arXiv subjects

Chang-shou Lin

Publications and source records attributed to Chang-shou Lin.

7 recordsLinked to original sources

arXiv2017

Degree counting for Toda system with simple singularity : one point blow up

In this paper, we study the degree counting formula of the rank two Toda system with simple singular source when $ρ_1\in(0,4π)\cup(4π,8π)$ and $ρ_2\notin 4π\mathbb{N}.$ The key step is to derive the degree formula of the shadow system, which arises from the bubbling solutions as $ρ_1$ tends to $4π$. In order to compute the topological degree of the shadow system, we need to find some suitable deformation. During this deformation, we shall deal with \textit{new} difficulty arising from the new phenomena: blow up does not necessarily imply concentration of mass. This phenomena occurs due to the collapsing of singularities. This is a continuation of the previous work Lee, Lin, Wei and Yang.

math.AP↗

arXiv2017

Sharp estimates for solutions of mean field equation with collapsing singularity

The pioneering work of Brezis-Merle [7], Li-Shafrir [27], Li [26] and Bartolucci-Tarantello [4] showed that any sequence of blow up solutions for (singular) mean field equations of Liouville type must exhibit a "mass concentration" property. A typical situation of blow-up occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case Lin-Tarantello in [30] pointed out that the phenomenon: "bubbling implies mass concentration" might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a "non-concentration" situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow up parameter to describe the bubbling properties of the solution-sequence. In this way we are able to establish accurate estimates around the blow-up points which we hope to use towards a degree counting formula for the shadow system (1.34) below.

math.AP↗

arXiv2015

Energy concentration and a priori estimates for $B_2$ and $G_2$ types of Toda systems

For Toda systems with Cartan matrix either $B_2$ or $G_2$, we prove that the local mass of blowup solutions at its blowup points converges to a finite set. Further more this finite set can be completely determined for $B_2$ Toda systems, while for $G_2$ systems we need one additional assumption. As an application of the local mass classification we establish a priori estimates for corresponding Toda systems defined on Riemann surfaces.

math.AP↗

arXiv2013

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.

math.AP↗

arXiv2013

On Liouville systems at critical parameters, Part 1: one bubble

In this paper we consider bubbling solutions to the general Liouville system: \label{abeq1} Δ_g u_i^k+\sum_{j=1}^n a_{ij}ρ_j^k(\frac{h_j e^{u_j^k}}{\int h_j e^{u_j^k}}-1)=0\quad\text{in}M, i=1,...,n (n\ge 2) where $(M,g)$ is a Riemann surface, and $A=(a_{ij})_{n\times n}$ is a constant non-negative matrix and $ρ_j^k\to ρ_j$ as $k\to \infty$. Among other things we prove the following sharp estimates. The location of the blowup point. The convergence rate of $ρ_j^k-ρ_j$, $j=1,..,n$. These results are of fundamental importance for constructing bubbling solutions. It is interesting to compare the difference between the general Liouville system and the SU(3) Toda system on estimates (1) and (2).

math.AP↗

arXiv2010

A Topological Degree Counting for some Liouville Systems of Mean Field Equations

Let $A=(a_{ij})_{n\times n}$ be an invertible matrix and $A^{-1}=(a^{ij})_{n\times n}$ be the inverse of $A$. In this paper, we consider the generalized Liouville system: \label{abeq1} Δ_g u_i+\sum_{j=1}^n a_{ij}ρ_j(\frac{h_j e^{u_j}}{\int h_j e^{u_j}}-1)=0\quad\text{in \,}M, where $0< h_j\in C^1(M)$ and $ρ_j\in \mathbb R^+$, and prove that, under the assumptions of $(H_1)$ and $(H_2)$\,(see Introduction), the Leray-Schauder degree of \eqref{abeq1} is equal to \frac{(-χ(M)+1)... (-χ(M)+N)}{N!} if $ρ=(ρ_1,..., ρ_n)$ satisfies 8πN\sum_{i=1}^nρ_i<\sum_{1\leq i,j\leq n}a_{ij}ρ_iρ_j<8π(N+1)\sum_{i=1}^nρ_i. Equation \eqref{abeq1} is a natural generalization of the classic Liouville equation and is the Euler-Lagrangian equation of Nonlinear function $\varPhi_ρ$: \varPhi_ρ(u)=1/2\int_M\sum_{1\leq i,j\leq n}a^{ij}\nabla_g u_i\cdot \nabla_g u_j+\sum_{i=1}^n\int_Mρ_iu_i -\sum_{i=1}^nρ_i\log \int_M h_i e^{u_i}. The Liouville system \eqref{abeq1} has arisen in many different research areas in mathematics and physics. Our counting formulas are the first result in degree theory for Liouville systems.

math.AP↗

arXiv2009

Profile of bubbling solutions to a Liouville system

In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in $\mathbb R^2$. Then we establish an uniform estimate for bubbling solutions of a locally defined Liouville system near an isolated blowup point. The uniqueness result, as well as the local uniform estimates are crucial ingredients for obtaining a priori estimate, degree counting formulas and existence results for Liouville systems defined on Riemann surfaces.

math.AP↗