Existence and Multiplicity of Normalized Solutions with Positive Energy for the Kirchhoff Equation

In this paper, we investigate the existence and multiplicity of normalized solutions for the following Kirchhoff equation, P \(\begin{aligned} \left\{ \begin{array}{l} -\left( a+b \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u-\lambda u=f(u), \quad \text{ in } \mathbb {R}^{3}, \\ \int _{\mathbb {R}^{3}}|u|^{2} d x=c, \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\left(a,+,b,{\int }_{{\mathbb{R}}^3},{|\mathrm{\nabla }u|}^2,d,x\right)\mathrm{\Delta }u-\lambda u=f(u),\text{in}{\mathbb{R}}^3,\hfill \\ {\int }_{{\mathbb{R}}^3}{|u|}^{2}dx=c,\hfill \end{array}\right)\end{array}$$ where a, b, c are positive constants and \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. Two normal solutions, manifesting as a local minimizer or mountain pass solution, have been obtained under the mass subcritical conditions on the nonlinearity f and some suitable mass c. Additionally, we employ the Symmetric Mountain Pass Theorem to establish the multiplicity of normalized solutions for problem (P). To the best of our knowledge, we extend and complement the research success in the Kirchhoff equation for a general nonlinearity with weaker subcritical mass growth.