\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
0,\; x\in\mathbb{R}^{N},\\ u(|x|)\to 0,\quad\text{as } |x|\to \infty, \end{gathered} \end{equation} where $N\geq3$, $1<\alpha
0, q>1$ and obtained the existence of radial ground states.
As $\lambda=0$, Ni and Serrin \cite{n2,n3} established that if
$1 0$, 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, $1<\alpha 1, s>1$, 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 $1<\alpha 0$ 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 $1 0$ 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, $1<\alpha -\frac{(N-1)}{(2p-1)}2p$.
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 $r>R$, $1<\alpha r_0>0$
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 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 $01$, 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 $t$, $1
0$, 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}