A Uniqueness Result for Strong Singular Kirchhoff-Type Fractional Laplacian Problems

• Published in: Applied mathematics & optimization Vol. 83; no. 3; pp. 1859 - 1875
• Main Authors: Wang, Li, Cheng, Kun, Zhang, Binlin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2021; Springer Nature B.V
• Subjects: Calculus of Variations and Optimal Control; Optimization; Control; Economic models; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Simulation; Systems Theory; Theoretical; Uniqueness; 35B38; Kirchhoff-type problem; 35J50; Uniqueness of solution; 35J60; 35B33; Fractional Laplacian; Strong singularity
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-019-09612-y

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.