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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 212, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/212\hfil Sublinear eigenvalue problems]
{Sublinear eigenvalue problems with singular weights related to the
critical Hardy inequality}

\author[M. Sano, F. Takahashi \hfil EJDE-2016/212\hfilneg]
{Megumi Sano, Futoshi Takahashi}

\address{Megumi Sano \newline
Department of Mathematics,
Graduate School of Science,
Osaka City University, \newline
Sumiyoshi-ku, Osaka, 558-8585, Japan}
\email{megumisano0609@st.osaka-cu.ac.jp}

\address{Futoshi Takahashi \newline
Department of Mathematics, 
Osaka City University \& 
OCAMI, Sumiyoshi-ku,
Osaka, 558-8585, Japan}
\email{futoshi@sci.osaka-cu.ac.jp}

\thanks{Submitted July 8, 2016. Published August 10, 2016.}
\subjclass[2010]{35A23, 35J62, 35J20}
\keywords{Critical Hardy inequality; sublinear reaction term;
eigenvalue problem}

\begin{abstract}
 In this article, we consider a weighted sublinear eigenvalue problem
 related to an improved critical Hardy inequality.
 We discuss to what extent the weights can be singular for the existence
 of weak solutions. Also we study the asymptotic behavior of the first
 eigenvalues as a parameter involved varies.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$, $N \ge 2$, with
$0 \in \Omega$. Here and henceforth, we put $R = \sup_{x \in \Omega} |x|$.
In this article, we consider the quasilinear eigenvalue problem with
singular weights
\begin{equation} \label{EP}
\begin{gathered}
-\Delta_N u - \mu \frac{|u|^{N-2}u}{ |x|^N ( \log \frac{Re}{|x|} )^N }
= \lambda f(x) |u|^{q-2}u  \quad \text{in } \Omega, \\
u = 0 \quad \text{on }  \partial \Omega,
	\end{gathered}
\end{equation}
where $\Delta_N u = \text{div} (|\nabla u|^{N-2} \nabla u)$ is the
$N$-Laplacian, $1 < q$,
$0 \leq \mu < (\frac{N-1}{N} )^N$, $\lambda \in \mathbb{R}$ and
$f \in L^{\infty}_{loc}(\Omega \setminus \{0\})$ is a positive weight
function which may be unbounded near the origin.
We assume that the weight function $f$ satisfies
$|\phi|^q f \in L^1(\Omega)$ for any $\phi \in W^{1,N}_0(\Omega)$.
This problem is related to the {\it critical Hardy inequality}
due to Adimurthi and Sandeep \cite{Adimurthi-Sandeep}:
\begin{equation} \label{H_N}
	\int_{\Omega} |\nabla u|^N dx \ge \big( \frac{N-1}{N} \big)^N
\int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx, \quad \forall
u \in W^{1,N}_0(\Omega).
\end{equation}
In the appendix, we provide a simple proof of \eqref{H_N} for the
sake of completeness. Thanks to \eqref{H_N}, the operator
\[
	L_\mu u = -\Delta_N u - \mu \frac{|u|^{N-2}u}{ |x|^N
( \log \frac{Re}{|x|} )^N }
\]
acting on $W^{1,N}_0(\Omega)$ is positive and coercive.
We call a function $u \in W^{1,N}_0(\Omega)$ a weak solution of the
problem \eqref{EP} if
\begin{equation*}
	\int_{\Omega} |\nabla u|^{N-2} \nabla u \cdot \nabla \phi dx
 = \mu \int_{\Omega} \frac{|u|^{N-2}u \phi}{|x|^N ( \log \frac{Re}{|x|} )^N } dx
+ \lambda \int_{\Omega} |u|^{q-2}u \phi f(x) dx
\end{equation*}
holds whenever $\phi \in W^{1,N}_0(\Omega)$.

When $q-1 = N-1$ case, $\eqref{EP}$ becomes a genuine eigenvalue problem
for $L_\mu$, and under suitable integrability assumptions of the indefinite
weight function $f$,
the existence of the positive first eigenvalue, its simplicity, and the
isolation property are obtained \cite{Sreenadh}.
Also in \cite{Zhang-Shao-Liu},
the authors obtain an unbounded sequence of minimax eigenvalues of
$L_{\mu}$ by the use of the cohomological index theory.

When $q-1 > N-1$ ($(N-1)$-superlinear case), since $|u|^{q-2}u$ is
subcritical from the view point of Trudinger-Moser inequality,
we find several references in which
the existence of (multiple) weak solutions is obtained, see for
example \cite{Tyagi}, \cite{Zhang-Shao-Liu}, and the reference therein.
See also \cite{Faria-Miyagaki-Pereira} for the critical growth case
and \cite{RaSr}, \cite{SaSr} for related results.

In this article, we focus on the $(N-1)$-sublinear case; $0 < q-1 < N-1$.
For $f$ in an appropriate class of weight functions,
we look for a weak solution $u \in W^{1,N}_0(\Omega)$ of \eqref{EP}
by a constrained minimization argument.
The solution obtained here corresponds to the first eigenvalue of
$\lambda_{\mu}(f)$ of the operator $L_\mu$: 
\[
	\lambda_{\mu} (f) = \inf_{u \in W^{1,N}_0(\Omega) \backslash \{ 0\}}
\frac{\int_{\Omega} |\nabla u|^N dx - \mu \int_{\Omega} \frac{|u|^N}{|x|^N
(\log \frac{Re}{|x|})^N} dx}{\big(\int_{\Omega} |u|^q f(x) dx \big)^{N/q}}.
\]
Furthermore we study the asymptotic behavior of $\lambda_{\mu}(f)$ as
$\mu \nearrow (\frac{N-1}{N})^N$. 

