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Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds. (English) Zbl 1100.53036
The main results of the paper are the following theorems: Theorem I. If $$(M,g)$$ is a compact Riemannian manifold without boundary, then $\sup\left\{\int_M e^{a_n|u|^{\frac{n}{n-1}}}\,dV_g:u\in H^{1,n}(M), \;\int_Mu\,dV_g=0,\;\int_M|\nabla u|^n\,dV_g=1\right\}$ and $\sup\left \{\int_Me^{a_n|u|^{\frac{n}{n-1}}}\,dV_g:u\in H^{1,n}(M),\;\int_M| \nabla u|^n+|u|^n\,dV_g=1\right\}$ are attained. Theorem II. If $$(N, g_N)$$ is a compact Riemannian manifold with boundary, then $\sup\left \{\int_Ne^{a_n|u|^{\frac{n}{n-1}}}\,dVg_N:u\in H^{1,n}(N),\;u|_{\partial N}=0,\;\int|\nabla u|^n\,dV_{g_N}=1\right\}$ is attained.

##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 26D15 Inequalities for sums, series and integrals
compact manifold
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