\section{Introduction and main results}

In this note we are interested in the existence versus non-existence of stable sub- and super-solutions of equations of the form
\begin{equation} \label{eq1}
-div( \omega_1(x) \nabla u ) = \omega_2(x) f(u) \qquad \mbox{in $ {\mathbb{R}}^N$,}
\end{equation} where $f(u)$ is one of the following non-linearities: $e^u$,  $ u^p$ where $ p>1$ and $ -u^{-p}$ where $ p>0$.  We assume that $ \omega_1(x)$ and $ \omega_2(x)$, which we call \emph{weights},  are smooth positive functions (we allow $ \omega_2$ to be zero at say a point) and which satisfy various growth conditions at $ \infty$.    Recall that we say that a solution $ u $ of $ -\Delta u = f(u)$ in $ {\mathbb{R}}^N$ is stable provided
\[ \int f'(u) \psi^2 \le \int | \nabla \psi|^2, \qquad \forall \psi \in C_c^2,\] where $ C_c^2$ is the set of $ C^2$ functions defined on $ {\mathbb{R}}^N$ with compact support.  Note that the stability of $u$ is just saying that the second variation at $u$ of the energy associated with the equation is non-negative.     In our setting this becomes:  We say a $C^2$ sub/super-solution $u$ of (\ref{eq1}) is \emph{stable} provided
\begin{equation} \label{stable}
\int \omega_2 f'(u) \psi^2 \le \int \omega_1 | \nabla \psi|^2 \qquad \forall \psi \in C_c^2.
\end{equation}
One should note that (\ref{eq1}) can be re-written as
\begin{equation*}
- \Delta u  + \nabla \gamma(x) \cdot \nabla u ={ \omega_2}/{\omega_1}\ f(u) \qquad \text{ in $ \mathbb{R}^N$},
\end{equation*}
where
$\gamma = - \log( \omega_1)$ and on occasion we shall take this point of view.

\begin{remark} \label{triv} Note that  if $ \omega_1$ has enough integrability then it is immediate that if $u$ is a stable solution  of (\ref{eq1}) we have $ \int \omega_2 f'(u) =0 $  (provided $f$ is increasing).   To see this let $ 0 \le \psi \le 1$ be supported in a ball of radius $2R$ centered at the origin ($B_{2R}$) with $ \psi =1$ on $ B_R$ and such that $  | \nabla \psi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$.   Putting this $ \psi$ into $ (\ref{stable})$ one obtains
\[ \int_{B_R} \omega_2 f'(u) \le \frac{C}{R^2} \int_{R < |x| <2R} \omega_1,\] and so if the right hand side goes to zero as $ R \rightarrow \infty$ we have the desired result.

\end{remark}

The existence versus non-existence of stable solutions of  $ -\Delta u =  f(u)$ in $ {\mathbb{R}}^N$ or $ -\Delta u = g(x) f(u)$ in $ {\mathbb{R}}^N$ is now quite well understood, see  \cite{dancer1, farina1, egg, zz, f2, f3, wei, ces, e1, e2}.  We remark that some of these results are examining the case where $ \Delta $ is replaced with $ \Delta_p$ (the $p$-Laplacian) and also in many cases the authors are interested in finite Morse index solutions or solutions which are stable outside a compact set.
  Much of the interest in these Liouville type theorems  stems from the fact that the non-existence of a stable solution is related to the existence of a priori estimates for stable solutions of a related equation on a bounded domain.

       In \cite{Ni}  equations similar to  $ -\Delta u = |x|^\alpha u^p$  where examined on the unit ball in $ {\mathbb{R}}^N$ with zero Dirichlet boundary conditions.  There it was shown that for $ \alpha >0$ that  one can obtain positive solutions for $ p $ supercritical with respect to Sobolev embedding and so one can view that the term $ |x|^\alpha$ is restoring some compactness.      A similar feature happens for equations of the form
\[ -\Delta u = |x|^\alpha f(u) \qquad \mbox{in $ {\mathbb{R}}^N$};\]    the value of $ \alpha$ can vastly alter the existence versus non-existence of a stable solution,  see \cite{e1, ces, e2, zz, egg}.

We now come to our main results and for this we need to define a few quantities:

\begin{eqnarray*}
I_G&:=& R^{-4t-2} \int_{ R < |x|<2R} \frac{ \omega_1^{2t+1}}{\omega_2^{2t}}dx , \\
 J_G&:=& R^{-2t-1} \int_{R < |x| <2R} \frac{| \nabla \omega_1|^{2t+1} }{\omega_2^{2t}} dx  ,\\I_L&:=& R^\frac{-2(2t+p-1)}{p-1}  \int_{R<|x|<2R }{ \left(   \frac{w_1^{p+2t-1}}{w_2^{2t}}    \right)^{\frac{1}{p-1} }  }  dx,\\ J_L&:= &R^{-\frac{p+2t-1}{p-1} }  \int_{R<|x|<2R }{ \left(   \frac{|\nabla w_1|^{p+2t-1}}{w_2^{2t}}     \right)^{\frac{1}{p-1} }  }  dx,\\
I_M  &:=&   R^{-2\frac{p+2t+1}{p+1} }  \int_{R<|x|<2R }{ \left(   \frac{w_1^{p+2t+1}}{w_2^{2t}}    \right)^{\frac{1}{p+1} }  } \ dx, \\
J_M  &:= &  R^{-\frac{p+2t+1}{p+1} }  \int_{R<|x|<2R }{ \left(   \frac{|\nabla w_1|^{p+2t+1}}{w_2^{2t}}     \right)^{\frac{1}{p+1} }  }  dx.
\end{eqnarray*}

The three equations we examine are
\[ -div( \omega_1 \nabla u ) = \omega_2 e^u \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (G), \]
\[  -div( \omega_1 \nabla u ) = \omega_2 u^p \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (L), \]
\[ -div( \omega_1 \nabla u ) = - \omega_2 u^{-p} \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (M),\]  and where we restrict $(L)$ to the case $ p>1$ and $(M)$ to $ p>0$.    By solution we always mean a $C^2$ solution.   We now come to our main results in terms of abstract $ \omega_1 $ and $ \omega_2$.    We remark that our approach to non-existence of stable solutions is the approach due to Farina, see \cite{f2,f3,farina1}.

\begin{thm} \label{main_non_exist} \begin{enumerate}

\item   There is no  stable sub-solution of $(G)$ if $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$   for some   $0<t<2$.
 \item    There is no positive   stable sub-solution (super-solution) of $(L)$  if  $ I_L,J_L \rightarrow 0$ as $ R \rightarrow \infty$   for some   $p- \sqrt{p(p-1)}   < t<p+\sqrt{p(p-1)} $  ($0<t<\frac{1}{2}$).
\item   There is no positive   stable super-solution of (M) if  $ I_M,J_M \rightarrow 0$ as $ R \rightarrow \infty$  for some $0<t<p+\sqrt{p(p+1)}$.

\end{enumerate}

\end{thm}     If we assume that $ \omega_1$ has some monotonicity we can do  better.   We will assume that the monotonicity conditions is satisfied for big $x$  but really all ones needs is for it to be satisfied on a suitable sequence of annuli.

\begin{thm} \label{mono}  \begin{enumerate} \item There is no  stable sub-solution of $(G)$ with $  \nabla \omega_1(x) \cdot  x \le 0$ for big $x$ if  $ I_G \rightarrow 0$ as $ R \rightarrow \infty$   for some   $0<t<2$.

\item There is no positive  stable sub-solution of  $(L)$ provided  $  I_L \rightarrow 0$ as $ R \rightarrow \infty$ for either:
\begin{itemize}
\item  some $  1 \le t < p + \sqrt{p(p-1)}$ and $  \nabla \omega_1(x) \cdot x \le 0$ for big $x$,  or  \\

\item some $ p - \sqrt{p(p-1)} < t \le 1$ and $ \nabla \omega_1(x) \cdot x \ge 0$ for big $ x$.
\end{itemize}
 There is no positive super-solution of $(L)$ provided $ I_L \rightarrow 0$ as $ R \rightarrow \infty$ for some $ 0 < t < \frac{1}{2}$ and $ \nabla \omega_1(x) \cdot x \le 0$ for big $x$.

\item  There is no positive stable super-solution of $(M)$ provided  $ I_M \rightarrow 0$ as $ R \rightarrow \infty$  for some  $0<t<p+\sqrt{p(p+1)}$.

\end{enumerate}

\end{thm}

\begin{cor}  \label{thing}  Suppose $ \omega_1 \le C  \omega_2$  for big $ x$, $ \omega_2 \in L^\infty$,  $ \nabla \omega_1(x) \cdot  x \le 0$ for big $ x$.
\begin{enumerate} \item There is no stable sub-solution of $(G)$ if $ N \le 9$.

