A critical phenomenon for sublinear \\ elliptic equations in cone--like domains
Vladimir Kondratiev
Vitali Liskevich
Vitaly Moroz and Zeev Sobol
We study positive supersolutions to an elliptic equation $(\ast)$ $-\Delta u=c|x|^{-s}u^p$, $\:p,s\in\R$, in cone--like domains in $\R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_\ast<1$ such that equation $(\ast)$ has a positive supersolution if and only if $-\infty<p<p_\ast$. The value of $p_\ast$ is determined explicitly by $s$ and the geometry of the cone.
klms-arXiv.tex
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\title{A critical phenomenon for sublinear \\ elliptic equations in cone--like domains}
%\author{Vladimir Kondratiev, Vitali Liskevich, Vitaly Moroz and Zeev Sobol}
\author{
{\Large Vladimir Kondratiev}\\
Department of Mathematics\\
and Mechanics\\
Moscow State University\\
Moscow 119 899, Russia\\
{\tt kondrat@vnmok.math.msu.su}\\
\and
{\Large Vitali Liskevich}\\
School of Mathematics\\
University of Bristol\\
Bristol BS8 1TW\\
United Kingdom\\
{\tt v.liskevich@bristol.ac.uk}\\
\smallskip\and
{\Large Vitaly Moroz}\\
School of Mathematics\\
University of Bristol\\
Bristol BS8 1TW\\
United Kingdom\\
{\tt v.moroz@bristol.ac.uk}\\
\and
{\Large Zeev Sobol}\\
Department of Mathematics\\
Imperial College, London\\
London SW7 2AZ\\
United Kingdom\\
{\tt z.sobol@imperial.ac.uk}}

\date{}

\begin{document}
\maketitle

\begin{abstract}
We study positive supersolutions to an elliptic equation $(\ast)$
$-\Delta u=c|x|^{-s}u^p$, $\:p,s\in\R$, in cone--like domains in $\R^N$ ($N\ge 2$).
We prove that in the sublinear case $p<1$ there exists a critical exponent
$p_\ast<1$ such that equation $(\ast)$ has a positive supersolution if and only if $-\infty<p<p_\ast$.
The value of $p_\ast$ is determined explicitly by $s$ and the geometry of the cone.
\end{abstract}


\section{Introduction}

We study the existence and nonexistence of positive solutions and supersolutions
to the equation
\begin{equation}\label{*}
-\Delta u=\frac{c}{|x|^{s}}u^p\quad\mbox{in }\:\C_\Omega^\rho.
\end{equation}
Here $p\in\R$, $s\in\R$, $c>0$ and $\C_\Omega^\rho\subset\R^N$ ($N\ge 2$)
is an unbounded cone--like domain
$$\C_\Omega^\rho:=\{(r,\omega)\in\R^N:\:\omega\in\Omega,\:r>\rho\},$$
where $(r,\omega)$ are the polar coordinates in $\R^N$, $\rho>0$ and
$\Omega\subseteq S^{N-1}$ is a subdomain (a connected open subset) of the unit sphere $S^{N-1}$ in $\R^N$.
We say that $u\in H^1_{loc}(\C_\Omega^\rho)$ is a {\em supersolution} ({\em subsolution})
to equation (\ref{*}) if
$$\int_{\C_\Omega^\rho}\nabla u\cdot\nabla\varphi\:dx\ge(\le)
\int_{\C_\Omega^\rho}\frac{c}{|x|^s}u^p\varphi\:dx\quad\mbox{for all }\:
0\le\varphi\in C^\infty_0(\C_\Omega^\rho).$$
If $u$ is a sub and supersolution to (\ref{*}) then $u$
is said to be a {\em solution} to (\ref{*}).
By the weak Harnack inequality any nontrivial nonnegative supersolution to (\ref{*})
is positive in $\C_\Omega^\rho$.

