On the nonlinear generalized Cauchy-Riemann system on the whole plane
The author proves an existence theorem for the nonlinear generalized Cauchy-Riemann system \[ w_{\bar{z}}+A(z,w,\bar{ w})w+B(z,w,\bar {w})\bar {w}=F(z) \] in the Lorentz space \({\mathcal L}_{2,1}(E)\) (for the definition of the Lorentz space, see \textit{H. Triebel}, Interpolation Theory, Function Spaces, Differential Operators, Leipzig: Barth (1995; Zbl 0830.46028)). The main conditions for solvability of the nonlinear Cauchy-Riemann system are as follows: \[ {\text{Re}} B(z,s,\bar {s})-\delta >|A(z,s,\bar{ s})|,\quad s\in E; \] \[ \sup_{|z-\eta|\leq 2} \sup_{\substack{|s_1-s_2|\leq G \\ |s_1|\leq G\\ s_1,s_2\in E}} \left|{B(z,s_1,\bar s_1)\over B(\eta,s_2,\bar s_2)}\right|\leq K(G),\quad K(G)\geq 1,G=\text{const}>0. \]
scientific article; zbMATH DE number 1037967