On coupled Schrödinger 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 Schrödinger system: $\begin{cases} -Δu-μ_1\frac{u}{|x|^2}-|u|^{2^{*}-2}u &=h(x)α|u|^{α-2}|v|^βu\quad \rm{in}\; \R^N, -Δv-μ_2\frac{v}{|x|^2}-|v|^{2^{*}-2}v &= h(x)β|u|^α|v|^{β-2}v\quad \rm{in}\; \R^N, \end{cases}$ where $α\geq 2, β\geq 2, α+β\leq 2^*$;\; $μ_1, μ_2\in [0, \frac{(N-2)^2}{4})$. 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 $α+β=2^*$ and obtain the existence of mountain pass solution. Besides, a nonexistence result of the ground state is given.

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