Multiplicity of positive solutions for the fractional Schrödinger–Poisson system with critical nonlocal term

Abstract

This paper deals with the following fractional Schrödinger–Poisson system: (−Δ)su + u − K(x)ϕ|u|2s∗−3u = f λ(x)|u|q−2u,x ∈ ℝ3,(−Δ)sϕ = K(x)|u|2s∗−1, x ∈ ℝ3 with multiple competing potentials and a critical nonlocal term, where [Formula: see text], [Formula: see text] or [Formula: see text], and [Formula: see text] is the fractional critical exponent. By combining the Nehari manifold analysis and the Ljusternik–Schnirelmann category theory, we establish how the coefficient [Formula: see text] of the nonlocal critical nonlinearity affects the number of positive solutions. We propose a new relation between the number of positive solutions and the category of the global maximal set of [Formula: see text].

Keywords

• Positive solutions
• critical nonlocal term
• fractional Schrödinger–Poisson system
• Ljusternik–Schnirelmann category
• variational method