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

• Published in: Bulletin of mathematical sciences Vol. 14; no. 2
• Main Authors: Dou, Xilin, He, Xiaoming, Rădulescu, Vicenţiu D.
• Format: Journal Article
• Language: English
• Published: World Scientific Publishing Company 01.08.2024; World Scientific Publishing
• Subjects: critical nonlocal term; fractional Schrödinger–Poisson system; Ljusternik–Schnirelmann category; Positive solutions; variational method; critical nonlocal term; Positive solutions; fractional Schrödinger–Poisson system; variational method; Ljusternik–Schnirelmann category
• ISSN: 1664-3607; 1664-3615
• DOI: 10.1142/S1664360723500121

This paper deals with the following fractional Schrödinger–Poisson system: ( − Δ ) s u + u − K ( x ) ϕ | u | 2 s ∗ − 3 u = f λ ( x ) | u | q − 2 u , x ∈ ℝ 3 , ( − Δ ) s ϕ = K ( x ) | u | 2 s ∗ − 1 , x ∈ ℝ 3 with multiple competing potentials and a critical nonlocal term, where s ∈ ( 0 , 1 ) , q ∈ ( 1 , 2 ) or q ∈ ( 4 , 2 s ∗ ) , and 2 s ∗ = 6 3 − 2 s is the fractional critical exponent. By combining the Nehari manifold analysis and the Ljusternik–Schnirelmann category theory, we establish how the coefficient K 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 K .