On a model semilinear elliptic equation in the plane

摘要

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)ζ, ζ〉 ≤ 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(ω)e~T in G. Here, the weight m(ω) is the Jacobian determinant of the inverse mapping ω~(-1)(ω).