Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)In this paper, we deal with a Kirchhoff type problem driven by a nonlocal fractional integrodifferential operator L-K, that is, -M(parallel to u parallel to(2))LKu = lambda f(x, u) [integral(Omega) F(x, u(x))d(x)](r) + vertical bar u vertical bar(2*-2)u in Omega, u = 0 in R-n \ Omega, where Omega is an open bounded subset of R-n, M and f are continuous functions, parallel to center dot parallel to is a functional norm, F(x, u(x)) = integral(0) (u(x)) f(x, tau)d tau, 2* is a fractional Sobolev exponent, lambda and r are real parameters. For this problem, we prove the existence of infinitely many solutions, through a suitable truncation argument and exploiting the genus theory introduced by Krasnoselskii.29 513530Coordenagdo de Aperfeiconamento de pessoal de nivel superior through fellowship PNPD [CAPES 33003017003P5]Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)