AbstractIn this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: urn:x-wiley:01704214:media:mma3000:mma3000-math-0001 where a, b > 0 are constants. Under certain assumptions on the sign‐changing function f(x,u), we prove the existence of positive solutions by variational methods.Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem urn:x-wiley:01704214:media:mma3000:mma3000-math-0002 where ϵ > 0 is a parameter, V (x) is a positive continuous potential, and with 4 < p < 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley & Sons, Ltd.