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Existence results for mean field equations. (English) Zbl 0937.35055
The authos study the variational problem in $$H^{1,2}_0(\Omega)$$ over a planar domain $$\Omega$$ for the functional $J_\beta (\psi)= {1\over 2}\int_\Omega |\nabla \psi|^2+ {1\over\beta} \log\int_\Omega e^{-\beta \psi}$ with $$\psi=0$$ on $$\partial\Omega$$; the problem arises in Onsager’s vortex model for turbulent Euler flows. If $$\beta> -8\pi$$, a minimizer is known to exist. On the unit disk, solutions blow up as $$\beta\to-8\pi$$; however for a circular annulus radially symmetric solutions have been shown to exist for any $$\beta$$; thus, the behavior of the functional depends on the topology of $$\Omega$$. The authors show in the present work that if the complement of $$\Omega$$ contains a bounded region, (as occurs with an annulus), then there is a solution of the Euler equation for any $$\beta\in (-16\pi,-8\pi)$$. The proof depends on an improved form of the Moser-Trudinger inequality. No such solution can minimize. The paper includes additionally a form of the result that applies to the somewhat more general functional $$J_c(u)={1\over 2}\int_\Sigma |\nabla u|^2 +c\int_\Sigma u-c\log \int_\Sigma Ke^u$$ with $$K$$ a positive function, on a compact Riemann surface $$\Sigma$$ of genus $$\geq 1$$: The Euler equation admits a non-minimizing solution for any $$c\in(8\pi,16\pi)$$. The authors point out related independent work relative to this problem by Struwe and Tarantello, who obtained a nontrivial solution in a particular case.
Reviewer: R.Finn (Stanford)

##### MSC:
 35J60 Nonlinear elliptic equations 35Q35 PDEs in connection with fluid mechanics 76F30 Renormalization and other field-theoretical methods for turbulence
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##### References:
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