Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities

In this paper, we are interested in the multiplicity of solutions for a non-homogeneous p-Kirchhoff-type problem driven by a non-local integro-differential operator. As a particular case, we deal with the following elliptic problem of Kirchhoff type with convex-concave nonlinearities: [a + b (integral integral(R2N) vertical bar u(x) - u(y)vertical bar(p)/vertical bar x - y vertical bar(N+sp) dx dy)(theta-1)](-Delta)(p)(s)u =lambda omega(1)(x)vertical bar u vertical bar(q-2)u + omega(2)(x)vertical bar u vertical bar(r-2)u + h(x) in R-N, where (-Delta)(p)(s) is the fractional p-Laplace operator, a + b > 0 with a, b is an element of R-0(+), lambda > 0 is a real parameter, 0 < s < 1 < p < infinity with sp < N, 1 < q < p <= theta p < r < Np/(N - sp), omega(1), omega(2), h are functions which may change sign in R-N. Under some suitable conditions, we obtain the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekeland's variational principle. A distinguished feature of this paper is that a may be zero, which means that the above-mentioned problem is degenerate. To the best of our knowledge, our results are new even in the Laplacian case.