Non-degeneracy of solution for critical Lane–Emden systems with linear perturbation

In this paper, we consider the following elliptic system \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |v|^{p-1}v +\epsilon (\alpha u + \beta _1 v), &{}\quad {\hbox {in}}\; \Omega , \\ -\Delta v = |u|^{q-1}u+\epsilon (\beta _2 u +\alpha v), &{}\quad {\hbox {in}}\;\Omega , \\ u=v=0,&{}\quad {\hbox {on}}\; \partial \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u={|v|}^{p-1}v+ϵ(\alpha u+{\beta }_{1}v),\hfill & \text{in}\mathrm{\Omega },\hfill \\ -\mathrm{\Delta }v={|u|}^{q-1}u+ϵ({\beta }_{2}u+\alpha v),\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=v=0,\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a smooth bounded domain in \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ , \(N\ge 3\) $N\ge 3$ , \(\epsilon \) $ϵ$ is a small parameter, \(\alpha \) $\alpha$ , \( \beta _1\) ${\beta }_1$ and \( \beta _2\) ${\beta }_2$ are real numbers, (p, q) is a pair of positive numbers lying on the critical hyperbola \(\begin{aligned} \begin{aligned} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}. \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.\end{array}\end{array}$$ We first revisited the blowing-up solutions constructed in Kim and Pistoia (J Funct Anal 281(2):58, 2021) and then we proved its non-degeneracy. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems involving critical Halmitonian system and the construction of new solutions.