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%\def\m{\mathfrak}
\def\sli{\sum\limits}
\def\ili{\int\limits}
%\def\bcl{\bigcup\limits}
%\def\bc{\bigcap}
%\def\px{\mathfrak{P}_{X}}
%\def\py{\mathfrak{P}_{Y}}
%\def\pz{\mathfrak{P}_{Z}}
%\def\g{\gamma}
%\def\de{\delta}
%\def\la{\lambda_1}
%\def\mb{\mathbb}
%\def\lp{L^{p(\cdot)}(\mb{R}^{n})}
%\def\a{\alpha}
\def\R{\mathbb{R}}
%\def\x{x^{\prime}}
%\def\y{y^{\prime}}
%\def\e{\varepsilon}
%\def\p{p^{\star}}
%\def\n{\nabla}
%\def\Om{\Omega}
%\def\vk{\varkappa}
\def\ep{\varepsilon}
\def\le{\leqslant}
\def\ge{\geqslant}
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%\newtheorem{hyp*}{Conjecture}
%\newtheorem{lemma}{Lemma}
%\newtheorem{defin}{Definition}
%\newtheorem{zamech}{Remark}
%\newtheorem*{th*}{Theorem}

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\begin{document}

%\title[Critical Sobolev Trace Embedding Theorem]
\begin{center}
 
 {\bf \large Attainability of infima in the critical Sobolev trace embedding theorem on manifolds}
\medskip

%\author{
Alexander Nazarov, 
%\address{St.-Petersburg State University}
%\email{al.il.nazarov@gmail.com}
%\author{
Alexander Reznikov
%\address{St.-Petersburg State University}
%\email{reznikov@ymail.com}
%\date{February 16, 2009}
%\dedicatory{Dedicated to Nina Nikolaevna Ural'tseva with admiration}
%\thanks
\footnote{This paper was partially supported by grant NSh.227.2008.1.
The first author was also supported by RFBR grant 08-01-00748.}
%\maketitle
\end{center}


\begin{abstract}
Sufficient conditions for the existence of extremal functions in the trace
Sobolev inequality and the trace Sobolev--Poincar\'e inequality on Riemannian
manifolds are established. It is shown that some of these conditions are sharp.
\end{abstract}


\section{Introduction}\label{intr}

Let $n\ge2$, and let $\Omega$ be an $n$-dimensional (${\mathcal C}^{3}$-smooth compact Riemannian)
manifold with ${\mathcal C}^3$-smooth boundary. For $1<p<n$ we denote by $p^\star=\frac {(n-1)p}{n-p}$
the trace Sobolev exponent for $p$ that is the critical exponent for the trace
embedding $W^1_p(\Omega)\hookrightarrow L_q(\partial\Omega)$.

Since the embedding operator $W^1_p(\Omega)\hookrightarrow L_{p^\star}
(\partial\Omega)$ is noncompact, the problem of attainability of the norm of
this operator (i.e. the problem of existence of an extremal function in the
trace embedding theorem) is nontrivial.
Corresponding problem for conventional embedding $W^1_p(\Omega)\hookrightarrow
L_{p^*}(\Omega)$ (here $p^*=\frac {np}{n-p}$ is the Sobolev conjugate of $p$)
was treated in many papers, see, e.g., the recent survey \cite{N} and further
references therein.

The problem for the trace embedding is considerably less investigated.
Let us consider the inequality
\begin{equation}\label{ccc}
K(n,p) \equiv\inf_{v\in \dot{\mathcal C}^{\infty}({\mathbb R}^n_+)
\setminus\{0\}}\frac{\|\nabla v\|_{p,{\mathbb R}^n_+}}
{\|v(\cdot,0)\|_{p^\star,{\mathbb R}^{n-1}}}>0,
\end{equation}
where $\dot{\mathcal C}^{\infty}({\mathbb R}^n_+)$ is the set of functions on
${\mathbb R}^n_+$ with bounded support.

Obviously, the functional in (\ref{ccc}) is invariant with respect to
translations and dilations of $v$. It was proved in the remarkable paper
\cite{Nt} that the infimum in (\ref{ccc}) is attained on the function with
{\it unbounded} support
\begin{equation}\label{cc}
\widetilde{w}_{\ep}(x)=|x-x^{\ep}|^{-\frac{n-p}{p-1}},
\end{equation}
with $x^{\ep}=(0,\dots,0,-\ep)$ (for the case $p=2$ this was established earlier in
\cite{E}). The result of \cite{Nt} implies
$$
K(n,p)=\Bigl(\frac{n-p}{p-1}\Bigr)^{\frac 1{p'}} \Bigl(\frac
{\omega_{n-2}}2\cdot{\mathcal B}\Bigl(\frac{n-1}{2},
\frac{n-1}{2(p-1)}\Bigr)\Bigr)^{\frac 1{(n-1)p'}}.
$$

In the paper \cite{NR} authors considered the critical trace embedding in bounded domain
$\Omega\Subset\mathbb R^n$, i.e. the inequality
$$\lambda_1(n,p,\Omega)=\inf_{v\in W^1_p({\Omega})\setminus\{0\}}\frac
{\|v\|_{W^1_p(\Omega)}}{\|v\|_{p^\star,\partial\Omega}}>0
\leqno(I)
$$
(the norm of the numerator is defined as $\|v\|^p_{W^1_p(\Omega)}= \|\nabla
v\|^p_{p,\Omega}+\|v\|^p_{p,\Omega}$). In \cite[Remark 4]{NR} it is noted that
the main result holds true for an arbitrary manifold $\Omega$ with a smooth
boundary, provided $\partial\Omega$ contains a point with positive mean
curvature (with respect to the inner normal). Namely, in this case the infimum in
{\rm(I)} is attained for $1<p<\frac{n+1}{2}+\beta$, where $\beta>0$ depends on
$\Omega$.\medskip

This paper deals with more complicated case where the mean curvature is non-positive.
Our first result reads as follows.

\begin{theorem}\label{Th1}
Let $n\ge5$, and let $\Omega$ be an $n$-dimensional manifold with ${\mathcal C}^3$-smooth 
boundary. Suppose that the mean curvature of $\partial\Omega$ with respect to the inner normal is non-positive everywhere. Finally, suppose that there exists a point $y^0\in\partial\Omega$ such that $\partial\Omega$ is a totally geodesic submanifold in $\Omega$ at $y^0$
and the scalar curvature of $\Omega$ is positive at $y^0$. Then for some $\beta>0$, for
$2<p<\frac{n+2}{3}+\beta$, the infimum in {\rm(I)} is attained.
\end{theorem}

\begin{zamech}
By standard argument it follows that under suitable normalization the
extremal function in {\rm(I)}, if it exists, is a positive solution to the
nonlinear Neumann problem
$$
-\Delta_pu+u^{p-1}=0 \quad \mbox {in}\ \
\Omega;\qquad |\nabla u|^{p-2}\frac {\partial u}{\partial {\bf n}}=
u^{p^\star-1}\quad
\mbox{on}\ \ \partial\Omega
$$
(here $\Delta_pu=\textup {div}(|\nabla u|^{p-2}\nabla u)$).
\end{zamech}

Observe that, in contrast to \cite{NR}, the assumptions of Theorem 1 contain non-trivial 
left border of the interval for $p$. We show that this left border is sharp.

\begin{theorem}\label{Th2}
Let $n\ge2$, and let $\Omega$ be an $n$-dimensional hemisphere
\begin{equation*}\label{ff}
{\Omega} =\{(\theta,\phi_1,\dots,\phi_{n-1})\in {\mathbb S}^n:\
0<\theta<\frac{\pi}{2}\}.
\end{equation*}
 Then for any $1<p<2$,
there exists $\varkappa^*>0$ such that for
$\varkappa > \varkappa^{*}$ the infimum in {\rm(I)} is not
attained on $\varkappa\Omega$. For $n\ge3$ this is true also for $p=2$.
\end{theorem}

\begin{zamech} It is easy to see that the non-attainability of the infimum in {\rm(I)} on
 $\varkappa\Omega$ for large values of $\varkappa$ is equivalent to the validity of the 
{\rm ``optimal trace Sobolev inequality''}
 $$
\|v\|_{p^\star,\partial\Omega}^p\le K^p(n,p)\cdot \|\nabla v\|^p_{p,\Omega}
+C(p,\Omega)\cdot\|v\|^p_{p,\Omega},\qquad v\in W^1_p(\Omega). 
$$
For the conventional optimal Sobolev inequality see \cite{Dr1} and references therein.
\end{zamech}

\begin{zamech}
Note that the fact that $\Omega$ is a hemisphere is used only for symmetrization arguments. Using the techniques of \cite{Dr1}, one can obtain the estimate (\ref{hryap1}) and prove Theorem 2 in general
case under assumption that $\partial\Omega$ is a totally geodesic submanifold in $\Omega$.
 \end{zamech}

Finally, we consider the trace Sobolev--Poincar\'e inequality
$$\lambda_2(n,p,\Omega)=\inf_{v\in W^1_p({\Omega})\setminus\{c\}}\frac
{\|\nabla v\|_{p,\Omega}}{\|v-\overline{v}\|_{p^\star,\partial\Omega}}>0
\leqno(II)
$$
(here we use the notation
$\overline{v}=|\partial\Omega|^{-1}\int\limits_{\partial\Omega}\!v\,d\Sigma$).

\begin{theorem}\label{Th3} Let $n\ge4$, and let $\Omega$ be as in Theorem \ref{Th1}.
Then for some $\beta(\Omega)>0$ and for $1<p<\frac{n+2}{3}+\beta$, the infimum in
{\rm(II)} is attained.
\end{theorem}

The structure of our paper is as follows. In \S\ref{T1} we establish
required integral estimates and prove Theorem \ref{Th1}. Theorem \ref{Th2} is proved
in \S\ref{T2}; inequality (II) is considered in \S\ref{poinc}.\medskip

We are grateful to Prof. S.V. Ivanov for important advise.\medskip

Let us recall some notations.  $x=(x_1,\dots, x_{n-1}, x_n)=(x',x_n)$ is a point in $\mathbb R^n$.
Put $|x'|=\sqrt{x_{1}^{2}+\ldots + x_{n-1}^{2}}$ and $Q_{\rho}=\{x: |x'|<\rho, 0<x_n<\rho\}$. 
By $y$ we denote points in $\Omega$.

