<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%~MoulinÃ© par MaN_auto v.0.33.8 (9c3f6e85) 2025-04-22 18:55:23
\documentclass[CRMATH,Unicode,XML,biblatex]{cedram}

\TopicFR{\'Equations aux dÃ©rivÃ©es partielles}
\TopicEN{Partial differential equations}

\addbibresource{CRMATH_Hauer_20240753.bib}

\newcommand*\Rde{{\mathbb{R}^d}}
\newcommand*\Ne{{\mathbb{N}}}
\renewcommand*\Re{{\mathbb R}}
\newcommand*{\dd}{\,\mathrm{d}}
\newcommand*\al{\alpha}
\newcommand*\be{\beta}
\newcommand*\sg{\sigma}
\newcommand*\f{\varphi}
\newcommand*\Om{\Omega}
\newcommand*\om{\omega}

\DeclareMathOperator*{\nuesssup}{{\nu}-ess\;sup}
\DeclareMathOperator*{\spt}{supp}
\DeclareMathOperator*{\dist}{dist}

\newcommand*\sT{\;:\;}

%%%---------------------------------------------------------------------------

%%% BAR &amp; OVERLINE

%%% Better widebar macro
\makeatletter
\let\save@mathaccent\mathaccent
\newcommand*\if@single[3]{%
  \setbox0\hbox{${\mathaccent"0362{#1}}^H$}%
  \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}%
  \ifdim\ht0=\ht2 #3\else #2\fi
  }
%The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath:
\newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna}
%If there's a superscript following the bar, then no negative kern may follow the bar;
%an additional {} makes sure that the superscript is high enough in this case:
\newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}}
%Use a separate algorithm for single symbols:
\newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}}
\newcommand*\wide@bar@[3]{%
  \begingroup
  \def\mathaccent##1##2{%
%Enable nesting of accents:
    \let\mathaccent\save@mathaccent
%If there's more than a single symbol, use the first character instead (see below):
    \if#32 \let\macc@nucleus\first@char \fi
%Determine the italic correction:
    \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}%
    \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}%
    \dimen@\wd\tw@
    \advance\dimen@-\wd\z@
%Now \dimen@ is the italic correction of the symbol.
    \divide\dimen@ 3
    \@tempdima\wd\tw@
    \advance\@tempdima-\scriptspace
%Now \@tempdima is the width of the symbol.
    \divide\@tempdima 10
    \advance\dimen@-\@tempdima
%Now \dimen@ = (italic correction / 3) - (Breite / 10)
    \ifdim\dimen@&gt;\z@ \dimen@0pt\fi
%The bar will be shortened in the case \dimen@&lt;0 !
    \rel@kern{0.6}\kern-\dimen@
    \if#31
      \overline{\rel@kern{-0.6}\kern\dimen@\macc@nucleus\rel@kern{0.4}\kern\dimen@}%
      \advance\dimen@0.4\dimexpr\macc@kerna
%Place the combined final kern (-\dimen@) if it is &gt;0 or if a superscript follows:
      \let\final@kern#2%
      \ifdim\dimen@&lt;\z@ \let\final@kern1\fi
      \if\final@kern1 \kern-\dimen@\fi
    \else
      \overline{\rel@kern{-0.6}\kern\dimen@#1}%
    \fi
  }%
  \macc@depth\@ne
  \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
  \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
  \macc@set@skewchar\relax
  \let\mathaccentV\macc@nested@a
%The following initialises \macc@kerna and calls \mathaccent:
  \if#31
    \macc@nested@a\relax111{#1}%
  \else
%If the argument consists of more than one symbol, and if the first token is
%a letter, use that letter for the computations:
    \def\gobble@till@marker##1\endmarker{}%
    \futurelet\first@char\gobble@till@marker#1\endmarker
    \ifcat\noexpand\first@char A\else
      \def\first@char{}%
    \fi
    \macc@nested@a\relax111{\first@char}%
  \fi
  \endgroup
}
\makeatother

\let\oldbar\bar
\renewcommand*{\bar}[1]{{\mathchoice{\widebar{#1}}{\widebar{#1}}{\widebar{#1}}{\oldbar{#1}}}}

% /!\ Il y a parfois des soucis avec cette macro lorsqu'il y a des caractÃ¨res en minuscule en subscript (et dans les captions). Dans ce cas, faire avec \overline et des espaces manuels.