To state the main result in this paper, for $0< q < N$, put
$\alpha^* = (\frac{N-1}{N})q +1$
and define a class of weight functions
\begin{align*}
	F_N = \Big\{& f:\Omega \to \mathbb{R}^+ :
 f \in L^{\infty}_{\rm loc}(\Omega \backslash \{ 0\} )  \text{ and }
\exists \alpha \in (\alpha^*, N]
\text{ such that}\\
&	\limsup_{|x| \to 0} f(x) |x|^N \big( \log \frac{Re}{|x|}
\big)^{\alpha} < \infty  \Big\}.
\end{align*}
Then the main result of the paper reads as follows:


\begin{theorem}\label{thmEP}
Let $0 < q-1 < N-1$.
Then for all $f \in F_N$ and $0 < \mu < (\frac{N-1}{N})^N$,
problem \eqref{EP} admits a positive weak solution $u \in W^{1,N}_0(\Omega)$
corresponding to $\lambda=\lambda_{\mu}(f) > 0$.
Furthermore, $\lambda_{\mu} (f) \to \lambda (f)$ as $\mu \nearrow (\frac{N-1}{N})^N$
for a limit $\lambda(f) > 0$.
\end{theorem}

For the proof of Theorem \ref{thmEP}, we need an improved version of
the critical Hardy inequality \eqref{H_N}.
It is known that the constant $(\frac{N-1}{N})^N$ in \eqref{H_N} is
optimal and never attained on any bounded domain $\Omega \subset \mathbb{R}^N$
 with $0 \in \Omega$, see Adimurthi and Sandeep \cite{Adimurthi-Sandeep}.
Therefore there is a possibility to add a nonnegative remainder term to the
 right-hand side of \eqref{H_N}.
In \cite{Adimurthi-Sandeep}, the authors claim that there exists $C > 0$
such that
\[
	\int_{\Omega} |\nabla u|^N dx \ge \big( \frac{N-1}{N} \big)^N \int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx
	+ C \int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N
(\log^{(2)} \frac{R_1}{|x|})^N} dx
\]
for any $u \in W^{1,N}_0(\Omega)$, where $R_1 \ge (e^e)^{2/N} R$.
Here for $k \in \mathbb{N}$,
$\log^{(k)}$ is defined inductively by
$\log^{(1)} (\cdot) = \log (\cdot)$,
$\log^{(k)} (\cdot) = \log ( \log^{(k-1)} (\cdot)  )$ for $k \geq 2$.
However, the proof of it is omitted in \cite{Adimurthi-Sandeep}.
Barbatis, Filippas and Tertikas \cite{BFT2} proved that, among other things,
the  improved critical Hardy inequality
\begin{align*}
&\int_{\Omega} |\nabla u|^N dx - \big( \frac{N-1}{N} \big)^N
 \int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx \\
&\ge \frac{1}{2} \big(\frac{N-1}{N}\big)^{N-1} \sum_{i=2}^{\infty}
\int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} X_2^2
( \frac{|x|}{R} ) \dots X_i^2( \frac{|x|}{R} ) dx
\end{align*}
where for $t \in (0,1)$ and $i = 2,3, \dots$,
\[
	X_1(t) = (1 - \log t)^{-1}
 = \frac{1}{\log (\frac{e}{t})}, \quad X_i(t) = X_1(X_{i-1}(t)).
\]
Note that
\[
	X_2( \frac{|x|}{R} ) = \frac{1}{\log ( e \log \frac{eR}{|x|} )}, \quad
X_3( \frac{|x|}{R} ) = \frac{1}{\log (e\log ( e \log \frac{eR}{|x|} ))}, \quad \dots.
\]
In \cite{BFT2}, the authors use a ``vector field approach" as in \cite{BFT1}.

In this paper, we obtain another kind of remainder terms for the critical
Hardy inequality \eqref{H_N} in much simpler way,
see Proposition \ref{Prop H_N R}.
We use a classical idea by Brezis and V\'azquez \cite{Brezis-Vazquez}
combined with a transformation of functions relevant to our study,
see \eqref{key scale} below.

The organization of this paper is as follows:
In \S 2, an improved critical Hardy inequality is proved.
In \S 3, the optimality of the weight in the improved critical
 Hardy inequality is discussed.
Finally in \S 4, Theorem \ref{thmEP} is proved.


\section{Improving the critical Hardy inequality with an idea of Brezis and V\'azquez}

In this section, we improve the critical Hardy inequality \eqref{H_N}
by adding a nonnegative term to the right hand side.
In the proof of Proposition \ref{Prop H_N R} below,
we utilize the well-known transformation of Brezis and V\'azquez
\cite{Brezis-Vazquez} combined with an appropriate change of variables.


\begin{proposition}\label{Prop H_N R}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N \ge 2$,
with $0 \in \Omega$, and $R=\sup_{x \in \Omega} |x|$.
For any $-1 < L < N-2$ and $0 < q < (\frac{N}{N-1})(N-2-L)$, put
\[
	\alpha = \alpha(q,L) = \frac{N-1}{N} q + L + 2.
\]
Then the inequality
\begin{equation} \label{H_N R}
\begin{aligned}
	\int_{\Omega} |\nabla u|^N dx
&\ge \big(\frac{N-1}{N}\big)^N \int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx \\
&\quad +\omega_N^{1-\frac{N}{q}} C(L,N,q)^{N/q}
\Big(\int_{\Omega} \frac{|u|^q}{|x|^N \big( \log \frac{Re}{|x|} \big)^{\alpha}}
 dx \Big)^{N/q}
\end{aligned}
\end{equation}
holds for all $u \in W_0^{1,N} (\Omega)$, where $\omega_N$ is the area of
the unit sphere in $\mathbb{R}^N$ and
\[
	C(L, N, q)^{-1} = \int_{0}^1 s^L ( \log \frac{1}{s} )^{\frac{N-1}{N} q} ds
= (L +1)^{-( \frac{N-1}{N} q + 1 )} \Gamma ( \frac{N-1}{N} q +1 ),
\]
here $\Gamma(\cdot)$ is the Gamma function.
\end{proposition}

\begin{remark} \label{rmk3} \rm
Inequality \eqref{H_N R} does not hold when $L \leq -1$
(see Theorem \ref{Optimality}).
Therefore we see that the weight function in the remainder term of \eqref{H_N R}
is optimal.
\end{remark}

First, we recall a simple lemma.

