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A blow-up phenomenon for a non-local Liouville-type equation

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Abstract

We consider the non-local Liouville equation

$${\left( { - \Delta } \right)^{{1 \over 2}}}u = {h_\varepsilon }{e^u} - 1\,\,\,\,\,{\rm{in}}\,\,{\mathbb{S}^1},$$

corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up, when hε approaches a function h as ε → 0, at a critical point of the harmonic extension of h provided some generic assumptions are satisfied.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo S. Leonardo Murialdo, 00146, Roma, Italy

    Luca Battaglia

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049, Madrid, Spain

    María Medina

  3. Dipartimento SBAI, “Sapienza” Università di Roma, via Antonio Scarpa 16, 00161, Roma, Italy

    Angela Pistoia

Authors
  1. Luca Battaglia
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  2. María Medina
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  3. Angela Pistoia
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Corresponding author

Correspondence to Angela Pistoia.

Additional information

M. Medina was partially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement N 754446 and UGR Research and Knowledge Transfer Fund — Athenea3i.

M. Medina also acknowledges the hospitality of Università La Sapienza di Roma, where this work was carried out during a long visit in the academic year 2019–2020.

A. Pistoia was partially supported by Fondi di Ateneo “Sapienza” Universita di Roma (Italy).

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Battaglia, L., Medina, M. & Pistoia, A. A blow-up phenomenon for a non-local Liouville-type equation. JAMA 149, 343–367 (2023). https://doi.org/10.1007/s11854-022-0260-1

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  • DOI: https://doi.org/10.1007/s11854-022-0260-1

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