\item  There is no positive stable sub-solution of $(L)$ if  $$N<2+\frac{4}{p-1} \left( p+\sqrt{p(p-1)}  \right).$$

\item  There is no positive stable super-solution of $(M)$ if $$N<2+\frac{4}{p+1} \left( p+\sqrt{p(p+1)}  \right).$$

\end{enumerate}

\end{cor}

If one takes $ \omega_1=\omega_2=1$ in the above corollary, the results obtained for $(G)$ and  $(L)$,  and for some values of $p$ in $(M)$, are optimal, see \cite{f2,f3,zz}.

We now drop all monotonicity conditions on $ \omega_1$.

\begin{cor} \label{po} Suppose  $ \omega_1 \le C \omega_2$ for big $x$, $ \omega_2 \in L^\infty$, $ | \nabla \omega_1| \le C \omega_2$ for big $x$.
\begin{enumerate} \item  There is no stable sub-solution of $(G)$ if $ N \le 4$.

\item  There is no positive stable sub-solution of $(L)$ if $$N<1+\frac{2}{p-1} \left( p+\sqrt{p(p-1)}  \right).$$

\item There is no positive super-solution of $(M)$ if $$N<1+\frac{2}{p+1} \left( p+\sqrt{p(p+1)}  \right).$$

\end{enumerate}

\end{cor}

Some of the conditions on $ \omega_i$ in Corollary \ref{po} seem somewhat artificial.  If we shift over to the advection equation (and we take $ \omega_1=\omega_2$  for simplicity)
\[ -\Delta u + \nabla \gamma \cdot \nabla u = f(u), \] the conditions on $ \gamma$ become: $ \gamma$ is bounded from below and has a bounded gradient.

In what follows we examine the case where $ \omega_1(x) = (|x|^2 +1)^\frac{\alpha}{2}$ and $ \omega_2(x)= g(x) (|x|^2 +1)^\frac{\beta}{2}$,  where $ g(x) $ is positive except at say a point, smooth and where $ \lim_{|x| \rightarrow \infty} g(x) = C \in (0,\infty)$.     For this class of weights we can essentially obtain optimal results.

\begin{thm} \label{alpha_beta}   Take $ \omega_1 $ and $ \omega_2$ as above.
\begin{enumerate}

\item If $ N+ \alpha - 2 <0$ then there is no stable sub-solution for $(G)$, $(L)$ (here we require it to be positive) and in the case of $(M)$ there is no positive  stable  super-solution. This case is the trivial case, see Remark \ref{triv}.  \\

\textbf{Assumption:} For the remaining cases we assume that $ N + \alpha -2 > 0$.

  \item If  $N+\alpha-2<4(\beta-\alpha+2)$ then there is no  stable sub-solution for $ (G)$.

\item If $N+\alpha-2<\frac{ 2(\beta-\alpha+2)   }{p-1} \left( p+\sqrt{p(p-1)}  \right)$ then there is  no positive stable sub-solution of $(L)$.

\item If $N+\alpha-2<\frac{2(\beta-\alpha+2)   }{p+1} \left( p+\sqrt{p(p+1)}  \right)$ then there is no positive stable super-solution of $(M)$.

\item Further more 2,3,4 are optimal in the sense if $ N + \alpha -2 > 0$ and the remaining inequality is not satisfied (and in addition we assume we don't have equality in the inequality) then we can find a suitable function $ g(x)$ which satisfies the above properties and a stable sub/super-solution $u$ for the appropriate equation.

\end{enumerate}

\end{thm}

\begin{remark}  Many of the above results can be extended to the case of equality in either the $ N + \alpha - 2 \ge 0$ and also the other inequality which depends on the equation we are  examining.  We omit the details because one cannot prove the results in a unified way.

\end{remark}

In showing that an explicit solution is stable we will need the   weighted Hardy inequality given in \cite{craig}.
\begin{lemma} \label{Har}
Suppose $ E>0$ is a smooth function.  Then one has
\[ (\tau-\frac{1}{2})^2 \int E^{2\tau-2} | \nabla E|^2 \phi^2 + (\frac{1}{2}-\tau) \int (-\Delta E) E^{2\tau-1} \phi^2 \le \int E^{2\tau} | \nabla \phi|^2,\] for all $ \phi \in C_c^\infty({\mathbb{R}}^N)$ and $ \tau \in {\mathbb{R}}$.
\end{lemma}  By picking an appropriate function $E$ this gives,

\begin{cor} \label{Hardy}
For all $ \phi \in C_c^\infty$ and $ t , \alpha \in {\mathbb{R}}$. We have
  \begin{eqnarray*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2 &\ge& (t+\frac{\alpha}{2})^2 \int |x|^2 (1+|x|^2)^{-2+\frac{\alpha}{2}}\phi^2\\
&&+(t+\frac{\alpha}{2})\int  (N-2(t+1)   \frac{|x|^2}{1+|x|^2}) (1+|x|^2)^{-1+\frac{\alpha} {2}} \phi^2.
\end{eqnarray*}
 \end{cor}

\section{Proof of main results}

\textbf{ Proof of Theorem \ref{main_non_exist}.}   (1). Suppose $ u$ is a stable sub-solution of $(G)$ with $ I_G,J_G \rightarrow 0$ as $ R \rightarrow \infty$ and  let $ 0 \le \phi \le 1$ denote a smooth compactly supported function.  Put $ \psi:= e^{tu} \phi$ into (\ref{stable}), where $ 0 <t<2$, to arrive at
\begin{eqnarray*}
\int \omega_2 e^{(2t+1)u} \phi^2 &\le & t^2 \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \\
&& +\int \omega_1 e^{2tu}|\nabla \phi|^2 + 2 t \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi.
\end{eqnarray*}  Now multiply $(G)$ by $ e^{2tu} \phi^2$ and integrate by parts to arrive at
\[ 2t \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \le \int \omega_2 e^{(2t+1) u} \phi^2 - 2 \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi,\]
and now if one equates like terms they arrive at
\begin{eqnarray} \label{start}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^2  & \le & \int \omega_1 e^{2tu} \left( | \nabla \phi |^2 - \frac{ \Delta \phi}{2} \right) dx \nonumber \\
&& - \frac{1}{2} \int e^{2tu} \phi \nabla \omega_1 \cdot \nabla \phi.
\end{eqnarray}   Now substitute $ \phi^m$ into this inequality for $ \phi$ where $ m $ is a big integer to obtain
\begin{eqnarray} \label{start_1}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m}  & \le & C_m \int \omega_1 e^{2tu} \phi^{2m-2}  \left( | \nabla \phi |^2 + \phi |\Delta \phi|  \right) dx \nonumber \\
&& - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi
\end{eqnarray} where $ C_m$ and $ D_m$ are positive constants just depending on $m$.   We now estimate the  terms on the right but we mention that when ones assume the appropriate monotonicity on $ \omega_1$ it is the last integral on the right which one is able to drop.

\begin{eqnarray*}
\int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2 & = & \int \omega_2^\frac{2t}{2t+1} e^{2tu} \phi^{2m-2}  \frac{ \omega_1 }{\omega_2^\frac{2t}{2t+1}} | \nabla \phi|^2  \\
& \le &  \left( \int \omega_2 e^{(2t+1) u} \phi^{(2m-2) \frac{(2t+1)}{2t}} dx \right)^\frac{2t}{2t+1}\\ &&\left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1}.
\end{eqnarray*}
Now, for fixed $ 0 <t<2$ we can take $ m $ big enough so $ (2m-2) \frac{(2t+1)}{2t} \ge 2m $ and since $ 0 \le \phi \le 1$ this allows us to replace the power on $ \phi$ in the first term on the right with $2m$   and hence we obtain
 \begin{equation} \label{three}
 \int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2  \le \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} \left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1}.
 \end{equation}    We now take the test functions $ \phi$ to be such that $ 0 \le  \phi \le 1$ with $ \phi $ supported in the ball $ B_{2R}$ with $ \phi = 1 $ on $ B_R$ and $ | \nabla \phi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$.   Putting this choice of $ \phi$ we obtain
 \begin{equation} \label{four}
 \int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi |^2 \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}.
 \end{equation}  One similarly shows that
 \[ \int \omega_1 e^{2tu} \phi^{2m-1} | \Delta \phi| \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}.\]
 So, combining the results we obtain

 \begin{eqnarray} \label{last} \nonumber \frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m} &\le& C_m \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}\\
 &&- D_m \int e^{2tu} \phi^{2m-1}  \nabla \omega_1 \cdot \nabla \phi.
 \end{eqnarray}
 We now estimate this last term.  A similar argument using H\"{o}lder's inequality shows that
 \[ \int e^{2tu} \phi^{2m-1} | \nabla \omega_1| | \nabla \phi| \le \left(  \int \omega_2 \phi^{2m} e^{(2t+1) u} dx \right)^\frac{2t}{2t+1} J_G^\frac{1}{2t+1}. \] Combining the results gives that
\begin{equation} \label{last}
(2-t) \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{1}{2t+1} \le I_G^\frac{1}{2t+1} + J_G^\frac{1}{2t+1},
\end{equation} and now we send $ R \rightarrow \infty$ and use the fact that $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$ to see that
\[ \int \omega_2 e^{(2t+1) u} =0, \] which is clearly a contradiction.   Hence there is no stable sub-solution of $(G)$.

