Existence of solutions for a bi-nonlocal fractional p -Kirchhoff type problem

In this paper, we are concerned with the existence of nonnegative solutions for a p -Kirchhoff type problem driven by a non-local integro-differential operator with homogeneous Dirichlet boundary data. As a particular case, we study the following problem M ( x , u s , p p ) ( - Δ ) p s u = f ( x , u , u s , p p ) in ? , u = 0 in R N ? ? , u s , p p = ? R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y , where ( - Δ ) p s is a fractional p -Laplace operator, ? is an open bounded subset of R N with Lipschitz boundary, M : ? × R 0 + ? R + is a continuous function and f : ? × R × R 0 + ? R is a continuous function satisfying the Ambrosetti-Rabinowitz type condition. The existence of nonnegative solutions is obtained by using the Mountain Pass Theorem and an iterative scheme. The main feature of this paper lies in the fact that the Kirchhoff function M depends on x ? ? and the nonlinearity f depends on the energy of solutions.