Normalized solutions to the Kirchhoff equation with a perturbation term

In this paper, we study the existence of solutions to the Kirchhoff equation \begin{equation*} - \Big ( {a + b\int_{{\mathbb{R}}^{3}} {|\nabla u{|^2}dx} } \Big ) \Delta u = \lambda u + |u{|^{p - 2}}u + \mu |u{|^{q - 2}}u~~{\rm in}~{\mathbb{R}}^{3}, \end{equation*} having prescribed mass $$ \int_{{\mathbb{R}}^{3}} {|u{|^2}dx} = c, $$ where $a,b > 0$, $\mu \in {\mathbb{R}}$, $2 < q < p < 6$. When $(p,q)$ belongs to a certain domain in ${{\mathbb{R}}^{2}}$, we prove the existence and nonexistence of normalized solutions by using constraint minimization and concentration compactness principle, our main results may be illustrated by the red areas and green areas shown in Figure 1. In particular, our results are closely related to the values of $\mu$ and prescribed mass $c$, and partially extend the results of Li et al. [10] and Soave [20].