end{equation}
where $\Omega$ is an open subset of
$\mathbb{R}^n(n\geq2)$ with Gauss measure less than one,
 $a:\Omega\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$ and
$H:\Omega\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}$
are Carath\'{e}odory functions such that for all
$(s,\xi)\in\mathbb{R}\times \mathbb{R}^n$ and for almost every $x\in\Omega$,
\begin{itemize}
\item[(A1)] $a(x,s,\xi)\xi\geq \varphi(x)|\xi|^p$, with $1<p<2$;
\item[(A2)] $|a(x,s,\xi)|\leq c_1\varphi(x)\left(k(x)+|s|^{p-1}+|\xi|^{p-1}\right)$,
with $c_1>0, k(x)\geq0$ and $k\in L^{p'}(\varphi,\Omega)$;
\item[(A3)] $|H(x,s,\xi)|\leq \varphi(x)\left(f(x)+\theta|\xi|^q\right)$, with
$\theta>0$, $p-1<q\leq p$, $f(x)\geq0$ and $f\in L^{\infty}(\Omega)$.
\end{itemize}

\begin{definition} \label{def3.1} \rm
We say that $u\in W_0^{1,p}(\varphi,\Omega)\cap
L^{\infty}(\Omega)$ is a solution of \eqref{P1}, if
\begin{equation} \label{e3.1}
\int_{\Omega}a(x,u,\nabla u)\nabla\psi\,dx=\int_{\Omega}H(x,u,\nabla
u)\psi\,dx,  \quad \forall\psi\in W_0^{1,p}(\varphi,\Omega)\cap
L^{\infty}(\Omega).
\end{equation}
\end{definition}

\begin{remark} \label{rmk3.1}\rm
 The assumption $1<p<2$ is necessary. As $p\geq2$, there may be no solution
to  \eqref{P3}  (see an example in the Appendix).
\end{remark}

First, let us turn our attention to the ``symmetrized" problem \eqref{P2},
 discussing the  existence and regularity of a unique symmetric solution,
which is a key step for the comparison results.

 \begin{theorem} \label{thm3.1}
Let $\gamma=\frac{q}{p-1}$, $\gamma'=\frac{\gamma}{\gamma-1}$
and
$M_0 =\theta\big(\sqrt{2\pi}\beta^{-1}(1-\log
\gamma_n(\Omega))^{-1/2}\big)^\gamma$.
If
\begin{equation}\label{1.2}
\|f\|_{L^{\infty}(\Omega)}\leq\frac{1}{\gamma'}
(\gamma M_0)^{\frac{1}{1-\gamma}},
\end{equation}
 then there exists a unique solution to \eqref{P2} such that
$v(x)=v^\sharp(x)$. Moreover,
$v\in C^{1}(\Omega^{\sharp}\backslash{x_1=+\infty})\cap W^{1,\infty}(\Omega^{\sharp})$
with provides the estimates
\begin{gather*}
\|\nabla v\|_{L^{\infty}(\Omega^{\sharp})}
\leq C_1(\beta, \gamma, \gamma_{n}(\Omega)), \\
\| v\|_{L^{\infty}(\Omega^{\sharp})}
\leq C_2(\beta, \gamma, \gamma_{n}(\Omega)),
\end{gather*}
where $C_1, C_2 $ are constants depending only on $\beta, \gamma$ and
$\gamma_{n}(\Omega)$.
\end{theorem}

In the case $f(x)\equiv f_0$, we have the following nonexistence result for
 \eqref{P2}.

\begin{theorem} \label{thm3.2} Let
\begin{equation}\label{1.4}
f_0\geq A(\gamma_n(\Omega)),
\end{equation}
where
\[
A(s)=\Big[\frac{\gamma^{\gamma'}}{\theta(\gamma-1)}
\big(\frac{\alpha}{\sqrt{2\pi}}\big)^{\gamma}\frac{s}{F_1(s)}
\Big]^{\frac{1}{\gamma-1}}, \quad
F_1(s)=\int_0^s(1-\ln\tau)^{-{\frac{\gamma}{2}}}d \tau, 
\]
for $s\in[0,\gamma_n(\Omega)]$.
 Then \eqref{P2} has no symmetric solution.
\end{theorem}

\begin{remark} \label{rmk3.2} \rm
By computations, it follows that $A(\gamma_n(\Omega))>\frac{1}{\gamma'}
(\gamma M_0)^{\frac{1}{1-\gamma}}$.
Thus, the above theorem proposes an example to show that \eqref{P2}
has no symmetric solution without the
assumption \eqref{1.2}.
\end{remark}

Now, the comparison results can be stated by
the following theorem.

\begin{theorem} \label{thm3.3}
Assume that {\rm (A1)--(A3)} hold. Let $u$ be a solution to \eqref{P3} and
 $v$ be a solution to  \eqref{P2} such that $v(x)=v^\sharp(x)$. Then
\begin{gather}\label{1.6}
u^{\sharp}(x)\leq v(x),  \quad  x\in\Omega^{\sharp}, \\
\label{1.7}\int_{\Omega}\eta(|\nabla u|^p)\varphi\,dx
\leq \int_{\Omega^\sharp}\eta(|\nabla v|^p)\varphi\,dx,
\end{gather}
where $\eta$ is a concave and nondecreasing function on
$[0,+\infty)$.
Moreover, if $f(x)\not\equiv 0$, the equality in \eqref{1.6} holds
if and only if
\begin{equation}\label{1.8}
\begin{gathered}
\Omega=\Omega^\sharp,\\
u(x)=\delta u^\sharp(x), \quad\text{a.e. } x\in\Omega^{\sharp},\\
a_1(x,u,\nabla u)=\delta\varphi|D_1 u^{\sharp}|^{p-2}D_1 u^{\sharp}, \quad
\text{a.e. }x\in\Omega^\sharp,\\
\sum_{i=2}^n D_ia_i(x,u,\nabla u)=0  \quad \text{in } \mathcal{D}'(\Omega^{\sharp}),\\
 H(x,u,\nabla u)=\delta\varphi(f^{\sharp}(x)+\theta|D_1
u^{\sharp}|^{q}), \quad \text{a.e. }x\in\Omega^\sharp
\end{gathered}
\end{equation}
modulo a rotation with $\delta=\pm 1$.
\end{theorem}

\begin{remark} \label{rmk3.3} \rm
Equality \eqref{1.8} implies that the comparison result \eqref{1.6} is
sharp in the sense that as the equality holds, problem \eqref{P3} is
equivalent to its ``symmetrized" problem \eqref{P2} modulo a rotation.
\end{remark}

The estimates we have found can be applied to prove the
existence result by using the well known
 approximation techniques \cite{Boccardo}.

 \begin{theorem} \label{thm3.4}
Let {\rm (A1)--(A3)} and \eqref{1.2} hold.
 Assume that
$$
[a(x,s,\xi_1)-a(x,s,\xi_2)]\cdot(\xi_1-\xi_2)>0,  \quad\text{for }
 \xi_1\neq\xi_2 .
$$
Then there exists at least one solution to \eqref{P3}.
\end{theorem}

\section{Results for  Volterra integral operators}

This section is devoted to study a class of Volterra
integral operator. We prove three results, i.e. the comparison principle,
the existence of fixed point and the nonexistence of fixed point for the Volterra
integral operator, which will be useful in proving the main results of this paper.

