text:
\section{Introduction and the main results}

In this paper, we examine weighted systems of integral equations involving Riesz potentials of the form
\begin{equation}\label{ie}
    u_{i}(x) = \displaystyle\int_{\mathbb{R}^{n}} \frac{1}{|x-y|^{n-\alpha}|y|^{\sigma_i}}f_{i}(y,u_{1}(y),u_{2}(y),\ldots,u_{L}(y))\,dy,~i=1,2,\ldots, L,
\end{equation}
where $n\geq 3$, $x \in \mathbb{R}^n$, $\alpha \in (0,n)$, $\sigma_i \in [0,\alpha)$ and $f_{i}$ is smooth with respect to its variables. System \eqref{ie} is closely related to the system of pseudo-differential equations involving fractional Laplacians
\begin{equation}\label{pde}
(-\Delta)^{\alpha/2} u_{i} = |x|^{-\sigma_i}f_{i}(x,u_1,u_2,\ldots, u_L) \text{ in } \mathbb{R}^n\backslash\{0\},
\end{equation}
and we shall determine the suitable conditions in which the two systems are equivalent. One reason for examining this general family of systems is to establish and extend results for the system of integral equations of the Hardy--Sobolev type with variable coefficients,
\begin{equation}\label{whls ie}
  \left\{\begin{array}{cl}
    u(x) = c_{1}(x)\displaystyle\int_{\mathbb{R}^{n}} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy, \\
    v(x) = c_{2}(x)\displaystyle\int_{\mathbb{R}^{n}} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy, \\
  \end{array}
\right.
\end{equation}
including its corresponding system of differential equations
\begin{equation}\label{whls pde}
  \left\{\begin{array}{cl}
    (-\Delta)^{\alpha/2} u(x) = \displaystyle c_{1}(x)\frac{v(x)^q}{|x|^{\sigma_1}} & 	\text{ in } \mathbb{R}^n\backslash\{0\}, \\
    (-\Delta)^{\alpha/2} v(x) = \displaystyle c_{2}(x)\frac{u(x)^p}{|x|^{\sigma_2}} & 	\text{ in } \mathbb{R}^n\backslash\{0\}.
  \end{array}
\right.
\end{equation}
Here the variable coefficients $c_{1}(x)$ and $c_{2}(x)$ are taken to be double bounded functions where a function $c(x)$ is said to be \textit{double bounded} if there exists a $C>0$ such that $1/C \leq c(x) \leq C$ for all $x\in \mathbb{R}^n$. In particular, we establish an optimal or sharp criterion for the existence and non-existence of positive solutions for the integral system \eqref{whls ie}, thereby obtaining an analogous sharp existence result for the differential system \eqref{whls pde}. Our pursuits are inspired by the classical result which states that the scalar equation
\begin{equation}\label{scalar lane-emden pde}
-\Delta u(x) = u(x)^{p},\,~x\,\in \mathbb{R}^n
\end{equation}
admits positive classical solutions whenever $p \geq \frac{n+2}{n-2}$ but has no positive solutions if $p \in (1, \frac{n+2}{n-2})$ (see \cite{CGS89,GS81apriori,GS81}). Basically, this asserts that the exponent $p = \frac{n+2}{n-2}$ is the dividing number that provides a sharp criterion for distinguishing between the existence and non-existence of solutions. Also, the qualitative analysis of such elliptic problems has many applications. The classification of solutions for \eqref{scalar lane-emden pde} has provided an important ingredient in the study of the Yamabe problem and the prescribing scalar curvature problem. In the critical case $p = \frac{n+2}{n-2}$, Gidas, Ni and Nirenberg \cite{GNN81} proved the radial symmetry and monotonicity of positive solutions for \eqref{scalar lane-emden pde} under the additional decay assumption $u = \mathrm{O}(|x|^{2-n})$. In \cite{CGS89}, Caffarelli, Gidas and Spruck removed this decay assumption and proved the same result. Chen and Li \cite{CL91} and Li \cite{Li96} later provided simplified proofs of these results using the method of moving planes and the Kelvin transform. We also mention that the significance of the Hardy--Sobolev systems lies in the fact that they serve as the ``blow-up" equations for many related nonlinear systems of partial differential equations (PDEs). Namely, results on the non-existence of positive entire solutions, often called Liouville type theorems, are essential in deriving a priori estimates and asymptotic and regularity properties of solutions for a large class of nonlinear elliptic equations.

Let us discuss some other notable examples within the family of Hardy--Sobolev type systems. If $\alpha = 2$, $\sigma_{i} \in \mathbb{R}$ and $c_{1},c_{2} \equiv 1$, then system \eqref{whls pde} reduces to the H\'{e}non--Lane--Emden equations, which provides a model for rotating stellar clusters in astrophysics \cite{Henon73} (see also \cite{CMS98,Phan12,PhanSouplet12} and the references therein). In the case where $\alpha \in (0,n)$, $\sigma_{1},\sigma_{2} = 0$ and $c_{1},c_{2} \equiv 1$, system \eqref{whls ie} becomes the well-known Hardy--Littlewood--Sobolev (HLS) system
\begin{equation}\label{hls ie}
  \left\{\begin{array}{cl}
    u(x) = \displaystyle\int_{\mathbb{R}^{n}} \frac{v(y)^q}{|x-y|^{n-\alpha}}\,dy, \\
    v(x) = \displaystyle\int_{\mathbb{R}^{n}} \frac{u(y)^p}{|x-y|^{n-\alpha}}\,dy, \\
  \end{array}
\right.
\end{equation}
which arises as the Euler--Lagrange equations for a functional related to the fundamental Hardy--Littlewood--Sobolev inequality (see \cite{HL,Lieb83,SW58}). Its corresponding system of psuedo-differential equations is the Lane--Emden type HLS system
\begin{equation}\label{hls pde}
  \left\{\begin{array}{cl}
    (-\Delta)^{\alpha/2} u(x) = v(x)^q & \text{ in } \mathbb{R}^n, \\
    (-\Delta)^{\alpha/2} v(x) = u(x)^p &	\text{ in } \mathbb{R}^n.
  \end{array}
\right.
\end{equation}
If $\alpha \in (0,n)$ the critical Sobolev hyperbola, $h(p,q):=\frac{1}{1+q} + \frac{1}{1 + p} = \frac{n-\alpha}{n}$, is known to play a key role in the existence criteria for the unweighted HLS systems. More precisely, the HLS conjecture \cite{Caristi2008} states that \eqref{hls ie} admits no positive classical solution under the subcritical case $h(p,q) > \frac{n-\alpha}{n}$. Even in the case where $\alpha = 2$, the conjecture, better known as the Lane--Emden conjecture, remains a long-standing open problem; however, partial results are known. For instance, the Lane--Emden conjecture has been verified for radial solutions (see \cite{Mitidieri96}; see also \cite{Caristi2008,LGZ06} for when $\alpha > 2$), for dimensions $n\leq 4$ (see \cite{PQS07,SZ96,Souplet09}), and for $n\geq 5$ but for certain regions below the Sobolev hyperbola (see \cite{BM02,DeFF94,Mitidieri96,Souplet09}). On the other hand, the existence of positive solutions in the critical case $h(p,q) = \frac{n-\alpha}{n}$ follows from a variational argument used in finding the best constant in the HLS inequality \cite{Lieb83}. In the non-subcritical case where $\alpha$ is an even integer, the authors in \cite{Li13} and \cite{LGZ06a} obtained the existence of positive solutions for the poly-harmonic HLS system \eqref{hls pde}. In \cite{Villavert:14b}, the author also obtained existence results for the weighted system \eqref{whls pde}.

In the scalar case where $u\equiv v$, $p=q$, $\sigma_1 = \sigma_2 = \sigma$ and $c_{1},c_{2}\equiv 1$, system \eqref{whls ie} reduces to the weighted integral equation
\begin{equation}\label{scalar whls ie}
u(x) = \displaystyle\int_{\mathbb{R}^{n}} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma}}\,dy.
\end{equation}
When $\alpha = 2$ and $p = \frac{n+\alpha - 2\sigma}{n-\alpha}$, this equation is closely related to the Euler--Lagrange equation for the extremal functions of the classical Hardy--Sobolev inequality (see \cite{CKN84,CW01}). In the unweighted case i.e. $\sigma = 0$, the Liouville type properties and the classification of positive solutions for \eqref{scalar whls ie} and its corresponding differential equation,
\begin{equation}\label{scalar hls pde}
(-\Delta)^{\alpha/2}u(x) = u(x)^{p},\,~x\, \in \mathbb{R}^n,
\end{equation}
was established in \cite{CLO05}--\cite{CLO06}, thus extending the aforementioned classical results for equation \eqref{scalar lane-emden pde}. For instance, it was shown that every positive regular solution of the equation with the critical exponent $p = \frac{n+\alpha}{n-\alpha}$ assumes the form
\begin{equation*}
u(x) = c_{n}\left( \frac{\lambda}{\lambda^2 + |x - x_{0}|^2} \right)^{\frac{n-\alpha}{2}}
\end{equation*}
for some positive constants $c_n$ and $\lambda$. We remark that the methods developed in the framework of integral equations have encouraged our studies here since they provide a generalization of the differential systems and generate novel ideas and alternative methods, perhaps leading to new and interesting directions for other related problems.

We are now ready to state our main results, but prior to doing so, let us first specify the assumptions we place on our general systems. Hereafter, we impose the following conditions on both systems \eqref{ie} and \eqref{pde}: Let $u = (u_1,u_2,\ldots,u_L)$ be a positive solution of either system and suppose $f_{i}(x,u) > 0$ and $f_{i}(x,0) = 0$ for $x \in \mathbb{R}^n$ and $i=1,2,\ldots ,L$. Then we assume that there exist $p_i > 1$, $\sigma \in [0,\alpha)$ and a positive constant $C$ such that
\begin{equation}\label{general condition}
\displaystyle\sum_{i=1}^{L} |x|^{-\sigma_i}f_{i}(x,u_1,u_2,\ldots,u_L)
\geq C|x|^{-\sigma}\sum_{i=1}^{L} u_{i}(x)^{p_i} \,\text{ for }\, x\neq 0.
\end{equation}

The main results of this paper are organized in the following theorems and corollary, the first of which concerns the equivalence between the system of integral equations and the system of differential equations.
\begin{theorem}\label{theorem1}
Let $\alpha = 2k$ be an even integer. Then the system of integral equations \eqref{ie} and the system of differential equations \eqref{pde} are equivalent. That is, every positive solution of \eqref{ie}, multiplied by a suitable constant if necessary, is a positive solution of \eqref{pde}, and vice versa.
\end{theorem}
\begin{remark}
When we refer here to positive solutions of either the integral system \eqref{ie} or the differential system \eqref{pde}, we mean classical solutions $u=(u_1,u_2,\ldots,u_L)$, whose positive components belong in $C^{2k}(\mathbb{R}^{n}\backslash\{0\})\cap C(\mathbb{R}^n)$, satisfying the system pointwise except at the origin.
\end{remark}
The next theorem establishes the necessary and sufficient conditions for the existence of positive solutions to the Hardy--Sobolev type system for some double bounded coefficients.
\begin{theorem}\label{theorem2}
Let $p,q > 0$ and $\sigma_{1},\sigma_{2} \in [0,\alpha)$. Then the system of integral equations \eqref{whls ie} admits a positive solution $(u,v)$ for some double bounded functions $c_{1}(x)$ and $c_{2}(x)$ if and only if $pq > 1$ and
\begin{equation}\label{criteria}
\max\Bigg\{ \frac{\alpha(q+1)-(\sigma_1 +\sigma_2 q)}{pq - 1}, \frac{\alpha(p+1)-(\sigma_2 + \sigma_1 p)}{pq - 1} \Bigg\} < n - \alpha.
\end{equation}
\end{theorem}
Theorems \ref{theorem1} and \ref{theorem2} imply the following.
\begin{corollary}\label{corollary1}
Let $\alpha = 2k$ be an even integer, $p,q > 1$ and $\sigma_{1},\sigma_{2} \in [0,\alpha)$. Then the system of differential equations \eqref{whls pde} admits a positive solution $(u,v)$ for some double bounded functions $c_{1}(x)$ and $c_{2}(x)$ if and only if
\begin{equation*}
\max\Bigg\{ \frac{2k(q+1)-(\sigma_1 +\sigma_2 q)}{pq - 1}, \frac{2k(p+1)-(\sigma_2 + \sigma_1 p)}{pq - 1} \Bigg\} < n - 2k.
\end{equation*}
\end{corollary}
Of course, these results hold for the scalar integral equation and its corresponding differential equation as well and we state them here for completeness sake. The following theorem, however, is essentially contained in \cite{Lei13}.
\begin{theorem}
Let $p>0$ and $\sigma \in [0,\alpha)$. Then there hold the following.
\begin{enumerate}[(i)]
\item The integral equation,
\begin{equation*}
u(x) = c(x)\int_{\mathbb{R}^n} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma}} \,dy,
\end{equation*}
admits a positive solution $u$ for some double bounded function $c(x)$ if and only if $p > \frac{n-\sigma}{n-\alpha}$.