%%% Double bar
% \overline{\overline{\!a}}

%\newcommand\bx{\,\overline{\!x}}
%\newcommand\brho{\,\overline{\!\rho}}
%better widebar a apparemment des soucis avec les \subcaptions pour les CR :shrug:

%%%---------------------------------------------------------------------------

%%% DELIMITERS

\usepackage{mathtools}
\DeclarePairedDelimiter{\parens}{\lparen}{\rparen}
\DeclarePairedDelimiter{\braces}{\{}{\}}
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
\DeclarePairedDelimiter{\bracks}{[}{]}
\DeclarePairedDelimiter{\angles}{\langle}{\rangle}
\DeclarePairedDelimiter{\aangles}{\langle\!\langle}{\rangle\!\rangle}
\DeclarePairedDelimiter{\pparens}{(\!(}{)\!)}

%%%---------------------------------------------------------------------------

%%% Changer la largeur des adresses
% Jouer avec \hsize :
\makeatletter
\def\@setafterauthor{%
  \vglue3mm%
%  \hspace*{0pt}%
\begingroup\hsize=11.2cm\advance\hsize\abstractmarginL\raggedright
\noindent
%\hspace*{\abstractmarginL}\begin{minipage}[t]{10cm}
   \leftskip\abstractmarginL
  \normalfont\Small
  \@afterauthor\par
\endgroup
\vskip2pt plus 3pt minus 1pt
}
\makeatother

%%%---------------------------------------------------------------------------


\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\title{Existence of an extremal function of critical Sobolev embedding with an $\alpha$-homogeneous weight}
\alttitle{Existence d'une fonction extr\'emale d'une inj\'ection de Sobolev critique avec des poids homog\`enes d'ordre $\alpha$}

\author{\firstname{Petr} \lastname{Gurka}\CDRorcid{0000-0002-0995-4711}}
\address{Department of Mathematics, Czech University of Life Sciences Prague, 165 21, Prague 6, Czech~Republic}
\address{Department of Mathematics, College of Polytechnics Jihlava, TolstÃ©ho 16, 586 01, Jihlava, Czech Republic}
\address{Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, SokolovskÃ¡ 83, 186 75 Praha 8, Czech Republic}
\email{gurka@tf.czu.cz}

\author{\firstname{Daniel} \lastname{Hauer}\CDRorcid{0000-0001-6210-7389}}
\address{Brandenburg University of Technology Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany}
\address{School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, 2006, Australia}
\email{daniel.hauer@b-tu.de}

\thanks{The research of the first author was supported by grant No.~GA23-04720S of the Czech Science Foundation. The second author's research was partially supported by the two Australian Research Council grant~DP200101065 and~DP220100067}
\CDRGrant[Czech Science Foundation]{GA23-04720S}
\CDRGrant[Australian Research Council]{DP200101065}
\CDRGrant[Australian Research Council]{DP220100067}

\subjclass{46E35, 46E30, 35A23, 26D10}

\keywords{\kwd{Trudinger--Moser inequality} \kwd{critical Sobolev embedding} \kwd{Orlicz exponential space} \kwd{$\alpha$-homogeneous weight} \kwd{extremal function}}
\altkeywords{\kwd{In\'egalit\'e de Trudinger--Moser} \kwd{injection de Sobolev critique} \kwd{espace d'Orlicz exponentiel} \kwd{poids homog\`enes d'ordre $\alpha$} \kwd{fonction extr\'emale}}

\begin{abstract}
In [\emph{Calc.\ Var.\ Partial Differ.\ Equ.}~\textbf{60} (2021), no.~1, article no.~16 (27 pages)], we revisited a critical Sobolev-type embedding for weighted Sobolev spaces as introduced in [\emph{J.\ Differ.\ Equations}~\textbf{255} (2013), no.~11, pp.~4312--4336] by Cabr\'e and Ros-Oton. In the same paper, we announced to explore the existence of an extremal function within this framework. In this current work, we not only provide a positive affirmation to this inquiry but extend it to a broader range of weights known as $\alpha$-\emph{homogeneous weights}.
\end{abstract}

\begin{altabstract}
Dans le papier [\emph{Calc.\ Var.\ Partial Differ.\ Equ.}~\textbf{60} (2021), no.~1, article no.~16 (27 pages)], nous avons revisit\'e une injection de Sobolev critique avec des poids comme introduit dans [\emph{J.\ Differ.\ Equations}~\textbf{255} (2013), no.~11, pp.~4312--4336] par Cabr\'e et Ros-Oton. Dans le m\^eme article, nous avons annonc\'e d'Ã©tablir l'existence d'une fonction extr\'emale sous cette hypoth\`ese. Dans ce papier, nous pr\'esentons un rÃ©sultat affirmatif et nous affaiblissons nos hypoth\`eses.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