Assume $h(s,\xi) :[0,T]\times\mathbb{R}\to\mathbb{R}$
is a Carath\'{e}odory function. Consider the following Volterra
integral operator (see \cite{pasic1,Korkut})
\begin{equation*}
K: D(K)\subseteq C([0,T])\to C([0,T]), \quad
K\psi(t)=\int_0^th(\tau,\psi(\tau))d\tau,\quad \forall\psi\in D(K).
\end{equation*}

\begin{definition} \label{def4.1}\rm
 We say that the operator $K$ has property
(m) if for all $\psi_1, \psi_2\in D(K)$ and $a\in[0,T)$, there exist
 constants $b\in (a,T]$ and $m(a,b,\psi_1, \psi_2)\in [0,1)$ such
that for any $t\in(a,b]$,
\begin{equation}\label{4.1}
\|h(\cdot,\psi_1(\cdot))-h(\cdot,\psi_2(\cdot))\|_{L^1(a,t)}
\leq t m(a,b,\psi_1, \psi_2)\big\|\frac{\psi_1 -\psi_2}{s}\big\|_{L^\infty(a,t)}.
\end{equation}
\end{definition}

 \begin{lemma} \label{lem4.1}
 Let the operator $K$ satisfy property (m) and $h(t,\cdot)$ be nondecreasing
for a.e. $t\in[0,T]$. If $ u, v \in D(K)$ are such that
$u\leq Ku$, $v\geq Kv$, then we have
\begin{equation}\label{3.4}
u\leq v.
\end{equation}
In particular, the equation $w=Kw$ possesses at most one solution in $D(K)$.
\end{lemma}

\begin{proof}
 We argue by contradiction. If \eqref{3.4} does not
hold, then there must exist $a\in[0,T)$ and $b_1\in(a,T]$ such that
$u(t)\leq v(t)$ for $t\in[0,a]$ and $u(t)>v(t)$ for $t\in(a,b_1]$. Set
$b_2=\min\{b_1,b\}$ with $b$ is the constant of property (m).
Then $u(t)>v(t)$ in $(a,b_2]$.
Since $u\leq Ku$, $v\geq Kv$ and $h(t,\cdot)$ is nondecreasing
for a.e. $t\in[0,T]$, it follows that for $\forall  t\in(a,b_2]$,
\begin{equation}\label{4.3}
\begin{split}
|u(t)-v(t)|&=u(t)-v(t)\leq
\int_0^t\left(h(\tau,u(\tau))-h(\tau,v(\tau))\right)d\tau
\\
&=\int_0^a\left(h(\tau,u(\tau))-h(\tau,v(\tau))\right)d\tau+
\int_a^t\left(h(\tau,u(\tau))-h(\tau,v(\tau))\right)d\tau
\\
&\leq\int_a^t\left(h(\tau,u(\tau))-h(\tau,v(\tau))\right)d\tau
\\
&\leq\|h\left(\cdot,u(\cdot)\right)
-h\left(\cdot,v(\cdot)\right)\|_{L^1(a,t)}.
\end{split}
\end{equation}
Using property (m), we obtain
\begin{equation}\label{4.31}
|u(t)-v(t)|\leq t m(a,b,u,
v)\|\frac{u-v}{s}\|_{L^\infty(a,t)}, \quad \forall  t\in(a,b_2].
\end{equation}
Taking the maximum over $t\in(a,b_2]$ and noting $m(a,b,u, v)\in [0,1)$, we have
$$
\|\frac{u-v}{s}\|_{L^\infty(a,b_2)}\leq m(a,b,u,v)
\|\frac{u -v}{s}\|_{L^\infty(a,b_2)}
<\|\frac{u -v}{s}\|_{L^\infty(a,b_2)},
$$
which is a contradiction. Thus
$$
u\leq v,   \forall  u, v \in D(K),
$$
and the uniqueness claim easily follows. Then the lemma is proved.
\end{proof}

Let
\begin{equation*}%\label{3.6}
K\psi(s)=\int_0^sf^\star(\tau)+\theta\Big(\sqrt{2\pi}
\exp\big(\frac{\Phi^{-1}(\tau)^{2}}{2}\big)\Big)^\gamma\psi^\gamma(\tau)d\tau,
\quad  s\in[0,\gamma_n(\Omega)].
\end{equation*}
We shall deal with two types of domains
\begin{gather*}%\label{3.7}
D_1(K)=\{\psi\in C([0,\gamma_n(\Omega)]):
M\geq0, 0\leq\psi(s)\leq Ms\},\\  %\label{3.8}
D_2(K)=\{\psi\in C([0,\gamma_n(\Omega)]):
M_{\psi}\geq0, 0\leq\psi(s)\leq M_{\psi}s\},
\end{gather*}
and let $R_i(K)$ be   the  range  of $K$ on  $D_i(K)$,  $i=1,2$.

