On the radial property of solutions to nonlinear elliptic equations in \({\mathbb{R}}^2\)
A rather simple proof of the following theorem is given. Theorem: Let \(\Omega \subseteq {\mathbb R}^{2}\) be any domain, and let \(\varphi :\Omega \rightarrow {\mathbb R} \) be a \(C^{\infty }\)-function such that \(u=\varphi _{x}+i\varphi _{y}\) satisfies in \(\Omega \) the equation \[ \Delta u=uF(|u|^{2}), \] where \(F(s)\) is a smooth \({\mathbb R}\) -valued function with \(F''(s)\neq 0\) at all \(s\) that belong to the image of \(\Omega \) under the mapping \(|u|^{2}:\Omega \rightarrow {\mathbb R}\). Then, up to translations and rotations of \(\Omega \), the function \(\varphi \) depends only on either \(x=\text{Re }z\) or \(|z|^{2}\). In the particular case \(\Omega ={\mathbb R}^{2}\) and \(F(s)=s-1\), i.e. the Ginzburg-Landau equation, the preceding theorem contains similar one of \textit{S. Chanillo} and \textit{M. K.-H. Kiessling} [C. R. Acad. Sci., Paris, Sér. I 321, No. 8, 1023-1026 (1995; Zbl 0843.35004)].
scientific article; zbMATH DE number 1557789