It is well-known that Trudinger--Moser type inequalities can be considered as the critical case of Sobolev embeddings. To be more precise, recall that the Sobolev space $W^{1,d}(\Rde)$, $d\in\Ne$, is embedded into the Lebesgue space $L^q(\Rde)$ for any $q\in[d,\infty)$, although $W^{1,d}(\Rde)$ is not embedded into $L^\infty(\Rde)$. It was proved in 1967 by Trudinger~\cite{Tr} that there is an embedding of this type of $W^{1,d}_0(\Om)$ into the Orlicz space $L^\Phi(\Om)$, with $\Om$ being a bounded domain in $\Rde$, and the Young function $\Phi(x)=\exp\parens[\big]{\alpha\abs{x}^{d/(d-1)}}$, $t&gt;0$, where $\al$ is a positive constant. However, this embedding was announced earlier by Yudovich~\cite{Yu} without a proof and, in a slightly weaker form, proved by Pokhozhaev~\cite{Po}. In 1971 Moser~\cite{Moser71} showed that if $\al\le\al_d=d\om_{d-1}^{1/(d-1)}$, where $\om_{d-1}$ denotes the surface area of the unit sphere in $\Rde$, then
\[
\sup_u\int_{\Om}\exp\parens[\big]{\alpha\abs{x}^{d/(d-1)}}\dd x&lt;\infty,
\]
where the supremum is taken over all functions $u$ from the unit ball of the space $W^{1,d}_0(\Om)$. When $\al&gt;\al_d$, this supremum becomes infinite.

A natural question arises as to whether there exists a function $u$ such that the supremum is attained. The first step towards solving this problem was made in 1986 by Carleson and Chang~\cite{CaCh}, who showed that an extremal function $u$ exists if $\Om$ is a ball in $\Rde$. In 1988, Struwe~\cite{MR970849} demonstrated that this result continues to hold for small perturbations of the ball in $\Re^2$, and in 1992, Flucher~\cite{Flu} extended the result to any bounded domain in $\Re^2$. Finally, in 1995, Lin~\cite{Lin} generalised the result to any dimension. Note that a key ingredient of his proof is the use of $d$-Green's functions, the singular solutions to the $d$-Laplacian.

In our previous paper, published in 2021~\cite{GuHa21}, we studied a weighted version of the Trudinger--Moser inequality and its relationship with the corresponding Sobolev-type embedding. This research was based on recent results by Cabr\'e\ and Ros-Oton~\cite{CaRoOt13} and Lam~\cite{Lam17}, whose function spaces involved the so-called \emph{monomial} weight. In our paper, we presented some weighted analogues of the classical results. Namely, we found the ``optimal'' Orlicz spaces (within the class of Orlicz spaces) of the corresponding critical Sobolev-type embedding and showed that this embedding is not compact. Moreover, we proved that an embedding into any properly larger Orlicz space is compact, and we also derived a corresponding \emph{concentrated compactness principle}. However, we left the question of the existence of an extremal function unsolved. In this paper, we aim to address this gap, at least for the case where $\Om$ is a ball in $\Rde$. We are working in a slightly more general context, as our weight is a \emph{homogeneous} weight (see Section~\ref{Prelim}).

\section{Preliminaries} \label{Prelim}

\subsection*{Weighted measure on a convex cone}

We denote by $\Sigma$ an open convex cone in $\Rde$, $d\in\Ne$, $d\ge 2$, with vertex at the origin. Following~\cite{CRoS16}, $w\colon \bar{\Sigma} \to [0, \infty)$ is a nonnegative (not identically zero) continuous function that is \emph{$\al$-homogeneous}, $\al&gt;0$, and such that $w^{1/\alpha}$ is concave in $\Sigma$. Recall that $w$ is $\al$-ho\-mo\-geneous if
\begin{equation}\label{Wfunc}
w(\kappa x)=\kappa^\al w(x)\quad\text{for any $x\in\bar{\Sigma}$ and all $\kappa&gt;0$}.
\end{equation}
Throughout the paper $\mu$ denotes the weighted measure on $\Sigma$ defined as
\[
\dd\mu(x)=w(x) \dd x.
\]
For future reference, we set
\[
D=d+\al\qquad\text{and}\qquad D'=\frac{D}{D-1}.
\]
Important examples of $\al$-homogeneous functions are the \emph{monomial weights}. Namely, let $\Sigma = \Sigma_1 \cap \dots \cap \Sigma_k$, $k\in\{1, \dots, d\}$, where $\Sigma_j\neq\Rde$, $j=1, \dots, k$, are open convex cones in $\Rde$ with vertex at the origin. Given $A_1, \dots, A_k&gt;0$, the weight $w\colon \bar{\Sigma} \to [0, \infty)$ defined as
\[
w(x) = \prod_{j = 1}^k \dist(x, \partial \Sigma_j)^{A_j}, \quad x\in \bar{\Sigma},
\]
satisfies the $\al$-homogeneity assumptions. For other interesting examples we refer the reader to~\cite[Section~2]{CRoS16}. In particular, when $\Sigma_j = \braces[\big]{x\in\Rde \sT x_j &gt; 0}$, $j=1, \dots, k$, the function (cf.\ also~\cite{CaRoOt13})
\[
w(x) = x_1^{A_1}\cdots x_k^{A_k}, \quad x\in \bar{\Sigma},
\]
we considered in our previous paper~\cite{GuHa21}.

