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Quantization for Palais-Smale sequences of the Liouville functional on closed surfaces
Abstract
In this paper we prove a quantization property for Palais-Smale sequences of functionals related to Liouville equations on compact surfaces. While these equations have been extensively studied over the past decades, such a quantization property had not been established so far, and has been often bypassed by the Struwe monotonicity trick. Our result, which relies on careful Lp estimates to obtain gradient bounds and to perform neck analysis, allows to directly apply variational methods to obtain existence of solutions in non-resonant regimes. We also give an example of clustering of blow-up points and a stronger assumption that, on the contrary, guarantees that the blow-up points are isolated.
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BibTeX
@misc{malchiodi2026quantization,
title = {{Quantization for Palais-Smale sequences of the Liouville functional on closed surfaces}},
author = {Andrea Malchiodi and Francesco Malizia and Luca Martinazzi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21851},
eprint = {2609.21851},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21851},
}
RIS
TY - UNPB
TI - Quantization for Palais-Smale sequences of the Liouville functional on closed surfaces
AU - Andrea Malchiodi
AU - Francesco Malizia
AU - Luca Martinazzi
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21851
UR - https://arxiv.org/abs/2609.21851
AB - In this paper we prove a quantization property for Palais-Smale sequences of functionals related to Liouville equations on compact surfaces. While these equations have been extensively studied over the past decades, such a quantization property had not been established so far, and has been often bypassed by the Struwe monotonicity trick. Our result, which relies on careful L^p estimates to obtain gradient bounds and to perform neck analysis, allows to directly apply variational methods to obtain existence of solutions in non-resonant regimes. We also give an example of clustering of blow-up points and a stronger assumption that, on the contrary, guarantees that the blow-up points are isolated.
ER -