Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity

In this paper we study the existence, multiplicity, and regularity of positive weak solutions for following Kirchhoff–Choquard problem: M ∬ ℝ 2 N | u ( x ) − y + s d Δ = λ γ ∫ Ω μ , ∗ in 0 \ where is open bounded domain with C2 boundary, > 2s ∈ (0, 1). models Kirchhoff-type coefficient particular, degenerate case Kirchhoff zero at zero. (− Δ)s fractional Laplace operator, a real parameter, 1) critical exponent sense Hardy–Littlewood–Sobolev inequality. We prove that each solution satisfy Hölder order s. Furthermore, using variational methods truncation arguments, existence two solutions.