A remark on fractional p-Kirchhoff problems involving multiple zeros

In this paper, we study the multiplicity of solutions for a p-Kirchhoff type problem driven by a nonlocal integro-differential operator. As a particular case, we consider the following problem: M (integral integral(R2N) vertical bar u(x) - u(y)vertical bar(p)/vertical bar x - y vertical bar(N+sp) dx dy + integral(RN) V(x)vertical bar u vertical bar(p) dx) ((-Delta)(s)(p)u + V(x)vertical bar u vertical bar(p-2)u) = f(lambda)(x, u) in R-N, where (-Delta)(s)(p) is the fractional p-Laplacian, 0 < s < 1 < p < infinity with sp < N, p*(s) = Np/(N - sp), M : [0, infinity) -> [0, infinity) is a continuous function vanishing in many different points, V : R-N -> (0, infinity) is a continuous function, and f(lambda): R-N x R -> R is a Caratheodory function for each lambda > 0. Under some suitable assumptions, we obtain the multiplicity of solutions for the above problem by applying the mountain pass theorem. Moreover, the asymptotic behavior of solutions is also investigated. A distinguished feature of this paper is that the Kirchhoff function M has multiple zeros.