Bifurcation and existence for Schrödinger–Poisson systems with doubly critical nonlinearities

This paper is concerned with the bifurcation properties of the standing wave solutions for the Schrödinger–Poisson system with doubly critical case The study of system ( \({\mathcal {P}}\) $\mathcal{P}$ ) is motivated by its important applications in many physical models, such as the quantum mechanical systems under external influences. Here, \(3\le N\le 6\) $3\le N\le 6$ , \(0<\alpha <N\) $0<\alpha <N$ , \(\lambda \in {\mathbb {R}}\) $\lambda \in \mathbb{R}$ , g is a nonnegative weight function, and \(2_\alpha ^\sharp \) $2_{\alpha }^{♯}$ and \(2_\alpha ^*\) $2_{\alpha }^{\ast }$ are the lower and upper Hardy–Littlewood–Sobolev critical exponents, respectively. Moreover, when \(N=6\) $N=6$ and \(0<\alpha <2\) $0<\alpha <2$ existence of the (weak) solutions of the system under consideration is also proved via the global bifurcation theorem due to Rabinowitz.