Kirchhoff-Type Fractional Laplacian Problems with Critical and Singular Nonlinearities

In the present paper, we are interested in the critical Kirchhoff-type fractional Laplacian problem involving strong singularity as shown below: a + b ‖ u ‖ 2 m - 2 - Δ s u = f ( x ) u - γ - h ( x ) u 2 s ∗ - 1 , in Ω , u > 0 , in Ω , u = 0 , in R N \ Ω , where Ω ⊂ R N is a bounded smooth domain, - Δ s is the fractional Laplace operator, s ∈ ( 0 , 1 ) , N > 2 s , a , b ⩾ 0 , a + b > 0 , m ⩾ 1 , γ > 1 , h ∈ L ∞ ( Ω ) is a nonnegative function, 2 s ∗ = 2 N / ( N - 2 s ) is the critical Sobolev exponent, and f ∈ L 1 ( Ω ) is positive almost everywhere in Ω . By the Nehari method and Ekeland’s variational principle, we overcome the shortage of compactness due to the critical nonlinearity and establish the existence and uniqueness of weak solution for the above problem. The novelties of our paper are that the Kirchhoff term M may vanish at zero and the considered fractional elliptic problem involves strong singularity and the critical exponent.