Multiple Blow-Up Phenomena for the Sinh-Poisson Equation

By M. Grossi, Angela Pistoia. 2012

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AbstractWe 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).

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0bc33435ecb021a03a33742936cde860ecacec19