\item Let $p>1$ and $\alpha = 2k$ be an even integer. Then the poly-harmonic equation,
\begin{equation*}
(-\Delta)^k u(x) = c(x)\frac{u(x)^p}{|x|^{\sigma}} \,\text{ in }\, \mathbb{R}^n \backslash \{0\},
\end{equation*}
admits a positive solution $u$ for some double bounded function $c(x)$ if and only if $p > \frac{n-\sigma}{n-2k}$.
\end{enumerate}
\end{theorem}

The remaining two results are Liouville type theorems concerning radially symmetric, decreasing solutions for Hardy--Sobolev type systems with constant coefficients. Here, the radial solutions are understood to belong to $C^{\lfloor \alpha \rfloor}(\mathbb{R}^n \backslash \{0\})\cap C(\mathbb{R}^n)$ where $\lfloor \,\cdot\, \rfloor$ is the greatest integer function. In view of the equivalence between the differential and integral equations, these theorems include the non-existence results for the unweighted Lane--Emden type differential system (see \cite{LGZ06,Mitidieri96}). Moreover, the results of \cite{Villavert:14b} suggest our non-existence results for the Hardy--Sobolev differential equation and system are indeed optimal as far as radial solutions are concerned.

\begin{theorem}\label{theorem4}
Let $\alpha \in [2,n)$ and $\sigma \in (-\infty,\alpha)$. Then the integral equation
\begin{equation}\label{hs eq}
u(x) = \displaystyle\int_{\mathbb{R}^{n}} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma}}\,dy,
\end{equation}
has no positive radial solution if
\begin{equation}\label{subcritical1}
0 < p < \frac{n + \alpha - 2\sigma}{n - \alpha}.
\end{equation}
\end{theorem}

\begin{theorem}\label{theorem5}
Let $p,q > 0$, $\alpha \in [2,n)$ and $\sigma_{1},\sigma_{2}\in (-\infty,\alpha)$. Then the system of integral equations
\begin{equation}\label{hs sys}
  \left\{\begin{array}{cl}
    u(x) = \displaystyle\int_{\mathbb{R}^{n}} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy, \\
    v(x) = \displaystyle\int_{\mathbb{R}^{n}} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy, \\
  \end{array}
\right.
\end{equation}
has no positive radial solution if $pq \in (0,1]$ or if $pq > 1$ and
\begin{equation}\label{subcritical2}
\frac{n-\sigma_1}{1 + q} + \frac{n-\sigma_2}{1 + p} > n-\alpha.
\end{equation}
\end{theorem}

The rest of this article is organized as follows. In Section \ref{section2}, we begin with several intermediate lemmas and basic results required in our proof of Theorem \ref{theorem1}, which we then provide at the end of the section. Section \ref{section3} starts off with a general non-existence result for the Hardy--Sobolev type integral system followed by the proofs of Theorem \ref{theorem2} and Corollary \ref{corollary1}. The paper concludes with Section \ref{section4}, which provides the proof of both Theorems \ref{theorem4} and \ref{theorem5}.
\begin{remark}
Throughout this paper, we adopt the convention that $C$ represents some constant in the inequalities which may change from line to line. At times, we append subscripts to $C$ to specify its dependence on the subscript parameters.
\end{remark}

\section{Some preliminaries and the proof of Theorem \ref{theorem1}}\label{section2}
A key idea in the proof of Theorem \ref{theorem1} is to multiply the PDEs by a suitable Greene's function, integrate on a ball domain of radius $R$, then send $R\rightarrow \infty$ to show the solutions of the PDE system satisfy the integral system. However, we need to address some technical issues such as the super poly-harmonic property of solutions and some integrability properties of solutions.
\subsection{Preliminaries}
We begin with the super poly-harmonic property for solutions of \eqref{pde}.
\begin{lemma}[Super poly-harmonic property]\label{super polyharmonic}
Let $\alpha = 2k$ be an even integer and suppose that $u=(u_1,u_2,\ldots,u_L)$ is a positive solution of \eqref{pde}. Then
$$(-\Delta)^{j}u_i > 0 \,\text{ for }\, i=1,2,\ldots,L,\, j = 1,2,\ldots,k-1.$$
\end{lemma}
\begin{proof}
We start by proving the result for a single differential inequality then prove it for the general systems, which follows naturally from the scalar case. Suppose $u$ is a positive solution of the differential inequality
\begin{equation}\label{single diff ineq}
(-\Delta)^{k}u(x) \geq C|x|^{-\sigma}u(x)^p, ~ x \in \mathbb{R}^n \backslash \{0\}.
\end{equation}
We assume $p>1$ and $\sigma \in (0,\alpha)$, since the unweighted case when $\sigma = 0$ was treated in \cite{CL13} (see also \cite{CFL13,WeiXu99}). Set
\begin{equation}\label{reduction}
u_{j} := (-\Delta)^{j-1}u \,\text{ for }\, j = 1,2,\ldots, k.
\end{equation}

\medskip

\noindent{\bf Step 1.} First, we show that $u_{k} > 0$. On the contrary, we can have two cases:
\begin{enumerate}[(a)]
\item $u_{k}(x_0) < 0$ at some non-zero point $x_0$;
\item $u_{k} \geq 0$ and $u_{k}(x_{0}) = 0$ at some non-zero point $x_0$.
\end{enumerate}
However, if case (b) holds then $x_0$ is a local minimum, but this contradicts with $-\Delta u_{k} > 0$. Thus, we only consider case (a). In addition, we define the average of a function $v$ on the ball of radius $r$ centered at $\overline{x_0}$ by
\begin{equation*}
\overline{v}(r) = \frac{1}{|\partial B_{r}(\overline{x_0})|}\int_{\partial B_{r}(\overline{x_0})} v(x) \,ds.
\end{equation*}
After a reduction of \eqref{single diff ineq} into a second-order system via \eqref{reduction}, then applying averaging centered at $x_0$ and using the well-known property $\overline{\Delta u} = \Delta\overline{u}$, we get
\begin{equation}\label{reduced system}
\left\{ \begin{aligned}
        &-\Delta \overline{u}_{1} = \overline{u}_{2}, \\
        &-\Delta \overline{u}_{2} = \overline{u}_{3},\\
        & \hspace{1.1cm} \vdots \\
        &-\Delta \overline{u}_{k-1} = \overline{u}_{k},\\
        &-\Delta \overline{u}_{k} > 0.
        \end{aligned} \right.
\end{equation}
From H\"{o}lder's inequality,
\begin{align*}
\overline{u}(r)^{p} = {} & \left(\frac{1}{|\partial B_{r}(x_0)|} \int_{\partial B_{r}(x_0)} u(x) \,ds\right)^p \\
\leq {} & \left(\frac{1}{|\partial B_{r}(x_0)|}\int_{\partial B_{r}(x_0)} |x|^{\frac{\sigma}{p-1}} \,ds \right)^{p-1} \left(\frac{1}{|\partial B_{r}(x_0)|}\int_{\partial B_{r}(x_0)} \frac{u(x)^p}{|x|^{\sigma}} \,ds \right) \\
\leq {} & a_r \cdot \frac{1}{|\partial B_{r}(x_0)|}\int_{\partial B_{r}(x_0)} \frac{u(x)^p}{|x|^{\sigma}} \,ds,
\end{align*}
where $a_r := (|x_0| + r)^{\sigma}$. Thus,
\begin{equation}\label{holder}
-\Delta \overline{u}_{k} = -\overline{\Delta u_{k}} \geq Ca_{r}^{-1} \overline{u}^{p}.
\end{equation}
We claim that after sufficient applications of averaging and re-centers, the resulting solution satisfies an alternating sign property.

\noindent{\bf Claim 1:} After sufficient re-centers of $u_{j}$ and denoting the resulting functions by $\tilde{u}_{j}$, we obtain
\begin{equation}\label{alternating}
\tilde{u}_{k} < 0, \tilde{u}_{k-1} > 0, \tilde{u}_{k-2} < 0,\tilde{u}_{k-3} > 0,\ldots.
\end{equation}
\noindent{\it Proof of Claim 1:} From the $k^{th}$ equation in \eqref{reduced system}, we get
$$\overline{u}_{k} '(r) < 0 \,\text{ and }\, \overline{u}_{k}(r) \leq \overline{u}_{k}(0) < 0 \,\text{ for all }\, r \geq 0. $$
Then the $(k-1)^{th}$ equation implies
\[ -(r^{n-1}\overline{u}_{k-1}'(r))' = r^{n-1}\overline{u}_{k}(r) \leq \overline{u}_{k}(0) r^{n-1} \equiv -c r^{n-1}. \]
Upon integrating we get
\[ \overline{u}'_{k-1}(r) > \frac{c}{n}r \,\text{ and }\, \overline{u}_{k-1}(r) \geq \overline{u}_{k-1}(0) + \frac{c}{2n}r^2 \,\text{ for }\, r \geq 0, \]
so that we can choose $r_1$ sufficiently large such that $\overline{u}_{k-1}(r_1) > 0$. Now, choose some $x_1 \in \partial B_{r_1}(0)$ to be the new point to re-center and apply averaging to the system to get
$$\overline{\overline{u}}_{k-1}(0) > 0,$$
and notice that $\overline{\overline{u}}_{j}$ for $j=1,2,\ldots,k$ satisfy \eqref{reduced system}. As before, we get $$\overline{\overline{u}}'_{k-1} > 0 \,\text{ and }\ \overline{\overline{u}}_{k-1}(r) \geq \overline{\overline{u}}_{k-1}(0) > 0 \,\text{ for all }\, r\geq 0.$$
Thus,
$$\overline{\overline{u}}_{k} < 0 \,\text{ and }\, \overline{\overline{u}}_{k-1} > 0.$$
From the $(k-2)^{th}$ equation, we get
\[ -(r^{n-1}\overline{\overline{u}}_{k-2}'(r))' = r^{n-1}\overline{\overline{u}}_{k-1}(r) \geq \overline{\overline{u}}_{k-1}(0) r^{n-1} \equiv c r^{n-1} \]
and integrating this yields
\[\overline{\overline{u}}'_{k-2}(r) > \frac{c}{n}r \,\text{ and }\, \overline{\overline{u}}_{k-2}(r) \leq \overline{\overline{u}}_{k-2}(0) - \frac{c}{2n}r^2 \,\text{ for }\, r \geq 0. \]
Then choose $r_2$ large enough such that $\overline{\overline{u}}_{k-2}(r_2) < 0$ and pick a point $x_2 \in \partial B_{r_2}(0)$ to be the new point to re-center to get $$\overline{\overline{\overline{u}}}_{k} < 0, \overline{\overline{\overline{u}}}_{k-1} > 0 \,\text{ and }\, \overline{\overline{\overline{u}}}_{k-2} < 0.$$
By repeating this procedure, we arrive at the alternating sign property after sufficient re-centers. This completes the proof of Claim 1.

From Claim 1, we must take $k$ to be an even integer, otherwise we get a contradiction with the positivity of $u$. Henceforth, we assume $k$ is even. Now, define the rescaling of $u$ by
\begin{equation}\label{rescale}
u_{\lambda}(x) := \lambda^{\frac{(2k - \sigma)}{{p-1}}}u(\lambda x)
\end{equation}
and notice that \eqref{single diff ineq} is invariant under this scaling i.e. for any $\lambda > 0$, $u_{\lambda}$ remains a solution of \eqref{single diff ineq}. By \eqref{alternating}, $-\Delta \tilde{u} < 0$, which implies $\tilde{u}'(r) > 0$. Thus,
$$ \tilde{u}(r) \geq \tilde{u}(0) = c_0 > 0. $$
By the scaling invariance, we can choose $u_{\lambda}$ to be as large as needed. Thus, for any $a_0$, we may assume
$$\tilde{u}(r)\geq a_0 \geq a_{0}r^{\beta_0} \,\,\text{ for all }\,\, 0 \leq r \leq 1$$
where the positive number $\beta_0$ is specified below. From \eqref{holder}, we have
\begin{equation*}
-(r^{n-1}\tilde{u}_{k}'(r))' \geq Ca_{0}^{p}r^{n-1 + \beta_{0}p}.
\end{equation*}
Integrating this twice with respect to $r$ and applying \eqref{alternating} yields
\begin{equation*}
\tilde{u}_{k}(r) \leq - \frac{Ca_{0}^{p}r^{2+p\beta_0}}{(n+p\beta_0)(2 + p\beta_0)}.
\end{equation*}
Then
\begin{equation*}
-(r^{n-1}\tilde{u}_{k-1}'(r))' = r^{n-1}\tilde{u}_{k} \leq -\frac{Ca_{0}^{p}r^{n-1 + 2+ \beta_{0}p}}{(n+p\beta_0)(2 + p\beta_0)},
\end{equation*}
and integrating this twice yields
\begin{align*}
\tilde{u}_{k-1}(r) \geq {} & \frac{Ca_{0}^{p} r^{2\cdot 2 + p\beta_0}}{(n+p\beta_0)(n+2 + p\beta_0)(2 + p\beta_0)(2\cdot 2 + p\beta_0)} \\
\geq {} & \frac{Ca_{0}^{p} r^{2\cdot 2 + p\beta_0}}{(n+2 + p\beta_0)^{2}(2\cdot 2 + p\beta_0)^{2}} \\
\geq {} & \frac{Ca_{0}^{p} r^{2\cdot 2 + \beta_{0}p}}{(n + 2\cdot 2 + p\beta_0)^{2\cdot 2}}.
\end{align*}
By continuing in this manner, we obtain
\begin{equation}\label{first step iteration}
\tilde{u}(r) \geq \frac{Ca_{0}^{p} r^{2k + p\beta_0}}{(n + 2k + p\beta_0)^{2k}}
\geq \frac{Ca_{0}^{p} r^{m + p\beta_0}}{(m + p\beta_0)^{m}},~ m = n+2k.
\end{equation}
Choose $\beta_0$ so that $\beta_{0}p \geq m$ and define
$$ \beta_{k+1} = 2\beta_{k}p \,\text{ and }\, a_{k+1} = \frac{a_{k}^{p}}{(2\beta_{k}p)^m}$$
so that $\beta_{k+1} \geq m + \beta_{k}p$ and $\beta_k \rightarrow \infty$ as $k\rightarrow \infty$. Notice that assertion \eqref{first step iteration} shows that $$\tilde{u}(r) \geq a_1 \geq a_{1}r^{\beta_1}\,\,\text{ for all }\,\, 0 \leq r \leq 1.$$
If we apply the previous argument successively, it is not too difficult to show that
\begin{equation}\label{lower bound iteration}
\tilde{u}(r) \geq a_{k}r^{\beta_k}\,\text{ for }\, k=0,1,2,\ldots.
\end{equation}

\noindent{\bf Claim 2:} Choose $l$ so that $l(p-1) > 2$ and further suppose $\beta_0 \geq 2^{1+l+p} p^l$. Then
\begin{equation}\label{claim 2}
a_{k}^{p} \geq (\beta_{k}p)^{m(l+1)},~k=0,1,2,\ldots.
\end{equation}
\noindent{\it Proof of Claim 2:} To prove \eqref{claim 2}, we proceed by induction. The initial case $k=0$ holds immediately since we are free to choose $a_0$ to be as large as needed. Now assume the $k^{th}$ case holds i.e. $$a_{k}^{p} \geq (\beta_{k}p)^{m(l+1)}.$$ Then
\begin{align*}
\frac{a_{k+1}^{p}}{(\beta_{k+1} p)^{m(l+1)}} = {} & \frac{\Big[\frac{a_{k}^{p}}{(2 \beta_{k}p)^{m}}\Big]^{p}}{(p \beta_{k+1})^{m(l+1)}}
\geq \frac{(\beta_{k} p)^{p m(l+1)}}{p^{m(l+1)}(2 \beta_{k} p)^{m p}\beta_{k+1}^{m (l+1)}} \\
= {} & \frac{(\beta_{k} p)^{p m(l+1)}}{p^{m (l+1)}(2\beta_{k} p)^{m p + m (l+1)}} = \frac{\beta_{k}^{ m(l(p-1)-1)}}{2^{m (1+ l + p)} p^{m (l+2 + l(1-p))}} \\
\geq {} & \Bigg[\frac{\beta_{k}}{2^{1+ l + p} p^{l+2 - l(p-1)}}\Bigg]^{m} \geq \Bigg[\frac{\beta_{k}}{2^{1+ l + p} p^l}\Bigg]^{m} \geq \Bigg[\frac{\beta_{0}}{2^{1+ l + p} p^l}\Bigg]^{m} \\
\geq {} & 1.
\end{align*}
Hence, the $(k+1)^{th}$ case holds and this completes the proof of Claim 2.

