\section{Introduction}
We consider the critical Lane-Emden system
\begin{equation}\label{eq-LEs}
\begin{cases}
-\Delta u = v^p &\text{in } \R^n,\\
-\Delta v = u^q &\text{in } \R^n,\\
u,v >0\ &\text{in } \R^n
\end{cases}
\end{equation}
where $n \ge 3$ and $p,q>0$  and $(p,q)$ belongs to the {\em critical hyperbola}
\begin{equation}\label{eq-hyp}
\frac{1}{p+1} + \frac{1}{q+1} = \frac{n-2}{n}.
\end{equation}

For $s \ge 1$, let $\mcd^{2,s}_0(\R^n)$ be the completion of $C^{\infty}_c(\R^n)$ with respect to the norm $\|\Delta \cdot\|_{L^s(\R^n)}$.
In \cite[Corollary I.2]{Li}, Lions found a positive ground state
\[(U,V) \in \mcd^{2, {p+1 \over p}}_0(\R^n) \times \mcd^{2, {q+1 \over q}}_0(\R^n)\]
of \eqref{eq-LEs}, by transforming it into an equivalent scalar equation
\begin{equation}\label{eq-LEsc}
(-\Delta) \(|\Delta u|^{{1 \over p}-1} (-\Delta u)\) = |u|^{q-1}u \quad \text{in } \R^n
\end{equation}
and employing a concentration-compactness argument to the associated minimization problem
\begin{equation}\label{eq-K_pq}
K_{p,q} = \inf \left\{\|\Delta u\|_{L^{p+1 \over p}(\R^n)}: \|u\|_{L^{q+1}(\R^n)} = 1 \right\}
= \inf_{u \in \mcd^{2, {p+1 \over p}}_0(\R^n) \setminus \{0\}} \frac{\int_{\R^n} |\Delta u|^{p+1 \over p}}{(\int_{\R^n} |u|^{q+1})^{\frac{p+1}{p(q+1)}}}.
\end{equation}
As shown by Alvino et al. \cite{ALT}, it is always radially symmetric and decreasing in $r = |x|$, after a suitable translation.
Moreover, Hulshof and Van der Vorst \cite{HV} proved that a positive radial solution of \eqref{eq-LEs} is unique up to scalings.

\medskip
The present paper deals with non-degeneracy of ground state solutions $(U,V)$ to \eqref{eq-K_pq}
(for which we may assume that $U(0)=1$ without loss of generality).
The invariance of the system under scaling and translations leads to natural solutions of the linearized system around the radial solution $(U,V)$. More precisely, the functions
$$ (U_{\delta,\xi}(x),V_{\delta,\xi}(x)) := \(\delta^{2(p+1) \over pq-1} U(\delta(x-\xi)), \delta^{2(q+1) \over pq-1} V(\delta(x-\xi))\) \quad \hbox{for any}\ \delta>0,\ \xi\in\R^n$$
are solutions to system \eqref{eq-LEs}. If we differentiate the system
$$\begin{cases}
-\Delta U_{\delta,\xi} = V^p_{\delta,\xi} &\text{in } \R^n,\\
-\Delta V_{\delta,\xi} = U^q_{\delta,\xi} &\text{in } \R^n
\end{cases}
$$
with respect to the parameters at $\delta=1$ and $\xi=0,$
we immediately see that the $(n+1)$ linearly independent functions
\[Z_0(x):= \(x\cdot\nabla U + \frac {2(p+1)}{pq-1} U, x\cdot\nabla V + \frac {2(q+1)}{pq-1} V\) \quad \hbox{and} \quad  Z_i(x):= \(\frac{\pa U}{\pa x_i}, \frac{\pa V}{\pa x_i}\)\]
for $i=1,\dots,n$ solves the linear system
\begin{equation}\label{lin}\begin{cases}
-\Delta \phi = pV^{p-1}\psi &\text{in } \R^n,\\
-\Delta \psi = qU^{q-1}\phi &\text{in } \R^n.\\
\end{cases}
\end{equation}
We say that $(U,V)$ is {\em non-degenerate} if all weak solutions to the linear system \eqref{lin} such that $\lim_{|x|\to\infty} (\phi(x),\psi(x)) = (0,0)$ are linear combinations of $Z_0,Z_1,\dots,Z_n.$

\medskip
The non-degeneracy of the solutions of system \eqref{eq-LEs} is a key ingredient in understanding the blow-up phenomena of solutions to the Lane-Emden systems with critical growth.
Therefore,  it is quite natural to ask the following question:
$$\hbox
{\em (Q) \quad  Are ground states $(U,V)$ non-degenerate?}$$

Here, we face the above question, and we give a positive answer in a perturbative setting.
It would be extremely interesting to prove or to disprove non-degeneracy of ground state solutions when $(p,q)$ ranges along all the critical hyperbola \eqref{eq-hyp}.

\medskip
Our main result is:
\begin{thm}\label{main}
There is a small number $\ep > 0$ such that if either $|p-1| \le \ep_0$ (with $n\ge 5$) or
$|p-{n+2\over n-2}| \le \ep$ (with $n\ge3$),
then the unique positive solution $(U,V)$ of \eqref{eq-LEs} (with $U(0) = 1$) is non-degenerate.
\end{thm}
Theorem \ref{main} follows immediately by Corollaries \ref{cor1} and \ref{cor2}, proved in Sections \ref{sec_p1} and \ref{sec_pn}, respectively.
The idea of the proof stems from the simple fact that if $p$ is close to $1$ or close to $n+2\over n-2$,
system \eqref{eq-LEs} is formally close to a single equation whose solutions are non-degenerate.
In particular, if $p$ is close to 1, then $q$ is close to
$ {n+4\over n-4} $ and system \eqref{eq-LEs} can be regarded as a perturbation of the Paneitz-Branson equation
\[(-\Delta)^2u=u^{n+4\over n-4} \quad \text{in } \R^n,\]
while if $p$ is close to ${n+2 \over n-2}$, then $q$ is also close to ${n+2 \over n-2}$ and system \eqref{eq-LEs} becomes a perturbation of the Yamabe equation
\[-\Delta u=u^{n+2\over n-2} \quad \text{in } \R^n.\]
Therefore, to prove our result, we will follow a perturbation argument, which has been successfully applied in various problems like the pseudo-relativistic Hartree equations \cite{Le},
the fractional Schr\"odinger equations \cite{FV} and the Choquard equations \cite{Xi}.
The most challenging part of the proof is to show rigorously that the linearized system \eqref{lin} is close to the corresponding linearized (single) equation,
because sophisticated uniform estimates in $p$ and $q$ of the decay of ground state solutions are required.

\medskip \noindent \textbf{Notations.}
\begin{itemize}
\item[-] For $a \in \R$, let $a_+ = \max\{a,0\}$.
\item[-] For any $x \in \R^n$ and $R > 0$, let $B_R(x) = \{y \in \R^n: |y-x| < R\}$.
\item[-] For a set $D \subset \R^n$, let $\chi_D$ be the characteristic function of $D$.
\item[-] The letters $C$ and $c$ denote positive numbers independent of $p$ that may vary from line to line and inside the same line.
\end{itemize}



\section{Non-degeneracy of the Lane-Emden system near $p = 1$}\label{sec_p1}
The main results of this section are Theorem \ref{thm-deg} and Corollary \ref{cor1}.
To prove them, we will use the following well-known uniqueness and non-degeneracy results about the fourth-order critical equation
\begin{equation}\label{eq-LEbi}
\begin{cases}
(-\Delta)^2 u = u^{n+4 \over n-4} &\text{in } \R^n,\\
u > 0 &\text{in } \R^n
\end{cases}
\end{equation}
for $n \ge 5$.
\begin{prop}\label{prop-lin-bi}
\textnormal{(1) (uniqueness)} Any smooth solution of \eqref{eq-LEbi} is expressed as
\begin{equation}\label{eq-w_lam}
w_{\delta,\xi}(x) := c_n \(\frac{\delta}{\delta^2+|x-\xi|^2}\)^{n-4 \over 2}
\end{equation}
for some $\delta > 0$, $\xi \in \R^n$ and $c_n = [n(n-4)(n-2)(n+2)]^{-{n-4 \over 8}}$.