\begin{lemma}[{\cite{Gazzola-Grunau-Mitidieri}[Lemma 1.1]}]
\label{lem1}
Let $N \ge 2$, and $\xi, \eta$ be real numbers such that
$\xi \ge 0$ and $\xi-\eta \ge 0$.
Then
\begin{equation}
\label{xi eta}
	(\xi - \eta)^N + N \xi^{N-1} \eta - \xi^N \geq |\eta |^N.
\end{equation}
\end{lemma}

\begin{proof}[Proof of Proposition \ref{Prop H_N R}] \quad
\smallskip

\noindent\textbf{Step 1:}
First we prove the inequality \eqref{H_N R} when $\Omega$ is a ball
 $B_R(0) \subset \mathbb{R}^N$ and for smooth nonnegative radially
non-increasing functions $u \in C_0^{\infty}(B_R(0))$.
We write $u(x) = u(r)$ with $r = |x|$ for radially symmetric functions $u$.
We define the transformation
\begin{equation} \label{key scale}
\begin{gathered}
v(s)=( \log \frac{Re}{r} )^{-\frac{N-1}{N}} u(r), \quad
\text{where } r=|x|, \; s=s(r)=\big( \log \frac{Re}{r} \big)^{-1} \in [0,1], \\
s'(r) =\frac{s(r)}{r \log \frac{Re}{r}} \geq 0.
\end{gathered}
\end{equation}
Note that $v(0) = v(1) = 0$ since $u(0)$ is finite and $u(R) = 0$, and
\begin{equation} \label{uprime}
	u'(r) = -\big( \frac{N-1}{N} \big) \big( \log \frac{Re}{r} \big)^{-1/N}
 \frac{v(s(r))}{r} + \big( \log \frac{Re}{r} \big)^{\frac{N-1}{N}} v'(s(r))
s' (r) \le 0.
\end{equation}
Now we observe that
\begin{align*} % \label{I 3}
	&I=\int_{B_R(0)} |\nabla u|^N dx - \big(\frac{N-1}{N}\big)^N
\int_{B_R(0)} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx \\
	&=\omega_N \int_{0}^{R} |u'(r)|^N r^{N-1} dr
- \big(\frac{N-1}{N}\big)^N \omega_N \int_{0}^{R} \frac{|u(r)|^N}{r
(\log \frac{Re}{r})^N} dr  \\
	&= \omega_N\int_{0}^{R} ( \frac{N-1}{N} ( \log \frac{Re}{r} )^{-1/N}
\frac{v(s(r))}{r} - \big( \log \frac{Re}{r} \big)^{\frac{N-1}{N}}
 v' (s(r)) s' (r) )^N r^{N-1} dr  \\
& \quad -\big(\frac{N-1}{N}\big)^N \omega_N \int_{0}^{R}
 \frac{|v(s(r))|^N}{r \log \frac{Re}{r}} dr.
\end{align*}
Here, we can apply Lemma \ref{lem1} with the choice
\begin{equation*}
	\xi = \frac{N-1}{N} \big( \log \frac{Re}{r} \big)^{-1/N}
\frac{v(s(r))}{r} \quad \text{and} \quad
\eta = \big( \log \frac{Re}{r} \big)^{\frac{N-1}{N}} v' (s(r)) s'(r).
\end{equation*}
By noticing the cancellation of the term $\xi^N$ in \eqref{xi eta}
and using the boundary conditions $v(0)=v(1)=0$, we obtain
\begin{equation} \label{I 1}
\begin{aligned}
I &\geq - \omega_N N \big(\frac{N-1}{N}\big)^{N-1} 
\int_{0}^{R} v(s(r))^{N-1} v'(s(r)) s'(r) dr \\
	&\quad +\omega_N \int_{0}^{R} | v' (s(r)) |^N ( s'(r) )^N ( r \log \frac{Re}{r} )^{N-1} dr  \\
&= - \omega_N N \big(\frac{N-1}{N}\big)^{N-1}
 \int_{0}^{1} v(s)^{N-1} v'(s)  ds + \omega_N \int_{0}^{1} | v' (s) |^N  s^{N-1} ds  \\
&=\omega_N \int_{0}^{1} | v' (s) |^N  s^{N-1} ds.
\end{aligned}
\end{equation}
When $N=2$, actually this inequality becomes the equality.
On the other hand, by using the estimate
\begin{align*}
|v(s)| &= \big| \int_{s}^{1} v' (t) dt \big|
= \big| \int_{s}^1 v' (t) t^{\frac{N-1}{N} - \frac{N-1}{N} } dt \big|\\
&\le \Big( \int_{0}^1 |v'(t)|^N  t^{N-1} \, dt \Big)^{1/N}
\Big( \log \frac{1}{s} \Big)^{\frac{N-1}{N}},
\end{align*}
we obtain
\[
	\int_0^1 |v(s)|^q s^L ds
\leq \Big( \int_0^1 |v' (s)|^N s^{N-1}  ds \Big)^{q/N}
 \int_0^1 s^L \big( \log \frac{1}{s} \big)^{\frac{N-1}{N} q}  ds.
\]
Note that the last integral is finite when $L > -1$ and $q > 0$.
Therefore, we have
\begin{equation}\label{I 2}
	\int_0^1 |v'(s)|^N s^{N-1} ds \ge C(L, N, q)^{N/q}
\Big( \int_0^1 |v(s)|^q s^L \, ds \Big)^{N/q}.
\end{equation}
Consequently, by \eqref{I 1} and \eqref{I 2}, we obtain
\begin{align*}
I
&\geq \omega_N C(L, N, q)^{N/q} \Big( \int_{0}^1 |v(s)|^q s^L \, ds \Big)^{N/q} \\
&= \omega_N C(L, N, q )^{N/q}
 \Big( \int_{0}^R \frac{|u(r)|^q}{r ( \log \frac{Re}{r} )^{\alpha}} dr \Big)^{N/q}\\
&= \omega_N^{1-\frac{N}{q}} C(L, N, q )^{N/q}
\Big( \int_{B_R(0)} \frac{|u|^q}{|x|^N ( \log \frac{Re}{|x|} )^{\alpha}} dx \Big)^{N/q}.
\end{align*}
where $\alpha = \alpha(q,L) = \frac{N-1}{N} q + L + 2$.
\smallskip

