On coupled Schr\"odinger systems with double critical exponents and indefinite weights

By using variational methods, we study the existence of mountain pass solution to the following doubly critical Schr"{o}dinger system: $$\{\begin{array}{lll}-\mathrm{\Delta }u-{\mu }_1\frac{u}{\mathrm{\mid }x{\mathrm{\mid }}^2}-\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u& =h(x)\alpha \mathrm{\mid }u{\mathrm{\mid }}^{\alpha -2}\mathrm{\mid }v{\mathrm{\mid }}^{\beta }u\mathrm{i}\mathrm{n}{\mathbb{R}}^N,-\mathrm{\Delta }v-{\mu }_2\frac{v}{\mathrm{\mid }x{\mathrm{\mid }}^2}-\mathrm{\mid }v{\mathrm{\mid }}^{2^{\ast }-2}v& =h(x)\beta \mathrm{\mid }u{\mathrm{\mid }}^{\alpha }\mathrm{\mid }v{\mathrm{\mid }}^{\beta -2}v\mathrm{i}\mathrm{n}{\mathbb{R}}^N,\end{array}$$ where $\alpha \ge 2,\beta \ge 2,\alpha +\beta \le 2^{\ast }$;; ${\mu }_1,{\mu }_2\in [0,\frac{(N-2{)}^2}{4})$. The weight function $h(x)$ is allowed to be sign-changing so that the nonlinearities include a large class of indefinite weights. We show that the $PS$ condition is satisfied at higher energy level when $\alpha +\beta =2^{\ast }$ and obtain the existence of mountain pass solution. Besides, a nonexistence result of the ground state is given.