Normalized solutions for $L^2$-supercritical Schr\"odinger equations with nonlinear point defects on noncompact metric graphs

Abstract

In this paper, we study the existence and multiplicity of normalized solutions for the following $L^2$-supercritical Schr\"odinger equation with nonlinear point defects on a noncompact metric graph $\G=(\V,\E)$ \begin{equation*} \begin{cases} u'' = \lambda u & \text{on every } \e \in \E, \\ \int_\G \abs{u}^2\, dx = \mu & \\ \displaystyle \sum_{\e \succ \vv} u'_\e(\vv) = -|u(\vv)|^{p-2}u(\vv) & \text{at every } \vv \in \V_0, \\ \displaystyle \sum_{\e \succ \vv} u'_\e(\vv) = 0 & \text{at every } \vv \in \V \setminus \V_0, \end{cases} \end{equation*} where $p>4$, $\G$ has finitely many edges and no self-loops, $\V_0 \subset \V$ is nonempty, $\mu>0$ is a given constant, $\lambda$ is an unknown 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 $\ell_\e$. We first prove the existence of a positive normalized solution for every prescribed mass and every nonempty set of defect vertices. We then establish a multiplicity result for normalized solutions when the prescribed mass is sufficiently small and sufficiently many half-lines are attached to the defect vertices.