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: [GRAPHIC] where L-k(p) is a non-local operator with singular kernel K, Omega is an open bounded subset of R-N with Lipshcitz boundary partial derivative Omega, M is a continuous function and f is a Caxatheodory 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. 2014 Elsevier Inc. All rights reserved.