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

Abstract

By using variational methods, we study the existence of mountain pass solution to the following doubly critical Schr\"{o}dinger system: $$ \begin{cases} -\Delta u-\mu_1\frac{u}{|x|^2}-|u|^{2^{*}-2}u &=h(x)\alpha|u|^{\alpha-2}|v|^\beta u\quad \rm{in}\; \R^N, -\Delta v-\mu_2\frac{v}{|x|^2}-|v|^{2^{*}-2}v &= h(x)\beta |u|^{\alpha}|v|^{\beta-2}v\quad \rm{in}\; \R^N, \end{cases} $$ where $\alpha\geq 2, \beta\geq 2, \alpha+\beta\leq 2^*$;\; $ \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 $\alpha+\beta=2^*$ and obtain the existence of mountain pass solution. Besides, a nonexistence result of the ground state is given.