On a Model Semilinear Elliptic Equation in the Plane

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• 作者：Vladimir Gutlyanskiĭ ; Olga Nesmelova ; Vladimir Ryazanov
• 关键词：Semilinear elliptic equations ; Bieberbach equation ; quasiconformal mappings ; Beltrami equation ; Keller–Osserman condition
• 刊名：Journal of Mathematical Sciences
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：220
• 期：5
• 页码：603-614
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer US
• ISSN：1573-8795
• 卷排序：220

文摘

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 ∈ ℝ2, 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).