(2).   Suppose that $u >0$ is a stable sub-solution (super-solution) of $(L)$.  Then a similar calculation as in (1) shows that for  $ p - \sqrt{p(p-1)} <t < p + \sqrt{p(p-1)}$,  $( 0 <t<\frac{1}{2})$ one has

\begin{eqnarray}   \label{shit}
(p  -\frac{t^2}{2t-1}   )\int \omega_2 u^{2t+p-1} \phi^{2m} & \le & D_m \int \omega_1 u^{2t} \phi^{2(m-1)} (|\nabla\phi|^2  +\phi |\Delta \phi |) \nonumber \\
&& +C_m \frac{(1-t)}{2(2t-1)} \int u^{2t} \phi^{2m-1}\nabla \omega_1 \cdot  \nabla \phi.
 \end{eqnarray}    One now applies H\"{o}lder's argument as in (1) but the terms $ I_L$ and $J_L$ will appear on the right hand side of the resulting
 equation.      This shift from a sub-solution to a super-solution depending on whether $ t >\frac{1}{2}$ or $ t < \frac{1}{2}$ is a result from the sign change of $ 2t-1$ at $ t = \frac{1}{2}$.   We leave the details for the reader.

(3).  This case is also similar to (1) and (2).

\hfill $ \Box$

 \textbf{Proof of Theorem \ref{mono}.}   (1).  Again we suppose there is a stable sub-solution $u$ of $(G)$.  Our starting point  is (\ref{start_1}) and we wish to be able to drop the term
 \[ - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi, \] from (\ref{start_1}).    We can choose $ \phi$ as in the proof of Theorem \ref{main_non_exist} but also such that $ \nabla \phi(x) = - C(x) x$ where $ C(x) \ge 0$.     So if we assume that $ \nabla \omega_1 \cdot x \le 0$ for big $x$ then we see that this last term is non-positive and hence we can drop the term.  The the proof is as before but now we only require that $ \lim_{R \rightarrow \infty} I_G=0$.

 (2).     Suppose that $ u >0$ is a stable sub-solution of $(L)$  and so (\ref{shit}) holds for all $  p - \sqrt{p(p-1)} <t< p + \sqrt{p(p-1)}$.   Now we wish to use monotonicity to drop the term from (\ref{shit}) involving the term $ \nabla \omega_1 \cdot \nabla \phi$.      $ \phi$ is chosen the same as in (1)  but here one notes that the co-efficient for this term changes sign at $ t=1$ and hence by restriction $t$ to the appropriate side of 1 (along with the above condition on $t$ and $\omega_1$) we can drop the last term depending on which monotonicity we have and hence to obtain a contraction we only require that $ \lim_{R \rightarrow \infty} I_L =0$.   The result for the non-existence of a stable super-solution is similar be here one restricts $ 0 < t < \frac{1}{2}$.

(3).  The proof here is similar to (1) and (2) and we omit the details.

 \hfill $\Box$

\textbf{Proof of Corollary \ref{thing}.}   We suppose  that $ \omega_1 \le C  \omega_2$  for big $ x$, $ \omega_2 \in L^\infty$,  $ \nabla \omega_1(x) \cdot  x \le 0$ for big $ x$.     \\
(1).    Since $ \nabla \omega_1 \cdot x \le 0$ for big $x$ we can apply Theorem \ref{mono} to show the non-existence of a stable solution to $(G)$.   Note that  with the above assumptions on $ \omega_i$ we have that
\[ I_G \le \frac{C R^N}{R^{4t+2}}.\]  For $ N \le 9$  we can take $ 0 <t<2$  but close enough to $2$ so the right hand side goes to zero as $ R \rightarrow \infty$.

Both (2) and (3) also follow directly from applying Theorem \ref{mono}.   Note that one can say more about (2) by taking the multiple cases as listed in Theorem \ref{mono} but we have choice to leave this to the reader.

\hfill $ \Box$

\textbf{Proof of Corollary \ref{po}.}   Since we have no monotonicity conditions now we will need both $I$ and $J$ to go to zero to show the non-existence of a stable solution.   Again the results are obtained immediately by applying Theorem \ref{main_non_exist}  and we prefer to omit the details.

\hfill $\Box$

\textbf{Proof of Theorem \ref{alpha_beta}.}  (1).  If $ N + \alpha -2 <0$ then using Remark \ref{triv}  one easily sees there is no stable sub-solution of $(G)$ and $(L)$ (positive for $(L)$) or a positive stable super-solution of $(M)$.   So we now assume that $ N + \alpha -2 > 0$.     Note that the monotonicity of $ \omega_1$ changes when $ \alpha $ changes sign and hence one would think that we need to consider separate cases if we hope to utilize the monotonicity results.   But a computation shows that in fact $ I$ and $J$ are just multiples of each other in all three cases so it suffices to show, say, that $ \lim_{R \rightarrow \infty} I =0$. \\
(2).   Note that for $ R >1$ one has
\begin{eqnarray*}
I_G & \le  & \frac{C}{R^{4t+2}} \int_{R <|x| < 2R} |x|^{ \alpha (2t+1) - 2t \beta} \\
& \le  &  \frac{C}{R^{4t+2}}  R^{N + \alpha (2t+1) - 2t \beta},
\end{eqnarray*} and so to show the non-existence we want to find some $ 0 <t<2$ such that
$  4t+2 > N  + \alpha(2t+1) - 2 t \beta$,   which is equivalent to  $ 2t ( \beta - \alpha +2) > (N + \alpha -2)$.    Now recall that we are assuming that $ 0 < N + \alpha -2 < 4 ( \beta - \alpha +2) $ and hence we have the desired result by taking $  t <2$ but sufficiently close.
The proof of the non-existence results for
(3) and (4) are similar and we omit the details.   \\
(5).  We now assume that $N+\alpha-2>0$.  In showing the existence of stable sub/super-solutions we need to consider  $ \beta - \alpha + 2 <0$ and $ \beta - \alpha +2 >0$ separately.

\begin{itemize} \item $(\beta - \alpha + 2 <0)$  Here we take $ u(x)=0$ in the case of $(G)$ and $ u=1$ in the case of $(L)$ and $(M)$. In addition we take $ g(x)=\E$.  It is clear that in all cases $u$ is the appropriate sub or super-solution.  The only thing one needs to check is the stability.    In all cases this reduces to trying to show that we have
\[ \sigma \int       (1+|x|^2)^{\frac{\alpha}{2}  -1}    \phi^2 \le \int      (1+|x|^2)^{\frac{\alpha}{2}}   | \nabla\phi  |^2,\]  for all $ \phi \in C_c^\infty$ where  $ \sigma $ is some small positive constant; its either $ \E$ or $ p \E$ depending on which equation were are examining.
To show this we use the result from Corollary \ref{Hardy} and we drop a few positive terms to arrive at
\begin{equation*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2\ge (t+\frac{\alpha}{2})\int \left (N-2(t+1)   \frac{|x|^2}{1+|x|^2}\right) (1+|x|^2)^{-1+\frac{\alpha} {2}}
\end{equation*} which holds for all $ \phi \in C_c^\infty$ and $ t,\alpha \in {\mathbb{R}}$.
  Now, since $N+\alpha-2>0$, we can choose $t$ such that $-\frac{\alpha}{2}<t<\frac{n-2}{2}$.  So, the integrand function in the right hand side is positive and since for small enough $\sigma$ we have
  \begin{equation*}
\sigma \le  (t+\frac{\alpha}{2})(N-2(t+1)   \frac{|x|^2}{1+|x|^2})  \ \ \ \text {for all} \ \ x\in \mathbb{R}^N
\end{equation*}
 we get stability.

\item ($\beta-\alpha+2>0$) In the case of $(G)$ we take   $u(x)=-\frac{\beta-\alpha+2}{2} \ln(1+|x|^2)$ and $g(x):= (\beta-\alpha+2)(N+(\alpha-2)\frac{|x|^2}{1+|x|^2})$. By a computation one sees that $u$ is a sub-solution of $(G)$ and hence we need now to only show the stability, which amounts to showing that
\begin{equation*}
\int \frac{g(x)\psi^2}{(1+|x|^{2   })^{-\frac{\alpha}{2}+1}}\le \int\frac{|\nabla\psi|^2}{    (1+|x|^2)^{-\frac{\alpha}{2}}     },
\end{equation*} for all $ \psi \in C_c^\infty$.  To show this we use  Corollary \ref{Hardy}.  So  we  need to choose an appropriate $t$ in   $-\frac{\alpha}{2}\le t\le\frac{N-2}{2}$  such that for all $x\in {\mathbb{R}}^N$ we have
 \begin{eqnarray*}
 (\beta-\alpha+2)\left(    N+  (\alpha-2)\frac{|x|^2}{1+|x|^2}\right)         &\le& (t+\frac{\alpha}{2})^2 \frac{ |x|^2 }{(1+|x|^2}\\
&&+(t+\frac{\alpha}{2}) \left(N-2(t+1)   \frac{|x|^2}{1+|x|^2}\right).
\end{eqnarray*}
With a  simple calculation one sees we need just to have
   \begin{eqnarray*}
 (\beta-\alpha+2)&\le& (t+\frac{\alpha}{2}) \\
  (\beta-\alpha+2)     \left(    N+  \alpha-2\right)      &   \le&  (t+\frac{\alpha}{2}) \left(N-t-2+\frac{\alpha}{2}) \right).
 \end{eqnarray*}     If one takes $ t= \frac{N-2}{2}$ in the case where $ N \neq 2$ and $ t $ close to zero in the case for $ N=2$ one easily sees the above inequalities both hold, after considering all the constraints on $ \alpha,\beta$ and $N$.

