Lipschitz continuity of quasiconformal solutions of the non-homogeneous Yukawa equations

The main aim of this paper is to establish the Lipschitz, coLipschitz and biLipschitz continuity of quasiconformal solutions of the non-homogeneous Yukawa equations $$f_{z{\overline{z}}}(z)=(\mu (z)+\tau (z)f_z(z))f(z)$$ with respect to the hyperbolic metric and the quasihyperbolic metric, respectively. As applications, the corresponding area distortion of measurable sets under the quasiconformal solutions of the mentioned Yukawa equations is also discussed.