We define {\em critical exponents} for equation (\ref{*}) by
$$p^\ast=p^\ast(\Omega,s)=
\inf\{p>1:\mbox{(\ref{*}) has a positive supersolution in $\C_\Omega^\rho$ for some $\rho>0$}\},$$
$$p_\ast=p_\ast(\Omega,s)=
\sup\{p<1:\mbox{(\ref{*}) has a positive supersolution in $\C_\Omega^\rho$ for some $\rho>0$}\}.$$
Set $p_\ast=-\infty$ if (\ref{*}) has no positive supersolution in $\C_\Omega^\rho$
for any $p<1$.

\begin{remark}
$(i)$
One can show that if $p<p_\ast$ or $p>p^\ast$ then (\ref{*}) has a positive solution in $\C_\Omega^\rho$
(see \cite{KLM} for the proof of the case $p>1$ and the proofs below for the case $p<1$).
The existence (or nonexistence) of positive (super) solutions
at the critical values $p_\ast$ and $p^\ast$ is a separate issue.

$(ii)$
Observe that in view of the scaling invariance of the Laplacian the critical
exponents $p_\ast$ and $p^\ast$ do not depend on $\rho>0$.
%Indeed, if $u>0$ is a solution to (\ref{*}) in $\C_\Omega^\rho$ then,
%by the scaling properties of the Laplacian,
%$\rho^{2}u(x/\rho)$ is a solution to (\ref{*}) in $\C_\Omega^1$.
%So in what follows we fix $\rho=1$.

$(iii)$
We do not make any assumptions on the smoothness of the domain $\Omega\subseteq S^{N-1}$.
\end{remark}


Let $\lambda_1=\lambda_1(\Omega)\ge 0$ be the principal eigenvalue
of the Dirichlet Laplace--Beltrami operator $-\Delta_\omega$ on $\Omega$.
Let $\alpha_+\ge 0$ and $\alpha_-<0$ be the roots of the quadratic equation
$$\alpha(\alpha+N-2)=\lambda_1(\Omega).$$
In the {\em superlinear} case $p>1$ the value of the critical exponent is $p^\ast=1-\frac{2-s}{\alpha_-}$.
Moreover, if $s<2$ then (\ref{*}) has no positive supersolutions in the critical case $p=p^\ast$.
This has been proved by Bandle and Levine \cite{Bandle-Levine}, Bandle and Essen \cite{Bandle}
and Berestycki, Capuzzo--Dolcetta and Nirenberg \cite{BCN}
(see also \cite{KLM} for yet another proof of this result and
for  equations with measurable coefficients).

The {\em sublinear} case $p<1$ has been studied in
\cite{Brezis-Kamin,KLS}. From the result of Brezis and Kamin \cite{Brezis-Kamin}
it follows that for $p\in(0,1)$ equation (\ref{*}) has a bounded
positive solution in $\R^N$ if and only if $s>2$.
It has been proved in \cite{KLS} (amongst other things) that
for any $p\in(-\infty,1)$ equation
(\ref{*}) has a positive supersolution outside a ball in $\R^N$ if and only if $s>2$.


In this note, we discover a new critical phenomenon. Namely, we show that in sublinear case
equation (\ref{*}) exhibits a "non-trivial" critical exponent ($p_\ast>-\infty$) in cone-like domains.
The main result of the paper reads as follows.

\begin{theorem}\label{Main}
For $p\le 1$, the critical exponent for equation (\ref{*}) is
$p_\ast=\min\{1-\frac{2-s}{\alpha_+},1\}$.
If $p_\ast<1$ then (\ref{*}) has no positive supersolutions in the critical case $p=p_\ast$.
\end{theorem}

\begin{figure}[t]
\input{pict-1.eepic}
\caption{Existence and nonexistence zones for equations (\ref{*}) (left) and (\ref{hat}) (right).
\label{fig}}
\end{figure}


\begin{remark}
$(i)$ If $\alpha_+=0$ then we set $p_\ast=-\infty$.

$(ii)$
If $s>2$ then $p_\ast=p^\ast=1$ and (\ref{*}) has positive solutions for any $p\in\R$ \cite{Brezis-Kamin,KLS}.
If $s=2$ then $p_\ast=p^\ast=1$.
In this critical case (\ref{*}) becomes a linear equation with the
potential $c|x|^{-2}$, which has a positive (super) solution
if and only if $c\le \frac{(N-2)^2}{4}+\lambda_1(\Omega)$.