\noindent\textbf{Step 2:}
Let $u^{\#}$ denote the symmetric decreasing rearrangement
(the Schwarz symmetrization) of $u \in C_0^{\infty}(\Omega)$:
\[
	u^{\#}(x) = u^{\#}(|x|) = \inf \{ \lambda > 0 :
\big| \{ x \in \Omega : |u(x)| > \lambda \} \big| \le |B_{|x|}(0)| \},
\]
where $|A|$ denotes the measure of the set $A \subset \mathbb{R}^N$.
Assume $|\Omega|=|B_{\tilde{R}}(0)|$ for some $\tilde{R} >0$.
Note that the function $r \mapsto \frac{1}{r^N (\log \frac{Re}{r})^{\alpha}}$
is monotonically decreasing on $[0, R]$ since $\alpha \leq N$.
Thus by using the symmetrization argument, we obtain
\begin{align*}
\int_{\Omega} |\nabla u|^N dx 
&\geq \int_{B_{\tilde{R}}(0)} |\nabla u^{\#}|^N dx \\
&\geq \big(\frac{N-1}{N}\big)^N \int_{B_{\tilde{R}}(0)}
 \frac{|u^{\#}|^N}{|x|^N (\log \frac{\tilde{R}e}{|x|})^N} dx \\
&\quad	+\omega_N^{1-\frac{N}{q}} C(L, N, q )^{N/q}
\Big( \int_{B_{\tilde{R}}(0)} \frac{|u^{\#}|^q}{|x|^N
( \log \frac{\tilde{R}e}{|x|} )^{\alpha}} dx \Big)^{N/q}
\\
	&\geq \big(\frac{N-1}{N}\big)^N \int_{B_{\tilde{R}}(0)}
\frac{|u^{\#}|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx \\
&\quad	+\omega_N^{1-\frac{N}{q}} C(L, N, q )^{N/q}
\Big(\int_{B_{\tilde{R}}(0)} \frac{|u^{\#}|^q}{|x|^N
( \log \frac{Re}{|x|} )^{\alpha}} dx \Big)^{N/q} \\
&\geq \big(\frac{N-1}{N}\big)^N \int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx\\
&\quad	+\omega_N^{1-\frac{N}{q}} C(L, N, q )^{N/q}
\Big( \int_{\Omega} \frac{|u|^q}{|x|^N ( \log \frac{Re}{|x|} )^{\alpha}} dx \Big)^{N/q}
\end{align*}
where the first inequality comes from the P\'{o}lya-Szeg\"{o} inequality,
the second one comes from Step 1, the third one comes from the fact that
$R \ge \tilde{R}$,
and the last one comes from the Hardy-Littlewood inequality:
$\int_{B_{\tilde{R}}(0)} f^{\#}g^{\#} \ge \int_{\Omega} f g$ for
nonnegative measurable functions $f$ and $g$.
Finally, a density argument assures \eqref{H_N R} holds true for all
$u \in W^{1,N}_0(\Omega)$.
The proof is complete.
\end{proof}


From Proposition \ref{Prop H_N R}, we easily have the following result.

\begin{corollary}[{Adimurthi-Sandeep \cite[Theorem 1.3]{Adimurthi-Sandeep}}]
\label{H_N not attain}
Let $N \geq 2$.
The best constant $\big(\frac{N-1}{N}\big)^N$ in the inequality
\eqref{H_N} is never attained in $W^{1,N}_0(\Omega)$.
\end{corollary}

\section{Optimality of weights}

In this section, we discuss the optimality of the weight function
in the improved critical Hardy inequality \eqref{H_N R}.

\begin{theorem} \label{Optimality}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N \geq 2$,
 $0 \in \Omega$, with $R = \sup_{x \in \Omega} |x|$.
For $0< q < N$, put
\[
	\alpha^* = \big(\frac{N-1}{N}\big)q +1
\]
and define
\begin{align*}
F_N = \Big\{& f:\Omega \to \mathbb{R}^+ :
f \in L^{\infty}_{\rm loc}(\Omega \backslash \{ 0\} )  \text{ and }
\exists \alpha \in (\alpha^*, N]   \text{ s.t. } \\
	& \limsup_{|x| \to 0} f(x) |x|^N \big( \log \frac{Re}{|x|} \big)^{\alpha}
< \infty  \Big\},
\end{align*}
and
\[
G_N = \Big\{ f:\Omega \to \mathbb{R}^+ :
 f \in L^{\infty}_{\rm loc}(\Omega \backslash \{ 0\} )  \text{ and }
	\liminf_{|x| \to 0} f(x) |x|^N ( \log \frac{Re}{|x|} )^{\alpha^*} > 0 \Big\}.
\]
If $f \in F_N$, then there exists $\lambda(f) > 0$ such that the inequality
\begin{equation} \label{f}
	\int_{\Omega} |\nabla u|^N dx \ge \big(\frac{N-1}{N}\big)^N
\int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx
+\lambda (f) ( \int_{\Omega} f(x) |u|^q\, dx )^{N/q}
\end{equation}
holds  for all $u \in W^{1,N}_0(\Omega)$.
If $f \in G_N$, then no inequality of type \eqref{f} can hold.
\end{theorem}

Especially, we cannot replace $\alpha$ in the remainder term of \eqref{H_N R}
 by $\alpha^*$. Also by Theorem \ref{Optimality}, we see
$\int_{\Omega} f(x) |u|^q\, dx < \infty$ for any $u \in W^{1,N}_0(\Omega)$ if $f \in F_N$.

\begin{remark} \label{rmk7} \rm
There exist functions $f$ with $f \notin F_N$ and $f \notin G_N$.
For example, $f_{\gamma}(x) = |x|^{-N} ( \log \frac{Re}{|x|} )^{-\alpha^*}
\big( \log |\log \frac{Re}{|x|}| \big)^{-\gamma}$ for $\gamma > 0$
are such functions.
\end{remark}

To prove Theorem \ref{Optimality}, we follow the argument of the proof
in Adimurthi-Chaudhuri-Ramaswamy \cite[Corollary 1.2]{Adimurthi-Chaudhuri-Ramaswamy}.

\begin{proof}[Proof of Theorem \ref{Optimality}]
If $f \in F_N$, then there exists $\alpha \in (\alpha^*, N]$ such that
\[
	\lim_{\varepsilon \to 0} \sup_{x \in B_{\varepsilon}} f(x) |x|^N
\Big( \log \frac{Re}{|x|} \Big)^{\alpha} < \infty.
\]
Hence for sufficiently small $\varepsilon>0$, there exists a constant $C>0$ such that
\begin{equation*}
	f(x) < \frac{C}{|x|^N ( \log \frac{Re}{|x|} )^{\alpha}} \quad \text{in }
 B_{\varepsilon}(0).
\end{equation*}
Outside of $B_{\varepsilon}$, $f$ is a bounded function and hence $C$ can be chosen
so that this inequality holds in the whole of $\Omega$.
Then, it is easy to check that \eqref{f} follows from the improved critical
 Hardy inequality \eqref{H_N R}.

