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Tuoxin Li

Publications and source records attributed to Tuoxin Li.

11 recordsLinked to original sources
arXiv2026

Sharp CR Beckner-Onofri inequalities under high-order holomorphic moment constraints

We establish sharp Beckner--Onofri on the CR sphere $\mathbb{S}^{2N-1}\subset\mathbb C^N$, $N\ge2$, under vanishing holomorphic moments through degree $m\ge1$. The optimal coefficients are $\frac{1}{m+1}$. The function inequality admits a stronger form involving only the energy above degree $m$. We also prove a quantitative concentration estimate in the $1$-Wasserstein distance and show that every maximizing sequence has diverging energy and approaches the family of uniform measures on regular $(m+1)$-polygons. The proof combines subcritical minimization with rigidity of the constrained Euler--Lagrange equation, obtained from an exact radial energy identity and a Hardy space projection estimate. Finally, weighted ball automorphisms yield sharp inequalities without moment constraints, with scalar conformal, barycenter, and holomorphic Gram determinant corrections.

math.AP↗
arXiv2026

Uniqueness and Symmetry Breaking for Singular $Q$-curvature Equations in $\R^n$

For $n\ge3$ and $γ>-1$, we study normal solutions of \begin{equation*} (-Δ)^{n/2}u=(n-1)!\,|x|^{nγ}e^{nu} \quad\hbox{in }\R^n. \end{equation*} We reformulate this problem on the cylinder via the Emden--Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove existence and uniqueness, up to dilation, of the radial normal solution and establish its radial nondegeneracy. Next, we carry out a detailed spectral analysis of the linearized operator and construct branches of nonradial normal solutions by the Crandall--Rabinowitz theorem. Finally, we prove that every normal solution is radial when $γ>0$ is sufficiently small.

math.AP↗
arXiv2026

Rigidity of a $Q$-curvature-type problem on $\mathbb{S}^N$ in every dimension $N\geq3$

We prove that for every $N\geq 3$, every solution to the $Q$-curvature-type problem $$ αP_N u + (N-1)!\left(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw}\right)=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ α\ge\frac{1}{2}$ and $α\not =1$. The proof consists of a spectral gap estimate and pointwise estimates of the normalized density $\left(\int_{\mathbb{S}^N} e^{Nu}dw\right)^{-1}e^{Nu}$.

math.AP↗
arXiv2026

A positive answer to the generalized Chang-Yang conjecture on $\mathbb{S}^N$

We prove that for every integer $N\geq 3$ and $α\geq \frac{1}{2}$, Beckner's inequality \[ \fracα{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\log\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \] holds for every $u\in H^{\frac{N}{2}}(\mathbb{S}^N)$ whose center of mass is at the origin. The proof is mainly based on an integral representation formula and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively for every integer $N\geq 3$.

math.AP↗
arXiv2026

Sharp Beckner's Inequalities for Axially Symmetric Functions on $\mathbb{S}^N$

We prove that for every integer $N\geq 3$ and $α\geq \frac{1}{2}$, Beckner's inequality \begin{equation*} \fracα{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\ln\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \end{equation*} holds for any axially symmetric $u\in H^{\frac{N}{2}}(\mathbb{S}^N)$ whose center of mass is at the origin. The proof is mainly based on a weighted $\ell ^2$ estimate on Gegenbauer coefficients and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively in the axially symmetric case for every integer $N\geq 3$.

math.AP↗
arXiv2026

Axially Symmetric Rigidity for a $Q$-Curvature-Type Equation on $ \mathbb{S}^N $

We prove that for every even integer $N\geq 4$ or $N\in\{3,5,7\}$, every axially symmetric solution to the $Q$-curvature-type problem $$ αP_N u + (N-1)!(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ α\ge\frac{1}{2}$ and $α\not =1$. The proof starts from a Gegenbauer expansion and its weighted $\ell ^2$ estimate on coefficients. Then one key estimate is the refined estimate of the seminorm $\lfloor G\rfloor^2$ using a specially designed integration-by-parts identity for general even integer $N\geq 4$. For $N\in\{3,5,7\}$, the seminorm estimates are established via a descent to a three--dimensional nonlocal difference quotient identity.