\begin{lemma} \label{lem4.2}
 Let $M=(\gamma M_0)^{\frac{1}{1-\gamma}}$ in $ D_1(K)$.
If \eqref{1.2} holds, then  the equation $w=Kw$ has a unique solution
in $ D_1(K)$. Furthermore, the solution is also
unique in $D_2(K)$.
\end{lemma}

 \begin{proof}
 First, we prove that
$R_1(K)\subseteq D_1(K)$. For $\forall\psi\in D_1(K)$ and
$s\in[0,\gamma_n(\Omega)]$,
 we have by \eqref{2.2} that
\begin{equation}\label{3.9}
\begin{split}
K\psi(s)&=\int_0^sf^\star(\tau)+\theta\Big(\sqrt{2\pi}
\exp\big(\frac{\Phi^{-1}(\tau)^{2}}{2}\big)\Big)^\gamma\psi^\gamma(\tau)d\tau
\\
&\leq s\|f\|_{L^{\infty}(\Omega)}+\theta\big(\sqrt{2\pi}\beta^{-1}\big)^\gamma
\int_0^s\Big(\frac{1}{\tau(1-\log
\tau)^{1/2}}\Big)^\gamma(M\tau)^\gamma d\tau\\
&\leq \left(\|f\|_{L^{\infty}(\Omega)}+\theta\left(\sqrt{2\pi}\beta^{-1}M(1-\log
\gamma_n(\Omega))^{-1/2}\right)^\gamma\right) s\\&
= \left(\|f\|_{L^{\infty}(\Omega)}+M_0M^{\gamma}\right) s.
\end{split}
\end{equation}
 Under the assumption \eqref{1.2}, noting that
$M=(\gamma M_0)^{\frac{1}{1-\gamma}}$, we obtain
$$
\|f\|_{L^{\infty}(\Omega)}+M_0M^{\gamma}\leq M.
$$
 Thus,
$$
0\leq K\psi(s)\leq Ms, \quad s\in[0,\gamma_n(\Omega)].
$$
This  implies that $R_1(K)\subseteq D_1(K)$.

Next, we verify that the operator $K$ is compact with
respect to the uniform topology of $C([0,\gamma_n(\Omega)])$.
To prove this, let us first show the
equicontinuity of $R_1(K)$ in $C([0,\gamma_n(\Omega)])$. Take any
$K\psi\in R_1(K)$. For any $0\leq a<b\leq \gamma_n(\Omega)$, it follows
\begin{equation}\begin{split}\label{3.10}
&|K\psi(b)-K\psi(a)|\\
&\leq \int_a^b\Big|f^\star(\tau)+\theta\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^\gamma\psi^\gamma(\tau)
\Big|d\tau\\
& \leq\|f\|_{L^{\infty}(\Omega)}(b-a)
 +\theta\big(\sqrt{2\pi}\beta^{-1}\big)^\gamma
\int_a^b\Big(\frac{1}{\tau(1-\log
\tau)^{1/2}}\Big)^\gamma(M\tau)^\gamma d\tau\\
& \leq(\|f\|_{L^{\infty}(\Omega)}+M_0M^\gamma)(b-a).
\end{split}
\end{equation}
Then, $R_1(K)$ is equicontinuous in $C([0,\gamma_n(\Omega)])$.
Also, the  family of functions from $R_1(K)$ is uniformly bounded
(see \eqref{3.9}), i.e.
\begin{equation}\label{3.11}0\leq
K\psi\leq\left(\|f\|_{L^{\infty}(\Omega)}+M_0M^{\gamma}\right)\gamma_n(\Omega).
\end{equation}
By Ascoli-Arzela theorem, 
$R_1(K)$ is relatively compact in $C([0,\gamma_n(\Omega)])$, and hence
the operator $K$ is compact.

Since the domain $D_1(K)$ is bounded, closed and
convex, by Schauder's fixed point theorem there exists at
least one solution $w\in D_1(K)$ to the equation
 $w=Kw$.

Now, we study the uniqueness of the solution $w$.
By   $D_1(K)\subseteq D_2(K)$,
it suffices to show that the solution $w$ is unique in $D_2(K)$.

Next we check that $K$ satisfies property (m) in $D_2(K)$.
For all $\psi_1, \psi_2\in D_2(K)$, $ 0\leq a< b\leq  \gamma_n(\Omega)$ and
$a<t\leq b$, we have
\begin{equation}\label{4.8}
\begin{split}
&\|h(\cdot,\psi_1(\cdot) )-h(\cdot,\psi_2(\cdot) )\|_{L^1(a,t)}\\
&\leq\int_a^t\theta\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^\gamma
|\psi_1^\gamma(\tau)-\psi_2^\gamma(\tau)|d\tau\\
& \leq\theta\gamma\big(\sqrt{2\pi}\beta^{-1}\big)^\gamma\int_a^t
\left(\frac{1}{\tau(1-\log \tau)^{1/2}}\right)^\gamma
\max\{\psi_1^{\gamma-1}(\tau),\psi_2^{\gamma-1}(\tau)\}\\
&\quad\times  |\psi_1(\tau)-\psi_2(\tau)|d\tau\\
& \leq\theta\gamma\Big(\sqrt{2\pi}\beta^{-1}\Big)^\gamma
\max\{M_{\psi_1},M_{\psi_2}\}^{\gamma-1}\int_a^t
\Big(\frac{1}{\tau(1-\log \tau)^{1/2}}\Big)^\gamma
\tau^{\gamma-1}\\
&\quad\times |\psi_1(\tau)-\psi_2(\tau)|d\tau\\
& \leq \theta\gamma\left(\sqrt{2\pi}\beta^{-1}(1-\log
\gamma_n(\Omega))^{-1/2}\right)^\gamma\widetilde{M}^{\gamma-1}
(t-a)\|\frac{\psi_1-\psi_2}{\tau}\|_{L^\infty(a,t)}\\
&\leq t\gamma M_0 \widetilde{M}^{\gamma-1}
\big(1-\frac{a}{b}\big)\|\frac{\psi_1-\psi_2}{\tau}\|_{L^\infty(a,t)}\\
&=tm(a,b,\psi_1,\psi_2)\|\frac{\psi_1-\psi_2}{\tau}\|_{L^\infty(a,t)},
\end{split}
\end{equation}
where
\[
m(a,b,\psi_1,\psi_2)=\gamma M_0 \widetilde{M}^{\gamma-1}
(1-\frac{a}{b})
\]
 with $\widetilde{M}=\max\{M_{\psi_1},M_{\psi_2}\}$.