For the proof of the latter half part of Theorem, let $f \in G_N$.
Then we can find $C >0$, $b > 0$ such that
$f(x) \geq \frac{C}{|x|^N ( \log (Re/|x|) )^{\alpha^*}}$ in
$0 \leq |x| \leq \frac{bRe}{2}$.
We may assume that $B_{bRe}(0) \subset \Omega$ ($\subset B_{R} (0)$).
Let $s < \frac{N-1}{N}$ be a positive parameter
and we define
\begin{align} \label{test}
	u_{s}(x) =
	   \begin{cases}
		( \log \frac{Re}{|x|} )^{s} &\text{if }  0 \leq |x| \leq \frac{bRe}{2} \\
		\text{smooth}   &\text{if }  \frac{bRe}{2} \leq |x| \leq bRe \\
		0  &\text{if }  bRe \leq |x|.
	   \end{cases}
\end{align}
Direct calculations show that
\begin{gather}
	\label{order 1}
\Big( \int_{\Omega} \frac{|u_{s}|^q}{|x|^N ( \log \frac{Re}{|x|} )^{\alpha^*} } dx 
\Big)^{N/q} = \Big( \omega_N \frac{1}{(\frac{N-1}{N} - s )q }
 \big( \log \frac{2}{b} \big)^{(s -\frac{N-1}{N})q} \Big)^{N/q}	+ O(1), \\
\label{order 2}
\int_{\Omega} |\nabla u_{s}|^N dx =\omega_N \frac{-s^N}{(s -1)N +1} 
\big( \log \frac{2}{b} \big)^{(s -1)N +1} + O(1), \\
\label{order 3}
\int_{\Omega} \frac{|u_{s}|^N}{|x|^N ( \log \frac{Re}{|x|} )^N } dx
 =\omega_N \frac{-1}{(s -1)N +1} \big( \log \frac{2}{b} \big)^{(s -1)N +1} + O(1)
\end{gather}
as $s \to \frac{N-1}{N}$.
By \eqref{order 1}, \eqref{order 2}, \eqref{order 3} and $N/q >1$,
we have
\begin{align*}
&\frac{\int_{\Omega} |\nabla u_{s}|^N dx  - \big(\frac{N-1}{N}\big)^N 
\int_{\Omega} \frac{|u_{s}|^N}{|x|^N ( \log \frac{Re}{|x|} )^N } dx}
{( \int_{\Omega} f(x) |u_{s}|^q dx )^{N/q}} \\
&\le \frac{\int_{\Omega} |\nabla u_{s}|^N dx  - \big(\frac{N-1}{N}\big)^N
 \int_{\Omega} \frac{|u_{s}|^N}{|x|^N ( \log \frac{Re}{|x|} )^N } dx}
{C ( \int_{\Omega}  \frac{|u_{s}|^q}{|x|^N (\log \frac{Re}{|x|} )^{\alpha^*}} dx )^{N/q}} \\
	&= C \big( \frac{N-1}{N} - s \big)^{\frac{N}{q} - 1} \to 0
\end{align*}
as $s \to \frac{N-1}{N}$.
Thus the inequality \eqref{H_N R} does not hold for $f$ as above.
\end{proof}


\section{Proof of Theorem \ref{thmEP}}


To prove the Theorem \ref{thmEP}, we need the following lemmas.

\begin{lemma}[{Boccardo-Murat \cite[Thm. 2.1]{Boccardo-Murat}}]\label{BM}
Let $\{ u_m \}_{m=1}^{\infty} \subset W^{1,p}_0(\Omega)$ be such that,
as $m \to \infty$,
$u_m \rightharpoonup u$ weakly in $W^{1,p}_0(\Omega)$ and satisfies
\begin{equation*}
	- \Delta_p u_m  = f_m +g_m \quad \text{in } \mathcal{D}'(\Omega),
\end{equation*}
where $f_m \to 0$ in $W^{-1,p'}_0(\Omega)$ and $g_m$ is bounded 
in $\mathcal{M} (\Omega)$, the space of Radon measures on $\Omega$, i.e.
\begin{equation*}
|\langle g_m, \phi \rangle | \leq C_K \| \phi \|_{\infty}
\end{equation*}
for all $\phi \in \mathcal{D} (\Omega)$ with $\operatorname{supp} \phi \subset K$.
Then there exists a subsequence $u_{m_k}$ such that
\[
	u_{m_k} \to u \quad \text{in } W^{1,\gamma}_0(\Omega ) \quad \forall \gamma < p.
\]
\end{lemma}

\begin{lemma}[Brezis-Lieb \cite{Brezis-Lieb}] \label{BL}
For $p \in (0, +\infty)$,
let $\{ g_m \}_{m=1}^{\infty} \subset L^p(\Omega, \mu)$ be a sequence of functions
on a measurable space $(\Omega, \mu)$ such that
\begin{itemize}
\item[(i)] $\| g_m \|_{L^p(\Omega, \mu)} \le \ ^{\exists} C < \infty$ for all 
$m \in \mathbb{N}$, and

\item[(ii)] $g_m(x) \to g(x)$ $\mu$ a.e. $x \in \Omega$ as $m \to \infty$.
\end{itemize}
Then 
\begin{equation*}
	\lim_{m \to \infty} ( \| g_m \|_{L^p(\Omega, \mu)}^p 
- \| g_m -g \|_{L^p(\Omega, \mu)}^p ) = \| g \|_{L^p(\Omega, \mu)}^p.
\end{equation*}
\end{lemma}