Consider the exponential map at $y^0\in\partial\Omega$ and recall (see, e.g., \cite[\S11]{BZ}) 
that there exists $\rho_0>0$ such that $\exp^{-1}_{y^0}$ is a diffeomorphism 
from the Riemannian ball $B_{2\rho_0}(y^0)\cap \Omega$ onto the set $B_{2\rho_0}(0)\cap \{x_{n}>F(x')\}\subset\mathbb R^n$. Moreover, one can suppose $F(0)=\nabla' F(0)=0$.
In what follows we always assume $\rho<\rho_0$.
For a function $f$ on $B_{2\rho_0}(y^0)\cap\Omega$ we 
define the ``transplanted'' function $\widetilde{f}(x)\equiv f(\exp_{y^0}(x))$.

We denote by $G=(g_{ij}(x))$ the Riemannian metric tensor on $\Omega$ and by $(g^{ij}(x))$ 
the inverse tensor. It is well known that for $f:\Omega \rightarrow \R$ the Riemannian length of 
$\nabla f$ is given by 
$|\nabla f|^{2}=\sum\limits_{ij} g^{ij}\widetilde{f}_{x_{i}}\widetilde{f}_{x_{j}}$.
%We shall also write $|\nabla \widetilde{f}|$ instead of
%$\sqrt{\sli_{i}\left(\frac{\partial \widetilde{f}}{\partial x_{i}}\right)^{2}}$.
Also $dy=\sqrt{\det (G)}dx$.

$\omega_{n-1}=\frac {2\pi^{n/2}} {\Gamma(\frac n2)}$ stands for the area of the unit sphere in 
${\mathbb R}^n$. We denote by $p'=\frac p{p-1}$ the H\"{o}lder conjugate exponent to $p$, and put $q=\frac{n-1}{2}p'$. $\Gamma$ is the Euler gamma-function, $\mathcal{B}$ is the Euler beta-function. 

We denote by $o_{\rho}(1)$ a quantity which tends to zero as $\rho\to0$.
We use letter $C$ to denote various positive constants. To indicate
that $C$ depends on some parameters, we write $C(\dots)$.


\section{Existence of the minimizer}\label{T1}

\begin{proof}[Proof of Theorem \ref{Th1}] 
Our main tool is the concentration-compactness principle of Lions (\cite{Ls}).
It is used in various forms; for the problem (I) it can be reformulated as follows (the proof is
verbatim repetition of \cite[Proposition 1]{NR}).

\begin{utv}\label{prop1} Let the infimum in {\rm(I)} satisfy the inequality
$$
\lambda_1(n,p,\Omega)<K(n,p).
$$
Then the infimum is attained.
\end{utv}

Thus, to prove the attainability of the infimum in (I) it is sufficient to
present a function such that the quotient in (I) is less than $K(n,p)$.
Following \cite{DN}, see also \cite{LPT}, we succeed, constructing a function
with a small support, simulating the behavior of $\widetilde w_{\ep}$ in Riemannian 
normal coordinates. Namely, for sufficiently small $\ep>0$ and $\rho>0$ we introduce the
function $u(y)$ such that
\begin{equation}
\widetilde{u}(x)=\widetilde{\varphi}(|x'| ,x_n) \widetilde{w}_{\ep}(x). \label{escobar-fun-def}
\end{equation}
Here $\widetilde{w}_{\ep}$ is defined in \eqref{cc}, while $\widetilde{\varphi}$ is a
smooth cut-off function such that
%\begin{equation}
$$
\widetilde{\varphi}=1\quad \mbox{in}\quad Q_{\frac{\rho}{2}};\qquad
\widetilde{\varphi}=0\quad \mbox{in}\quad \mathbb R^n\setminus Q_\rho,
\label{trunc-def}
$$%\end{equation}
and $|\nabla \widetilde\varphi|\le \frac{C}{\rho}$.

\subsection{Auxiliary relations}\label{aux}

It is well known (see, e.g., \cite[\S12.6]{BZ}) that 
\begin{equation}\label{partial0}
g_{ij}(0)=\delta_{ij}, \; \frac{\partial g_{ij}}{\partial x_{k}}(0)=0.
\end{equation}
Further, from \cite[\S\S 14.5 and 19.5]{BZ} we conclude that for $i\ne j$
\begin{equation}\label{partial1}
\frac{\partial^{2} g_{ii}}{\partial x_{j}^{2}}(0)=-\frac{{\mathcal R}_{ij}}{3};\qquad
\frac{\partial^{2} g_{ij}}{\partial x_{i}\partial x_{j}}(0)=\frac{2{\mathcal R}_{ij}}{3}.
\end{equation}
%and from \cite[pages 45, 93]{BZ} and from \eqref{partial1} we have for $i\not= j$
%\begin{equation}\label{partial2}
%\frac{\partial^{2} g_{ij}}{\partial x_{i}\partial x_{j}}(0)=\frac{2}{3}K_{\sigma}(e_{i}, e_{j}).
%\end{equation}
Here ${\mathcal R}_{ij}=K_{\sigma}(e_{i}, e_{j})$ is a sectional curvature of $\Omega$ at $y^0$. 
We write ${\mathcal R}_{ii}=0$ so the formulas (\ref{partial1}) hold true even for $i=j$.

Also we denote by
$${\mathcal R}_i=\sum\limits_j{\mathcal R}_{ij}=Ric (e_i)
$$
a Ricci curvature at $y^0$ and introduce the notation  
$${\mathcal R}'_i=\sum\limits_{j<n}{\mathcal R}_{ij},\qquad i<n.
$$

Next, consider the function $F$ giving the local representation of $\partial\Omega$ at $y^0$. Since 
the boundary is ${\mathcal C}^{3}$-smooth, we have 
\begin{equation}\label{F}
F(x')=\sum\limits_{i,j<n}a_{ij}x_ix_i+\sum\limits_{i,j,k<n} b_{ijk}x_{i}x_{j}x_{k}+o(|x'|^{3})
\quad \mbox{as} \quad x'\to0.
\end{equation}
Moreover, the assumption on $\partial\Omega$ to be totally geodesic at $y^0$ means that $a_{ij}=0$.


\subsection {Estimate of $\|\nabla u\|_{p,\Omega}$ for $1<p\le\frac{n+2}{3}$}\label{est1}
%\label{numerator}

We apply the estimate
\begin{equation}\label{gradient}
|\nabla (fg)|^p\le|f\,\nabla g|^p+C\,\left(|\nabla f\,g|^p+|\nabla
f\,g| \cdot|f\,\nabla g|^{p-1}\right)
\end{equation}
to $\nabla u$. Since $\nabla \widetilde\varphi$ does not vanish only in
$Q_\rho\setminus Q_{\frac\rho2}$, one obtains
\begin{multline}\label{eq-sect1-0}
\int\limits_{\Omega}\left(|\nabla\varphi\cdot w_{\ep}|^p+ |\nabla
\varphi\cdot w_{\ep}|\cdot |\varphi\,\nabla
w_{\ep}|^{p-1}\right)\,dy
\le\\
\le C \bigg(\int\limits_0^{\rho/2}\int\limits_{\rho/2}^\rho+
\int\limits_{\rho/2}^\rho\int\limits_0^\rho\bigg)\ \frac
{\rho^{-p}r^{n-2}\,dx_ndr}
{(r^{2}+x_n^2)^{\frac{p(n-p)}{2(p-1)}}}\cdot\Big[1+\Big(\frac {\rho}
{\sqrt{r^{2}+x_n^2}}\Big)^{p-1}\Big]\le C\,\rho^{n-2q}.
\end{multline}
The inequalities \eqref{gradient} and \eqref{eq-sect1-0} imply
\begin{equation}\label{qq}
\ili_{\Omega}|\nabla u|^{p}dy\le\! \ili_{\Omega\cap \exp_{y^0}(Q_{\rho})}\!\!
|\nabla w_{\ep}|^{p}dy+C\rho^{n-2q}.
\end{equation}


