Existence and multiplicity of normalized solutions for $L^2$-supercritical Schrödinger equations on noncompact metric graphs with nonlinear point defects

In this paper, we study the existence and multiplicity of normalized solutions for the following $L^2$-supercritical Schrödinger equation on noncompact metric graph $\text{G}=(\text{V},\text{E})$ with nonlinear point defects \begin{equation*} \begin{cases} u’’ = λu & $\text{on every }$\e \in \E, \ |u|{L^2($\mathcal{G}$)}^2 = μ& \ \displaystyle\sum{\e \succ \vv} u’_\e(\vv) = -|u(\vv)|^{p-2}u(\vv) & $\text{at every }$\vv \in \V, \end{cases} \end{equation*} where $p>4$, $\text{G}$ has finitely many edges, $\mu >0$ is a given constant, the parameter $\lambda$ is a part of the unknown which arises as a Lagrange multiplier, $\text{e}\succ \text{vv}$ means that the edge $\text{e}$ is incident at $\text{vv}$, and the notation $u_{\text{e}}^{\mathrm{\prime }}(\text{vv})$ stands for $u_{\text{e}}^{\mathrm{\prime }}(0)$ or $-u_{\text{e}}^{\mathrm{\prime }}({\mathrm{\ell }}_{\text{e}})$, according to whether the vertex $\text{vv}$ is identified with $0$ or ${\mathrm{\ell }}_{\text{e}}$. This work complements the study initiated by Boni, Dovetta, and Serra [J. Funct. Anal. 288 (2025), 110760], which addressed only the existence of normalized solutions for the $L^2$-subcritical ($2<p<4$) Schrödinger equation on metric graphs with nonlinear point defects.