\medskip \noindent \textnormal{(2) (non-degeneracy)}
The solution space of the linear equation
\begin{equation}\label{eq-lin-bi}
(-\Delta)^2 \phi = \(\frac{n+4}{n-4}\) u^{8 \over n-4} \phi \quad \text{in } \R^n, \quad \phi \in \mcd^{2,2}_0(\R^n)
\end{equation}
is spanned by
\[\frac{\pa u}{\pa x_1},\, \cdots,\, \frac{\pa u}{\pa x_n} \quad \text{and} \quad x \cdot \nabla u + \(\frac{n-4}{2}\)u.\]
\end{prop}
\begin{proof}
Results (1) and (2) have been proved by Lin \cite{Lin} and Lu and Wei \cite{LW}, respectively.
\end{proof}

\subsection{A compactness result}\label{subsec-cpt}
The following is our main result in this subsection.
\begin{prop}\label{prop-cpt}
Suppose that $n \ge 5$. Let $\{p_k\}_{k=1}^{\infty}$ be a sequence such that $p_k \in (\frac{2}{n-2}, \frac{n}{n-2})$ for all $k \in \N$ and $p_k \to 1$ as $k \to \infty$,
Also, let $\{(U_{p_k}, V_{p_k})\}_{k=1}^{\infty}$ be a sequence of the unique positive ground states of \eqref{eq-LEs} with $p = p_k$ such that $U_{p_k}(0) = 1$.
Then we have that
\[(U_{p_k}, V_{p_k}) \to (U_1, V_1) \quad \text{in } \mcd^{2,2}_0(\R^n) \times \mcd^{2,{2n \over n+4}}_0(\R^n) \quad \text{as } k \to \infty.\]
Here $U_1$ is the unique positive solution of \eqref{eq-LEbi} with $U_1(0) = 1$ and $V_1 = -\Delta U_1$ in $\R^n$.
In other words, $(U_1, V_1) = (w_{a_n,0}, -\Delta w_{a_n,0})$ in $\R^n$ where $a_n := c_n^{2 \over n-4}$.
\end{prop}
As we will see, the proofs of the above proposition and Theorem \ref{thm-deg} require uniform upper bound of $(U_{p_k}, V_{p_k})$'s.
It is useful to recall the asymptotic profile of ground state solutions. In \cite{HV}, it has been shown that there exists a pair of positive constants $(\alpha_p, \beta_p)$ such that
\begin{equation}\label{eq-dec}
\lim_{r \to \infty} r^{n-2}\, v(r) = \beta_p\ \hbox{and}\ \begin{cases}
\lim\limits_{r \to \infty} r^{n-2}\, u(r) = \alpha_p &\text{if } p \in (\frac{n}{n-2}, \frac{n+2}{n-2}],\\
\lim\limits_{r \to \infty} \dfrac{r^{n-2}}{\log r}\, u(r) = \alpha_p &\text{if } p = \frac{n}{n-2} ,\\
\lim\limits_{r \to \infty} r^{p(n-2)-2}\, u(r) = \alpha_p &\text{if } p \in (\frac{2}{n-2}, \frac{n}{n-2}).
\end{cases}
\end{equation}

Even though \eqref{eq-dec} depicts the precise asymptotic behavior of $(U_{p_k}, V_{p_k})$ for each $k \in \N$,
it does not readily imply the uniform bound, because the arguments in \cite{HV} do not describe how the sequence $\{(\alpha_{p_k}, \beta_{p_k})\}_{k=1}^{\infty}$ behaves.
In the next two lemmas, we will obtain it by using potential theory. It is not sharp but enough for our purpose.

Without loss of generality, we can assume that $|p_k - 1| \le \ep_0$ for all $k \in \N$ and a small fixed number $\ep_0 > 0$.
Let $q_k$ be the number $q$ determined by \eqref{eq-hyp} with $p = p_k$.

\begin{lemma}\label{lem-cpt-11}
There exist a constant $C > 0$ depending only on $n$ and $\ep_0$ such that
\begin{equation}\label{eq-cpt-11}
U_{p_k}(x) \le 1 \quad \text{and} \quad V_{p_k}(x) \le C \quad \text{for all } x \in \R^n \text{ and } k \in \N
\end{equation}
provided $\ep_0 > 0$ small enough.
\end{lemma}
\begin{proof}
We present the proof by dividing it into 2 steps.

\medskip
\noindent \textsc{Step 1: Uniform boundedness of $K_{p_k, q_k}$.}
Using $w_{a_n,0}$ as a test function of the minimization problem \eqref{eq-K_pq}, we obtain
\begin{equation}\label{eq-K_pq-1}
K_{p_k, q_k} \le \frac{\int_{\R^n} |\Delta w_{a_n,0}|^{p_k+1 \over p_k}}{(\int_{\R^n} w_{a_n,0}^{q_k+1})^{\frac{p_k+1}{p_k(q_k+1)}}}.
\end{equation}
Exploiting the explicit form of $w_{a_n,0}$ and applying the dominated convergence theorem on the right-hand side, we easily deduce that $K_{p_k, q_k}$ is uniformly bounded. In particular,
\begin{equation}\label{eq-K_pq-2}
K_{p_k, q_k} = \(\int_{\R^n} U_k^{q_k+1}\)^{1-\frac{p_k+1}{p_k(q_k+1)}} \le C
\quad \text{or } \quad
\int_{\R^n} U_k^{q_k+1} \le C.
\end{equation}
Note that the second inequality holds, since $q_k \to \frac{n+4}{n-4}$ and so $\frac{(p_k+1)}{p_k(q_k+1)} \to \frac{n-4}{n} < 1$ as $k \to \infty$.

\medskip
\noindent \textsc{Step 2: Uniform boundedness of $(U_{p_k}, V_{p_k})$.}
Because $U_{p_k}(0) = 1$ and $U_{p_k}$ is decreasing in $r$, it holds that $\|U_{p_k}\|_{L^{\infty}(\R^n)} \le 1$ for all $k \in \N$.
By \eqref{eq-LEs}, the Green's representation formula, H\"older's inequality and Step 1, we have
\begin{align*}
V_{p_k}(0) &= \ga_n \int_{B_1(0)} \frac{1}{|y|^{n-2}} U_{p_k}^{q_k}(y) dy
+ \ga_n \int_{\R^n \setminus B_1(0)} \frac{1}{|y|^{n-2}} U_{p_k}^{q_k}(y) dy \\
&\le \ga_n \int_{B_1(0)} \frac{1}{|y|^{n-2}} dy + \ga_n \(\int_{\R^n \setminus B_1(0)} \frac{1}{|y|^{(n-2)(q_k+1)}} dy\)^{1 \over q_k+1}
\(\int_{\R^n \setminus B_1(0)} U_{p_k}^{q_k+1}\)^{q_k \over q_k+1} \\
&\le C
\end{align*}
where $\ga_n := (n(n-2)|B_1(0)|)^{-1}$. The last inequality holds because of \eqref{eq-K_pq-2} and the relation that $(n-2) \frac{2n}{n-4} > n$.
As before, since $V_{p_k}$ is decreasing in $r$, we see that $\|V_{p_k}\|_{L^{\infty}(\R^n)} \le C$ for all $k \in \N$.
\end{proof}

\begin{lemma}\label{lem-cpt-12}
Suppose that an arbitrarily small number $\eta_0 > 0$ is given.
Reducing the size of $\ep_0$ if necessary, one can find a constant $C > 0$ depending only on $n$, $\ep_0$ and $\eta_0$ such that
\[U_{p_k}(x) \le \frac{C}{1+|x|^{n-4-\eta_0}} \quad \text{and} \quad V_{p_k}(x) \le \frac{C}{1+|x|^{n-2}}\]
for all $x \in \R^n$ and $k \in \N$.
\end{lemma}
\begin{proof}
The arguments used here is motivated by the ones in \cite[Section 4]{CK}.
Our proof is relatively simple since we make use of good qualitative properties of the ground states $(U_{p_k}, V_{p_k})$.
We present the proof by dividing it into 2 steps.

