arXiv subjects

Xiaobao Zhu

Publications and source records attributed to Xiaobao Zhu.

17 recordsLinked to original sources

arXiv2025

Critical mean field equations for equilibrium turbulence with sign-changing prescribed functions

Let $(M,g)$ be a compact Riemann surface with unit area. We investigate the mean field equation for equilibrium turbulence: \begin{align} \begin{cases} -Δu = ρ_1\left(\frac{h_1e^{u}}{\int_Mh_1e^udv_g}-1\right) - ρ_2\left(\frac{h_2e^{-u}}{\int_Mh_2e^{-u}dv_g}-1\right), \\ \int_Mudv_g=0, \end{cases} \end{align} where $ρ_1=8π$ and $ρ_2\in(0,8π]$ are parameters, and $h_1, h_2$ are smooth functions on $M$ that are positive somewhere. By employing a refined Brezis-Merle type analysis, we establish sufficient conditions of Ding-Jost-Li-Wang type for the existence of solutions to this equation in critical cases, particularly when $h_1$ and $h_2$ may change signs. Our results extend Zhou's existence theorems (Nonlinear Anal. 69 (2008), no.~8, 2541--2552) for the case $h_1=h_2\equiv 1$.

math.AP↗

arXiv2024

Existence results for Toda systems with sign-changing prescribed functions: Part II

Let $(M, g)$ be a compact Riemann surface with area $1$. We investigate the Toda system \begin{align} \begin{cases} -Δu_1 = 2ρ_1(h_1e^{u_1}-1) - ρ_2(h_2e^{u_2}-1),\\ -Δu_2 = 2ρ_2(h_2e^{u_2}-1) - ρ_1(h_1e^{u_1}-1), \end{cases} \end{align} on $(M, g)$ where $ρ_1, ρ_2 \in (0,4π]$, and $h_1$ and $h_2$ are two smooth functions on $M$.When some $ρ_i$ equals $4π$, the Toda system becomes critical with respect to the Moser-Trudinger inequality for it, making the existence problem significantly more challenging. In their seminal article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558), Jost, Lin, and Wang established sufficient conditions for the existence of solutions the Toda system when $ρ_1=4π$, $ρ_2 \in (0,4π)$ or $ρ_1=ρ_2=4π$, assuming that $h_1$ and $h_2$ are both positive. In our previous paper we extended these results to allow $h_1$ and $h_2$ to change signs in the case $ρ_1=4π$, $ρ_2 \in (0,4π)$. In this paper we further extend the study to prove that Jost-Lin-Wang's sufficient conditions remain valid even when $h_1$ and $h_2$ can change signs and $ρ_1=ρ_2=4π$. Our proof relies on an improved version of the Moser-Trudinger inequality for the Toda system, along with edicated analyses similar to Brezis-Merle type and the use of Pohozaev identities.

math.AP↗

arXiv2024

Existence results for Toda systems with sign-changing prescribed functions: Part I

Let $(M,g)$ be a compact Riemann surface with area $1$, we shall study the Toda system $$ \begin{cases} -Δu_1 = 2ρ_1(h_1e^{u_1}-1) - ρ_2(h_2e^{u_2}-1),\\ -Δu_2 = 2ρ_2(h_2e^{u_2}-1) - ρ_1(h_1e^{u_1}-1), \end{cases} $$ on $(M,g)$ with $ρ_1=4π$, $ρ_2\in(0,4π)$, $h_1$ and $h_2$ are two smooth functions on $M$. In Jost-Lin-Wang's celebrated article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558), they obtained a sufficient condition for the existence of this Toda system when $h_1$ and $h_2$ are both positive. In this paper, we shall improve this result to the case $h_1$ and $h_2$ can change signs. We shall pursue a variational method and use the standard blowup analysis. Among other things, the main contribution in our proof is to show the blowup can only happen at one point where $h_1$ is positive.

