Multiplicity of solutions for a class of fractional p-Kirchhoff system with sign-changing weight functions

• Published in: Boundary value problems 2018-05, Vol.2018 (1), p.1-18, Article 78
• Main Authors: Wei, Yunfeng, Chen, Caisheng, Yang, Hongwei, Song, Hongxue
• Format: Article
• Language: English
• Subjects: Analysis; Approximations and Expansions; Difference and Functional Equations; Fractional p-Kirchhoff system; Mathematics; Mathematics and Statistics; Mountain pass theorem; Multiplicity; Nehari manifold; Ordinary Differential Equations; Partial Differential Equations; Sign-changing weight functions
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-018-0998-7

In this paper, we investigate the fractional p -Kirchhoff -type system: { M ( ∫ R 2 N | u ( x ) − u ( y ) | p | x − y | N + p s d x d y ) ( − Δ ) p s u = μ g ( x ) | u | β − 2 u + a a + b h ( x ) | u | a − 2 u | v | b , in  Ω , M ( ∫ R 2 N | v ( x ) − v ( y ) | p | x − y | N + p s d x d y ) ( − Δ ) p s v = σ f ( x ) | v | β − 2 v + b a + b h ( x ) | v | b − 2 v | u | a , in  Ω , u = v = 0 , in  R N ∖ Ω , where Ω ⊂ R N is a smooth bounded domain, ( − Δ ) p s is the fractional p -Laplacian operator with 0 < s < 1 < p and p s < N . a > 1 , b > 1 satisfy 2 < a + b < p s ∗ . 1 < β < p s ∗ , p s ∗ = N p N − p s is the fractional critical exponent. μ , σ are two real parameters. M ( t ) = k + λ t τ , k > 0 , λ , τ ≥ 0 , τ = 0 if and only if λ = 0 . The weight functions g , f , h change sign in Ω and satisfy suitable conditions. By using the Nehari manifold method, it is proved that the system has at least two solutions provided that 2 < a + b < p ≤ p ( τ + 1 ) < β < p s ∗ and ( μ , σ ) belongs to a certain subset of R 2 . Also, by using the mountain pass theorem, we prove that there exist λ 1 ≥ λ 0 such that the system admits at least a nontrivial solution for λ ∈ ( 0 , λ 0 ) and no nontrivial solution for λ > λ 1 under the assumptions μ = σ = 0 and p < a + b < min { p ( τ + 1 ) , p s ∗ } .