Multiplicity of positive solutions for the fractional Schrödinger-Poisson system with critical nonlocal term (Q6611249)

This article discusses the fractional Schrödinger-Poisson coupled equations \((-\Delta)^s u+u-K(x)\phi |u|^{2_s^*-3}u=f_\lambda(x) |u|^{q-2}u\) in \(\mathbb{R}^3\), \((-\Delta)^s \phi=K(x)\phi |u|^{2_s^*-1}\) in \(\mathbb{R}^3\) where \(0<s<1\), \(q\in (1,2)\cup (4, 2_s^*)\). The two main results of the article establish the existence of multiple positive solutions provided some growth conditions on \(f\) and \(K\) are imposed. The approach is variational and relies on the study of the Nehari manifold and the Ljusternik-Schnirelmann category theory.