Infinitely many solutions for the stationary fractional p -Kirchhoff problems in RN

In the present paper, we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Kirchhoff equation M(∬R2N|u(x)-u(y)|p|x-y|N+psdxdy+∫RNV(x)|u|pdx)×((-Δ)psu+V(x)|u|p-2u)=f(x,u)inRN,where (-Δ)ps is the fractional p-Laplacian operator, 0 < s< 1 < p< ∞ with sp< N, M:R0+→R0+ is a nonnegative, continuous and increasing Kirchhoff function, the nonlinearity f: RN× R→ R is a Carathéodory function that obeys some conditions which will be stated later and V∈ C(RN, R+) is a non-negative potential function. We first establish the Bartsch–Pankov–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the variational method, (S+) mapping theory and Krasnoselskii’s genus theory.