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Multiple Blow-Up Phenomena for the Sinh-Poisson Equation

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Abstract

We consider the sinh-Poisson equation

$$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$

, where Ω is a smooth bounded domain in \({\mathbb{R}^2}\) and λ is a small positive parameter. If \({0 \in \Omega}\) and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) λ , which blows up at the origin, whose positive mass is 4πk(k−1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008).

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Authors and Affiliations

  1. Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le Aldo Moro 5, 00185, Rome, Italy

    Massimo Grossi

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

    Angela Pistoia

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  1. Massimo Grossi
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  2. Angela Pistoia
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Correspondence to Massimo Grossi.

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Communicated by F. Otto

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Grossi, M., Pistoia, A. Multiple Blow-Up Phenomena for the Sinh-Poisson Equation. Arch Rational Mech Anal 209, 287–320 (2013). https://doi.org/10.1007/s00205-013-0625-9

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  • DOI: https://doi.org/10.1007/s00205-013-0625-9

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