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Genggeng Huang

Publications and source records attributed to Genggeng Huang.

At least 19 recordsLinked to original sources
arXiv2025

Weighted Eigenvalue Problem for a Class of Singular/Degenerate $k$-Hessian Equations

In this paper, we study the existence and uniqueness of solutions to the weighted eigenvalue problem for $k$-Hessian equation. To achieve this, we establish the uniform a priori estimates for gradient and second derivatives of solutions to Hessian equation with weight $|x|^{2sk}$ on the right-hand-side. We also prove that the eigenfunction is a minimizer of the corresponding functional among all $k$-admissible functions vanishing on the boundary.

math.AP↗
arXiv2024

Monge-Ampère equation with Guillemin boundary condition in high dimension

The Guillemin boundary condition naturally appears in the study of Kähler geometry of toric manifolds. In the present paper, the following Guillemin boundary value problem is investigated \begin{align} \label{eq1} &\det D^2 u=\frac{h(x)}{\prod_{i=1}^N l_i(x)},\quad\text{in}\quad\quad P\subset\mathbb R^n, \quad\quad \quad \quad\quad \quad \quad \quad\quad (1)\\ \label{bdy1} &u(x)-\sum_{i=1}^N l_i(x)\ln l_i(x)\in C^\infty(\overline{P}). \quad\quad\quad\quad \quad \quad\quad \quad \quad \quad\quad\quad (2) \end{align} Here \begin{equation*} 0 0\} \end{equation*} is a simple convex polytope in $\mathbb R^n$. The solvability of (1)-(2) is given under the necessary and sufficient condition. The key issue in the proof is to obtain the boundary regularity of $u(x)-\displaystyle \sum_{i=1}^N l_i(x)\ln l_i(x)$. Due to the difficulty caused by the structure of the equation itself and the singularity of $\partial P$, special attention is required to understand the influence of different singularity types at various positions on $\partial P$ and how these impact the behavior of $u$ in its vicinity.

math.AP↗
arXiv2024

Regularity of the $p$-Gauss curvature flow with flat side

We study the regularity of the $p$-Gauss curvature flow with flat side. In our previous paper(arxiv:2403.12292), we obtained the regularity of the interface, namely the boundary of the flat part. In this paper, we study the regularity of the convex hypersurface near the interface.

math.DG↗
arXiv2024

Mixed type boundary value problem of elliptic equation in a thin domain

In this paper, we prove the a priori estimates for two-dimensional second order homogeneous linear elliptic equations in a narrow region. In a crescent-shaped area, part of the boundary is subject to an oblique derivative boundary condition, while the other part of the boundary is subject to a Dirichlet boundary condition. We show that, as the crescent-shaped area collapses into a segment under suitable conditions, the boundary value problem obeys uniform Schauder estimates and induces an asymptotic estimate.

math.AP↗
arXiv2023

Classification of solutions of higher order critical Choquard equation

In this paper, we classify the solutions of the following critical Choquard equation \[ (-Δ)^{\frac{n}{2}} u(x) = \int_{\mathbb{R}^n} \frac{e^{\frac{2n- μ}{2}u(y)}}{|x-y|^μ}dy e^{\frac{2n- μ}{2}u(x)}, \ \text{in} \ \mathbb{R}^n, \] where $ 0<μ< n$, $ n\ge 2$. Suppose $ u(x) = o(|x|^2) \ \text{at} \ \infty $ for $ n \geq 3$ and satisfies \[ \int_{\mathbb{R}^n}e^{\frac{2n- μ}{2}u(y)} dy < \infty, \ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{e^{\frac{2n- μ}{2}u(y)}}{|x-y|^μ} e^{\frac{2n- μ}{2}u(x)} dy dx < \infty. \] By using the method of moving spheres, we show that the solutions have the following form \[ u(x)= \ln \frac{C_1(\varepsilon)}{|x-x_0|^2 + \varepsilon^2}. \]

math.AP↗
arXiv2022

Classification of solutions for some mixed order elliptic system

In this paper, we classify the solution of the following mixed-order conformally invariant system with coupled nonlinearity in $ \mathbb{R}^4$: \begin{equation}\left\{ \begin{aligned} & -Δu(x) = u^{p_1}(x) e^{q_1v(x)}, \quad x\in \mathbb{R}^4,\\ & (-Δ)^2 v(x) = u^{p_2}(x) e^{q_2v(x)}, \quad x\in \mathbb{R}^4, \end{aligned} \right. \end{equation} where $ 0\leq p_1 < 1$, $ p_2 >0$, $ q_1 > 0$, $ q_2 \geq 0$, $ u>0$ and satisfies $$ \int_{\mathbb{R}^4} u^{p_1}(x) e^{q_1v(x)} dx < \infty,\quad \int_{\mathbb{R}^4} u^{p_2}(x) e^{q_2 v(x)} dx < \infty.$$ Under additional assumptions (H1) or (H2), we study the asymptotic behavior of the solutions to the system and we establish the equivalent integral formula for the system. By using the method of moving spheres, we obtain the classification results of the solutions in the system.

