Critical Schrödinger systems in $\mathbb R^N$ with indefinite weight and Hardy potential

Abstract

By using variational methods, we study the following doubly critical elliptic system: \[ \begin{cases} -\Delta u-\mu_1\frac{u}{|x|^2}-|u|^{2^{*}-2}u= h(x)\alpha|u|^{\alpha-2}|v|^\beta u\quad & \rm{in}\; \mathbb R^N,\\ -\Delta v-\mu_2\frac{u}{|x|^2}-|v|^{2^{*}-2}v= h(x)\beta |u|^{\alpha}|v|^{\beta-2}v\quad & \rm{in}\; \mathbb R^N, \\ \end{cases} \] where $\alpha >1, \beta >1$ satisfying $\alpha+\beta \leq 2^{*}:=\frac{2N}{N-2}\; (N\geq 3)$, and either $\mu_1=\mu_2=0$ or $\mu_1,\mu_2\in (0,\frac{(N-2)^2}{4})$. The function $h(x)$ is allowed to be sign-changing and satisfies weaker conditions which permit the nonlinearities to include a large class of indefinite weights. We obtain the existence of the ground state solutions. The indefiniteness of $h$ makes the problem intrinsically complicated. However, our assumptions on $h$ are almost optimal.