Consequently, estimates \eqref{lower bound iteration} and \eqref{claim 2} imply
\begin{equation*}
\tilde{u}(1) \geq a_k \geq (\beta_{k}p)^{(m(l+1))/p} \rightarrow \infty \,\text{ as }\, k \rightarrow \infty,
\end{equation*}
which is a contradiction. This proves $u_k > 0$.

\medskip

\noindent{\bf Step 2.} We prove $u_j > 0$ for $j= 2,3,\ldots,k-1$.

On the contrary, assume that $u_{j_0} < 0$ at some point for $j_0 \neq k$. Since $u_k > 0$, we can assume that $u_{j_0 + 1},u_{j_0 + 2},\ldots, u_{k} > 0$. By adopting the same arguments in the proof of Claim 1 with several re-centers and denoting the resulting functions by $\tilde{u}_{j}$, we obtain the alternating sign property $$\tilde{u}_{1} > 0, \tilde{u}_{2}<0,\ldots, \tilde{u}_{j_0 -1} > 0, \tilde{u}_{j_0} < 0 \,\text{ and }\, \tilde{u}_{j_0 + 1},\tilde{u}_{j_0 + 2},\ldots, \tilde{u}_{k} > 0. $$
Thus, by the positivity of $u$ we obtain $-\Delta \tilde{u} < 0$ and $\tilde{u} > 0$, which implies
\begin{equation}\label{positivity}
\tilde{u}(r) \geq \tilde{u}(0) \equiv c_0 > 0 \,\text{ for }\, r \geq 0.
\end{equation}
Then from the $k^{th}$ equation we get $$-(r^{n-1}\tilde{u}_{k}'(r))' \geq Ca^{-1}_{r}\tilde{u}(r)^{p}r^{n-1}\geq Ca^{-1}_{r}r^{n-1},$$
where $a_{r} := (C + r)^{\sigma}$. Integrating this with respect to $r$ yields $$-r\tilde{u}_{k}'(r) \geq Ca^{-1}_{r} r^{2},$$
then applying the standard identity (see Lemma 3.1 in \cite{Mitidieri93})
\begin{equation}\label{standard id}
(n-2)\tilde{u}_{k}(r) + r \tilde{u}_{k}'(r) \geq 0
\end{equation}
yields
\begin{equation}\label{u_k}
\tilde{u}_{k}(r) \geq Ca_{r}^{-1}r^{2}.
\end{equation}
Then integrating the $(k-1)^{th}$ equation, $-(r^{n-1}\tilde{u}_{k-1}'(r))' = r^{n-1}\tilde{u}_{k}(r) $, yields $$-r\tilde{u}_{k-1}'(r) \geq C\tilde{u}_k(r)r^2.$$
Combining this estimate with \eqref{u_k} gives us
$$ \tilde{u}_{k-1}(r) \geq -Cr\tilde{u}_{k-1}'(r) \geq Ca_{r}^{-1}r^{2\cdot 2}.$$
By continuing in this way up to the $(j_{0}+1)^{th}$ equation, we obtain
$$ \tilde{u}_{j_{0}+1}(r) \geq Ca^{-1}_{r}r^{2(k-j_{0})}. $$
From here, we basically mimic the steps at the end of Claim 1; that is, from the $j_{0}^{th}$ equation, $-\Delta \tilde{u}_{j_0}(r) = \tilde{u}_{j_{0}+1}(r) \geq Ca_{r}^{-1}r^{2(k-j_{0})}$, and since $\tilde{u}_{j_{0}} < 0$, we can easily obtain
$$ \tilde{u}_{j_0}(r) \leq -Ca_{r}^{-1}r^{2(k-j_0) + 2}. $$
Likewise, the $(j_{0}-1)^{th}$ equation and $\tilde{u}_{j_{0}-1} > 0$ will imply
$$ \tilde{u}_{j_0 - 1}(r) \geq Ca_{r}^{-1}r^{2(k-j_0) + 2\cdot 2}.  $$
We continue in this way using the alternating sign property with the positivity of $u$ to arrive at the improved lower bound
\begin{equation}\label{lower u}
\tilde{u}(r) \geq Ca_{r}^{-1}r^{2k}.
\end{equation}
Therefore, if $2 - \sigma + (2k - \sigma)p > 0$, we can integrate the resulting inequality from the $k^{th}$ equation,
$$-(r^{n-1}\tilde{u}'_{k}(r))' \geq Ca_{r}^{-1}r^{n-1}\tilde{u}(r)^p \geq Ca_{r}^{-1}r^{n-1}(a^{-1}_{r}r^{2k})^p, $$
to get for some fixed $r_0 > 0$
$$\tilde{u}_{k}(r) \leq \tilde{u}_{k}(0) - Cr^{2-\sigma + (2k -\sigma)p} \,\text{ for }\, r \geq r_0.$$
Then we can choose $R$ suitably large so that $\tilde{u}_{k}(R) \leq 0$, but this is impossible. Otherwise, if $2 - \sigma + (2k - \sigma)p \leq 0$, we can repeat the procedure for obtaining \eqref{lower u} to find a suitably large $m > 1$ depending on $p$ for which $2 - \sigma + m(2k-\sigma)p > 0$ and $\tilde{u}(r) \geq C(a_{r}^{-1}r^{2k})^{m}$. By applying this to the $k^{th}$ equation and integrating, we will arrive at the same contradiction. This completes the proof of Step 2.

\medskip

\noindent{\bf Step 3.} We are ready to prove the theorem for a system of poly-harmonic equations. Assume the contrary; that is, assume for some $i_0,j_0$,
\begin{equation}\label{negative}
(-\Delta)^{j_0}u_{i_0} < 0 \,\text{ at some point.}\,
\end{equation}
Set $v = u_1 + u_2 + \ldots + u_{L}$ and $v_{\epsilon} = u_{i_0} + \epsilon\sum_{i\neq i_0} u_{i}$ for each small $\epsilon > 0$. In view of condition \eqref{general condition}, we can find $p>1$, $\sigma \in [0,\alpha)$ and a suitable constant $C_{\delta} >0$ such that for $v \geq \delta$,
\begin{align*}
(-\Delta)^{k}v_{\epsilon}(x) \geq {} & |x|^{-\sigma_{i_0}}f_{i_0}(x,u_{1},\ldots,u_{L}) + \epsilon\sum_{i\neq i_0} |x|^{-\sigma_i}f_{i}(x,u_{1},\ldots,u_{L}) \\
\geq {} & \epsilon C_{\delta}|x|^{-\sigma}v_{\epsilon}(x)^{p}.
\end{align*}
Here we can apply the same procedures in steps 1 and 2 that we used to derive the super poly-harmonic property for \eqref{single diff ineq} to arrive at
$$ (-\Delta)^{j}v_{\epsilon} > 0, \,\text{ for }\, j=1,2,\ldots,k-1.$$
Then, by choosing $\epsilon$ sufficiently small, we can arrive at a contradiction with \eqref{negative} thereby completing the proof of the theorem.
\end{proof}

Next are two lemmas concerning integrability properties of solutions whose proofs require the following basic results on a fundamental boundary value problem. For each $r> 0$, let $\varphi_{r}(x)$ be the solution of
\begin{align*}
  \left\{\begin{array}{lll}
    (-\Delta)^{k}\varphi(x) = \delta_{x_0}(x) & \text{ in } & B_{r}(x_0), \\
    \varphi = \Delta \varphi = \ldots = \Delta^{k-1}\varphi = 0 & \text{ on } & \partial B_{r}(x_0),
  \end{array}
\right.
\end{align*}
where $\delta_{x_0}(x) = \delta(x-x_0)$ is the centered Dirac delta function. Then
\begin{equation}\label{Hopf}
\frac{\partial}{\partial n}\Big[ (-\Delta)^{k-j}\varphi_{r}(x) \Big] \leq 0 \,\text{ on }\, \partial B_{r}(x_0),\,\text{ for }\, j=1,2,\ldots, k;
\end{equation}
\begin{equation}\label{boundary estimate}
\Big | \frac{\partial}{\partial n}[(-\Delta)^{k-j}\varphi_{r}(x)] \Big |\leq \frac{C}{r^{n+1-2j}} \,\text{ on }\, \partial B_{r}(x_0),\,\text{ for }\, j=1,2,\ldots,k;
\end{equation}
\begin{equation}\label{delta convergence}
\varphi_{r}(x) \rightarrow \frac{c}{|x - x_0|^{n-2k}} \,\text{ as }\, r \rightarrow \infty;
\end{equation}
and
\begin{equation}\label{higher delta convergence}
(-\Delta)^{j-1}\varphi_{r}(x) \rightarrow \frac{c}{|x - x_0|^{n-2(k-j+1)}} \,\text{ as }\, r \rightarrow \infty
\end{equation}
for $j = 2,3,\ldots,k$. Another basic result we often invoke is the following.
\begin{lemma}\label{basic}
If $u\in L^{1}(\mathbb{R}^n)$ and $x_0$ is some point in $\mathbb{R}^n$, then we can find a sequence $\{ r_{\ell} \} \rightarrow \infty$ such that
\begin{equation*}
r_{\ell}\int_{\partial B_{r_{\ell}}(x_0)} |u(x)| \,ds \rightarrow 0.
\end{equation*}
\end{lemma}
For more details on these basic properties and Lemma \ref{basic}, we refer the reader to \cite{CL13,CLO06} and \cite{LL13}.
\begin{lemma}\label{integrable}
Let $\alpha = 2k$ be some positive integer and let $u$ be a solution of \eqref{pde}. Write
$$ u_{ij} = (-\Delta)^{j-1} u_i,~i=1,2,\ldots,L,~j = 1,2,\ldots, k.$$
Then
\begin{equation}\label{integrable 1}
c\int_{\mathbb{R}^n} \frac{1}{|x - x_0|^{n-2k}|x|^{\sigma_i}}f_{i}(x,u_{1}(x),\ldots,u_{L}(x)) \,dx \leq u_{i}(0) < \infty,
\end{equation}
and
\begin{equation}\label{integrable 2}
\int_{\mathbb{R}^n} \frac{u_{ij}(x)}{|x - x_0|^{n+2-2j}} \,dx < \infty \,\text{ for }\, i=1,2,\ldots, L,~j = 2,3,\ldots, k.
\end{equation}
\end{lemma}
\begin{proof}
Multiply both sides of the $i^{th}$ equation,
\begin{equation}\label{ith equation}
(-\Delta)^{k}u_{i}(x) = |x|^{-\sigma_i}f_{i}(x,u_{1}(x),\ldots,u_{L}(x)),
\end{equation}
by $\varphi_{r}(x)$ and integrate on $B_{r}(x_0)$. After successive integration by parts, using the super poly-harmonic property of solutions and \eqref{Hopf}, we get
\begin{align*}
\int_{B_{r}(x_0)} |x|^{-\sigma_i}f_{i}(x,u_{1}(x), {} & \ldots,u_{L}(x))\varphi_{r}(x) \,dx \\
= {} & u_{i}(x_0) + \sum_{j=1}^{k} \int_{\partial B_{r}(x_0)} u_{ij}(x)\frac{\partial}{\partial n}\Big[(-\Delta)^{k-j}\varphi_{r}(x)\Big]\,ds \\
\leq {} & u_{i}(x_0).
\end{align*}
Using this and \eqref{delta convergence}, we can send $r \rightarrow \infty$ to get \eqref{integrable 1}. The proof of \eqref{integrable 2} follows from similar calculations by considering the equation $u_{ij} = (-\Delta)^{j-1}u_{i}$ instead of \eqref{ith equation}, multiplying this equation by $(-\Delta)^{k-j+1}\varphi_r$, integrating on $B_{r}(x_0)$ and then applying \eqref{higher delta convergence}.
\end{proof}
A consequence of Lemma \ref{integrable} is the following.
\begin{lemma}\label{vanishing boundary integral}
There exists a sequence $\{ r_{\ell} \} \rightarrow \infty$ for which
\begin{equation}\label{vbi}
\displaystyle\int_{\partial B_{r_{\ell}}(x_0)} \frac{u_{ij}}{r_{\ell}^{n+1-2j}} \,ds \rightarrow 0 \,\text{ for }\, i=1,2,\ldots,L,\, j=1,2,\ldots,k.
\end{equation}
\end{lemma}
\begin{proof}
For each $i=1,2,\ldots,L$ and $j=2,3,\ldots,k$, the result follows directly from \eqref{integrable 2} and Lemma \ref{basic}, so it remains to verify the lemma for $j=1$. Observe that from \eqref{integrable 1} we can find a sequence $\{ r_{\ell} \} \rightarrow \infty$ for which
\begin{equation}\label{vbu}
\int_{\partial B_{r_{\ell}}(x_0)} \sum_{m=1}^{L} \frac{f_{m}(x,u_{1}(x),\ldots,u_{L}(x))}{|x-x_0|^{n-1-2k}|x|^{\sigma_m}} \,ds \rightarrow 0.
\end{equation}
Then H\"{o}lder's inequality and \eqref{general condition} imply
\begin{align*}
\int_{\partial B_{r}(x_0)} {} & \frac{u_{i}(x)}{|x-x_0|^{n-1}} \,ds \\
\leq {} & \frac{C}{r^{n-1}} r^{(n-1-2k + \sigma + (n-1)(p_i -1))/p_i} \Big(\int_{\partial B_{r}(x_0)} \frac{u_{i}(x)^{p_{i}}}{|x-x_0|^{n-1-2k}|x|^{\sigma}} \,ds \Big)^{\frac{1}{p_i}} \\
\leq {} & \frac{C}{r^{(2k-\sigma)/p_i}}\Big(\int_{\partial B_{r}(x_0)} \frac{u_{i}(x)^{p_i}}{|x-x_0|^{n-1-2k}|x|^{\sigma}} \,ds \Big)^{\frac{1}{p_i}}\\
\leq {} & \frac{C}{r^{(2k-\sigma)/p_i}}\Big( \int_{\partial B_{r}(x_0)} \sum_{m=1}^{L}\frac{f_{m}(x,u_{1}(x),\ldots,u_{L}(x))}{|x-x_0|^{n-1-2k}|x|^{\sigma_m}} \,ds \Big)^{\frac{1}{p_i}}.
\end{align*}
This estimate and \eqref{vbu} imply that we can find a sequence $\{ r_{\ell} \} \rightarrow \infty$ such that
$$\frac{1}{r_{\ell}^{n-1}}\int_{\partial B_{r_{\ell}}(x_0)} u_{i}(x)\,ds \rightarrow 0, \,~\, i=1,2,\ldots,L.$$
This completes the proof.
\end{proof}

