Asymptotic radial symmetry for solutions of Δu+eu= 0 in a punctured disc

Chou, Kai Seng; Wan, Tom Yau-heng

doi:10.2140/pjm.1994.163.269

Cited by 84 publications

(76 citation statements)

References 4 publications

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“…It is easy to show (see the Appendix) that there is no solution to (23) if λ ≤ 0, so λ > 0. On the other hand, due to the classification of all solutions of (23) (see, for example, [19,21] and [15]), we know that v is the function given in (10). It follows that…”

Section: Compactness and Existence For λ ∈ (−∞ 8π)mentioning

confidence: 99%

Harnack Type Inequality: the Method of Moving Planes

Li¹

1999

Communications in Mathematical Physics

273

271

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Abstract: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. IntroductionLet (M, g) be a compact Riemann surface without boundary, V be a positive function on M , W be a function with M W dv g = 1. Throughout the paper dv g denotes the volume element of g, g denotes the Laplace Beltrami operator with respect to g. For λ ∈ R, we seek a solution ofClearly M W dv g = 1 is a necessary condition for (E u ) λ to have a solution. If we set ξ = u − log M V e u dv g for a solution of (E u ) λ , then ξ satisfies

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Section: Compactness and Existence For λ ∈ (−∞ 8π)mentioning

confidence: 99%

Harnack Type Inequality: the Method of Moving Planes

Li¹

1999

Communications in Mathematical Physics

273

271

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“…It has also been proven in [14], and subsequently in [10,16] by using alternate techniques, that (13) has only one regular family of solutions, given by…”

Section: The Classical Limit (Bennett Theory)mentioning

confidence: 99%

Symmetry Results for Finite-Temperature,¶Relativistic Thomas-Fermi Equations

Kiessling

2002

Communications in Mathematical Physics

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In the semi-classical limit the relativistic quantum mechanics of a stationary beam of counter-streaming (negatively charged) electrons and one species of positively charged ions is described by a nonlinear system of finite-temperature Thomas-Fermi equations. In the high temperature / low density limit these Thomas-Fermi equations reduce to the (semi-)conformal system of Bennett equations discussed earlier by Lebowitz and the author. With the help of a sharp isoperimetric inequality it is shown that any hypothetical particle density function which is not radially symmetric about and decreasing away from the beam's axis would violate the virial theorem. Hence, all beams have the symmetry of the circular cylinder. * In celebration of the 70 th birthday of Joel L. Lebowitz.

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“…The question as to when such solutions u satisfy (1.2) or (1. 4) was left open in that paper (see [11, open question at the bottom of p. 1887 and conjecture on p. 1889]). Many authors (see for example [1], [2], [3], [4], [5], [6], [7]) have studied the asymptotic behavior at an isolated singularity of solutions of the differential equation −∆u = f (u) under various conditions on the positive function f .…”

Section: U(x) H(x)mentioning

confidence: 99%

“…4) was left open in that paper (see [11, open question at the bottom of p. 1887 and conjecture on p. 1889]). Many authors (see for example [1], [2], [3], [4], [5], [6], [7]) have studied the asymptotic behavior at an isolated singularity of solutions of the differential equation −∆u = f (u) under various conditions on the positive function f . Of particular relevance to our results is a result of Lions [8] which states that every C 2 positive solution of −∆u = u p in a punctured neighborhood of the origin in R R R n is asymptotically harmonic as |x| → 0 + provided p < n/(n − 2) (if n = 2, p < ∞).…”

Section: U(x) H(x)mentioning

confidence: 99%