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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 85, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/85\hfil Ground states]
{Ground states for a modified capillary surface equation
in weighted Orlicz-Sobolev space}

\author[G. Zhang, H. Fu \hfil EJDE-2015/85\hfilneg]
{Guoqing Zhang, Huiling Fu}

\address{Guoqing Zhang \newline
College of Sciences, University of Shanghai for Science and Technology,
Shanghai 200093,  China}
\email{shzhangguoqing@126.com}

\address{Huiling Fu \newline
College of Sciences, University of Shanghai for Science and Technology,
Shanghai 200093,  China}
\email{fuhuiliing80@163.com}

\thanks{Submitted August 6, 2014. Published March 7, 2015.}
\subjclass[2000]{35J65, 35J70}
\keywords{Compact theorem; modified capillary surface equation;
\hfil\break\indent weighted Orlicz-Sobolev space; ground state}

\begin{abstract}
 In this article, we prove a compact embedding theorem for the weighted
 Orlicz-Sobolev space of radially symmetric functions. Using the embedding
 theorem and critical points theory, we prove the existence of multiple
 radial solutions and radial ground states for the following modified capillary
 surface equation
\begin{gather*}
-\operatorname{div}\Big(\frac{|\nabla u|^{2p-2}\nabla u}{\sqrt{1+|\nabla u|^{2p}}}\Big)
+T(|x|)|u|^{\alpha-2}u=K(|x|)|u|^{s-2}u,\quad u>0,\; x\in\mathbb{R}^{N},\\
u(|x|)\to 0,\quad\text{as } |x|\to \infty,
\end{gather*}
where $N\geq3$, $1<\alpha<p<2p<N$, $s$ satisfies some suitable conditions,
$K(|x|)$ and $T(|x|)$ are continuous, nonnegative functions.
\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 study the following modified capillary surface equation
in a weighted Orlicz-Sobolev space,
\begin{equation} \label{e1.1}
\begin{gathered}
-\operatorname{div}\Big(\frac{|\nabla u|^{2p-2}\nabla u}{\sqrt{1+|\nabla u|^{2p}}}
\Big)+T(|x|)|u|^{\alpha-2}u=K(|x|)|u|^{s-2}u,\quad u>0,\; x\in\mathbb{R}^{N},\\
u(|x|)\to 0,\quad\text{as } |x|\to \infty,
\end{gathered}
\end{equation}
where $N\geq3$, $1<\alpha<p<2p<N$, $s$ satisfies some suitable conditions,
$\nabla u$ denotes the gradient of $u$, $T$ and $K$ are continuous, nonnegative
and measurable functions,
i.e., $T, K:(0,+\infty)\to [0,+\infty]$ and may be unbounded, decaying
and vanishing.

 Recently, these type equations have attracted much attention. As $p=1$,
the problem \eqref{e1.1} becomes known as the prescribed mean curvature equation
 or the capillary surface equation. Peletier and Serrin \cite{p2} studied the following problem
\begin{equation} \label{e1.2}
\begin{gathered}
-\operatorname{div}\Big(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\Big)
=-\lambda u+u^{q},\quad x\in\mathbb{R}^{N},\\
u(x)\to 0,\quad\text{as } x\to \infty,
\end{gathered}
\end{equation}
where $\lambda>0, q>1$ and obtained the existence of radial ground states.
As $\lambda=0$, Ni and Serrin \cite{n2,n3} established that if
 $1<q\leq\frac{N}{N-2}$, no positive solutions exist, on the contrary,
if  $q\geq\frac{N+2}{N-2}$, there is a continuum of solutions.
del Pino and Guerra \cite{d2} proved the existence of large finite number of ground states,
provided that $q$ lies below but close enough to the critical exponent
$\frac{N+2}{N-2}$. Moreover, existence, nonexistence and multiplicity of
solutions decaying to zero at infinity have been proved by \cite{b2,c1,h1,k1,s1}.

 As $p>1$, using minimization sequence method and Mountain Pass Lemma,
Narukawa and Suzuki \cite{n1} discussed the existence of nonzero solutions for the
modified capillary surface equation
\begin{equation} \label{e1.3}
\begin{gathered}
-\operatorname{div}\Big(\frac{|\nabla u|^{2p-2}\nabla u}{\sqrt{1+|\nabla u|^{2p}}}
\Big)=\lambda f(x,u),\quad u\geq0,\; x\in\Omega,\\
u=0,\quad x\in\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary,
$\lambda$ is a positive parameter; Liang \cite{l1} investigated the following modified
capillary equation
\begin{equation} \label{e1.4}
\begin{gathered}
-\operatorname{div}\Big(\frac{|\nabla u|^{2p-2}\nabla u}{\sqrt{1+|\nabla u|^{2p}}}
\Big)= f(x,u),\quad x\in\Omega,\\
u=0,\quad x\in\partial\Omega,
\end{gathered}
\end{equation}
and obtained a negative and a positive solution by variational methods.
In particular, Azzollini, d'Avenia and Pomponio \cite{a1} studied the quasilinear
elliptic problems
\begin{equation} \label{e1.5}
\begin{gathered}
-\nabla [\phi'(|\nabla u|^2)\nabla u]+|u|^{\alpha-2}u=|u|^{s-2}u,
\quad x\in\mathbb{R}^{N},\\
u(x)\to 0,\quad \text{as } |x|\to \infty,
\end{gathered}
\end{equation}
where $\phi(t)$ behaves like $t^{\frac{q}{2}}$ for small $t$ and $t^{\frac{p}{2}}$
for large $t$, $1<p<q<N$, and obtained some existence results in Orlicz-Sobolev
space by using critical points theory.

 On the other hand, some authors studied the semilinear (quasilinear) elliptic
equations with unbounded or decaying radial potentials. Su, Wang and
 Willem \cite{s2,s3} proved some embedding results for the weighted Sobolev spaces
of radially symmetric functions. Zhang \cite{z1} obtained some Strauss-type decay
estimates and obtained some continuous and compact embedding theorems.

 In this article, we prove the existence of multiple radial solutions and radial
ground states for the problem \eqref{e1.1}. Firstly, we obtain a compact embedding
theorem for the weighted Orlicz-Sobolev space of radially symmetric functions.
Secondly, we obtain the existence of radial ground states for the problem \eqref{e1.1}
with unbounded or decaying radial potentials by using this compact embedding
theorem and critical points theory.

 Consider the functional
\begin{equation}
J(u)=\frac{1}{p}\int_{\mathbb{R}^{N}}(\sqrt{1+|\nabla u|^{2p}}-1)dx
+\frac{1}{\alpha}\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx
-\frac{1}{s}\int_{\mathbb{R}^{N}}K(|x|)|u|^{s}dx, \label{e1.6}
\end{equation}
where
\begin{equation}
 \sqrt{1+|\nabla u|^{2p}}-1
\sim \begin{cases}
|\nabla u|^{p},& \text{as }|\nabla u|\to  \infty,\\
\frac{1}{2}|\nabla u|^{2p}, &\text{as }|\nabla u|\to 0.
\end{cases} \label{e1.7}
\end{equation}
Solutions of \eqref{e1.1} are, at least formally, critical points of the functional
$J(u)$. By \eqref{e1.7}, we obtain that this different growth at zero and at
infinity of the function $\sqrt{1+|\nabla u|^{2p}}-1$ and the whole space
$\mathbb{R}^{N}$ suggest us not to use classical Sobolev spaces. Hence, we should
define a class of weighted Orlicz-Sobolev space with respect to the functional
\eqref{e1.6} is well defined and $C^{1}$. For dealing with the compact properties
of the functional $J(u)$, we would like to get compactness lies in the fact that
the group of translation constitutes an obstruction to compact embedding in
$\mathbb{R}^{N}$, and examine the affects of the unbounded or decaying potentials
$T(|x|)$ and $K(|x|)$. Hence, we restrict the domain of the functional $J(u)$
to the suitable Orlicz-Sobolev space.

 Now we state our main theorems in this paper. Let $|x|=r$, $T(|x|)$, $K(|x|)$
be continuous nonnegative functions in $(0,\infty)$, and
\begin{itemize}
\item[(T1)] There exist real number $a$ and $a_0$, such that
$\liminf_{r\to \infty} T(r)/r^a>0$, and
$\liminf_{r\to 0} T(r)/r^{a_0}>0$;

\item[(K1)] There exist real number $b$ and $b_0$, such that
 $\limsup_{r\to \infty} K(r)/r^{b}<\infty$, and
$\limsup_{r\to 0} K(r/r^{b_0}<\infty$,
$K(r)>0$.
\end{itemize}

 The existence and embedding results depend on the potentials $T,K$ near
$0$ and $\infty$. We define the  following relations between $p,2p$, and $a,b$ or
$a_0, b_0$:
\begin{equation}
s_{*}=\begin{cases}
\frac{(2p)\alpha(N-1+b)-a\alpha}{2p(N-1)+a(2p-1)},& b\geq a>-p,\\[4pt]
\frac{2p(N+b)}{(N-2p)},& b\geq-p, a\leq-p,\\[4pt]
\alpha, & b\leq \max \{a,-p\},
\end{cases} \label{e1.8}
\end{equation}
and
\begin{equation}
s^{*}=\begin{cases}
\frac{2p(N+b_0)}{(N-2p)}, & b_0\geq-p, \; a_0\geq-p,\\[4pt]
\frac{(2p)a(N-1+b_0)-a_0\alpha}{2p(N-1)+a_0(2p-1)},
&-p> a_0>-\frac{(N-1)}{(2p-1)}2p, \; b_0\geq a_0,\\[4pt]
\infty, & a_0\leq-\frac{(N-1)}{(p-1)}p, \; b_0\geq a_0.
\end{cases}
\label{e1.9}
\end{equation}

\begin{remark} \label{rmk1.1} \rm
 The idea which for establishing conditions \eqref{e1.8} and \eqref{e1.9} comes
from Su, Wang and Willem \cite{s2,s3}.
In this article, we not only develop the methods in \cite{s2,s3,z1} to the modified
capillary surface equation, but also improve and extend the results in classical
Sobolev space to the Orlicz-Sobolev space.
\end{remark}

\begin{theorem}[Multiplicity Result] \label{thm1.2}
 Assume that {\rm (T1)} and {\rm (K1)} hold, $1<\alpha<p<2p<N$, $s_{*}<s<s^{*}$,
then there exist infinitely many radially symmetric solutions for \eqref{e1.1}.
\end{theorem}

\begin{theorem}[Ground States] \label{thm1.3}
 Assume that {\rm (T1)} and {\rm (K1)} hold, $1<\alpha<p<2p<N$, $s_{*}<s<s^{*}$,
 then there exists a radial ground states for \eqref{e1.1}.
\end{theorem}

This article is organized as follows.
In Section 2, we introduce a weighted Orlicz-Sobolev space of radially
symmetric function and recall some important lemmas.
In Section 3, we prove some inequalities with radial functions,
extending some inequalities in classic Sobolev space to the Orlicz-Sobolev space,
 and establish a new compact embedding theorem (i.e. Theorem \ref{thm3.1}).
Section 4 is devoted to the proof of Theorems \ref{thm1.2} and Theorem \ref{thm1.3}.

