arXiv subjects

Yifan Tao

Publications and source records attributed to Yifan Tao.

2 recordsLinked to original sources

arXiv2026

Normalized solutions for $L^2$-supercritical Schrödinger equations with nonlinear point defects on noncompact metric graphs

In this paper, we study the existence and multiplicity of normalized solutions for the following $L^2$-supercritical Schrödinger equation with nonlinear point defects on a noncompact metric graph $\G=(\V,\E)$ \begin{equation*} \begin{cases} u'' = λu & \text{on every } \e \in \E, \\ \int_\G \abs{u}^2\, dx = μ& \\ \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, $μ>0$ is a given constant, $λ$ 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.

math.AP↗

arXiv2025

Existence of normalized solutions to nonlinear Schrödinger equations on lattice graphs

In this paper, using a discrete Schwarz rearrangement on lattice graphs developed in \cite{DSR}, we study the existence of global minimizers for the following functional $I:H^1\left(\mathbb{Z}^N\right)\to \R$, $$I(u)=\frac{1}{2} \int_{\mathbb{Z}^N}|\nabla u|^2 \,dμ-\int_{\mathbb{Z}^N} F(u)\, dμ,$$ constrained on $S_m:=\left\{u \in H^1\left(\mathbb{Z}^N\right) \mid\|u\|_{\ell^2\left(\mathbb{Z}^N\right)}^2=m\right\}$, where $N \geq 2$, $m>0$ is prescribed, $f \in C(\mathbb{R}, \mathbb{R})$ satisfying some technical assumptions and $F(t):=\int_0^t f(τ) \,dτ$. We prove the following minimization problem $$ \inf_{u \in S_m} I(u) $$ has an excitation threshold $m^*\in [0,+\infty]$ such that \begin{equation*} \inf_{u \in S_m} I(u)<0 \quad \text{if and only if } m>m^*. \end{equation*} Based primarily on $m^* \in (0,+\infty)$ or $m^*=0$, we classify the problem into three different cases: $L^2$-subcritical, $L^2$-critical and $L^2$-supercritical. Moreover, for all three cases, under assumptions that we believe to be nearly optimal, we show that $m^*$ also separates the existence and nonexistence of global minimizers for $I(u)$ constrained on $S_{m}$.

math.AP↗