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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)

35J60 Nonlinear elliptic equations
35Q35 PDEs in connection with fluid mechanics
76F30 Renormalization and other field-theoretical methods for turbulence
Full Text: DOI Numdam EuDML
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