math.AP↗
arXiv2021

Regularity of free boundary for the Monge-Ampère obstacle problem

In this paper, we prove the regularity of the free boundary in the Monge-Ampère obstacle problem $\det D^2 v= f(y)χ_{\{v>0\}}. $ By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of the solution to the singular Monge-Ampère equation $\det D^2 u = 1/f (Du)+δ_0$ at the origin. We first establish an asymptotic estimate for the solution $u$ near the singular point, then use a partial Legendre transform to change the Monge-Ampère equation to a singular, fully nonlinear elliptic equation, and establish the regularity of solutions to the singular elliptic equation.

math.AP↗
arXiv2020

Analyticity of the solutions to degenerate Monge-Ampère equations

This paper is devoted to study the following degenerate Monge-Ampère equation: \begin{eqnarray}\label{ab1} \begin{cases} \det D^2 u=Λ_q (-u)^q \quad \text{in}\quad Ω,\\ u=0 \quad\text{on}\quad \partialΩ\end{cases} \end{eqnarray} for some positive constant $Λ_q$. Suppose $Ω\subset\subset \mathbb R^n$ is uniformly convex and analytic. Then the solution of the degenerate Monge-Ampère equation is analytic in $\barΩ$ provided $q\in \mathbb Z^+$.

math.AP↗
arXiv2019

Coexisting Vortices and Antivortices Generated by Dually Gauged Harmonic Maps

In this paper we first formulate a dually gauged harmonic map model, suggested from a product Abelian Higgs field theory arising in impurity-inspired field theories, and obtain a new BPS system of equations governing coexisting vortices and antivortices, which are topologically characterized by the first Chern class of the underlying Hermitian bundle and the Thom class of the associated dual bundle. We then establish existence and uniqueness theorems for such vortices. For the equations over a compact surface, we obtain necessary and sufficient conditions for the existence of solutions. For the equations over the full plane, we obtain all finite-energy solutions. Besides, we also present precise expressions giving the values of various physical quantities of the solutions, including magnetic charges and energies, in terms of the total numbers of vortices and antivortices, of two species, and the coupling parameters involved.

math-ph↗
arXiv2016

A Liouville theorem for subcritical Lane-Emden system

In this paper, we present a necessary and sufficient condition to the Lane-Emden conjecture. This condition is an energy type of integral estimate on solutions to subcritical Lane-Emden system. To approach the long standing and interesting conjecture, we believe that one plausible path is to refocus on establishing this energy type estimate.

math.AP↗
arXiv2015

On the Hardy-Littlewood-Sobolev type systems

In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.

math.AP↗
arXiv2014

Uniqueness of topological solutions of self-dual Chern-Simons equation with collapsing vortices

We consider the following Chern-Simons equation, \begin{equation} \label{0.1} Δu+\frac 1{\varepsilon^2} e^u(1-e^u)=4π\sum_{i=1}^N δ_{p_i^\varepsilon},\quad \text{in}\quad Ω, \end{equation} where $Ω$ is a 2-dimensional flat torus, $\varepsilon>0$ is a coupling parameter and $δ_p$ stands for the Dirac measure concentrated at $p$. In this paper, we proved that the topological solutions of \eqref{0.1} are uniquely determined by the location of their vortices provided the coupling parameter $\varepsilon$ is small and the collapsing velocity of vortices $p_i^\varepsilon$ is slow enough or fast enough comparing with $\varepsilon$. This extends the uniqueness results of Choe \cite{Choe2005} and Tarantello \cite{Tarantello2007}. Meanwhile, for any topological solution $ψ$ defined in $\mathbb R^2$ whose linearized operator is non-degenerate, we construct a sequence topological solutions $u_\varepsilon$ of \eqref{0.1} whose asymptotic limit is exactly $ψ$ after rescaling around $0$. A consequence is that non-uniqueness of topological solutions in $\mathbb R^2$ implies non-uniqueness of topological solutions on torus with collapsing vortices.

math.AP↗
arXiv2014

A Liouville theorem for high order degenerate elliptic equations

In this paper, we apply the moving plane method to the following high order degenerate elliptic equation,\begin{equation*} (-A)^p u=u^α\text{ in } \mathbb R^{n+1}_+,n\geq 1, \end{equation*}where the operator $A=y\partial_y^2+a\partial_y+Δ_x,a\geq 1$. We get a Liouville theorem for subcritical case and classify the solutions for the critical case.

math.AP↗
arXiv2014

Compactness of Alexandrov-Nirenberg Surfaces

We study a class of compact surfaces in $\mathbb R^3$ introduced by Alexandrov and generalized by Nirenberg and prove a compactness result under suitable assumptions on induced metrics and Gauss curvatures.

math.DG↗
arXiv2014

Existence of non-topological solutions for a skew-symmetric Chern-Simons system

We investigate the existence of non-topological solutions $(u_1,u_2)$ satisfying $$u_{i}(x)=-2β_i\ln|x|+O(1),\quad\text{as }|x|\rightarrow +\infty,$$ such that $β_i>1$ and $$(β_1-1)(β_2-1)>(N_1+1)(N_2+1),$$ for a skew-symmetric Chern-Simons system. By the bubbling analysis and the Leray-Schauder degree theory, we get the existence results except for a finite set of curves: $$\frac{N_1}{β_1+N_1}+\frac{N_2}{β_2+N_2}=\frac{k-1}{k},k=2,\cdots,\max(N_1,N_2).$$ This generalizes a previous work by Choe-Kim-Lin \cite{ChoeKimLin2011}.

math.AP↗