Multiple positive solutions for a fractional Kirchhoff type equation with logarithmic and singular nonlinearities

Abstract

In this paper, we study the following fractional Kirchhoff type equation { ( a + b ∫ R N ∫ R N | u ( x ) − u ( y ) | p | x − y | N + p s d x d y ) ( − Δ ) p s u = | u | q − 2 u ln ⁡ | u | 2 + λ u γ , i n Ω , u > 0 , i n Ω , u = 0 , i n R N ∖ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, 0 < s < 1 < p , 0 < γ < 1 , a > 0 , b ≥ 0 , N > p s , 2 p < q < q + 2 < p s ∗ , p s ∗ = N p N − p s is the fractional critical exponent, λ > 0 is a real parameter. By using the critical point theory for nonsmooth functionals and analytic techniques, the existence and multiplicity of positive solutions are obtained.