Existence results for mean field equations

Jiayu Li; Guofang Wang; Weiyue Ding; Jürgen Jost

doi:10.1016/s0294-1449(99)80031-6

JournalsaihpcVol. 16, No. 5pp. 653–666

Existence results for mean field equations

Jiayu Li
Institute of Mathematics, Academia Sinica, 100080 Beijing, China
Guofang Wang
Institute of Mathematics, Academia Sinica, 100080 Beijing, China
Weiyue Ding
Institute of Mathematics, Academia Sinica, Beijing 100080, P. R. China
Jürgen Jost
Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig, Germany

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Abstract

Let $Ω$ be an annulus. We prove that the mean field equation

{- Δ ψ = \frac{e ^{- β ψ}}{\int _{Ω} e ^{- β ψ}} ψ = 0 in Ω on \partial Ω

admits a solution for $β \in (- 16 π, - 8 π)$ . This is a supercritical case for the Moser–Trudinger inequality.

Résumé

On montre que l’équation de champ moyen

{- Δ ψ = \frac{e ^{- β ψ}}{\int _{Ω} e ^{- β ψ}} ψ = 0 dans Ω sur \partial Ω

pour $Ω$ étant un anneau, admet une solution pour $β \in (- 16 π, - 8 π)$ . Celà represente un cas supercritique pour l’inegalité de Moser–Trudinger.

Cite this article

Jiayu Li, Guofang Wang, Weiyue Ding, Jürgen Jost, Existence results for mean field equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), no. 5, pp. 653–666

DOI 10.1016/S0294-1449(99)80031-6