From  $\lim_{b\to a^+}m(a,b,\psi_1,\psi_2)=0$, it follows that there exists $b_0>a$
such that $0\leq m(a,b_0,\psi_1,\psi_2)<1$. Thus \eqref{4.8} implies that
$K$ satisfies property (m) in $D_2(K)$. Since $ h(t,\psi)$ is nondecreasing
for $t\in[0,\gamma_n(\Omega)]$, we have by Lemma \ref{lem4.1} that the solution
of $w=Kw$ is unique in $D_2(K)$. Thus the proof is complete.
\end{proof}

For $f(x)\equiv f_0$, we have the following nonexistence result.

\begin{lemma} \label{lem4.3}
 Let \eqref{1.4} hold. Then the equation
$z=Kz$ has no solution in $D_1(K)$.
\end{lemma}

\begin{proof}
 We argue by contradiction. Assume that there exists
a solution $z\in D_1(K)$ of $z=Kz$.
Let $z_0(s)=f_0s$ and
\begin{equation}\label{3.32}
z_{m}(s)=\theta\int_0^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^\gamma
z_{m-1}^\gamma(\tau)d\tau, \quad s\in[0,\gamma_n(\Omega)].
\end{equation}
First, we claim that
\begin{equation}\label{3.34}
z_m(s)\geq sA(s)\Big(\frac{f_0}{A(s)}\Big)^{\gamma^m}\gamma^{\frac{m}{\gamma-1}},
\quad m\geq1
\end{equation}
where
\[
A(s)=\Big[\frac{\gamma^{\gamma'}}{\theta(\gamma-1)}
\left(\frac{\alpha}{\sqrt{2\pi}}\right)^{\gamma}\frac{s}{F_1(s)}
\Big]^{\frac{1}{\gamma-1}},\quad
F_1(s)=\int_0^s(1-\ln\tau)^{-{\frac{\gamma}{2}}}d \tau.
\]

 To prove \eqref{3.34}, set
\begin{equation*}%\label{3.36}
F_{m+1}(s)=\int_0^s \left( 1-\ln \tau\right)^{-\gamma/2}
F^{\gamma}_m(\tau)d\tau.
\end{equation*}
Thus,
\begin{equation}\label{3.35}
z_m(s)\geq\theta^{\frac{\gamma^m-1}{\gamma-1}}
\Big(\frac{\sqrt{2\pi}}{\alpha}\Big)^{\frac{\gamma^{m+1}-\gamma}{\gamma-1}}
 f_0^{\gamma^m}
s^{-\frac{\gamma^m-\gamma}{\gamma-1}}F_m(s).
\end{equation}
Indeed, by \eqref{2.1} and \eqref{3.32}, we have
\begin{align*}
z_{1}(s)&=\theta \int_0^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^\gamma
z_0^\gamma(\tau)d\tau\\
& \geq\theta \Big(\frac{\sqrt{2\pi}}{\alpha}\Big)^{\gamma}f^{\gamma}_0
\int_0^s\Big(\frac{1}{\tau(1-\ln\tau)^{1/2}}\Big)^\gamma
{\tau}^\gamma d\tau\\
&=\theta \Big(\frac{\sqrt{2\pi}}{\alpha}\Big)^{\gamma}f^{\gamma}_0F_1(s),
\end{align*}
which proves \eqref{3.35} for $m=1$. Assume it holds for $m-1$. Then
\eqref{3.35} can be established by induction on $m$. Next, treat the term
$F_m(s)$ in \eqref{3.35}. Actually,
\begin{equation}\label{3.37}
F_m(s)\geq \prod_{\delta=1}^{m}
\Big(\frac{\gamma-1}{\gamma^\delta-1}\Big)^{\gamma^{m-\delta}}
F_1^{\frac{\gamma^m-1}{\gamma-1}}(s), m\geq1.
\end{equation}
The case $m=1$ is obvious. Suppose \eqref{3.37} holds for some $m$. Then
\begin{align*}
F_{m+1}(s)&=\int_0^s \left( 1-\ln
\tau\right)^{-\gamma/2} F^{\gamma}_m(\tau)d\tau\\
& \geq\prod_{\delta=1}^{m}
 \Big(\frac{\gamma-1}{\gamma^\delta-1}\Big)^{\gamma^{m+1-\delta}}
\int_0^s \left( 1-\ln \tau\right)^{-\gamma/2}
 F^{\frac{\gamma^{m+1}-\gamma}{\gamma-1}}_1(\tau)d\tau\\
&= \prod_{\delta=1}^{m}\Big(\frac{\gamma-1}{\gamma^\delta-1}\Big)^{\gamma^{m+1-\delta}}
\int_0^s
F^{\frac{\gamma^{m+1}-\gamma}{\gamma-1}}_1(\tau)dF_1(\tau)\\
&= \prod_{\delta=1}^{m+1}\Big(\frac{\gamma-1}{\gamma^\delta-1}
 \Big)^{\gamma^{m+1-\delta}}
 F^{\frac{\gamma^{m+1}-1}{\gamma-1}}_1(s),
\end{align*}
which proves \eqref{3.37}.
Moreover,
\begin{equation} \label{3.39}
\begin{split}
\prod_{\delta=1}^{m}\Big(\frac{\gamma-1}{\gamma^\delta-1}\Big)^{\gamma^{m-\delta}}
&\geq \prod_{\delta=1}^{m}
 \Big(\frac{\gamma-1}{\gamma^\delta}\Big)^{\gamma^{m-\delta}}
=\left(\gamma-1\right)^{\sum_{\delta=1}^{m}\gamma^{m-\delta}}
\gamma^{-\sum_{\delta=1}^{m}\delta\gamma^{m-\delta}}\\
&=\left(\gamma-1\right)^{\frac{\gamma^m-1}{\gamma-1}}
\gamma^{\frac{m}{\gamma-1}-\frac{\gamma(\gamma^m-1)}{(\gamma-1)^2}}.
\end{split}
\end{equation}
Combining \eqref{3.35}, \eqref{3.37} and \eqref{3.39},
 we know that \eqref{3.34} holds.

 On the other hand, $A(s)$ is a decreasing continuous function on
$(0,\gamma_n(\Omega)]$ and $\lim_{s\to 0^+}A(s)=+\infty$.
Then the range of $A$ is $[A(\gamma_n(\Omega)), +\infty)$.
Recalling that $f_0\geq A(\gamma_n(\Omega)))$, thus there must exist a constant
$s^\ast\in(0,\gamma_n(\Omega)]$ such that $f_0=A(s^\ast)$.
As $s\geq s^\ast$, it follows that $\frac{f_0}{A(s)}\geq1$ and
\begin{equation*}
z_m(s)\geq sA(s)\gamma^{\frac{m}{\gamma-1}}.
 \end{equation*}
Note that
 $\lim_{m\to\infty}\gamma^{\frac{m}{\gamma-1}}=+\infty$.
  We conclude that $\sum_{m=0}^{\infty}z_m(s)=+\infty$ for
 $s\in [s^\ast,\gamma_n(\Omega)]$. On the other hand, by inducing on
$k$, it is easy to prove that
\begin{equation}\label{3.33}
z(s)\geq\sum_{m=0}^kz_m(s),  \quad \forall k\in \mathbb{N}.
\end{equation}
Indeed, since $z(s)=Kz(s)\geq f_0s=z_0(s)$, we obtain
\[
z(s)=f_0s+\theta\int_0^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^\gamma
z^\gamma(\tau)d\tau
   \geq z_0(s)+z_1(s),
\]
which proves the claim for $k=1$. Now assume \eqref{3.33} holds for some $k\in N$.
Then
\begin{align*}
z(s)&\geq
z_0(s)+\theta\int_0^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^\gamma
\Big(\sum_{m=0}^kz_m(\tau)\Big)^\gamma d\tau\\
& \geq z_0(s)+\theta\sum_{m=0}^k\int_0^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^\gamma
z_m^\gamma(\tau)d\tau\\
&=z_0(s)+\sum_{m=0}^kz_{m+1}(s)
=\sum_{m=0}^{k+1}z_m(s).
\end{align*}
Thus for $k\in \mathbb{N}$, \eqref{3.33} holds.
 However, recalling that $z\in D_1(K)$,
 we obtain
$$
Ms\geq z(s)\geq \sum_{m=0}^{\infty}z_m(s)=+\infty, \quad
 s\in [s^\ast,\gamma_n(\Omega)],
$$
which is a contradiction. Thus the lemma is proved.
\end{proof}

