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

TL;DR

This paper investigates the existence and multiplicity of normalized solutions for a supercritical Schrödinger equation on noncompact metric graphs with nonlinear point defects, extending previous work on subcritical cases.

Contribution

It introduces new results on the existence and multiplicity of solutions for the $L^2$-supercritical case on metric graphs with nonlinear point defects.

Findings

Established existence of solutions for $p>4$

Proved multiple solutions under certain conditions

Extended previous subcritical results to supercritical regime

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.