A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern–Simons Higgs model and have been studied recently by various authors

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Harnack type inequality : the method of moving planes

In this paper, we consider a sequence of multibubble solutions u k of the equation (0.1) Δ 0 u k + ρ k (he uk /f M he u k d μ -1)=0 in M, where h is a C 2,β positive function in a compact Riemann surface M, and ρ k is a constant satisfying lim k→+ ∞ ρ k = 8mπ for some positive integer m ≥ 1. We prove among other things that ρ k - 8mπ = 2/m m Σ/j=1h -1 (p k,j )(Δ 0 log h(p k,j ) + 8mπ - 2K(p k,j ))λ k,j e -λk,j + O(e -λk,j ), where p k,j are centers of the bubbles of u k and λ k,j are the local maxima of u k after adding a constant. This yields a uniform bound of solutions as ρ k converges to 8mπ from below provided that Δ 0 log h(p k,j ) + 8mπ - 2K(p k,j ) > 0. It generalizes a previous result, due to Ding, Jost, Li, and Wang [18] and Nolasco and Tarantello [31], which says that any sequence of minimizers u k is uniformly bounded if ρ k 0 for any maximum point p of the sum of 2 log h and the regular part of the Green function, where K is the Gaussian curvature of M. The analytic work of this paper is the first step toward computing the topological degree of (0.1), which was initiated by Li [24].

Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces

We consider the following mean field equations: 


(0.1)



where M is a compact Riemann surface with volume 1, h is a positive continuous function on M, ρ is a real number, 


(0.2)



and where Ω is a bounded smooth domain, h is a C1 positive function on Ω, and ρ ∈ ℝ. Based on our previous analytic work [14], we prove, among other things, that the degree-counting formula for (0.1) is given by () for ρ ∈ (8mπ, 8(m + 1)π). © 2003 Wiley Periodicals, Inc.

Topological degree for a mean field equation on Riemann surfaces

http://ucce.ucdavis.edu/files/datastore/234-16.pdf

Effects of superatmospheric oxygen levels on postharvest physiology and quality of fresh fruits and vegetables

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of [35].

/pdf/existence-of-conformal-metrics-with-constant-q-curvature-2a368t2pr2.pdf

Existence of conformal metrics with constant Q-curvature

Let Ω be an annulus. We prove that the mean field equation 
−δψ=e−βψ∫Ωe−βψ in Ω

ψ=0 on∂Ω
admits a solution for β ∈ (−16π, −8π). This is a supercritical case for the Moser-Trudinger inequality.

/pdf/existence-results-for-mean-field-equations-1m2661mgas.pdf

Existence results for mean field equations

Let $\Omega$ be an annulus. We prove that the mean field equation $-\Delta\psi=\frac{e\sp{-\beta\psi}}{\int\sb{\Omega}e\sp{-\beta\psi}} $ admits a solution with zero boundary for $\beta\in (-16\pi,-8\pi)$. This is a supercritical case for the Moser-Trudinger inequality.

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1997/0001/0002/AJM-1997-0001-0002-a003.pdf

The differential equation $\Delta u = {8 \pi - 8 \pi he}^u$ on a compact Riemann surface

We introduce a functional that couples the nonlinear sigma model with a spinor field: In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem.

Dirac-harmonic maps

Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for two-vortex solutions of the periodic Chern-Simons Higgs model and analyze the asymptotic behavior of these solutions as the parameter coupling the gauge field with the scalar field tends to 0.

Jiayu Li

Papers

Existence results for mean field equations

Existence results for mean field equations

The differential equation $\Delta u = {8 \pi - 8 \pi he}^u$ on a compact Riemann surface

Dirac-harmonic maps

An analysis of the two-vortex case in the Chern-Simons Higgs model