\subsection{Proof of Theorem \ref{theorem1}}
We begin by showing the solutions of the PDE system are, up to a multiplicative constant, solutions of the integral system. First, multiply both sides of the $i^{th}$ equation, $$(-\Delta)^{k}u_{i}(x) = |x|^{-\sigma_i}f_{i}(x,u_{1}(x),\ldots,u_{L}(x)),$$ by $\varphi_{r}(x)$ then integrate on $B_{r}(x_0)$. As before, we calculate
\begin{align}\label{main id}
\int_{B_{r}(x_0)} |x|^{-\sigma_i}f_{i}(x,u_{1}(x), {} & \ldots,u_{L}(x)) \varphi_{r}(x) \,dx \notag \\
= {} & u_{i}(x_0) + \sum_{j=1}^{k} \int_{\partial B_{r}(x_0)} u_{ij}(x)\frac{\partial}{\partial n}\Big[(-\Delta)^{k-j}\varphi_{r}(x)\Big]\,ds.
\end{align}
From \eqref{boundary estimate} and Lemma \ref{vanishing boundary integral}, there exists a sequence $r:=r_{\ell} \rightarrow \infty$ in which the boundary integrals in \eqref{main id} vanish. By virtue of \eqref{delta convergence}, \eqref{integrable 1} and the Lebesgue dominated convergence theorem, we conclude that
\begin{equation*}
u_{i}(x_0) = c\int_{\mathbb{R}^n} \frac{1}{|x_0 - y|^{n-2k}|y|^{\sigma_i}}f_{i}(y,u_{1}(y),\ldots,u_{L}(y))\,dy,~i=1,2,\ldots,L.
\end{equation*}

Conversely, showing that the solutions of \eqref{ie} are solutions of \eqref{pde} follows from more elementary arguments. Namely, since $c|x|^{\alpha-n}$ is the fundamental solution of $(-\Delta)^{k}u = \delta_0$, differentiating the integral equations with respect to $x$ and using the convolution properties of the Dirac delta function will show that positive solutions of the integral equations satisfy \eqref{pde}. This completes the proof of the theorem. \qed

\section{Proof of Theorem \ref{theorem2}}\label{section3}
Prior to proving Theorem \ref{theorem2}, we establish a general Liouville type theorem for the Hardy--Sobolev type system with any pair of double bounded coefficients.
\begin{theorem}\label{non-existence}
Let $p,q>0$. Then the integral system \eqref{whls ie} has no positive solution for any double bounded coefficients $c_{1}(x)$ and $c_{2}(x)$ whenever $pq \leq 1$ or if $pq > 1$ and
\begin{equation}\label{non-existence criteria}
\max\Bigg\{ \frac{\alpha(q+1)-(\sigma_1 +\sigma_2 q)}{pq - 1}, \frac{\alpha(p+1)-(\sigma_2 + \sigma_1 p)}{pq - 1} \Bigg\} \geq n - \alpha.
\end{equation}
\end{theorem}

\begin{remark}
We also refer the reader to the earlier papers \cite{Caristi2008,DAmbrosioMitidieri14} for more Liouville type theorems and other interesting results related to the Hardy--Sobolev type systems.
\end{remark}

\begin{proof}[Proof of Theorem \ref{non-existence}]
We proceed by contradiction. That is, assume there is a positive solution $(u,v)$. Let $|x| > R$ for some suitable $R>0$. Note that $|x-y|\leq 2|x|$ for $y\in B_{R}(0)$ and there holds
\begin{equation*}
u(x) \geq C\int_{B_{R}(0)} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \geq \frac{C}{|x|^{a_0}},
\end{equation*}
where $a_0 = n-\alpha$. This estimate and the integral equations imply that for $|x| > R$,
\begin{equation*}
v(x) \geq C\int_{B_{|x|/2}(x)} \frac{|y|^{-p a_0}}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy \geq \frac{C}{|x|^{b_0}},
\end{equation*}
where $b_0 = p a_0 - \alpha + \sigma_2$. Hence, for $|x| > R$
\begin{equation*}
u(x) \geq C\int_{B_{|x|/2}(x)} \frac{|y|^{-q b_0}}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \geq \frac{C}{|x|^{a_1}},
\end{equation*}
where $a_1 = qb_0 - \alpha + \sigma_1$.
By induction, we obtain
\begin{equation*}
u(x) \geq \frac{C}{|x|^{a_j}} \,\text{ and }\, v(x) \geq \frac{C}{|x|^{b_j}} \,\text{ for }\, |x| > R,
\end{equation*}
where $$a_{k+1} = q b_k - \alpha + \sigma_1 \,\text{ and }\, b_{k} = p a_{k} - \alpha + \sigma_2 \,\text{ for }\, k=0,1,2,\ldots.$$
It follows that
\begin{align}\label{sequence}
a_j = {} & q b_{j-1} - \alpha + \sigma_1 = pq a_{j-1} - \alpha(1+q) \sigma_1 + \sigma_2 q \notag \\
= {} & pq(q b_{j-2} - \alpha + \sigma_1) - \alpha(1+q) + \sigma_1 + \sigma_2 q \notag \\
= {} & (pq)^2 a_{j-2} - (\alpha(1+q) - (\sigma_1 + \sigma_2 q))(pq + 1) \notag \\
= {} & (pq)^3 a_{j-3} - (\alpha(1+q) - (\sigma_1 + \sigma_2 q))((pq)^2 + pq + 1)
 \notag \\
\vdots \notag \\
= {} & (pq)^j a_{0} - (\alpha(1+q) - (\sigma_1 + \sigma_2 q))((pq)^{j-1} + \cdots + (pq)^2 + pq + 1).
\end{align}
There are two cases to consider: when $pq \in (0,1]$ and when $pq > 1$.
\medskip

\noindent{\bf Case 1: } Suppose $pq \in (0,1]$. If $pq = 1$, then \eqref{sequence} implies that
\begin{equation*}
a_{j} = a_{0} - j(\alpha(1+q) - (\sigma_1 + \sigma_2 q)) \rightarrow -\infty
\end{equation*}
and $b_j \rightarrow -\infty$ as $j\rightarrow \infty$. If $pq \in (0,1)$, then \eqref{sequence} implies that
\begin{align*}
a_j = {} & (pq)^{j}(n-\alpha) - (\alpha(1+q) - (\sigma_1 + \sigma_2 q))\frac{(pq)^j - 1}{pq-1} \\
= {} & (pq)^{j}\left(n-\alpha  -\frac{\alpha(1+q) - (\sigma_1 + \sigma_2 q)}{pq-1}\right) + \frac{\alpha(1+q) - (\sigma_1 + \sigma_2 q)}{pq-1} \\
{} & \rightarrow a := -\frac{\alpha(1+q) - (\sigma_1 + \sigma_2 q)}{1-pq} < 0 \,\text{ as }\, j\rightarrow \infty.
\end{align*}
Thus, $b_j \rightarrow a - \alpha + \sigma_2 < 0$. In any case, we can choose a sufficiently large $j_0$ so that $a_{j_0},b_{j_0} < 0$ and
\begin{align*}
u(x) \geq {} & C\displaystyle\int_{\mathbb{R}^n\backslash B_{R}(0)} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy
\geq C\int_{\mathbb{R}^{n}\backslash B_{R}(0)} \frac{|y|^{-q b_{j_0}}}{|x-y|^{n-\alpha}|y|^{\sigma_1}} \,dy \\
\geq {} & C\int_{R}^{\infty} r^{\alpha - \sigma - pb_{j_0}} \,\frac{dr}{r} = \infty.
\end{align*}
Thus, $u(x) = \infty$, which is impossible.
\medskip

\noindent{\bf Case 2: } Suppose $pq > 1$.

\noindent 1. Let
\begin{equation*}
n-\alpha < M := \max\Bigg\{ \frac{\alpha(q+1)-(\sigma_1 +\sigma_2 q)}{pq - 1}, \frac{\alpha(p+1)-(\sigma_2 + \sigma_1 p)}{pq - 1} \Bigg\}.
\end{equation*}
First, assume $M = \frac{\alpha(q+1)-(\sigma_1 +\sigma_2 q)}{pq - 1}$. As in Case 1, \eqref{sequence} implies
\begin{equation*}
a_j = (pq)^{j}\left(n-\alpha -\frac{\alpha(1+q) - (\sigma_1 + \sigma_2 q)}{pq-1}\right) + \frac{\alpha(1+q) - (\sigma_1 + \sigma_2 q)}{pq-1} \rightarrow -\infty
\end{equation*}
as $j \rightarrow \infty$. Thus, $u(x) = \infty$, which is impossible. On the other hand, if $pq>1$ and $M = \frac{\alpha(p+1)-(\sigma_2 +\sigma_1 p)}{pq - 1}$, then we can apply the same iteration procedure as above to obtain a contradiction. \\

\noindent 2. Lastly, we consider when $pq>1$ and $M = n-\alpha$. Without loss of generality, we assume $n-\alpha = \frac{\alpha(p+1)-(\sigma_2 +\sigma_1 p)}{pq - 1}$. We see that
\begin{equation*}
u(x) \geq \frac{1}{(R + |x|)^{n-\alpha}}\int_{B_{R}(0)} \frac{v(y)^q}{|y|^{\sigma_1}}\,dy \,\text{ and }\, v(x) \geq \frac{1}{(R + |x|)^{n-\alpha}}\int_{B_{R}(0)} \frac{u(y)^p}{|y|^{\sigma_2}}\,dy.
\end{equation*}
From this, we obtain
\begin{equation*}
\int_{B_{R}(0)} \frac{u(x)^p}{|x|^{\sigma_2}}\,dx \geq \frac{C}{R^{(n-\alpha)p +\sigma_2 - n}}\left(\int_{B_{R}(0)} \frac{v(y)^q}{|y|^{\sigma_1}}\,dy\right)^p,
\end{equation*}
and
\begin{equation*}
\int_{B_{R}(0)} \frac{v(x)^q}{|x|^{\sigma_1}}\,dx \geq \frac{C}{R^{(n-\alpha)q + \sigma_1 - n}}\left(\int_{B_{R}(0)} \frac{u(y)^p}{|y|^{\sigma_2}}\,dy\right)^q.
\end{equation*}
Then
\begin{align}\label{Lp}
\int_{B_{R}(0)} \frac{u(x)^p}{|x|^{\sigma_2}} \,dx \geq {} & \frac{C}{R^{(n-\alpha)p +\sigma_2 - n+ p(q(n-\alpha) +\sigma_1 - n)}}\left(\int_{B_{R}(0)} \frac{u(x)^p}{|x|^{\sigma_2}} \,dy \right)^{pq} \notag \\
\geq {} & C\left(\int_{B_{R}(0)} \frac{u(x)^p}{|x|^{\sigma_2}} \,dy \right)^{pq},
\end{align}
where we are using the fact that
\begin{align*}
(n-\alpha)p +\sigma_2 - n + {} & p(q(n-\alpha) +\sigma_1 - n) \\
= {} & (n-\alpha)p + \sigma_2 - n + pq(n-\alpha) + \sigma_1 p - np \\
= {} & (pq-1)(n-\alpha) - \lbrace \alpha(1+p) - (\sigma_2 + \sigma_1 p)\rbrace \\
= {} & (pq-1)(n-\alpha - M) \\
= {} & 0.
\end{align*}
Thus, the constant $C>0$ in \eqref{Lp} is independent of $R$. By sending $R\rightarrow \infty$ we get $|x|^{-\sigma_2}u(x)^p \in L^{1}(\mathbb{R}^n)$. By applying similar calculations used in the derivation of \eqref{Lp}, we obtain
\begin{equation*}
\int_{B_{2R}(0)\backslash B_{R}(0)} \frac{u(x)^p}{|x|^{\sigma_2}} \,dx \geq C\left(\int_{B_{R}(0)} \frac{u(x)^p}{|x|^{\sigma_2}} \,dy \right)^{pq},
\end{equation*}
where $C>0$ is independent of $R$. Sending $R\rightarrow \infty$ yields
\begin{equation*}
\int_{\mathbb{R}^n} \frac{u(x)^p}{|x|^{\sigma_2}} \,dx = 0,
\end{equation*}
which implies that $u \equiv 0$ and we arrive at a contradiction. This completes the proof of the theorem.
\end{proof}

\begin{proof}[{\bf Proof of Theorem \ref{theorem2}}]
It suffices to show the existence of positive solutions for system \eqref{whls ie} when $pq > 1$ and \eqref{criteria} holds, since Theorem \ref{non-existence} basically states that this existence result must then be sharp. We show that candidates for solutions are the radial functions
\begin{equation}\label{radial}
u(x) = \frac{1}{(1 + |x|^2)^{\theta_1}} \,\text{ and }\, v(x) = \frac{1}{(1 + |x|^2)^{\theta_2}},
\end{equation}
where
\begin{equation*}
2\theta_1 = \frac{\alpha(1+q) - (\sigma_1 + \sigma_2 q)}{pq-1} \,\text{ and }\, 2\theta_2 = \frac{\alpha(1+p) - (\sigma_2 + \sigma_1 p)}{pq-1}.
\end{equation*}
Notice that \eqref{criteria} implies
\begin{equation}\label{sandwich}
\alpha < 2\theta_1 p + \sigma_2 < n \,\text{ and }\, \alpha < 2\theta_2 q + \sigma_1 < n.
\end{equation}
Note that if $|x| < 2R$ for some suitable $R>0$, $u(x)$ and $v(x)$ are proportional to \\ $\int_{\mathbb{R}^n} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy$ and $\int_{\mathbb{R}^n} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy$, respectively. Thus, we only consider $|x| > 2R$.

