Multiple Blow-Up Phenomena for the Sinh-Poisson Equation

2012/10/21 by Massimo Grossi, Angela Pistoia · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Bounded function #Combinatorics #Domain (mathematical analysis) #Geometric Analysis and Curvature Flows #Integer (computer science) #Lambda #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Partial Differential Equations #Omega #Physics #Poisson's equation #Quantum mechanics #math.AP

paper · pdf · doi:10.1007/s00205-013-0625-9

arxiv created 2012/10/21 · openalex publication_date 2013/04/02 · arxiv updated 2015/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We consider the sinh-Poisson equation (P)_λ -Δu=\la\sinh u \hboxin Ω, u=0 \hboxon ∂Ω, where Ω is a smooth bounded domain in \rr² and λ is a small positive parameter. If 0∈Ω and Ω is symmetric with respect to the origin, for any integer k if \la is small enough, we construct a family of solutions to (P)_\la which blows-up at the origin whose positive mass is 4πk(k-1) and negative mass is 4πk(k+1). It gives a complete answer to an open problem formulated by Jost-Wang-Ye-Zhou in [Calc. Var. PDE (2008) 31: 263-276].

Citations

• About \( L^p \) estimates for the spatially homogeneous Boltzmann equation
• Counterexample to a conjecture of H. Hopf
• The blow up analysis of solutions of the elliptic sinh-Gordon equation
• Lane Emden problems with large exponents and singular Liouville equations
• Statistical hydrodynamics

Cited by

• New blow-up phenomena for SU(n+1) Toda system

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