Multiplicity results for the non-homogeneous fractionalp-Kirchhoff equations with concave–convex nonlinearities

In this paper, we are interested in the multiplicity of solutions for a non-homogeneousp -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 ∬ R 2 N | u ( x ) − u ( y ) | p | x − y | N + s p   d x   d y θ − 1 ( − Δ ) p s u = λ ω 1 ( x ) | u | q − 2 u + ω 2 ( x ) | u | r − 2 u + h ( x ) in   R N , where( − Δ ) p s is the fractionalp -Laplace operator,a +b >0 witha , b ∈ R 0 + , λ>0 is a real parameter,0 < s < 1 < p < ∞ withsp <N , 1<q <p ≤θp <r <Np /(N −sp ),ω 1 ,ω 2 ,h are functions which may change sign inR 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 thata 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.