$(iii)$
Let $S_k=\{x\in S^{N-1}:x_1>0,\dots x_k>0\}$.
Then $\lambda_1(S_k)=k(k+N-2)$ and $\alpha_+(S_k)=k$, $\alpha_-(S_k)=2-N-k$.
Hence $p_\ast(S_k,s)=1-\frac{2-s}{k}$ and  $p^\ast(S_k,s)=1-\frac{2-s}{2-N-k}$.
In particular, in the case of the halfspace $S_1$ we have
$p_\ast(S_1,s)=s-1$ and $p^\ast(S_1,s)=\frac{N+1-s}{N-1}$.
\end{remark}

Applying the Kelvin transformation $y=y(x)=\frac{x}{|x|^2}$
we see that if $u$ is a positive solution to (\ref{*}) in $\C_\Omega^1$
then $\hat{u}(y)=|y|^{2-N}u(x(y))$ is a positive solution to
\begin{equation}\label{hat}
-\Delta\hat{u}=\frac{c}{|y|^{\sigma}}\hat{u}^p\quad\mbox{in }\:\widehat{\C}_\Omega^1,
\end{equation}
where $\sigma=(N+2)-p(N-2)-s$ and
$\widehat{\C}_\Omega^{\:1}:=\{(r,\omega)\in\R^N:\:\omega\in\Omega,\:0<r<1\}$.
We define the critical exponents $\widehat{p}^\ast=\widehat{p}^\ast(\Omega,s)$ and $\widehat{p}_\ast=\widehat{p}_\ast(\Omega,s)$
for equation (\ref{hat}) similarly to $p^\ast(\Omega,s)$ and $p_\ast(\Omega,s)$.
In the superlinear case $p>1$, Bandle and Essen \cite{Bandle} proved that if
$\sigma>2$ then
$\widehat{p}^\ast=1-\frac{2-\sigma}{\alpha_+}$ and (\ref{hat})
has no positive supersolutions when $p=\widehat{p}^\ast(\Omega)$.
In the sublinear case $p<1$ by an easy computation we derive from Theorem \ref{Main} the following result.

\begin{theorem}\label{Kelvin}
For $p\le 1$, the critical exponent for equation (\ref{hat}) is
$\widehat{p}_\ast=\min\{1-\frac{2-\sigma}{\alpha_-},1\}$.
If $\widehat{p}_\ast<1$ then (\ref{hat}) has no positive supersolutions
in the critical case $p=\widehat{p}_\ast$.
\end{theorem}


In the remaining part of the paper we prove Theorem \ref{Main}.



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


\paragraph{Existence.}
In the polar coordinates equation (\ref{*}) reads as follows
\begin{equation}\label{polar}
-u_{rr}-\frac{N\!-\!1}{r}u_r-\frac{1}{r^2}\Delta_\omega u = \frac{c}{r^{s}}u^p
\quad\mbox{in }\:\C_\Omega^1.
\end{equation}
Let $s\le 2, \, p< 1 - \frac{2-s}{\alpha_+}$.
Let $0<\psi\in H^1_{loc}(\Omega)$ be a positive solution to the equation
\begin{equation}\label{psi}
-\Delta_\omega\psi-\alpha(\alpha+N-2)\psi=\psi^{p}\quad\mbox{in }\:\Omega,
\end{equation}
where $\alpha:=\frac{2-s}{1-p}$.
Then  it is readily seen that
$u:=c^\frac{1}{1-p}r^{\alpha}\psi\in H^1_{loc}(\C_\Omega^1)$
is a positive solution to $(\ref{polar})$ in $\C_\Omega^1$.
Thus the problem reduces to the existence of
positive solutions to (\ref{psi}).

Note that
$0<\alpha(\alpha+N-2)<\lambda_1(\Omega)$.
Hence the operator $-\Delta_\omega-\alpha(\alpha+N-2)$
is coercive on $H^1_0(\Omega)$ and satisfies the maximum principle.
We consider separately the cases $p\in[0,1)$ and $p<0$.

\paragraph{\it Case $p\in[0,1)$.} %This case is standard see, e.g. \cite{Brezis-Kamin}.
Let $\phi_1>0$ be the principal Dirichlet eigenfunction of $-\Delta_\omega$ on $\Omega$.
Let $\ol\phi>0$ be the unique solution to the problem
$$-\Delta_\omega\phi-\alpha(\alpha+N-2)\phi=1,\qquad\phi\in H^1_0(\Omega).$$
Observe that $\phi_1, \ol\phi \in L^\infty$.

