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 ∈ ℝ 2, 1 ≤ K < ∞. We study the blow-up problem for a model semilinear equation div (A(z)∇u) = eu 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).
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 13, No. 1, pp. 91–105, January–March, 2016.
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Gutlyanskiĭ, V., Nesmelova, O. & Ryazanov, V. On a Model Semilinear Elliptic Equation in the Plane. J Math Sci 220, 603–614 (2017). https://doi.org/10.1007/s10958-016-3203-5
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DOI: https://doi.org/10.1007/s10958-016-3203-5