\section{Proofs of the main results}

First, let us enunciate the following lemma, the proof of which is not
supplied here since it follows the same lines as in \cite{Ferone1,Messano1}.


\begin{lemma} \label{lem5.1}
 Let $u$ be a solution of \eqref{P1} and $v$ be
a solution of \eqref{P2} such that $v(x)=v^\sharp(x)$. Then
\begin{equation} \label{3.12}
\begin{split}
&\big(-u^{\star'}(s)\big)^{{p-1}}
\Big(\frac{1}{\sqrt{2\pi}}
\exp\Big(-\frac{\Phi^{-1}(s)^{2}}{2}\Big)\Big)^p\\
&\leq\int_0^sf^\star(\tau)\exp
\Big[\theta\int_\tau^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\sigma)^{2}}{2}\Big)\Big)^{p-q}
\left(-u^{\star'}(\sigma)\right)^{q-p+1}d\sigma\Big] d\tau
\end{split}
\end{equation}
and
\begin{equation} \label{3.13}
\begin{split}
&\big(-v^{\star'}(s)\big)^{{p-1}}
\Big(\frac{1}{\sqrt{2\pi}}
\exp\Big(-\frac{\Phi^{-1}(s)^{2}}{2}\Big)\Big)^p\\
&=\int_0^sf^\star(\tau)\exp
\Big[\theta\int_\tau^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\sigma)^{2}}{2}\Big)\Big)^{p-q}
\big(-v^{\star'}(\sigma)\big)^{q-p+1}d\sigma\Big] d\tau,
\end{split}
\end{equation}
 a.e. $s\in(0,\gamma_n(\Omega))$.
\end{lemma}


\begin{proof}[Proof of Theorem \ref{thm3.1}]
Let
\begin{equation}\label{3.14}
v(x)=V(\Phi(x_1))
=\int_{\Phi(x_1)}^{\gamma_n(\Omega)}\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^{p'}w^{\frac{1}{p-1}}(\tau)d
\tau, x\in\Omega^\sharp,
\end{equation}
where $w$ is the unique solution of $w=Kw$ obtained in Lemma \ref{lem4.2}.
 Clearly, $v(x)=v^{\sharp}(x)$, $x\in\Omega^{\sharp}$.
 By \eqref{3.14} and \eqref{2.2},
\begin{align*}
|\nabla v(x)|
&=D_1v(x)=-\frac{1}{\sqrt{2\pi}}V'(\Phi(x_1))
\exp\big(-\frac{x_1^{2}}{2}\big)\\
&= \Big(\sqrt{2\pi}\exp\big(\frac{x_1^{2}}{2}\big)\Big)^{\frac{1}{p-1}}
w^{\frac{1}{p-1}}(\Phi(x_1))\\
&\leq\big(\sqrt{2\pi}\big)^\frac{1}{p-1}\left(\beta\Phi(x_1)(1-\log
\Phi(x_1))^{1/2}\right)^{-\frac{1}{p-1}}(M\Phi(x_1))^{\frac{1}{p-1}}\\
& \leq \Big(\sqrt{2\pi}\beta^{-1} M(1-\log
\gamma_n(\Omega))^{-1/2}\Big)^{\frac{1}{p-1}}.
\end{align*}
Moreover, since $1<p<2$,
\begin{align*}
\|v\|_{L^{\infty}(\Omega^{\sharp})}
&=V(0)=\int_0^{\gamma_n(\Omega)}\Big(\sqrt{2\pi}
\exp\big(\frac{\Phi^{-1}(\tau)^{2}}{2}\big)\Big)^{p'}w^{\frac{1}{p-1}}(\tau)d
\tau\\
&\leq \big(\sqrt{2\pi}\beta^{-1}\big)^{p'}M^{\frac{1}{p-1}}
 \int_0^{\gamma_n(\Omega)}\Big(\frac{1}{\tau(1-\log
\tau)^{1/2}}\Big)^{p'}\tau^{\frac{1}{p-1}}d \tau\\
& =\big(\sqrt{2\pi}\beta^{-1}\big)^{p'}M^{\frac{1}{p-1}}
\int_0^{\gamma_n(\Omega)}(1-\log \tau)^{-\frac{p'}{2}}\tau^{-1}d \tau\\
& =\frac{2}{p'-2}\big(\sqrt{2\pi}\beta^{-1}\big)^{p'}M^{\frac{1}{p-1}}
(1-\log\gamma_n(\Omega))^{-\frac{p'}{2}+1}.
\end{align*}
Then $v\in W^{1,p}_0(\varphi,\Omega^{\sharp})\cap L^{\infty}(\Omega^{\sharp})$.

