Bifurcation problems for double critical Schrödinger–Poisson systems

This paper focuses on the following double critical Schrödinger–Poisson system: [Formula: see text] where [Formula: see text] and [Formula: see text] is a nonnegative weight function. Moreover, by the celebrated global bifurcation theorem due to Rabinowitz [Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971) 487–513], we prove existence of (weak) solutions for the above system. To the best of our knowledge, it seems to be the first time contribution to considering the bifurcation and existence of solutions for Schrödinger–Poisson systems with doubly critical growth by the global bifurcation theorem. To some extent, our main theorems complement some results established in [Y. Meng and X. He, Normalized solutions for the Schrödinger–Poisson system with doubly critical growth, Topol. Methods Nonlinear Anal. 62 (2023) 509–534; Y. Meng and X. He, Multiplicity of normalized solutions for the fractional Schrödinger–Poisson system with doubly critical growth, Acta Math. Sci. Ser. B Engl. Ed. 44 (2024) 997–1019; P. Pucci, L. Wang and B. Zhang, Bifurcation and existence for Schrödinger–Poisson systems with doubly critical nonlinearities, Z. Angew. Math. Phys. 75 (2024) 170; P. Pucci, L. Wang and B. Zhang, Bifurcation and regularity analysis of the Schrödinger–Poisson equation, Nonlinearity 37 (2024) 035011; L. Wang and Y. Xing, Bifurcation analysis of fractional Kirchhoff–Schrödinger–Poisson systems in [Formula: see text], Electron. J. Qual. Theory Differ. Equ. 3 (2024) 1–17].