A classification result for the quasi-linear Liouville equation

$\mathbb R^N$ 中拟线性椭圆方程的刘维尔定理

Liouville theorem for quasilinear elliptic equations in $\mathbb R^N$

本文证明了 $\mathbb R^N$ 中一类拟线性椭圆方程在临界与次临界情形下的刘维尔定理，核心方法基于积分恒等式与杨不等式。

A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms

In this article we study local and global properties of positive solutions of $-Δ_mu=|u|^{p-1}u+M|\nabla u|^q$ in a domain $Ω$ of $\mathbb R^N$, with $m>1$, $p,q>0$ and $M\in\mathbb R$. Following some ideas used in \cite{BV,Vron1}, and by using a direct Bernstein method combined with Keller-Osserman's estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin's classical results.

Classification of stable solutions to a non-local Gelfand-Liouville equation

We study finite Morse index solutions to the non-local Gelfand-Liouville problem $$ (-Δ)^su=e^u\quad\mathrm{in}\quad \mathbb{R}^n,$$ for every $s\in(0,1)$ and $n>2s$. Precisely, we prove non-existence of finite Morse index solutions whenever the singular solution $$u_{n,s}(x)=-2s\log|x|+\log \left(2^{2s}\frac{Γ(\frac{n}{2})Γ(1+s)}{Γ(\frac{n-2s}{2})}\right)$$ is unstable.

Liouville方程的分类结果与Riemann面的刚性

Classification results of Liouville equations and rigidity of Riemannian surfaces

研究Liouville方程在非负曲率Riemann面上的解分类及 ambient 曼-fold 的刚性结果，假设条件最优且不同于有限总曲率假设。

Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term

We consider the elliptic quasilinear equation --$Δ$ m u = u p |$\nabla$u| q in R N with q $\ge$ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also give their asymptotic behaviour : all the solutions in an exterior domain R N \B r0 are bounded. The solutions in B r0 \ {0} can be extended as a continuous functions in B r0. The solutions in R N \ {0} has a finite limit l $\ge$ 0 as |x| $\rightarrow$ $\infty$. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a p…

Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the $m$-Laplacian operator \begin{equation*} -Δ_{m}u=u^q|\nabla u|^p\ \ \ \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq1$, $m>1$ and $p,q\geq0$. The technique of Bernstein gradient estimates is ultilized to study the case $p<m$. Moreover, a Liouville-type theorem for supersolutions under subcritial range of exponents \begin{equation*} q(N-m)+p(N-1)<N(m-1) \e…

上半空间上的尖锐Onofri迹不等式与带Neumann边界的拟线性Liouville方程

Sharp Onofri trace inequality on the upper half space and quasi-linear Liouville equation with Neumann boundary

本文在上半空间建立尖锐Onofri迹不等式并分类极值函数，同时利用Serrin-Zou型和Pohozaev型恒等式给出带Neumann边界的拟线性Liouville方程解的分类。

Classification of Radial Solutions to Liouville Systems with Singularities

Let $A=(a_{ij})_{n\times n}$ be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity: $$\{{array}{ll} Δu_i+\sum_{j=1}^n a_{ij}|x|^{β_j}e^{u_j(x)}=0,\quad \mathbb R^2, \quad i=1,...,n \int_{\mathbb R^2}|x|^{β_i}e^{u_i(x)}dx<\infty, \quad i=1,...,n {array}. $$ where $β_1,...,β_n$ are constants greater than -2. If all $β_i$s are negative we prove that all solutions are radial and the linearized system is non-degenerate.

Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians

In this paper, we are concerned with equations \eqref{PDE} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to \eqref{PDE} (Theorem \ref{Thm0}). Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to fractional higher-order equations \eqref{PDE} with general nonlineari…

Harnack inequalities and quantization properties for the n-Liouville equation

We consider a quasilinear equation involving the $n-$Laplacian and an exponential nonlinearity, a problem that includes the celebrated Liouville equation in the plane as a special case. For a non-compact sequence of solutions it is known that the exponential nonlinearity converges, up to a subsequence, to a sum of Dirac measures. By performing a precise local asymptotic analysis we complete such a result by showing that the corresponding Dirac masses are quantized as multiples of a given one, related to the mass of limiting profiles after rescaling according to the classification result obtain…

Isolated singularities for the n-Liouville equation

In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in arXiv:1609.03608 and establish a quantization result for entire solutions of the singular n-Liouville equation.