\medskip
\noindent \textsc{Step 1: Tail estimate for $U_{p_k}$.}
For any fixed number $R > 0$, we define the functions
\[U_{p_ki} = \chi_{B_R(0)} U_{p_k} \quad \text{and} \quad U_{p_ko} = \chi_{\R^n \setminus B_R(0)} U_{p_k} \quad \text{in } \R^n.\]
We assert that for any given number $\zeta > 0$, there exists a number $R > 0$ depending only on $n$, $\ep_0$ and $\zeta$ such that
\begin{equation}\label{eq-dec-3}
\int_{\R^n \setminus B_R(0)} U_{p_k}^{q_k+1} = \int_{\R^n} U_{p_ko}^{q_k+1} \le \zeta
\end{equation}
for all $k \in \N$.

By Lemma \ref{lem-cpt-11} and elliptic regularity, there exists a pair $(\wtu_1, \wtv_1) \in (C^2(\R^n))^2$ of nonnegative radial functions such that
\begin{equation}\label{eq-conv}
(U_{p_k}, V_{p_k}) \to (\wtu_1, \wtv_1) \quad \text{in } (C^2_{\text{loc}}(\R^n))^2 \quad \text{as } k \to \infty
\end{equation}
along a subsequence. In particular, $(\wtu_1, \wtv_1)$ is a classical solution of \eqref{eq-LEs} with $(p,q) = (1,\frac{n+4}{n-4})$.
Also, since $\wtu_1$ is superharmonic and $\wtu_1(0) = 1$, the maximum principle implies that $\wtu_1 > 0$ in $\R^n$.
In view of Proposition \ref{prop-lin-bi} (1), it holds that $\wtu_1 = w_{a_n,0}$ in $\R^n$ and the convergence in \eqref{eq-conv} is valid for the entire sequence (not just for a subsequence). Summing up,
\begin{equation}\label{eq-conv-2}
(U_{p_k}, V_{p_k}) \to (U_1, V_1) \quad \text{in } (C^2_{\text{loc}}(\R^n))^2 \quad \text{as } k \to \infty
\end{equation}
where we write $(U_1, V_1) = (w_{a_n,0}, -\Delta w_{a_n,0})$ as in the statement of Proposition \ref{prop-cpt}.

Taking the limit $k \to \infty$ on the both sides of \eqref{eq-K_pq-1}, and employing Fatou's lemma, \eqref{eq-K_pq-2} and \eqref{eq-conv-2}, we obtain
\begin{equation}\label{eq-conv-3}
\begin{aligned}
\int_{\R^n} w_{a_n,0}^{\frac{2n}{n-4}} &\le \liminf_{k \to \infty} \int_{\R^n} U_{p_k}^{q_k+1} \le \limsup_{k \to \infty} \int_{\R^n} U_{p_k}^{q_k+1} \\
&= \limsup_{k \to \infty} K_{p_k, q_k}^{\(1-\frac{p_k+1}{p_k(q_k+1)}\)^{-1}}
\le \left[ \frac{\int_{\R^n} |\Delta w_{a_n,0}|^2}{(\int_{\R^n} w_{a_n,0}^{2n \over n-4})^{n-4 \over n}} \right]^{n \over 4} = \int_{\R^n} w_{a_n,0}^{\frac{2n}{n-4}}.
\end{aligned}
\end{equation}
Therefore, all the inequalities must be the equalities. Consequently, applying \eqref{eq-conv-2} and \eqref{eq-conv-3}, we can select $R > 0$ so large that
\[\int_{\R^n} U_{p_ko}^{q_k+1} = \int_{\R^n} U_{p_k}^{q_k+1} - \int_{\R^n} U_{p_ki}^{q_k+1} \to \int_{\R^n} w_{a_n,0}^{\frac{2n}{n-4}}
- \int_{B_R(0)} w_{a_n,0}^{\frac{2n}{n-4}} \le \frac{\zeta}{2} \quad \text{as } k \to \infty.\]
This proves the assertion \eqref{eq-dec-3}.

\medskip
\noindent \textsc{Step 2: Completion of the proof.}
By Green's representation formula, it holds that
\begin{equation}\label{eq-V_po}
\begin{aligned}
\|V_{p_k}\|_{L^{a_1}(\R^n)} &= \ga_n \left\| |\cdot|^{-(n-2)} \ast U_{p_k}^{q_k} \right\|_{L^{a_1}(\R^n)} \\
&\le \ga_n \(\left\| |\cdot|^{-(n-2)} \ast U_{p_ko}^{q_k} \right\|_{L^{a_1}(\R^n)}
+ \left\| |\cdot|^{-(n-2)} \ast U_{p_ki}^{q_k} \right\|_{L^{a_1}(\R^n)}\)
\end{aligned}
\end{equation}
provided that the rightmost side is finite.
Its finiteness is guaranteed for $a_1 > \frac{n}{n-2}$, since \eqref{eq-dec} and Lemma \ref{lem-cpt-11} imply
\begin{equation}\label{eq-dec-2}
(|\cdot|^{-(n-2)} \ast U_{p_ko}^{q_k})(x) \le \frac{C\alpha_{p_k}}{1+|x|^{n-2}}
\quad \text{and} \quad
(|\cdot|^{-(n-2)} \ast U_{p_ki}^{q_k})(x) \le \frac{C}{1+|x|^{n-2}}
\end{equation}
for all $x \in \R^n$ and some constant $C > 0$ depending only on $n$, $\ep_0$ and $R$.\footnote{The inequalities in \eqref{eq-dec-2} are well-known and can be proved as in the proof of \cite[Lemma B.2]{WY}.
We note that the right-hand side of the first inequality in \eqref{eq-dec-2} depends on $k \in \N$, while that of the second one does not.}

On the other hand, the Hardy-Littlewood-Sobolev inequality, H\"older's inequality and \eqref{eq-dec-3} yield
\begin{equation}\label{eq-V_po-2}
\begin{aligned}
\left\| |\cdot|^{-(n-2)} \ast U_{p_ko}^{q_k} \right\|_{L^{a_1}(\R^n)} &\le \|U_{p_ko}^{q_k}\|_{L^{a_2}(\R^n)}
\le \Big\|U_{p_ko}^{p_kq_k-1 \over p_k}\Big\|_{L^{p_k(q_k+1) \over p_kq_k-1}(\R^n)} \Big\|U_{p_ko}^{1 \over p_k}\Big\|_{L^{a_3}(\R^n)} \\
&\le \|U_{p_ko}\|^{p_kq_k-1 \over p_k}_{L^{q_k+1}(\R^n)} \|U_{p_k}\|^{1 \over p_k}_{L^{a_3 \over p_k}(\R^n)}
 \\
&\le C\zeta \||\cdot|^{-(n-2)} \ast V_{p_k}^{p_k}\|^{1 \over p_k}_{L^{a_3 \over p_k}(\R^n)} \\
&\le C\zeta \|V_{p_k}^{p_k}\|^{1 \over p_k}_{L^{a_4}(\R^n)} = C\zeta \|V_{p_k}\|_{L^{a_4p_k}(\R^n)}
\end{aligned}
\end{equation}
where
\[\frac{1}{a_2} = \frac{1}{a_1} + \frac{2}{n}, \quad \frac{1}{a_3} + \frac{p_kq_k-1}{p_k(q_k+1)} = \frac{1}{a_2}, \quad \frac{1}{a_4} = \frac{p_k}{a_3} + \frac{2}{n}\]
and $\zeta > 0$ is an arbitrarily small number.
If we take $\ep_0 > 0$ sufficiently small, then
\[\min\left\{a_2,\, \frac{p_k(q_k+1)}{p_kq_k-1},\, a_3,\, \frac{a_3}{p_k},\, a_4\right\} > 1.\]
Furthermore, we infer from \eqref{eq-hyp} and \eqref{eq-dec} that $a_1 = a_4p_k$ and all the quantities in \eqref{eq-V_po-2} are finite.