We have
$$\ili_{\Omega\cap \exp_{y^0}(Q_{\rho})}\!\!|\nabla w_{\ep}|^{p}dy=
\ili_{|x'|<\rho}\ili^{\rho}_{F(x')}
\bigg(\sum\limits_{ij} g^{ij}
\frac{\partial\widetilde{w_{\ep}}}{\partial x_{i}}
\frac{\partial\widetilde{w_{\ep}}}{\partial x_{j}}\bigg)
^{\frac p2}\cdot\sqrt{\det(G)}\,dx_ndx'.
$$
Expanding the metric tensor by the Taylor formula and using \eqref{partial0}, we obtain 
%$$
%g_{ij}=\delta_{ij}+\frac{1}{2}\sum_{k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial %x_{l}}(0)x_{k}x_{l}+o(|x|^{2}),
%$$
%so
%$$
%g^{ij}=\delta_{ij}-\frac{1}{2}\sum_{k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial %x_{l}}(0)x_{k}x_{l}+o(|x|^{2}).
%$$
%Now from the rule of derivation of determinants we conclude
%$$
%\frac{\partial \det (G)}{\partial x_{i}}(0)=0,
%$$
%$$
%\frac{\partial^{2} \det (G)}{\partial x_{i}x_{j}}(0)=\sli_{k, i, j} \frac{\partial^{2} g_{kk}}{\partial %x_{i}\partial x_{j}}(0)
%$$
%so
%$$
%\sqrt{\det (G)}=1+\frac{1}{4}\sli_{k, i, j} \frac{\partial^{2} g_{kk}}{\partial x_{i}\partial x_{j}}(0)x_{i}x_{j}+o(|x|^{2}).
%$$
%Above formulas imply
%\begin{multline}
%$$
%|\nabla w_{\ep}|^{p}=\left( \sli_{i,j} %\left(\delta_{ij}-\frac{1}{2}\sli_{k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}(0)x_{k}x_{l}+o(|x|^{2})\right)\frac{\partial \widetilde{w}_{\ep}}{\partial x_{i}}\frac{\partial %\widetilde{w}_{\ep}}{\partial x_{j}}\right)^{\frac{p}{2}}=\\=\left(|\nabla \widetilde{w}_{\ep}|^{2}-\left(\frac{1}{2}\sli_{i,j,k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}(0)x_{k}x_{l}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{i}}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{j}}\right)(1+o(1))\right)^{\frac{p}{2}}=\\=|\nabla \widetilde{w}_{\ep}|^{p}\left(1-\left(\frac{1}{2} \sli_{i,j,k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}(0)x_{k}x_{l}\frac{\frac{\partial \widetilde{w}_{\ep}}{\partial x_{i}}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{j}}}{|\nabla w_{\ep}|^{2}}\right)(1+o(1))\right)^{\frac{p}{2}} = \\=|\nabla \widetilde{w}_{\ep}|^{p}-\left(\frac{p}{4}\sli_{i,j,k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}x_{k}x_{l}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{i}}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{j}}|\nabla \widetilde{w}_{\ep}|^{p-2}\right)(1+o_{\rho}(1)),
%$$
%\end{multline}
%so
%\begin{multline}
%$$
%|\nabla w_{\ep}|_{g}^{p}dy=|\nabla \widetilde{w}_{\ep}|^{p}dx-\frac{p}{4}\left(\sli_{i,j,k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}x_{k}x_{l}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{i}}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{j}}|\nabla \widetilde{w}|^{p-2} \right)(1+o_{\rho}(1))\cdot\\ \cdot \left(1+\left(\frac{1}{4}\sli_{k, i, j} \frac{\partial^{2} g_{kk}}{\partial x_{i}\partial x_{j}}(0)x_{i}x_{j}\right)(1+o_{\rho}(1))\right)dx=\\=|\nabla \widetilde{w}_{\ep}|^{p}+\left(-\frac{p}{4}\sli_{i,j,k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}x_{k}x_{l}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{i}}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{j}}|\nabla \widetilde{w}_{\ep}|^{p-2}+\frac{1}{4}\sli_{k, i, j} \frac{\partial^{2} g_{kk}}{\partial x_{i}\partial x_{j}}(0)x_{i}x_{j}|\nabla \widetilde{w}_{\ep}|^{p}\right)(1+o_{\rho}(1))dx.
%$$
%\end{multline}
%Now we shall count
\begin{multline}\label{int1}
\ili_{\Omega\cap \exp_{y^0}(Q_{\rho})}\!|\nabla w_{\ep}|^pdy =\\
={\mathcal I}_1+
\frac 14 \sli_{ijk} \frac{\partial^{2} g_{kk}}{\partial x_{i}\partial x_{j}}(0)
\cdot{\mathcal I}^{(ij)}_2
-\frac p4 \sli_{ijkl}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}(0)
\cdot{\mathcal I}^{(ijkl)}_3
+{\mathcal I}_4-{\mathcal I}_5,
\end{multline}
where
$${\mathcal I}_1= \ili_{|x'|<\rho}\ili_{0}^{\rho}|\nabla\widetilde{w}_{\ep}|^{p}dx_ndx';$$
$${\mathcal I}^{(ij)}_2=\ili_{|x'|<\rho}\ili_{0}^{\rho}
x_{i}x_{j}|\nabla \widetilde{w}_{\ep}|^{p}dx_ndx';
$$
$${\mathcal I}^{(ijkl)}_3=\ili_{|x'|<\rho}\ili_{0}^{\rho}
x_{k}x_{l}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{i}}
\frac{\partial \widetilde{w}_{\ep}}{\partial x_{j}}|\nabla \widetilde{w}_{\ep}|^{p-2}dx_ndx';
$$
$${\mathcal I}_4=\ili_{|x'|<\rho}\ili_{0}^{\rho}|x|^2|\nabla \widetilde{w}_{\ep}|^{p}dx_ndx'
\cdot o_\rho(1);
$$
$${\mathcal I}_5=\ili_{|x'|<\rho}\ili_{0}^{F(x')}
|\nabla\widetilde{w}_{\ep}|^{p}\cdot\big(1+o(\rho^2)\big)\,dx_ndx'.
$$

It is easy to see that 
\begin{equation}\label{I1}
 {\mathcal I}_1= \Big(\frac{n-p}{p-1}\Big)^{p}\cdot\big(\ep^{n-2q}{\mathcal E}_1-O(\rho^{n-2q})\big),
\end{equation}
where
\begin{equation}\label{E_1}
{\mathcal E}_1=\omega_{n-2}\ili_{0}^{\infty}\ili_{0}^{\infty}
\frac{r^{n-2}drdx_{n}}{(r^{2}+(x_{n}+1)^{2})^{q}}=
\frac{\omega_{n-2}}{2(2q-n)}\cdot {\mathcal B}\big({\textstyle\frac{n-1}{2},\frac{n-1}{2(p-1)}}\big).
\end{equation}

 In a similar way we get
\begin{equation}\label{vich2-1}
{\mathcal I}^{(ij)}_2=
\Big(\frac{n-p}{p-1}\Big)^{p}\delta_{ij}\cdot\left\{\begin{array}{rclc}
\frac{\ep^{n-2q+2}}{2q-n-2_{\textstyle\vphantom{1}}}\cdot{\mathcal E}_{2}^{(j)}&-&O(\rho^{n-2q+2}),
&\ \ p<\frac {n+2}3;\\
\ln(\frac{1}{\ep})\cdot{\mathcal E}_{2}^{(j)}&-& O(\ln(\frac{1}{\rho})),&\ \ p=\frac {n+2}3,
\end{array}\right.
\end{equation}
where
\begin{equation}%\label{E2}
{\mathcal E}_{2}^{(j)}=\frac{\omega_{n-2}}{2q-n-1}\cdot {\mathcal B}
\big({\textstyle\frac{n-1}{2},\frac{n-1}{2(p-1)}}\big)\cdot\left\{
\begin{array}{cc} \frac 12, &j<n; \\ \frac{1^{\textstyle\vphantom1}}{2q-n}, &j=n. 
\end{array}\right.
\end{equation}



To calculate ${\mathcal I}_3$, we consider three cases.\medskip

{\bf 1}. $i<n$, $j<n$. Then
\begin{multline}
{\mathcal I}^{(ijkl)}_3=\Big(\frac{n-p}{p-1}\Big)^{p}\ili_{|x'|<\rho}\ili_{0}^{\rho}
\frac{x_{i}x_{j}x_{k}x_{l}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\\
=(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}){\mathcal F}^{(ij)}
+\delta_{ij}\delta_{kl}{\mathcal F}^{(ik)},
\end{multline}
where, for $i\ne k$,
\begin{multline}
{\mathcal F}^{(ik)}=\Big(\frac{n-p}{p-1}\Big)^{p}\ili_{|x'|<\rho}\ili_{0}^{\rho}
\frac{x_{i}^{2}x_{k}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\\
=\Big(\frac{n-p}{p-1}\Big)^{p}\cdot\frac 1{2q}\cdot\left\{\begin{array}{rclc}
\frac{\ep^{n-2q+2}}{2q-n-2_{\textstyle\vphantom{1}}}\cdot{\mathcal E}_{2}^{(k)}&+&O(\rho^{n-2q+2}),
&\ \ p<\frac {n+2}3;\\
\ln(\frac{1}{\ep})\cdot{\mathcal E}_{2}^{(k)}&+& O(\ln(\frac{1}{\rho})),&\ \ p=\frac {n+2}3
\end{array}\right.
\end{multline}
(note that the case $i=k$ is not interesting for us due to ${\mathcal R}_{ii}=0$).\smallskip

%Now if $i\not=k$ and $i,k<n$ then
%\begin{multline}\label{vich7}
%\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}x_{k}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx
%\le\\ \le  
%\frac{\ep^{n-2q+2}}{(n-1)(n-2)}\left(\ili_{\R^{n-1}}\ili_{\R_{+}}\frac{|x'|^{4}dx}{(|x'|^{2}+(x_{n}+1)^{2})^{q+1}}-\sli_{i}\ili_{\R^{n-1}}\ili_{R_{+}}\frac{x_{i}^{4}dx}{(|x'|^{2}+(x_{n}+1)^{2})^{q+1}}\right)\le\\ 
%\le \ep^{n-2q+2}\left(\frac{1}{(n-1)(n-2)}\ili_{0}^{\infty}\ili_{0}^{\infty}\frac{r^{4}r^{n-2}\omega_{n-2}drdx_{n}}{(r^{2}+(x_{n}+1)^{2})^{q+1}}-\frac{1}{n-2}\ili_{-\infty}^{\infty}\ili_{0}^{\infty}\frac{s^{4}r^{n-3}\omega_{n-3}drdsdx_{n}}{(r^{2}+s^{2}+(x_{n}+1)^{2})^{q+1}}\right)=\\
%= \frac{\omega_{n-2}}{8}\frac{\Gamma(\frac{n-1}{2})\Gamma(\frac{2q-n-1}{2})}{\Gamma(q+1)}.
%\end{multline}

%\begin{multline}\label{E_10}
%\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}x_{n}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx
%=\ep^{n-2q+2}\frac{\omega_{n-2}}{2(2q-n)(2q-n-1)}
%\frac{\Gamma(\frac{n-1}{2})\Gamma(\frac{2q-n+1}{2})}{\Gamma(q+1)}
%\end{multline}

{\bf 2}. $i<n$, $j=n$. Then
\begin{equation}
{\mathcal I}^{(ijkl)}_3=\Big(\frac{n-p}{p-1}\Big)^{p}\ili_{|x'|<\rho}\ili_{0}^{\rho}
\frac{x_{i}(x_{n}+\ep)x_{k}x_{l}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx%=\\
=(\delta_{ik}\delta_{ln}+\delta_{il}\delta_{kn}){\mathcal F}^{(i)}_1,
\end{equation}
where
\begin{multline}%\label{vich8}
{\mathcal F}^{(i)}_1=\Big(\frac{n-p}{p-1}\Big)^{p}\ili_{|x'|<\rho}\ili_{0}^{\rho}
\frac{x_{i}^{2}(x_{n}+\ep)x_{n}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\\ 
%\le\ep^{n-2q+2}\frac{1}{n-1}\ili_{0}^{\infty}\ili_{0}^{\infty}
%\frac{r^{2}(x_{n}+1)x_{n}r^{n-2}\omega_{n-2}drdx_{n}}{(r^{2}+(x_{n}+\ep)^{2})^{q+1}}drdx_{n}=\\
%=\ep^{2q-n+2}\frac{\omega_{n-2}}{4(2q-n-1)(2q-n-2)}
%\frac{\Gamma(\frac{n-1}{2})\Gamma(\frac{2q-n+1}{2})}{\Gamma(q+1)},
=\Big(\frac{n-p}{p-1}\Big)^{p}\cdot\frac 1{2q}\cdot\left\{\begin{array}{rclc}
\frac{\ep^{n-2q+2}}{2q-n-2_{\textstyle\vphantom{1}}}\cdot{\mathcal E}_{2}^{(i)}&+&O(\rho^{n-2q+2}),
&\ \ p<\frac {n+2}3;\\
\ln(\frac{1}{\ep})\cdot{\mathcal E}_{2}^{(i)}&+& O(\ln(\frac{1}{\rho})),&\ \ p=\frac {n+2}3.
\end{array}\right.
\end{multline}
Obviously, similar result is true for $i=n$, $j<n$.\smallskip 