\section{Weighted Orlicz-Sobolev spaces}

As a first step, we recall some well known facts on the sum of Lebesgue spaces
and introduce some notation of function space.

\begin{definition}[\cite{b1}] \label{def2.1} \rm
Let $1<p<q$ and $\Omega\subset\mathbb{R}^{N}$. We denote with
$L^{p}(\Omega)+L^{q}(\Omega)$ the completion of
$\mathcal{C}_{c}^{\infty}(\Omega,\mathbb{R}^{N})$ in the norm
\begin{equation}
\|u\|_{L^{p}(\Omega)+L^{ q}(\Omega)}
=\inf \big\{\|v\|_{p}+\|w\|_{q}: v\in L^{p}(\Omega), w\in L^{ q}(\Omega),
 u=v+w\big\}.\label{e2.1}
\end{equation}
In this article, we set $q=2p$ and
$\|u\|_{p,2p}=\|u\|_{L^{p}(\Omega)+L^{ 2p}(\Omega)}$. Moreover, from \cite{b1},
we obtain that $L^{p}(\Omega)+L^{ 2p}(\Omega)$ are Orlicz spaces.
\end{definition}

 For $\alpha>1, s>1$, we define
$$
L^{\alpha}(\mathbb{R}^{N};T)=\big\{u:\mathbb{R}^{N}\to \mathbb{R}:
u\text{ is Lebesgue measurable},
\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx<\infty\big\},
$$
and
$$
L^{s}(\mathbb{R}^{N};K)=\big\{u:\mathbb{R}^{N}\to \mathbb{R}:
u\text{ is Lebesgue measurable},
\int_{\mathbb{R}^{N}}K(|x|)|u|^{s}dx<\infty\big\}.
$$
The corresponding norms in $L^{\alpha}(\mathbb{R}^{N};T)$ and
$L^{s}(\mathbb{R}^{N};K)$ are respectively
\begin{equation}
\begin{gathered}
\|u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}
=\Big(\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx\Big)^{1/\alpha}, \\
\|u\|_{L_{K}^{s}(\mathbb{R}^{N})}
=\Big(\int_{\mathbb{R}^{N}}K(|x|)|u|^{s}dx\Big)^{1/s}.
\end{gathered}\label{e2.2}
\end{equation}

From \cite{b1}, we have a list of properties of the Orlicz spaces
$L^{p}(\Omega)+L^{ 2p}(\Omega)$.

\begin{proposition}[\cite{b1}] \label{prop2.2}
Let $\Omega\subset\mathbb{R}^{N}$, $u\in L^{p}(\Omega)+L^{2p}(\Omega)$ and
$\Lambda_{u}=\{x\in\Omega|~|u(x)|>1\}$. We have
\begin{itemize}
 \item[(i)] if $\Omega'\subset\Omega$ is such that
$|\Omega'|<+\infty$, then $u\in L^{p}(\Omega')$;

 \item[(ii)] if $\Omega'\subset\Omega$ is such that $u\in L^{\infty}(\Omega')$,
then $u\in L^{2p}(\Omega')$;

 \item[(iii)] $|\Lambda _{u}|<+\infty$;

 \item[(iv)] $u\in L^{p}(\Lambda_{u})\cap L^{2p}(\Lambda_{u}^{c})$;

 \item[(v)] the infimum in \eqref{e2.1} is attained;

 \item[(vi)] $L^{p}(\Omega)+L^{2p}(\Omega)$ is reflexive and
$(L^{p}(\Omega)+L^{2p}(\Omega))'=L^{p^{'}}(\Omega)\cap L^{(2p)^{'}}(\Omega)$;

 \item[(vii)] $\|u\|_{L^{p}(\Omega)+L^{ 2p}(\Omega)}
\leq \max \{\|u\|_{L^{p}(\Lambda_{u})}, \|u\|_{L^{2p}(\Lambda_{u}^{c})}\}$;

 \item[(viii)] if $B\subset \Omega$, then
$\|u\|_{L^{p}(\Omega)+L^{ 2p}(\Omega)}
\leq\|u\|_{L^{p}(B)+L^{ 2p}(B)}+\|u\|_{L^{p}(\Omega\setminus B)
+L^{ 2p}(\Omega\setminus B)}$.
\end{itemize}
\end{proposition}

 Let $\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N},\mathbb{R})$ denote the collection
of smooth functions with compact support and
$$
(\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N},\mathbb{R}))_{\rm rad}
=\{u\in \mathcal{C}_{c}^{\infty}(\mathbb{R}^{N},\mathbb{R}): u\text{ is radial}\}.
$$

\begin{definition} \label{def2.3} \rm
Let $\alpha>1$, $\mathcal{W}$ be the completion of
$\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N},\mathbb{R})$ in the norm
\begin{equation}
\|u\|_{\mathcal{W}}=\|u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}
+\|\nabla u\|_{p,2p},\label{e2.3}
\end{equation}
$\mathcal{W}_{\rm rad}$ be the completion of
 $(\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N},\mathbb{R}))_{\rm rad}$ in the norm
$\|\cdot\|$, namely
$$
\mathcal{W}_{\rm rad}=\overline{(\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N},
\mathbb{R}))_{\rm rad}}^{\|\cdot\|}.
$$
\end{definition}

\begin{lemma} \label{lem2.4}
The space  $(\mathcal{W}_{\rm rad},\|\cdot\|)$ is a reflexive Banach space.
\end{lemma}

\begin{proof}
 Firstly, we prove that $(\mathcal{W}_{\rm rad},\|\cdot\|)$ is a Banach space.
In fact, since $L^{\alpha}(\mathbb{R}^{N};T)$ and
$L^{p}(\mathbb{R}^{N})+L^{2p}(\mathbb{R}^{N})$ are completed.
Let $\{u_n\}_n$ be a Cauchy sequence in $\mathcal{W}_{\rm rad}$, then
$\{u_n\}_n$ is a Cauchy sequence in $L^{\alpha}(\mathbb{R}^{N};T)$,
and there exists $u\in L^{\alpha}(\mathbb{R}^{N};T)$, such that
$\|u_n-u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}\to 0$, as $n\to \infty$.
Also $\{\nabla u_n\}_n$ is a Cauchy sequence in
$L^{p}(\mathbb{R}^{N})+L^{2p}(\mathbb{R}^{N})$, there exists
$\delta\in L^{p}(\mathbb{R}^{N})+L^{2p}(\mathbb{R}^{N})$, such that
$\|\nabla u_n-\delta\|_{p,2p}\to 0$, as $n\to \infty$.

 Sufficiently, for every $\xi\in \mathcal{C}_{c}^{\infty}(\mathbb{R}^{N})$,
$n\in\mathbb{N}$, we have
$$
\lim_{n\to \infty}\int_{\mathbb{R}^{N}}T(|x|)u_n\nabla\xi dx
=\int_{\mathbb{R}^{N}}T(|x|)u\nabla\xi dx, \quad
\lim_{n\to \infty}\int_{\mathbb{R}^{N}}\xi\nabla u_ndx
=\int_{\mathbb{R}^{N}}\xi \delta dx.
$$
In fact, by H\"{o}lder inequality and  Proposition \ref{prop2.2} (v), by 
considering $(\mathbf{v}_n,\mathbf{w}_n)$ in $in L^{p}(\mathbb{R}^{N})
\times L^{2p}(\mathbb{R}^{N})$ such that
$$
\nabla u_n-\delta=\mathbf{v}_n+\mathbf{w}_n\,, \quad
\|\nabla u_n-\delta\|_{p,2p}=\|\mathbf{v}_n\|_{p}+\|\mathbf{w}_n\|_{2p},
$$
we have 
$$
\Big|\int_{\mathbb{R}^{N}}T(|x|)(u_n-u)\nabla\xi dx\Big|
\leq\|\nabla \xi\|_{\alpha^{'}}\|u_n-u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}\to 0,
$$
and
\begin{align*}
\big|\int_{\mathbb{R}^{N}}\xi(\nabla u_n-\delta)dx\big|
&=\big|\int_{\mathbb{R}^{N}}\xi \mathbf{v}_ndx
 +\int_{\mathbb{R}^{N}}\xi \mathbf{w}_ndx\big|\\
&\leq\|\xi\|_{p'}\|\mathbf{v}_n\|_{p}+\|\xi\|_{(2p)'}\|\mathbf{w}_n\|_{2p}\to 0.
\end{align*}
 Obviously, by the definition of weak derivatives, we have
$$
\int_{\mathbb{R}^{N}}T(|x|)u_n\nabla\xi dx
=-\int_{\mathbb{R}^{N}}T(|x|)\xi\nabla u_ndx.
$$
Hence, we obtain
$$
\int_{\mathbb{R}^{N}}T(|x|)u\nabla\xi dx
=-\int_{\mathbb{R}^{N}}T(|x|)\xi \delta dx;
$$
that is, $\nabla u=\delta$.

