Inequalities for second-order elliptic equations with applications to unbounded domains. I
In recent papers, the authors have studied symmetry and monotonicity properties for positive solutions \(u\) of elliptic equations of the form \[ u>0,\;\Delta u+f(u)=0 \text{ in } \Omega, \quad u=0 \text{ on } \partial\Omega \tag{1} \] in several classes of unbounded domains \(\Omega\) in \(\mathbb{R}^n\). Here they continue this program by considering another type of domain, \(\Omega= \mathbb{R}^{n-j} \times\omega\), where \(\omega\) is a smooth bounded domain in \(\mathbb{R}^j\). Denote by \(x=(x_1, \dots, x_{n-j})\) the coordinates in \(\mathbb{R}^{n-j}\), and by \(y=(y_1, \dots, y_j)\) the coordinates in \(\omega\). The goal is to establish symmetry of solutions of (1) corresponding to symmetries of \(\omega\). For example, if \(\omega\) is a ball \(\{|y |<R\}\), they prove that any solution of (1) depends only on \(|y|\) and \(x\), and is decreasing in \(|y|\). Note that \(u\) is not assumed to be bounded. Throughout the paper it is assumed that \(f\) is Lipschitz continuous, with Lipschitz constant \(k\), on \(\mathbb{R}^+\) (or on \([0, \sup u]\) in the case where \(u\) is bounded).
scientific article; zbMATH DE number 908678
scientific article