Plugging \eqref{eq-V_po-2} into \eqref{eq-V_po} and choosing any $\eta_0' > 0$ small, we find a constant $C > 0$ depending only on $n$, $\ep_0$, $R$ and $\eta_0'$ such that
\[\|V_{p_k}\|_{L^{{n \over n-2} + \eta_0'}(\R^n)} \le C.\]
From the radial symmetry and the decay property of $V_{p_k}$, we deduce
\[V_{p_k}^{{n \over n-2} + \eta_0'}(r)\, r^n \le C\int_{B_r(0)} V_{p_k}^{{n \over n-2} + \eta_0'} \le C\]
where $r = |x| \ge 1$. By combining this with \eqref{eq-cpt-11}, we see that
\[V_{p_k}(x) \le \frac{C}{1+|x|^{n-2-\eta_0'}}\]
for all $x \in \R^n$ and $k \in \N$. As a consequence, we obtain
\[U_{p_k}(x) = \ga_n (|\cdot|^{-(n-2)} \ast V_{p_k}^{p_k})(x) \le C \(|\cdot|^{-(n-2)} \ast \frac{1}{1+|\cdot|^{n-2-\eta_0}}\)(x)
\le \frac{C}{1+|x|^{n-4-\eta_0}},\]
and so
\[V_{p_k}(x) = \ga_n (|\cdot|^{-(n-2)} \ast U_{p_k}^{q_k})(x) \le C \(|\cdot|^{-(n-2)} \ast \frac{1}{1+|\cdot|^{n+4-\eta_0''}}\)(x)
\le \frac{C}{1+|x|^{n-2}}\]
for all $x \in \R^n$ for small $\eta_0,\, \eta_0'' > 0$. This completes the proof.
\end{proof}

\begin{proof}[Completion of the proof of Proposition \ref{prop-cpt}]
By Lemma \ref{lem-cpt-12}, there exists a constant $C > 0$ depending only on $n$ and $\ep_0$ such that
\[\|U_{p_k}\|_{L^{\frac{2n}{n+4} \cdot q_k}(\R^n)} + \|V_{p_k}\|_{L^{2p_k}(\R^n)} \le C.\]
This together with \eqref{eq-LEs} implies uniform boundedness of the sequence $\{(U_{p_k}, V_{p_k})\}_{k=1}^{\infty}$ in the space $\mcd^{2,2}_0(\R^n) \times \mcd^{2,\frac{2n}{n+4}}_0(\R^n)$.

In addition, the uniform decay estimate of $\{(U_{p_k}, V_{p_k})\}_{k=1}^{\infty}$ presented in Lemma \ref{lem-cpt-12} leads
\[\|\Delta U_{p_k}\|_{L^2(\R^n)} = \|V_{p_k}\|_{L^{2p_k}(\R^n)}^{p_k} \to \|V_1\|_{L^2(\R^n)} = \|\Delta U_1\|_{L^2(\R^n)}\]
and
\[\|\Delta V_{p_k}\|_{L^{2n \over n+4}(\R^n)} = \|U_{p_k}\|_{L^{{2n \over n+4} \cdot q_k}(\R^n)}^{q_k} \to \|U_1\|_{L^{2n \over n-4}(\R^n)}^{n+4 \over n-4} = \|\Delta V_1\|_{L^{2n \over n+4}(\R^n)}\]
as $k \to \infty$. As a result, we can invoke \eqref{eq-conv-2} to conclude that
\[(U_{p_k}, V_{p_k}) \to (U_1, V_1) \quad \text{in } \mcd^{2,2}_0(\R^n) \times \mcd^{2,{2n \over n+4}}_0(\R^n) \quad \text{as } k \to \infty,\]
finishing the proof.
\end{proof}

We end this subsection, providing two estimates which will used later.
\begin{lemma}\label{lem-cpt-13}
Suppose that an arbitrarily small number $\eta_2 > 0$ is given.
Reducing the size of $\ep_0$ if necessary, one can find a constant $C > 0$ depending only on $n$, $\ep_0$ and $\eta_2$ such that
\[U_{p_k}(x) \ge \frac{C}{1+|x|^{n-4+\eta_2}} \quad \text{and} \quad V_{p_k}(x) \ge \frac{C}{1+|x|^{n-2}}\]
for all $x \in \R^n$ and $k \in \N$.
\end{lemma}
\begin{proof}
According to \eqref{eq-conv-2}, there exists $C > 0$ depending only on $n$ such that
\[U_{p_k}^{q_k}(x) \ge C \quad \text{for all } x \in B_1(0).\]
Applying the argument in the proof of \cite[Proposition 2]{Vi}, we obtain
\[V_{p_k}(x) \ge \int_{B_1(0)} \frac{\ga_n}{|x-y|^{n-2}} U_{p_k}^{q_k}(y) dy
\ge \frac{C}{1+|x|^{n-2}} \int_{B_1(0)} U_{p_k}^{q_k}(y) dy \ge \frac{C}{1+|x|^{n-2}}\]
and
\[U_{p_k}(x) \ge \int_{B_{\frac{|x|}{2}}(x)} \frac{\ga_n }{|x-y|^{n-2}} V_{p_k}^{p_k}(y) dy \ge \frac{C}{|x|^{n-2}} \int_{B_{\frac{|x|}{2}}(x)} \frac{1}{1+|y|^{n-2+\eta_2}} dy \ge \frac{C}{1+|x|^{n-4+\eta_2}}\]
for $x \in \R^n \setminus B_1(0)$.
\end{proof}

\begin{cor}\label{cor-cpt-12}
Suppose that an arbitrarily small number $\eta_1 > 0$ is given.
Reducing the size of $\ep_0$ if necessary, one can find a constant $C > 0$ depending only on $n$, $\ep_0$ and $\eta_1$ such that
\[|\nabla^l U_{p_k}(x)| \le \frac{C}{1+|x|^{n-4+l-\eta_1}} \quad \text{and} \quad |\nabla^l V_{p_k}(x)| \le \frac{C}{1+|x|^{n-2+l}}\]
for all $x \in \R^n$, $k \in \N$ and $l = 1, 2, 3$.
\end{cor}
\begin{proof}
It immediately follows from Lemmas \ref{lem-cpt-12} and \ref{lem-cpt-13}, and the standard rescaling argument based on elliptic regularity.
\end{proof}


\subsection{Non-degeneracy results near $p = 1$}
Employing the compactness result and pointwise estimates of the sequence $\{(U_{p_k},V_{p_k})\}_{k=1}^{\infty}$ of the unique positive ground states of \eqref{eq-LEs} derived in the previous subsection,
we first deduce a non-degeneracy result of equation \eqref{eq-LEsc} for $p$ near $1$.
\begin{thm}\label{thm-deg}
There exists a small number $\ep_1 \in (0,\ep_0]$ such that if $|p-1| \le \ep_1$ and $U_p$ is the unique positive ground state of \eqref{eq-LEsc} with $U_p(0) = 1$,
then the solution space of the linear equation
\begin{equation}\label{eq-lin}
(-\Delta) \((-\Delta U_p)^{{1 \over p}-1} (-\Delta \phi) \) = pq U_p^{q-1} \phi \quad \text{in } \R^n, \quad \phi \in \mcd^{2,2}_0(\R^n)
\end{equation}
is spanned by
\begin{equation}\label{eq-lin-sol}
\frac{\pa U_p}{\pa x_1},\, \cdots,\, \frac{\pa U_p}{\pa x_n} \quad \text{and} \quad x \cdot \nabla U_p + \(n-2 - \frac{n}{p+1}\) U_p.
\end{equation}
\end{thm}
\begin{proof}
Thanks to Lemma \ref{lem-cpt-12}, Corollary \ref{cor-cpt-12} and elliptic regularity,
the functions listed in \eqref{eq-lin-sol} belong to the space $\mcd^{2,2}_0(\R^n) \cap C^{\infty}(\R^n)$.

