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Inequalities for second-order elliptic equations with applications to unbounded domains. I. (English) Zbl 0860.35004
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).
Reviewer: V.Mustonen (Oulu)

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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