{\bf 2}. $i=j=n$. Then
\begin{equation}%\label{vich9}
{\mathcal I}^{(ijkl)}_3=\Big(\frac{n-p}{p-1}\Big)^{p}\ili_{|x'|<\rho}\ili_{0}^{\rho}
\frac{(x_{n}+\ep)^{2}x_{k}x_{l}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\delta_{kl}{\mathcal F}^{(k)}_2,
%\ge\ep^{n-2q+2}\frac{1}{n-1}\ili_{0}^{\infty}\ili_{0}^{\infty}
%\frac{r^{2}(x_{n}+1)^{2}r^{n-2}drdx_{n}}{(r^{2}+(x_{n}+1)^{2})^{q+1}}-C(\rho)=\\=
\end{equation}
where, for $k\ne n$,
\begin{multline}
{\mathcal F}^{(k)}_2=\Big(\frac{n-p}{p-1}\Big)^{p}\ili_{|x'|<\rho}\ili_{0}^{\rho}
\frac{x_{k}^2(x_{n}+\ep)^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\\
=\Big(\frac{n-p}{p-1}\Big)^{p}\cdot\frac {2q-n-1}{2q}\cdot\left\{\begin{array}{rclc}
\frac{\ep^{n-2q+2}}{2q-n-2_{\textstyle\vphantom{1}}}\cdot{\mathcal E}_{2}^{(k)}&+&O(\rho^{n-2q+2}),
&\ \ p<\frac {n+2}3;\\
\ln(\frac{1}{\ep})\cdot{\mathcal E}_{2}^{(k)}&+& O(\ln(\frac{1}{\rho})),&\ \ p=\frac {n+2}3.
\end{array}\right.
\end{multline}

Further, it is obvious that
\begin{equation}{\mathcal I}_4={\mathcal I}^{(nn)}_2\cdot o_\rho(1).
\end{equation}
Finally, the last term in (\ref{int1}) can be estimated using (\ref{F}):
\begin{multline}\label{E_10}
{\mathcal I}_5
%=\left(\frac{n-p}{p-1}\right)^{p}\ili_{|x'|<\rho}\frac{F(x')dx'}
%{(|x'|^{2}+\ep^{2})^{q}}(1+o_{\rho}(1))=\\
=\Big(\frac{n-p}{p-1}\Big)^{p}\ili_{|x'|<\rho}\sli_{i,j,k<n}\frac{b_{ijk}x_{i}x_{j}x_{k}}
{(|x'|^{2}+\ep^{2})^{q}}\,dx'+\\
+\ili_{|x'|<\rho}\frac{|x'|^{3}\,dx'}{(|x'|^{2}+\ep^{2})^{q}}\cdot o_{\rho}(1)=
{\mathcal I}^{(nn)}_2\cdot o_\rho(1).
\end{multline}

%It is also obvious that
%$$
%\ili_{|x'|<\rho}\ili_{0}^{F(x')}\frac{p}{4}\sli_{i,j,k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}x_{k}x_{l}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{i}}\frac{\partial \widetilde{w}_{\ep}}{\partial x_{j}}|\nabla \widetilde{w}_{\ep}|^{p-2}+\frac{1}{4}\sli_{k, i, j} \frac{\partial^{2} g_{kk}}{\partial x_{i}\partial x_{j}}(0)x_{i}x_{j}|\nabla \widetilde{w}_{\ep}|^{p}dx=o(\ep^{n-2q+2}).
%$$

%\begin{multline}\label{vich3-1}
%\Big(\frac{n-p}{p-1}\Big)^{-p}\ili_{|x'|<\rho}\ili_{0}^{\rho}\sli_{i,j,k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}(0) x_{k}x_{l}\frac{\partial \widetilde{w_{\ep}}}{\partial x_{i}}\frac{\partial
%\widetilde{w}_{\ep}}{\partial x_{j}}|\nabla \widetilde{w}_{\ep}|^{p-2}dx=\\= \sli_{i,j<n}\sli_{k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}(0)\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}x_{j}x_{k}x_{l}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx + \\+ 2\sli_{i<n}\sli_{k,l}\frac{\partial^{2}g_{in}}{\partial x_{k}\partial x_{l}}(0)\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}(x_{n}+\ep)x_{k}x_{l}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx+\\+ \sli_{k,l}\frac{\partial^{2}g_{nn}}{\partial x_{k}\partial x_{l}}(0)\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{(x_{n}+\ep)^{2}x_{k}x_{l}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx.
%\end{multline}
%We shall simplify each item of above formula. We will use the simple observation. Since $Q_{\rho}$ is symmetrical with respect to $x_{i}$ for $i<n$ so if for some $i<n$ integral contains $x_{i}$ or $x_{i}^{3}$ then this integral vanishes. So,
%\begin{multline}\label{vich4}
%$$
%\sli_{i,j<n}\sli_{k,l}\frac{\partial^{2}g_{ij}}{\partial x_{k}\partial x_{l}}(0)\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}x_{j}x_{k}x_{l}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\\=
%\sli_{i<n}\sli_{k}\frac{\partial^{2}g_{ii}}{\partial x_{k}^{2}}(0)\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}x_{k}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx+
%2\sli_{i<n}\sli_{j<n}\frac{\partial^{2}g_{ij}}{\partial x_{i}\partial x_{j}}(0)\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}x_{j}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\\=
%\sli_{i<n, k<n}-\frac{1}{3}K_{\sigma}(e_{i}, e_{k})\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}x_{k}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx + \sli_{i<n}-\frac{1}{3}K_{\sigma}(e_{i}, e_{n})\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}x_{n}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx+\\+2\sli_{i<n, k<n}\frac{2}{3}K_{\sigma}(e_{i}, e_{k})\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}x_{k}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\\=\sli_{i,k<n}K_{\sigma}(e_{i}, e_{k})\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}x_{k}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx - \frac{1}{3}\sli_{i<n}K_{\sigma}(e_{i}, e_{n})\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}x_{n}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx,
%$$
%\end{multline}
%\begin{multline}\label{vich5}
%$$
%\sli_{i<n}\sli_{k,l}\frac{\partial^{2}g_{in}}{\partial x_{k}\partial x_{l}}(0)\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}(x_{n}+\ep)x_{k}x_{l}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=2\sli_{i<n}\frac{\partial^{2}g_{in}}{\partial x_{i}\partial x_{n}}(0)\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}(x_{n}+\ep)x_{n}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\\=\frac{4}{3}\sli_{i<n}K_{\sigma}(e_{i}, e_{n})\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{x_{i}^{2}(x_{n}+\ep)x_{n}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx,
%$$
%\end{multline}
%\begin{multline}\label{vich6}
%$$
%\sli_{k,l}\frac{\partial^{2}g_{nn}}{\partial x_{k}\partial x_{l}}(0)\ili_{Q_{\rho}}\frac{(x_{n}+\ep)^{2}x_{k}x_{l}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\sli_{k}\frac{\partial^{2}g_{nn}}{\partial x_{i}^{2}}(0)\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{(x_{n}+\ep)^{2}x_{k}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx=\\=-\frac{1}{3}\sli_{i<n}K_{\sigma}(e_{i}, e_{n})\ili_{|x'|<\rho}\ili_{0}^{\rho}\frac{(x_{n}+\ep)^{2}x_{k}^{2}dx}{(|x'|^{2}+(x_{n}+\ep)^{2})^{q+1}}dx.
%$$
%\end{multline}


Substituting relations \eqref{int1}-\eqref{E_10} into (\ref{qq}), we obtain with regards to (\ref{partial1}), 
\begin{equation}\label{chislitel}
\ili_{\Omega}|\nabla u|^{p}dy\le 
\Big(\frac{n-p}{p-1}\Big)^{p}\times\left\{\begin{array}{lc}
\ep^{n-2q}{\mathcal E}_1-{\mathcal E}_2\frac{\ep^{n-2q+2}}{2q-n-2_{\textstyle\vphantom{1}}}
+O(\rho^{n-2q}),&p<\frac {n+2}3;\\
\ep^{n-2q}{\mathcal E}_1-{\mathcal E}_2\ln(\frac{1}{\ep})+O(\rho^{n-2q}),&p=\frac {n+2}3,
\end{array}\right.
\end{equation}
where
\begin{equation}\label{E_2}
{\mathcal E}_2=\frac {{\mathcal E}_{2}^{(1)}}{12(2q-n+1)}\cdot
\big(12{\mathcal R}_n+(2q-n+4)\sum\limits_{i<n}{\mathcal R}'_i+o_\rho(1)\big).
\end{equation}


\subsection{Estimate of $\|u\|_{p^{\star},\partial\Omega}$ for $1<p<\frac {n+1}2$}\label{est2}