For $j = 1, \cdots, n$, each $\frac{\pa U_p}{\pa x_j}$ clearly solve \eqref{eq-lin}. Also, if we set
\[U_{p,\delta}(x) = \delta^{2(p+1) \over pq-1} U_p(\delta x) \quad \text{in } \R^n\]
for each $\delta > 0$, every $U_{p,\delta}$ is a solution of \eqref{eq-LEsc}. Therefore
\begin{equation}\label{eq-U_pmu-2}
\left. \frac{\pa U_{p,\delta}}{d\delta} \right|_{\delta=1} = x \cdot \nabla U_p + \(n-2 - \frac{n}{p+1}\) U_p,
\end{equation}
whose equality holds due to \eqref{eq-hyp}, solves \eqref{eq-lin}.

\medskip
Suppose that there exist
\begin{itemize}
\item[-] a sequence $\{p_k\}_{k=1}^{\infty} \subset [1-\ep_0, 1+\ep_0]$ of numbers tending to $1$ as $k \to \infty$;
\item[-] a sequence $\{U_{p_k}\}_{k=1}^{\infty}$ of the unique positive ground states of \eqref{eq-LEsc} with $p = p_k$ such that $U_{p_k}(0) = 1$;
\item[-] a sequence $\{\phi_k\}_{k=1}^{\infty} \subset \mcd^{2,2}_0(\R^n)$ of solutions of \eqref{eq-lin} with $p = p_k$ and $u = U_{p_k}$
    which cannot be written as a linear combination of the functions
\[\qquad Z_{1p_k} = \frac{\pa U_{p_k}}{\pa x_1},\, \cdots,\, Z_{np_k} = \frac{\pa U_{p_k}}{\pa x_1} \quad \text{and} \quad Z_{0p_k} = x \cdot \nabla U_{p_k} + \(n-2 - \frac{n}{p_k+1}\) U_{p_k}.\]
\end{itemize}
We may assume further that $\|\Delta \phi_k\|_{L^2(\R^n)} = 1$ and
\begin{equation}\label{eq-lin-0}
\int_{\R^n} \Delta \phi_k\, \Delta Z_{0p_k} = \int_{\R^n} \Delta \phi_k\, \Delta Z_{1p_k} = \cdots = \int_{\R^n} \Delta \phi_k\, \Delta Z_{np_k} = 0.
\end{equation}

The rest of the proof is split into 4 steps.

\medskip
\noindent \textsc{Step 1: Uniform boundedness of $\phi_k$'s and $\Delta \phi_k$'s.}
We claim that there exists a constant $C > 0$ depending only on $n$ and $\ep_0$ such that
\begin{equation}\label{eq-phi}
\|\phi_k\|_{L^{\infty}(\R^n)} + \|\Delta \phi_k\|_{L^{\infty}(\R^n)} \le C
\end{equation}
for all $k \in \N$.

Define
\begin{equation}\label{eq-psi_k}
\psi_k = \frac{1}{p_k} (-\Delta U_{p_k})^{{1 \over p_k}-1} (-\Delta \phi_k) = - \frac{1}{p_k} V_{p_k}^{1-p_k} \Delta \phi_k \quad \text{in } \R^n
\end{equation}
for each $k \in \N$. Then \eqref{eq-lin} is rewritten as the linearized equation of system \eqref{eq-LEs}
\begin{equation}\label{eq-lins}
\begin{cases}
-\Delta \phi_k = p_k V_{p_k}^{p_k-1} \psi_k &\text{in } \R^n,\\
-\Delta \psi_k = q_k U_{p_k}^{q_k-1} \phi_k &\text{in } \R^n.
\end{cases}
\end{equation}
Fix any $x_0 \in \R^n$ such that $|x_0| \ge 2$ and set $R = |x_0|$. For any $0 < r' < r \le 1$ and $l \in \N \cap \{0\}$ such that $n > 4l$, it holds that
\begin{equation}\label{eq-CZ-1}
\|\psi_k\|_{W^{2,{2n \over n-4l}}(B_{r'}(x_0))} \le C\(\|\psi_k\|_{L^{2n \over n-4l}(B_r(x_0))} + \|U_{p_k}^{q_k-1} \phi_k\|_{L^{2n \over n-4l}(B_r(x_0))} \)
\end{equation}
and
\begin{equation}\label{eq-CZ-2}
\|\phi_k\|_{W^{2,{2n \over n-4l}}(B_{r'}(x_0))} \le C\(\|\phi_k\|_{L^{2n \over n-4l}(B_r(x_0))} + \|V_{p_k}^{p_k-1} \psi_k\|_{L^{2n \over n-4l}(B_r(x_0))}\)
\end{equation}
provided that the right-hand sides are finite.

By \eqref{eq-psi_k}, Lemmas \ref{lem-cpt-12} and \ref{lem-cpt-13}, $\|\Delta \phi_k\|_{L^2(\R^n)} = 1$ and the Sobolev inequality, we find
\begin{align*}
\|\psi_k\|_{L^2(B_1(x_0))} &\le C \|V_{p_k}^{1-p_k} \Delta \phi_k\|_{L^2(B_1(x_0))} \\
&\le CR^{(n-2)(p_k-1)} \|\Delta \phi_k\|_{L^2(B_1(x_0))} \le CR^{(n-2)(p_k-1)}
\end{align*}
and
\[\|U_{p_k}^{q_k-1} \phi_k\|_{L^2(B_1(x_0))} \le CR^{-(n-4-\eta_0)(q_k-1)} \|\phi_k\|_{L^{2n \over n-4}(B_1(x_0))} \le C R^{-7} \|\Delta \phi_k\|_{L^2(\R^n)} = CR^{-7}.\]
Here $C > 0$ is a constant depending only on $n$, $\ep_0$ and $\eta_0$, and particularly, independent of $x_0$ and $R$.
Hence \eqref{eq-CZ-1} with $l = 0$ shows that
\begin{equation}\label{eq-CZ-3}
\|\psi_k\|_{L^{2n \over n-4}(B_{1/2}(x_0))} \le C \|\psi_k\|_{W^{2,2}(B_{1/2}(x_0))} \le CR^{(n-2)(p_k-1)}.
\end{equation}
On the other hand, it follows from \eqref{eq-CZ-3} that
\[\|V_{p_k}^{p_k-1} \psi_k\|_{L^{2n \over n-4}(B_{1/2}(x_0))} \le R^{(n-2)(1-p_k)} \|\psi_k\|_{L^{2n \over n-4}(B_{1/2}(x_0))} \le C.\]
Thus \eqref{eq-CZ-2} with $l = 1$ gives
\begin{equation}\label{eq-CZ-4}
\|\phi_k\|_{W^{2,{2n \over n-4}}(B_{1/3}(x_0))} \le C\(\|\Delta \phi_k\|_{L^2(\R^n)} + \|V_{p_k}^{p_k-1} \psi_k\|_{L^{2n \over n-4}(B_{1/2}(x_0))} \) \le C.
\end{equation}
Putting \eqref{eq-CZ-3} and \eqref{eq-CZ-4} into \eqref{eq-CZ-1} with $l = 1$, we obtain
\begin{equation}\label{eq-CZ-5}
\|\psi_k\|_{W^{2,{2n \over n-4}}(B_{1/4}(x_0))} \le CR^{(n-2)(p_k-1)}.
\end{equation}
If $5 \le n \le 7$, we have that $W^{2,{2n \over n-4}}(B_r(x_0)) \hookrightarrow L^{\infty}(B_r(x_0))$ for $r > 0$.
Therefore, by means of \eqref{eq-psi_k}, \eqref{eq-CZ-4} and \eqref{eq-CZ-5}, we deduce
\begin{equation}\label{eq-CZ-6}
\|\phi_k\|_{L^{\infty}(\{|x| \ge 2\})} + \|\Delta \phi_k\|_{L^{\infty}(\{|x| \ge 2\})} \le C.
\end{equation}
For higher dimensional case, we repeat the above process to improve integrability of $\psi_k$'s and $\phi_k$'s.
After a finite number of iterations, we obtain \eqref{eq-CZ-6}.
The uniform boundedness of $\phi_k$'s and their Laplacians on the set $\{|x| \le 2\}$ is easier to deduce.
Our claim \eqref{eq-phi} is justified.