关于盘上奇异李乌维尔方程爆破解分类的研究

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

本文研究了奇异李乌维尔方程的爆破行为，分类了所有爆破解，并确定了解的数量。对于固定α>0和λ∈(0,λ_α)，方程有确切的解数，包括最小能量解、奇异序列和多峰序列。

利鲁维叶型结果：涉及梯度项的准线性椭圆不等式在加权图上的研究

Liouville Type Results for Quasilinear Elliptic Inequalities Involving Gradient Terms on Weighted Graphs

本文研究了加权图上的准线性椭圆不等式Δ_m u +u^p |∇u|^q ≤ 0，通过参数范围确定非平凡正解的不存在性，扩展了黎曼流形上的结果，揭示了离散加权图框架下的独特行为和挑战。

Classification of finite Morse index solutions of higher-order Gelfand-Liouville equation

We classify finite Morse index solutions of the following Gelfand-Liouville equation \begin{equation*} (-Δ)^{s} u= e^u \ \ \text{in} \ \ \mathbb{R}^n, \end{equation*} for $1<s<2$ and $s=2$ via a novel monotonicity formula and technical blow-down analysis. We show that the above equation does not admit any finite Morse index solution with $(-Δ)^{s/2} u$ vanishes at infinity provided $n>2s$ and \begin{equation*} \label{1.condition} \frac{ Γ(\frac{n+2s}{4})^2 }{ Γ(\frac{n-2s}{4})^2} < \frac{Γ(\frac{n}{2}) Γ(1+s)}{ Γ(\frac{n-2s}{2})}, \end{equation*} where $Γ$ is the classical Gamma function. The …

具有非负 Ricci 曲率的紧流形上拟线性方程的梯度界与 Liouville 定理

Gradient Bounds and Liouville theorems for Quasi-linear equations on compact Manifolds with nonnegative Ricci curvature

本文研究非负 Ricci 曲率紧 Riemannian 流形上的拟线性椭圆方程，建立梯度界、Liouville 型定理、等号情形的局部分裂定理，以及梯度的 Harnack 型不等式和 ABP 估计。

$\mathbb{R}^N$ 中各向异性 $N$-Liouville 方程解的分类

Classification of solutions to the anisotropic $N$-Liouville equation in $\mathbb{R}^N$

本文在有限质量条件下完全分类了各向异性N-Liouville方程的解，并针对N=2情形肯定回答了Wang和Xia提出的猜想。

黎曼曲面上的刘维尔方程：体积增长在分类和刚性结果中的作用

The Liouville equation on Riemannian surfaces: the role of volume growth in classification and rigidity results

研究具有非负里奇曲率的黎曼曲面上的刘维尔方程，通过假设解的渐近下界，建立解与环境流形的分类结果，揭示流形体积增长与分类的紧密联系。

非负Ricci曲率流形上的拟线性Liouville方程

Quasilinear Liouville equation on manifolds with nonnegative Ricci curvature

研究非负Ricci曲率完备非紧Riemann流形上n-Laplace拟线性Liouville方程的刚性与分类，证明在尖锐对数下界条件下流形等距于欧氏空间且解为标准气泡，并构造非平坦流形上的解表明假设的尖锐性。

Blow--up Solutions of Liouville's Equation and Quasi--Normality

We prove that the family $\mathcal{F}_C(D)$ of all meromorphic functions $f$ on a domain $D\subseteq \mathbb{C}$ with the property that the spherical area of the image domain $f(D)$ is uniformly bounded by $C π$ is quasi--normal of order $\le C$. We also discuss the close relations between this result and the well--known work of Brézis and Merle on blow--up solutions of Liouville's equation. These results are completely in the spirit of Gromov's compactness theorem, as pointed out at the end of the paper.