Secondly, we prove that $(\mathcal{W}_{\rm rad},\|\cdot\|)$ is reflexive.
Indeed, we consider the  norm
$$
\|u\|_{p,2p}^{*}=\inf \{(\|v\|_{p}^2+\|w\|_{2p}^2)^{\frac{1}{2}}|\,
v\in L^{p}(\mathbb{R}^{N}), w\in L^{2p}(\mathbb{R}^{N}), u=v+w\},
$$
and then, on $\mathcal{W}_{\rm rad}$, the norm
$$
\|u\|_{\mathcal{W}_{\rm rad}}^{*}
=\|u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}+\|\nabla u\|_{p,2p}^{*},
$$
is equivalent to the norm $\|u\|_{\mathcal{W}_{\rm rad}}$.
 Moreover, by \cite[Proposition 2.6]{b1}, the norm
$\|u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}$ and the norm $\|\cdot\|^{*}$
are uniformly convex. So, on $\mathcal{W}_{\rm rad}$, we consider uniformly
convex norm $\|\nabla.\|_{p,2p}^{*}$ and the norm
$\|\cdot\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}$. By a well known result, also the norm
$$
\|\cdot\|_{\mathcal{W}_{\rm rad}}^{\sharp}
=\sqrt{\|\cdot\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}^2+(\|\nabla.\|_{p,2p}^{*}})^2,
$$
is uniformly convex and then $(\mathcal{W}_{\rm rad},\|\cdot\|^{\sharp})$
is reflexive. Hence the norm $\|\cdot\|_{\mathcal{W}_{\rm rad}}^{\sharp}$
is equivalent to $\|\cdot\|_{\mathcal{W}_{\rm rad}}$. Then, we obtain that
$(\mathcal{W}_{\rm rad},\|\cdot\|)$ is also reflexive.
\end{proof}

\begin{remark} \label{rmk2.5} \rm
 Similar to \cite[Theorem 2.8]{a1}, we obtain that $\mathcal{W}_{\rm rad}$
coincides with the set of radial functions of $\mathcal{W}$. Hence, using
the principle of symmetric criticality in \cite{p1}, we only consider the
functional $J(u)$ in \eqref{e1.6} restricted to the weighted Orlicz-Sobolev
 space $\mathcal{W}_{\rm rad}$.
\end{remark}

\section{Embedding theorem}
 To obtain the compactness of the functional $J(u)$, we prove a compact embedding
theorem (Theorem \ref{thm3.1}). Denote by $B_{r}$ the ball in $\mathbb{R}^{N}$
centered at 0 with radius $r$.

\begin{theorem} \label{thm3.1}
 Let $1<\alpha<p<2p<N$. Assume {\rm (T1)} and {\rm (K1)} hold, then we 
have the continuous embedding
$$
\mathcal{W}_{\rm rad}\hookrightarrow L^{s}(\mathbb{R}^{N};K)
$$
for $s_{*}\leq s\leq s^{*}$ when $s^{*}<\infty$, and for $s_{*}\leq s<\infty$
when $s^{*}=\infty$.
Furthermore, the embedding is compact for $s_{*}<s<s^{*}$.
\end{theorem}

 Firstly, we prove some inequalities on radial functions which are
interesting.

\begin{lemma} \label{lem3.2}
 If $1<p<2p<N$, there exists $\widehat{M}>0$ such that for every
 $u\in \mathcal{W}_{\rm rad}$,
\begin{equation}
|u(x)|\leq\begin{cases}
\widehat{M}|x|^{-(\frac{N-2p}{2p})}\|\nabla u\|_{p,2p}, &\text{for }|x|\geq1,\\
\widehat{M}|x|^{-(\frac{N-p}{p})}\|\nabla u\|_{p,2p}, &\text{for }0<|x|<1.
\end{cases} \label{e3.1}
\end{equation}
\end{lemma}

 The proof of the above lemma is similar to that of \cite[Lemma 2.13]{a1}
 and of \cite[Lemma 1]{s3}.

\begin{lemma} \label{lem3.3}
 Let $1<p<2p<N$. Assume $2p<s<\infty$ and write $s=\frac{2p(N+c)}{(N-2p)}$,
for some $-p\leq c<\infty$. Then there exists $\widetilde{M}>0$ such that
for all $u\in \mathcal{W}_{\rm rad}$
\begin{equation}
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}dx\Big)^{1/s}
\leq \widetilde{M}\max \big(\|\nabla u\|_{p,2p},\|\nabla u\|^2_{p,2p}\big).
\label{e3.2}
\end{equation}
\end{lemma}

\begin{proof}
By denseness, it is sufficient to prove that
$u\in (\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N},\mathbb{R}))_{\rm rad},
(\mathbf{v},\mathbf{w})\in L^{p}(\mathbb{R}^{N})\times L^{2p}(\mathbb{R}^{N})$,
such that $\nabla u=\mathbf{v}+\mathbf{w}$. By using Lemma \ref{lem3.2}, and
$s=\frac{2p(N+c)}{(N-2p)}$, we have
\begin{align*}
&\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}dx\\
&=\omega_{N}\int_0^{\infty}r^{(N-1+c)}|u(r)|^{s}dr\\
&=-\frac{s\omega_{N}}{(N+c)}\int_0^{\infty}r^{(N+c)}|u(r)|^{(s-2)}u(r)u'(r)dr \\
&\leq\frac{(2p)\omega_{N}}{(N-2p)}\int_0^{\infty}r^{(N+c)}|u(r)|^{(s-1)}|u'(r)|dr\\
&=\frac{2p}{(N-2p)}\int_{\mathbb{R}^{N}}|x|^{(c+1)}|u|^{(s-1)}|\nabla u|dx\\
&\leq\frac{2p}{(N-2p)}\Big(\int_{\mathbb{R}^{N}}|x|^{(c+1)}|u|^{(s-1)}|\mathbf{v}|dx
+\int_{\mathbb{R}^{N}}|x|^{(c+1)}|u|^{(s-1)}|\mathbf{w}|dx\Big)\\
&\leq\frac{2p}{(N-2p)}\Big[\Big(\int_{\mathbb{R}^{N}}|\mathbf{v}|^{p}dx
\Big)^{1/p}
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}|x|^{\frac{(p+c)}{(p-1)}}|u|
^{\frac{(s-p)}{(p-1)}}dx\Big)^{(\frac{p-1}{p})}\\
&\quad +\Big(\int_{\mathbb{R}^{N}}|\mathbf{w}|^{2p}dx\Big)^{\frac{1}{2p}}
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}|x|^{\frac{(2p+c)}{(p-1)}}
|u|^{\frac{(s-2p)}{(2p-1)}}dx\Big)^{(\frac{2p-1}{2p})}\Big]\\
&\leq M'\frac{2p}{(N-2p)}\Big[\|\mathbf{v}\|_{L^{p}(\mathbb{R}^{N})}\|
 \nabla u\|_{p,2p}^{(\frac{s-p}{p})}
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}dx\Big)^{(\frac{p-1}{p})}\\
&\quad +\|\mathbf{w}\|_{L^{2p}(\mathbb{R}^{N})}\|\nabla u\|_{p,2p}
^{(\frac{s-2p}{2p})}
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}dx\Big)^{(\frac{2p-1}{2p})}\Big]\\
&\leq M'\frac{2p}{(N-2p)}\max \Big(\|\nabla u\|_{p,2p}^{(\frac{s-p}{p})},
\|\nabla u\|_{p,2p}^{(\frac{s-2p}{2p})}\Big)
\Big[\|\mathbf{v}\|_{L^{p}(\mathbb{R}^{N})}
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}dx\Big)^{(\frac{2p-1}{2p})}\\
&\quad +\|\mathbf{w}\|_{L^{2p}(\mathbb{R}^{N})}
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}dx\Big)^{(\frac{2p-1}{2p})}\Big]\\
&\leq M'\frac{2p}{(N-2p)}\max \Big(\|\nabla u\|_{p,2p}^{(\frac{s-p}{p})},
\|\nabla u\|_{p,2p}^{(\frac{s-2p}{2p})}\Big)\|\nabla u\|_{p,2p}
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}dx\Big)^{(\frac{2p-1}{2p})}\\
&\leq \widetilde{M}\max \Big(\|\nabla u\|_{p,2p}^{\frac{s}{p}},
\|\nabla u\|_{p,2p}^{\frac{s}{2p}}\Big)
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}dx\Big)^{(\frac{2p-1}{2p})},
\end{align*}
where $\omega_{N}$ is the volume of the unit sphere in $\mathbb{R}^{N}$.
 It follows that
$$
\Big(\int_{\mathbb{R}^{N}}|x|^{c}|u|^{s}dx\Big)^{1/s}\leq \widetilde{M}
\max \Big(\|\nabla u\|_{p,2p},\|\nabla u\|^2_{p,2p}\Big).
$$
\end{proof}

\begin{lemma} \label{lem3.4}
 Assume {\rm (T1)} holds, $1<\alpha<p<2p<N$, and $a>-\frac{(N-1)}{(2p-1)}2p$.
 Then there exists $\widehat{M}_0>0$ such that for all $u\in \mathcal{W}_{\rm rad}$,
\begin{equation}
|u(x)|\leq \widehat{M}_0|x|^{-(\frac{2p(N-1)+a(2p-1)}{\alpha (2p)})}
\|u\|_{\mathcal{W}_{\rm rad}}, \quad \text{for }|x|\gg1.\label{e3.3}
\end{equation}
\end{lemma}