\subsection*{Basic definitions}

%\begin{defi}[Measure spaces]
%By the symbol $(X,\nu)$ we denote a measure space~$X$ with a nonnegative $\sigma$-finite measure~$\nu$. If $\Om$ is a $\nu$-measurable subset of $X$, we denote by $\nu(\Om)$ the $\nu$-measure of $\Om$, that is, $\nu(\Om)=\int_\Om\dd\nu(x)$. If $X=\Rde$ with its $n$-dimensional Lebesgue measure $\dd\nu(x)=\dd x$ and $M\subset\Rde$ is a Lebesgue measurable set in $\Rde$, we use the notation $\abs{M}=\int_\Om\dd x$.
%\end{defi}
%
%\begin{defi}[Lebesgue spaces]
%If $(X,\nu)$ is a measure space, we denote by $L^p(\Om,\nu)$, $p\in[1,\infty]$, the \emph{Lebesgue space}, of all $\nu$-measurable functions~$f$ on a $\nu$-measurable set $\Om$ in $X$, equipped with the norm
%\[
%\norm{f}_{p,\Om,\nu}=
%\begin{cases}
%\parens[\bigg]{\int_\Om \abs[\big]{f(x)}^p\dd \nu(x)}^{1/p}	&amp; \text{if $p&lt;\infty$}, \\
%\nuesssup_{x\in\Om}\abs[\big]{f(x)} 					&amp; \text{if $p=\infty$}.
%\end{cases}
%\]
%If $X=\Rde$ and $\nu$ is the $d$-dimensional Lebesgue measure, then we write $\norm{\,\cdot\,}_{p,\Om}$ instead of $\norm{\,\cdot\,}_{p,\Om,\nu}$. Moreover, we simply write $\norm{\,\cdot\,}_{p,\nu}$ (or $\norm{\,\cdot\,}_{p}$) when $\Om=X$ (or $\Om=\Rde$). For $p\in[1,\infty]$ we define the \emph{H\"older conjugate} number $p'\in[1,\infty]$ by the equality $\frac1p+\frac1{p'}=1$.
%\end{defi}
%
%\begin{defi}[Sobolev spaces]
%Let $\Om$ be a domain in a measure space $(\Rde,\nu)$ with a nonnegative Borel measure~$\nu$ and let $p\in[1,\infty]$. The \emph{Sobolev space} $W^{1,p}_0(\Om,\nu)$ is defined as the closure of $C_c^\infty(\Om)$ (the space of infinitely differentiable functions with compact supports in $\Om$) with respect to the norm
%\[
%\norm{u}=\norm{\nabla u}_{p,\Om,\nu},
%\]
%where $\nabla u$ is the gradient of $u$ and $\abs{\nabla u}$ is its Euclidean length, that is,
%\[
%\nabla u=\parens[\Bigg]{\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}},
%\qquad \abs{\nabla u}=\parens[\Bigg]{\sum_{j=1}^{n}({\partial u}/{\partial x_j})^2}^{1/2}
%\]
%(to simplify the notation we are writing $\norm{\nabla u}_{p,\Om,\nu}$ instead of $\norm[\big]{\abs{\nabla u}}_{p,\Om,\nu}$).
%\end{defi}
%
%\begin{defi}[Distributions and rearrangements]
%For a measurable function $f$ in $\Sigma$ we define its \emph{distribution function with respect to $\mu$} as
%\[
%f_{*\mu}(\tau)=\mu\parens[\bigg]{\braces[\bigg]{x\in \Sigma \sT \abs[\big]{f(x)}&gt;\tau}}, \quad \tau&gt;0,
%\]
%and its \emph{nonincreasing rearrangement with respect to $\mu$} as
%\[
%f^*_\mu(t)=\inf\braces[\big]{\tau&gt;0 \sT f_{*\mu}(\tau)\le t},\quad t&gt;0.
%\]
%The function $f^\ast_\mu$ defined as
%\begin{equation}\label{SymmRearr}
%f^\ast_\mu(x) = f^*_\mu\parens[\big]{C_D \abs{x}^D},\quad x\in\Rde,
%\end{equation}
%where
%\begin{equation}\label{unit_ball_in_sigma_measure}
%C_D = \mu(B_1\cap \Sigma),
%\end{equation}
%is the \emph{radial rearrangement} of $f$ \emph{with respect to} $\mu$. We use the notation
%\[
%B_r=\braces[\big]{x\in\Rde \sT \abs{x}&lt;r},\quad r&gt;0.
%\]
%Thanks to the $\al$-homogeneity of $w$, we have
%\[
%\mu(B_r\cap \Sigma) = C_D r^D, \quad r&gt;0.
%\]
%Note that the function $f^\ast_\mu$ is nonnegative and radially nonincreasing. Though it is defined on the whole $\Rde$, it depends only on function values of $f$ in $\Sigma$. Furthermore, the functions $f$ and $f^\ast_\mu$ are equimeasurable with respect to $\mu$, that is,
%\[
%\mu\parens[\bigg]{\braces[\bigg]{x\in \Sigma \sT \abs[\big]{f(x)}&gt;\tau}} =
%\mu\parens[\bigg]{\braces[\bigg]{x\in\Sigma \sT \abs[\big]{f^\ast_\mu(x)}&gt;\tau}}, \quad \tau&gt;0.
%\]
%\end{defi}
\begin{proof}[Measure spaces]
By the symbol $(X,\nu)$ we denote a measure space~$X$ with a nonnegative $\sigma$-finite measure~$\nu$. If $\Om$ is a $\nu$-measurable subset of $X$, we denote by $\nu(\Om)$ the $\nu$-measure of $\Om$, that is, $\nu(\Om)=\int_\Om\dd\nu(x)$. If $X=\Rde$ with its $n$-dimensional Lebesgue measure $\dd\nu(x)=\dd x$ and $M\subset\Rde$ is a Lebesgue measurable set in $\Rde$, we use the notation $\abs{M}=\int_\Om\dd x$.
\let\qed\relax
\end{proof}

