On coupled Schrödinger 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 Schrödinger system: $$\{\begin{array}{llc}-\Delta u-{\mu }_1\frac{u}{|x{|}^2}-|u{|}^{2^{\ast }-2}u& =h(x)\alpha |u{|}^{\alpha -2}|v{|}^{\beta }u\mathrm{in}{\mathbb{R}}^{\mathrm{N}},-\mathrm{\Delta v}-{\mu }_2\frac{\mathrm{v}}{|x{|}^2}-|v{|}^{2^{\ast }-2}\mathrm{v}& =h(x)\beta |u{|}^{\alpha }|v{|}^{\beta -2}v\mathrm{in}{\mathbb{R}}^{\mathrm{N}},\end{array}$$ 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.

Cited by in corpus (1)

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