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Concentration compactness of Moser functionals on manifolds. (English) Zbl 1133.58013
The author proves the concentration compactness principle of the Moser functional on a compact Riemannian manifold without boundary \((M, g)\). Let \(\Omega\) be a bounded domain in \(\mathbb R^n\). J. Moser [Indiana Univ. Math. J. 20, 1077–1092 (1971; Zbl 0213.13001)] showed the Moser-Trudinger inequality
\[ \sup_{u\in H_{0}^{1,n} (\Omega),||\nabla u || L^{n}(\Omega)= 1}\int_\Omega {e}^{\alpha_{n}| {u}| ^\frac{n}{n-1}}\, dx < \infty \] where \(\alpha_n = n \omega^\frac {1}{n-1}_{n-1}\), and \(\omega_{n-1}\) is the measure of the unit sphere in \(\mathbb R^n\). A very interesting result on the Moser-Trudinger inequality, which was proved by P.-L. Lions [Rev. Mat. Iberoam. 1, No. 1, 145–201 (1985; Zbl 0704.49005)], is the so-called concentration compactness principle for the Moser functional
\[ {\mathcal F}(u) = \int_{\Omega} e^{\alpha_{n}| u |^\frac {n}{n-1}}\, dx, \] which is defined on the unit ball in \(H^{1,n}_{0} (\Omega)\). The principle says that if \({\mathcal F}(u)\) is not compact, then \(u_k \rightharpoondown 0\) and \(| \nabla u_k |^{n} dV_g \rightharpoondown \delta_{x_0}\) for some \(x_0 \in \overline {\Omega}\). The Moser-Trudinger type inequality on \((M, g)\) was proved by L. Fontana [Comment. Math. Helv. 68, No. 3, 415–454 (1993; Zbl 0844.58082)]. As a special case of Fontana’s result, we have
\[ \sup_{u \in {\mathcal H}} \int_{M} e^{\alpha_{n} |u|^\frac{n} {n-1}}\, dV_g < \infty, \] where
\[ {\mathcal H}= \left\{u \in H^{ 1,n} (M ) : \int_{M} u\,dV_{g} = 0, \int_{M} |\nabla u|^{n} \,dV_{g} = 1\right\}. \] Using the ideas in [Y. Li, Sci. China, Ser. A 48, No. 5, 618–648 (2005; Zbl 1100.53036)], the author proves that
\[ \sup\left\{ \lim_{k \rightarrow \infty} \int_{M} \biggl( e^{\alpha_{n} |u|^\frac {n} {n-1}} - 1 \biggr) \,dV_{g} : v_k \in {\mathcal H}, |\nabla v_k |^{n} \,dV_{g} \rightharpoondown \delta_{p}\right\} =\frac{\omega_{n-1}} {n} e^{\alpha_{n}S_{p} + \sum^{n-1}_{j = 1}\frac{1}{j}}, \] which extends Lin’s result [K. C. Lin, Trans. Am. Math. Soc. 348, No. 7, 2663–2671 (1996; Zbl 0861.49001)] on \(\Omega\) to the result on \((M, g)\). The construction is presented in a precise and elegant way.

MSC:
58J05 Elliptic equations on manifolds, general theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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