\begin{proof} By assumption (T1), there exists $R>1$ such that for some $M_0>0$,
$$
T(|x|)\geq M_0|x|^a, \quad |x|>R>1.
$$
For $u\in \mathcal{W}_{\rm rad}$, as $\theta>-(N-1)$, we have
\begin{equation}
\begin{aligned}
\frac{d}{dr}(r^{(\theta+N-1)}|u|^{\alpha})
&=\alpha r^{(\theta+N-1)}|u|^{(\alpha-2)}u\frac{du}{dr}
 +(\theta+N-1)|u|^{\alpha}r^{(\theta+N-2)}\\
&\geq\alpha r^{(\theta+N-1)}|u|^{(\alpha-2)}u\frac{du}{dr}.
\end{aligned}\label{e3.4}
\end{equation}
Next we only consider $|u|\geq1$, when $|u|\leq1$, set $|u'|=\frac{1}{|u|}$,
then $|u'|\geq1$. For all $u\in \mathcal{W}_{\rm rad}$,
 $(\mathbf{v},\mathbf{w})\in L^{p}(\mathbb{R}^{N})\times L^{2p}(\mathbb{R}^{N})$,
such that $\nabla u=\mathbf{v}+\mathbf{w}$. Since, $a>-\frac{(N-1)}{(2p-1)}2p$,
so take $\theta=\text{min}\{\frac{a(p-1)}{p},\frac{a(2p-1)}{2p}\}$,
then $\theta>-(N-1)$. For $r>R$, $1<\alpha<p<2p<N$, we have
\begin{align*}
|u|^{\alpha}r^{(\theta+N-1)}
&\leq\alpha \int_{r}^{\infty}|u|^{(\alpha-1)}t^{(\theta+N-1)}|u'(t)|dt\\
&=\frac{\alpha}{\omega_{N}}\int_{B_{r}^{c}}|x|^{\theta}|u|^{(\alpha-1)}|\nabla u|dx\\
&\leq\frac{\alpha}{\omega_{N}}\Big(\int_{B_{r}^{c}}|x|^{\theta}|u|
^{(\alpha-1)}|\mathbf{v}|dx
+\int_{B_{r}^{c}}|x|^{\theta}|u|^{(\alpha-1)}|\mathbf{w}|dx\Big)\\
&\leq\frac{\alpha}{\omega_{N}}\Big[\|\mathbf{v}\|_{L^{p}(B_{r}^{c})}
\Big(\int_{B_{r}^{c}}|x|^{\frac{\theta p}{(p-1)}}|u|
^{(\frac{(\alpha-1)p}{p-1})}dx\Big)^{(\frac{p-1}{p})}\\
&\quad +   \|\mathbf{w}\|_{L^{2p}(B_{r}^{c})}
\Big(\int_{B_{r}^{c}}|x|^{\frac{\theta(2p)}{(2p-1)}}|u|
^{(\frac{(\alpha-1)(2p)}{(2p-1)})}dx\Big)
^{(\frac{2p-1}{2p})}\Big]\\
&\leq \frac{\alpha}{\omega_{N}}
\Big[\|\mathbf{v}\|_{L^{p}(B_{r}^{c})}
\Big(\int_{B_{r}^{c}}|x|^a|u|^{(\frac{(\alpha-1)p}{p-1})}dx\Big)
^{(\frac{p-1}{p})}\\
&\quad +\|\mathbf{w}\|_{L^{2p}(B_{r}^{c})}
\Big(\int_{B_{r}^{c}}|x|^a|u|^{(\frac{(\alpha-1)2p}{(2p-1)})}dx
\Big)^{(\frac{2p-1}{2p})}\Big]\\
&\leq M_{(\alpha,p,N)}\Big[\|\mathbf{v}\|_{L^{p}(B_{r}^{c})}
\Big(\int_{B_{r}^{c}}T(|x|)|u|^{(\frac{(\alpha-1)p}{p-1})}dx\Big)
^{(\frac{p-1}{p(\alpha-1)}(\alpha-1))} \\
&\quad +\|\mathbf{w}\|_{L^{2p}(B_{r}^{c})}
\Big(\int_{B_{r}^{c}}T(|x|)|u|^{(\frac{(\alpha-1)2p}{2p-1})}dx\Big)
^{(\frac{2p-1}{2p(\alpha-1)}(\alpha-1))} \\
&\leq M_{(\alpha,p,N)}\|\nabla u\|_{p,2p}\|u\|_{L_{T}
^{\frac{(\alpha-1)p}{(p-1)}}(\mathbb{R}^{N})}^{(\alpha-1)}\\
&\leq M_{(\alpha,p,N)}\|\nabla u\|_{p,2p}\|u\|_{L_{T}^{\alpha}
(\mathbb{R}^{N})}^{(\alpha-1)}.
\end{align*} %\label{e3.5}
By Young inequality, and $|x|=r$, we obtain
\begin{align*}
|u|\,|x|^{(\frac{\theta+N-1}{\alpha})}
&\leq M_{(\alpha,p,N)}^{1/\alpha}\|\nabla u\|_{p,2p}^{1/\alpha}
\|u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}^{(\frac{\alpha-1}{\alpha})}\\
&\leq \widehat{M}_0(\|\nabla u\|_{p,2p}+\|u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})});
\end{align*}
i.e.,
$$
|u|\leq \widehat{M}_0|x|^{-(\frac{2p(N-1)+a(2p-1)}{\alpha (2p)})}
\|u\|_{\mathcal{W}_{\rm rad}},
$$
where the constant $\widehat{M}_0=\widehat{M}_{0(\alpha,p,N)}$.
\end{proof}

\begin{lemma} \label{lem3.5}
Assume {\rm (T1)} holds, $1<\alpha<p<2p<N$. Then there exist $1>r_0>0$
and $\widetilde{M}_0>0$ such that for all $u\in\mathcal{W}_{\rm rad}$,
\begin{equation}
|u(x)|\leq \widetilde{M}_0|x|^{-(\frac{2p(N-1)
+a_0(2p-1)}{\alpha (2p)})}\|u\|_{\mathcal{W}_{\rm rad}},\quad
 \text{for } 0<|x|\leq r_0<1,\label{e3.6}
\end{equation}
where $\widetilde{M}_0=\widetilde{M}_0(a_0,r_0,\alpha,N)$.
\end{lemma}