We may apply Lemma \ref{BL} to $\mu(dx) = f(x) dx$, where $f$ is any 
nonnegative $L^1(\Omega)$ function.
Next we have a compactness theorem for the embedding $W^{1,N}_0(\Omega)$ 
into a weighted Lebesgue space 
$L^q(\Omega, f) = \{ u \in L^1_{loc}(\Omega) : \int_{\Omega} |u|^q f(x) dx < \infty \}$.

\begin{lemma}\label{Lemma:Hardy_q}
For any $0<q<N$ and any $\alpha > \alpha^* = \frac{N-1}{N} q + 1$, 
there exists $C > 0$ such that the inequality
\begin{equation}
\label{Hardy_q}
	\int_{\Omega} |\nabla u|^N dx \ge C 
\Big( \int_{\Omega} \frac{|u|^q}{|x|^N ( \log \frac{Re}{|x|} )^{\alpha}} dx \Big)^{N/q}
\end{equation}
holds  for all $u \in W^{1,N}_0(\Omega)$.
Moreover, for 
\[
f_{\alpha}(x) = \frac{1}{|x|^N ( \log (Re/|x|) )^{\alpha}},
\]
the embedding $W^{1,N}_0(\Omega) \hookrightarrow L^q(\Omega, f_{\alpha})$ 
is compact for $1 \le q < N$.
\end{lemma}

Recently,  inequality \eqref{Hardy_q} was proved by Machihara-Ozawa-Wadade \cite{MOW}.
In the following, we provide a simpler proof of \eqref{Hardy_q} 
than the one in \cite{MOW}.

\begin{proof}
By H\"older inequality and the critical Hardy inequality \eqref{H_N}, we have
\begin{align*}
&\int_{\Omega} \frac{|u|^q}{|x|^N (\log \frac{Re}{|x|})^{\alpha}} dx \\
&\le \Big( \int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx \Big)^{q/N}
\Big( \int_{\Omega} \frac{1}{|x|^N (\log \frac{Re}{|x|})^{\frac{N}{N-q}(\alpha - q)}} dx
\Big)^{1 - \frac{q}{N}} \\
&\le \Big( \big(\frac{N-1}{N}\big)^{-N} \int_{\Omega} |\nabla u|^N dx \Big)^{q/N}
\Big( \int_{\Omega} \frac{1}{|x|^N (\log \frac{Re}{|x|})^{\frac{N}{N-q}(\alpha - q)}} dx
 \Big)^{1 - \frac{q}{N}}.
\end{align*}
Since $\alpha > \alpha^* = \frac{N-1}{N} q + 1$, the exponent 
$\frac{N}{N-q}(\beta - q) > 1$, so the last integral is finite.
Thus we have \eqref{Hardy_q}.

For the proof of the latter half part, we follow the argument by 
Chaudhuri-Ramaswamy \cite[Proposition 2.1]{Chaudhuri-Ramaswamy}.
The continuous embedding $W^{1,N}_0(\Omega) \hookrightarrow L^q(\Omega, f_{\alpha})$ 
comes from the inequality \eqref{Hardy_q}.
To prove that this embedding is compact,
let $\{ u_m \}$ be a bounded sequence in $W^{1,N}_0(\Omega)$.
Then we have a subsequence $\{ u_{m_k} \}$ such that
\begin{gather*}
u_{m_k} \rightharpoonup u \quad \text{weakly in }W^{1,N}_0(\Omega ) 
\text{ as }  k \to \infty, \\
u_{m_k} \to u \quad \text{strongly in }L^{\gamma} ( \Omega )  \text{ as }
 k \to \infty \; \forall  1\leq \gamma < \infty .
\end{gather*}
Take $\beta$ such that $\alpha > \beta > \alpha^*$ and note that 
$\lim_{|x| \to 0}|x|^N  ( \log \frac{Re}{|x|} )^{\beta} f_{\alpha}(x) = 0$.
Then for any $\varepsilon > 0$ we can find $\delta > 0$ such that
\[
	\sup_{B_{\delta}(0)} |x|^N  \Big( \log \frac{Re}{|x|} \Big)^{\beta} 
f_{\alpha}(x) \le \varepsilon \quad \text{and} \quad
 \| f_{\alpha} \|_{L^{\infty}(\Omega \backslash B_{\delta}(0))} < \infty.
\]
Thus
\begin{align*}
	\| u_{m_k} - u \|^q_{L^q (\Omega, f_{\alpha})} 
&= \int_{\Omega \setminus B_{\delta}(0)} |u_{m_k} - u|^q f_{\alpha}(x) dx 
 + \int_{B_{\delta}(0)} |u_{m_k} - u|^q f_{\alpha}(x) dx \\
&\leq \| f_{\alpha} \|_{L^{\infty}(\Omega \backslash B_{\delta}(0) ) } 
 \| u_{m_k} - u\|^q_{L^q(\Omega )} + \varepsilon \int_{\Omega} \frac{|u_{m_k} - u|^q}{|x|^N 
 ( \log \frac{Re}{|x|} )^{\beta}} dx \\
&\leq \| f_{\alpha} \|_{L^{\infty}(\Omega \backslash B_{\delta}(0) ) }  
\| u_{m_k} - u\|^q_{L^q(\Omega )} + \varepsilon C \| \nabla ( u_{m_k} - u ) 
\|^q_{L^N (\Omega )} \\
&= o(1) + \varepsilon O(1) \quad \text{as }  k \to \infty,
\end{align*}
here the second inequality comes from \eqref{Hardy_q}.
Finally, letting $\varepsilon \to 0$, we obtain 
$\| u_{m_k} - u \|^q_{L^q (\Omega, f_{\alpha})} \to 0$ 
and the proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thmEP}]
We  use the methods similar to the proof in
\cite[Theorem 1.2]{Adimurthi-Chaudhuri-Ramaswamy}.
We look for a minimizer of the functional
\begin{equation*}
J_{\mu} (u) = \int_{\Omega} |\nabla u|^N \,dx 
- \mu \int_{\Omega} \frac{|u|^N}{( |x| \log \frac{Re}{|x|} )^N} \,dx \quad
\forall u \in W^{1,N}_0(\Omega ) 
\end{equation*}
over the manifold $M=\{ u \in W^{1,N}_0(\Omega) : \int_{\Omega} |u|^q f(x) dx=1 \}$.
Since $f \in F_N$, $M$ is well-defined and non empty by Theorem \ref{Optimality}.
Note that $J_{\mu}$ is continuous, G\v{a}teaux differentiable and 
coercive on $W^{1,N}_0(\Omega)$ for any $\mu \in [0, \big(\frac{N-1}{N}\big)^N )$ 
thanks to the Hardy inequality \eqref{H_N}.
Thus it is clear that $\lambda_{\mu} (f)=\inf_{u \in M} J_{\mu} (u)$ is positive.
Let $\{ u_m \}_{m=1}^{\infty} \subset M$ be a minimizing sequence of 
$\lambda_{\mu} (f)$.
By Ekeland's Variational Principle, we may assume $J'_{\mu} (u_m) \to 0$ 
in $W^{-1,N'}_0(\Omega)$ as $m \to \infty$ without loss of generality.
The coercivity of $J_{\mu}$ implies that $\{ u_m \}_{m=1}^{\infty}$ 
is a bounded sequence in $W^{1,N}_0(\Omega)$,
hence we have a subsequence $\{ u_{m_k} \}_{k=1}^{\infty}$ and 
$u \in W^{1,N}_0(\Omega)$ such that
\begin{gather}
\label{weak 1}
u_{m_k} \rightharpoonup u \quad \text{weakly in }W^{1,N}_0(\Omega ) \text{ as } 
k \to \infty, \\
\label{weak 2}
u_{m_k} \rightharpoonup u \quad \text{weakly in } L^N 
\Big( \Omega, ( |x| \log \frac{Re}{|x|} )^{-N} \Big)  \text{ as }  k \to \infty, \\
\label{strong}
u_{m_k} \to u \quad \text{strongly in } L^{\gamma} ( \Omega )  \text{ as } 
 k \to \infty \; \forall  1\leq \gamma < \infty ), \\
\label{a.e. 1}
u_{m_k} \to u \quad \text{a.e. in } \Omega \text{ as }  k \to \infty
\end{gather}
for some $u \in W^{1,N}_0(\Omega)$.
Note that the second convergence \eqref{weak 2} comes from the 
fact that
\[
\Big( L^N ( \Omega, ( |x| \log \frac{Re}{|x|} )^{-N} ) \Big)^* 
\subset W^{-1,N'} (\Omega) = ( W^{1,N}_0(\Omega) )^*,
\]
which is a consequence of the Hardy inequality \eqref{H_N}, and \eqref{weak 1}.
Recall that for $f \in F_N$, there exist $C > 0$ and $\alpha \in (\alpha^*, N]$ 
such that
\[
	f(x) \le \frac{C}{|x|^N \big( \log \frac{Re}{|x|} \big)^{\alpha}} 
\quad \text{in }  \Omega.
\]
Thus $W^{1,N}_0(\Omega)$ is compactly embedded in $L^q(\Omega, f)$ 
by Lemma \ref{Lemma:Hardy_q}.
Hence $M$ is weakly closed in $W^{1,N}_0(\Omega)$ and $u \in M$.