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arXiv2026

On the Classification of blow-up solutions of a singular Liouville equation on the disk

We study the blow-up behavior of solutions to the singular Liouville equation \[ Δ\tilde u+λe^{\tilde u}=4παδ_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] where $α>0$, $λ>0$ and $B\subset\mathbb R^2$ is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed $α>0$ and $λ\in(0,λ_α)$, the singular Liouville equation has exactly $\lceil α\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil α\rceil$, a unique $m$-peak sequence whose blow-up points are the vertices of a regular $m$-gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco \cite{Bartolucci-Montefusco06} and Bartolucci \cite{Bartolucci10}. We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk.

math.AP↗
arXiv2025

Infinitely many solutions for the prescribed scalar curvature problem with volcano-like curvature

In this paper, we consider the following prescribed scalar curvature problem: \begin{equation*} -Δu = K(x) u^{\frac{n+2}{n-2}}, \quad u>0\quad\hbox{in}\quad \mathbb{R}^n, \quad u \in D^{1,2}(\mathbb{R}^n), \end{equation*} where $K(x)$ is a volcano-like positive function such that $$ K(x)= K(r_0)- c_0 | |x|- r_0|^m + O( | |x|- r_0|^{m+θ}),\quad r_0- δ<|x| 0, θ>2, \min \{\frac{n-2}{2}, 2\} < m< n-2$. We first prove the existence of infinitely many positive solutions. A consequence of our proof yields that the infinitely many solutions constructed in \cite{WY} are non-degenerate in the whole $D^{1, 2}(\mathbb{R}^{n})$ space. To our knowledge, it seems to be the first result of infinitely many solutions of prescribed scalar curvature problem when the potential function $K(x)$ is not radial. Our non-degeneracy results are also more complete and improve the result in \cite{GMPS}.

math.AP↗
arXiv2023

On Beckner's Inequality for Axially Symmetric Functions on $\mathbb{S}^6$

We prove that axially symmetric solutions to the $Q$-curvature type problem $$ αP_6 u + 120(1-\frac{e^{6u}}{\int_{\mathbb{S}^6} e^{6u}})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^6 $$ must be constants, provided that $ \frac{1}{2}\leq α<1$. In view of the existence of non-constant solutions obtained by Gui-Hu-Xie \cite{GHW2022} for $\frac{1}{7}<α<\frac{1}{2}$, this result is sharp. This result closes the gap of the related results in \cite{GHW2022}, which proved a similar uniqueness result for $α\geq 0.6168$. The improvement is based on two types of new estimates: one is a better estimate of the semi-norm $\lfloor G\rfloor^2$, the other one is a family of refined estimates on Gegenbauer coefficients, such as pointwise decaying and cancellations properties.

math.AP↗
arXiv2022

Infinitely many nonradial positive solutions for multi-species nonlinear Schrödinger systems in ${\mathbb R}^N$

In this paper, we consider the multi-species nonlinear Schrödinger systems in $\bbr^N$: \begin{equation*} \left\{\aligned&-Δu_j+V_j(x)u_j=μ_ju_j^3+\sum_{i=1;i\not=j}^dβ_{i,j} u_i^2u_j\quad\text{in }\bbr^N, &u_j(x)>0\quad\text{in } {\mathbb R}^N, &u_j(x)\to0\quad\text{as }|x|\to+\infty,\quad j=1,2,\cdots,d,\endaligned\right. \end{equation*} where $N=2,3$, $μ_j>0$ are constants, $β_{i,j}=β_{j,i}\not=0$ are coupling parameters, $d\geq2$ and $V_j(x)$ are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials $V_j(x)$ and coupling parameters $\{β_{i,j}\}$, {\it without any symmetric assumptions on the limit case of the above system}. Our result, giving a positive answer to the conjecture in Pistoia and Vaira \cite{PV22} and extending the results in \cite{PW13,PV22}, reveals {\it new phenomenon} in the case of $N=2$ and $d=2$ and is {\it almost optimal} for the coupling parameters $\{β_{i,j}\}$.

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