Hence
$\tau\ol\phi$ is a supersolution to (\ref{psi}) for a large $\tau>0$,
and $\epsilon\phi_1$ is a subsolution to (\ref{psi}) for a small $\epsilon>0$.
Thus by the sub and supersolutions argument equation (\ref{psi})
has a solution $\psi\in H^1_0(\Omega)$ such that $\epsilon\phi_1<\psi\le\tau\ol\phi$.

\paragraph{\it Case $p<0$.}
Consider the problem
\begin{equation}\label{psi-1}
-\Delta_\omega\phi-\alpha(\alpha+N-2)(\phi+1)=
(\phi+1)^{p},
\qquad\phi\in H^1_0(\Omega).
\end{equation}
Let $\ol\phi>0$ be the unique solution to the problem
$$-\Delta\phi-\alpha(\alpha+N-2)(\phi+1)=1,\qquad\phi\in H^1_0(\Omega).$$
It is clear that $\ol\phi$ is a supersolution to (\ref{psi-1}) and
$\ul\phi\equiv 0$ is a subsolution to (\ref{psi-1}).
We conclude that (\ref{psi-1}) has a positive solution $\phi\in H^1_0(\Omega)$
such that $0<\phi\le\ol\phi$.
Then $\psi:=\phi+1\in H^1_{loc}(\Omega)$ is a positive solution to (\ref{psi}).
This completes the proof of the existence part of Theorem \ref{Main}.


\paragraph{Nonexistence.}
In what follows we set $\delta:=1$ if $p<0$ and $\delta:=0$ if
$p\in[0,1)$. Let $G\subset\R^N$ be a domain, $0\not\in G$. Observe
that equation (\ref{*}) has a positive supersolution in $G$ if and
only if the equation
\begin{equation}\label{**}
-\Delta w=\frac{c}{|x|^{s}}(w+\delta)^p\quad\mbox{in }\:G
\end{equation}
has a positive supersolution.
Indeed, if $u>0$ is a supersolution to (\ref{*}) in $G$ then $u$
is a supersolution to (\ref{**}). If $w>0$ is a supersolution to (\ref{**})
then $u=w+\delta$ is a supersolution to (\ref{*}).
The main argument of the proof nonexistence rests upon the following two lemmas.


The next lemma is an adaptation a comparison principle
by Ambrosetti, Brezis and Cerami \cite[Lemma 3.3]{ABC}.
\begin{lemma}\label{Brezis}
Let $G\subset\R^N$ be a bounded domain, $0\not\in G$.
Let $0\le \ul{w}\in H^1_0(G)$ be a subsolution
and $0\le \ol{w}\in H^1_{loc}(G)$ a supersolution to (\ref{**}).
Then $\ul{w}\le\ol{w}$ in $G$.
\end{lemma}


\proof
In \cite[Lemma 3.3]{ABC} the result was proved for a smooth bounded domain $G$
and $\ul{w},\ol{w}\in H^1_0(G)$ (and more general nonlinearities).
The proof given in \cite{ABC} carries over literally to the case of
an arbitrary bounded domain $G$ and $\ul{w},\ol{w}\in H^1_0(G)$,
or a smooth bounded domain $G$,
$\ul{w}\in H^1_0(G)$  and $0\le\ol{w}\in H^1(G)$.
Thus we only need to extend the lemma to an arbitrary bounded domain $G$ and $\ol{w}\in H^1_{loc}(G)$.

Let $\ol w\in H^1_{loc}(G)$ be a supersolution to (\ref{**}) in $G$.
Let $(G_n)_{n\in\N}$ be an exhaustion of $G$, that is a sequence of bounded smooth domains
such that $\ol G_n\subset G_{n+1}\subset G$ and $\cup_{n\in\N}G_n=G$.
Analogously to the argument given above in the existence part of the proof,
one can readily see that, for each $n\in\N$,
there exists a solution $0<w_n\in H^1_0(G_n)$ to (\ref{**})
(e.g., by constructing appropriate sub and supersolutions).
Moreover, $w_n\le w_{n+1}$.
Observe that $w_n\le\ol w$ in $G_n$ by \cite[Lemma 3.3]{ABC}.