It is obvious that the area on $\partial\Omega$ satisfies the relation 
$$d\Sigma=\sqrt{1+|\nabla F(x')|^{2}}\cdot\sqrt{\det (G')}\,dx'\ge \sqrt{\det (G')}\,dx',$$ 
where $G'=(g_{ij})_{i,j<n}$. Thus, we have
\begin{multline}\label{ppp}
\int\limits_{\partial\Omega}u^{p^\star}d\Sigma%\ge\\
%\int\limits_{|x'|<\frac{\rho}{2}}\frac {\sqrt{1+|\nabla F(x')|^{2}}\sqrt{\det G'}\,dx'}
%{(|x'|^{2}+(\ep+F(x'))^{2})^{q}}\ge\\
\ge\int\limits_{|x'|<\frac{\rho}{2}}\frac {\sqrt{\det (G')}\, dx'}{(|x'|^{2}+(\ep+F(x'))^{2})^{q}}=\\
=\int\limits_{|x'|<\frac{\rho}2}\frac {\sqrt{\det (G')}\,dx'}
{(|x'|^{2}+\ep^2)^{q}}%\times\\
\times\left[1-2q\frac {\ep F(x')}{|x'|^{2}+\ep^2}-\frac{F^2(x')}{|x'|^{2}+\ep^2}\cdot O(1)\right].
\end{multline}
%One can see that if $|x'|<\frac{\rho}{2}$ then
%\begin{multline}
%|x'|^{2}+(\ep+F(x'))^{2}=(|x'|^{2}+\ep^{2})\Big(1+\frac{2\ep F(x')+F(x')^{2}}{|x'|^{2}+\ep^{2}}\Big)\le\\
%\le (|x'|^{2}+\ep^{2})\Big(1+\frac{|2\ep F(x')+F(x')^{2}|}{|x'|^{2}+\ep^{2}}\Big)\le \\ %\le(|x'|^{2}+\ep^{2})\Big(1+\frac{|2\ep F(x')+F(x')^{2}|}{|x'|^{2}}\Big)=
%(|x'|^{2}+\ep^{2})(1+o_{\rho}(1)).
%\end{multline}
%Therefore,
%$$
%\ili_{|x'|<\frac{\rho}{2}}\frac{\sqrt{\det (G')}\,dx'}{(|x'|^{2}+(\ep+F(x'))^{2})^{q}}\ge \ili_{|x'|<\frac{\rho}{2}}\frac{\sqrt{\det (G')}\,dx'}{(|x'|^{2}+\ep^{2})^{q}}\cdot(1+o_{\rho}(1)).
%$$
Expanding the metric tensor by the Taylor formula and using \eqref{partial0}, we obtain 
%$$
%\sqrt{\det (G')}=1+\left(\frac{1}{4}\sli_{i,j,k<n}\frac{\partial^{2} g_{kk}}{\partial x_{i}\partial x_{j}}(0)x_{i}x_{j}\right)(1+o_{\rho}(1)).
%$$
%Therefore,
\begin{equation}\label{qqq}
\ili_{\partial\Omega}u^{p^{\star}}d\Sigma\ge{\mathcal J}_1+
\frac 14 \sli_{i,j,k<n} \frac{\partial^{2} g_{kk}}{\partial x_{i}\partial x_{j}}(0)
\cdot{\mathcal J}^{(ij)}_2-{\mathcal J}_3-2q{\mathcal J}_4-{\mathcal J}_5,
\end{equation}
where
%\begin{equation}
$${\mathcal J}_1=\ili_{|x'|<\frac{\rho}{2}}\frac{dx'}{(|x'|^{2}+\ep^{2})^{q}};
$$%\end{equation}
%\begin{equation}
$${\mathcal J}_2^{(ij)}=\ili_{|x'|<\frac{\rho}{2}}\frac{x_{i}x_{j}dx'}{(|x'|^{2}+\ep^{2})^{q}};
$$%\end{equation}
$${\mathcal J}_3=\ili_{|x'|<\frac{\rho}{2}}\frac{|x'|^2\,dx'}{(|x'|^{2}+\ep^{2})^{q}}
\cdot o_\rho(1);
$$
%\begin{equation}
$${\mathcal J}_4=\int\limits_{|x'|<\frac{\rho}2}\frac {\ep F(x')\big(1+o(\rho^2)\big)\,dx'}
{(|x'|^{2}+\ep^2)^{q+1}};%\times\\
%\times\left[1-2q\frac {\ep F(x')}{|x'|^{2}+\ep^2}+\frac
%{F^2(x')}{|x'|^{2}+\ep^2}\cdot O(1)\right].
$$
$${\mathcal J}_5=\int\limits_{|x'|<\frac{\rho}2}\frac {F^2(x')\,dx'}
{(|x'|^{2}+\ep^2)^{q+1}}\cdot O(1).
$$

It is easy to see that 
\begin{equation}\label{J1}
 {\mathcal J}_1= \ep^{n-2q-1}{\mathcal E}'_1-O(\rho^{n-2q-1}),
\end{equation}
where
\begin{equation}\label{E_1'}
{\mathcal E}'_1=\omega_{n-2}\ili_{0}^{\infty}
\frac{r^{n-2}dr}{(r^{2}+1)^{q}}=
\frac{\omega_{n-2}}{2}\cdot {\mathcal B}\big({\textstyle\frac{n-1}{2},\frac{n-1}{2(p-1)}}\big).
\end{equation}
Similarly,
\begin{equation}\label{J2}
 {\mathcal J}_2^{(ij)}= \delta_{ij}\ep^{n-2q+1}{\mathcal E}^{\prime\, (i)}_2-O(\rho^{n-2q+1}),
\end{equation}
where
\begin{equation}\label{E_2'i}
{\mathcal E}^{\prime\, (i)}_2=\ili_{\mathbb R^{n-1}}\frac{x_{i}^2\,dx'}{(|x'|^{2}+1)^{q}}=
\frac{\omega_{n-2}}{2(2q-n-1)}
\cdot{\mathcal B}\big({\textstyle\frac{n-1}{2},\frac{n-1}{2(p-1)}}\big).
\end{equation}

Further, using (\ref{F}) we obtain
\begin{multline}\label{J4}
{\mathcal J}_4
%=\left(\frac{n-p}{p-1}\right)^{p}\ili_{|x'|<\rho}\frac{F(x')dx'}
%{(|x'|^{2}+\ep^{2})^{q}}(1+o_{\rho}(1))=\\
=\ili_{|x'|<\rho}\sli_{i,j,k<n}\frac{\ep b_{ijk}x_{i}x_{j}x_{k}}
{(|x'|^{2}+\ep^{2})^{q+1}}\,dx'+\\
+\ili_{|x'|<\rho}\frac{\ep|x'|^{3}\,dx'}{(|x'|^{2}+\ep^{2})^{q+1}}\cdot o_{\rho}(1)=
{\mathcal J}^{(11)}_2\cdot o_\rho(1).
\end{multline}
Finally, it is obvious that
\begin{equation}\label{J3-5}
 {\mathcal J}_3= {\mathcal J}^{(11)}_2\cdot o_\rho(1);\qquad
{\mathcal J}_5={\mathcal J}^{(11)}_2\cdot o_\rho(1).
\end{equation}

Substituting relations \eqref{J1}-\eqref{J3-5} into (\ref{qqq}), we obtain with regards to (\ref{partial1}), 
\begin{equation}\label{znamen}
\ili_{\partial\Omega}u^{p^{\star}}d\Sigma\ge
\ep^{n-2q-1}{\mathcal E}'_1-\ep^{n-2q+1}{\mathcal E}'_2-O(\rho^{n-2q-1}),
\end{equation}
where 
\begin{equation}\label{E_2'}
{\mathcal E}'_2=\frac {{\mathcal E}^{\prime\, (1)}_2}{12}
\cdot\big(\sum\limits_{i<n}{\mathcal R}'_i+o_\rho(1)\big).
\end{equation}

\subsection{Estimate of $\|u\|_{p,\Omega}$}\label{est3}

Since $dy\le C dx$ and $F(x')=O(|x'|^{3})$  for $|x'|<\rho_{0}$, we have
\begin{multline}\label{dobavka}
\ili_{\Omega}u^{p}dy\le C\ili_{|x'|<\rho}dx'\!\!\ili_{-C|x'|^{3}}^{\infty}
\frac{dx_n}{(|x'|^{2}+(x_{n}+\ep)^{2})^{\frac{p(n-p)}{2(p-1)}}}\le\\ \le\begin{cases}C\ep^{\frac{p^{2}-n}{p-1}}, &p<\sqrt{n}; \\ C(1+\ln\frac{\rho}{\ep}), &p=\sqrt{n}; \\ C, &p>\sqrt{n}. \end{cases}
\end{multline}


\subsection{Completion of the proof}


%where
%$$
%E_{1}=\frac{2q-n}{(2q-n-2)(2q-n-1)(2q-n+1)},
%$$
%$$
%E_{2}=\frac{(2q-n)(2q-n+4)}{12(2q-n-2)(2q-n-1)(2q-n+1)}.
%$$
%$$=\ep^{n-2q}\Big(\frac{n-p}{p-1}\Big)^{p}\frac{\omega_{n-2}}{2(2q-n)}\frac{\Gamma(\frac{n-1}{2})\Gamma(\frac{2q-n+1}{2})}{\Gamma(q)}\left[1-\ep^{2}\left(E_{1}R(n)+E_{2}\sli_{k<n}R_{0}(k) \right)(1+o_{\rho}(1))+o(\ep^{2})\right]+C(\rho),
%$$

Estimates (\ref{chislitel}) and (\ref{dobavka}) imply
%\begin{multline}\label{W1p}
$$\|u\|^p_{W^1_p(\Omega)}%\le\\
\le{\textstyle\begin{cases} 
\big(\frac{n-p}{p-1}\big)^{p}\ep^{n-2q}{\mathcal E}_1-
\ep^{n-2q+2}\frac{{\mathcal E}_2}{2q-n-2_{\textstyle\vphantom{1}}}
+ C\,\varepsilon^{\frac{p^2-n}{p-1}}+C(\rho),
                          & 1{<}p{<}\sqrt{n}; \\
\big(\frac{n-p}{p-1}\big)^{p}\ep^{n-2q}{\mathcal E}_1-
\ep^{n-2q+2}\frac{{\mathcal E}_2}{2q-n-2_{\textstyle\vphantom{1}}}
+ C\ln(\frac 1\varepsilon)+C(\rho),
                          & p=\sqrt{n}; \\
\big(\frac{n-p}{p-1}\big)^{p}\ep^{n-2q}{\mathcal E}_1-
\ep^{n-2q+2}\frac{{\mathcal E}_2}{2q-n-2_{\textstyle\vphantom{1}}}
+C(\rho), &\sqrt{n}{<}p{<}\frac{n+2}{3}; \\
\big(\frac{n-p}{p-1}\big)^{p}\ep^{n-2q}{\mathcal E}_1-
{\mathcal E}_2\ln(\frac 1\varepsilon)+C\,\rho^{\frac{p-n}{p-1}},
                          & p=\frac{n+2}{3},
\end{cases}}
$$%\end{multline}
where ${\mathcal E}_1$, ${\mathcal E}_2$ are defined in (\ref{E_1}) and (\ref{E_2}).

