Chang-Shou Lin

ESTIMATES OF THE CONFORMAL SCALAR CURVATURE EQUATION VIA THE METHOD OF MOVING PLANES

In this paper we derive a local estimate of a positive singular solution u near its singular set Z of the conformal equation where K(x) is a positive continuous function, Z is a compact subset of , and g satisfies that is nonincreasing for t > 0. Assuming that the order of flatness at critical points of K on Z is no less than , we prove that, through the application of the method of moving planes, the inequality holds for any solution of (0.1) with Cap(Z) = 0. By the same method, we also derive a Harnack-type inequality for smooth positive solutions. Let u satisfy Assume that the order of flatness at critical points of K is no less than n - 2; then the inequality holds for R ≤ 1. We also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality (0.4).

Profile of Blow-up Solutions to Mean Field Equations with Singular Data

Abstract Motivated by the study of selfdual vortices in gauge field theory, we consider a class of Mean Field equations of Liouville-type on compact surfaces involving singular data assigned by Dirac measures supported at finitely many points (the so called vortex points). According to the applications, we need to describe the blow-up behavior of solution-sequences which concentrate exactly at the given vortex points. We provide accurate pointwise estimates for the profile of the bubbling sequences as well as “sup + inf” estimates for solutions. Those results extend previous work of Li [Li, Y. Y. (1999). Harnack type inequality: The method of moving planes. Comm. Math. Phys. 200:421–444] and Brezis et al. [Brezis, H., Li, Y. Shafrir, I. (1993). A sup + inf inequality for some nonlinear elliptic equations involving the exponential nonlinearities. J. Funct. Anal. 115: 344–358] relative to the “regular” case, namely in absence of singular sources.

Interpolation inequalities with weights

We use elementary methods to prove a sufficient and necessary condition for a Sobolev interpolation inequalities with weight [ILLM0001] where [ILLM0001] are real numbers, and [ILLM0001]

A counterexample to the nodal domain conjecture and a related semilinear equation

In this paper we first establish a nonuniqueness result for a semilinear Dirichlet problem of which the nonlinearity is of super-critical growth. We then apply this result to construct a Schr6dinger operator on a domain Q such that the second eigenfunctions of this operator (with zero Dirichlet boundary data) have their nodal sets completely contained in the interior of the domain 0.

Topological degree for mean field equations on S2

Equation (1.1) ρ is called the mean field equation because it often arises in the context of statistical mechanics of point vortices in the mean field limits. Recently, there has been interest in (1.1) ρ because it also arises from the Chern-Simons-Higgs model vortex theory when some parameter tends to zero. (For these recent developments, we refer the readers to [5], [2], [3], [10], [11], [13], [14], [18], [19], [21], [22], and the references therein.) Clearly, equation (1.1) ρ is the Euler-Lagrange equation of the nonlinear functional

Profile of bubbling solutions to a Liouville system

Abstract 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 R 2 . Then we establish a 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.

On the symmetry of blowup solutions to a mean field equation

In this article, we consider the mean field equation Δu+ρeu∫eu−1A=0inΣ, where Σ is a flat torus and A is the area of Σ. This paper is concerned with the symmetry induced by the phenomenon of concentration. By using the method of moving planes, we prove that blowup solutions often possess certain symmetry. In this paper, we consider cases when solutions blowup at one or two points. We also consider related problems for annulu domains of R2.

Uniqueness of least energy solutions to a semilinear elliptic equation in ℝ2

SummaryIn this paper, we prove that solutions minimizing the nonlinear functional

The symmetry of least-energy solutions for semilinear elliptic equations

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