\medskip
\noindent \textsc{Step 2: Rough decay estimates of $\phi_k$'s and $\Delta \phi_k$'s.}
We assert that there exists a constant $C > 0$ depending only on $n$ and $\ep_0$ such that
\begin{equation}\label{eq-dec-phi-1}
|\phi_k(x)| \le \frac{C}{1+|x|^{n-4 \over 2}} \quad \text{and} \quad |\Delta \phi_k(x)| \le \frac{C}{1+|x|^{n \over 2}}
\end{equation}
for all $x \in \R^n$ and $k \in \N$. The arguments in this and the next steps are inspired by the proof of \cite[Lemma 3.3]{DKP}.

Fix any $x_0 \in \R^n$ such that $|x_0| \ge 2$ and $R = |x_0|$. Define also
\[\phi_{kR}(x) = R^{n-4 \over 2} \phi_k(Rx) \quad \text{and} \quad \psi_{kR}(x) = R^{n \over 2}\psi_k(Rx) \quad \text{in } \R^n\]
for each $k \in \N$. They solve
\[\begin{cases}
-\Delta \phi_{kR} = p_k (V_{p_k}(Rx))^{p_k-1} \psi_{kR} &\text{in } \R^n,\\
-\Delta \psi_{kR} = q_k R^4 (U_{p_k}(Rx))^{q_k-1} \phi_{kR} &\text{in } \R^n.
\end{cases}\]
For each $t > 1$, set $A_t = \{x \in \R^n: 1/t < |x| < t\}$.
For any $r > r' > 1$ and $l \in \N \cap \{0\}$ such that $n > 4l$, it holds that
\[\|\psi_{kR}\|_{W^{2,{2n \over n-4l}}(A_{r'})} \le C\(\|\psi_{kR}\|_{L^{2n \over n-4l}(A_r)} + R^4 \|(U_{p_k}(R\cdot))^{q_k-1} \phi_{kR}\|_{L^{2n \over n-4l}(A_r)} \)\]
and
\[\|\phi_{kR}\|_{W^{2,{2n \over n-4l}}(A_{r'})} \le C\(\|\phi_{kR}\|_{L^{2n \over n-4l}(A_r)} + \|(V_{p_k}(R\cdot))^{p_k-1} \psi_{kR}\|_{L^{2n \over n-4l}(A_r)}\)\]
provided that the right-hand sides are finite. Besides, we have that $\|\Delta \phi_{kR}\|_{L^2(\R^n)} = 1$.
Hence, arguing as in Step 1, we obtain
\[\|\phi_{kR}\|_{L^{\infty}(\{|x| \ge 2\})} + \|\Delta \phi_{kR}\|_{L^{\infty}(\{|x| \ge 2\})} \le C.\]
Combining this with \eqref{eq-phi}, we conclude that \eqref{eq-dec-phi-1} is true.

\medskip
\noindent \textsc{Step 3: Almost sharp decay estimates of $\phi_k$'s and $\Delta \phi_k$'s.}
Let $\eta_3 > 0$ be any small number.
We will show that there exists a constant $C > 0$ depending only on $n$, $\ep_0$ and $\eta_3$ such that
\begin{equation}\label{eq-dec-phi-2}
|\phi_k(x)| \le \frac{C}{1+|x|^{n-4-\eta_3}} \quad \text{and} \quad |\Delta \phi_k(x)| \le \frac{C}{1+|x|^{n-2-\eta_3}}
\end{equation}
for all $x \in \R^n$ and $k \in \N$.

Fix $k \in \N$, and let $\mu = \frac{n-4}{2}$ and $\nu = \frac{n}{2}$.
For an arbitrary number $\eta > 0$, we define
\[F_{k, \mu, \eta}(x) = \phi_k(x) - \frac{M_{\mu, \eta}}{|x|^{\mu+\eta}}
\quad \text{and} \quad
G_{k, \nu, \eta}(x) = \psi_k(x) - \frac{m_{\nu, \eta}}{|x|^{\nu+\eta}}
\quad \text{in } \{|x| \ge 1\},\]
where $M_{\mu, \eta}$ and $m_{\nu, \eta}$ are large positive numbers determined by their subscripts.
If $R > 1$ is given, we get from \eqref{eq-dec-phi-1} that
\[-\Delta G_{k, \nu, \eta}(x) = q_k U_{p_k}^{q_k-1} \phi_k - \frac{m_{\nu, \eta} (\nu+\eta)(n-2-(\nu+\eta))}{|x|^{\nu+\eta+2}} \le 0 \quad \text{in } \{1 < |x| < R\}\]
and
\[G_{k, \nu, \eta}(x) \le 0 \quad \text{on } \{|x| = 1\}\]
provided that $\nu+\eta < \min\{\mu+5, n-2\}$ and $m_{\nu, \eta}$ is chosen to be large enough.
Hence the maximum principle yields that
\[G_{k, \nu, \eta}(x) \le \max_{\{|x| = R\}} (G_{k, \nu, \eta})_+ \quad \text{in } \{1 < |x| < R\}.\]
Taking $R \to \infty$ and applying \eqref{eq-dec-phi-1} again, we deduce that $G_{k, \nu, \eta}(x) \le 0$, or equivalently,
\[\psi_k(x) \le \frac{m_{\nu, \eta}}{|x|^{\nu+\eta}} \quad \text{in } \{|x| \ge 1\}.\]
Similarly, one can show that $-\psi_k$ has the same upper bound in $\{|x| \ge 1\}$.
Therefore, we improve the decay rate of $\psi_k$ as follows:
\[|\psi_k(x)| \le \frac{m_{\nu, \eta}}{|x|^{\nu+\eta}} \quad \text{in } \{|x| \ge 1\}.\]
Resetting $\nu$ as $\nu+\eta$, we repeat the above procedure with the function $F_{k, \mu, \eta}$ to improve the decay rate of $\phi_k$'s.
This information can be used in further improvement of the decay rate of $\psi_k$'s.
We iterate such a process until we reach \eqref{eq-dec-3}.

