Summary: In this paper, we consider the existence of solutions of the Kirchhoff problem \[ -\left(a + b \int_{\mathbb{R}^N} | \nabla u |^2 d x\right) \Delta u + u = k f(u) + | u |^{2^\ast - 2} u,\quad \text{in } \mathbb{R}^N, \tag{\(\mathcal K\)} \] where \(a, b > 0\), \(k > 0\) and \(N \geq 3\). We transform it into an equivalent system with respect to \((u, \lambda)\), which is easier to solve, \[ \begin{cases} - \Delta u + u = kf ( u ) + | u |^{2^\ast - 2} u , \\ \lambda - a - b \lambda^{\frac{ N - 2}{ 2}} \int_{\mathbb{R}^N} | \nabla u |^2 d x = 0 , \end{cases} \text{ in } \mathbb{R}^N \times \mathbb{R}^+ . \tag{\(\mathcal{S}\)} \] With the equivalence of \((\mathcal{K})\) to \((\mathcal{S})\), we obtain the existence of solutions for \((\mathcal{K})\) by solving \((\mathcal{S})\). Existence of solutions of \((\mathcal{S})\) is verified mainly by mountain pass theorem without (PS) condition. Our method works without the restriction of the (AR) condition in spaces of three or more dimensions.