\begin{proof}
 By assumption (T1), there exists $1>r_0>0$ such that for some constant $M_0>0$,
$$
T(|x|)\geq M_0|x|^{a_0}, \quad 0<|x|\leq r_0<1.
$$
For $u\in \mathcal{W}_{\rm rad}$, we have
$$
\frac{d}{dr}(r^{(\beta+N-1)}|u|^{\alpha})
=\alpha r^{(\beta+N-1)}|u|^{(\alpha-2)}u\frac{du}{dr}
+(\beta+N-1)|u|^{\alpha}r^{(\beta+N-2)}.
$$
Thus, for $0<r\leq r_0<1$,
\begin{equation}
r^{(\beta+N-1)}|u|^{\alpha}\leq\alpha\int_{r}^{r_0}|u|^{(\alpha-1)}
t^{(\beta+N-1)}|u'(t)|dt+(\beta+N-1)\int_{r}^{r_0}
|u|^{\alpha}t^{(\beta+N-2)}dt.
\label{e3.7}
\end{equation}
As $\beta\geq a_0+1$, we have
\begin{equation}
\begin{aligned}
\int_{r}^{r_0}|u|^{\alpha}t^{(\beta+N-2)}dt
&=\int_{r}^{r_0}t^{(a_0+N-1)}|u|^{\alpha}t^{(\beta-a_0-1)}dt\\
&\leq \omega_{N}^{-1}r_0^{(\beta-a_0-1)}
 \int_{B_{r_0}(0)\backslash B_{r}(0)}|x|^{a_0}|u|^{\alpha}dx\\
&\leq \omega_{N}^{-1} M_0^{-1}r_0^{(\beta-a_0-1)}
 \int_{B_{r_0}(0)\backslash B_{r}(0)}T(|x|)|u|^{\alpha}dx \\
&\leq \omega_{N}^{-1} M_0^{-1}r_0^{(\beta-a_0-1)}
\|u\|_{L_{T}^{\alpha}}^{\alpha}.
\end{aligned} \label{e3.8}
\end{equation}
Let $\beta=\max \{\frac{(2p-1)}{2p}a_0,\frac{(p-1)}{p}a_0\}$, for all
$u\in \mathcal{W}_{\rm rad}$,
 $(\mathbf{v},\mathbf{w})\in L^{p}(\mathbb{R}^{N})\times L^{2p}(\mathbb{R}^{N})$,
such that $\nabla u=\mathbf{v}+\mathbf{w}$, we only consider $|u|\geq1$.
If $|u|\leq1$, set $|u'|=1/|u|$, then we have $|u'|\geq1$. Hence, we have
\begin{align*}
&\int_{r}^{r_0}|u|^{(\alpha-1)}t^{(\beta+N-1)}|u'(t)|dt\\
&=\omega_{N}^{-1}\int_{B_{r_0}(0)\backslash B_{r}(0)}
 |x|^{\beta}|u|^{(\alpha-1)}|\nabla u|dx\\
&\leq\omega_{N}^{-1}\Big(\int_{B_{r_0}(0)\backslash B_{r}(0)}
 |x|^{\beta}|u|^{(\alpha-1)}|\mathbf{v}|dx
 +\int_{B_{r_0}(0)\backslash B_{r}(0)}|x|^{\beta}|u|^{(\alpha-1)}
 |\mathbf{w}|dx\Big)\\
&\leq\omega_{N}^{-1}\Big[\|\mathbf{v}\|_{L^{p}(B_{r_0}(0)
\backslash B_{r}(0))}\Big(\int_{B_{r_0}(0)\backslash B_{r}(0)}|x|
^{\frac{\beta p}{(p-1)}}|u|^{\frac{(\alpha-1)p}{(p-1)}}dx\Big)^{\frac{(p-1)}{p}}\\
&\quad+ \|\mathbf{w}\|_{L^{2p}(B_{r_0}(0)\backslash B_{r}(0))}
\Big(\int_{B_{r_0}(0)\backslash B_{r}(0)}|x|^{\frac{\beta (2p)}{(2p-1)}}
|u|^{\frac{(\alpha-1)(2p)}{(2p-1)}}dx\Big)^{\frac{(2p-1)}{2p}}\Big]
\\
&\leq \omega_{N}^{-1}\|\mathbf{v}\|_{L^{p}(B_{r_0}(0)\backslash B_{r}(0))}
\Big(\int_{B_{r_0}(0)\backslash B_{r}(0)}|x|^{a_0}|u|^{(\frac{(\alpha-1)p}{p-1})}dx
 \Big)^{(\frac{p-1}{p(\alpha-1)}(\alpha-1))}\\
&\quad +\omega_{N}^{-1}\|\mathbf{w}\|_{L^{2p}(B_{r_0}(0)\backslash B_{r}(0))}
\Big(\int_{B_{r_0}(0)\backslash B_{r}(0)}|x|^{a_0}|u|
^{(\frac{2p(\alpha-1)}{2p-1})}dx\Big)
^{(\frac{2p-1}{2p(\alpha-1)}(\alpha-1))}\\
&\leq \omega_{N}^{-1}\|\nabla u\|_{p,2p(B_{r_0}(0)\backslash B_{r}(0))}
\Big(\int_{B_{r_0}(0)\backslash B_{r}(0)}|x|^{a_0}|u|
^{(\frac{(\alpha-1)p}{p-1})}dx\Big)
^{(\frac{p-1}{p(\alpha-1)}(\alpha-1))}\\
& \leq \omega_{N}^{-1} M_0^{-(\frac{p-1}{p})}
\|\nabla u\|_{p,2p(B_{r_0}(0)\backslash B_{r}(0))}\\
&\quad\times \Big(\int_{B_{r_0}(0)\backslash B_{r}(0)}T(|x|)|u|^{(\frac{(\alpha-1)p}{p-1})}dx\Big)
^{(\frac{p-1}{p(\alpha-1)}(\alpha-1))}\\
&\leq \omega_{N}^{-1} M_1\|\nabla u\|_{p,2p(B_{r_0}(0)\backslash B_{r}(0))}
\Big(\int_{B_{r_0}(0)\backslash B_{r}(0)}T(|x|)|u|^{\alpha}dx\Big)
^{(\frac{\alpha-1}{\alpha})}\\
&= \omega_{N}^{-1} M_1\|\nabla u\|_{p,2p}\|u\|_{L_{T}^{\alpha}(B_{r_0}(0)
\backslash B_{r}(0))}^{(\alpha-1)}.
\end{align*} %\label{e3.9}
Since
$$
\beta+N-1\geq 0 \Longleftrightarrow a_0>-\frac{(N-1)}{(2p-1)}2p.
$$
It follows that $\beta+N-1\leq0$ implies
$\beta-a_0-1\geq(\frac{N-p}{p-1})$. Hence, from the above arguments, we have
$$
|u(x)|\leq \widetilde{M}_0|x|^{-(\frac{2p(N-1)+a_0(2p-1)}{\alpha (2p)})}
\|u\|_{\mathcal{W}_{\rm rad}}, \quad 0<|x|\leq r_0<1,
$$
where the constant $\widetilde{M}_0=\widetilde{M}_0(a_0,r_0,\alpha,N)$.
\end{proof}

\begin{lemma} \label{lem3.6}
Let $1<\alpha<p<2p<N$, $2p<s\leq\infty$. Then for any $0<r<1<R<\infty$,
the following embedding is compact
$$
\mathcal{W}_{\rm rad}(B_{R}\backslash B_{r})\hookrightarrow L^{s}(B_{R}
\backslash B_{r};K).
$$
\end{lemma}

The proof of the above lemma is similar to \cite[Lemma 6]{s3}.

\begin{proof}[Proof of Theorem \ref{thm3.1}]
First we prove that the embedding is continuous. It is sufficient to show
\begin{equation}
S_{\rm rad}(T,K)=\inf_{u\in \mathcal{W}_{\rm rad}(\mathbb{R}^{N})}
\frac{\|\nabla u\|_{p,2p}+\|u\|_{L_{T}^{\alpha}
(\mathbb{R}^{N})}}{\|u\|_{L_{K}^{s}(\mathbb{R}^{N})}}>0.
\label{e3.10}
\end{equation}
If not, assume that there exists $\{u_n\}\subset \mathcal{W}_{\rm rad}$ such that
\begin{gather}
\|\nabla u\|_{p,2p}+\|u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}=o(1),
\quad  \text{as } n\to \infty,\label{e3.11}\\
\|u\|_{L_{K}^{s}(\mathbb{R}^{N})}=1, \quad \text{for all }
n\in\mathbb{N}.\label{e3.12}
\end{gather}
It is a contradiction, if we have
\begin{equation}
\|u\|_{L_{K}^{s}(\mathbb{R}^{N})}=0.\label{e3.13}
\end{equation}
By (T1) and (K1), there exist $R_0>1>r_0>0$, for some $M_0$,
\begin{equation}
\begin{gathered}
K(|x|)\leq M_0|x|^{b}, \quad T(|x|)\geq M_0|x|^a, \quad\text{for }|x|\geq R_0,\\
K(|x|)\leq M_0|x|^{b_0}, \quad T(|x|)\geq M_0|x|^{a_0}, \quad \text{for }
0<|x|\leq r_0.
\end{gathered} \label{e3.14}
\end{equation}
For $R>R_0$ and $0<r<r_0$, we estimate the integrals
$\big(\int_{B_{r}}K(|x|)|u_n|^{s}dx\big)^{1/s}$ and
$\big(\int_{B^{c}_{R}}K(|x|)|u_n|^{s}dx\big)^{1/s}$ in different cases according
to the definitions of $s^{*}$ and $s_{*}$, $B^{c}_{R}$ denotes the complement
of $B_{R}$.

 Firstly, we estimate the term $\Big(\int_{B_{r}}K(|x|)|u_n|^{s}dx\Big)^{1/s}$.

\noindent\textbf{Case 1.1:} For $a_0\geq-p$, $b_0\geq-p$. Let
$s=\frac{2p(N+c)}{(N-2p)}$, by $s\leq s^{*}$, we obtain $\eta_1=b_0-c\geq0$.
Hence by Lemma \ref{lem3.3} and \eqref{e3.11}, we have
\begin{equation}
\begin{aligned}
\Big(\int_{B_{r}}K(|x|)|u_n|^{s}dx\Big)^{1/s}
&\leq M_0^{1/s}\Big(\int_{B_{r}}|x|^{b_0}|u_n|^{s}dx\Big)^{1/s}\\
&\leq M_0^{1/s} r^{(\frac{b_0-c}{s})}
\Big(\int_{B_{r}}|x|^{c}|u_n|^{s}dx\Big)^{1/s}\\
&\leq M_0^{1/s} r^{(\frac{b_0-c}{s})}
\max\Big(\|\nabla u_n\|_{p,2p},\|\nabla u_n\|^2_{p,2p}\Big)\\
&= r^{(\frac{b_0-c}{s})} o(1), \quad \text{as } n\to \infty.
\end{aligned} \label{e3.15.1}
\end{equation}

\noindent\textbf{Case 1.2:} For $-p>a_0>-\frac{(N-1)}{(2p-1)}2p, b_0\geq a_0$.
From $s\leq s^{*}$, we obtain
$$
\eta_2=b_0-a_0-(s-\alpha)\frac{2p(N-1)+a_0(2p-1)}{\alpha (2p)}\geq0.
$$
We choose a cut-off function $\phi$ such that $\phi=1$ for
$0\leq|x|\leq\frac{r_0}{2}$, and $\phi=0$ for $|x|\geq r_0$.
Then by Lemma \ref{lem3.5}, for $r<\frac{r_0}{2}$, we have
\begin{align}
&\Big(\int_{B_{r}}K(|x|)|u_n|^{s}dx\Big)^{1/s} \nonumber \\
&\leq M_0^{1/s}\Big(\int_{B_{r}}|x|^{b_0}|\phi u_n|^{s}dx\Big)^{1/s}\nonumber \\
&=M_0^{1/s}\Big(\int_{B_{r}}|x|^{(b_0-a_0)}|\phi u_n|^{(s-\alpha)}
|x|^{a_0}|\phi u_n|^{\alpha}dx\Big)^{1/s} \nonumber \\
&\leq M_2\|\phi u_n\|_{\mathcal{W}}^{(\frac{s-\alpha}{s})}
\Big(\int_{B_{r}}|x|^{(b_0-a_0-(s-\alpha)\frac{2p(N-1)
+a_0(2p-1)}{\alpha (2p)})}T(|x|)|u_n|^{\alpha}dx\Big)^{1/s} \nonumber \\
&\leq M_{3} r^{(\frac{b_0-a_0}{s}-(s-\alpha)\frac{2p(N-1)+a_0(2p-1)}{s\alpha(2p)})}
\|u_n\|_{\mathcal{W}_{\rm rad}}
^{(\frac{s-\alpha}{s})}\|u_n\|_{L_{T}^{\alpha}(B_{r})}
^{\alpha/s} \nonumber \\
&\leq M'_{3}r^{(\frac{b_0-a_0}{s}-(s-\alpha)
\frac{2p(N-1)+a_0(2p-1)}{s\alpha(2p)})}\|u_n\|_{\mathcal{W}_{\rm rad}}
\nonumber \\
&=r^{\big(\frac{b_0-a_0}{s}-(s-\alpha)\frac{2p(N-1)
+a_0(2p-1)}{s\alpha(2p)}\big)}o(1), \quad \text{as } n\to \infty.
\label{e3.15.2}
\end{align}