math.AP↗

arXiv2023

Another remark on a result of Ding-Jost-Li-Wang

Let $(M,g)$ be a compact Riemann surface, $h$ be a positive smooth function on $M$. It is well known the functional $$J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8π\int_M udv_g-8π\log\int_Mhe^{u}dv_g$$ achieves its minimum under Ding-Jost-Li-Wang condition. This result was generalized to nonnegative $h$ by Yang and the author. Later, Sun and Zhu (arXiv:2012.12840) showed Ding-Jost-Li-Wang condition is also sufficient for $J$ achieves its minimum when $h$ changes sign, which was reproved later by Wang and Yang (J. Funct. Anal. 282: Paper No. 109449, 2022) and Li and Xu (Calc. Var. 61: Paper No. 143, 2022) respectively using flow approach. The aim of this note is to give a new proof of Sun and Zhu's result. Our proof is based on the variational method and the maximum principle.

math.DG↗

arXiv2023

Liouville Theorem for Harmonic Maps from Riemannian Manifold with Compact Boundary

In this note we will provide a gradient estimate for harmonic maps from a complete noncompact Riemannian manifold with compact boundary (which we call "Kasue manifold") into a simply connected complete Riemannian manifold with non-positive sectional curvature. As a consequence, we can obtain a Liouville theorem. We will also show the nonexistence of positive solutions to some linear elliptic equation on Kasue manifold.

math.DG↗

arXiv2023

A singular Kazdan-Warner problem on a compact Riemann surface

Let $(M,g)$ be a compact Riemann surface with unit area, $h\in C^{\infty}(M)$ a function which is positive somewhere, $ρ>0$, $p_i\in M$ and $α_i\in(-1,+\infty)$ for $i=1,\cdots,\ell$, we consider the mean field equation \begin{align*} Δv + 4π\sum_{i=1}^{\ell}α_i\left(1-δ_{p_i}\right) = ρ\left(1-\frac{he^v}{\int_Mhe^vdμ}\right), \end{align*} on $M$, where $Δ$ and $dμ$ are the Laplace-Beltrami operator and the area element of $(M,g)$ respectively. Using variational method and blowup analysis, we prove some existence results in the critical case $ρ=8π(1+\min\{0,\min_{1\leq i\leq\ell}α_i\})$. These results can be seen as partial generalizations of works of Chen-Li (J. Geom. Anal. 1: 359--372, 1991), Ding-Jost-Li-Wang (Asian J. Math. 1: 230--248, 1997), Mancini (J. Geom. Anal. 26: 1202--1230, 2016), Yang-Zhu (Proc. Amer. Math. Soc. 145: 3953--3959, 2017), Sun-Zhu (arXiv:2012.12840) and Zhu (arXiv:2212.09943). Among other things, we prove that the blowup (if happens) must be at the point where the conical angle is the smallest one and $h$ is positive, this is the most important contribution of our paper.

math.AP↗

arXiv2022

A singular Moser-Trudinger inequality for mean value zero functions in dimension two

Let $Ω\subset\mathbb{R}^2$ be a smooth bounded domain with $0\in\partialΩ$. In this paper, we prove that for any $β\in(0,1)$, the supremum $$\sup_{u\in W^{1,2}(Ω), \int_Ωu dx=0, \int_Ω|\nabla u|^2dx\leq1}\int_Ω\frac{e^{2π(1-β) u^2}}{|x|^{2β}}dx$$ is finite and can be attained. This partially generalizes a well-known work of Alice Chang and Paul Yang (J. Differential Geom. 27 (1988), no. 2, 259-296) who have obtained the inequality when $β=0$.

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arXiv2018

Mean field equations on a closed Riemannian surface with the action of an isometric group

Let $(Σ,g)$ be a closed Riemannian surface, $\textbf{G}=\{σ_1,\cdots,σ_N\}$ be an isometric group acting on it. Denote a positive integer $\ell=\inf_{x\inΣ}I(x)$, where $I(x)$ is the number of all distinct points of the set $\{σ_1(x),\cdots,σ_N(x)\}$. A sufficient condition for existence of solutions to the mean field equation $$Δ_g u=8π\ell\left(\frac{he^u}{\int_Σhe^udv_g}-\frac{1}{{\rm Vol}_g(Σ)}\right)$$ is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when $\ell=1$ or equivalently $\textbf{G}=\{Id\}$, where $Id$ is the identity map.

