In this paper, for given \(c>0\) , we study the existence of a couple of solution \((u_c,\lambda _c)\in H^1({\mathbb R}^N)\times {\mathbb R}_+\) to the following Kirchhoff type problem: \( \left\{ \begin{array}{ll} -\left( a+b \int _{{\mathbb R}^N}|\nabla u|^2\right) \Delta u+\lambda u=f(u),\,\,\, & ~x\in {\mathbb R}^N, \\ (\int _{{\mathbb R}^N}|u|^2)^{\frac{1}{2}}=c,\,\,\, & \end{array} \right. \) where \(N\le 3\) , a, \(b>0\) are constants, \(f(u)\sim |u|^{\frac{8}{N}}u\) is a \(L^2\) -critical general nonlinearity. By using the scaling method and a new version of global compactness lemma, we prove that there exists \(c_*>0\) such that the problem admits no solution for \(0<c\le c_*\) and the problem admits at least one solution with a minimax characterization for \(c>c_*\) . Our main results can be viewed as an extension of [He et. al. JDE, 356:375–406, (2023)] concerning the \(L^2\) -supercritical nonlinearity.
