Hong-Ge Chen

A Pfaffian Proof and Generalization of a Conjecture of Zhi-Wei Sun

Hong-Ge Chen, Fei Liu

cn, hk (code pays fourni par la source)

On Brezis' open problem 2.2

Hong-Ge Chen, Yong Liu, Juncheng Wei, Wen Yang

We prove that the global minimizer of the Ginzburg-Landau energy in the disk of radius $R$ with boundary value $ u(x)=\frac{x}{|x|}$ is the degree-one radial solution of the planar Ginzburg--Landau equation. This gives an affirmative answer to Open Problem~2.2 in Brezis' open-problem …

On Sirakov's equal-frequency uniqueness conjecture

Hong-Ge Chen, Yong Liu, Juncheng Wei, Wen Yang

Let $N\in\{2,3\}$, $0<μ_1\leqμ_2$, and $0<β<μ_1$. We prove that the equal-frequency two-component cubic Schrödinger system \[ -Δu+u=μ_1u^3+βuv^2, \qquad -Δv+v=μ_2v^3+βu^2v \quad\text{in }\mathbb{R}^N \] has exactly one positive solution in $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ modulo simultaneous translations. More precisely, every positive solution is a simultaneous translate of …

On Sirakov's equal-frequency uniqueness conjecture

Hong-Ge Chen, Yong Liu, Juncheng Wei, Wen Yang

Let $N\in\{2,3\}$, $0<μ_1\leqμ_2$, and $0<β<μ_1$. We prove that the equal-frequency two-component cubic Schrödinger system \[ -Δu+u=μ_1u^3+βuv^2, \qquad -Δv+v=μ_2v^3+βu^2v \quad\text{in }\mathbb{R}^N \] has exactly one positive solution in $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ modulo simultaneous translations. More precisely, every positive solution is a simultaneous translate of …

cn, hk, mo (code pays fourni par la source)

A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei

Hong-Ge Chen, Fei Liu

Let $p$ be an odd prime, let $n=(p-1)/2$, and let $χ=(\frac{\cdot}{p})$, with $χ(0)=0$. For $a\in\mathbb F_p^\times$ define \[ D_a(x)=\det_{1\le i,j\le n}(x+χ(i^2-aj)), \qquad D_a^{(0)}(x)=\det_{0\le i,j\le n}(x+χ(i^2-aj)). \] We prove \[ D_a(0)=0 \quad\Longleftrightarrow\quad p\equiv 3 \pmod 4 \quad\text{and}\quad χ(a n!)=1. \] For $p\equiv3\pmod4$ we …

A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei

Hong-Ge Chen, Fei Liu

Let $p$ be an odd prime, let $n=(p-1)/2$, and let $χ=(\frac{\cdot}{p})$, with $χ(0)=0$. For $a\in\mathbb F_p^\times$ define \[ D_a(x)=\det_{1\le i,j\le n}(x+χ(i^2-aj)), \qquad D_a^{(0)}(x)=\det_{0\le i,j\le n}(x+χ(i^2-aj)). \] We prove \[ D_a(0)=0 \quad\Longleftrightarrow\quad p\equiv 3 \pmod 4 \quad\text{and}\quad χ(a n!)=1. \] For $p\equiv3\pmod4$ we …