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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.)
##### Keywords:
Nirenberg problem; compact Riemann surface
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