Since $n-2q+2<\frac{p^{2}-n}{p-1}$ for $p>2$, we get for $2<p<\frac{n+2}{3}$, with regard to 
%(\ref{W1p}) and 
(\ref{znamen}),
%\begin{equation}
$$\frac{\|u\|_{W^1_p(\Omega)}^{p}}{\|u\|_{p^{\star},\partial\Omega}^{p}} \le \left(\frac{n-p}{p-1}\right)^{p}\frac{{\mathcal E}_1\phantom{^{\frac{p}{p^\star}}}}
{({\mathcal E}'_1)^{\frac{p}{p^\star}}}\cdot
\bigg(1+\ep^2\bigg(\frac{p}{p^\star}\frac{{\mathcal E}'_2}{{\mathcal E}'_1}-
\frac{{\mathcal E}_2}{(2q-n-2){\mathcal E}_1}\bigg)+o(\ep^2)\bigg).
$$%\end{equation}
Direct computation shows that for $2<p<\frac{n+2}{3}$
\begin{equation}\label{znak}
\frac{p}{p^\star}\frac{{\mathcal E}'_2}{{\mathcal E}'_1}-
\frac{{\mathcal E}_2}{(2q-n-2){\mathcal E}_1}=
-\frac{(2q-n)\big(2{\mathcal R}_n+\sum_{i<n}{\mathcal R}'_i+o_\rho(1)\big)}
{2(2q-n+1)(2q-n-1)(2q-n-2)}.
\end{equation}
 The last quantity is negative for sufficiently small $\rho$ since
$$
2{\mathcal R}_{n}+\sli_{i<n}{\mathcal R}'_{i}=\sli_{i}{\mathcal R}_i=
{\mathcal R}(y^0)
$$
is the scalar curvature of $\Omega$ at $y^0$ which is positive by assumption.

Hence, for $\ep$ and $\rho$ sufficiently small, one has
\begin{equation} \label{qqqq}
\frac {\|u\|_{W^1_p(\Omega)}}{\|u\|_{p^\star,\partial\Omega}}< 
\frac{n-p}{p-1}\frac{{\mathcal E}_1^{\frac{1}{p}}}{({\mathcal E}'_1)^{\frac{1}{p^\star}}}.
\end{equation}

Similarly, in the case $p=\frac{n+2}{3}$ the relation (\ref{qqqq}) holds true for $\ep$ and
$\rho$ sufficiently small. By continuity, for some $\beta>0$ and
$p<\frac{n+2}{3}+\beta$
$$\lambda_1(n,p,\Omega)<\frac{n-p}{p-1}\frac{{\mathcal E}_1^{\frac{1}{p}}}
{({\mathcal E}'_1)^{\frac{1}{p^\star}}}=K(n,p).$$
The application of Proposition \ref{prop1} completes the proof.
\end{proof}


\section{Nonexistence of the minimizer}\label{T2}

\begin{proof}[Proof of theorem \ref{Th2}] 
We proceed by contradiction. Suppose there exists an unbounded
sequence of $\varkappa$'s such that the infimum in (I) is attained on
$\varkappa\Omega$. Then, by a standard argument,
the corresponding minimizers $u_{\varkappa}$ are positive in
$\varkappa\Omega$.

Consider the functions
$v_{\varkappa}(y)=C(\varkappa)u_{\varkappa}(\frac y{\varkappa})$ in $\Omega$,
with $C(\varkappa)$ given by the normalization condition
$\|v_{\varkappa}\|_{p^\star,\partial\Omega}=1$. Then the weak Euler equation
for the $v_{\varkappa}$ reads as follows:
\begin{equation}\label{euler}
\int\limits_{\Omega}|\nabla v_{\varkappa}|^{p-2}\langle\nabla v_{\varkappa}, \nabla h\rangle\,dy+
\varkappa^{p}\int\limits_{\Omega}v_{\varkappa}^{p-1}h\,dy=
\lambda_{\varkappa}^{p}\int\limits_{\partial\Omega}v_{\varkappa}^{p^{\star}-1}h\,d\Sigma,
\end{equation}
where $\lambda_{\varkappa}=\lambda_1(n, p, \varkappa\Omega)$. Since, given $\varkappa$,
for the sequence considered in the proof of Theorem \ref{Th1} the quotient (I) tends to $K(n,p)$,
one has
\begin{equation}\label{lambda}
 \lambda_{\varkappa}\le K(n,p).
\end{equation}

Put $h=v_{\varkappa}$ in (\ref{euler}). Then
\begin{equation}\label{norms}
\|\nabla
v_{\varkappa}\|_{p,\Omega}^{p}+\varkappa^{p}\|v_{\varkappa}\|_{p,\Omega}^{p}=
\lambda_{\varkappa}^{p}.
\end{equation}

Without loss of generality, $v_{\varkappa}\rightharpoondown v$ in
$W^1_p(\Omega)$. From (\ref{norms}) we conclude that $v=0$.
Arguing as in the proof of Theorem 2.2 \cite{LPT} one deduces that
there exists a point $y^{0}\in\partial\Omega$ such that
$|v_{\varkappa}|^{p^\star}\rightharpoondown \delta(y-y^0)$ in the
sense of measures on $\partial\Omega$. Therefore, for a given $\rho>0$,
\begin{equation}\label{bryak}
\|v_{\varkappa}\|_{p^{\star}, \partial\Omega\setminus
B_{\frac{\rho}{2}}(y^{0})}\to 0.
\end{equation}

\begin{zamech} In fact, the stronger property
$\sup_{\widetilde{\Omega}\setminus B_{\rho}(y^{0})}v_{\varkappa} \to 0$ can be
proved, but we do not need it.
\end{zamech}


Now we use spherical symmetrization (see, for instance, \cite{PS}) in ``planes''
$\theta=const$. Since this operation retains the denominator in (I) and does not
increase the numerator, we can assume that $v_\varkappa\big|_{\partial\Omega}$ is
symmetrically decreasing around the point $y^0$. Since the normalization of
$v_{\varkappa}$ implies $\|v_{\varkappa}\|_{p^\star,weak,\partial\Omega}\le C$,
this symmetry provides the estimate
\begin{equation}\label{hryap1}
{\rm dist}(y,y^0)^{\frac np-1}\cdot v_{\varkappa}(y)\le C,\qquad y\in\partial\Omega.
\end{equation}


By $\eta$ we denote a smooth cut-off function such that
$$
\eta=0\quad \mbox{in}\quad B_{\frac{\rho}{2}}(y^{0});\qquad
\eta=1\quad \mbox{in}\quad \mathbb R^n\setminus B_{\rho}(y^{0}).
$$

We substitute $h=\eta^{p}v_{\varkappa}$ into the equation
(\ref{euler}):
$$
\int\limits_{\Omega}\eta^{p}|\nabla v_{\varkappa}|^{p}\,dy+p\int\limits_{\Omega}\eta^{p-1}
|\nabla v_{\varkappa}|^{p-2}\langle\nabla v_{\varkappa}, \nabla \eta\rangle\,dy+
\varkappa^{p}\int\limits_{\Omega}v_{\varkappa}^{p}\,dy\le
\lambda_{\varkappa}^p\int\limits_{\partial\Omega}\eta^{p}v_{\varkappa}^{p^{\star}}d\Sigma.
$$
Recalling (\ref{lambda}), we arrive at
\begin{equation}\label{tryam}
\|\eta\nabla v_{\varkappa}\|_{p,\Omega}^{p}+
\varkappa^{p}\|\eta v_{\varkappa}\|_{p,\Omega}^{p}\le
K^p(n,p)\int\limits_{\partial\Omega}\eta^{p}v_{\varkappa}^{p^{\star}}d\Sigma +
C\|v_{\varkappa}\|_{p,\Omega}\cdot\|\eta\nabla v_{\varkappa}\|_{p,\Omega}^{p-1}.
\end{equation}
The H\"older inequality and (\ref{bryak}) give us
$$
\int\limits_{\partial\Omega}\eta^{p}v_{\varkappa}^{p^{\star}}d\Sigma
%=\!\!\int\limits_{\partial\Omega\setminus B_{\frac{\rho}{2}}(x^{0})}\!\!
%\eta^{p}v_{\varkappa}^{p^{\star}}d\Sigma
\le \|\eta v_\varkappa\|_{p^{\star},\partial\Omega}^{p}\cdot
\|v_{\varkappa}\|_{p^{\star},\partial\Omega\setminus
B_{\frac{\rho}{2}}(x^{0})}^{p^{\star}-p}=
\|\eta v_\varkappa\|_{p^{\star},\partial\Omega}^{p}\cdot o_\varkappa(1).
$$
We apply the trace embedding theorem to the function
$\eta v_{\varkappa}$:
\begin{equation}\label{tryam2}
\|\eta v_{\varkappa}\|_{p^{\star}, \partial\Omega}^{p}\le C(\|\nabla
(\eta v_{\varkappa})\|_{p,\Omega}^{p}+\|\eta v_{\varkappa}\|_{p,\Omega}^{p})
\le C(\|\eta\nabla
v_{\varkappa}\|_{p,\Omega}^{p}+\|v_{\varkappa}\|_{p,\Omega}^{p}).
\end{equation}

Substituting these relations into (\ref{tryam}), we obtain
$$
\left(\frac {\|\eta\nabla v_{\varkappa}\|_{p,\Omega}}
{\|v_{\varkappa}\|_{p,\Omega}}\right)^p\le
C\left(\frac {\|\eta\nabla v_{\varkappa}\|_{p,\Omega}}
{\|v_{\varkappa}\|_{p,\Omega}}\right)^{p-1}+o_\varkappa(1)\cdot
\bigg(1+\bigg(\frac {\|\eta\nabla v_{\varkappa}\|_{p,\Omega}}
{\|v_{\varkappa}\|_{p,\Omega}}\bigg)^p\bigg).
$$
Therefore, for $\varkappa$ sufficiently large one has
\begin{equation}\label{tryam3}
\frac {\|\eta\nabla v_{\varkappa}\|_{p,\Omega}}
{\|v_{\varkappa}\|_{p,\Omega}}\le C.
\end{equation}

To obtain accurate estimates of $v_{\varkappa}$ at the neighborhood of $y^0$, we
use the idea of \cite{Dr1}. Recall that $\exp_{y^0}$ is a diffeomorphism from the half-ball
$B^+_{2\rho_0}\subset\mathbb R^n$ onto Riemannian half-ball $B^+_{2\rho_0}(y^0)$.
 Without loss of generality, one can assume that $\rho<\rho_0$.