math.AP↗

arXiv2018

A Trudinger-Moser inequality for conical metric in the unit ball

In this note, we prove a Trudinger-Moser inequality for conical metric in the unit ball. Precisely, let $\mathbb{B}$ be the unit ball in $\mathbb{R}^N$ $(N\geq 2)$, $p>1$, $g=|x|^{\frac{2p}{N}β}(dx_1^2+\cdots+dx_N^2)$ be a conical metric on $\mathbb{B}$, and $λ_p(\mathbb{B})=\inf\left\{\int_\mathbb{B}|\nabla u|^Ndx: u\in W_0^{1,N}(\mathbb{B}),\,\int_\mathbb{B}|u|^pdx=1\right\}$. We prove that for any $β\geq 0$ and $α<(1+\frac{p}{N}β)^{N-1+\frac{N}{p}}λ_p(\mathbb{B})$, there exists a constant $C$ such that for all radially symmetric functions $u\in W_0^{1,N}(\mathbb{B})$ with $\int_\mathbb{B}|\nabla u|^Ndx-α(\int_\mathbb{B}|u|^p|x|^{pβ}dx)^{N/p}\leq 1$, there holds $$\int_\mathbb{B}e^{α_N(1+\frac{p}{N}β)|u|^{\frac{N}{N-1}}}|x|^{pβ}dx\leq C,$$ where $|x|^{pβ}dx=dv_g$, $α_N=Nω_{N-1}^{1/(N-1)}$, $ω_{N-1}$ is the area of the unit sphere in $\mathbb{R}^N$; moreover, extremal functions for such inequalities exist. The case $p=N$, $-1<β<0$ and $α=0$ was considered by Adimurthi-Sandeep \cite{A-S}, while the case $p=N=2$, $β\geq 0$ and $α=0$ was studied by de Figueiredo-do Ó-dos Santos \cite{F-do-dos}.

math.AP↗

arXiv2018

Blow-up analysis concerning singular Trudinger-Moser inequalities in dimension two

In this paper, we derive a sharp version of the singular Trudinger-Moser inequality, which was originally established by Adimurthi and Sandeep (Nonlinear Differ. Equ. Appl. 2007). Moreover, extremal functions for those singular Trudinger-Moser inequalities are also obtained. Our method is the blow-up analysis. Compared with our previous work (J. Differential Equations 2015), the essential difficulty caused by the presence of singularity is how to analyse the asymptotic behaviour of certain maximizing sequence near the blow-up point. We overcome this difficulty by combining two different classification theorems of Chen and Li (Duke Math. J. 1991; Duke Math. J. 1995) to get the desired bubble.

math.AP↗

arXiv2017

Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface

Let $(Σ,g)$ be a compact Riemannian surface without boundary and $λ_1(Σ)$ be the first eigenvalue of the Laplace-Beltrami operator $Δ_g$. Let $h$ be a positive smooth function on $Σ$. Define a functional $$J_{α,β}(u)=\frac{1}{2}\int_Σ(|\nabla_gu|^2-αu^2)dv_g-β\log\int_Σhe^udv_g$$ on a function space $\mathcal{H}=\left\{u\in W^{1,2}(Σ): \int_Σudv_g=0\right\}$. If $α<λ_1(Σ)$ and $J_{α,8π}$ has no minimizer on $\mathcal{H}$, then we calculate the infimum of $J_{α,8π}$ on $\mathcal{H}$ by using the method of blow-up analysis. As a consequence, we give a sufficient condition under which a Kazdan-Warner equation has a solution. If $α\geq λ_1(Σ)$, then $\inf_{u\in\mathcal{H}}J_{α,8π}(u)=-\infty$. If $β>8π$, then for any $α\in\mathbb{R}$, there holds $\inf_{u\in\mathcal{H}}J_{α,β}(u)=-\infty$. Moreover, we consider the same problem in the case that $α$ is large, where higher order eigenvalues are involved.

math.AP↗

arXiv2017

Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case

The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(Σ,β)$ be a closed Riemann surface with a divisor $β$, and $K_λ=K+λ$, where $K:Σ\rightarrow\mathbb{R}$ is a Hölder continuous function satisfying $\max_ΣK= 0$, $K\not\equiv 0$, and $λ\in\mathbb{R}$. If the Euler characteristic $χ(Σ,β)$ is negative, then by a variational method, it is proved that there exists a constant $λ^\ast>0$ such that for any $λ\leq 0$, there is a unique conformal metric with the Gaussian curvature $K_λ$; for any $λ$, $0<λ<λ^\ast$, there are at least two conformal metrics having $K_λ$ its Gaussian curvature; for $λ=λ^\ast$, there is at least one conformal metric with the Gaussian curvature $K_{λ^\ast}$; for any $λ>λ^\ast$, there is no certain conformal metric having $K_λ$ its Gaussian curvature. This result is an analog of that of Ding and Liu \cite{Ding-Liu}, partly resembles that of Borer, Galimberti and Struwe \cite{B-G-Stru}, and generalizes that of Troyanov \cite{Troyanov} in the negative case.