First, it is clear that
\begin{equation*}
\int_{\mathbb{R}^n} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \geq \frac{C}{|x|^{n-\alpha}}\int_{B_{|x|/2}(0)} \frac{v(y)^q}{|y|^{\sigma_1}}\,dy \geq \frac{C}{(1+|x|^2)^{q\theta_{2} + \sigma_{1}/2-\alpha/2}}.
\end{equation*}
This implies
\begin{equation*}
u(x)\left( \int_{\mathbb{R}^n} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy\right)^{-1} \leq \frac{C}{(1+|x|^2)^{\theta_{1} - q\theta_{2} - \sigma_{1}/2+\alpha/2}} \leq C,
\end{equation*}
since $\theta_{1} - q\theta_{2} - \sigma_{1}/2+\alpha/2 = 0$. On the other hand,
\begin{align*}
\int_{\mathbb{R}^n} {} & \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \\
= {} & \left( \int_{B_{R}(0)} + \int_{B_{|x|/2}(x)} + \int_{B_{R}(0)^{C}\backslash B_{|x|/2}(x)} \right) \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \\
:= {} & I_1 + I_2 + I_3,
\end{align*}
where
\begin{align*}
I_1 = \int_{B_{R}(0)} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \leq {} & \frac{C}{(1+|x|^2)^{(n-\alpha)/2}}\int_{B_{R}(0)} \frac{1}{|y|^{2q\theta_2 + \sigma_{1}}}\,dy \\
\leq {} & \frac{C}{(1+|x|^2)^{(n-\alpha)/2}};
\end{align*}
\begin{align*}
I_2 = \int_{B_{|x|/2}(x)} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \leq {} & \frac{C}{(1 + |x|^2)^{q\theta_2 + \sigma_{1}/2}}\int_{B_{|x|/2}(x)} \frac{1}{|x-y|^{n-\alpha}}\,dy \\
\leq {} & \frac{C}{(1 + |x|^2)^{q\theta_2 +\sigma_{1}/2 -\alpha/2}};
\end{align*}
\begin{align*}
I_3 = \int_{B_{R}(0)^{C}\backslash B_{|x|/2}(x)} \frac{v(y)^q}{|x\!-\!y|^{n-\alpha}|y|^{\sigma_1}}\,dy \leq {} & C\int_{B_{R}(0)^{C}\backslash B_{|x|/2}(x)} \frac{1}{|x\!-\!y|^{n-\alpha}|y|^{2q\theta_2 + \sigma_1}} \\
\leq {} & C\int_{|x|/2}^{\infty} r^{n-(n-\alpha + 2q\theta_2 + \sigma_1)}\, \frac{dr}{r} \\
\leq {} & \frac{C}{(1 + |x|^2)^{q\theta_2 +\sigma_{1}/2 -\alpha/2}}.
\end{align*}
These estimates imply
\begin{align*}
\int_{\mathbb{R}^n} {} & \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \leq C\Bigg\lbrace \frac{1}{(1 + |x|^2)^{q\theta_2 + \sigma_{1}/2 -\alpha/2}} + \frac{1}{(1+|x|^2)^{(n-\alpha)/2}} \Bigg\rbrace \\
\leq {} & C\Bigg\lbrace \frac{1}{(1 + |x|^2)^{q\theta_2 + \sigma_{1}/2 -\alpha/2 - \theta_1}} + \frac{1}{(1+|x|^2)^{(n-\alpha)/2 -\theta_1}}\Bigg\rbrace\frac{1}{(1+|x|^{2})^{\theta_1}} \\
\leq {} & Cu(x),
\end{align*}
since  \eqref{criteria} implies $(n-\alpha)/2 - \theta_1 > 0$.
Therefore,
\begin{equation*}
\frac{1}{C}\int_{\mathbb{R}^n} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \leq u(x) \leq C\int_{\mathbb{R}^n} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy
\end{equation*}
for some suitable constant $C>0$. Hence,
\begin{equation*}
u(x) = c_{1}(x)\int_{\mathbb{R}^n} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy
\end{equation*}
for some double bounded function $c_{1}(x)$. Similar calculations show
\begin{equation*}
v(x) = c_{2}(x)\int_{\mathbb{R}^n} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy
\end{equation*}
for some double bounded function $c_{2}(x)$. This completes the proof of the theorem.
\end{proof}

\begin{remark}
Instead of solutions with the slow rates, we can also find solutions of the form \eqref{radial} with the fast rate $2\theta_{1} = 2\theta_{2} = n-\alpha$ provided the following stronger conditions hold: $p>\frac{n-\sigma_2}{n-\alpha}$ and $q> \frac{n-\sigma_1}{n-\alpha}$. The proof of this involves similar calculations as before and so we omit the details.
\end{remark}

\begin{proof}[{\bf Proof of Corollary \ref{corollary1}}]
This follows immediately from Theorem \ref{theorem2} provided \eqref{whls ie} and \eqref{whls pde} are equivalent in the following sense.

\medskip

\noindent{\bf Equivalence:} From Theorem \ref{theorem1}, the classical solution of \eqref{whls pde} satisfies
\begin{equation*}
  \left\{\begin{array}{cl}
    u(x) = a_{1}\displaystyle\int_{\mathbb{R}^{n}} \frac{c_{1}(y)v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy, \\
    v(x) = a_{2}\displaystyle\int_{\mathbb{R}^{n}} \frac{c_{2}(y)u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy, \\
  \end{array}
\right.
\end{equation*}
for some positive constants $a_1$ and $a_2$. Clearly,
\begin{equation*}
\frac{1}{C_1}\int_{\mathbb{R}^{n}} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \leq u(x) \leq C_{1}\int_{\mathbb{R}^{n}} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy,
\end{equation*}
and
\begin{equation*}
\frac{1}{C_2}\int_{\mathbb{R}^{n}} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy \leq v(x) \leq C_{2}\int_{\mathbb{R}^{n}} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy,
\end{equation*}
since $c_{1}(x)$ and $c_{2}(x)$ are double bounded. Thus,
\begin{equation}\label{inint}
  \left\{\begin{array}{cl}
    u(x) = a_{1}(x)\displaystyle\int_{\mathbb{R}^{n}} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy, \\
    v(x) = a_{2}(x)\displaystyle\int_{\mathbb{R}^{n}} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy, \\
  \end{array}
\right.
\end{equation}
for the double bounded functions
\begin{equation*}
a_{1}(x) = u(x)\Bigg[ \displaystyle\int_{\mathbb{R}^{n}} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \Bigg]^{-1}
\end{equation*}
and
\begin{equation*}
a_{2}(x) = v(x)\Bigg[ \int_{\mathbb{R}^{n}} \frac{v(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy \Bigg]^{-1}.
\end{equation*}

On the contrary, suppose $u$ and $v$ solve \eqref{inint} for some double bounded functions $a_{1}(x)$ and $a_{2}(x)$. Set $w_{1}(x) = u(x)/a_{1}(x)$ and $w_{2}(x) = v(x)/a_{2}(x)$. Then for any pair of constants $\tilde{c}_{1}$ and $\tilde{c}_{2}$, $(w_{1},w_{2})$ satisfies
\begin{equation*}
  \left\{\begin{array}{cl}
    w_{1}(x) = \tilde{c}_{1}\displaystyle\int_{\mathbb{R}^{n}} \frac{c_{1}(y)w_{2}(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy, \\
    w_{2}(x) = \tilde{c}_{2}\displaystyle\int_{\mathbb{R}^{n}} \frac{c_{2}(y)w_{1}(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy, \\
  \end{array}
\right.
\end{equation*}
where $c_{1}(x) = a_{2}(x)^p/\tilde{c}_1$ and $c_{2}(x) = a_{1}(x)^p/\tilde{c}_2$. Hence, by the equivalence result of Theorem \ref{theorem1}, we can choose $\tilde{c}_1$ and $\tilde{c}_2$ such that
\begin{equation*}
  \left\{\begin{array}{l}
    (-\Delta)^{\alpha/2} w_{1}(x) = \displaystyle c_{1}(x)\frac{w_{2}(x)^q}{|x|^{\sigma_1}}, \\
    (-\Delta)^{\alpha/2} w_{2}(x) = \displaystyle c_{2}(x)\frac{w_{1}(x)^p}{|x|^{\sigma_2}},
  \end{array}
\right.
\end{equation*}
for double bounded functions $c_{1}(x)$ and $c_{2}(x)$. This completes the proof of the corollary.
\end{proof}

\section{Proof of Theorems \ref{theorem4} and \ref{theorem5}}\label{section4}
\begin{remark} Hereafter, we deliberately neglect the case when $p \in (0,1]$ for \eqref{hs eq} in Theorem \ref{theorem4} and when $pq \in (0,1]$ for \eqref{hs sys} in Theorem \ref{theorem5}, since the non-existence of positive (radial or non-radial) solutions for these follows from Theorem \ref{non-existence} and its scalar counterpart.
\end{remark}
As discussed earlier, these two Liouville type theorems stem from two essential ingredients: the decay estimates for radial solutions and integral forms of Pohozaev type identities. The following lemma addresses the former ingredient.
\begin{lemma}[Decay of radial solutions]\label{decay of u}
Suppose $u=u(r)~(r=|x|)$ is a positive radial solution of \eqref{hs eq}, then there holds the decay property $$u(r) \leq C r^{-\frac{\alpha-\sigma}{p-1}}, \,~\, r > 0.$$
\end{lemma}
\begin{proof}
From the integral equation and the decreasing property of radial solutions, we have
\begin{equation*}
u(r) \geq \int_{B_{r}(0)} \frac{u(y)^p}{|x-y|^{n-\alpha}|y|^{\sigma}}\,dy \geq C r^{n-(n-\alpha +\sigma)}u(r)^{p}
\geq Cr^{\alpha-\sigma}u(r)^p.
\end{equation*}
The result follows by directly solving for $u$ in this inequality.
\end{proof}
As a result of the decay properties of radial solutions, we have the following.
\begin{lemma}\label{finite energy 1}
If $u$ is a positive radial solution of \eqref{hs sys}, then the integrals
\begin{equation*}
\int_{\mathbb{R}^n} \frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx \,\text{ and }\, \int_{\mathbb{R}^n} \frac{u(x)^p}{|x|^{\sigma}}(x\cdot \nabla u(x))\,dx
\end{equation*}
are finite.
\end{lemma}
\begin{proof}
Since $\sigma < n$, the integrals $\int_{B_{R}(0)} \frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx \,\text{ and }\, \int_{B_{R}(0)} \frac{u(x)^p}{|x|^{\sigma}}(x\cdot \nabla u(x))\,dx$ are finite for each $R>0$. Thus, it suffices to show that the following integrals converge:
\begin{equation*}
A_{1}:= \int_{B_{R}(0)^{C}} \frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx \,\text{ and }\, A_{2}:= \int_{B_{R}(0)^{C}} \frac{u(x)^p}{|x|^{\sigma}}(x\cdot \nabla u(x))\,dx.
\end{equation*}
From Lemma \ref{decay of u} and the standard identity we invoked in \eqref{standard id}, we obtain
\begin{align*}
|A_{1}| + |A_{2}| \leq {} & C\int_{R}^{\infty} u(r)^{p}|ru'(r)| r^{n-\sigma}\frac{dr}{r} + C\int_{R}^{\infty} u(r)^{p+1} r^{n-\sigma}\,\frac{dr}{r} \\
\leq {} & C\int_{R}^{\infty} u(r)^{p+1} r^{n-\sigma}\frac{dr}{r} \leq C \int_{R}^{\infty} r^{n-\sigma - (\alpha - \sigma)\frac{p+1}{p-1}} \,\frac{dr}{r},
\end{align*}
where the improper integrals are convergent since $1 < p < \frac{n + \alpha -2\sigma}{n-\alpha}$ implies
\begin{equation*}
n-\sigma - \frac{(\alpha-\sigma)(p+1)}{p-1} = \frac{p(n-\alpha) - (n+\alpha -2\sigma)}{p-1} < 0.
\end{equation*}
\end{proof}

