A Uniqueness Result for Strong Singular Kirchhoff-Type Fractional Laplacian Problems - Wang, Li; Cheng, Kun; Zhang, Binlin (Applied mathematics & optimization, 2021)
In this paper, we study the following Kirchhoff-type fractional Laplacian problem with strong singularity: ( a + b ‖ u ‖ 2 ) ( - Δ ) s u = f ( x ) u - γ - k ( x ) u q in Ω , u > 0 in Ω , u = 0 in R 3 \ Ω , where ( - Δ ) s is the fractional Laplace operator, a , b ≥ 0 , a + b > 0 , Ω is a bounded smooth domain of R 3 , k ∈ L ∞ ( Ω ) is a non-negative function, q ∈ ( 0 , 1 ) , γ > 1 and f ∈ L 1 ( Ω ) is positive almost everywhere in Ω . Using variational method and Nehari method, we obtain a uniqueness result. A novelty is that the Kirchhoff coefficient may vanish at zero.