Multiplicity of solutions of Kirchhoff-type fractional Laplacian problems with critical and singular nonlinearities

Abstract

In this ariticle, the following Kirchhoff-type fractional Laplacian problem with singular and critical nonlinearities is studied: { ( a + b ‖ u ‖ 2 μ − 2 ) ( − Δ ) s u = λ l ( x ) u 2 s ∗ − 1 + h ( x ) u − γ , in   Ω , u > 0 , in   Ω , u = 0 , in   R N ∖ Ω , where s ∈ ( 0 , 1 ) , N > 2 s , ( − Δ ) s is the fractional Laplace operator, 2 s ∗ = 2 N / ( N − 2 s ) is the critical Sobolev exponent, Ω ⊂ R N is a smooth bounded domain, l ∈ L ∞ ( Ω ) is a non-negative function and max { l ( x ) , 0 } ≢ 0 , h ∈ L 2 s ∗ 2 s ∗ + γ − 1 ( Ω ) is positive almost everywhere in Ω , γ ∈ ( 0 , 1 ) , a > 0 , b > 0 , μ ∈ [ 1 , 2 s ∗ / 2 ) and parameter λ is a positive constant. Here we utilize a special method to recover the lack of compactness due to the appearance of the critical exponent. By imposing appropriate constraint on λ , we obtain two positive solutions to the above problem based on the Ekeland variational principle and Nehari manifold technique.