\noindent\textbf{Case 1.3:} For $a_0\leq -\frac{(N-1)}{(p-1)}p$, $b_0\geq a_0$,
in the case $s^{*}=\infty$. For $\infty>s>\alpha$, it  holds
$$
\eta_{3}=b_0-a_0-(s-\alpha)\frac{2p(N-1)+a_0(2p-1)}{\alpha (2p)}\geq0.
$$
With the same function $\phi$ given in Case 1.2, and $r<\frac{r_0}{2}$,
by Lemma \ref{lem3.5}, we have
\begin{equation}
\begin{aligned}
&\Big(\int_{B_{r}}K(|x|)|u_n|^{s}dx\Big)^{1/s}\\
&\leq M_0^{1/s}\Big(\int_{B_{r}}|x|^{b_0}|\phi u_n|^{s}dx\Big)^{1/s}\\
&=M_0^{1/s}\Big(\int_{B_{r}}|x|^{(b_0-a_0)}|\phi u_n|^{(s-\alpha)}|x|^{a_0}
|\phi u_n|^{\alpha}dx\Big)^{1/s}\\
&\leq M_{4}\|\phi u_n\|_{\mathcal{W}_{\rm rad}}
^{(\frac{s-\alpha}{s})}
\Big(\int_{B_{r}}|x|^{(b_0-a_0-(s-\alpha)\frac{2p(N-1)
+a_0(2p-1)}{\alpha (2p)})}T(|x|)|u_n|^{\alpha}dx\Big)^{1/s}\\
&\leq M_{5}r^{(\frac{b_0-a_0}{s}-(s-\alpha)\frac{2p(N-1)
+a_0(2p-1)}{s\alpha(2p)})}\|u_n\|_{\mathcal{W}_{\rm rad}}
^{(\frac{s-\alpha}{s})}\|u_n\|_{L_{T}^{\alpha}(B_{r})}
^{\frac{\alpha}{s}}\\
& \leq M'_{5}r^{(\frac{b_0-a_0}{s}-(s-\alpha)
\frac{2p(N-1)+a_0(2p-1)}{s\alpha(2p)})}\|u_n\|_{\mathcal{W}_{\rm rad}} \\
&=r^{(\frac{b_0-a_0}{s}-(s-\alpha)\frac{2p(N-1)
+a_0(2p-1)}{s\alpha(2p)})}o(1), \quad \text{as } n\to \infty.
\end{aligned}\label{e3.15.3}
\end{equation}

 Secondly, we estimate the term
$\Big(\int_{B_{R}^{c}}K(|x|)|u_n|^{s}dx\Big)^{1/s}$.

\noindent\textbf{Case 2.1:} For $-p<a\leq b$, by $s\geq s_{*}$, we obtain
$$
\lambda_1=b-a-(s-\alpha)\frac{2p(N-1)+a(2p-1)}{\alpha (2p)}\leq0.
$$
Hence by Lemma \ref{lem3.4} and \eqref{e3.11}, for $R>R_0>1$, we have
\begin{equation}
\begin{aligned}
&\Big(\int_{B_{R}^{c}}K(|x|)|u_n|^{s}dx\Big)^{1/s}\\
&\leq M_0^{1/s}\Big(\int_{B_{R}^{c}}|x|^{b}|u_n|^{s}dx\Big)^{1/s}\\
&=M_0^{1/s}\Big(\int_{B_{R}^{c}}|x|^{(b-a)}|u_n|^{(s-\alpha)}|x|^a
|u_n|^{\alpha}dx\Big)^{1/s}\\
&\leq M_{6}\|u_n\|_{\mathcal{W}_{\rm rad}}
^{(\frac{s-\alpha}{s})}\Big(\int_{B_{R}^{c}}|x|
^{(b-a-(s-\alpha)\frac{2p(N-1)+a(2p-1)}{\alpha (2p)})}
T(|x|)|u_n|^{\alpha}dx\Big)^{1/s}\\
&\leq M_{7}R^{\frac{1}{s}(b-a-(s-\alpha)
\frac{2p(N-1)+a(2p-1)}{\alpha (2p)})}
\|u_n\|_{\mathcal{W}_{\rm rad}}^{(\frac{s-\alpha}{s})}
\|u_n\|_{L_{T}^{\alpha}(B_{r}^{c})}^{\frac{\alpha}{s}}\\
&\leq M'_{7}R^{\frac{1}{s}(b-a-(s-\alpha)\frac{2p(N-1)
+a(2p-1)}{\alpha (2p)})}\|u_n\|_{\mathcal{W}_{\rm rad}} \\
&= R^{\frac{1}{s}(b-a-(s-\alpha)\frac{2p(N-1)+a(2p-1)}{\alpha (2p)})}o(1),
 \quad \text{as } n\to \infty.
\end{aligned}\label{e3.16.1}
\end{equation}

\noindent\textbf{Case 2.2:} For $b\geq-p, a\leq-p$, let $s=\frac{2p(N+c)}{(N-2p)}$,
by $s\geq s_{*}$, we obtain $\lambda_2=b-c\leq0$.
Hence by Lemma \ref{lem3.3}, for $R>R_0>1$, we have
\begin{equation}
\begin{aligned}
\Big(\int_{B_{R}^{c}}K(|x|)|u_n|^{s}dx\Big)^{1/s}
&\leq M_0^{1/s}\Big(\int_{B_{R}^{c}}|x|^{(b-c)}|x|^{c}|u_n|^{s}dx\Big)^{1/s}\\
&\leq M_{8}R^{(\frac{b-c}{s})}
\max \big(\|\nabla u_n\|_{p,2p},\|\nabla u_n\|^2_{p,2p}\big)\\
&\leq M'_{8}R^{(\frac{b-c}{s})}\|u_n\|_{\mathcal{W}_{\rm rad}}
=R^{(\frac{b-c}{s})}o(1), \quad \text{as }  n\to \infty.
\end{aligned}
\label{e3.16.2}
\end{equation}

\noindent\textbf{Case 2.3:} For $b\leq \max \{a,-p\}$, $s>\alpha=s_{*}$.
As for $R>R_0>1$, when $a>-p, b\leq a$, it always holds
$$
\lambda_{3}=b-a-(s-\alpha)\frac{2p(N-1)+a(2p-1)}{\alpha (2p)}<0,
$$
so similar to Case 2.1, we have
\begin{equation}
\begin{aligned}
\Big(\int_{B_{R}^{c}}K(|x|)|u_n|^{s}dx\Big)^{1/s}
&\leq M'_{7}R^{\frac{1}{s}(b-a-(s-\alpha)\frac{2p(N-1)+a(2p-1)}
{\alpha (2p)})}\|u_n\|_{\mathcal{W}_{\rm rad}} \\
&= R^{\frac{1}{s}(b-a-(s-\alpha)\frac{2p(N-1)+a(2p-1)}{\alpha (2p)})}o(1),
 \quad \text{as }  n\to \infty.
\end{aligned} \label{e3.16.3}
\end{equation}
and when $a\leq-p, b\leq-p\leq c$, let $s=\frac{2p(N+c)}{(N-2 p)}$,
 we obtain $(b-c)\leq0$, we have similar to Case 2.2 that
\begin{equation}
\Big(\int_{B_{R}^{c}}K(|x|)|u_n|^{s}dx\Big)^{1/s}
\leq M'_{8}R^{(\frac{b-c}{s})}\|u_n\|_{\mathcal{W}_{\rm rad}}
=R^{(\frac{b-c}{s})}o(1), \quad \text{as }  n\to \infty.
\label{e3.16'3}
\end{equation}
Now we write
\begin{align*}
\int_{\mathbb{R}^{N}}K(|x|)|u_n|^{s}dx
&=\int_{B_{r}}K(|x|)|u_n|^{s}dx
 +\int_{B_{R}^{c}}K(|x|)|u_n|^{s}dx \\
&\quad +\int_{B_{R}\backslash B_{r}}K(|x|)|u_n|^{s}dx.
\end{align*}
As $s^{*}$ is finite and $s_{*}\leq s\leq s^{*}$, by
\eqref{e3.15.1}, \eqref{e3.15.2}, \eqref{e3.16.1}, \eqref{e3.16.2}, \eqref{e3.16.3}
and Lemma \ref{lem3.6}, we obtain that \eqref{e3.13} holds.
As $s^{*}$ is infinite and $s_{*}\leq s<\infty$,
by \eqref{e3.15.3}, \eqref{e3.16.1}, \eqref{e3.16.2}, \eqref{e3.16.3}
and Lemma \ref{lem3.6}, we obtain that \eqref{e3.13} holds.
Therefore the embedding is continuous in each case.