具有非线性Neumann边界条件的Liouville方程解的分类及其在尖锐不等式中的应用

Classification of solutions to the Liouville equation with a nonlinear Neumann boundary condition and applications to sharp inequalities

本文通过边界自适应P-函数方法，对具有非线性Neumann边界条件的Liouville方程所有光滑解进行分类，并应用于建立涵盖Lebedev-Milin不等式的尖锐Sobolev迹型不等式族及亏损估计。

A Liouville theorem for the $p$-Laplacian and related questions

We prove several classification results for $p$-Laplacian problems on bounded and unbounded domains, and deal with qualitative properties of sign-changing solutions to $p$-Laplacian equations on $\mathbb R^N$ involving critical nonlinearities. Moreover, on radial domains we characterise the compactness of possibly sign-changing Palais-Smale sequences.

The Liouville theorem for $p$-harmonic functions and quasiminimizers with finite energy

We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite $p$th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global $p$-Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line $\mathbf{R}$, we characterize all locally doubling measures, sup…

二阶分类对奇异李乌维尔方程的第二类分类

Second order classification for singular Liouville equations with a coefficient function

本文研究了奇异李乌维尔方程在单位球内解的发散性，通过分析势函数$V$的第二阶分类，确定了当参数$λ$趋近于零时解序列发散的条件。

单位圆盘上带非线性Robin边界条件的Liouville方程解的分类

Classification of solutions to the Liouville equation with a nonlinear Robin boundary condition on the unit disk

本文研究单位圆盘上带非线性Robin边界条件的Liouville方程，通过Hardy-Wronskian边界刚性方法，对$0<\lambda\le2$给出光滑解完全分类，对$2<\lambda\leq3$证明径向与非径向解的二分性。

关于一类N-利乌维勒方程在R^N中解的分类

On the classification of solutions to a class of $N$-Liouville equations in $\mathbb{R}^N$

本文通过P函数方法对N=2时的N-利乌维勒方程在α>-1范围内的解进行分类，并推广至N≥3的情况，揭示了径向解在α≤0时的唯一性及α>0时的退化特性。

非线性Neumann边界问题的n-拉普拉斯李乌利维尔方程在半空间上的解

Nonlinear Neumann boundary problems for $n$-Laplacian Liouville equation on a half space

本文研究了n-拉普拉斯李乌利维尔方程在半空间上正非线性Neumann边界条件下的解的分类，将二次阶结果推广到一般n≥2，并将临界p-拉普拉斯方程结果推广到p=n。

Classification of solutions to the higher order Liouville's equation on R^{2m}

We classify the solutions to the equation (- Δ)^m u=(2m-1)!e^{2mu} on R^{2m} giving rise to a metric g=e^{2u}g_{R^{2m}} with finite total $Q$-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu(x) as |x|\to \infty. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric e^{2u}g_{R^{2m}} at infinity, and we observe that the pull-back of this metric to $S^{2m}$ via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is …

与N Finsler拉普拉斯算子相关的奇解分类问题

Classification of solutions to the singular Liouville's equation associated with the $N$ Finsler Laplacian

本文在有限质量条件下，对与Finsler-N拉普拉斯算子相关的奇Liouville方程进行分类，放宽了[39]中所需的质量条件。

Liouville type theorems for fractional elliptic problems

In this paper, we establish Liouville type theorems for stable solutions on the whole space $\mathbb R^N$ to the fractional elliptic equation $$(-Δ)^su=f(u)$$ where the nonlinearity is nondecreasing and convex. We also obtain a classification of stable solutions to the fractional Lane-Emden system $$\begin{cases} (-Δ)^s u = v^p \mbox{ in }\mathbb R^N (-Δ)^s v = u^q \mbox{ in }\mathbb R^N \end{cases}$$ with $p>1$ and $ q>1$. In our knowledge, this is the first classification result for stable solutions of the fractional Lane-Emden system in literature.