\medskip
\noindent \textsc{Step 4: Completion of the proof.}
Eq. \eqref{eq-lin} is read as
\begin{equation}\label{eq-lin-1}
\int_{\R^n} (-\Delta U_{p_k})^{{1 \over p_k}-1} \Delta \phi_k\, \Delta \vph = p_k q_k \int_{\R^n} U_{p_k}^{q_k-1} \phi_k \vph \quad \text{for any } \vph \in C^{\infty}_c(\R^n).
\end{equation}
Also, there exists a function $\phi_{\infty} \in \mcd^{2,2}_0(\R^n)$ such that
\[\phi_k \rightharpoonup \phi_{\infty} \quad \text{in }\mcd^{2,2}_0(\R^n)
\quad \text{and} \quad
\phi_k \to \phi_{\infty},\, \Delta \phi_k \to \Delta \phi_{\infty} \quad \text{a.e. in } \R^n\]
as $k \to \infty$, passing to a subsequence.

Invoking Lemmas \ref{lem-cpt-12} and \ref{lem-cpt-13}, Proposition \ref{prop-cpt} and the dominated convergence theorem, we infer
\begin{equation}\label{eq-lin-2}
\int_{\R^n} (-\Delta U_{p_k})^{{1 \over p_k}-1} \Delta \phi_k\, \Delta \vph
= \int_{\R^n} V_{p_k}^{1-p_k} \Delta \phi_k\, \Delta \vph
\to \int_{\R^n} \Delta \phi_{\infty}\, \Delta \vph
\end{equation}
and
\begin{equation}\label{eq-lin-3}
p_k q_k \int_{\R^n} U_{p_k}^{q_k-1} \phi_k \vph \to \(\frac{n+4}{n-4}\) \int_{\R^n} U_1^{8 \over n-4} \phi_{\infty} \vph
\end{equation}
as $k \to \infty$.

Putting \eqref{eq-lin-1}, \eqref{eq-lin-2} and \eqref{eq-lin-3} together, we conclude that $\phi_{\infty}$ is a solution of \eqref{eq-lin-bi}.
By Proposition \ref{prop-lin-bi} (2), it follows that
\[\phi_{\infty} = \sum_{j=1}^n c_j \frac{\pa U_1}{\pa x_j} + c_0 \left[ x \cdot \nabla U_1 + \(\frac{n-4}{2}\)U_1 \right] \quad \text{in } \R^n.\]

We now assert that $\phi_{\infty} \ne 0$.
By \eqref{eq-dec-phi-2}, we can take $\vph = \phi_k$ in \eqref{eq-lin-1}. Applying the mean value theorem, we observe
\[\int_{\R^n} (-\Delta U_{p_k})^{{1 \over p_k}-1} (\Delta \phi_k)^2 =
\|\Delta \phi_k\|_{L^2(\R^n)}^2 + \int_{\R^n} \(V_{p_k}^{1-p_k}-1\) (\Delta \phi_k)^2 = 1 + o(1)\]
where $o(1) \to 0$ as $k \to \infty$. Since
\[p_k q_k \int_{\R^n} U_{p_k}^{q_k-1} \phi_k^2 \to \(\frac{n+4}{n-4}\) \int_{\R^n} U_1^{8 \over n-4} \phi_{\infty}^2 \quad \text{as } k \to \infty,\]
it holds that
\[\int_{\R^n} U_1^{8 \over n-4} \phi_{\infty}^2 = \frac{n-4}{n+4} \ne 0.\]
Consequently, we have that $\phi_{\infty} \ne 0$ and so $\sum_{j=0}^n |c_j| \ne 0$.
However, \eqref{eq-lin-0} and Corollary \ref{cor-cpt-12} imply
\[\int_{\R^n} \Delta \phi_{\infty}\, \Delta \(\frac{\pa U_1}{\pa x_j}\)
= \int_{\R^n} \Delta \phi_{\infty}\, \Delta \left[ x \cdot \nabla U_1 + \(\frac{n-4}{2}\)U_1 \right] = 0\]
for $j = 1, \cdots, n$. Hence $c_0 = \cdots = c_n = 0$, a contradiction.
This completes the proof of the theorem.
\end{proof}

Theorem \ref{thm-deg} and the equivalence between system \eqref{eq-LEs} and equation \eqref{eq-LEsc} yield the following non-degeneracy result for \eqref{eq-LEs} with $p$ near $1$.
\begin{cor}\label{cor1}
There exists a small number $\ep_1 \in (0,\ep_0]$ such that if $|p-1| \le \ep_1$ and $(U_p,V_p)$ is the unique positive ground state of \eqref{eq-LEs} with $U_p(0) = 1$,
then the solution space of the linear equation
\begin{equation}\label{eq-lins-2}
\begin{cases}
-\Delta \phi = p V_p^{p-1} \psi &\text{in } \R^n,\\
-\Delta \psi = q U_p^{q-1} \phi &\text{in } \R^n,
\end{cases}
\quad \lim_{|x|\to\infty} (\phi(x),\psi(x)) = (0,0)
\end{equation}
is spanned by
\[\(\frac{\pa U_p}{\pa x_1}, \frac{\pa V_p}{\pa x_1}\),\, \cdots,\, \(\frac{\pa U_p}{\pa x_n}, \frac{\pa V_p}{\pa x_n}\)\]
and
\[\(x \cdot \nabla U_p + \(n-2 - \frac{n}{p+1}\) U_p, x \cdot \nabla V_p + \(n-2 - \frac{n}{q+1}\) V_p\).\]
\end{cor}
\begin{proof}
The relationship between \eqref{eq-lin} and \eqref{eq-lins-2} was already explored in \eqref{eq-psi_k} and \eqref{eq-lins}.
Moreover, with the help of the decay assumption of $(\phi,\psi)$ and elliptic regularity, one can argue as in Step 3 of the proof of Theorem \ref{thm-deg} to verify that $\phi \in \mcd^{2,2}_0(\R^n)$.
Therefore, the necessary condition to apply Theorem \ref{thm-deg} is fulfilled and so
\[\phi = \frac{\pa U_p}{\pa x_1},\, \cdots,\, \frac{\pa U_p}{\pa x_n} \quad \text{and} \quad x \cdot \nabla U_p + \(n-2 - \frac{n}{p+1}\) U_p.\]
Suppose that $\phi = \frac{\pa U_p}{\pa x_j}$ for some $j = 1, \cdots, n$. Differentiating the first equation in \eqref{eq-LEs} with respect to $x_j$ and using \eqref{eq-lins-2}, we find
\[\psi = - \frac{1}{p} V_p^{1-p} \Delta \(\frac{\pa U_p}{\pa x_j}\) = \frac{1}{p} V_p^{1-p} \cdot p V_p^{p-1} \frac{\pa V_p}{\pa x_j} = \frac{\pa V_p}{\pa x_j}.\]
Set
\[V_{p,\delta}(x) = \delta^{2(q+1) \over pq-1} V_p(\delta x) \quad \text{in } \R^n.\]
If $\phi = x \cdot \nabla U_p + (n-2 - \frac{n}{p+1}) U_p$, then \eqref{eq-U_pmu-2} shows
\begin{align*}
\psi &= - \frac{1}{p} V_p^{1-p} \left. \Delta \(\frac{\pa U_{p,\delta}}{\pa \delta}\) \right|_{\delta=1}
= \frac{1}{p} V_p^{1-p} \cdot \left. p V_{p,\delta}^{p-1} \frac{\pa V_{p,\delta}}{\pa \delta} \right|_{\delta=1} \\
&= x \cdot \nabla V_p + \(n-2 - \frac{n}{q+1}\) V_p.
\end{align*}
The proof is done.
\end{proof}


\section{Non-degeneracy of the Lane-Emden system near $p = \frac{n+2}{n-2}$}\label{sec_pn}
The main results of this section are Theorem \ref{thm-deg-so} and Corollary \ref{cor2},
whose proof depends on arguments similar to those used in the previous section.
In this time, we use the following well-known uniqueness and non-degeneracy results about the second-order critical equation
\begin{equation}\label{eq-LE-so}
\begin{cases}
-\Delta u = u^{n+2 \over n-2} &\text{in } \R^n,\\
u > 0 &\text{in } \R^n
\end{cases}
\end{equation}
for $n \ge 3$.
\begin{prop
\textnormal{(1) (uniqueness)} Any smooth solution of \eqref{eq-LE-so} is expressed as
\[w_{\delta,\xi}^*(x) := c_n^* \(\frac{\delta}{\delta^2+|x-\xi|^2}\)^{n-2 \over 2}\]
for some $\delta > 0$, $\xi \in \R^n$ and $c_n^* = [n(n-2)]^{{n-2 \over 4}}$.