\begin{proof}[\bf Proof of Theorem \ref{theorem4}]
Assume that $u$ is a positive radial solution of the integral equation. We rewrite the integral equation as
\begin{equation*}
u(\lambda x) = \int_{\mathbb{R}^n} \frac{u(y)^{p}}{|\lambda x - y|^{n-\alpha}|y|^{\sigma}}\,dy = \lambda^{\alpha - \sigma}\int_{\mathbb{R}^n} \frac{u(\lambda z)^p}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz.
\end{equation*}
Differentiating this identity with respect to $\lambda$ on both sides yields
\begin{equation*}
x\cdot \nabla u(\lambda x) = (\alpha - \sigma)\lambda^{\alpha - \sigma -1}\int_{\mathbb{R}^n}\frac{u(\lambda z)^p}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz + \lambda^{\alpha}\int_{\mathbb{R}^n} \frac{p u(\lambda z)^{p-1}(z\cdot \nabla u)}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz.
\end{equation*}
Set $\lambda = 1$ to get
\begin{align}\label{xdotwithu}
x\cdot \nabla u(x) = {} & (\alpha - \sigma)\int_{\mathbb{R}^n}\frac{u(z)^p}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz + \int_{\mathbb{R}^n} \frac{p u(z)^{p-1}(z\cdot \nabla u)}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz \notag \\
= {} & (\alpha - \sigma)u(x) + \int_{\mathbb{R}^n} \frac{z\cdot \nabla u(z)^{p}}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz.
\end{align}
To handle the last term in \eqref{xdotwithu}, an integration by parts yields
\begin{align*}
\int_{B_{R}(0)} {} & \frac{z\cdot \nabla u(z)^{p}}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz = R\int_{\partial B_{R}(0)} \frac{u(z)^{p}}{|x-z|^{n-\alpha}|z|^{\sigma}} \,ds \\
{} & - (n-\sigma)\int_{B_{R}(0)} \frac{u(z)^{p}}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz - (n-\alpha)\int_{B_{R}(0)} \frac{(z\cdot (x-z))u(z)^{p}}{|x-z|^{n-\alpha + 2}|z|^{\sigma}}\,dz,
\end{align*}
where the boundary integral vanishes as $R\rightarrow \infty$ since $\int_{\mathbb{R}^n} \frac{u(z)^{p}}{|x-z|^{n-\alpha}|z|^{\sigma}} \,dz < \infty.$ With this, we obtain the identity
\begin{equation*}
x\cdot \nabla u(x) = -(n-\alpha)u(x) + (n-\alpha)\int_{\mathbb{R}^n} \frac{(z\cdot (x-z))u(z)^{p}}{|x-z|^{n-\alpha + 2}|z|^{\sigma}}\,dz.
\end{equation*}
If we multiply this by $\frac{u(x)^p}{|x|^{\sigma}}$ and integrate over $\mathbb{R}^n$ we get
\begin{align}\label{int on ball}
\int_{\mathbb{R}^n} {} & \frac{u(x)^p}{|x|^{\sigma}}(x\cdot \nabla u(x))\,dx \notag \\
= {} & -(n-\alpha)\int_{\mathbb{R}^n}\frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx - (n-\alpha)\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(z\cdot (x-z))u(x)^{p}u(z)^{p}}{|x-z|^{n-\alpha + 2}|x|^{\sigma}|z|^{\sigma}}\,dz dx.
\end{align}
Noticing that $z\cdot(x-z) + x\cdot(z-x) = -|x-z|^2$, we have
\begin{equation*}
\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(z\cdot (x-z))u(x)^{p}u(z)^{p}}{|x-z|^{n-\alpha + 2}|x|^{\sigma}|z|^{\sigma}}\,dz dx
= -\frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{u(x)^{p}u(z)^{p}}{|x-z|^{n-\alpha}|x|^{\sigma}|z|^{\sigma}}\,dz dx.
\end{equation*}
Thus,
\begin{align}\label{pohozaev}
\int_{\mathbb{R}^n} \frac{u(x)^p}{|x|^{\sigma}} {} &  (x\cdot \nabla u(x))\,dx \notag \\
= {} & -(n\!-\!\alpha)\int_{\mathbb{R}^n} \frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx\! -\! (n\!-\!\alpha)\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(z\cdot (x\!-\!z)) u(x)^{p}u(z)^{p}}{|x\!-\!z|^{n-\alpha + 2}|x|^{\sigma}|z|^{\sigma}}\,dz dx \notag \\
= {} & -(n-\alpha)\int_{\mathbb{R}^n} \frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx + \frac{n-\alpha}{2}\int_{\mathbb{R}^n} \frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx \notag \\
= {} & -\frac{n-\alpha}{2}\int_{\mathbb{R}^n}\frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx.
\end{align}
In view of Lemma \ref{finite energy 1}, integration by parts yields
\begin{align*}
\int_{B_{R}(0)} \frac{u(x)^p}{|x|^{\sigma}} (x\cdot \nabla u(x))\,dx = {} & \frac{1}{1+p}\int_{B_{R}(0)} \frac{x}{|x|^{\sigma}} \cdot \nabla (u(x)^{p+1})\,dx \notag \\
= {} &\! -\!\frac{(n\!-\!\sigma)}{1\!+\!p}\int_{B_{R}(0)} \frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx
\!+\! \underbrace{R\int_{\partial B_{R}(0)} \frac{u(x)^{p+1}}{|x|^{\sigma}} \,ds}_{ = \mathrm{o}(1) \text{ as } R\rightarrow \infty}.
\end{align*}
Thus,
\begin{align*}
\int_{\mathbb{R}^n} \frac{u(x)^p}{|x|^{\sigma}} (x\cdot \nabla u(x))\,dx = -\frac{(n-\sigma)}{1+p}\int_{\mathbb{R}^n} \frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx.
\end{align*}
Inserting this into \eqref{pohozaev} yields
\begin{equation*}
\Bigg\lbrace \frac{n-\sigma}{1+p} - \frac{n-\alpha}{2} \Bigg\rbrace \int_{\mathbb{R}^n} \frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx
= 0,
\end{equation*}
but this contradicts with \eqref{subcritical1}. This completes the proof of the theorem.
\end{proof}
The proof of Theorem \ref{theorem5} is similar to the proof of Theorem \ref{theorem4}. First, we need the following lemma on the decay estimates for radial solutions.
\begin{lemma}\label{decay of uv}
Suppose $(u,v)=(u(r),v(r))$ is a positive radial solution pair of \eqref{hs sys}, then
\begin{equation*}
u(r) \leq C r^{-\frac{\alpha(1+q) - (\sigma_1 + \sigma_{2}q)}{pq-1}} \,\text{ and }\,~ v(r) \leq C r^{-\frac{\alpha(1+p) - (\sigma_2 + \sigma_{1}p)}{pq-1}}, \,~\, r > 0.
\end{equation*}
\end{lemma}
\begin{proof}
From the integral equations and the monotonicity of radial solutions, we obtain
\begin{align*}
u(r) \geq {} & \int_{B_{r}(0)} \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \geq Cr^{n-(n-\alpha + \sigma_1)}v(r)^{q}, \\
v(r) \geq {} & \int_{B_{r}(0)} \frac{u(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_2}}\,dy \geq Cr^{n-(n-\alpha + \sigma_2)}u(r)^{p}.
\end{align*}
Thus, $u(r) \geq C r^{\alpha - \sigma_{1}}v(r)^q$ and $v(r) \geq C r^{\alpha - \sigma_{2}}u(r)^p$, and the result follows by solving for $u$ and $v$ in the estimates.
\end{proof}

As a result of the decay property of radial solutions, we obtain the following.
\begin{lemma}\label{finite energy 2}
Let $(u,v)$ be a positive radial solution of \eqref{hs sys}. Then the following integrals
\begin{align*}
&\int_{\mathbb{R}^n} \frac{u(x)^{p}}{|x|^{\sigma_2}}(x\cdot \nabla u(x)) \,dx,\,~\, \int_{\mathbb{R}^n} \frac{u(x)^{p+1}}{|x|^{\sigma_2}} \,dx, \\
&\int_{\mathbb{R}^n} \frac{v(x)^{q}}{|x|^{\sigma_1}}(x\cdot \nabla v(x)) \,dx, \,\text{ and }\, \int_{\mathbb{R}^n} \frac{v(x)^{q+1}}{|x|^{\sigma_1}} \,dx,
\end{align*}
are finite.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{finite energy 1}, it suffices to show the following integrals converge for any finite $R>0$:
\begin{enumerate}[(a)]
\item $B_1 := \displaystyle\int_{B_{R}(0)^{C}} \frac{u(x)^{p+1}}{|x|^{\sigma_2}} \,dx$,
\item $B_2 := \displaystyle\int_{B_{R}(0)^{C}} \frac{v(x)^{q+1}}{|x|^{\sigma_1}} \,dx$,
\item $B_3 := \displaystyle\int_{B_{R}(0)^{C}} \frac{u(x)^{p}}{|x|^{\sigma_2}}(x\cdot \nabla u(x)) \,dx$,
\item $B_4 := \displaystyle\int_{B_{R}(0)^{C}} \frac{v(x)^{q}}{|x|^{\sigma_1}}(x\cdot \nabla v(x)) \,dx$.
\end{enumerate}

From Lemma \ref{decay of uv} and the standard identity we used in \eqref{standard id}, we obtain
\begin{align*}
|B_{1}| + |B_{3}| \leq {} & C\int_{R}^{\infty} r^{n-\sigma_2}u(r)^{p}|ru'(r)|\,\frac{dr}{r} + C\int_{R}^{\infty} r^{n-\sigma_2}u(r)^{p+1} \,\frac{dr}{r} \\
\leq {} & C\int_{R}^{\infty} r^{n-\sigma_2}u(r)^{p+1}\,\frac{dr}{r} \leq C\int_{R}^{\infty} r^{n-\sigma_2 - \frac{\alpha(1+q) - (\sigma_1 + \sigma_{2}q)}{pq-1}(p+1)} \,\frac{dr}{r} \\
\leq {} & C\int_{R}^{\infty} r^{n-\alpha - \frac{\alpha(1+q) - (\sigma_1 + \sigma_{2}q) + \alpha(1+p) - (\sigma_2 + \sigma_{1}p)}{pq-1}}\,\frac{dr}{r} < \infty,
\end{align*}
since the subcritical condition \eqref{subcritical2} is equivalent to
\begin{equation*}
\frac{\alpha(1+q) - (\sigma_1 + \sigma_{2}q) + \alpha(1+p) - (\sigma_2 + \sigma_{1}p)}{pq-1} > n-\alpha.
\end{equation*}
This proves the convergence of the integrals in parts (a) and (c). Parts (b) and (d) follow similarly.
\end{proof}

\begin{proof}[\bf Proof of Theorem \ref{theorem5}]
On the contrary, assume $(u,v)$ is a positive radial solution pair.
As before, we obtain
\begin{equation*}
x\cdot \nabla u(x) = \frac{d}{d\lambda}u(\lambda x)\Big|_{\lambda = 1} = -(n - \alpha)u(x) - (n-\alpha)\int_{\mathbb{R}^n} \frac{(z\cdot (x-z))v(z)^{q}}{|x-z|^{n-\alpha + 2}|z|^{\sigma_1}}\,dz.
\end{equation*}
If we multiply this by $|x|^{-\sigma_2}u(x)^{p}$ and integrate over $\mathbb{R}^n$, we get
\begin{align}\label{int u final}
\int_{\mathbb{R}^n} \frac{u(x)^{p}}{|x|^{\sigma_2}}(x\cdot \nabla u(x))\,dx
= {} & -(n-\alpha)\int_{\mathbb{R}^n} \frac{u(x)^{p+1}}{|x|^{\sigma_2}}\,dx \notag \\
{} & - (n-\alpha)\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(z\cdot (x-z))u(x)^{p}v(z)^{q}}{|x-z|^{n-\alpha + 2}|x|^{\sigma_2}|z|^{\sigma_1}}\,dz dx.
\end{align}
Analogous calculations on the second integral equation will lead to
\begin{align}\label{int v final}
\int_{\mathbb{R}^n} \frac{v(x)^{q}}{|x|^{\sigma_1}}(x\cdot \nabla v(x))\,dx
= {} & -(n-\alpha)\int_{\mathbb{R}^n} \frac{v(x)^{q+1}}{|x|^{\sigma_1}}\,dx \notag \\
{} & - (n-\alpha)\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(z\cdot (x-z))u(z)^{p}v(x)^{q}}{|x-z|^{n-\alpha + 2}|z|^{\sigma_2}|x|^{\sigma_1}}\,dz dx.
\end{align}
By summing \eqref{int u final} and \eqref{int v final} together and recalling that $z\cdot(x-z) + x\cdot(z-x) = -|x-z|^2$, we get
\begin{equation}\label{pohozaev sys}
\int_{\mathbb{R}^n} \frac{v(x)^{q}}{|x|^{\sigma_1}}(x\cdot \nabla v(x))\,dx + \int_{\mathbb{R}^n} \frac{u(x)^{p}}{|x|^{\sigma_2}}(x\cdot \nabla u(x))\,dx = -(n-\alpha)\int_{\mathbb{R}^n} \frac{v(x)^{q+1}}{|x|^{\sigma_1}}\,dx.
\end{equation}
In view of Lemma \ref{finite energy 2}, integration by parts implies
\begin{align*}
\int_{B_{R}(0)} {} & \frac{v(x)^q}{|x|^{\sigma_1}} (x\cdot \nabla v(x)) \,dx + \int_{B_{R}(0)} \frac{u(x)^p}{|x|^{\sigma_2}} (x\cdot \nabla u(x)) \,dx \notag \\
= {} & -\frac{n-\sigma_1}{1+q}\int_{B_{R}(0)} \frac{v(x)^{q+1}}{|x|^{\sigma_1}}\,dx - \frac{n-\sigma_2}{1+p}\int_{B_{R}(0)} \frac{u(x)^{p+1}}{|x|^{\sigma_2}}\,dx \notag \\
+ {} & \underbrace{\frac{R}{1+q}\int_{\partial B_{R}(0)} \frac{v(x)^{q+1}}{|x|^{\sigma_1}}\,ds}_{ = \mathrm{o}(1) \text{ as } R\rightarrow \infty} + \underbrace{\frac{R}{1+p}\int_{\partial B_{R}(0)} \frac{u(x)^{p+1}}{|x|^{\sigma_2}}\,ds}_{ = \mathrm{o}(1) \text{ as } R\rightarrow \infty}.
\end{align*}
Sending $R\rightarrow \infty$ yields
\begin{align}\label{intparts sys}
\int_{\mathbb{R}^n} \frac{v(x)^q}{|x|^{\sigma_1}} (x\cdot \nabla v(x)) \,dx + {} & \int_{\mathbb{R}^n} \frac{u(x)^p}{|x|^{\sigma_2}} (x\cdot \nabla u(x)) \,dx \notag \\
= {} & -\Bigg\lbrace \frac{n-\sigma_1}{1+q} + \frac{n-\sigma_2}{1+p}\Bigg\rbrace\int_{\mathbb{R}^n} \frac{v(x)^{q+1}}{|x|^{\sigma_1}}\,dx,
\end{align}
where we used the fact that
\begin{equation*}
\int_{\mathbb{R}^n} \frac{u(x)^{p+1}}{|x|^{\sigma_2}}\,dx = \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{u(x)^{p}v(z)^q}{|x-z|^{n-\alpha}|x|^{\sigma_2}|z|^{\sigma_1}}\,dz dx = \int_{\mathbb{R}^n} \frac{v(x)^{q+1}}{|x|^{\sigma_1}}\,dx.
\end{equation*}
Hence, \eqref{pohozaev sys} and \eqref{intparts sys} imply that
\begin{equation*}
\Bigg\lbrace \frac{n-\sigma_1}{1+q} + \frac{n-\sigma_2}{1+p} - (n-\alpha) \Bigg\rbrace \int_{\mathbb{R}^n} \frac{v(x)^{q+1}}{|x|^{\sigma_1}}\,dx
= 0,
\end{equation*}
but this contradicts with \eqref{subcritical2}. This completes the proof of the theorem.
\end{proof}

\small
\noindent Acknowledgements: This work was completed during a visiting position at the University of Oklahoma. The author would like to thank the university and the Department of Mathematics, especially Professor R\"{u}diger Landes and Professor Meijun Zhu, for their hospitality.