\begin{proof}[Lebesgue spaces]
If $(X,\nu)$ is a measure space, we denote by $L^p(\Om,\nu)$, $p\in[1,\infty]$, the \emph{Lebesgue space}, of all $\nu$-measurable functions~$f$ on a $\nu$-measurable set $\Om$ in $X$, equipped with the norm
\[
\norm{f}_{p,\Om,\nu}=
\begin{cases}
\parens[\bigg]{\int_\Om \abs[\big]{f(x)}^p\dd \nu(x)}^{1/p}	&amp; \text{if $p&lt;\infty$}, \\
\nuesssup_{x\in\Om}\abs[\big]{f(x)} 					&amp; \text{if $p=\infty$}.
\end{cases}
\]
If $X=\Rde$ and $\nu$ is the $d$-dimensional Lebesgue measure, then we write $\norm{\,\cdot\,}_{p,\Om}$ instead of $\norm{\,\cdot\,}_{p,\Om,\nu}$. Moreover, we simply write $\norm{\,\cdot\,}_{p,\nu}$ (or $\norm{\,\cdot\,}_{p}$) when $\Om=X$ (or $\Om=\Rde$). For $p\in[1,\infty]$ we define the \emph{H\"older conjugate} number $p'\in[1,\infty]$ by the equality $\frac1p+\frac1{p'}=1$.
\let\qed\relax
\end{proof}

\begin{proof}[Sobolev spaces]
Let $\Om$ be a domain in a measure space $(\Rde,\nu)$ with a nonnegative Borel measure~$\nu$ and let $p\in[1,\infty]$. The \emph{Sobolev space} $W^{1,p}_0(\Om,\nu)$ is defined as the closure of $C_c^\infty(\Om)$ (the space of infinitely differentiable functions with compact supports in $\Om$) with respect to the norm
\[
\norm{u}=\norm{\nabla u}_{p,\Om,\nu},
\]
where $\nabla u$ is the gradient of $u$ and $\abs{\nabla u}$ is its Euclidean length, that is,
\[
\nabla u=\parens[\Bigg]{\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}},
\qquad \abs{\nabla u}=\parens[\Bigg]{\sum_{j=1}^{n}({\partial u}/{\partial x_j})^2}^{1/2}
\]
(to simplify the notation we are writing $\norm{\nabla u}_{p,\Om,\nu}$ instead of $\norm[\big]{\abs{\nabla u}}_{p,\Om,\nu}$).
\let\qed\relax
\end{proof}