具有非负Ricci曲率流形上拟线性Liouville方程的尖锐刚性

Sharp rigidity for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature

研究具有非负Ricci曲率流形上的拟线性Liouville方程，通过解的最优对数下界得出流形与欧氏空间等距且解为标准泡解，证明中关键是对数下界与总体积上界联系，还提出并证明了关于解与几何相互作用的猜想。

黎曼流形上Lane-Emden方程的最优Liouville定理

Optimal Liouville theorems for the Lane-Emden equation on Riemannian manifolds

本文在具有非负Ricci曲率的Riemannian流形上，通过严格证明亚临界Lane-Emden-Fowler方程正解的不存在性，扩展了Gidas-Spruck和Serrin-Zou的工作。

The $C^0$-convergence at the Neumann boundary for Liouville equations

In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li \cite{Li} gave a uniform asymptotic estimate. Later, Zhang \cite{Zhang} and Gluck \cite{Gluck} improved Li's estimate in the sense of $C^0$-convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao-Wang-Zhou \cite{Bao-Wang-Zhou} proved a similar asymptotic estima…

拟线性Hénon型N-拉普拉斯刘维尔方程的非径向解

Non-radial solutions for the quasi-linear Hénon type $N$-Laplacian Liouville equation

研究拟线性加权N-拉普拉斯刘维尔方程，通过研究线性化问题、应用逼近方法和分歧理论，在特定参数条件下证明存在非径向解，推广了相关存在性结果，克服了诸多关键困难。

Asymptotic and optimal Liouville properties for Wolff type integral systems

This article examines the properties of positive solutions to fully nonlinear systems of integral equations involving Hardy and Wolff potentials. The first part of the paper establishes an optimal existence result and a Liouville type theorem for the integral systems. Then, the second part examines the decay rates of positive bound states at infinity. In particular, a complete characterization of the asymptotic properties of bounded and decaying solutions is given by showing that such solutions vanish at infinity with two principle rates: the slow decay rates and the fast decay rates. In fact,…

流形上拟线性椭圆方程的对数梯度估计与刘维尔性质的Nash-Moser迭代方法

Nash-Moser iteration approach to the logarithmic gradient estimates and Liouville Properties of quasilinear elliptic equations on manifolds

本文提出采用Nash-Moser迭代技术分析完备黎曼流形上拟线性椭圆方程正解的局部与全局性质，推导出普适的Cheng-Yau型对数梯度估计，并据此获得刘维尔型定理、Harnack不等式及奇点局部估计。

Liouville type theorems for solutions of the weighted fractional Lane-Emden system

In this paper, we prove Liouville type theorems for stable solutions to the weighted fractional Lane-Emden system \begin{align*} (-Δ)^s u = h(x)v^p,\quad (-Δ)^s v= h(x)u^q, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N, \end{align*} where $1<q\leq p$ and $h$ is a positive continuous function in $\mathbb{R}^N$ satisfying $\displaystyle{\liminf_{|x|\to \infty}}\frac{h(x)}{|x|^\ell} > 0$ with $\ell > 0.$ Our results generalize the results established in \cite{HHM16} for the Laplacian case (correspond to $s=1$) and improve the previous work \cite{TuanHoang21}. As a consequence, we prove classification…

三维空间中速度无需可积性条件的分数阶Navier-Stokes方程的Liouville型定理

Liouville type theorems for the fractional Navier-Stokes equations without the integrability condition of velocity in $\mathbb{R}^3$

本文在速度无需可积性条件下，利用Lizorkin乘子定理和Coifman-McIntosh-Meyer型交换子估计，证明了三维分数阶Navier-Stokes方程在无穷远处一致趋于非零常向量的光滑解必为平凡解。

Solution of the Sturm-Liouville and the Korteweg-de-Vries equations with periodic and quasi-periodic parameters using theory of vessels

