In this paper, we investigate the multiplicity of solutions for a p-Kirchhoff system driven by a nonlocal integro-differential operator with zero Dirichlet boundary data. As a special case, we consider the following fractional p-Kirchhoff system {(∑i=1k[ui]s,pp)θ-1(-Δ)psuj(x)=λj|uj|q-2uj+∑i≠jβij|ui|m|uj|m-2uj in Ω, uj=0 in ℝN\Ω, where (∫∫ℝ2N|uj(x) uj(y)|p/|x-y|N+psdxdy)1/p, j = 1,2⋯, k, k≥2, θ ≤ 1, ω is an open bounded subset of ℝN with Lipschitz boundary ∂ω, N > ps with s ∈(0,1), (-Δ)ps is the fractional p-Laplacian, λj > 0 and βij = βji for i ≠ j, j = 1,2, ⋯,k. When 1 < q < θp < 2m < Ps∗ and βij > 0 for all 1 ≤ i < j ≤ k, two distinct solutions are obtained by using the Nehari manifold method. When 1 < θp < 2m ≤ q < ps∗ and βij ∈ ℝ for all 1 ≤ i < j ≤ k or 1 < θp < q < 2m < ps∗ and βij > 0 for all 1 ≤ i < j ≤ k, the existence of infinitely many solutions is obtained by applying the symmetric mountain pass theorem. To our best knowledge, our results for the above system are new in the study of Kirchhoff problems.
