Supercritical fractional Kirchhoff type problems

In this paper we deal with the following fractional Kirchhoff problem [M(Rn×Rn|u(x)-u(y)|p|x-y|n+spdxdyp-1(-Δ)psu=f(x,u)+λ|u|r-2u in ω,u=0 in Rnω. Here ω ω ⊂ n is a smooth bounded open set with continuous boundary δω, p &insin; (1, +∞), s &insin; (0, 1), n > sp, (-Δ)ps is the fractional p-Laplacian, M is a Kirchhoff function, f is a continuous function with subcritical growth, λ is a nonnegative parameter and r > p*s where ps*-=np/n-sp is the fractional critical Sobolev exponent. By combining variational techniques and a truncation argument, we prove two existence results for this problem, provided that the parameter λ is sufficiently small.