Furthermore since $\| J'_{\mu} (u_m) \|_{W^{-1,N'} (\Omega)} \to 0$, 
$u_m$ satisfies
\begin{equation*}
	-\Delta_N u_m = \mu \frac{|u_m|^{N-2}u_m}
{ \big( |x| \log \frac{Re}{|x|} \big)^N } + \lambda_m |u_m|^{q-2}u_m f + f_m
\end{equation*}
in $\mathcal{D}' (\Omega )$,
where $f_m \to 0$ in $W^{-1,N'}(\Omega)$ and $\lambda_m \to \lambda$ as $m \to \infty$.
Putting 
\[
g_m=\mu \frac{|u_m|^{N-2}u_m}{ ( |x| \log \frac{Re}{|x|} )^N } 
+ \lambda_m |u_m|^{q-2}u_m f,
\]
one can check that $g_m$ is bounded in $\mathcal{M}(\Omega)$.
Thus we have
\begin{equation}\label{a.e. 2}
	\nabla u_{m_k} \to \nabla u \quad \text{a.e. in } \Omega
\end{equation}
from Lemma \ref{BM}.
By using Lemma \ref{BL}, \eqref{weak 1}, \eqref{weak 2}, \eqref{a.e. 1},
 \eqref{a.e. 2}, and the Hardy inequality \eqref{H_N}, we obtain
\begin{align*}
\lambda_{\mu}(f) 
&= \| \nabla u_{m_k} \|_N^N - \mu \| u_{m_k} 
\|_{L^N ( \Omega, ( |x| \log \frac{Re}{|x|} )^{-N} )}^{N}  + o (1) \\
	&=\| \nabla ( u_{m_k} -u ) \|_N^N - \mu \| u_{m_k} -u 
\|_{L^N ( \Omega, ( |x| \log \frac{Re}{|x|} )^{-N} )}^{N}  \\
&\quad + \| \nabla u \|_N^N
	- \mu \| u \|_{L^N ( \Omega, ( |x| \log \frac{Re}{|x|} )^{-N} )}^{N} + o (1) \\
&\geq \Big( \big(\frac{N-1}{N}\big)^N - \mu \Big) \| u_{m_k} -u 
\|_{L^N ( \Omega, ( |x| \log \frac{Re}{|x|} )^{-N} )}^{N} + \lambda_{\mu} (f) + o(1)
\end{align*}
where $o(1) \to 0$ as $k \to \infty$.
As $\mu < \big(\frac{N-1}{N}\big)^N$, we conclude that
\begin{equation} \label{nabla u_m}
\begin{gathered}
\| u_{m_k} - u \|^N_{L^N ( \Omega, ( |x| \log \frac{Re}{|x|} )^{-N} )} \to 0
 \quad \text{as } k \to \infty,  \\
\| \nabla ( u_{m_k} -u ) \|_N^N \to 0 \quad \text{as } k \to \infty.
\end{gathered}
\end{equation}
Hence we have the strong convergence of $\{ u_{m_k} \}$ which implies
$J_{\mu}(u) = \lambda_{\mu} (f)$ and $\lambda=\lambda_{\mu} (f)$.
Since $J_{\mu}(|u|)=J_{\mu}(u)$ and the strong maximum principle of $\Delta_N$,
we can take $u >0$ in $\Omega$.
Then using Lemma \ref{BM} and \eqref{nabla u_m}, we assure that $u$
is a distributional solution of \eqref{EP} corresponding to $\lambda=\lambda_{\mu} (f)$.
Moreover $u$ is a weak solution of \eqref{EP} from density argument.