 We now consider the case of $(L)$.  Here one takes $g(x):=\frac {\beta-\alpha+2}{p-1}(    N+  (\alpha-2-\frac{\beta-\alpha+2}{p-1})
\frac{|x|^2}{1+|x|^2})$ and $ u(x)=(1+|x|^2)^{ -\frac  {\beta-\alpha+2}{2(p-1)} }$.   Using essentially the same approach as in $(G)$ one shows that $u$ is a stable sub-solution of $(L)$ with this choice of $g$.   \\
For the case of $(M)$ we take   $u(x)=(1+|x|^2)^{ \frac  {\beta-\alpha+2}{2(p+1)}   }$ and $g(x):=\frac {\beta-\alpha+2}{p+1}(    N+  (\alpha-2+\frac{\beta-\alpha+2}{p+1})
\frac{|x|^2}{1+|x|^2})$.

\end{itemize}

\hfill  $  \Box$


\section{Introduction}

Chlorine is a minor constituent of the Universe, with a solar abundance of only $\sim 3 \times10^{-7}$
relative to hydrogen \citep{Asplund2009}, that is orders of magnitude smaller than the elemental abundances of carbon, nitrogen, and oxygen. Nevertheless, chlorine has a strong chemical tendency to form hydrides (e.g., \H2Cl+, HCl) whose abundances can be as high as those of CH and H$_2$O, for example, in diffuse molecular clouds \citep{Neufeld2010, Sonnentrucker2010, Lis2010}. This is because of its unique thermo-chemistry properties: first, chlorine has an ionization potential of 12.97 eV, which is lower than that of hydrogen, such that it is predominantly in its ionized stage, Cl$^+$, in the atomic phase of the interstellar medium (ISM), and second, Cl$^+$ can react exothermically with H$_2$ triggering an active chemistry.

In fact, the interstellar chemical network of chlorine is thought to be relatively simple, dominated by a handful of hydrides \citep{Neufeld2009}, though the vast majority of interstellar chlorine will be in its neutral form in the presence of even small amounts of H$_2$. The reaction of Cl$^+$ with H$_2$ readily forms HCl$^+$, which has been detected in absorption in Galactic diffuse clouds \citep{DeLuca2012} harboring a significant amount (3--5\%) of the gas-phase chlorine. In turn, HCl$^+$ can further react exothermically with H$_2$ to form chloronium (\H2Cl+; first observed by \citealp{Lis2010}). Chloronium itself can react with free electrons to either form hydrogen chloride \citep[HCl; historically the first chlorine-bearing molecule detected in the ISM;][]{Blake1985} or release neutral chlorine. There are several pathways to destroy HCl through photodissociation, photoionization, or reactions with He$^+$, H$_3^+$, and C$^+$.
Methyl chloride, CH$_3$Cl, was also recently identified in the young stellar object IRAS $16293-2422$ and the gaseous coma of comet 67P/Churyumov-Gerasimenko \citep{Fayolle2017}, suggesting that chlorine chemistry in space also extends to more complex species (see, e.g., \citealp{Acharyya2017} for gas-grain chemical models of chlorine chemistry).

From the nucleosynthetic point of view, elemental chlorine is produced on both short timescales in core-collapse supernovae, and long timescales in Asymptotic Giant Branch (AGB) stars and Type Ia supernovae \citep[p.164,][]{Clayton2003}. Chlorine is produced during oxygen burning, from fast reactions with the more abundant alpha elements, in stars massive enough to ignite it. In core-collapse supernovae excess neutrons cause a great increase in chlorine abundance, and hence they produce the majority of chlorine in the universe through explosive nucleosynthesis. Type Ia supernovae produce smaller amounts of chlorine through explosive oxygen burning.

Chlorine can be found in two stable isotopes, \hbox{$^{35}{\rm Cl}$}\ and \hbox{$^{37}{\rm Cl}$}, with different sources producing different relative abundances. Low metallicity core-collapse supernovae, like the ones that likely enriched the pre-Solar nebula, produce a \hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ ratio of $\sim$3 as measured in the Solar system \citep{Asplund2009}.
Higher metallicity core-collapse supernovae can produce lower ratios \citep{Kobayashi2006,Kobayashi2011}. The isotopic ratios produced in AGB stars also depend heavily on metallicity, with low metallicities producing lower \hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ values \citep{Cristallo2011}.

Molecular absorbers at intermediate redshift ($z\sim1$) are powerful probes of the physico-chemical properties of distant galaxies. They offer a cosmological perspective on the chemical evolution of the Universe, for example with measurements of isotopic ratios at look-back times on the order of half or more of the present age of the Universe, but unfortunately, only a handful of such absorbers have been identified so far \citep[e.g.,][]{Combes2008,Wiklind2018}.
In this paper, we collect observations of chlorine-bearing species in the two best studied and most molecule-rich redshifted radio molecular absorbers, with the goal of investigating their chemical properties and measuring the $^{35}$Cl/$^{37}$Cl isotopic ratio in different absorbing sightlines.

The molecular absorber located in front of the quasar \hbox{PKS\,1830$-$211}\ is a face-on spiral galaxy at a redshift z=0.88582 (hereafter labeled MA0.89). The intervening galaxy lenses the quasar into two bright and compact images, leading to two independent sightlines in the southwest and northeast where absorption is detected (hereafter MA0.89~SW and MA0.89~NE), with impact parameters of $\sim$2 kpc and 4 kpc, respectively, in the absorber. These two compact, lensed images are embedded in a faint structure, reminiscent of an Einstein ring, seen at low radio frequencies \citep{Jauncey1991}.
A wealth of molecules and their rare isotopologs have been detected in this absorber, especially in MA0.89 SW, which is characterized by a large H$_2$ column density ($\sim 2 \times 10^{22}$~cm$^{-2}$) and moderate gas density $\sim 10^3$~cm$^{-3}$ \citep[e.g.][]{Wiklind1996, Muller2011, Muller2013, Muller2014}. MA0.89 NE is characterized by a lower H$_2$ column density ($\sim 1 \times 10^{21}$~cm$^{-2}$) and more diffuse gas composition as shown by the enhancement of the relative abundances of hydrides such as OH$^+$, H$_2$O$^+$, and H$_2$Cl$^+$ \citep[e.g.,][]{Muller2016b}.

The second molecular absorber (hereafter labeled MA0.68), located in front of the quasar \hbox{B\,0218$+$357}, shares similar properties with MA0.89. It is a nearly face-on spiral galaxy at an intermediate redshift z=0.68466, also lensing the background quasar into two main compact images and an Einstein ring, seen at low radio frequencies and centered on the faintest image. Molecular absorption has only been detected in one line of sight (toward the brightest image), at an impact parameter of $\sim$2 kpc from the absorber's center \citep{Wiklind1995, Muller2007}. A number of molecules and their isotopologs have also been observed in MA0.68 \citep[e.g.][]{Wallstrom2016}, which has an H$_2$ column density ($\sim 8 \times 10^{21}$~cm$^{-2}$) that is intermediate between MA0.89 SW and MA0.89 NE.

With these two absorbers, we are thus exploring three independent lines of sight, with different redshifts, galactocentric distances, H$_2$ column densities, visual extinctions, metallicities, local radiation fields, cosmic-ray ionization rates, etc. Our observations and results are presented in Sections \ref{sec:obs} and \ref{sec:results}. We discuss our findings in terms of chemical properties derived from chlorine-bearing species and $^{35}$Cl/$^{37}$Cl isotopic ratios in Section~\ref{sec:discussion}.

