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11.
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). © 1997 John Wiley & Sons, Inc. 相似文献
12.
13.
Jann-Long Chern Zhi-You Chen Chang-Shou Lin 《Communications in Mathematical Physics》2010,296(2):323-351
The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce
and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. In
this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and
give a complete study of the radial solutions, in particular, the existence of some non-topological solutions. 相似文献
14.
Let S
2 be the 2-dimensional unit sphere and let J
α
denote the nonlinear functional on the Sobolev space H
1(S
2) defined by
$J_\alpha(u) = \frac{\alpha}{16\pi}\int_{S^2}|\nabla u|^2\, d\mu_0 + \frac{1}{4\pi} \int_{S^2} u\, d \mu_0 -{\rm ln} \int_{S^2} e^{u}
\, \frac{d \mu_0}{4\pi},$J_\alpha(u) = \frac{\alpha}{16\pi}\int_{S^2}|\nabla u|^2\, d\mu_0 + \frac{1}{4\pi} \int_{S^2} u\, d \mu_0 -{\rm ln} \int_{S^2} e^{u}
\, \frac{d \mu_0}{4\pi}, 相似文献
15.
We study the existence of bubbling solutions for the the following Chern–Simons–Higgs equation: $$\Delta u +\frac1{\varepsilon^2} {\rm e}^u(1-{\rm e}^u) = 4\pi \sum_{i=1}^{2k}\delta_{p_i},\quad \text{in}\,\Omega,$$ where Ω is a torus. If k = 1, for any critical point q of the associated sum of the Green functions, we introduce a quantity D(q) (see (1.11) below). We show that for any non-degenerate critical point q with D(q) < 0, the above problem has a solution u ε satisfying that ε → 0, u ε blows up at q. The calculations in this paper also show that, if a sequence of solutions u ε blows up at q as ε → 0, then q must be a critical point of the associated sum of the Green functions, and ${D(q) \leqq 0}$ . So, the condition D(q) < 0 is almost necessary to obtain our result. We also construct solutions with k bubbles for ${k \geqq 2}$ . 相似文献
16.
In this paper, we obtain sharp estimates of fully bubbling solutions of SU(3) Toda system in a compact Riemann surface. In geometry, the SU(n?+?1) Toda system is related to holomorphic curves, harmonic maps or harmonic sequences of the Riemann surface to ${\mathbb{CP}^n}$ . In order to compute the Leray?CSchcuder degree for the Toda system, we have to obtain accurate approximations of the bubbling solutions. Our main goals in this paper are (i) to obtain a sharp convergence rate, (ii) to completely determine the locations, and (iii) to derive the ${\partial _z^2}$ condition, a unexpected and important geometric constraint. 相似文献
17.
Kuo-Shung Cheng Chang-Shou Lin 《Calculus of Variations and Partial Differential Equations》2000,11(2):203-231
In this paper, we consider the equation
where is a nonpositive function in . A solution u is said to be complete if the conformal metric is complete in . Let
Assuming only that , we prove that equation (0.1) possesses infinitely many complete solutions. If in addition, K is assumed to satisfy
for some positive constant m, then is also necessary for equation (0.1) to have a complete solution with finite total curvature. We are also able to classify
the solution set of equation (0.1) for a wider class of the curvature function K than those considered in [5, 6].
Received October 1, 1997 / Revised version August 10, 1999 / Published online April 6, 2000 相似文献
18.
We prove the existence of topological vortices in a relativistic self-dual Abelian Chern-Simons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R2. We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R2 limit from the bounded-domain solutions to obtain a topological solution in R2. 相似文献
19.
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 R2. 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. 相似文献
20.
In this paper we consider the problem
|