Finally, Theorem \ref{Optimality} implies
\[
	\lambda (f) = \inf_{u \in W^{1,N}_0(\Omega) \backslash \{ 0\}} 
\frac{\int_{\Omega} |\nabla u|^N dx - \big(\frac{N-1}{N}\big)^N \int_{\Omega} 
\frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx}{\big(\int_{\Omega} |u|^q f(x) dx \big)^{N/q}} 
 > 0
\]
if $f \in F_N$.
Since it is trivial that $\lambda_{\mu}(f) \to \lambda(f)$ as 
$\mu \nearrow \big(\frac{N-1}{N}\big)^N$, this completes the proof.
\end{proof}

\begin{remark} \label{rmk1.1} \rm
By using the test function $u_s$ defined by \eqref{test},
we check that
\[
	\inf_{u \in W^{1,N}_0(\Omega), u \ne 0} 
\frac{\int_{\Omega} |\nabla u|^N dx}{\big( \int_{\Omega} \frac{|u|^q}{|x|^N
 (\log \frac{Re}{|x|})^{\alpha^*}} dx \big)^{N/q}} = 0.
\]
Thus we cannot replace $\alpha$ in the inequality \eqref{Hardy_q} by $\alpha^*$.
By this reason, if we define the class of weight functions
\begin{align*}
	\mathcal{F}_N =\big\{ f:\Omega \to \mathbb{R}^+ :
 f \in L^{\infty}_{\rm loc} (\Omega \backslash \{ 0\} )  \text{ and } 
	\limsup_{|x| \to 0} f(x) |x|^N \big( \log \frac{Re}{|x|} \big)^{\alpha^*} 
< \infty  \big\},
\end{align*}
then we do not know the solvability of  \eqref{EP} for $f \in \mathcal{F}_N$.
\end{remark}



\section{Appendix}

In this appendix, we prove the following result.

\begin{lemma}
Let $\Omega \subset \mathbb{R}^N$, $N\geq 2$, be a bounded domain.
Then the inequality
\begin{equation}
\label{ASH}
	\int_{\Omega} \big| \frac{x}{|x|} \cdot \nabla u \big|^N dx 
\ge \big(\frac{N-1}{N}\big)^N \int_{\Omega} \frac{|u|^N}{|x|^N (\log \frac{Re}{|x|})^N} dx
\end{equation}
holds for all $u \in W_0^{1,N}(\Omega)$.
\end{lemma}

\begin{proof}
We argue as in \cite{TF}.
It is sufficient to prove \eqref{ASH} for $u \in C_0^{\infty}(\Omega)$.
By the identity
\[
	\operatorname{div} \Big( \frac{x}{|x|^N ( \log \frac{Re}{|x|} )^{N-1}} \Big)
 = \frac{N-1}{|x|^N ( \log \frac{Re}{|x|} )^N},
\]
integration by parts and H\"older's inequality yield 
\begin{align*}
&\int_{\Omega} \frac{|u(x)|^N}{|x|^N ( \log \frac{Re}{|x|} )^N} dx \\
&= \Big| \frac{1}{N-1} \int_{\Omega} |u|^N \operatorname{div} 
\Big( \frac{x}{|x|^N ( \log \frac{Re}{|x|} )^{N-1}} \Big) dx \Big| \\
&= \Big| - \frac{1}{N-1} \int_{\Omega} \nabla (|u|^N ) \cdot \frac{x}{|x|^N 
( \log \frac{Re}{|x|} )^{N-1}} dx \Big| \\
&= \Big| - \frac{N}{N-1} \int_{\Omega} |u|^{N-2} u \nabla u  \cdot 
\frac{x/|x|}{|x|^{N-1} ( \log \frac{Re}{|x|} )^{N-1}} dx \Big| \\
&\le | \frac{N}{N-1} | \Big( \int_{\Omega} \frac{|u|^N}{|x|^N ( \log \frac{Re}{|x|} )^N} 
dx \Big)^{(N-1)/N}
\Big( \int_{\Omega} | \frac{x}{|x|} \cdot \nabla u |^N dx \Big)^{1/N}.
\end{align*}
After some manipulations, we obtain \eqref{ASH}.
\end{proof}

\begin{remark} \label{rmk13} \rm
The same proof as above yields
\emph{the critical Hardy inequality in a sharp form}:
\begin{equation}\label{SH_N}
	\int_{\Omega} \big| \frac{x}{|x|} \cdot \nabla u \big|^N dx 
\ge \big(\frac{N-1}{N}\big)^N \int_{\Omega} \frac{|u|^N}{|x|^N
 (\log \frac{R}{|x|})^N} dx, \quad \forall u \in W_0^{1,N}(\Omega),
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ $(N\geq 2)$.
Note that the weight function $\frac{1}{|x|^N (\log \frac{R}{|x|})^N}$ 
is singular both on the origin and on the boundary.
When $\Omega = B_R(0)$ case, Ioku and Ishiwata \cite{Ioku-Ishiwata} 
showed that the constant $(\frac{N-1}{N})^N$ in the inequality \eqref{SH_N} 
is optimal and never attained in $W^{1,N}_0(B_R(0))$.
Furthermore in \cite{Sano-Takahashi}, the current authors provide a remainder 
term for the inequality \eqref{SH_N} when $\Omega = B_R(0)$.
\end{remark}

\subsection*{Acknowledgments}
This work was supported in part by 
JSPS Grant-in-Aid for Fellows (DC2), No. 16J07472 (MS),
by JSPS Grant-in-Aid for Scientific Research (B), No. 23340038,
and by JSPS Grant-in-Aid for Challenging Exploratory Research, No. 26610030 (FT).

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