We claim that $\sup\|\nabla w_n\|_{L^2}<\infty$.
This is clear for $p<0$, since $(w+1)^p\le 1$.
For $p\in[0,1)$,
using $w_n$ as a test function in (\ref{**}), we have
$$\int_G|\nabla w_n|^2dx=\int_G\frac{c}{|x|^{s}}w_n^{p+1}\,dx\le
c_1\left(\int_G|\nabla w_n|^2 dx\right)^{(p+1)/2},$$
which implies the claim.
It follows that $w_n$ converges pointwise in $G$, strongly in $L^2(G)$ and weakly in $H^1_0(G)$
to a positive $w_\ast\in H^1_0(G)$.
Clearly $w_\ast>0$ is a solution to (\ref{**}) in $G$ and $0<w_\ast\le\ol w$ in $G$.

Now let $0\le \ul w\in H^1_0(G)$ be a subsolution to (\ref{**}) in $G$.
By \cite[Lemma 3.3]{ABC} we conclude that $\ul w\le w_\ast$ in $G$.
\qed


Next, consider the initial value problem
\begin{equation}\label{ODE}
-v_{rr}-\frac{N-1}{r}v_r+\frac{\lambda_1}{r^2}v=\frac{c}{r^s}v^p\quad\mbox{for }\:r>1;
\qquad v(1)=\delta,\quad v_r(1)=K;
\end{equation}
where $p<1$, $s\in\R$, $c>0$, $K>1$ and $\delta$ as above.
Let $(1,R)$, $R=R(\delta,K)\le\infty$,
be the maximal right interval of existence of the solution
$v$ to (\ref{ODE}) in the region $\{(r,v)\in(1,+\infty)\times(\delta,+\infty)\}$.


\begin{lemma}\label{L-ODE}
Let $s<2$ and $p\in [1-\frac{2-s}{\alpha_+},1)$.
Then for any interval $[r_\ast,r^\ast]\subset(1,+\infty)$ there exists $K_0>1$ such that
\begin{itemize}
\item[$i)$]
for all $K>K_0$ one has $r^\ast<R<+\infty$ and $v(r)\to\delta$ as $r\nearrow R$;
\item[$ii)$]
for any $M>\delta$ there exists $K>K_0$ such that $\min_{[r_\ast,r^\ast]}v\ge M$.
\end{itemize}
\end{lemma}

% Maple graphics

%\begin{figure}[t]
%\includegraphics[width=220pt,height=180pt]{pict-2.eps}
%\hfill
%\includegraphics[width=220pt,height=180pt]{pict-3.eps}
%\caption{\footnotesize Solutions $v(r)$ to
%$-v_{rr}-\frac{3}{r}v_r+\frac{24}{r^2}v=\sqrt{v}$ (left) and
%$-v_{rr}-\frac{3}{r}v_r+\frac{3}{r^2}v=\frac{1}{v}$ (right)
%[using Maple's {\tt "with(DEtools):phaseportrait"} procedure]}
%\end{figure}


\proof
Set $\alpha:=\alpha_+$, $v:=wr^{\alpha}$, $t=r^{2-N-2\alpha}$.
Then $w$ solves the following problem
$$w_{tt}+c_1t^{-\sigma}w^p=0\quad\mbox{for }\:t\in(T,1);\qquad w(1)=\delta,\quad w_t(1)=-L,$$
where $\sigma=\frac{2N-2+\alpha(p+3)-s}{N-2+2\alpha}\ge 2$, $c_1>0$, $0\le T=R^{2-N-2\alpha}<1$
and $L=\frac{K-\alpha\delta}{N-2+2\alpha}\to\infty$ as $K\to\infty$. Choose $K_0$ such that $L>\delta$.
Observe that $w(t)$ is concave, hence
$$\delta<w(t)\le w(1)-w_t(1)(1-t)\le \delta+L\quad\mbox{for }\:t\in(T,1).$$
To see that $T>0$ let $\tilde w:=w$ for $p<0$, otherwise let $\tilde w:=w^{1-p}$.
Then $\tilde w$ satisfies the inequality
$$\tilde w_{tt} + c_2 t^{-2}\tilde w^{q}\le 0\quad\mbox{for }\:t\in(T,1),$$
with $c_2>0$ and $q:=\min\{p,0\}$.
Integrating $\tilde w_{tt}$ twice one can easily see that such inequality
has no positive solutions in any neighborhood of zero.
Thus we conclude that $T>0$, hence $w(t)\to\delta$ as $t\searrow T$.
In particular, $w(t)$ attains its maximum on $(T,1)$.