\section{Observations} \label{sec:obs}

Observations of the two absorbers were obtained with the Atacama Large Millimeter/submillimeter Array (ALMA) between 2014 and 2017. A summary of these observations is given in Table~\ref{tab:obs}. Details of the observational setups and data analysis are described below for each absorber separately.

\begin{table*}[ht]
\caption{Summary of ALMA observations.}
\label{tab:obs}
\begin{center} \begin{tabular}{ccccccccc}
\hline \hline
Target & Species & Rest freq. $^{(a)}$ & Sky freq. $^{(b)}$ & $\delta v$ $^{(c)}$ & Dates of observations & Project ID \\
        &        &  (GHz)      &  (GHz) & (\hbox{km\,s$^{-1}$})&              & \\
\hline
\hbox{PKS\,1830$-$211} & ortho-H$_2^{35}$Cl$^+$ & 189.225  & 100.341 & 1.5 & 2014, Jul. 21 and Aug. 26 & 2013.1.00020.S \\
         & ortho-H$_2^{37}$Cl$^+$ & 188.428  &  99.919 && observed simultaneously with previous &  \\

         & para-H$_2^{35}$Cl$^+$ & 485.418  & 257.404 & 1.1& 2014, Jun. 06 and Jul. 29 & 2013.1.00020.S \\
         & para-H$_2^{37}$Cl$^+$ & 484.232  & 256.775 && observed simultaneously with previous & \\

         & H$^{35}$Cl & 625.919 & 331.908 & 0.9 & 2014, May 3, 6 & 2012.1.00056.S \\
         &            & &  && 2014 Jun 30 & 2013.1.00020.S \\
         & H$^{37}$Cl & 624.978 & 331.409 &&  observed simultaneously with previous &  \\
\hline
\hbox{B\,0218$+$357} & ortho-H$_2^{35}$Cl$^+$ & 189.225 & 112.322 & 2.6 & 2016 Oct 22, 2017 May 02 & 2016.1.00031.S \\
       & ortho-H$_2^{37}$Cl$^+$ & 188.428 & 111.849 && observed simultaneously with previous & \\
\hline
\end{tabular}
\tablefoot{
$a)$ Rest frequencies of the strongest hyperfine component, taken from the Cologne Database for Molecular Spectroscopy \citep[CDMS; ][]{Muller2001}.
$b)$ Sky frequencies are given taking redshifts $z_{abs}=0.88582$ and 0.68466 for MA0.89 and MA0.68, respectively, in the heliocentric frame.
$c)$ Velocity resolution.
}
\end{center} \end{table*}

\paragraph{MA0.68:} Observations of \H2Cl+ were obtained on 2016 October 22 and 2017 May 02. During the first run, the precipitable water vapor content was $\sim 2.4$~mm. The array was in a configuration where the longest baseline was 1.7~km. The bandpass calibrator was J\,0237+2848 ($\sim 2.2$~Jy at 100~GHz) and the gain calibrator was J\,0220+3241 ($\sim 0.2$~Jy at 100~GHz). During the second run, the precipitable water vapor content was $< 1$~mm. The array's longest baseline was 1.1~km. The bandpass calibrator was J0510+1800 ($\sim 2.6$~Jy at 100~GHz) and the gain calibrator J\,0205+3212 ($\sim 0.9$~Jy at 100~GHz).
The data calibration was done with the CASA package following standard procedures. We further improved the data quality with a step of self-calibration of the phase, using a model of two point-sources for MA0.68. When combined, the two datasets result in a synthesized beam of $\sim 1.0 \arcsec \times 0.6 \arcsec$ (P.A.$=-176^\circ$) at 100~GHz, slightly larger than the separation of $0.3\arcsec$ between the two lensed images of the quasar. However, we extracted the spectra along each line of sight by directly fitting each visibility set separately using the CASA-Python task UVMULTIFIT \citep{Marti-Vidal2014}, with a model of two point-sources. This is possible in the Fourier plane, thanks to the high signal-to-noise ratio of the data and the known and simple geometry of the source. The final spectrum was the weighted average of the individual spectra.

\paragraph{MA0.89:} We complemented previous observations of the ortho and para forms of \H2Cl+\ toward MA0.89 presented by \cite{Muller2014} and \cite{LeGal2017} with new observations of HCl obtained with ALMA during three observing runs: in 2014 May 03, 2014 May 06, and 2014 Jun 30. Both HCl isotopologs were observed in the same spectral window, and within the same receiver tuning as the para-H$_2$O$^+$ data presented by \cite{Muller2017}. The data reduction followed the same method as for MA0.68 data described above (self-calibration on the quasar, extraction of the spectra from visibility fitting, and weighted-averaged final spectrum).

\section{Results} \label{sec:results}

\begin{table*}[ht]
\caption{Summary of results and line-of-sight properties.}\label{tab:ncol}
\begin{center} \begin{tabular}{cccccccc}
\hline \hline

Sightline & $z_{\rm abs}$ & A$_{V,tot}$ & N(H$_2$) & N(H) &  N(H$_2$Cl$^+$) $^a$ &  N(HCl) $^a$ & $^{35}$Cl/$^{37}$Cl \\
                  &             & (mag)       & (cm$^{-2}$) &  (cm$^{-2}$) &  (cm$^{-2}$)       & (cm$^{-2}$) &  \\
\hline
MA0.89 SW  & 0.88582 & 20  & $2 \times 10^{22}$ & $1 \times 10^{21}$ & $(2.3 \pm 0.3) \times 10^{13}$ & $(2.7 \pm 0.5) \times 10^{13}$ & $2.99 \pm 0.05$ $^b$ \\
             &         &     &                   & &                     &                     & $3.28 \pm 0.08$ $^c$  \\
MA0.89 NE  & 0.88582 &  2  & $1 \times 10^{21}$ & $2 \times 10^{21}$ & $(5.2 \pm 0.5) \times 10^{12}$   & $< 2.4 \times 10^{11}$ & $3.3 \pm 0.3$ $^b$ \\
\hline
MA0.68    & 0.68466 &  8 & $8 \times 10^{21}$ & $4 \times 10^{20}$ & $(1.4 \pm 0.2) \times 10^{13}$   & --                  & $2.2 \pm 0.3$ $^b$ \\

\hline
\end{tabular}
\tablefoot{ $a)$ Total column density, including both $^{35}$Cl and $^{37}$Cl isotopologs, and ortho and para forms (assuming OPR=3) of \H2Cl+; $b)$ Calculated from H$_2$Cl$^+$ isotopologs; $c)$ Calculated from HCl isotopologs.}
\end{center} \end{table*}

\begin{figure*}[ht!] \begin{center}
\includegraphics[width=\textwidth]{fig-spec-all.eps}
\caption{Absorption spectra of chlorine-bearing species observed toward MA0.89~SW (top three panels), MA0.89~NE (middle three panels), and MA0.68 (bottom panel). Hyperfine structure is indicated in green for $^{35}$Cl-isotopologs of each line toward MA0.89~SW. Hyperfine structure was deconvolved by fitting multiple Gaussian velocity components; these fits are shown in red on top of spectra, with residuals shown in blue. On the right, deconvolved opacity profiles (assuming optically thin lines with $f_c=1$, as explained in Section~\ref{sec:results}), normalized to their peak opacity, are shown, together with opacity profiles of some other species for comparison. The profile of p-H$_2$O$^+$ is of the strongest hyperfine component of the $1_{10}$-$1_{01}$ transition, at rest frequency of 607~GHz. The profile of o-H$_2$S is the $1_{10}$-$1_{01}$ transition, at rest frequency of 168.8~GHz, which was observed simultaneously with o-\H2Cl+\ in the same tuning.}
\label{fig:spec}
\end{center} \end{figure*}

The spectra observed along the three lines of sight are shown in Fig.~\ref{fig:spec}. H$_2$Cl$^+$ is detected in all sightlines, unsurprisingly since it is ubiquitous in the Galactic diffuse and translucent gas \citep{Lis2010,Neufeld2012}. On the other hand, HCl is detected in only one of the two sightlines toward \hbox{PKS\,1830$-$211}\ (MA0.89~SW), with a stringent upper limit toward the other. It was not observed toward MA0.68.

The \H2Cl+\ absorptions reach at most a few percent of the continuum level for all sightlines, so we must determine whether the absorption is optically thin and whether the source covering factor, $f_c$, is less than unity.
In MA0.89~SW, the ground state transition of species like CH$^+$ and H$_2$O is strongly saturated and reaches a depth of nearly 100\% of the continuum level, implying that $f_c \approx 1$ for these species \citep{Muller2014,Muller2017}. In MA0.89~NE, the situation is not so clear, but the deepest absorption reaches a depth of several tens of percent of the continuum level \citep{Muller2017}. For MA0.68, the covering factor of the 557~GHz water line is known to be large and close to unity \citep{Combes1997}, as confirmed by recent ALMA data (Muller, private communication).
The HCl absorption reaches a depth of $\sim$10\% of the continuum level toward MA0.89~SW, so without further evidence we consider $f_c=1$ for all species and all sightlines -- hence an optically thin line regime -- and our derived column densities (Table~\ref{tab:ncol}) can be viewed, strictly speaking, as lower limits.

When detected, both H$_2$Cl$^+$ and HCl show both their \hbox{$^{35}{\rm Cl}$}- and \hbox{$^{37}{\rm Cl}$}-isotopologs, observed simultaneously in the same spectral window (see a summary of the spectroscopic parameters of both species in Table~\ref{tab:hfs}). Considering optically thin lines and given the very small difference in their relative weight, the abundance ratio of the $^{35}$Cl- and $^{37}$Cl-isotopologs should reflect the $^{35}$Cl/$^{37}$Cl isotopic ratio (see Section~\ref{sec:35Cl/37Cl}).