On the other hand, for all
$\psi\in W_0^{1,p}(\varphi,\Omega^{\sharp})\cap L^{\infty}(\Omega^{\sharp})$,
by \eqref{3.14} and the fact that $w=Kw$ in $D_1(K)$,
we have
\begin{align*}
&\int_{\Omega^{\sharp}}\varphi (D_1v)^{p-1} D_1\psi\,dx\\
&=\int_{\Omega^{\sharp}}\varphi\Big(\frac{1}{\sqrt{2\pi}}
\exp\big(-\frac{x_1^2}{2}\big)\Big)^{p-1}
\left(-V'\left(\Phi(x_1)\right)\right)^{p-1}D_1\psi\,dx\\
& =\int_{\Omega^{\sharp}} w(\Phi(x_1))D_1\psi\,dx\\
& =\int_{\Omega^{\sharp}}\int_0^{\Phi(x_1)}
\Big[f^\star(\tau)+\theta\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^\gamma
w^\gamma(\tau)\Big]d\tau D_1\psi\,dx.
\end{align*}
Integrating by part on the right-hand side of the third equality to obtain
\begin{align*}
&\int_{\Omega^{\sharp}}\varphi\left( D_1v\right)^{p-1} D_1\psi\,dx\\
&=\int_{\Omega^{\sharp}}\Big[f^{\sharp}(x)+\theta\Big(\sqrt{2\pi}
\exp\big(\frac{x_1^{2}}{2}\big)\Big)^\gamma
w^\gamma(\Phi(x_1))\Big] \psi \varphi\,dx\\
&= \int_{\Omega^{\sharp}}f^{\sharp} \psi \varphi\,dx
 +\int_{\Omega^{\sharp}}\theta\Big(-V'(\Phi(x_1))\frac{1}{\sqrt{2\pi}}
\exp\Big(-\frac{x_1^{2}}{2}\Big)\Big)^q \psi \varphi\,dx\\
&=\int_{\Omega^{\sharp}}f^{\sharp} \psi \varphi\,dx
 +\int_{\Omega^{\sharp}}\theta|D_1v|^q \psi \varphi\,dx.
\end{align*}
Hence, $v$ is a symmetric solution to  \eqref{P2}.
\end{proof}

Next, we show that $v$ is the unique symmetric
solutions to  \eqref{P2}. Indeed, assume that there exists another
symmetric solution $v_1$. Let
\begin{equation} \label{3.20}
\begin{aligned}
w_1(s)&=\int_0^sf^\star(\tau)\exp
\Big[\theta\int_\tau^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\sigma)^{2}}{2}\Big)\Big)^{p-q}\\
&\quad   \times\big(-v_1^{\star'}(\sigma)\big)^{q-p+1}d\sigma\Big] d\tau.
\end{aligned}
\end{equation}
It follows from \eqref{3.13} that
\begin{equation}\label{3.21}
-v_1^{\star'}(s)=\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(s)^{2}}{2}\Big)\Big)^{p'}w_1^{\frac{1}{p-1}}(s),
\quad \text{a.e. }  s\in (0,\gamma_n(\Omega)).
\end{equation}
Furthermore, a simple computation shows that
\[
w_1'(s)=f^\star(s)+\theta\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(s)^{2}}{2}\Big)\Big)^{\gamma}w_1^{\gamma},
\quad \text{a.e. }s\in (0,\gamma_n(\Omega)),
\]
which gives $w_1=Kw_1$.
By \eqref{3.20}, \eqref{2.2} and H\"{o}lder inequality,
\begin{align*}
w_1(s)&\leq \int_0^sf^\star(\tau)\exp
\Big[\theta\big(\sqrt{2\pi}\beta^{-1}\big)^{p-q}
\Big(\int_\tau^s\frac{1}{\sigma(1-\log \sigma)^{1/2}}d \sigma\Big)^{p-q}
\\
&\quad  \times\Big(-\int_\tau^sv_1^{\star'}(\sigma)d\sigma\Big)^{q-p+1}\Big] d\tau\\
&\leq \int_0^sf^\star(\tau)\exp
\Big[\theta\big(\sqrt{2\pi}\beta^{-1}\big)^{p-q}
 \| v_1\|_{L^\infty(\Omega^\sharp)}^{q-p+1}
\Big(\int_\tau^s\frac{1}{\sigma(1-\log \sigma)^{1/2}}d
\sigma\Big)^{p-q}\Big] d\tau \\
&\leq\int_0^sf^\star(\tau)\exp
\Big[C_1 \Big(\int_\tau^s\sigma^{-1}(1-\log
\sigma)^{-\frac{3}{2}}d\sigma\Big)^\frac{{p-q}}{3}
\Big(\int_\tau^s\sigma^{-1}d\sigma\Big)^\frac{{2(p-q)}}{3}\Big] d\tau,
\end{align*}
where  $C_1=\theta\big(\sqrt{2\pi}\beta^{-1}\big)^{p-q}
\| v_1\|_{L^\infty(\Omega^\sharp)}^{q-p+1}$. Since
\[
\int_{\tau}^{s} \sigma^{-1}(1-\log \sigma)^{-\frac{3}{2}}
 d\sigma\leq 2(1-\log \gamma_n(\Omega))^{-1/2},
\]
 by Young's inequality with $\varepsilon$, the above
 inequality becomes
\begin{align*}
w_1(s)&\leq \int_0^sf^\star(\tau)\exp
\Big[C_2 \Big(\int_\tau^s\sigma^{-1}d\sigma\Big)^\frac{{2(p-q)}}{3}\Big] d\tau
\\
&\leq \int_0^sf^\star(\tau)\exp
\Big( C_{1\varepsilon}C_2+\varepsilon C_2
\int_\tau^s\sigma^{-1}d\sigma\Big) d\tau  \\
&=C_{2\varepsilon}\int_0^sf^\star(\tau)
(\frac{s}{\tau})^{\varepsilon C_2}d\tau,
\end{align*}
where
\[
C_2=\theta\big(\sqrt{2\pi}\beta^{-1}\big)^{p-q}
\| v_1\|_{L^\infty(\Omega^\sharp)}^{q-p+1}
\left(2(1-\log \gamma_n(\Omega)\right)^{-1/2})^{\frac{p-q}{3}},
\]
 $C_{1\varepsilon}$ and $C_{2\varepsilon}$ are positive constants depending
on $\varepsilon$.
Take $\varepsilon$ small enough such that $\varepsilon C_2<1$. Then
\[
0\leq w_1(s)\leq C_{3\varepsilon}\|f\|_{L^\infty(\Omega)}s.
\]
 Thus, $w_1\in D_2(K)$ satisfying $w_1=Kw_1 $. By Lemma \ref{lem4.2}, we know $w_1=w$.
Noting
$v_1^{\star}(\gamma_n(\Omega))=0$, from \eqref{3.21} we have
\begin{equation*}
v_1^{\star}(s)=\int_s^{\gamma_n(\Omega)}\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^{p'}w^{\frac{1}{p-1}}(\tau)d
\tau.
\end{equation*}
Hence $v_1=v$ and then the uniqueness is proved.


