Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian

The purpose of this paper is to investigate the existence of weak solutions for a Kirchhoff type problem driven by a non-local integro-differential operator of elliptic type with homogeneous Dirichlet boundary conditions as follows:{M(∫R2N|u(x)−u(y)|pK(x−y)dxdy)LKpu=f(x,u)inΩ,u=0inRN∖Ω, where LKp is a non-local operator with singular kernel K, Ω   is an open bounded subset of RNRN with Lipshcitz boundary ∂Ω, M is a continuous function and f is a Carathéodory function satisfying the Ambrosetti–Rabinowitz type condition. We discuss the above-mentioned problem in two cases: when f satisfies sublinear growth condition, the existence of nontrivial weak solutions is obtained by applying the direct method in variational methods; when f satisfies suplinear growth condition, the existence of two nontrivial weak solutions is obtained by using the Mountain Pass Theorem.