On a model semilinear elliptic equation in the plane

• Published in: Український математичний вісник
• Date: 2016
• Main Authors: Gutlyanskii, V.Y., Nesmelova, O.V., Ryazanov, V.I.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2016
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/140893
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: On a model semilinear elliptic equation in the plane / V.Y. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2016. — Т. 13, № 1. — С. 91-105. — Бібліогр.: 18 назв. — англ.

Assume that Ω is a regular domain in the complex plane C and A(z) is symmetric 2 × 2 matrix with measurable entries, det A = 1 and such that 1/K|ξ|² ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|², ξ ∊ R², 1 ≤ K < ∞. We study the blow-up problem for a model semilinear equation div (A(z)∇u) = e^u in Ω and show that the well-known Liouville–Bieberbach function solves the problem under an appropriate choice of the matrix A(z). The proof is based on the fact that every regular solution u can be expressed as u(z) = T(ω(z)) where ω : Ω → G stands for quasiconformal homeomorphism generated by the matrix A(z) and T is a solution of the semilinear weihted Bieberbach equation ∆T = m(w)e^T in G. Here the weight m(w) is the Jacobian determinant of the inverse mapping ω⁻¹(w).