In MA0.89, the ground-state transitions of both the ortho and para forms of \H2Cl+\ were observed. It should be noted that, from the same data, \cite{LeGal2017} determined ortho-to-para ratios (OPR) in agreement with the spin statistical weight 3:1 for both sightlines, within the uncertainties. In MA0.68, only ortho-\H2Cl+\ was observed. We adopted an OPR=3 for the calculations of \H2Cl+\ total column densities in all sightlines.

Finally, both \H2Cl+\ and HCl harbor a hyperfine structure that needs to be deconvolved before comparing intrinsic absorption profiles. This was done by fitting an intrinsic normalized profile, composed by the sum of individual Gaussian velocity components and convolved with the hyperfine structure (see Table~\ref{tab:hfs}), using $\chi^2$ minimization and taking the errors to be 1$\sigma$. Both the $^{35}$Cl- and $^{37}$Cl-isotopologs were treated in a common fit, with the $^{35}$Cl/$^{37}$Cl ratio as a free parameter.
In order to assess whether the hyperfine structure deconvolution was robust, we fitted the ortho and para lines of \H2Cl+\ separately for MA0.89 SW. We obtained a good solution, with residuals at the noise level, with four Gaussian velocity components for this line of sight.
For MA0.89 NE and MA0.68 the detections have lower signal-to-noise ratios, so we limited the number of Gaussian components to one for sake of simplicity. These fits still yield residuals consistent with the noise level.
All the velocity profiles in Fig.~\ref{fig:spec} were normalized to the peak opacity to enable a straightforward comparison.

There is a striking difference in the opacity profiles of \H2Cl+\ and HCl toward MA0.89~SW: the former has prominent line wings, at velocities $| v | = 10-50$~\hbox{km\,s$^{-1}$}, while the latter is mostly a narrow component centered at $v=0$~\hbox{km\,s$^{-1}$}\ with FWHM $\sim 20$~\hbox{km\,s$^{-1}$}. In addition to the chlorine-bearing species of interest here, we also show in Fig.~\ref{fig:spec} the opacity profiles for some other species observed at the same epoch (to prevent effects from time variations of the absorption profile, see \citealp{Muller2008,Muller2014}). In MA0.89~SW, we find that the profile of \H2Cl+\ best matches that of H$_2$O$^+$, while the profile of HCl best matches that of H$_2$S.
Similarly, in MA0.68, we observe that the FWHM of the \H2Cl+\ opacity is noticeably larger than that of H$_2$S, suggesting again that the two species trace different gas components.

Using the column density ratios between the two lines of sight of MA0.89, $\gamma_{SW/NE}$, \citet{Muller2016b,Muller2017} could classify the different species in two simple categories: those with low $\gamma_{SW/NE}$ ($\lesssim 5$), for example ArH$^+$, OH$^+$, H$_2$Cl$^+$, and H$_2$O$^+$, which are known to trace gas with low molecular hydrogen fraction, \hbox{$f_{\rm H_2}$}\footnote{ \hbox{$f_{\rm H_2}$} = 2$\times$n(H$_2$)/[n(H)+2$\times$n(H$_2$)]},
of 1--10\% (see, e.g., \citealt{Neufeld2016}), and those with $\gamma_{SW/NE}$ above 20, for example CH, HF, HCO$^+$, HCN, and H$_2$S, tracing gas with much higher molecular hydrogen fraction, $>30$\% and up to 100\%. Since MA0.89~NE is dominated by gas with low \hbox{$f_{\rm H_2}$}\ the non-detection of HCl in this line of sight, with a large $\gamma_{SW/NE} > 90$, suggests that HCl is a tracer of high-\hbox{$f_{\rm H_2}$} gas, in agreement with chemical predictions (see Section~\ref{sec:discussion}).
The line of sight towards MA0.68 is not yet fully characterized, but the estimated column density of H$_2$ and relative abundance of \H2Cl+\ suggest that it is intermediate between MA0.89~NE and SW in term of molecular hydrogen fraction.

\section{Discussion} \label{sec:discussion}

\subsection{Chemistry}

\begin{table*}[ht!]
\caption{Abundance ratios and derived parameters for diffuse gas in MA0.89 SW and MA0.89 NE}
\label{tab:relat}
\begin{center}
\begin{tabular}{c|c|c|c}
\hline\hline
       &    MA0.89 SW &  MA0.89 NE  & Reference/comment \\
\hline
[\HHClp] / [HCl] & 0.8 & $>$ 17  & present work \\
$[\rm{OH}^+]$ / [H$_2$O$^+$]  &  5.9  &  11  & \cite{Muller2016b} \\
\hbox{$f_{\rm H_2}$}\ & $\sim$ 0.04 & $\sim$ 0.02 & \cite{Muller2016b} \\
$\zeta$ (s$^{-1}$) & $2 \times 10^{-14}$ & $3 \times 10^{-15}$ & \cite{Muller2016b} \\
$x_e$ & $2.5 \times 10^{-4}$ & $2.4 \times 10^{-4}$ & using [OH$^+$]/[H$_2$O$^+$] (Eq.\ref{eq:xefromOH+/H2O+})  \\
G$_0$ & $\sim$ 0.5 $^\dagger$ & $>$ 10 & using [H$_2$Cl$^+$]/[HCl] and $x_e$ (Eq.\ref{eq:G0fromH2Cl+/HCl})  \\
\hline
\end{tabular}
\tablefoot{$\dagger$ Likely underestimated due to extinction.}
\end{center}
\end{table*}

Given the lack of observational constraints other than molecular abundances in the lines of sight through MA0.68 and MA0.89, and the fact that they sample different clouds and potentially different gas components, we have limited ourselves to a simple analytical model to explore the chlorine chemistry and how it varies with a few key parameters.
For diffuse cloud conditions, HCl and \HHClp\ are linked in a simple way, as HCl results from one specific channel of the dissociative recombination of \HHClp\ and is principally destroyed by photodissociation:

\begin{equation}
    \mathrm{\HHClp + e^- \rightarrow  HCl + H}
\end{equation}
\begin{equation}
    \mathrm{HCl + \gamma  \rightarrow   H + Cl}
\end{equation}
\noindent with reaction rates $k_1$ =  7.5 $\times$ 10$^{-8}$ (T/300)$^{-0.5}$ cm$^3$ s$^{-1}$ (see Appendix~\ref{app:ChemicalConsiderations}) and $k_2$ = 1.8 $\times 10^{-9}$ G$_0$ exp(-2.9 $A_V$) s$^{-1}$, where G$_0$ is the scaling factor of the interstellar UV radiation field (ISRF), expressed in Draine units \citep{Heays2017}. Under these conditions HCl is almost completely destroyed.

From these two reactions, we can write, at steady-state:
\begin{equation}
    [\HHClp ]/ [{\rm{HCl}}]  = \frac{k_2}{k_1 n_{e^-}}
\end{equation}

\noindent and, introducing the fractional ionization $x_e$ = $n(e^-) /n_{\rm{H}}$ and neglecting the $A_V$ dependence of the photodissociation rate as in a diffuse line of sight $A_V << 1$, we get

\begin{equation} \label{eq:G0fromH2Cl+/HCl}
    \frac{[\HHClp]}{[{\rm{HCl}}]} \sim 1.4 \times 10^{-2} \frac{G_0}{n_{\rm{H}} x_e} \left(\frac{T}{100}\right)^{0.5}.
\end{equation}
\noindent A large [\HHClp]/[HCl] ratio is obtained for large values of  G$_0$/$n_{\mathrm{H}}$ as suggested by \citet{Neufeld2009}.

We can perform a similar analysis for the [\OHp]/[\HHOp] ratio by considering the main formation process of \HHOp\ via the \OHp\ + \HH\ reaction and its destruction by dissociative recombination and reaction with molecular hydrogen:
\begin{equation}
    \mathrm{\OHp + \HH \rightarrow  \HHOp + H}
\end{equation}
\begin{equation}
    \mathrm{\HHOp + e^- \rightarrow  products}
\end{equation}
\begin{equation}
    \mathrm{\HHOp + \HH \rightarrow  H_3O^+ + H}
\end{equation}
\noindent with reaction rates $k_5 = 1.7 \times 10^{-9}$ cm$^3$ s$^{-1}$,  $k_6  = 4.3 \times 10^{-7}$ (T/300)$^{-0.5}$ cm$^3$ s$^{-1}$, and $k_7 = 6.4 \times 10^{-10}$ cm$^3$ s$^{-1}$.
Substituting the fractional ionization $x_e$ and the molecular fraction \hbox{$f_{\rm H_2}$}\ we get
\begin{equation*}
\frac{[\OHp]}{[\HHOp]} = \frac{k_6 [e^-] + k_7 [\HH]} {k_5 [\HH]}  = \frac{2 k_6 x_e + k_7 \hbox{$f_{\rm H_2}$}}{k_5 \hbox{$f_{\rm H_2}$}},
\end{equation*}
\begin{equation} \label{eq:xefromOH+/H2O+}
    \sim 8.8 \times 10^2 \frac{x_e}{\hbox{$f_{\rm H_2}$}}  \left(\frac{100}{T}\right)^{0.5} + 0.38 .
\end{equation}

Using the derived molecular fractions from \cite{Muller2016b}, we can estimate the fractional ionization from the [\OHp]/[\HHOp] ratio in the two MA0.89 lines of sight, assuming T = 100~K.
The [\HHClp]/[HCl] ratio then allows for the derivation of G$_0$, again following \cite{Muller2016b} in the assumption of $n_{\rm{H}}$ = 35 cm$^{-3}$.

