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Tuoxin Li

Publications and source records attributed to Tuoxin Li.

7 recordsLinked to original sources
arXiv2026

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

We prove that for every integer $N\geq 3$ and $\alpha\geq \frac{1}{2}$, Beckner's inequality \[ \frac{\alpha}{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\log\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \] 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 $\alpha\geq \frac{1}{2}$, Beckner's inequality \begin{equation*} \frac{\alpha}{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

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 \[ \Delta \tilde u+\lambda e^{\tilde u}=4\pi\alpha\delta_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] where $\alpha>0$, $\lambda>0$ 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 $\alpha>0$ and $\lambda\in(0,\lambda_\alpha)$, the singular Liouville equation has exactly $\lceil \alpha\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil \alpha\rceil$, 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*} -\Delta u = 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+\theta}),\quad r_0- \delta <|x| 0, \theta >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 $$ \alpha P_6 u + 120(1-\frac{e^{6u}}{\int_{\mathbb{S}^6} e^{6u}})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^6 $$ must be constants, provided that $ \frac{1}{2}\leq \alpha <1$. In view of the existence of non-constant solutions obtained by Gui-Hu-Xie \cite{GHW2022} for $\frac{1}{7}<\alpha<\frac{1}{2}$, this result is sharp. This result closes the gap of the related results in \cite{GHW2022}, which proved a similar uniqueness result for $\alpha \geq 0.6168$. 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 Schr\"odinger systems in ${\mathbb R}^N$

In this paper, we consider the multi-species nonlinear Schr\"odinger systems in $\bbr^N$: \begin{equation*} \left\{\aligned&-\Delta u_j+V_j(x)u_j=\mu_ju_j^3+\sum_{i=1;i\not=j}^d\beta_{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 $N=2,3$, $\mu_j>0$ are constants, $\beta_{i,j}=\beta_{j,i}\not=0$ 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 $\{\beta_{i,j}\}$, {\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 $\{\beta_{i,j}\}$.

math.AP↗