Multiplicity and Concentration of Solutions for a Fractional Magnetic Kirchhoff Equation with Competing Potentials

• Published in: Annales Henri Poincaré 2024-07, Vol.25 (7), p.3499-3528
• Main Authors: Deng, Shengbing, Luo, Wenshan
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Dynamical Systems and Ergodic Theory; Elementary Particles; Mathematical and Computational Physics; Mathematical Methods in Physics; Original Paper; Physics; Physics and Astronomy; Quantum Field Theory; Quantum Physics; Relativity Theory; Theoretical; Variational methods
• ISSN: 1424-0637; 1424-0661
• DOI: 10.1007/s00023-023-01372-4

This paper is concerned with the following fractional electromagnetic Kirchhoff equation with competing potentials and critical nonlinearity a ε 2 s + b ε 4 s - 3 [ u ] A / ε 2 ( - Δ ) A / ε s u + V ( x ) u = f ( | u | 2 ) u + K ( x ) | u | 2 s ∗ - 2 u in R 3 , where ε > 0 is a small parameter, A ∈ C 0 , α ( R 3 , R 3 ) with exponent α ∈ ( 0 , 1 ] , ( - Δ ) A / ε s is the fractional magnetic operator with s ∈ ( 3 4 , 1 ) , 2 s ∗ = 6 3 - 2 s is the fractional critical exponent, and a , b > 0 are fixed constants. Assuming that V , K and f satisfy some suitable conditions, we establish the multiplicity and concentration of solutions by variational methods and Ljusternik–Schnirelmann theory.