Nonexistence for Lane-Emden system involving Hardy potentials with singularities on the boundary

Our purpose of this article is to study nonexistence of positive super solutions for Lane-Emden system involving inverse-square potentials 0.1 \(\begin{aligned} -\Delta u+\frac{\mu _1}{|x|^2} u= v^p \ \ \textrm{in}\ \, \Omega ,\qquad -\Delta v+\frac{\mu _2}{|x|^2} v= u^q \ \ \textrm{in}\ \, \Omega , \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+\frac{{\mu }_1}{{|x|}^2}u=v^p\text{in}\mathrm{\Omega },-\mathrm{\Delta }v+\frac{{\mu }_2}{{|x|}^2}v=u^q\text{in}\mathrm{\Omega },\end{array}$$ where \(p,q>0\) $p,q>0$ , \(\mu _1,\mu _2\ge -N^2/4\) ${\mu }_1,{\mu }_2\ge -N^2/4$ , \(\Omega \) $\mathrm{\Omega }$ is a bounded smooth domain in \(\mathbb {R}^N\) ${\mathbb{R}}^N$ with \(N\ge 3\) $N\ge 3$ such that \(0\in \partial \Omega \) $0\in \partial \mathrm{\Omega }$ and \(B^+_2(0):=\{x=(x',x_N)\in \mathbb {R}^{N-1}\times \mathbb {R}: x_N>0,\, |x|<2\}\subset \Omega \) $B_2^{+}(0):=\{x=(x^{\prime },x_N)\in {\mathbb{R}}^{N-1}\times \mathbb{R}:x_N>0,|x|<2\}\subset \mathrm{\Omega }$ . Sharp critical curves of (q, p) are derived for nonexistence of positive super solutions to system (0.1) in the case that \(-N^2/4\le \mu _1,\mu _2<1-N\) $-N^2/4\le {\mu }_1,{\mu }_2<1-N$ and \(-N^2/4\le \mu _1<1-N\le \mu _2\) $-N^2/4\le {\mu }_1<1-N\le {\mu }_2$ . Our method is to iterate an initial singularities at the origin to improve the blowing-up rate until the nonlinearities are not admissible in some weighted \(L^1\) $L^1$ space.