 \begin{remark} \label{rmk5.1}
From the above proof, we find that $v^{\star}$ and
 $w$ can be expressed by each other via the following equations:
\begin{equation}\label{3.27}
v^{\star}(s)=\int_s^{\gamma_n(\Omega)}\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^{p'}
w^{\frac{1}{p-1}}(\tau)d \tau
\end{equation}
and
\begin{equation} \label{3.28}
w(s)=\int_0^sf^\star(\tau)\exp
\Big[\theta\int_\tau^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\sigma)^{2}}{2}\Big)\Big)^{p-q}
\big(-v^{\star'}(\sigma)\big)^{q-p+1}d\sigma\Big] d\tau.
\end{equation}
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm3.2}]
 Assume that there exists a symmetric solution $v$ to
 \eqref{P2}. From Remark \ref{rmk5.1}, we see that $w$ defined by
\eqref{3.28} is a solution of the equation
\[
z=Kz, \quad  z\in D_1(K),
\]
 which contradict with Lemma \ref{lem4.3}. Thus the proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.3}] \quad \smallskip

\noindent\textbf{Step 1.}
 First we verify \eqref{1.6}. Take
\[
\rho(s)=\int_0^sf^\star(\tau)\exp
\Big[\theta\int_\tau^s\Big(\sqrt{2\pi}
\exp\Big(\frac{\Phi^{-1}(\sigma)^{2}}{2}\Big)\Big)^{p-q}
 \big(-u^{\star'}(\sigma)\big)^{q-p+1}d\sigma\Big] d\tau.
\]
Then $\rho\in D_2(K)$ and $\rho(s)\leq K\rho(s)$ for $s\in[0,\gamma_n(\Omega)]$.
Now remembering that $K$ has property (m) (see the proof of Lemma \ref{lem4.2}) and
$w=Kw$ in $D_2(K)$,
by Lemma \ref{lem4.1} we obtain
\begin{equation*}
\rho(s)\leq w(s),\quad  s\in[0,\gamma_n(\Omega)].
\end{equation*}
Moreover, \eqref{3.12} and \eqref{3.13} imply that
\begin{equation*}
\big(-u^{\star'}(s)\big)^{{p-1}}
\Big(\frac{1}{\sqrt{2\pi}}
\exp\Big(-\frac{\Phi^{-1}(s)^{2}}{2}\Big)\Big)^p\leq
\rho(s)  \quad \text{a.e. }s\in(0,\gamma_n(\Omega))
\end{equation*}
and
\begin{equation*}
\big(-v^{\star'}(s)\big)^{{p-1}}
\Big(\frac{1}{\sqrt{2\pi}}
\exp\Big(-\frac{\Phi^{-1}(s)^{2}}{2}\Big)\Big)^p
=w(s)  \quad \text{a.e. }s\in(0,\gamma_n(\Omega)).
\end{equation*}
Thus,
$-u^{\star'}(s)\leq -v^{\star'}(s)$,  a.e. $ s\in(0,\gamma_n(\Omega))$.
Note that $u^{\star}(\gamma_n(\Omega))=v^{\star}(\gamma_n(\Omega))=0$.
We have
$$
u^{\star}(s)\leq v^{\star}(s),  s\in[0,\gamma_n(\Omega)].
$$
The proof of \eqref{1.7} can be done by repeating the proof of
\cite[Theorem 4.1]{Ferone1}.
\smallskip

\textbf{Step 2.} The equality case of \eqref{1.6} will be studied.
Sufficiency is obvious. Let us prove the necessity.
Assume that $u^\star(s)= v^\star(s)$ for
$s\in[0,\gamma_n(\Omega)]$. By Polya-Sz\"{e}go principle
and \eqref{1.7}, we have
\[
\int_{\Omega^{\sharp}}|\nabla v|^p\varphi\,\,dx
=\int_{\Omega}|\nabla u^{\sharp}|^p\varphi\,dx
\leq\int_{\Omega}|\nabla u|^p\varphi\,\,dx
\leq\int_{\Omega^{\sharp}}|\nabla v|^p\varphi \,dx.
\]
 Thus,
\[
\int_{\Omega^{\sharp}}|\nabla u^{\sharp}|^p\varphi
\,dx=\int_{\Omega}|\nabla u|^p\varphi\,dx
\]
 and then
$\Omega=\Omega^\sharp$ and $|u|=u^\sharp$ modulo  a  rotation,
which implies
$u=\delta u^\sharp$ and $\Omega^\sharp$ modulo  a  rotation
with $\delta=\pm 1$.
 Now  for $h>0$, we take
\[
\psi(x)=\begin{cases}
\operatorname{sign} u(x)   &\text{if }     |u(x)|>t+h,\\
\frac{(|u(x)|-t)\operatorname{sign} u(x)}{h}  &\text{if } t<|u(x)|\leq t+h,\\
0 &\text{otherwise}
\end{cases}
\]
as the test function in \eqref{e3.1} and let $h\to 0$.
 Since $\delta u=|u|=u^{\sharp}=v^{\sharp}=v$, we have by Hardy-Littlewood
inequality that
\begin{equation} \label{3.51}
\begin{aligned}
&-\frac{d}{dt}\int_{u^{\sharp}>t}|\nabla u^{\sharp}|^p\varphi
\,dx \\
&=-\frac{d}{dt}\int_{|u|>t}|\nabla u|^p\varphi
\,dx\leq-\frac{d}{dt}\int_{|u|>t}a(x,u,\nabla u)\nabla u\,dx\\
&=\int_{|u|>t}H(x,u,\nabla u)udx
 \leq\int_{|u|>t}fu\varphi\,dx +\theta\int_{|u|>t}|\nabla u|^qu\varphi\,dx\\
&\leq \int_{u^{\sharp}>t}f^{\sharp}u^{\sharp}\varphi\,dx
 +\theta\int_{u^{\sharp}>t}|\nabla u^{\sharp}|^qu^{\sharp}\varphi\,dx\\
&=\int_{v>t}f^{\sharp}v\varphi\,dx+\theta\int_{v>t}|\nabla v|^qv\varphi\,dx\\
& =-\frac{d}{dt}\int_{v>t}|\nabla v|^p\varphi\,dx
 =-\frac{d}{dt}\int_{u^{\sharp}>t}|\nabla u^{\sharp}|^p\varphi\,dx.
\end{aligned}
\end{equation}
Thus, the above equalities hold. In particular,
\begin{gather}\label{3.54}
-\frac{d}{dt}\int_{|u|>t}a(x,u,\nabla u)\nabla u\,dx
=-\frac{d}{dt}\int_{u^{\sharp}>t}|\nabla u^{\sharp}|^p\varphi\,dx, \\
\label{3.53}
\int_{|u|>t}H(x,u,\nabla u)udx
=\int_{u^{\sharp}>t}f^{\sharp}u^{\sharp}\varphi \,dx
+\theta\int_{u^{\sharp}>t}|\nabla u^{\sharp}|^qu^{\sharp}\varphi\,dx,\\
\label{3.52}
\int_{u^{\sharp}>t}fu^{\sharp}\varphi \,dx
=\int_{u^{\sharp}>t}f^{\sharp}u^{\sharp}\varphi \,dx.
\end{gather}
By \cite[Lemma 4.3]{Tian3}, from \eqref{3.52} on has
$f=f^{\sharp}$  a.e. $x\in\Omega^{\sharp}$.
Then combine \eqref{3.53} and assumption (iii) to discover that
\[
H(x,u,\nabla u)=\delta(f^{\sharp}\varphi+\theta|D_1
u^{\sharp}|^q\varphi),  \quad \text{a.e. }x\in\Omega^{\sharp}
\]
Similarly,  from \eqref{3.54} and assumption (i) we have
\[
\delta a(x,u,\nabla u)\nabla u^{\sharp}=|\nabla u^{\sharp}|^p\varphi.
\]
Recalling \eqref{3.27}, we have
\begin{gather*}
 D_1 u^{\sharp}(x)=D_1 v^{\sharp}(x)
=\Big(\sqrt{2\pi} \exp\big(\frac{x_1^2}{2}\big)\Big)^{\frac{1}{p-1}}
w^{\frac{1}{p-1}}(\Phi(x_1))>0, \quad \text{a.e. } x\in\Omega^{\sharp},\\
D_iu^\sharp(x)=0, \quad i=2,3,\dots,n.
\end{gather*}
Hence,
\[
a_1(x,u,\nabla u)=\delta|D_1
u^{\sharp}|^{p-2}D_1 u^{\sharp}\varphi, \quad \text{a.e. }
x\in\Omega^{\sharp}.
\]
From the definition of solution it follows that
for all $\psi\in W_0^{1,p}(\varphi,\Omega^\sharp)\cap L^{\infty}(\Omega^\sharp)$,
\begin{align*}
&\int_{\Omega^\sharp}\delta|D_1
u^{\sharp}|^{p-2}D_1 u^{\sharp} D_1\psi \varphi\,dx
+\int_{\Omega^\sharp}\sum_{i=2}^n a_i(x,u,\nabla u)D_i\psi\,dx\\
&=\int_{\Omega}a(x,u,\nabla u)\nabla \psi \,dx
 =\int_{\Omega}H(x,u,\nabla u)\psi \,dx\\
&=\int_{\Omega^\sharp}\delta(f^{\sharp}\varphi+\theta|D_1
 u^{\sharp}|^q\varphi)\psi \,dx
 =\int_{\Omega^\sharp}\delta(f^{\sharp}\varphi
 +\theta|D_1 v|^q\varphi)\psi\,dx\\
&=\int_{\Omega^\sharp}\delta|D_1 v|^{p-2}D_1 vD_1\psi \varphi\,dx
=\int_{\Omega^\sharp}\delta|D_1 u^{\sharp}|^{p-2}
 D_1 u^{\sharp}D_1\psi \varphi \,dx.
\end{align*}
Then
\[
\int_{\Omega^\sharp}\sum_{i=2}^n a_i(x,u,\nabla u)D_i\phi
\,dx=0,  \quad \forall \psi\in W_0^{1,p}(\varphi,\Omega^\sharp)\cap
L^{\infty}(\Omega^\sharp),
\]
which completes the proof.
\end{proof}

