High Degeneracy of Effective Hamiltonian in Two Dimensions

具有可变退化性和Hamilton低阶项的全非线性方程的梯度正则性

Gradient Regularity for Fully Nonlinear Equations with Variable Degeneracy and Hamiltonian Lower-Order Terms

研究具有可变梯度退化和Hamilton型低阶项的全非线性椭圆方程的粘性解，通过紧致性估计和扁平化改进迭代，证明了内梯度Hölder估计、源项和Hamilton系数消失点的逐点改进以及极值点的Schauder型估计。

Existence and non-degeneracy of positive multi-bubbling solutions to critical elliptic systems of Hamiltonian type

This paper deals with the following critical elliptic systems of Hamiltonian type, which are variants of the critical Lane-Emden systems and analogous to the prescribed curvature problem: \begin{equation*} \begin{cases} -Δu_1=K_1(y)u_2^{p},\ y\in \mathbb{R}^N,\\ -Δu_2=K_2(y)u_1^{q}, \ y\in \mathbb{R}^N,\\ u_1,u_2>0, \end{cases} \end{equation*} where $N\geq 5, p,q\in(1,\infty)$ with $\frac1{p+1}+\frac1{q+1}=\frac{N-2}N$, $K_1(y)$ and $K_2(y)$ are positive radial potentials. At first, under suitable conditions on $K_1,K_2$ and the certain range of the exponents $p,q$, we construct an unbounded s…

带哈密顿项的退化完全非线性非局部方程的正则性理论

Regularity theory for degenerate fully nonlinear nonlocal equations with a Hamiltonian term

针对带哈密顿项的退化完全非线性非局部椭圆方程，通过刻画退化规律与哈密顿项的作用关系，结合Ishii-Lions方法与扰动技术，建立解的Lipschitz正则性、梯度Hölder连续性并探究其$C^1$可微性。

带哈密顿项的退化完全非线性椭圆方程的高阶Hölder正则性

Higher Hölder regularity for degenerate fully nonlinear elliptic equations with Hamiltonian terms

针对带变指数双相退化律和哈密顿项的完全非线性椭圆方程，在哈密顿系数和源项消失的点处获得梯度Hölder正则性改进，并在局部极值处建立关于消失率的Sharp Schauder型估计。

带哈密顿项的退化/奇异规范化 $p$-Laplacian 方程的正则性

Regularity for degenerate/singular normalized $p$-Laplacian equations with Hamiltonian terms

针对变指数双相型退化/奇异和哈密顿项的规范化 $p$-Laplacian 方程，基于新的振荡型估计和局部分析，统一建立了尖锐的 $C^{1,α}$ 正则性估计，并在退化情形下得到了梯度 Hölder 正则性的改进结果。

Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two

We are concerned with the Lane-Emden problem \begin{equation*} \begin{cases} -Δu=u^{p} &{\text{in}~Ω},\\[0.5mm] u>0 &{\text{in}~Ω},\\[0.5mm] u=0 &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $Ω\subset \mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently large. Improving some known asymptotic estimates on the solutions, we prove the non-degeneracy and local uniqueness of the multi-spikes positive solutions for general domains. Our methods mainly use ODE's theory, various local Pohozaev identities, blow-up analysis and the properties of Green's function.