Abstract: In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions $(\lambda_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the general Kirchhoff problem $-M\left(\int_{\mathbb{R} ^N}|\nabla u|^2 {\rm d}x\right)\Delta u +\lambda u=g(u) \hbox{in} \mathbb{R} ^N, u\in H^1(\mathbb{R} ^N),N\geq 1,$ satisfying the normalization constraint $\int_{\mathbb{R}^N}u^2{\rm d}x=c,$ where $M\in C([0,\infty))$ is a given function satisfying some suitable assumptions. Our argument is not by the classical variational method, but by a global branch approach developed by Jeanjean \textit{et al}. [J Math Pures Appl, 2024, 183: 44-75] and a direct correspondence, so we can handle in a unified way the nonlinearities $g(s)$, which are either mass subcritical, mass critical or mass supercritical.

Key words: normalized solution, Kirchhoff type equations, general nonlinearities, asymptotic behavior