\begin{proof}[Distributions and rearrangements]
For a measurable function $f$ in $\Sigma$ we define its \emph{distribution function with respect to $\mu$} as
\[
f_{*\mu}(\tau)=\mu\parens[\big]{\braces[\big]{x\in \Sigma \sT \abs[\big]{f(x)}&gt;\tau}}, \quad \tau&gt;0,
\]
and its \emph{nonincreasing rearrangement with respect to $\mu$} as
\[
f^*_\mu(t)=\inf\braces[\big]{\tau&gt;0 \sT f_{*\mu}(\tau)\le t},\quad t&gt;0.
\]
The function $f^\ast_\mu$ defined as
\begin{equation}\label{SymmRearr}
f^\ast_\mu(x) = f^*_\mu\parens[\big]{C_D \abs{x}^D},\quad x\in\Rde,
\end{equation}
where
\begin{equation}\label{unit_ball_in_sigma_measure}
C_D = \mu(B_1\cap \Sigma),
\end{equation}
is the \emph{radial rearrangement} of $f$ \emph{with respect to} $\mu$. We use the notation
\[
B_r=\braces[\big]{x\in\Rde \sT \abs{x}&lt;r},\quad r&gt;0.
\]
Thanks to the $\al$-homogeneity of $w$, we have
\[
\mu(B_r\cap \Sigma) = C_D r^D, \quad r&gt;0.
\]
Note that the function $f^\ast_\mu$ is nonnegative and radially nonincreasing. Though it is defined on the whole $\Rde$, it depends only on function values of $f$ in $\Sigma$. Furthermore, the functions $f$ and $f^\ast_\mu$ are equimeasurable with respect to $\mu$, that is,
\[
\mu\parens[\big]{\braces[\big]{x\in \Sigma \sT \abs[\big]{f(x)}&gt;\tau}} =
\mu\parens[\big]{\braces[\big]{x\in\Sigma \sT \abs[\big]{f^\ast_\mu(x)}&gt;\tau}}, \quad \tau&gt;0.
\]
\let\qed\relax
\end{proof}

\subsection*{P\'olya--Szeg\"o inequality}

\begin{theo}[{\cite[Theorem~3.1]{GLM23}}]\label{theo:PS}
Let $1\le p &lt; \infty$. For every function $u\in\mathcal C_c^1(\Rde)$, its radial rearrangement $u^\ast_\mu$ with respect to $\mu$ is locally absolutely continuous and
\begin{equation}\label{theo:PS:ineq}
\norm{\nabla u^\ast_\mu}_{p,\mu} \le \norm{\nabla u}_{p,\mu}.
\end{equation}
In particular, $\spt u^\ast_\mu = \bar{B}_R$ where $R$ is such that $\mu(B_R \cap \Sigma) = \mu(\spt u \cap \Sigma)$.
\end{theo}

We need another obvious result. Here, a function $\Psi \colon (0,\infty)\to(0,\infty)$ is called \emph{nondecreasing} if for every $x$, $y\in (0,\infty)$ with $x\le y$, one has that $\Psi(x)\le \Psi(y)$.

\begin{lemm}\label{RearrReduc}
Let\/ $\Psi \colon (0,\infty)\to(0,\infty)$ be a nondecreasing and measurable function. Then
\[
\int_{\Rde}\Psi\parens[\big]{u_\mu^\ast(x)}\dd\mu(x) =
\int_{\Rde}\Psi\parens[\big]{\abs[\big]{u(x)}}\dd\mu(x).
\]
\end{lemm}

\subsection*{Reduction to the one-dimensional problem}

\begin{prop}\label{PropAAA}
Let $u\in \mathcal{C}_c^1(\Rde)$ be a nonnegative function with $\spt u\subset\Om$ such that $\mu(\Om)&lt;\infty$. Assume that\/ $0&lt;a\le c_D$ and $R&gt;0$ is the number satisfying $\mu(B_R \cap \Sigma) = \mu(\Om \cap \Sigma)$. Let~$u_\mu^\ast$ be the radial rearrangement of $u$ with respect to $\mu$. Set
\begin{equation}\label{fuU}
U(\abs{x})=u^\ast(x),\quad x\in\Rde\cap\Sigma,
\end{equation}
and
\begin{equation}\label{ustar03}
\f(t)=C_D^{1/D'}\,U(Re^{-t/D}),\quad t\in(0,\infty).
\end{equation}
Then $\f$ satisfies that $\f'\ge0$ on $(0,\infty)$ and $\f(0)=0$. Moreover,
\[
\int_{\Om\cap\Sigma}\abs[\big]{\nabla u(x)}^D\dd\mu(x)
\ge
\int_{B_R\cap\Sigma}\abs[\big]{\nabla u^\ast_\mu(x)}^D\dd\mu(x) =
\int_{0}^{\infty}\parens[\big]{\f'(t)}^D\dd t
\]
and
\[
\begin{split}
\frac{1}{\mu(\Om\cap\Sigma)}\int_{\Om\cap\Sigma}\exp\parens[\big]{a\parens[\big]{u(x)}^{D'}}\dd\mu(x) &amp; =
\frac{1}{\mu(B_R\cap\Sigma)}\int_{B_R\cap\Sigma}\exp\parens[\big]{a\parens[\big]{u^\ast_\mu(x)}^{D'}}\dd\mu(x) \\
&amp; = \int_{0}^{\infty}\exp\parens[\big]{\be \f^{D'}(t)-t}\dd t,
\end{split}
\]
where $\be=a/C_D\in(0,1]$.
\end{prop}

