In this paper we study the existence, multiplicity, and regularity of positive weak solutions for the following Kirchhoff–Choquard problem: M()∬ℝ2Nfalse|ufalse(xfalse)−ufalse(yfalse)false|2false|x−yfalse|N+2s0.1emdxdyfalse(−normalΔfalse)su=λuγ+()∫normalΩfalse|ufalse(yfalse)false|2μ,s∗false|x−yfalse|μ0.1emdyfalse|ufalse|2μ,s∗−2u0.51emin0.51emnormalΩ,3emu=00.51emin0.51emℝN\normalΩ, where Ω is open bounded domain of ℝN with C2 boundary, N > 2s and s ∈ (0, 1). M models Kirchhoff‐type coefficient in particular, the degenerate case where Kirchhoff coefficient M is zero at zero. (− Δ)s is fractional Laplace operator, λ > 0 is a real parameter, γ ∈ (0, 1) and 2μ,s∗=2N−μN−2s is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We prove that each positive weak solution is bounded and satisfy Hölder regularity of order s. Furthermore, using the variational methods and truncation arguments, we prove the existence of two positive solutions.

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