The calculated values are reported in Table \ref{tab:relat}. As the derivation does not consider any dependence on the visual extinction, the derived values of the radiation field correspond to lower limits.
We note that the SW line of sight has a significantly higher molecular fraction, and hence a larger total visual extinction ($A_V$; see Section~\ref{subsec:sightlines}), than the NE line of sight, so its ISRF is likely to be underestimated in our reasoning. The results depend strongly on the
value of $A_V$, but as an example: for a cloud of $A_V$ = 1 in MA0.89~SW, we calculate G$_0$ = 9.
This suggests that both MA0.89 SW and MA0.89 NE have more intense radiation fields than in the Solar neighborhood.

Our simple model is broadly consistent with the results of \citet{Neufeld2009}, though we note that the chlorine chemical network still has some uncertainties, as discussed in Appendix~\ref{app:ChemicalConsiderations}, which can significantly impact the model results.
\citeauthor{Neufeld2009} find a column density ratio [\HHClp]/[HCl]~$\sim1$, as we measure in MA0.89, for a two-sided slab model with G$_0$ = 10 and $n_{\rm{H}}$ = 315 cm$^{-3}$ at $A_V\sim1$. However, the predicted column densities of these molecules are significantly smaller than the observed $\sim$10$^{13}$~cm$^{-2}$ in our lines of sight.

\begin{figure}[ht] \begin{center}
\includegraphics[width=8.8cm]{fig-ratio-CHLORINEBearingSpecies.eps}
\caption{Abundance ratio of H$_2$Cl$^+$ (derived from ortho and para forms, shown in top and bottom panels, respectively) over HCl across absorption profile toward MA0.89~SW.
We use measured ortho-to-para ratio of three to obtain total H$_2$Cl$^+$ column density. Abundance ratio is calculated only for channels where signal-to-noise ratio of normalized profile of both \H2Cl+\ and HCl is higher than five.}
\label{fig:abundanceRatio-H2Cl+-HCl}
\end{center} \end{figure}

\subsubsection{Properties of each sightline} \label{subsec:sightlines}

Here we discuss more specifically our results towards MA0.68 and MA0.89 and compare the physical and chemical properties in the different sightlines.

\paragraph{MA0.89 SW:}
This sightline is rich in molecular species, with almost 50 different species detected so far \citep[e.g.,][]{Muller2011,Muller2014}, and the H$_2$ column density is relatively high, at $\sim 10^{22}$~cm$^{-2}$.
We can estimate the total visual extinction $A_{V,tot}$ in MA0.89 SW using Bohlin's law \citep{Bohlin1978} and the H and H$_2$ column densities \citep{Chengalur1999,Muller2014}. We find $A_{V,tot} \approx 20$ mag, which is shared among a number of individual clouds or kinematical structures. Obviously, there is at least one, most likely a few, cloud(s) with $A_V$ high enough ($\sim 1$) in the line of sight for HCl to be detectable.
Indeed, the line profile of HCl indicates two peaks close to $v \sim 0$~\hbox{km\,s$^{-1}$}. Those two velocity components were previously noticed as two distinct clouds with different chemical setups, as they show opposite peaks in their CF$^+$ vs CH$_3$OH absorption \citep{Muller2016}, suggesting chemical segregation.

The multi-phase composition of the absorbing gas in this sightline has been clearly revealed by the recent observations of hydrides with ALMA \citep{Muller2014,Muller2016,Muller2017}.
Accordingly, the absorption from \H2Cl+\ and HCl can be simply understood
as tracing gas with different molecular fraction. In Fig.~\ref{fig:abundanceRatio-H2Cl+-HCl}, we see a clear difference between the core and wings of the line profiles: in the line center the gas is denser or better shielded and HCl dominates, with a [\H2Cl+]/[HCl] ratio of $\sim 0.5$, while in the wings the ratio increases to $\gtrsim 1$ and \H2Cl+\ dominates. This is consistent with different gas properties (i.e., \hbox{$f_{\rm H_2}$}) between the core and wings.

\paragraph{MA0.89 NE:}
Only \H2Cl+\ is detected in this line of sight, which is characterized by gas at low \hbox{$f_{\rm H_2}$}.
The [H$_2$Cl$^+$]/[HCl] abundance ratio has a lower limit of $\sim$17, more than one order of magnitude higher than in the SW line of sight.
Following the method outlined above, we estimate a total $A_V \approx 2$ mag. Based on a statistical analysis, \cite{Muller2008} argued that the large time variations of the opacity profiles\footnote{due to the intrinsic activity of the background quasar, causing morphological changes, hence changing illumination to the foreground absorber.} imply a relatively small number of individual clouds in the line of sight, $\lesssim 5$, and that the bulk of the absorption should arise from clouds not much smaller than the background continuum source, $\sim 1$~pc. ALMA observations of the CH$^+$ absorption \citep{Muller2017} reveal a complex velocity profile which can be fitted with $\sim 10$ different Gaussian velocity components, some with widths as small as a few \hbox{km\,s$^{-1}$}, spanning a total velocity interval between $-300$~\hbox{km\,s$^{-1}$}\ and $-100$~\hbox{km\,s$^{-1}$}. Taking a simplified model of $\sim$10 individual clouds along the line of sight results in each having an average $A_V$ of $\sim$0.2. From the chemical predictions, we indeed expect some production of \H2Cl+\ at this level of $A_V$ but very little HCl, consistent with the non-detection of HCl.

\paragraph{MA0.68}
Here, we estimate a total $A_V \approx 8$ magnitudes. The overall absorption profile is narrow compared to that of MA0.89 and we estimate the number of clouds to be $\sim 3$ based on the number of distinct velocity features in the profile \citep[e.g.][]{Wallstrom2016}, suggesting characteristics intermediate between MA0.89~SW and MA0.89~NE. The abundance of \H2Cl+\ relative to H$_2$ is also intermediate between the MA0.89 sightlines.

\subsubsection{Elemental chlorine and abundances of chlorine-bearing molecules}

Models of chlorine chemistry \citep[e.g.,][]{Neufeld2009} find that in the presence of H$_2$ most chlorine resides in the form of the neutral atom Cl. Observationally, a strong correlation is found between the column densities of Cl and H$_2$, in both the local ISM \citep{Jura1974,Sonnentrucker2006,Moomey2012} and in high-redshift damped Lyman-$\alpha$ absorption systems \citep{Balashev2015} over three orders of magnitude in H$_2$ column densities, with a relation\footnote{from an independent least-squares fit of the data from \citet{Moomey2012} and \citet{Balashev2015}.}

\begin{equation}
\mathrm{log[N_{Cl}]} = (0.79 \pm0.06) \times \mathrm{log[N_{H_2}]} - (2.13 \pm1.15) .
\label{eq:ClH2corr}
\end{equation}

\noindent This trend is found to be independent of the overall gas metallicity, suggesting that neutral chlorine could be an excellent tracer of H$_2$.

Assuming that the correlation in Eq.~\ref{eq:ClH2corr} also holds for our molecular absorbers at intermediate redshifts, we can infer the column density of neutral chlorine corresponding to the column density of H$_2$ in the different sightlines.
With the observed column densities of H$_2$Cl$^+$ and HCl, we can then estimate the fraction of chlorine in these molecules (assuming neutral chlorine is vastly dominant and summing the \hbox{$^{35}{\rm Cl}$}\ and \hbox{$^{37}{\rm Cl}$}-isotopologs). Accordingly, we estimate that about 0.7\%, 1.8\%, and 0.9\% of chlorine is in the form of H$_2$Cl$^+$ for MA0.89~SW, MA0.89~NE, and MA0.68, respectively. For MA0.89~SW, we also estimate that about 0.9\% of Cl is in HCl.
These values are uncertain by about an order of magnitude (due to dispersion of the $\mathrm{N_{Cl}-N_{H_2}}$ correlation and uncertainty in the H$_2$ column density), and averaged over the whole sightline.

\subsubsection{HCl$^+$ and CH$_3$Cl}

Besides \H2Cl+\ and HCl, we here discuss briefly some other chlorine-bearing species, already detected or potentially present in the ISM in our intermediate-redshift absorbers.
As previously mentioned, HCl$^+$ is the first chlorine-bearing species to be formed in the chlorine-chemistry network from the reaction of Cl$^+$ and H$_2$, in gas with a low enough \hbox{$f_{\rm H_2}$} to maintain some Cl$^+$, and as such, HCl$^+$ is an excellent tracer of the diffuse gas component. HCl$^+$ has a complex spectrum, with multiple ground-state hyperfine transitions around 1.44~THz \citep{DeLuca2012}. Unfortunately, those transitions fall out of ALMA bands for a redshift of 0.89 and HCl$^+$ cannot be observed toward \hbox{PKS\,1830$-$211}. It could in principle be observed toward \hbox{B\,0218$+$357} in ALMA Band~10, although with the observational challenge of a weak background continuum source at low elevation.