\medskip \noindent \textnormal{(2) (non-degeneracy)}
The solution space of the linear equation
\begin{equation}\label{eq-lin-so}
-\Delta \phi = \(\frac{n+2}{n-2}\) u^{4 \over n-2} \phi \quad \text{in } \R^n, \quad \phi \in \mcd^{1,2}_0(\R^n)
\end{equation}
is spanned by
\[\frac{\pa u}{\pa x_1},\, \cdots,\, \frac{\pa u}{\pa x_n} \quad \text{and} \quad x \cdot \nabla u + \(\frac{n-2}{2}\)u.\]
Here, $\mcd^{1,2}_0(\R^n)$ be the completion of $C^{\infty}_c(\R^n)$ with respect to the norm $\|\nabla \cdot\|_{L^2(\R^n)}$.
\end{prop}
\begin{proof}
Result (1) has been proved by Aubin \cite{A}, Talenti \cite{T} and Caffarelli et al. \cite{CGS}.
A proof of (2) can be found in Rey  \cite{R}.
\end{proof}
We will assume that $p$ is {\em slightly smaller} than $\frac{n+2}{n-2}$.
By interchanging the role of $U$ and $V$ and of $p$ and $q$, we can also cover the case that $p$ is {\em slightly bigger} than $\frac{n+2}{n-2}$.
Adapting the arguments in Subsection \ref{subsec-cpt}, we obtain the next two results.
\begin{prop}\label{prop-cpt-so}
Suppose that $n \ge 3$ and $2^* = \frac{n+2}{n-2}$.
Let $\{p_k\}_{k=1}^{\infty}$ be a sequence such that $p_k \in [\frac{n}{n-2}, 2^*]$ for all $k \in \N$ and $p_k \to 2^*$ as $k \to \infty$,
Also, let $\{(U_{p_k}, V_{p_k})\}_{k=1}^{\infty}$ be the sequence of the unique positive ground states of \eqref{eq-LEs} with $p = p_k$ such that $U_{p_k}(0) = 1$.
Then we have that
\[(U_{p_k}, V_{p_k}) \to (U_{2^*}, V_{2^*}) \quad \text{in } \mcd^{1,2}_0(\R^n) \times \mcd^{1,2}_0(\R^n) \quad \text{as } k \to \infty.\]
Here $U_{2^*}$ is the unique positive radial solution of \eqref{eq-LE-so} with $U_{2^*}(0) = 1$ and $V_{2^*} = U_{2^*}$ in $\R^n$.
In other words, $(U_{2^*}, V_{2^*}) = (w_{b_n,0}^*, w_{b_n,0}^*)$ in $\R^n$ where $b_n := c_n^{2 \over n-2}$.
\end{prop}
\begin{proof}
The fact that $V_{2^*} = U_{2^*}$ comes from \cite[Lemma 2.7 and Remark 2.1 (a)]{Sou}.
\end{proof}
\begin{lemma
Given $\ep_2 > 0$ small enough, we assume that $|p_k - \frac{n+2}{n-2}| \le \ep_2$ for all $k \in \N$.
Then one can find a constant $C > 0$ depending only on $n$, $\ep_2$ and $\eta_1$ such that
\[|\nabla^l U_{p_k}(x)| + |\nabla^l V_{p_k}(x)| \le \frac{C}{1+|x|^{n-2+l}}\]
for all $x \in \R^n$, $k \in \N$ and $l = 1, 2, 3$.
\end{lemma}

By employing the above results and slightly modifying the proof of Theorem \ref{thm-deg}, one can deduce the following theorem.
\begin{thm}\label{thm-deg-so}
There exists a small number $\ep_3 \in (0,\ep_2]$ such that if $|p-\frac{n+2}{n-2}| \le \ep_3$ and $U_p$ is the unique positive ground state of \eqref{eq-LE-so} with $U_p(0) = 1$,
then the solution space of the linear equation
\[(-\Delta) \((-\Delta U_p)^{{1 \over p}-1} (-\Delta \phi) \) = pq U_p^{q-1} \phi \quad \text{in } \R^n, \quad \phi \in \mcd^{2,2}_0(\R^n)\]
is spanned by
\[\frac{\pa U_p}{\pa x_1},\, \cdots,\, \frac{\pa U_p}{\pa x_n} \quad \text{and} \quad x \cdot \nabla U_p + \(n-2 - \frac{n}{p+1}\) U_p.\]
\end{thm}
\begin{proof}
The proof goes along the same way to the proof of Theorem \ref{thm-deg} except Step 4.

In order to facilitate Step 4, we have to prove that $\phi_{\infty}$ is a solution of \eqref{eq-lin-so}.
Taking $k \to \infty$ in \eqref{eq-lins} and using $U_{2^*} = V_{2^*} = w_{b_n,0}^*$ (which was confirmed in Proposition \ref{prop-cpt-so}), we obtain
\[\int_{\R^n} \nabla \phi_{\infty} \cdot \nabla \vph = \(\frac{n+2}{n-2}\) \int_{\R^n} U_{2^*}^{4 \over n-2} \psi_{\infty} \vph\]
and
\[\int_{\R^n} \nabla \psi_{\infty} \cdot \nabla \vph = \(\frac{n+2}{n-2}\) \int_{\R^n} U_{2^*}^{4 \over n-2} \phi_{\infty} \vph.\]
We subtract the second equation from the first equation, and then put $\vph = \phi_{\infty} - \psi_{\infty}$. Then we get
\[0 \le \int_{\R^n} |\nabla (\phi_{\infty} - \psi_{\infty})|^2 = - \(\frac{n+2}{n-2}\) \int_{\R^n} U_{2^*}^{4 \over n-2} (\phi_{\infty} - \psi_{\infty})^2 \le 0.\]
Therefore, $\phi_{\infty} = \psi_{\infty}$ solves \eqref{eq-lin-so}. The rest of the proof remains the same.
\end{proof}

Arguing as in the proof of Corollary \ref{cor1}, we derive the following result from the previous theorem.
\begin{cor}\label{cor2}
There exists a small number $\ep_3 \in (0,\ep_2]$ such that if $|p-{n+2\over n-2}| \le \ep_3$  and $(U_p,V_p)$ is the unique positive ground state of \eqref{eq-LEs} with $U_p(0) = 1$,
then the solution space of the linear equation
\[\begin{cases}
-\Delta \phi = p V_p^{p-1} \psi &\text{in } \R^n,\\
-\Delta \psi = q U_p^{q-1} \phi &\text{in } \R^n,
\end{cases}
\quad \lim_{|x|\to\infty} (\phi(x),\psi(x)) = (0,0)\]
is spanned by
\[\(\frac{\pa U_p}{\pa x_1}, \frac{\pa V_p}{\pa x_1}\),\, \cdots,\, \(\frac{\pa U_p}{\pa x_n}, \frac{\pa V_p}{\pa x_n}\)\]
and
\[\(x \cdot \nabla U_p + \(n-2 - \frac{n}{p+1}\) U_p, x \cdot \nabla V_p + \(n-2 - \frac{n}{q+1}\) V_p\).\]
\end{cor}