\footnotesize

redpajama_set_name: RedPajamaArXiv


text:
Senior Essay: Meet Carlie Layton of Bloomfield
By Carlie Layton
The Daily Messenger has been asking area seniors to reflect on losing their final spring season due to the coronavirus pandemic
For my entire high school career at Bloomfield, I have been a three-sport athlete.
Coming into my senior year, I was really excited to continue that streak. Tennis, basketball and track and field were a very big part of my life. Knowing that this would be my last year participating, it pushed me even harder to reach my goals.
After a difficult winter season, I was unsure about what I was going to do in the spring. I was thinking about trying something new, like playing softball with my best friend and a coach that had made a huge impact on my life, or if I would continue with track.
Little did I know at that time, my decision of whether or not to play would ultimately be made for me.
Track and field season has always been a very exciting time for me. Over the last four years, along with having so much fun, I was able to see my skills increase. My events included throwing both shotput and discus.
This year, when I was having an unsure feeling about what I wanted to do, I was very lucky to have such a supportive coach to talk to and get advice from, like Coach Lindsay. She helped me come to the decision that I would continue my throwing career.
Due to my success from the previous year, I was very excited to see where this season would bring me.
I was also really looking forward to getting our Bloomfield-Honeoye Pride team back together! Our merged Bloomfield-Honeoye track and field team gave me the opportunity to make many new friends.
As the only female thrower from Bloomfield, I was very lucky to have Danielle from Honeoye join me on the throwing team. She had become a very good friend, and teammate, which makes me miss the sport even more.
Although I am sad that the pandemic brought an early end to my high school athletic career, one thing that the pandemic can't take away from me, and my fellow senior teammates are the friendships, and lessons we have all experienced together.
I am forever grateful for the athletic opportunities that Bloomfield has given me, and I will never forget all of the amazing memories I have made.
If you're a senior athlete missing your spring sport at an Ontario County school, including Palmyra-Macedon, and you want to submit a Senior Essay, e-mail rchavez@messengerpostmedia.com with at least 350 words and a spring sport photo of yourself.

redpajama_set_name: RedPajamaCommonCrawl


SUMMER TIME IS ALMOST HERE, HIP HIP HOORAY!
School is winding down, 8th grade graduation is this coming Monday, and Fr. Kevin and I couldn't be any happier! It is true, we are going to miss the kids but we are both looking forward to the slower pace of summer to recharge our batteries per se! Summer is a time to leave our busy winter schedules behind, it is a time when the craziness of life gets to be put on hold. Summer is a time of relaxation with family, going on vacations, hanging out with friends and spending quality down time with God.
Yes, believe it or not, God is found in our relaxation, He is found on vacation, He is found at family reunions, and He is even found amongst us when we are hanging out with our friends. Matt. 18:20 "For where two or three gather in my name, there am I with them."
Writing this article made me reminisce about a slower time in my life, when time seemed to have stood still. When I was a boy many, many, yes many years ago, summers seemed to last forever. It is funny when I think back to those days, I seem to remember them a little different than when I was living them out. For instance, we had ton's of chores to do, like mowing grass, hoeing the garden and picking apples off the tall trees and praying hard that the ladder wouldn't fall. I remember my hands being stuck by the thorns as I would pick blackberries and raspberries off the fence line so mom could make her special summer jams. What kid wants to do that work?
But the chores and labor under the hot sun would not compare to the fun my brothers and I would have playing baseball every single day. Baseball was my favorite sport when I was a young boy. I would do the Pete Rose belly slide into home base every day whether I needed to slide or not. I remember scraping up my chest and stomach every time. My brothers and I had a blast as we would play a long season of baseball just like the big leaguers. At night we would play hide-and-seek, eat watermelon and chase fire flies. Everything in the world was right!
Even though my family wasn't very religious at this point in our lives, I know that Jesus was present with us, without our realizing it. See, Jesus is present anyplace where family and friends get together. Jesus is present wherever love is being shown!
Summer time is an opportunity to think back to the fun times we had when we were younger and to create new memories for our children and ourselves. Summer time, is a time to relax, take it easy, have a beverage of choice with some close friends and worry a little less. And if we invite God into the long days of summer, then we will truly be blessed and rejuvenated.
YES!!! SUMMER TIME IS ALMOST HERE! HIP HIP HOORAY!


text:
Mental health of young adults with lesbian parents is same as their peers
New findings from the 32-year U.S. National Longitudinal Lesbian Study, published in The New England Journal of Medicine
The longest-running prospective study on sexual minority parent families found that 25-year-olds raised by lesbian parents do as well on multiple measures of psychological health as adults from a population-based sample. The researchers compared relationships, educational/job performance, and behavioral, emotional and mental health problems in the two samples.
The 25-year-olds are participants in the ongoing U.S. National Longitudinal Lesbian Family Study (NLLFS), which has followed this cohort of offspring from conception to adulthood. The NLLFS, now in its 32nd year, has a 92 percent retention rate. This is the first NLLFS report based on data collected when the offspring were legal adults.
"When I began this study in 1986, there was considerable speculation about the future mental health of children conceived through donor insemination and raised by sexual minority parents," said lead author Nanette Gartrell, M.D., Visiting Distinguished Scholar at the Williams Institute at UCLA School of Law. "We have followed these families since the mothers were inseminating or pregnant and now find that their 25-year-old daughters and sons score as well on mental health as other adults of the same age."
The study focused on mental health because the peak incidence of many psychiatric disorders occurs during young adulthood. The researchers matched the 77 adult offspring in the NLLFS with a population-based sample of 77 adults of comparable age, sex, race/ethnicity and education to examine any disparities in their mental health.
The researchers specifically assessed adaptive functioning, the presence of behavioral or emotional problems, scores on the mental health diagnostic scales, and percentages of scores in the borderline or clinical range. Results showed no significant difference between the two groups for any of the measures.
"These findings demonstrate that claims that it is harmful for children to be raised by same-sex couples are completely unfounded," said co-author Henny Bos, Ph.D., Professor of Child Development and Education, and Endowed Chair in Sexual and Gender Diversity in Families and Youth at the University of Amsterdam. "There is no justification to restrict child custody or placement, or access to reproductive technologies, based on the parents' sexual orientation."
There are an estimated 114,000 same-sex couples raising children in the United States, including 86,000 female couples. Ten states, including Alabama, Kansas, Michigan, Mississippi, North Dakota, Oklahoma, South Carolina, South Dakota, Texas, Virginia, allow state-licensed child welfare agencies to refuse to place and provide services to children and families if it conflicts with their religious beliefs.
This is the 23rd publication from data collected in the NLLFS. Previous research found that the development of psychological well-being in these offspring over a 7-year period from childhood through adolescence was the same for those conceived through known or unknown sperm donors. In addition, the absence of male role models did not adversely affect the psychological adjustment of 17-year-olds raised in lesbian households. None of these 17-year-olds had been abused by a parent or caregiver. In contrast, 26 percent of 17-year-olds nationally report physical abuse and 8 percent report sexual abuse by a parent or caregiver.
The report, "National Longitudinal Lesbian Family Study—Mental Health of Adult Offspring," appears in The New England Journal of Medicine and is co-authored by Nanette Gartrell, M.D., Visiting Distinguished Scholar, along with Henny Bos, Ph.D., former Visiting International Scholar at the Williams Institute, and Audrey Koh, M.D., Associate Professor, Department of Obstetrics, Gynecology, and Reproductive Sciences, University of California, San Francisco.
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I've been pleasantly surprised to experience a spirit of volunteerism here in Jordan. Last weekend after a tasty juice of lemon and mint on Rainbow Street, I was so surprised to see a large group of kids in orange vests carrying trash bags, I had to take a picture of it.
Lillie, an friend who was in Morocco with me noticed this picture and said she would have had a heart attack if she saw volunteer kids picking up trash in Fez. I tried to explain this to my new American friend here in Jordan when he asked me why I thought the volunteers were photoworthy: I just never experienced much of a motivation for community involvement and volunteerism in Morocco, and was happy to see it alive and well here in Amman.
This Saturday I got to do a little volunteering myself. I went with Hamza, his friends, and nearly 100 other young Jordanian volunteers to plant 1,000 trees north of Amman with volunteers from the Ministry of Agriculture.
The initiative is to plant 10,000 total trees this year.
Only 0.9% of Jordan is covered with forest, and has big problems with deforestation and desertification.
I planted around 20 trees with Ziad, a volunteer with the Ministry of Agriculture who also works as a night guard in Amman. He has invited me to his house next weekend for a meal of mansaf with his family: a traditional Jordanian dish made with lamb and rice!
Laura and Ziad, combating deforestation one tree at a time.


text:
Confirmed: Putin will lead UN General Assembly in NYC
By Joaquin Flores Last updated Feb 19, 2018
Rg.ru – translated for Fort Russ by Soviet Bear
Russian delegation at the 70th session of the UN General Assembly in New York
in September, will be led by President Vladimir Putin. The presidential decree was
published on the official Internet portal of legal information.
The delegation, in particular, will be
composed of the Chairman of the Federation Council Committee on international
Affairs Konstantin Kosachev, the head of Russian foreign Ministry Sergei
Lavrov, Chairman of the Duma Committee on international Affairs Alexei Pushkov,
assistant to the President Yuri Ushakov.
As deputy delegates to the list were included:
Deputy foreign Minister Gennady Gatilov, Russia's Ambassador to the U.S. Sergey
Kislyak, the Director of the Department of international organizations of the
Russian foreign Ministry Alexander Pankin, Director of the Department for
non-proliferation and arms control of the Russian foreign Ministry Mikhail
Ulyanov and Russia's permanent representative to the UN Vitaly Churkin.
After the President's participation in the
work of the 70th session of the UN General Assembly Sergey Lavrov is going to
perform duties of the head of the delegation
The Russian foreign Ministry, according to the
order, will approve the composition of the advisors and experts, as well as the
necessary technical personnel.
NYCPutinUN
Joaquin Flores 2138 posts 0 comments
Joaquin Flores is Editor-in-Chief of Fort Russ News, as well as the Director of the Belgrade based think-tank, the Center for Syncretic Studies. Educated at California State University, Los Angeles, in the field of International Relations, he previously served as Chief Negotiator and Internal Organizer in several jurisdictions for the SEIU labor union in California. Flores has twenty years experience in community, labor, and anti-war organizing. Flores has appeared regularly on Iran's 'PressTV' and Russia's 'RT' news to share his expert opinion and analysis on current geopolitical matters. As a Thought Leader, he has spoken publicly internationally at numerous forums, published in over 10 languages, his conceptual and ideological frameworks, approaches to branding and aesthetics have influenced others in his field.
Army of Russia, Belarus and Serbia will hold exercises to suppress Color Revolutions
Zakharchenko blasts Poroshenko's "Farce" reforms
The British Hand Behind America's Color Revolution: Soros and Lord Malloch…
Why China is NOT your Enemy (and what really controls Canada and the US Deep State)
The Nuremberg Tribunal: 75 Years Later and Still the Basis for Humanity's Survival
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text:
Mass graves: 'No Malaysians caught as all were foreigners'
August 9, 2017 August 10, 2017 Nur Farzana Md Yusof
KUALA LUMPUR: Deputy Prime Minister Ahmad Zahid Hamidi says there was no compromise in the detention of those involved in the Wang Kelian mass graves discovered in 2015.
He added that no Malaysian agents had been detained as those involved were all foreigners.
"Why were no Malaysians accused? The ones who smuggled (the Rohingya into the country) were all foreigners.
"There is no compromise on crime," he told Charles Santiago (DAP-Klang), adding that this was a trans-boundary matter and the agent involved had been taken to court.
Another four foreigners were also charged with human trafficking, he said.
"Please understand how I work. I am serious about the issue," he said when winding up the debate on the Malaysian Border Security Agency Bill (2017), which was passed at the Dewan Rakyat today.
Santiago had asked why the police in Wang Kelian were not aware of the mass graves in which 106 Rohingya migrants had been buried. Read more →
Human Rights & Wrongs	Ahmad Zahid Hamidi, Malaysian Border Security Agency Bill (2017), Wang Kelian
Putrajaya still pursuing TPP, says Mustapa
August 9, 2017 August 11, 2017 admin
Source: The Malaysian Insight
Demonstrators hold up placards during a protest against the Trans-Pacific Partnership Agreement (TPPA), outside the U.S. embassy in Kuala Lumpur on August 23, 2013. The Malaysian Insider/Najjua Zulkefli
THE government will still pursue the Trans-Pacific Partnership (TPP) despite the Regional Comprehensive Economic Partnership Agreement (RCEP) showing progress.
International Trade and Industry Minister Mustapa Mohamed said the TPP is still relevant to the country and discussions are moving in tandem with the RCEP.
"We have spent a lot of time and invested a lot of money in the TPP and also reached a final agreement (for the pact) before it was abandoned.
"We are still looking at alternatives and discussing with other TPP countries on the agreement. There will be a meeting to be hosted by Australia regarding this matter next month," he said today. Read more →
Human Rights & Wrongs	Trans-Pacific Partnership Agreement (TPPA)
Suhakam chief: Minister should not have called for atheists to be hunted down
Source: The Malay Mail Online
KUALA LUMPUR, Aug 9 — The chief commissioner of the National Human Rights Commission (Suhakam) has censured a federal minister for calling on the authorities to clamp down and prosecute Malay atheists.
Tan Sri Razali Ismail called the suggestion inflammatory, and said that Datuk Seri Shahidan Kassim as a Cabinet member should have been more careful when addressing a highly sensitive issue.
"I don't think a Cabinet Minister should have said that," Razali told Malay Mail Online.
"He shouldn't have made a statement that would invite emotional argument," he added.
But despite the criticism, Razali did not make his position on the issue clear, nor did he wish to state if Suhakam was against the persecution of atheists.
When queried if the commission was of the view that atheism — or the right not to believe in religion — is a human right and therefore must be respected, Razali replied:
"We don't want to make (things worse) than what it is…we don't want to say anything.
"Suhakam is studying the thing. But we really don't want to say anything at the moment that doesn't really help," he said. Read more →
Human Rights & Wrongs	Razali Ismail, rights of atheists, Suhakam - Human Rights Commission of Malaysia
Suhakam to hold public inquiry into missing pastor, others
August 9, 2017 October 11, 2017 Nur Farzana Md Yusof
PETALING JAYA: The Human Rights Commission of Malaysia (Suhakam) will conduct a public inquiry into the disappearances of Pastor Raymond Koh and three others who went missing last year.
In a statement today, Suhakam chairman Razali Ismail said the hearings were tentatively scheduled for October.
The inquiry will consider, among others, whether the cases of Koh, Amri Che Mat, Pastor Joshua Hilmy and his wife Ruth, were cases of enforced or involuntary disappearances as defined under the International Convention for the Protection of All Persons from Enforced Disappearance.
The convention defines an enforced disappearance as the arrest, detention, abduction or any other form of deprivation of liberty by agents of the state or by persons or groups of persons acting with the authorisation, support or acquiescence of the state.
Razali said the inquiry will also consider whether the authorities, specifically the Royal Malaysia Police (PDRM), took adequate steps to investigate the cases.
"Suhakam will continue to gather information from all stakeholders, including the police and others who have been cooperative towards finding some truth to the matter," he said. Read more →
Human Rights & Wrongs	Amri Che Mat, Joshua Hilmy, Raymond Koh, Razali Ismail, Ruth Hilmy, Suhakam - Human Rights Commission of Malaysia
Minister violating human rights by 'hunting' atheists, says humanist group
KUALA LUMPUR, Aug 9 — Criminalising apostasy contradicts the right to freedom of thought and belief, an international humanist group said amid a government crackdown on Malaysian atheists.
International Humanist and Ethical Union (IHEU) president Andrew Copson also said Minister in the Prime Minister's Department Datuk Seri Shahidan Kassim was "absolutely wrong" in claiming that atheism violated human rights.
"Non-religious people have freedom of thought, freedom of expression and freedom of association, just like the religious, and it is his talk of 'hunting' human beings simply for joining together in celebration of their views and values which represents a grave human rights violation," he said in a statement yesterday.
Shahidan claimed yesterday that atheism was unconstitutional because it was not mentioned in the Federal Constitution and suggested that "we hunt them down vehemently".
Deputy minister in charge of Islamic affairs Datuk Dr Asyraf Wajdi Dusuki has said the religious authorities will investigate the Kuala Lumpur chapter of international non-profit group Atheist Republic, after a photo of its gathering caused uproar among some Muslims online. Read more →
Human Rights & Wrongs	freedom of religion, International Humanist and Ethical Union (IHEU), rights of atheists
Court of Appeal has no authority to hear legality of unlawful assembly
PUTRAJAYA: The Court of Appeal today allowed the government's objection that it has no jurisdiction to hear constitutional issues raised by four men convicted of violating the Peaceful Assembly Act.
A three-man bench led by Mohd Zawawi Salleh said the High Court had erred in deciding that Section 4(2)(b) of the Peaceful Assembly Act was a valid provision.
"We are not seized with the jurisdiction under the Courts of Judicature Act 1964 to determine if that provision is constitutional," he said.
Zawawi said the case would be remitted to the High Court in Johor Baru and a case management would be held on Aug 16.
Government lawyer Awang Armadajaya Awang Mahmud explained to reporters that the High Court judge would have to send the case straight to the Federal Court to determine the constitutionality of the provision.
He said the case of the four men started in the Sessions Court and the High Court went on to decide the legality of the section, which was disallowed based on a Federal Court ruling. Read more →
Human Rights & Wrongs	Court of Appeal, Peaceful Assembly Act (PAA) 2012


