Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian

Abstract

The aim of this paper is to establish the multiplicity of weak solutions for a Kirchhoff-type problem driven by a fractional p-Laplacian operator with homogeneous Dirichlet boundary conditions:{M(∬R2N⁡|u(x)−u( y)|p|x−y|N+psdxdy) (−Δ)psu(x)=f(x,u)in Ωu=0in RN\\Ω, ?>where is an open bounded subset of with Lipshcitz boundary , is the fractional p-Laplacian operator with 0 < s < 1 < p < N such that sp < N, M is a continuous function and f is a Carathéodory function satisfying the Ambrosetti–Rabinowitz-type condition. When f satisfies the suplinear growth condition, we obtain the existence of a sequence of nontrivial solutions by using the symmetric mountain pass theorem; when f satisfies the sublinear growth condition, we obtain infinitely many pairs of nontrivial solutions by applying the Krasnoselskii genus theory. Our results cover the degenerate case in the fractional setting: the Kirchhoff function M can be zero at zero.