We prove the existence of solutions to the Sturm-Liouville (SL) equation -y"(x)+q(x)y(x) = s^2 y(x) with periodic and quasi-periodic potential q(x) using theory of SL vessels, implementing a Backlund transformation of SL equation. In this paper quasi-periodic means a finite sum of periodic integrable functions. The solutions for a general s are explicitly constructed in terms of the solutions zn(x), satisfying the SL equation with initial conditions zn(0)=0, zn'(0)=1 for a discrete Levinson set of numbers s=sn, n-natural number. The tau function tau(x) of the corresponding vessel realizes the …

Asymptotic behavior of entire solutions for degenerate partial differential inequalities on Carnot-Carathéodory metric spaces and Liouville type results

This article is devoted to the study of the behavior of generalized entire solutions for a wide class of quasilinear degenerate inequalities modeled on the following prototype with p-Laplacian in the main part \begin{equation*} {\underset{m}{\overset{i=1}{\sum}}} X_i^*(|\mathbf{X}u|^{p-2} X_i u)\geq |u|^{q-2}u, \ \ x\in {\mathbb{R}}^{n},\ q>1,\ p>1, \end{equation*} where ${\mathbb{R}}^{n}$ is a Carnot-Carathéodory metric space, generated by the system of vector fields $\mathbf{X}=(X_1,X_2,..,X_m)$ and $X_i^*$ denotes the adjoint of $X_i$ with respect to Lebesgue measure. For the case where $p$…

Effect of weights on stable solutions of a quasilinear elliptic equation

In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasilinear elliptic equation $-Δ_p u= f(x) F(u) $ in $\mathbb{R}^n$ where $p\ge 2$, $0\le f\in C(\mathbb{R}^n)$ and $F\in C^1(\mathbb{R})$. We refer to $f(x)$ as {\it weight} and to $F(u)$ as {\it nonlinearity}. The remarkable fact is that if the weight function is bounded from below by a strict positive constant that is $0<C\le f$ then it does not have much impact on the stable solutions, however, a nonnegative weight that is $0\le f$ will push certain critical dimensions. This analytical ob…

具单调系数的临界p-拉普拉斯方程：刘维尔分类与Schoen型哈纳克不等式

Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality

本文研究具单调系数的临界p-拉普拉斯方程，完成其正整体解的刘维尔分类，建立Schoen型哈纳克不等式，为经典半线性方法提供拟线性对应及替代证明。

Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations

We study the quasilinear elliptic equation \begin{equation*} -Qu=e^u \ \ \text{in} \ \ Ω\subset \mathbb{R}^{N} \end{equation*} where the operator $Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{ξ_{i}}(\nabla u)),$$ where $F_{ξ_{i}}=\frac{\partial F}{\partialξ_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $ C^{2}(\mathbb{R}^{N}\setminus\{0\})$, that satisfies certain assumptions. For bounded domain $Ω$ and for a stable weak solution of the above equation, we prove that the H…

闭4-流形上的Liouville型方程与Onofri型不等式

The Liouville-type equation and an Onofri-type inequality on closed 4-manifolds

本文针对满足$\textrm{Ric}\geqslant 3κg$且$κ>0$的闭4-流形，研究Liouville型方程，用不变张量法推导微分恒等式分类参数范围内的解，通过连续性方法建立二阶导数估计，还推导了4-球面上的Onofri型不等式并证明其刚性。

Extremal solution and Liouville theorem for anisotropic elliptic equations

We study the quasilinear Dirichlet boundary problem \begin{equation}\nonumber \left\{ \begin{aligned} -Qu&=λe^{u} \quad \mbox{in}\quadΩ\\ u&=0 \quad \mbox{on}\quad\partialΩ,\\ \end{aligned} \right. \end{equation} where $λ>0$ is a parameter, $Ω\subset\mathbb{R}^{N}$ with $N\geq2$ be a bounded domain, and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{ξ_{i}}(\nabla u)). $$ Here, $F_{ξ_{i}}=\frac{\partial F}{\partialξ_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $…

Positive Liouville theorem and asymptotic behaviour for $(p,A)$-Laplacian type elliptic equations with Fuchsian potentials in Morrey space

We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $ζ\in\partialΩ\cup\{\infty\}$ of the quasilinear elliptic equations $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{in } Ω\setminus\{ζ\},$$ where $Ω$ is a domain in $\mathbb{R}^d$ ($d\geq 2$), and $A=(a_{ij})\in L_{\rm loc}^{\infty}(Ω;\mathbb{R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. The potential $V$ lies in a certain local Morrey space (depending on $p$) and has a Fuchsian-type isolated singularity at $ζ$.

