On a Model Semilinear Elliptic Equation in the Plane
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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) = 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).
KeywordsSemilinear elliptic equations Bieberbach equation quasiconformal mappings Beltrami equation Keller–Osserman condition
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- 3.C. Bandle, Yu. Cheng, and G. Porru, Boundary Blow-Up in Semilinear Elliptic Problems with Singular Weights at the Boundary, Mittag-Leffler Institute, Stockholm, 1999/2000, Report No. 39.Google Scholar