This past weekend, we took a trip to Philly to check out the Philadelphia Auto Show at the Philadelphia Convention Center. We were invited* to come see the wide range of cars and exhibitors that are here from January 26th to February 4th. Car shows are always a lot of fun to go to so you can see the latest and greatest automobiles there are on the market. But I will admit, I know very little about cars. No matter, because you don't have to know a lot about cars to enjoy yourself at the show.
We did happen to be looking for some information on 2018 cars and going to the car show was a great place to see the cars we were looking for all under one roof. We were able to see all the models from numerous companies and compare and contrast. For someone who doesn't have a ton of free time to run around from dealership to dealership, this is very helpful!
In between checking out the cars we were interested in, we were treated to a lot of special exhibits. There was a Hollywood section where we got to see some fun blasts from the past. Recognize these vehicles?
The 80s child inside me flipped out! The Mystery Machine, the A-Team van, and Kit from Knight Rider all in person to see. It was fun to show my kids special memories from my childhood.
Moving on, we saw some really cool collectors cars, all of which were very unique and special to see. However, this one stood out to me especially, as I was a huge Back to the Future fan. Have you ever seen a gold DeLorean? Incredible!
I loved seeing the exotic collection of cars. They are really gorgeous!
There was something for every one at the show. Hands on exhibits (Camp Jeep hosted an off-road adventure course for those who wanted to get up close and personal with the capabilities of their 4 wheel drives), giveaways, and vendors could be found all throughout the convention center. It was a great way to spend an evening with the family.
You can find out more about the PhillyAutoShow online and plan a visit for yourself. You'll love all it has to offer!
*I was invited to visit the show to facilitate a feature, all opinions are my own.


text:
A Gábor-érem (angolul: Gabor Medal) egyike a Royal Society érmeinek. A biológia, azon belül leginkább molekuláris biológia és a genomika terén elért jelentős felfedezésekért ítélik oda. 1989-ben osztották ki először, és páratlan években ítélik oda (2010-ben kivételesen adtak át díjat a társulat 350 éves évfordulóján). A magyar Nobel-díjas Gábor Dénes tiszteletére alapították.

A díjazottak

 2020 David Stuart
 2019 Alison Noble
 2017 Richard Michael Durbin
 2015 Benjamin Simons
 2013 Christofer Toumazou
 2011 Angela McLean
 2010 Gideon Davies
 2009 Gregory Challis
 2007 Richard Roberts
 2005 Lionel Crawford
 2003 Jean Begg
 2001 M. Azim Surani
 1999 Adrian Bird
 1997 Kenneth Holmes
 1995 David Hopwood
 1993 Charles Weissmann
 1991 Alan Fersht
 1989 Noreen Elizabeth Murray

Jegyzetek

Források
 Royal Society: Gabor Medal

A Royal Society érmei
Biológia

redpajama_set_name: RedPajamaWikipedia


If a new version is found, it will be downloaded and stored in ~/.local/share/heroku/cli. This background check will happen at most once every 4 hours.
heroku.com: get started with nodejs.
# When you create an app, a git remote (called heroku) is also created and associated with your local git repository.
heroku config:set TIMES=2 TWITTER_KEY="aaa" TWITTER_SECRET="bbb"
Heroku treats web apps just like any other app and doesn't allow you to assign listening ports directly. The port to bind to is assigned by Heroku as the PORT environment variable.
The web process type is special as it's the only process type that will receive HTTP traffic from Heroku's routers. Other process types can be named arbitrarily.
Colorful code examples powered by highlight.js.
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Norlight: VoIP, POTS, and a range of networking services.
Part 44 of Phone for Rent: Understanding Hosted PBX Services — Norlight is a regional provider, serving both urban and rural areas in the Midwest and Southeast.
Norlight Telecommunications, Inc., headquartered in Evansville, Indiana, was founded in 1972 as a Wisconsin-based voice and data services company and since that time has grown to become a predominant telecommunications provider throughout the Midwest and Southeastern United States.
Norlight customers represent a broad spectrum of business interests, including local, state, and federal government, education, healthcare, and large enterprises. These customers are served by an extensive fiber optic backbone that reaches from the Gulf States on the south to the Great Lakes on the north, and from Oklahoma and Texas on the west to Virginia and Pennsylvania on the east.
Much of this expansion has been the result of a series of strategic mergers and acquisitions that began in the 1990s, with the present network offering a carrier-class, Cisco-routed and DWDM/SONET-based infrastructure that serves over 200 markets and nearly 650 POPs.
Norlight currently has a presence in many tier-1 metropolitan areas, and is continuing to expand its fiber connectivity into the rural tier-2 and tier-3 markets that other carriers have largely bypassed. This strategy includes locating its existing POPs in LEC central offices, thus allowing its customers easier connections to local service, plus the ability to custom-build fiber access all the way to a customer's premises.
In addition, the Norlight network is supported by two redundant data centers, operated in Evansville, Indiana, and Brookfield, Wisconsin, that include earthquake-reinforced ceilings and 12-inch thick poured concrete walls.
Norlight's extensive fiber network serves as a solid foundation that provides a variety of telecommunication services for both residential and business applications.
The residential offerings include local phone service, dialup Internet access, DSL in selected cities, and a wireless network that provides high-speed Internet access to rural areas of western Kentucky. Also available in selected areas of Indiana and Kentucky is the firm's Broadband Digital Telephone service, providing unlimited local and long distance calling for $19.95 per month.
On the business side, Norlight's Hosted VBX for Business offering bundles local phone, long distance, and dedicated Internet access over a single, private IP connection, that is provided for a single monthly fee. This hosted offering is available with two feature sets: Basic and Professional.
The Basic set includes a specified number of either pooled or non-pooled long distance minutes, DID (Direct Inward Dial) numbers, 4-digit dialing, call forward always/busy/no answer; remote access to call forwarding; voicemail with storage for messages up to two minutes in length, with notifications via e-mail, pager, or telephone; call logs for incoming, outgoing, and missed calls; multi-line call park and retrieve; a night mode to re-route calls after business hours, and more.
The Professional feature set adds integrated messaging with voicemail retrieval via e-mail, find me/follow me for mobile phone integration, the ability to make calls from anywhere using remote phone integration, conference calling with up to four parties, and a fax-to-e-mail service. Both packages include a Web portal that allows the end users to view call logs, change settings, and perform other key administrative tasks. The system supports Cisco IP phones, and can also integrate existing analog phones for non-office environments, such as warehouses.
In addition to hosted IP telephony for business, Norlight offers voice services that range from POTS services delivered over copper pairs, to voice service for large businesses that require hundreds of lines. Q-Link Plus is a bundled service that consolidates local and long distance services, plus T-1 Internet access, over a single, private, IP-based connection, which Norlight claims results in a higher quality of service and higher Internet access speeds than comparable services that use a public Internet connection. In addition, this service can be provided with their Metro Ethernet access, providing even higher bandwidth options.
Other Norlight telecoms services for businesses include Web Services (Web hosting and development) and an e-mail service called CleanMail that filters messages for threats. Network Services includes a managed security offering that can tailor managed firewall, antivirus, intrusion detection, and prevention, Web site blocking, and content filtering, as well as managed VPN services.
Norlight also offers Metro Ethernet connectivity that provides both a committed and symmetrical rate for high speed network access and Wide Area Networking services that can include VPN and private line connectivity, plus a MPLS offering that creates a fully meshed private network that prioritizes voice, video, and data with a specified quality of service. The Business Internet Services build on Norlight's experience as an ISP since 1994, and provide access at rates that vary from dialup to DS-3; while the company's business DSL service is offered in select cities in Indiana, Kentucky, and Tennessee.
Further information on the Norlight solutions can be found at www.norlight.com. Our next tutorial will continue our review of various service providers hosted voice solutions.


text:
Q: Standard deviation of the sum of a discrete uniform If I randomly generate a number between 1 and 10.... 10 times, and then total all the numbers, what will the standard deviation of that total be?
I'm pretty sure the mean of the total will be 55.5, but what about the average distance from that mean?

A: You're right about the average.
Using the expectation operator $E$, this is a consequence of the linearity of expectation.  If we write $X$ for a random integer between one and ten drawn inclusively and uniformly (this is what the function call random.randint is modeling), then
$$ E(X) = 1 \times 0.1 + 2 \times 0.1 + \cdots + 10 * 0.1 = 5.5 $$
You're asking about the sum of ten draws from $X$, so using the linearity of expectation
$$ E \left( \sum_{i = 1}^{10} X \right) = \sum_{i = 1}^{10} E(X) = 10 E(X) = 5.5 $$
The variance calculation is a little more tricky, but goes through with a standard trick.  For clarity, let's assign each of your ten draws to a different symbol, $X_1, X_2, \ldots, X_{10}$ are the ten draws from $X$.  Now recall that for any two random variables $A$ and $B$
$$ Var(A + B) = Var(A) + Var(B) + 2Cov(A, B) $$
The variance is not linear, its failure to be linear is summarized in the $Cov(A, B)$ term that appears.
In your case, you need to know that when you call random.randint multiple times, the draws you get are independent.  This is really important, the random number generator would be useless if this was not true.  For $A$ and $B$ independent random variables
$$ Cov(A, B) = 0 $$
Again, super important, must be independent or this step probably fails.
Given all this, you can break down the variance
$$ Var \left( \sum_{i = 1}^{10} X_i \right) = \sum_{i=1}^n Var(X_i) + 2 \sum_{i = 1}^n \sum_{j = 1}^n Cov(X_i, X_j) $$
You should play around with that until you convince yourself it's true.  The index manipulation is a little weird until you get used to it.  Anyway, in our situation, the covariance terms are zero, so this reduces to
$$ Var \left( \sum_{i = 1}^{10} X_i \right) = \sum_{i=1}^{10} Var(X_i) $$
But each $X_i$ has the same distribution, so this is really just
$$ Var \left( \sum_{i = 1}^{10} X_i \right) = \sum_{i=1}^n Var(X) = 10 Var(X) $$
And my guess is you can finish it off from here.

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Hi, Please inform possibility of editing GR & how.
Please inform possibility of editing GR & how.
Hi, Please inform possibility of editing GR & how. ..
1) Inventory Management 2) accounting document.
only solution to reversal & cancle Gr document.
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