\begin{proof}
By Lemma~\ref{RearrReduc} with $\Psi(t)=\exp(a t^{D'})$ we have
\begin{equation}\label{ustar01}
\frac{1}{\mu(\Om\cap\Sigma)}
\int_{\Om\cap\Sigma}\exp\parens[\big]{a\abs[\big]{u(x)}^{D'}}\dd\mu(x) =
\frac{1}{\mu(B_R\cap\Sigma)}
\int_{B_R\cap\Sigma}\exp\parens[\big]{a\parens[\big]{u^\ast(x)}^{D'}}\dd\mu(x).
\end{equation}
Since $u^\ast$ is radially symmetric, we obtain
\[
\begin{split}
\int_{B_R\cap\Sigma}\exp\parens[\big]{a\parens[\big]{u^\ast(x)}^{D'}}\dd\mu(x)	&amp; =
\int_{0}^{R}\parens[\Bigg]{\int_{x\in\Sigma, \, \abs{x}=r}\exp\parens[\big]{a\parens[\big]{u^\ast(x)}^{D'}}w(x)\dd\sg(x)}\dd r \\
&amp; =
\int_{0}^{R}\exp\parens[\big]{a\parens[\big]{U(r)}^{D'}}\parens[\Bigg]{\int_{x\in\Sigma, \, \abs{x}=r}w(x)\dd\sg(x)}\dd r \\
&amp; = P_w(B_1\cap\Sigma)\int_{0}^{R}\exp\parens[\big]{a\parens[\big]{U(r)}^{D'}}\,r^{D-1}\dd r,
\end{split}
\]
where $U(\abs{x})=u^\ast(x)$ and $ P_w(B_1\cap\Sigma)=\int_{x\in\Sigma, \, \abs{x}=1}w(x)\dd\sg(x)$ ($\sg$ is the Hausdorff measure on the unit sphere centred at the origin). We make the change of variables
\begin{equation}\label{ChgVAR}
r=Re^{-t/D}
\end{equation}
in the last integral to obtain
\[
\int_{0}^{R}\exp\parens[\big]{a\parens[\big]{U(r)}^{D'}}\,r^{D-1}\dd r = D^{-1}R^D\int_{0}^{\infty}\exp\parens[\big]{a\parens[\big]{U(Re^{-t/D})}^{D'}-t}\dd t.
\]
Setting $\be=a/C_D\in(0,1]$ and using the fact that
\[
C_D =
\mu(B_R\cap\Sigma) =\int_{B_R\cap\Sigma}w(x)\dd x=D^{-1} P_w(B_1\cap\Sigma)R^D,
\]
we arrive at
\begin{equation}\label{ustar02}
\frac{1}{\mu(B_R\cap\Sigma)}
\int_{B_R\cap\Sigma}\exp\parens[\big]{a\parens[\big]{u^\ast(x)}^{D'}}\dd\mu(x) =
\int_{0}^{\infty}\exp\parens[\big]{\be\parens[\big]{\f(t)}^{D'}-t}\dd t,
\end{equation}
where the function $\f$ is defined by~\eqref{ustar03}. Similarly, since $\abs[\big]{\nabla u^\ast(x)}=\abs[\big]{U'(\abs{x})}$, $x\in\Rde\cap\Sigma$, we have
\[
\int_{B_R\cap\Sigma}\abs[\big]{\nabla u^\ast(x)}^D\dd\mu(x) = P_w(B_1\cap\Sigma)\int_{0}^{R}\abs[\big]{U'(r)}^D\,r^{D-1}\dd r.
\]
Differentiating~\eqref{ustar03} we have, for all $t\in(0,\infty)$,
\[
\f'(t)=-D^{-1}C_D^{1/D'}\,U'(Re^{-t/D})Re^{-t/D} = -D^{-1}C_D^{1/D'}\,U'(r)r.
\]
Observe that, since $U$ is nonincreasing, the right-hand side is nonnegative. Change of variables~\eqref{ChgVAR} gives
\[
\begin{split}
\int_{0}^{R}\abs[\big]{U'(r)}^D\,r^{D-1}\dd r &amp; = D^{-1}(D^{-1}C_D^{1/D'})^{-D}\,\int_{0}^{\infty}\abs[\bigg]{D^{-1}C_D^{1/D'}
\,U'(Re^{-t/D})Re^{-t/D}}^D\dd t \\
&amp; = D^{-1}(D^{-1}C_D^{1/D'})^{-D}\,\int_{0}^{\infty}\abs[\big]{\f'(t)}^D\dd t.
\end{split}
\]
That is,
\[
\int_{B_R\cap\Sigma}\abs[\big]{\nabla u^\ast(x)}^D\dd\mu(x) = P_w(B_1\cap\Sigma)D^{-1}(D^{-1}C_D^{1/D'})^{-D}\,\int_{0}^{\infty}\abs[\big]{\f'(t)}^D\dd t =
\int_{0}^{\infty}\abs[\big]{\f'(t)}^D\dd t.
\]
The proof is completed.
\end{proof}