凸锥中的各向异性Finsler $N$-Laplacian Liouville方程

Anisotropic Finsler $N$-Laplacian Liouville equation in convex cones

本文研究凸锥中各向异性Finsler $N$-Laplacian Liouville方程，完全分类了有限质量解，将已有结果推广到一般凸锥，并证明了径向Poincaré型不等式作为关键工具。

A Liouville property for eternal solutions to a supercritical semilinear heat equation

We are concerned with solutions to the nonlinear heat equation $u_t=Δu+|u|^{p-1}u$, $x\in \mathbb{R}^N$, that are defined for all positive and negative time. If the exponent $p$ is greater or equal to the Joseph-Lundgren exponent $p_c$ and $|u|$ stays below some positive radially symmetric steady state, under a mild condition on the behaviour of $u$ as $|x|\to \infty$, we show that $u$ is independent of time. Our method of proof uses Serrin's sweeping principle, based on the strong maximum principle, applied to the linearized equation for $u_t$. Our result covers that of Poláčik and Yanagida […

关于图上正拟线性算子的Landis猜想

On the Landis Conjecture for Positive Quasi-linear Operators on Graphs

本文证明了图上正拟线性算子的Landis型唯一延拓定理，通过衰减准则和简化能量方法，应用于模型图与正则树。

维数在稳态准地转系统唯一性结果中的作用

The role of the dimension in uniqueness results for the stationary quasi-geostrophic system

本文针对二维、三维和四维稳态分数阶准地转方程，在特定Lebesgue可积性条件下证明了弱解的唯一性，并阐明了维数与Laplace算子分数幂次对唯一性的影响。

带半线性项的各向异性 $p$-Laplace 方程的 Liouville 定理

Liouville theorems for anisotropic $p$-Laplace equations with a semilinear term

针对带半线性项的各向异性 $p$-Laplace 方程，结合 Serrin 技巧与不变张量方法，建立正解、超/子解及稳定变号解的 Liouville 不存在性与分类结果。

黎曼流形上 Φ-Laplacian 方程的梯度估计与Liouville定理

Gradient estimates and Liouville theorems for the \(Φ\)-Laplacian equations on Riemannian manifolds

本文通过非线性 Φ-Bochner 公式和 Nash-Moser 迭代，在 Ricci 曲率下界和 Φ、G 的适当条件下，建立了 Φ-Laplacian 方程的局部梯度界，进而得到全局解的 Liouville 定理；对于抛物情形，利用最大值原理在紧流形上导出梯度估计并获得 Liouville 型结果。

海森堡群中的一个Liouville定理

A Liouville theorem in the Heisenberg group

本文分类了海森堡群中临界半线性椭圆方程的正解，证明其为Jerison-Lee气泡，并推广了Caffarelli-Gidas-Spruck定理。

基于Liouville比较原理与临界性理论的Landis猜想

The Landis conjecture via Liouville comparison principle and criticality theory

该研究借助Liouville比较原理与临界性理论，针对两类线性二阶椭圆算子在全维度下给出Landis猜想的部分肯定答案，提出非负Schrödinger方程平凡解的精确衰减判据，还拓展研究了拟线性问题的对应猜想。

A Non-Local Quasi-Linear Ground State Representation and Criticality Theory

We study energy functionals associated with non-local quasi-linear Schrödinger operators, and develop a ground state representation. Our main focus is on infinite graphs but we also consider non-local quasi-linear Schrödinger operators in the Euclidean space. Using the representation, we develop a criticality theory for quasi-linear Schrödinger operators on general weighted graphs, and show characterisations for a Hardy inequality to hold true. As an application, we show a Liouville comparison principle on graphs.