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Journal of Mathematical Sciences

, Volume 220, Issue 5, pp 603–614 | Cite as

On a Model Semilinear Elliptic Equation in the Plane

  • Vladimir Gutlyanskiĭ
  • Olga Nesmelova
  • Vladimir Ryazanov
Article

Abstract

Assume that Ω is a regular domain in the complex plane ℂ, and A(z) is a 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 ∈  2, 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 a quasiconformal homeomorphism generated by the matrix A(z), and T is a solution of the semilinear weihted Bieberbach equation △T = m(w)e in G. Here, the weight m(w) is the Jacobian determinant of the inverse mapping ω −1(w).

Keywords

Semilinear elliptic equations Bieberbach equation quasiconformal mappings Beltrami equation Keller–Osserman condition 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Vladimir Gutlyanskiĭ
    • 1
  • Olga Nesmelova
    • 1
  • Vladimir Ryazanov
    • 1
  1. 1.Institute of Applied Mathematics and Mechanics, National Academy of Sciences of UkraineSlavyanskUkraine

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