arXiv subjects
Publications and source records attributed to Genggeng Huang.
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.
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}.
In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, \begin{equation} \sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-α}}\leq C_{r,s,α} |f|_{l^r} |g|_{l^s}, \end{equation}where $i,j\in \mathbb Z^n$, $r,s>1$, $0<α 2$.
In this paper, we mainly discuss a priori bounds of the following degenerate elliptic equation, {equation}\label{000} a^{ij}(x)\partial_{ij}u+b^i(x)\partial_i u +f(x,u)=0,\text{in}Ω\subset\subset R^n, {equation} where $a^{ij}\partial_i ϕ\partial_j ϕ=0$ on $\partial Ω$, $ϕ$ is the defining function of $\partial Ω$. Imposing suitable conditions on the coefficients and $f(x,u)$, one can get the $L^\infty$-estimates of \eqref{000} via blow up method.
In this paper, we apply the moving plane method to some degenerate elliptic equations to get a Liouville type theorem. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic equations.