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

TL;DR

This paper proves the existence of mountain pass solutions for a class of doubly critical Schr"odinger systems with indefinite weights using variational methods, and also establishes nonexistence of ground states under certain conditions.

Contribution

It introduces new existence results for doubly critical Schr"odinger systems with sign-changing weights, extending variational methods to this complex setting.

Findings

Existence of mountain pass solutions at higher energy levels when $\,\alpha+\beta=2^*$.

The Palais-Smale condition is satisfied at these higher energy levels.

A nonexistence result for ground states is established.

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.