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

Abstract

In this paper, we study the existence and multiplicity of normalized solutions for the following $L^2$-supercritical Schr\"odinger equation on noncompact metric graph $\G=(\V,\E)$ with nonlinear point defects \begin{equation*} \begin{cases} u'' = \lambda u & \text{on every }\e \in \E, \\ \|u\|_{L^2(\mathcal{G})}^2 = \mu & \\ \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 $p>4$, $\G$ 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, $\e \succ \vv$ means that the edge $\e$ is incident at $\vv$, and the notation $u'_\e(\vv)$ stands for $u'_\e(0)$ or $-u'_\e(\ell_\e)$, according to whether the vertex $\vv$ is identified with $0$ or $\ell_\e$. 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 ($2<p<4$) Schr\"{o}dinger equation on metric graphs with nonlinear point defects.