math.AP↗

arXiv2016

A generalized Trudinger-Moser inequality on a compact Riemannian surface

Let $(Σ, g)$ be a compact Riemannian surface. Let $ψ$, $h$ be two smooth functions on $Σ$ with $\int_Σψdv_g \neq 0$ and $h\geq0$, $h\not\equiv0$. In this paper, using a method of blowup analysis, we prove that the functional \begin{align}\label{functional_J} J^{ψ,h}(u)=\frac{1}{2}\int _Σ|\nabla_g u|^2dv_g + 8π\frac{1}{\int_Σψdv_g}\int_Σψudv_g-8π\log\int _Σhe^{u}dv_g \end{align} is bounded from below in $W^{1,2}(Σ,g)$. Moreover, we obtain a sufficient condition under which $J^{ψ, h}$ attains its infimum in $W^{1,2}(Σ,g)$. These results generalize the main results in \cite{DJLW97} and \cite{YZ2016}.

math.DG↗

arXiv2016

A remark on a result of Ding-Jost-Li-Wang

Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space, $J: W^{1,2}(M)\rightarrow \mathbb{R}$ be the functional defined by $$J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8π\int_M udv_g-8π\log\int_Mhe^udv_g,$$ where $h$ is a positive smooth function on $M$. In an inspiring work (Asian J. Math., vol. 1, pp. 230-248, 1997), Ding, Jost, Li and Wang obtained a sufficient condition under which $J$ achieves its minimum. In this note, we prove that if the smooth function $h$ satisfies $h\geq 0$ and $h\not\equiv 0$, then the above result still holds. Our method is to exclude blow-up points on the zero set of $h$.

math.AP↗

arXiv2015

An improved Hardy-Trudinger-Moser inequality

Let $\mathbb{B}$ be the unit disc in $\mathbb{R}^2$, $\mathscr{H}$ be the completion of $C_0^\infty(\mathbb{B})$ under the norm $$\|u\|_{\mathscr{H}}=\left(\int_\mathbb{B}|\nabla u|^2dx-\int_\mathbb{B}\frac{u^2}{(1-|x|^2)^2}dx\right)^{1/2},\quad\forall u\in C_0^\infty(\mathbb{B}).$$ Denote $λ_1(\mathbb{B})=\inf_{u\in \mathscr{H},\,\|u\|_2=1}\|u\|_{\mathscr{H}}^2$, where $\|\cdot\|_2$ stands for the $L^2(\mathbb{B})$-norm. Using blow-up analysis, we prove that for any $α$, $0\leq α<λ_1(\mathbb{B})$, $$\sup_{u\in\mathscr{H},\,\|u\|_{\mathscr{H}}^2-α\|u\|_2^2\leq 1}\int_\mathbb{B} e^{4πu^2}dx<+\infty,$$ and that the above supremum can be attained by some function $u\in \mathscr{H}$ with $\|u\|_{\mathscr{H}}^2-α\|u\|_2^2= 1$. This improves an earlier result of G. Wang and D. Ye [28].

math.AP↗

arXiv2013

Trudinger-Moser embedding on the hyperbolic space

In [16], we established Trudinger-Moser inequalities for complete noncompact Riemannian manifold on which the Ricci curvature has lower bound and the injectivity radius is strictly positive. In this note, we improve those inequalties when the manifold is the hyperbolic space. The method we used here is still gluing local estimates.

math.DG↗

arXiv2012

A new proof of subcritical Trudinger-Moser inequalities on the whole Euclidean space

In this note, we give a new proof of subcritical Trudinger-Moser inequality on $\mathbb{R}^n$. All the existing proofs on this inequality are based on the rearrangement argument with respect to functions in the Sobolev space $W^{1,n}(\mathbb{R}^n)$. Our method avoids this technique and thus can be used in the Riemannian manifold case and in the entire Heisenberg group.

math.AP↗