Classification of solutions to the higher order Liouville’s equation on $${\mathbb{R}^{2m}}$$
By L. Martinazzi. 2008
76 citations recorded by the source corpus.
We classify the solutions to the equation (−Δ)mu = (2m − 1)!e2mu on $${\mathbb{R}^{2m}}$$ giving rise to a metric $${g=e^{2u}g_{\mathbb{R}^{2m}}}$$ with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric $${e^{2u}g_{\mathbb{R}^{2m}}}$$ at infinity, and we observe that the pull-back of this metric to S2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/51fd8d789522ffb16c878072359701603de0842f
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In Applied Mathematics
- Francesca Da Lio
- L. Martinazzi
- T. Rivière
- Armin Schikorra
- Michael Struwe
- Katarzyna Mazowiecka
- Adina Ciomaga
- Yann Bernard
In Computational Theory and Mathematics
Citations in the corpus
- Blow-up analysis of a nonlocal Liouville-type equation (2015, background): “MR 2009m:35153 Zbl 1194.35154 [Martinazzi 2009a] L. Martinazzi, “Classification of solutions to the higher order Liouville’s equation on R2m”, Math.”
- Concentration–compactness phenomena in the higher order Liouville's equation☆ (2008, methodology): “An important tool in the proof of Theorem 1 is the following estimate, proved by Br´ezis and Merle [BM] in dimension 2. For the proof in arbitrary dimension see [ Mar1 ].”