Denote by $\eta_1$ a smooth cut-off function such that
$$
\eta_1=1\quad \mbox{in}\quad B_{\rho}(y^{0});\qquad
\eta_1=0\quad \mbox{in}\quad \mathbb R^n\setminus B_{2\rho}(y^{0}).
$$

By (\ref{ccc}),
\begin{equation}\label{hryap}
\|\widetilde\eta_{1}\widetilde v_{\varkappa}\|_{p^{\star}, \mathbb R^{n-1}}^p\le 
\frac 1{K^p(n,p)}\cdot
\|\nabla(\widetilde\eta_1\widetilde v_{\varkappa})\|_{p,\mathbb R^n_+}^p.
\end{equation}

By \cite[\S19.5]{BZ},
$$|\nabla(\widetilde\eta_1\widetilde v_{\varkappa})|(x)= |\nabla(\eta_1v_{\varkappa})|(y)\cdot
\big(1+O({\rm dist}^2(y,y^0))\big)
$$
and
$$dx=dy\cdot\big(1+O({\rm dist}^2(y,y^0))\big).$$
This gives
\begin{multline}\label{manifold1}
\|\nabla(\widetilde\eta_1\widetilde v_{\varkappa})\|_{p,\mathbb R^n_+}^p=\!
\int\limits_{B_{2\rho}(0)}\!|\nabla(\widetilde\eta_1\widetilde v_{\varkappa})|^p(x)
\,dx\le\\
\le\!\int\limits_{B_{2\rho}(y^{0})}\!|\nabla(\eta_1v_{\varkappa})|^p(y)\cdot
\big(1+C{\rm dist}^2(y,y^0)\big)\,dy
\end{multline}
and
\begin{multline*}
\|\widetilde\eta_{1}\widetilde v_{\varkappa}\|_{p^{\star}, \mathbb R^{n-1}}^{p^{\star}}=\!\!
\int\limits_{B_{2\rho}(0)\cap\mathbb R^{n-1}}\!\!
(\widetilde\eta_{1}\widetilde v_{\varkappa})^{p^{\star}}(x')
\,dx'\ge\\
\ge\!\!\int\limits_{B_{2\rho}(y^{0})\cap\partial\Omega}\!\!
(\eta_{1}v_{\varkappa})^{p^{\star}}(y)\cdot\big(1-C{\rm dist}^2(y,y^0)\big)\,dy\ge\\
\ge\|v_{\varkappa}\|_{p^\star,B_{\rho}(y^{0})\cap\partial\Omega}^{p^\star}-
C\int\limits_{\partial\Omega}v_{\varkappa}^{p^{\star}}(y)\cdot{\rm dist}^2(y,y^0)\,dy.
\end{multline*}
Moreover, with regard to the normalization of $v_\varkappa$,
\begin{equation}\label{manifold2}
\|\widetilde\eta_{1}\widetilde v_{\varkappa}\|_{p^{\star}, \mathbb R^{n-1}}^{p}\ge
\|v_{\varkappa}\|_{p^\star,B_{\rho}(y^{0})\cap\partial\Omega}^{p^\star}-
C\int\limits_{\partial\Omega}v_{\varkappa}^{p^{\star}}(y)\cdot{\rm dist}^2(y,y^0)\,dy.
\end{equation}

Substituting (\ref{manifold1}) and (\ref{manifold2}) into (\ref{hryap}), we obtain
by (\ref{gradient})
\begin{multline}\label{manifold3}
\|v_{\varkappa}\|_{p^{\star},\partial\Omega\cap B_{\rho}(y^{0})}^{p^{\star}}\le\\
\le\frac{1}{K^{p}(n,p)}\cdot\|\nabla v_{\varkappa}\|_{p,\Omega}^{p}+
C\Big[\|v_{\varkappa}\|_{p,\Omega}^{p}+\|v_{\varkappa}\|_{p,\Omega}\cdot
\|\nabla v_{\varkappa}\|_{p, \Omega\setminus B_{\rho}(y^{0})}^{p-1}\Big]+\\
+C\int\limits_{\Omega}|\nabla v_{\varkappa}|^p\,{\rm dist}^2(y,y^0))\,dy+
C\int\limits_{\partial\Omega}v_{\varkappa}^{p^{\star}}\,{\rm dist}^2(y,y^0)\,dy.
\end{multline}

We estimate the last term in the right-hand side of (\ref{manifold3}) by (\ref{hryap1}):
\begin{multline*}
\int\limits_{\partial\Omega}v_{\varkappa}^{p^{\star}}\,{\rm dist}^2(y,y^0)\,dy=\\
=\int\limits_{\partial\Omega}\big(v_{\varkappa}^{\frac p{n-p}}{\rm dist}(y,y^0)\big)^{p-1}
\cdot v_{\varkappa}^p\,{\rm dist}^{3-p}(y,y^0) \,dy\le\\
\le C\int\limits_{\partial\Omega}v_{\varkappa}^p\,{\rm dist}^{3-p}(y,y^0) \,dy.
\end{multline*}
Using the trace embedding theorem and the H\"older inequality, we obtain
\begin{multline}\label{manifold4}
\int\limits_{\partial\Omega}v_{\varkappa}^{p^{\star}}\,{\rm dist}^2(y,y^0)\,dy\le\\
\le C\biggl(\int\limits_{\Omega}|\nabla(v_{\varkappa}^p\,{\rm dist}^{3-p}(y,y^0))| \,dy+
\int\limits_{\Omega}v_{\varkappa}^p\,{\rm dist}^{3-p}(y,y^0) \,dy\biggr)\le \\
\le C\biggl(\int\limits_{\Omega}|\nabla v_{\varkappa}|^p\,{\rm dist}^{(3-p)p}(y,y^0)\,dy\biggr)
^\frac 1p\cdot\|v_{\varkappa}\|_{p,\Omega}^{p-1}+
C\int\limits_{\Omega}v_{\varkappa}^p\,{\rm dist}^{2-p}(y,y^0) \,dy\le\\
\le C\biggl(\int\limits_{\Omega}|\nabla v_{\varkappa}|^p\,{\rm dist}^2(y,y^0)\,dy\biggr)
^\frac 1p\cdot\|v_{\varkappa}\|_{p,\Omega}^{p-1}+C\|v_{\varkappa}\|_{p,\Omega}^p
\end{multline}
(the last inequality is due to $1<p\le2$).

Further, we substitute $h=v_{\varkappa}\,{\rm dist}^2(y,y^0)$ into (2). Canceling a positive term results in
$$\int\limits_{\partial\Omega}|\nabla v_{\varkappa}|^p\,{\rm dist}^2(y,y^0)\,dy\le
K^p(n,p)\int\limits_{\partial\Omega}v_{\varkappa}^{p^{\star}}\,{\rm dist}^2(y,y^0)\,dy.
$$
Substituting this relation into (\ref{manifold4}), we arrive at
$$
\int\limits_{\partial\Omega}|\nabla v_{\varkappa}|^p\,{\rm dist}^2(y,y^0)\,dy
%{\|v_{\varkappa}\|_{p,\Omega}^p
\le C \bigg(\int\limits_{\partial\Omega}|\nabla v_{\varkappa}|^p\,{\rm dist}^2(y,y^0)\,dy\bigg)
^{\frac 1p}\|v_{\varkappa}\|_{p,\Omega}^{p-1}+C\|v_{\varkappa}\|_{p,\Omega}^p,
$$
which gives
\begin{equation*}\label{hryap2}
 %\frac
{\int\limits_{\partial\Omega}|\nabla v_{\varkappa}|^p\,{\rm dist}^2(y,y^0)\,dy}
\le C{\|v_{\varkappa}\|_{p,\Omega}^p};\qquad
%\frac
{\int\limits_{\partial\Omega}v_{\varkappa}^{p^{\star}}\,{\rm dist}^2(y,y^0)\,dy}
\le C{\|v_{\varkappa}\|_{p,\Omega}^p}.
\end{equation*}

We substitute these relation into (\ref{manifold3}). Expressing
$\|\nabla v_{\varkappa}\|_{p,\Omega}^{p}$ from (\ref{norms}) and
recalling (\ref{lambda}), we get
$$
\frac {\varkappa^{p}}{\gamma^{p}}\cdot\|v_{\varkappa}\|_{p,\Omega}^{p}
\le 1-\|v_{\varkappa}\|_{p^{\star}, \partial\Omega\cap B_{\rho}(y^{0})}^{p^{\star}}
+C\Big[\|v_{\varkappa}\|_{p,\Omega}^{p}+\|v_{\varkappa}\|_{p,\Omega}\cdot
\|\nabla v_{\varkappa}\|_{p, \Omega\setminus B_{\rho}(y^{0})}^{p-1}\Big].
$$
Thus,
\begin{equation}\label{yes}
\frac{\varkappa^{p}}{\gamma^{p}}\le
\frac{\|v_{\varkappa}\|_{p^{\star},
\partial \Omega\setminus
B_{\rho}(y^{0})}^{p^{\star}}}{\|v_{\varkappa}\|_{p,\Omega}^{p}}+
C\left(1+\left(\frac{\|\nabla v_{\varkappa}\|_{p, \Omega\setminus
B_{\rho}(y^{0})}}{\|v_{\varkappa}\|_{p,\Omega}}\right)^{p-1}\right).
\end{equation}
Using normalization of $v_\varkappa$, (\ref{tryam2}) and (\ref{tryam3}), we
obtain
$$\frac{\|v_{\varkappa}\|_{p^{\star},\partial \Omega\setminus
B_{\rho}(y^{0})}^{p^{\star}}}{\|v_{\varkappa}\|_{p,\Omega}^{p}}\le
\frac{\|v_{\varkappa}\|_{p^{\star},\partial \Omega\setminus
B_{\rho}(y^{0})}^{p}}{\|v_{\varkappa}\|_{p,\Omega}^{p}}\le
\frac{\|\eta v_{\varkappa}\|_{p^{\star},\partial \Omega}^{p}}
{\|v_{\varkappa}\|_{p,\Omega}^{p}}\le C
\bigg(1+\frac {\|\eta\nabla v_{\varkappa}\|_{p,\Omega}^p}
{\|v_{\varkappa}\|_{p,\Omega}^p}\bigg)\le C,
$$
while the second term in (\ref{yes}) is bounded by (\ref{tryam3}):
$$\frac{\|\nabla v_{\varkappa}\|_{p, \Omega\setminus
B_{\rho}(y^{0})}}{\|v_{\varkappa}\|_{p,\Omega}}\le
\frac {\|\eta\nabla v_{\varkappa}\|_{p, \Omega}}{\|v_{\varkappa}\|_{p,\Omega}}
\le C.
$$
Therefore, the right-hand side in (\ref{yes}) is bounded. This gives a
contradiction for $\varkappa$ large enough.
\end{proof}


