Symmetry results for semilinear elliptic systems in the whole space (Q1566827)

By means of the ``moving plane'' method, the symmetry of classical solutions for the elliptic system: \[ \begin{cases} \Delta u_{i}+f_{i}(r,u_{1},....,u_{n})=0 & \text{ in }\mathbb{R}^{N},\;i=1,...,n, \\ u_{i}>0 & \text{ in }\mathbb{R}^{N}, \\ u_{i}(x)\rightarrow 0 & \text{ as } r=|x|\rightarrow \infty ,\end{cases} \] is proved, where \(n\geq 1\) and \(N\geq 2\). It generalizes related results from the case of bounded domains. For instance: \textit{W. C. Troy} [J. Differ. Equations 42, 400-413 (1981; Zbl 0486.35032)], \textit{D. G. de Figueiredo} [NoDEA, Nonlinear Differ. Equ. Appl. 1, No.~2, 119-123 (1994; Zbl 0822.35039)], and \textit{A. W. Shaker} [Appl. Anal. 41, No.~1-4, 1-9 (1991; Zbl 0724.35022)]. Furthermore, a few simple examples for the nonexistence of positive solutions are given when one of the main assumptions to \(f_{i}\) is not satisfied.