Another chlorine-bearing molecule, recently detected in the ISM, is methyl chloride (CH$_3$Cl). It has several hyperfine transitions which were covered in a deep spectral scan in the 7~mm band with the Australia Telescope Compact Array toward \hbox{PKS\,1830$-$211} \citep{Muller2011}. No features related to CH$_3$Cl (rest frequency near 79.8~GHz) are detected down to a few per mil of the continuum level.
Simulated absorption spectra for the conditions appropriate for the MA0.89 absorber yield a peak optical depth of $\tau = 1.95\times 10^{-3}$ for the hyperfine-structure blend, with an upper limit on the column density of $N({\rm CH}_3{\rm Cl}) = 5\times 10^{12}$ cm$^{-2}$.
CH$_3$Cl was recently identified in the young stellar object IRAS $16293-2422$ by \citet{Fayolle2017}, who infer a CH$_3$Cl column density from these high-resolution ALMA data under the assumption of a fixed excitation temperature of 102 K of $4.6\times 10^{14}$ cm$^{-2}$ over a $0\farcs 5$ beam. A previous observation of HCl toward the same source by \citet{Peng2010} yielded well-constrained optical depths in the $J=1-0$ transition but referred to a much larger beam size of $13\farcs 5$. If the HCl result is re-interpreted for a fixed excitation temperature of 102 K, the resulting column density of HCl would be $8.9\times 10^{14}$ cm$^{-2}$, and if the ratio of column densities is simply scaled by the ratio of beam areas, then the abundance ratio would be [CH$_3$Cl]/[HCl] $= 7\times 10^{-4}$. This is likely to be an underestimate, because the HCl-emitting source probably does not fill the $13\farcs 5$ beam. Even so, the [CH$_3$Cl]/[HCl] ratio in IRAS $16293-2422$ is not inconsistent with the value $(4\pm 2)\times10^{-3}$ determined from the Rosetta Orbiter Spectrometer for Ion and Neutral Analysis (ROSINA) measurements of the gaseous coma of comet 67P/Churyumov-Gerasimenko \citep{Fayolle2017}.
This [CH$_3$Cl]/[HCl] ratio is consistent with our upper limit on CH$_3$Cl in MA0.89~SW.

\subsection{$^{35}$Cl/$^{37}$Cl isotopic ratio} \label{sec:35Cl/37Cl}

The $^{35}$Cl/$^{37}$Cl isotopic ratio has been studied in a range of Galactic sources ranging from circumstellar envelopes to molecular clouds and star-forming regions. The measured ratios vary between $\sim$1 and 5, though most fall around 2.5 \citep{Maas2018} with uncertainties large enough to be consistent with the solar system ratio of 3.13 \citep{Lodders2009}. This value is measured from meteorites, and hence is indicative of the conditions at the formation of the solar system. Galactic chemical evolution models by \citet{Kobayashi2011} predict that the chlorine isotopic ratio in the solar system neighborhood has decreased, with increasing metallicity, since its formation to about 1.8 at the present day and Solar metallicity.

From \H2Cl+\ absorption observed in 2012 in MA0.89~SW, \cite{Muller2014b} found a ratio $^{35}$Cl/$^{37}$Cl~=~$3.1 _{-0.2} ^{0.3}$, consistent with the Solar system value.
With the better quality of these new 2014 data, we revise this ratio ($2.99 \pm 0.05$) and we are also able to measure it in MA0.89~NE ($3.3 \pm 0.3$). These two measurements, at galactocentric radii of 2 and 4 kpc, respectively, are consistent each other, suggesting there are only small effects from metallicity or stellar population differences in the disk of MA0.89. In addition, we obtain another measurement from the HCl isotopologs in MA0.89~SW, $3.28 \pm 0.08$, relatively consistent with previous values and suggesting a good mixing between the different gas components with low and high \hbox{$f_{\rm H_2}$}.

In MA0.68, we find a $^{35}$Cl/$^{37}$Cl ratio of $2.2 \pm 0.3$, from H$_2$Cl$^+$ isotopologs. \citet{Wallstrom2016} found overall very similar C, N, O, and S isotopic ratios between MA0.68 and MA0.89~SW, which show clear evolution effects compared to the ratios found in the Solar neighborhood. The $^{35}$Cl/$^{37}$Cl ratio is the first to deviate between MA0.89~SW and MA0.68, and it would be interesting to confirm this difference with further observations.

A $^{35}$Cl/$^{37}$Cl ratio around three reflects nucleosynthesis products mainly from massive stars and core-collapse supernovae \citep{Kobayashi2011}. On the other hand, a \hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ ratio of $\sim 2$ requires nucleosynthesis by either low-metallicity AGB stars or higher-metallicity Type II supernovae. As AGB stars affect their environment on long timescales, the first possibility requires MA0.68 to be a relatively old galaxy. For the second possibility, higher-metallicity Type II supernovae require short timescales and an intrinsically more metal-rich galaxy. The simplest explanation of the difference in chlorine isotopic ratios between MA0.89 and MA0.68 might be that MA0.68 has an intrinsically higher metallicity.

\section{Summary and conclusions} \label{sec:conclusions}

We investigate the absorption of chlorine-bearing molecules in two molecular absorbers (three independent lines of sight with different properties) at intermediate redshift, MA0.89 located toward \hbox{PKS\,1830$-$211}\ ($z_{abs}=0.89$) and MA0.68 towards \hbox{B\,0218$+$357}\ ($z_{abs} = 0.68$). \H2Cl+\ was observed, and detected, toward all sightlines. HCl was observed only toward \hbox{PKS\,1830$-$211}, but detected only in one of the two sightlines.

Our results and conclusions are summarized as follows:
\begin{itemize}
\item The comparison of the absorption spectra of chlorine-bearing species between the different sightlines and with that of other species, namely H$_2$O$^+$ and H$_2$S, provides us with a simple classification based on molecular fraction (\hbox{$f_{\rm H_2}$}). \H2Cl+\ and HCl trace the gas component with low and high \hbox{$f_{\rm H_2}$}\ gas, respectively. This picture is consistent with predictions from chemical modeling, where HCl requires higher visual extinction ($A_V \sim 1$) than \H2Cl+\ to form.

\item These two chlorine-bearing species can hence be used together to characterize the column of absorbing gas. In particular, toward MA0.89~SW, we find that the [\H2Cl+]/[HCl] abundance ratio varies from $\sim$0.5 at the line center, to $\sim$2 in the line wings, reflecting the multi-phase composition of the gas along the line of sight.
The same ratio has a lower limit more than one order of magnitude higher toward MA0.89~NE.

\item The detection of the \hbox{$^{35}{\rm Cl}$}- and \hbox{$^{37}{\rm Cl}$}- isotopologs allows us to measure the \hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ isotopic ratio at look-back times of about half the present age of the Universe. We find basically the same values at galactocentric radii of 2~kpc (\hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ $= 2.99 \pm 0.05$ from \H2Cl+\ and $3.28 \pm 0.08$ from HCl) and 4~kpc (\hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ $= 3.3 \pm 0.3$) in the disk of MA0.89 and in the Solar neighborhood (\hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ $\sim 3$), while we find a lower ratio (\hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ $= 2.2 \pm 0.3$) in MA0.68. This could be interpreted as MA0.68 having an intrinsically higher metallicity.
It would be interesting to obtain more measurements of the \hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ isotopic ratio, including at higher redshifts, to investigate whether this isotopic ratio is a useful tracer of evolution.

\item The comparison of the observed abundances for \H2Cl+\ and HCl with that predicted from chemical modeling suggests the need for a stronger interstellar radiation field in the disk of these absorbers than in the Solar neighborhood. Evidence for a slightly increased cosmic-ray ionization rate of atomic hydrogen was also found in MA0.89 \citep{Muller2016b}, potentially related to the higher star formation activity at these intermediate redshifts.

\end{itemize}

In short, the two chlorine-bearing species \H2Cl+\ and HCl are sensible probes of the interstellar medium in redshifted absorbers. They offer diagnostics of the molecular fraction, gas composition, and nucleosynthesis enrichment via the \hbox{$^{35}{\rm Cl}$}/\hbox{$^{37}{\rm Cl}$}\ ratio. Their ground-state transitions are readily observable, for instance with ALMA, in a wide range of redshifts ($z \sim 0-6$), making them interesting tools to probe physico-chemical properties and evolution effects in the young Universe.

\begin{acknowledgement}
This paper uses the following ALMA data:\\
ADS/JAO.ALMA\#2012.1.00056.S \\
ADS/JAO.ALMA\#2013.1.00020.S \\
ADS/JAO.ALMA\#2016.1.00031.S. \\
ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.

S.H.J.W. acknowledges support by the Ministry of Science and Technology of Taiwan under grants MOST104-2628-M-001-004-MY3 and MOST107-2119-M-001-031-MY3, and from Academia Sinica under AS-IA-106-M03.
\end{acknowledgement}