Normalized solutions for the general Kirchhoff type equations

In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions (λ_c, u_c) ∈ ℝ × H¹ (ℝ^N) to the general Kirchhoff problem

\(-M\left(\int_{\mathbb{R}^N}\vert\nabla u\vert^2 {\rm d}x\right)\Delta u +\lambda u=g(u)~\hbox{in}~\mathbb{R}^N, u\in H^1(\mathbb{R}^N),N\geq 1,\) $$-M({\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{\Delta }u+\lambda u=g(u)\text{in}{\mathbb{R}}^N,u\in H^1({\mathbb{R}}^N),N\ge 1,$$

satisfying the normalization constraint \(\int_{\mathbb{R}^N}u^2{\rm d}x=c\) $${\int }_{{\mathbb{R}}^N}{u}^2\mathrm{d}x=c$$ , where M ∈ C([0, ∞)) 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 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.