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arXiv · 2609.21851

Quantization for Palais-Smale sequences of the Liouville functional on closed surfaces

Andrea Malchiodi·Francesco Malizia·Luca Martinazzi
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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 $L^p$ 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.

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Andrea Malchiodi, Francesco Malizia, Luca Martinazzi. 2026-09-18. Quantization for Palais-Smale sequences of the Liouville functional on closed surfaces. https://arxiv.org/abs/2609.21851

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