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The differential equation \(\Delta u=8\pi-8\pi he^u\) on a compact Riemann surface. (English) Zbl 0955.58010
Summary: Let \(M\) be a compact Riemann surface, \(h(x)\) a positive smooth function on \(M\).
We consider the functional \[ J(u)= \textstyle {1\over 2}\int_M|\nabla u|^2 +8\pi\int_M u-8\pi\log \int_M he^u. \] We give a sufficient condition under which \(J\) achieves its minimum.

MSC:
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
35J60 Nonlinear elliptic equations
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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