\section{Appendix}

In this section give an example to Remark \ref{rmk3.1}.
Assume that $v$ is a solution of the problem
\begin{equation}  \label{P4}
\begin{gathered}
-D_1(\varphi|D_1v|^{p-2}D_1v)=\varphi+\theta|D_1v|^q\varphi \quad
\text{in } \Omega^\sharp,\\
v\in W^{1,p}_0(\varphi,\Omega^\sharp)\cap L^{\infty}(\Omega^\sharp)
\end{gathered}
\end{equation}
and  $Y$ is the solution of the problem
\begin{equation}  \label{P5}
\begin{gathered}
-D_1(\varphi|D_1Y|^{p-2}D_1Y)=\varphi \quad   \text{in } \Omega^\sharp,\\
Y=0   \quad\text{on } \partial\Omega^\sharp.
\end{gathered}
\end{equation}
For any given $k>0$, let
$$
T_k(s)=\begin{cases}
k   & \text{if } s>k,\\
s   & \text{if } |s|\leq k,\\
-k  & \text{if } s<-k.
\end{cases}
$$
Take $T_k((v-Y)^-)$ as the test function in  \eqref{P4} and \eqref{P5},
and subtract the two results. We obtain
\[
0\geq\int_{\{0<(v-Y)^-\leq k\}}\varphi(|D_1
v|^{p-2}D_1 v-|D_1 Y|^{p-2}D_1 Y)(Y-v)dx\geq 0.
\]
Thus, $\gamma_n(\{0<(v-Y)^-\leq k\})=0$.
Let $k\to +\infty$. Then
$\gamma_n(\{(v-Y)^->0\})=0$, which implies $v\geq Y$ a.e. in
$\Omega^{\sharp}$. However, if $p\geq 2$,
\begin{align*}
\|v\|_{L^\infty(\Omega^{\sharp})}
&\geq\|Y\|_{L^\infty(\Omega^{\sharp})}=Y^\star(0)\\
&=\int_0^{\gamma_n(\Omega)}
\Big(\sqrt{2\pi}\exp\Big(\frac{\Phi^{-1}(\tau)^{2}}{2}\Big)\Big)^{p'}
\tau^{\frac{1}{p-1}} d\tau\\
&\geq C \int_0^{\gamma_n(\Omega)}
\Big(\frac{1}{\tau(1-\log \tau)^{1/2}}\Big)^{p'}
\tau^{\frac{1}{p-1}} d\tau\\
&= C \int_0^{\gamma_n(\Omega)} (1-\log
\tau)^{-\frac{p'}{2}} \tau^{-1} d\tau=+\infty.
\end{align*}
This contradicts with $v\in L^{\infty}(\Omega^\sharp)$.
As $p\geq 2$, problem \eqref{P4} has no solution.

\subsection*{Acknowledgements}

This research was supported by the National Natural Science Foundation
of China (Grant No. 11501333), by the Tianyuan Special Funds of the National
Natural Science Foundation of China (Grant No. 11426146), by the
Promotive Research Fund for Excellent Young and Middle-Aged Scientists
 of Shandong Province (Grant No. BS2012SF026), by the Natural Science
Foundation of Shandong Province (Grant No. ZR2015AL015) and by the
Doctoral Fund of University of Jinan (Grant No. XBS1337).

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\end{document}