Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities

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. (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. (2)

   The existence of a mountain-pass type solution under explicit conditions on c when \(\frac{10}{3}\le q<\frac{14}{3}\).

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.