Normalized solutions for nonlinear Kirchhoff type equations in high dimensions

We study the normalized solutions for nonlinear Kirchhoff equation with Sobolev critical exponent in high dimensions ${\mathbb{R}}^N(N⩾4)$. In particular, in dimension $N=4$, there is a special phenomenon for Kirchhoff equation that the mass critical exponent $2+\frac{8}{N}$ is equal to the energy critical exponent $\frac{2N}{N-2}$, which leads to the fact that the equation no longer has a variational structure in dimensions $N⩾4$ if we consider the mass supercritical case, and remains unsolved in the existing literature. In this paper, by using appropriate transform, we first get the equivalent system of Kirchhoff equation. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions by variational methods, Cardano's formulas and Pohožaev identity.