Normalized solutions of a critical Kirchhoff equation in high dimensional spaces

The following well-known critical Kirchhoff equation with prescribed mass, \[\begin{cases}\displaystyle{-\Big(a+b\int_{\mathbb{R}^N} |\nabla u|^2dx\Big) \Delta u+\lambda u =\mu |u|^{q-2}u+|u|^{2^*-2}u} &\text{in } \mathbb{R}^N, \; N\geq 4, \\ \displaystyle{\,\int_{\mathbb{R}^N}|u|^2dx=c}, & \end{cases}\] has been extensively studied, where \(a\), \(b\), \(c\), \(\mu\), and \(q\) are constants with \(a,c\gt 0\) and \(2\lt q\lt 2^*\), while \(\lambda\) is a real Lagrange multiplier. When \(\mu=0\), we obtain complete results on the existence, nonexistence, and multiplicity of normalized solutions. When \(\mu\gt 0\) and \(N\geq 5\), we investigate the existence of positive normalized solutions, partly filling a gap in the existing literature. Lastly, when \(\mu\gt 0\) and \(N=4\), we obtain a mountain-pass solution by using a new threshold for the energy functional. The main results of this paper broaden and improve the existing results on normalized solutions of the Kirchhoff equation in high-dimensional spaces.