Let $T_0\in(T,1)$ be such that $w_t(T_0)=-\frac{L-\delta}{2}$.
Since $\delta\le w(t)\le\delta + L$ for $t\in(T_0,1)$, it follows that
$$\frac{L+\delta}{2}=w_t(T_0)-w_t(1)=-\int_{T_0}^{1}w_{tt}d\tau=
c_1\int_{T_0}^{1}\frac{w^p}{\tau^\sigma}d\tau\le c_3\left(\frac{1}{T_0^{\sigma-1}}-1\right)
\quad\mbox{for }\:t\in(T_0,1).$$
Hence $T_0\to 0$ as $L\to+\infty$.
Therefore for any given $t^\ast<1$ there exists $L_0>1$ such that for any $L>L_0$ one has $0<T<T_0<t^\ast$.
Thus, $(i)$ follows with $r^\ast=(t^\ast)^{\frac{1}{N-2+2\alpha}}$.

Observe now that for any $L>L_0$ we have
$$-\frac{L-\delta}2\ge w_t(t)\ge -L \quad\mbox{for }\:t\in(t^\ast,1),$$
since $w$ is concave. Hence for any $t\in(t^\ast,1)$ we obtain
$$w(t)=w(1)-\int_{t}^{1}w_t\,d\tau\ge\delta+(1-t)\frac{L-\delta}{2}\to\infty\quad
\mbox{as }\:L\to\infty.$$
Thus $(ii)$ follows.
\qed


\paragraph{\it Nonexistence -- completed.}

Let $p\in[1-\frac{2-s}{\alpha_+},1)$.
Fix a compact $K\subset \C^1_\Omega$ and $M>1$. There exists an interval
$[r_\ast,r^\ast]\subset(1,+\infty)$ such that $K\subset \C_\Omega^{(r_\ast,r^\ast)}$,
where $\C_\Omega^{(r_1,r_2)}$ denotes the set $\{x\in\C_\Omega^1\,|\,r_1\le |x|\le r_2\}$.
Then by  Lemma \ref{L-ODE}
there exists
$v:(1,R)\to(\delta,+\infty)$ solving (\ref{ODE}) such that $R>r^\ast$ and $\inf_{[r_\ast,r^\ast]}v\ge M+\delta$.

Let $\phi_1>0$ be the principal Dirichlet eigenvalue of $-\Delta_\omega$ on $\Omega$
with $\|\phi_1\|_\infty =1$.
Set $w_M:=(v-\delta)\phi_1$. Then $0<w_M\in H^1_0(\C_\Omega^{(1,R)})$,
and direct computation shows that $w_M$ is a subsolution to (\ref{**}) in $\C_\Omega^{(1,R)}$.
Now assume that $w>0$ is a supersolution to (\ref{**}) in $\C_\Omega^1$.
By Lemma \ref{Brezis} it follows that that $w\ge w_M$ in $\C_\Omega^{(1,R)}$.
By the weak Harnack inequality we have
$$\inf_K w\ge c_H\int_K w\,dx\ge c_H\int_K w_M\,dx\ge c_2M.$$
Since $M$ was arbitrary, we conclude that $w\equiv+\infty$ in $K$.
\qed
\medskip

\begin{small}
{\bf Acknowledgement.}
The support of the MFI Oberwolfach, Nuffield Foundation
and Institute of Advanced Studies of Bristol University is gratefully acknowledged.
\end{small}



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\end{small}



\end{document}

math0307183.extracted_bib
\begin{thebibliography}{99}

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{\sc A. Ambrosetti, H. Brezis and G. Cerami},
{\it Combined effect of concave and convex nonlinearities in some elliptic problems.}
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\bibitem{Bandle-Levine}
{\sc C. Bandle and H. A. Levine},
{\it On the existence and nonexistence of global solutions
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Trans. Amer. Math. Soc. {\bf 316} (1989), 595--622.

\bibitem{BCN}
{\sc H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg},
{\it Superlinear indefinite elliptic problems and nonlinear Liouville theorems.}
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