Qualitative properties for semilinear elliptic systems with non-Lipschitz nonlinearity
This paper is devoted to study symmetry properties for elliptic systems, the model problem of which has the form: \[ \begin{cases} \Delta u-u^p+ v^q=0 \quad & \text{in }\mathbb{R}^N\\ \Delta v-v^p+ u^q=0 \quad & \text{in }\mathbb{R}^N, \end{cases} \tag{1} \] where \(0<p< 1<q\), \(u,v\geq 0\), \(u,v\to 0\) as \(|x|\to \infty\). The authors present a symmetry result in the case of non-Lipschitzian nonlinearities. A proof of the main symmetry result for (1) is based on an adaptation of the well-known moving planes method, which crucially relies on some integral form of the maximum principle, developed by the authors in this paper. A more general class of nonlinearities is investigated as well.
scientific article; zbMATH DE number 1788012