 Now we show that the embedding obtained above is compact.
 Let $\{u_n\}\subset \mathcal{W}_{\rm rad}$ be such that
\begin{equation}
\|u_n\|_{\mathcal{W}_{\rm rad}}
=\|\nabla u_n\|_{p,2p}+\|u_n\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}\leq M.
\label{e3.17}
\end{equation}
Without loss of generality, we consider
\begin{equation}
u_n\rightharpoonup0, \quad \text{in $\mathcal{W}_{\rm rad}$  as $n\to  \infty$}.
\label{e3.18}
\end{equation}
To obtain the compactness, we only need to show that
\begin{equation}
\lim_{n\to \infty}\Big(\int_{\mathbb{R}^{N}}K(|x|)|u_n|^{s}dx\Big)^{1/s}=0.
\label{e3.19}
\end{equation}
As $s_{*}<s<s^{*}$, the exponents $\eta_{i}$ of $r$ in the estimates
\eqref{e3.15.1}, \eqref{e3.15.2}, \eqref{e3.15.3}  are strictly positive,
and the exponents $\lambda_{j}$ of $R$ in the estimates
\eqref{e3.16.1}, \eqref{e3.16.2}, \eqref{e3.16.3}  are strictly negative,
we obtain the following estimates by similar arguments as above
\begin{gather}
\Big(\int_{B_{r}}K(|x|)|u_n|^{s}dx\Big)^{1/s}
 \leq Mr^{\eta_{i}}\|u_n\|_{\mathcal{W}_{\rm rad}}, \quad i=1,2,3,\label{e3.20}\\
\Big(\int_{B_{R}^{c}}K(|x|)|u_n|^{s}dx\Big)^{1/s}
 \leq MR^{\lambda_{j}}\|u_n\|_{\mathcal{W}_{\rm rad}}, \quad j=1,2,3,\label{e3.21}
\end{gather}
By \eqref{e3.17}, \eqref{e3.20}, \eqref{e3.21} and Lemma \ref{lem3.6},
we obtain \eqref{e3.19}. Hence the embedding is compact in each case.
 In conclusion, the proof of Theorem \ref{thm3.1} is complete.
\end{proof}

\section{Proof of Theorems \ref{thm1.2} and  \ref{thm1.3}}

 In this section, we prove our main theorems. Now, let us define the
functional $J:\mathcal{W}_{\rm rad}\to \mathbb{R}$ as:
\begin{equation}
J(u)=\frac{1}{p}\int_{\mathbb{R}^{N}}(\sqrt{1+|\nabla u|^{2p}}-1)dx
+\frac{1}{\alpha}\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx
-\frac{1}{s}\int_{\mathbb{R}^{N}}K(|x|)|u|^{s}dx.
\label{e4.1}
\end{equation}
 Obviously, by \cite[Lemma 2.2]{d1}, the functional $J$ is well defined
and it is of class $\mathcal{C}^{1}$. We obtain that solutions of
 \eqref{e1.1} are critical points of the functional $J$.
By Remark 2.5 and using the standard Palais' result \cite{p1}, we infer that
$\mathcal{W}_{\rm rad}$ is a natural constraint for the functional $J$.

In the following propositions and lemmas, we show that the functional $J$
 satisfies the geometrical assumptions $\mathbb{Z}_2$-symmetric version
of the Mountain Pass Lemma \cite{r1}. More precisely, we have the following result.

\begin{proposition} \label{prop4.1}
 The functional $J$ satisfies the following properties:
\begin{itemize}
\item[(i)] $J(0)=0$;

\item[(ii)] there exist $\rho, \overline{c}$ such that $J(u)\geq\overline{c}$,
for any $u\in \mathcal{W}_{\rm rad}$ with $\|u\|_{{W}_{\rm rad}}=\rho$;

\item[(iii)] there exists $\overline{u}\in \mathcal{W}_{\rm rad}$ such that
$J(\overline{u})\leq0$.
\end{itemize}
\end{proposition}

\begin{proof}
(i) Trivially, $J(0)=0$.
(ii) As there exists a positive constant $c$ such that
\begin{gather*}
c|\nabla u|^{p} \leq  
\sqrt{1+|\nabla u|^{2p}}-1, \quad \text{if }|\nabla u|\geq1,\\
c|\nabla u|^{2p} \leq
\sqrt{1+|\nabla u|^{2p}}-1, \quad \text{if }0\leq |\nabla u|\leq 1.
\end{gather*}
Then, if $\|u\|_{\mathcal{W}_{\rm rad}}$ is sufficiently small, by $\alpha<p<2p<s$,
Proposition \ref{prop2.2} (iv),  and since
 $\mathcal{W}_{\rm rad}\hookrightarrow L^{s}(\mathbb{R}^{N};K)$, we have that
\begin{align*}
 J(u)
&\geq c_1\int_{\Lambda_{\nabla u}^{c}}|\nabla u|^{2p}dx
 +c_2\int_{\Lambda_{\nabla u}}|\nabla u|^{p}dx
 +\frac{1}{\alpha}\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx\\
&\quad -\frac{1}{s}\int_{\mathbb{R}^{N}}K(|x|)|u|^{s}dx\\
&\geq c\max \Big(\int_{\Lambda_{\nabla u}^{c}}|\nabla u|^{2p}dx,
 \int_{\Lambda_{\nabla u}}|\nabla u|^{p}dx\Big)
  +\frac{1}{\alpha}\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx\\
&\quad   -\frac{1}{s}\int_{\mathbb{R}^{N}}K(|x|)|u|^{s}dx\\
&\geq c\big[\|\nabla u\|_{p,2p}^{2p}
 +\|u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}^{\alpha}-\|u\|^{s}_{L_{K}^{s}
 (\mathbb{R}^{N})}\big]\\
&\geq c\big[\|u\|_{\mathcal{W}_{\rm rad}}^{2p}
 -\|u\|_{\mathcal{W}_{\rm rad}}^{s}\big]\geq\overline{c}.
\end{align*}

(iii) Let $u\in C_{c}^{\infty}(\mathbb{R}^{N},\mathbb{R})$, as there exists
a positive constant $C$ such that
\[
\begin{cases}
(\sqrt{1+|\nabla u|^{2p}}-1)
\leq  C|\nabla u|^{p}, & \text{if }|\nabla u|\geq1,\\
 C|\nabla u|^{2p},& \text{if }  0\leq |\nabla u|\leq1;
\end{cases}
\]
then for all $t>0$, we obtain
\begin{align*}
 J(tu)
&\leq C_1\int_{\Lambda_{\nabla (tu)}^{c}}|\nabla (tu)|^{2p}dx
 +C_2\int_{\Lambda_{\nabla (tu)}}|\nabla (tu)|^{p}dx\\
&\quad +\frac{1}{\alpha}\int_{\mathbb{R}^{N}}T(|x|)|tu|^{\alpha}dx
 -\frac{1}{s}\int_{\mathbb{R}^{N}}K(|x|)|tu|^{s}dx\\
&\leq C\Big[t^{2p}\int_{\mathbb{R}^{N}}|\nabla u|^{2p}dx
 +t^{p}\int_{\mathbb{R}^{N}}|\nabla u|^{p}dx \\
&\quad +t^{\alpha}\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx
 -t^{s}\int_{\mathbb{R}^{N}}K(|x|)|u|^{s}dx\Big].
\end{align*}
Therefore, for $t$ sufficiently large, there exists $u_0=tu$ such that
$J(u_0)=J(tu)<0$.
\end{proof}

\begin{proposition} \label{prop4.2}
 The functional $J|_{\mathcal{W}_{\rm rad}}$ satisfies the (PS) condition.
\end{proposition}

