This paper focuses on the existence of normalized solutions for the following Kirchhoff equation: \(\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2\textrm{d}x\right) \Delta u+\lambda u=u^5+\mu |u|^{q-2}u, & x\in {\mathbb {R}}^3, \\ \int _{{\mathbb {R}}^3}u^2\textrm{d}x=c, \\ \end{array} \right. \end{aligned}\) where \(a,b,c>0\) , \(\mu \in {\mathbb {R}}\) and \(2<q<6\) , \(\lambda \in {\mathbb {R}}\) will arise as a Lagrange multiplier that is not a priori given. By using new analytical techniques, the paper establishes several existence results for the case \(\mu >0\) : (1) The existence of two solutions, one being a local minimizer and the other of mountain-pass type, under explicit conditions on c when \(2<q<\frac{10}{3}\) .
(2) The existence of a mountain-pass type solution under explicit conditions on c when \(\frac{10}{3}\le q<\frac{14}{3}\) .
(3) The existence of a ground state solution for all \(c>0\) when \(\frac{14}{3}\le q<6\) .
Furthermore, the paper presents the first non-existence result for the case \(\mu \le 0\) and \(2<q<6\) . In particular, refined estimates of energy levels are proposed, suggesting a new threshold of compactness in the \(L^2\) -constraint. This study addresses an open problem for \(2<q<\frac{10}{3}\) and fills a gap in the case \(\frac{10}{3}\le q<\frac{14}{3}\) . We believe that our approach can be applied to a broader range of nonlinear terms with Sobolev critical growth, and the underlying ideas have potential for future development and applicability.
