Summary: We shall show an existence result of a positive solution for a Kirchhoff problem type in a bounded domain of \(\mathbb R^N\), that is, for the problem \[ -M\left(\int_\Omega|\nabla u|^2dx\right)\Delta u=\lambda f(x,u)+|u|^{2^*-2}u\;\text{in}\;\Omega,\;u=0\;\text{on}\;\partial\Omega. \] We shall study the asymptotic behavior of this solution when \(\lambda\) converges to infinity. Our approach is based on the variational method, an appropriated truncated argument, and a priori estimates to obtain the solution.