\begin{proof}
Let $\{u_n\}_n\subset \mathcal{W}_{\rm rad}$ be a (PS)-sequence for the $J$,
 namely for a suitable $\overline{c}\in \mathbb{R}$
$$
J(u_n)\to \overline{c}\quad\text{and}\quad
J'(u_n)\to 0\quad \text{in }\mathcal{W}'_{\rm rad}.
$$
Let us check that $\{u_n\}_n$ is bounded. In fact, as there exists
$0<\mu<1$ such that
$$
\frac{|\nabla u|^{2p}}{\sqrt{1+|\nabla u|^{2p}}}
\leq\frac{s\mu}{2}(\sqrt{1+|\nabla u|^{2p}}-1), \quad \text{for all }t\geq0,
$$
then we have
$$
\overline{c}+o_n(1)\|u_n\|=J(u_n)-\frac{1}{s}J'(u_n)u_n;
$$
i.e.,
\begin{align*}
&\overline{c}+o_n(1)\|u_n\|\\
&=\int_{\mathbb{R}^{N}}\big[\frac{1}{p}(\sqrt{1+|\nabla u_n|^{2p}}-1)
 -\frac{1}{s}\frac{|\nabla u|^{2p}}{\sqrt{1+|\nabla u|^{2p}}}\big]dx
 +\big(\frac{1}{\alpha}-\frac{1}{s}\big)\int_{\mathbb{R}^{N}}T(|x|)
 |u_n|^{\alpha}dx\\
&\geq\frac{(2-\mu p)}{2p}\int_{\mathbb{R}^{N}}(\sqrt{1+|\nabla u_n|^{2p}}-1)dx
 +\big(\frac{1}{\alpha}-\frac{1}{s}\big)\int_{\mathbb{R}^{N}}T(|x|)
 |u_n|^{\alpha}dx\\
&\geq c\big[\text{min}\big(\|\nabla u_n\|_{p,2p}^{2p},
 \|\nabla u_n\|_{p,2p}^{p}\big)+\|u_n\|_{L_{T}^{\alpha}
 (\mathbb{R}^{N})}^{\alpha}\big].
\end{align*}
Therefore, by Theorem \ref{thm3.1}, there exists $u_0\in \mathcal{W}_{\rm rad}$ such that
\begin{gather}
u_n\rightharpoonup u_0, ~\text{weakly~in}~\mathcal{W}_{\rm rad},\label{e4.2}\\
u_n\to  u_0, \quad \text{strongly in } L^{s}(\mathbb{R}^{N};K),\label{e4.3}\\
u_n\to  u_0, \quad  \text{a.e. in } \mathbb{R}^{N}.
\end{gather}
Inspired by \cite{n1}, we write $J(u)=A(u)-B(u)$, where $A(u)=A_1(u)+A_2(u)$ and
\begin{gather*}
A_1(u)=\frac{1}{p}\int_{\mathbb{R}^{N}}(\sqrt{1+|\nabla u|^{2p}}-1)dx, \quad
A_2(u)=\frac{1}{\alpha}\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx, \\
B(u)=\frac{1}{s}\int_{\mathbb{R}^{N}}K(|x|)|u|^{s}dx.
\end{gather*}
Then, we have
$$
A(u_n)-B(u_n)\to \overline{c},\quad
A'(u_n)-B'(u_n)\to 0,\quad  \text{in }\mathcal{W}'_{\rm rad}.
$$
By \eqref{e4.3}, we infer that
$$
B(u_n)\to  B(u_0), \quad B'(u_n)\to  B'(u_0), \quad\text{in }\mathcal{W}'_{\rm rad}.
$$
Therefore,
\begin{equation}
A'(u_n)\to  B'(u_0) \quad \text{in }\mathcal{W}'_{\rm rad}.\label{e4.4}
\end{equation}
Since $A_1(u)$ and $A_2(u)$ are convex, so $A(u)$ is convex, we have
$$
A(u_0)\geq A(u_n)+A'(u_n)(u_0-u_n),
$$
namely
$$
A(u_n)\leq A(u_0)+A'(u_n)(u_n-u_0).
$$
So, by \eqref{e4.2} and \eqref{e4.4}, we obtain
$\limsup_{n\to \infty}A(u_n)\leq A(u_0)$.
Since $A$ is convex and continuous, we obtain $A$ is lower weak semicontinuity
$$
A(u_0)\leq\liminf_{n\to \infty}A(u_n);
$$
therefore,
\begin{equation}
A(u_n)\to  A(u_0), \quad \text{as }n\to \infty.\label{e4.5}
\end{equation}
By \eqref{e4.2} and arguing as in \cite[page 208]{l2}, we have
\begin{gather}
\nabla u_n\rightharpoonup\nabla u_0,\quad \text{weakly in }
L^{p}(\mathbb{R}^{N})+L^{2p}(\mathbb{R}^{N}),\label{e4.6}\\
u_n\rightharpoonup u_0,\quad \text{weakly in }
L^{\alpha}(\mathbb{R}^{N};T),\label{e4.7}
\end{gather}
and $A_1$ and $A_2$ are lower weak semicontinuity, we have
$$
A_1(u_0)\leq\liminf_{n\to \infty}A_1(u_n),\quad
A_2(u_0)\leq\liminf_{n\to \infty}A_2(u_n).
$$
Thus, together with \eqref{e4.5}, we obtain
\begin{gather}
A_1(u_0)=\liminf_{n\to \infty}A_1(u_n),\label{e4.8}\\
A_2(u_0)=\liminf_{n\to \infty}A_2(u_n).\label{e4.9}
\end{gather}
Then \eqref{e4.7} and \eqref{e4.9}, imply
$$
u_n\to  u_0, \quad \text{in } L^{\alpha}(\mathbb{R}^{N};T).
$$
Moreover, by \eqref{e4.6} and \eqref{e4.8} and by  \cite[Lemma 2.3]{d1}, we have
$$
\nabla u_n\to \nabla u_0, \quad\text{in }
L^{p}(\mathbb{R}^{N})+L^{2p}(\mathbb{R}^{N}).
$$
Therefore, $u_n\to  u_0$ in $\mathcal{W}_{\rm rad}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 By the $\mathbb{Z}_2$-symmetric version of the Mountain Pass Lemma,
 we  only need to prove that there exist $\{V_n\}_n$, a sequence
of finite dimensional subspaces of $\mathcal{W}_{\rm rad}$ with
$\operatorname{dim} V_n=n$ and $V_n\subset V_{n+1}$, and $\{R_n\}_n$,
a sequence of positive numbers, such that $J(u)\leq0$ for all
 $u\in V_n\backslash B_{R_n}$.

 Let $\{\phi_n\}_n$ be a sequence of radially symmetric test functions such that,
for any $n\geq1$, the functions $\phi_1,\phi_2,\dots ,\phi_n$ are linearly independent.
 Denote by $V_n=\text{span}\{\phi_1,\phi_2,\dots ,\phi_n\}
\subset(C_{c}^{\infty}(\mathbb{R}^{N},\mathbb{R}))_{\rm rad}
\subset \mathcal{W}_{\rm rad}$.

By the proof of  Proposition \ref{prop4.1} (iii), and since $V_n$ is a finite dimensional
space of test functions, so the norms in $V_n$ are equivalent, and we conclude
 observing that, if $u\in V_n\backslash B_{R_n}$ and $R_n$ is sufficiently large,
\begin{align*}
J(u)
&\leq C\big[\|\nabla u\|_{p,2p}^{2p}+\|T(|x|)u\|_{\alpha}^{\alpha}
-\|K(|x|)u\|_{s}^{s}\big]\\
&\leq C\big[\|u\|_{\mathcal{W}_{\rm rad}}^{2p}
 +\|u\|_{\mathcal{W}_{\rm rad}}^{\alpha}-\|u\|_{\mathcal{W}_{\rm rad}}^{s}\big]\\
&\leq C[R_n^{2p}+R_n^{\alpha}-R_n^{s}]\leq 0.
\end{align*}
So $J$ satisfies the $\mathbb{Z}_2$-symmetric version of the Mountain
Pass Lemma \cite{r1}, and  problem \eqref{e1.1} has infinitely many radially
symmetric solutions.
\end{proof}

To obtain a ground state solution in $\mathcal{W}_{\rm rad}$, we need the
following lemmas.

 Let us denote with $\mathcal{M}$ the set of all nontrivial solutions of \eqref{e1.1}
in $\mathcal{W}_{\rm rad}$, namely
$$
\mathcal{M}=\{u\in \mathcal{W}_{\rm rad}\backslash\{0\}|J'(u)=0\}.
$$
Obviously, we know that $\mathcal{M}\neq\emptyset$.

\begin{lemma} \label{lem4.3}
 There exists a positive constant $\overline{c}>0$, such that
$\|u\|\geq\overline{c}$, for all $u\in \mathcal{M}$.
\end{lemma}

\begin{proof} As $J'(u)=0$, namely
$$
\int_{\mathbb{R}^{N}}\frac{|\nabla u|^{2p}}{\sqrt{1+|\nabla u|^{2p}}}dx
+\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx
-\int_{\mathbb{R}^{N}}K(|x|)|u|^{s}dx=0.
$$
Since there exists a positive constant $c$ such that
\[
 c|\nabla u|^{(p-2)}\leq\begin{cases}
\frac{|\nabla u|^{(2p-2)}}{\sqrt{1+|\nabla u|^{2p}}},
&  \text{if } |\nabla u|\geq1,\\[4pt]
\frac{|\nabla u|^{(2p-2)}}{\sqrt{1+|\nabla u|^{2p}}},
& \text{if } 0\leq |\nabla u|\leq1;
\end{cases}
\]
we have
\begin{align*}
\|u\|_{L_{K}^{s}(\mathbb{R}^{N})}^{s}
&=\int_{\mathbb{R}^{N}}\frac{|\nabla u|^{2p}}{\sqrt{1+|\nabla u|^{2p}}}dx+
\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx\\
&\geq c~\max \Big(\int_{\Lambda_{\nabla u}^{c}}|\nabla u|^{2p}dx,
\int_{\Lambda_{\nabla u}}|\nabla u|^{p}dx\Big)
+\int_{\mathbb{R}^{N}}T(|x|)|u|^{\alpha}dx\\
&\geq c\big[\|\nabla u\|_{p,2p}^{2p}+\|u\|_{L_{T}^{\alpha}
(\mathbb{R}^{N})}^{\alpha}\big]\\
&\geq c\|u\|_{{W}_{\rm rad}}^{2p}
\geq c\|u\|_{L_{K}^{s}(\mathbb{R}^{N})}^{2p}.
\end{align*}
\end{proof}

\begin{lemma} \label{lem4.4}
 There exists a positive constant $\overline{c}>0$, such that
$J(u)\geq\overline{c}$, for all $u\in \mathcal{M}$
\end{lemma}

\begin{proof}
Let $u\in \mathcal{M}$. Repeating the arguments of the proof of
Proposition \ref{prop4.2}
and by Lemma \ref{lem4.3}, we have
$$
J(u)=J(u)-\frac{1}{s}J'(u)u
\geq c\big[\text{min}(\|\nabla u\|_{p,2p}^{2p},
\|\nabla u\|_{p,2p}^{p})+\|u\|_{L_{T}^{\alpha}(\mathbb{R}^{N})}^{\alpha}\big]
\geq \overline{c}.
$$
\end{proof}

\begin{remark} \label{rmk4.5} \rm
By Lemma \ref{lem4.4}, we infer that
$$
\tau=\inf_{u\in \mathcal{M}}J(u)>0,
$$
and by Theorem \ref{thm1.3}, we obtain that this infimum is achieved.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
Let $\{u_n\}_n\subset \mathcal{M}$ be a minimizing sequence, namely
$$
J(u_n)\to \tau\quad \text{and}\quad J'(u_n)=0.
$$
Then  $\{u_n\}_n$ is a (PS)-sequence for the functional $J$ and we obtain
the result by means of Proposition \ref{prop4.2}.
\end{proof}

\begin{remark} \label{rmk4.6} \rm
As special case, our result can be applied to mean curvature equation or
the capillary equation
\begin{gather*}
-\operatorname{div}\Big(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\Big)
+T(|x|)|u|^{\alpha-2}u=K(|x|)|u|^{s-2}u,\quad  u>0,\; x\in\mathbb{R}^{N},\\
u(|x|)\to 0,\quad\text{as } |x|\to \infty.
\end{gather*}
\end{remark}

\subsection*{Acknowledgments}
This research was supported by the Shanghai Natural Science Foundation Project
(No. 15ZR1429500), by the Shanghai Leading Academic Discipline Project 
(No. XTKX 2012), and by the National Project Cultivate Foundation of
USST (No. 15HJPY-MS03).

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\end{document}