\section{The trace Sobolev--Poincar\'e inequality}\label{poinc}

\begin{proof}[Proof of Theorem \ref{Th3}]
We use the following analog of Proposition \ref{prop1}.

\begin{utv}\label{prop3} 
Let the infimum in {\rm(II)} satisfy the inequality
$\lambda_2(n,p,\Omega)<K(n,p)$.
Then the infimum is attained.
\end{utv}

To construct a function with zero trace mean-value having the quotient (II) less
then $K(n,p)$ we modify the function (\ref{escobar-fun-def}). Similarly to \cite[Theorem 5.1]{DN},
for large $p$ we proceed by subtracting a suitable function with a small support
while for small $p$ we will subtract a constant.\medskip

We need the follownig estimate for the function $u$ defined in (\ref{escobar-fun-def}):
\begin{equation}\label{est-fun-1}
\ili_{\partial\Omega}u(y)d\Sigma \le C\ili_{|x'|<\rho}\frac{dx'}{(|x'|^{2}+\ep^{2})^{\frac{n-p}{2(p-1)}}}\le \begin{cases} C\ep^{n-1-\frac{n-p}{p-1}}, &p<\frac{2n-1}{n}; \\ 
C\ln(\frac 1{\ep})+C(\rho), &p=\frac{2n-1}{n}; \\ 
C, &p>\frac{2n-1}{n}.\end{cases}
\end{equation}


{\bf 1.} First, consider the case $p>\frac{2n-1}{n}$. Let $u_{1}$ be a smooth non-negative function such that
$u_1\big|_{\partial\Omega}\not\equiv0$ and ${\rm supp}\,u_1\cap {\rm supp}\, u=\emptyset$.
We define the function with zero trace mean-value
$$
\widehat u(y)=u(y)-u_{1}(y)\,\frac{\int\limits_{\partial\Omega}\!u\,d\Sigma}
{\int\limits_{\partial\Omega}^{\vphantom {_1}}\!u_{1}d\Sigma}.
$$
The estimates (\ref{chislitel}), (\ref{znamen}) and (\ref{znak}) yield for $p<\frac{n+2}{3}$
$$\frac{\|\nabla u\|_{p,\Omega}^{p}}{\|u\|_{p^{\star},\partial\Omega}^{p}} \le K^p(n,p)\cdot
\bigg(1-\ep^2\frac{(2q-n){\mathcal R}(y^0)}{2(2q-n+1)(2q-n-1)(2q-n-2)}+o(\ep^2)\bigg)
$$%\end{equation}
and therefore, for small $\rho$ and $\ep$
\begin{equation}\label{qqqqq}
\frac {\|\nabla \widehat u\|_{p,\Omega}}{\|\widehat u\|_{p^\star,\partial\Omega}}<K(n,p).
\end{equation}

For $p=\frac{n+2}{3}$ this relation also holds true for $\ep$ and
$\rho$ sufficiently small. By continuity, for some $\beta>0$ and
$p<\frac{n+2}{3}+\beta$
\begin{equation}\label{qqqqqq}
\lambda_2(n,p,\Omega)<K(n,p).
\end{equation}
%\smallskip


{\bf 2.} Now we consider the case $1<p\le \frac{2n-1}{n}$. Define $u$ by (\ref{escobar-fun-def})
and put $\rho=\ep^{\gamma}$ with some $\gamma\in(0,\gamma^*)$,
where $\gamma^*=\frac{n+2-3p}{n-p}\in(0,1)$.

The estimate (\ref{znamen}) yields
$$
\int\limits_{\partial\Omega}u^{p^{\star}}d\Sigma\ge 
\ep^{n-2q-1}{\mathcal E}'_{1}-\ep^{n-2q+1}{\mathcal E}'_{2}-C\ep^{\gamma(n-2q-1)}.
$$
Immediate computation shows that
%\begin{equation}\label{qwe}
$$\gamma^*(n-2q-1)>n-2q+1.
$$%\end{equation}
Therefore, for $\gamma\in(0, \gamma^*)$
$$
\int\limits_{\partial\Omega}u^{p^{\star}}d\Sigma\ge
\ep^{n-2q-1}{\mathcal E}'_{1}-\ep^{n-2q+1}({\mathcal E}'_{2}+o_\ep(1)).
$$
Furthermore, the estimate (\ref{chislitel}) implies
$$
\int\limits_{\Omega} |\nabla u|^p\,dy\le
\Big(\frac{n-p}{p-1}\Big)^{p}\cdot\big(\ep^{n-2q}{\mathcal E}_1-
\ep^{n-2q+2}\frac{{\mathcal E}_2}{2q-n-2}+C\ep^{\gamma(n-2q)}\big).
 $$

Direct calculations result in
$$
\gamma^{*}(2q-n)=2q-n-2.
$$
Hence, for $\gamma\in(0, \gamma^*)$
\begin{equation}\label{gradient1}
\int\limits_\Omega |\nabla u|^{p}\,dy\le  
\Big(\frac{n-p}{p-1}\Big)^{p}\cdot\Big(\ep^{n-2q}{\mathcal E}_1-
\ep^{n-2q+2}\frac{{\mathcal E}_2-o_\ep(1)}{2q-n-2}\Big).
\end{equation}

%so there exists $\gamma\in(0, \gamma^{*})$ such that $-\gamma(2q-n)>n-2q+2$ and
%$$
%\ili |\nabla u|^{p}dy\le \\ \le \ep^{n-2q}\Big(\frac{n-p}{p-1}\Big)^{p}\frac{\omega_{n-2}}{2(2q-n)}\frac{\Gamma(\frac{n-1}{2})\Gamma(\frac{2q-n+1}{2})}{\Gamma(q)}\left[1-\ep^{2}\left(E_{1}R(n)+E_{2}\sli_{k<n}R_{0}(k) \right)(1+o_{\rho}(1))+o(\ep^{2})\right].
%$$
%Furhter,
%$$
%\ili_{\partial \Omega}u^{p^{\star}}d\Sigma \ge \ep^{n-2q-1}\frac{\omega_{n-2}}{2}\frac{\Gamma(\frac{2q-n+1}{2})\Gamma(\frac{n-1}{2})}{\Gamma(q)}\left(1-\ep^{2}E_{3}\sli_{k<n}R_{0}(k)(1+o_{\rho}(1))\right)-C\ep^{\gamma(n-1-2q)}.
%$$
%It is easy to see that
%$$
%\gamma^{*}(2q-n+1)=<2q-n-1,
%$$
%so there exists $\gamma\in(0,\gamma^{*})$ such that $-\gamma(2q-n+1)>n-2q+1$ so
%$$
%\ili_{\partial \Omega}u^{p^{\star}}d\Sigma \ge \ep^{n-2q-1}\frac{\omega_{n-2}}{2}\frac{\Gamma(\frac{2q-n+1}{2})\Gamma(\frac{n-1}{2})}{\Gamma(q)}\left(1-\ep^{2}E_{3}\sli_{k<n}R_{0}(k)(1+o_{\rho}(1))\right).
%$$

Now we define the function with zero trace mean-value
$$
\widehat u(y)=u(y)-\frac 1{|\partial\Omega|}\int\limits_{\partial\Omega}\!u\,d\Sigma.
$$
Using (\ref{est-fun-1}) and the Minkowski inequality we conclude that
\begin{multline}\label{minkowski}
\|\widehat u\|_{p^{\star}, \partial\Omega}\ge
\biggl(\int\limits_{{\rm supp}\,u\cap \partial\Omega}
|\widehat u|^{p^{\star}}d\Sigma\biggr)^{\frac{1}{p^{\star}}}\ge\\
\ge\|u\|_{p^{\star}, \partial\Omega}
-\frac {|{\rm supp}\,u\cap \partial\Omega|^{\frac{1}{p^{\star}}}}{|\partial\Omega|}
\int\limits_{\partial\Omega}u d\Sigma\ge 
\ep^{\frac{n-2q-1}{p^\star}}({\mathcal E}'_1)^{\frac{1}{p^{\star}}}\times\\
\times\Big(1-\ep^2\,\frac{{\mathcal E}'_2+o_\ep(1)}{p^{\star}{\mathcal E}'_1} - C\ep^{\gamma\frac{n-1}{p^{\star}}-\frac{n-2q-1}{p^\star}}\Phi(\ep)\Big),
\end{multline}
where
$$\Phi(\ep)=\begin{cases}\ep^{n-1-\frac{n-p}{p-1}},&  \mbox{if\quad} 1<p<\frac{2n-1}{n};\\
1+\ln(\frac 1\ep),&  \mbox{if\quad} p=\frac{2n-1}{n}.
\end{cases}
$$

Immediate calculations show that for $n\ge4$ and $1<p\le\frac{2n-1}{n}$, the inequality
$$
\gamma^*\frac{n-1}{p^\star}+\frac{n-2q-1}{p^\star}+n-1-\frac{n-p}{p-1}>2
$$
holds true for $\gamma=\gamma^*$ and, therefore, for some $\gamma\in(0,\gamma^*)$. Choose such a $\gamma$, then (\ref{minkowski}) implies
\begin{equation}\label{granica}
\|\widehat u\|^p_{p^{\star}, \partial\Omega}\ge \ep^{n-2q}({\mathcal E}'_1)^{\frac{p}{p^{\star}}}
\left(1-\ep^2\frac{p}{p^{\star}}\frac{{\mathcal E}'_2+o_{\ep}(1)}{{\mathcal E}'_1}\right).
\end{equation}

The relations (\ref{gradient1}), (\ref{granica}) and (\ref{znak}) imply (\ref{qqqqq}) for $\ep$ small enough. Thus, the inequality (\ref{qqqqqq}) is valid in this case. The application of Proposition \ref{prop3} completes the proof.
\end{proof}


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\end{thebibliography}

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