\begin{lemm}\label{L01}
Let $q\in(1,\infty)$. Consider nonnegative functions $\f\in C^1[0,\infty)$ such that $\f(0)=0$, $\f'\ge0$ on $(0,\infty)$ and $\int_{0}^{\infty}\parens[\big]{\f'(t)}^q\dd t\le1$. Then
\begin{equation}\label{MoserSup}
A(q)=\sup_{\f}\int_{0}^{\infty}\exp\parens[\big]{\parens[\big]{\f(t)}^{q'}-t}\dd t&lt;\infty.
\end{equation}
\end{lemm}

\begin{proof}
The proof for $q\ge2$ was given by Moser~\cite{Moser71}, for $q\in(1,2)$ it was extended by Jodeit~\cite{Jod}.
\end{proof}

\begin{lemm}\label{L02}
The supremum in~\eqref{MoserSup} is attained for every $q&gt;1$.
\end{lemm}

\begin{proof}
The result for $q\in\Ne$, $q\ge2$ was proved first by Carleson and Chang~\cite{CaCh}, it was extended to real $q&gt;1$ by Hudson and Leckband~\cite{HuLe02}.
\end{proof}

\section{Main result}

Now we formulate the main theorem.

\begin{theo}
Assume that $p\in (1,\infty)$, $p=D$, and for $\alpha&gt;0$, $w$ is an $\alpha$-homogeneous function such that $w^{1/\alpha}$ is concave in $\Sigma$. Then
\[
S=\sup_u\frac{1}{\mu(\Om\cap\Sigma)}
\int_{\Om\cap\Sigma}\exp\parens[\big]{a\abs[\big]{u(x)}^{D'}}\dd\mu(x) &lt;\infty,
\]
where the supremum is taken over all $u\in \mathcal{C}_c^1(\Rde)$ such that $\spt u\subset\Om$ with $\mu(\Om)&lt;\infty$ and $\norm{\nabla u}_{p,\mu}\le1$. If\/ $\Om=B_R$ with some $R&gt;0$, then the supremum is attained.
\end{theo}

\begin{proof}
To prove that $S$ is finite we apply Proposition~\ref{PropAAA} to obtain
\[
1\ge\int_{\Om\cap\Sigma}\abs[\big]{\nabla u(x)}^D\dd\mu(x)
\ge
\int_{0}^{\infty}\parens[\big]{\f'(t)}^D\dd t,
\]
and so, by Lemma~\ref{L01} and Proposition~\ref{PropAAA},
\[
\sup_u\frac{1}{\mu(\Om\cap\Sigma)}
\int_{\Om\cap\Sigma}\exp\parens[\big]{a\abs[\big]{u(x)}^{D'}}\dd\mu(x) =
\sup_\f\int_{0}^{\infty}\exp\parens[\big]{\parens[\big]{\f(t)}^{D'}-t}\dd t&lt;\infty.
\]
For proving the second part it is enough to take the function $u(x)=U(\abs{x})$, $x\in B_R\cap\Sigma$, where $U$ is such that~\eqref{fuU} holds and $\f$ is the extremal function from Lemma~\ref{L02} when $q=D$.
\end{proof}

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

\printbibliography

\end{document}</pre></body></html>