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Further qualitative properties for elliptic equations in unbounded domains. (English) Zbl 1079.35513
Summary: This article is one in a series by the authors [Commun. Pure Appl. Math. 50, 1089–1112 (1997; Zbl 0906.35035), Duke Math. J. 81, 467–494 (1996; Zbl 0860.35004)] to study some qualitative properties of positive solutions of elliptic second order boundary value problems of the type \[ \begin{aligned} \Delta u+f(u) & =0\text{ in }\Omega,\quad u>0\text{ in }Q,\\ u& =0\text{ on }\partial\Omega\tag{1}\end{aligned} \] in various kinds of unbounded domains \(\Omega\) of \(\mathbb R^n\). Typically, we are interested in features like monotonicity in some directions and symmetry. In some cases, the positive solutions we consider are supposed to be bounded while in other cases boundedness is not assumed. The function \(f\) appearing in (1.1) will always be assumed to be (globally) Lipschitz continuous: \(\mathbb R^+\to\mathbb R\).
The present paper is devoted to the investigation of three main configurations. We consider a half space \(\Omega = \{x = (x_1,\dots,x_n)\), \(x_n > 0\}\), infinite cylindrical or slab-like domains \(\Omega = \mathbb R^{n-1}\times(0, h)\) and also the case when \(\Omega\) is the whole plane. In the case of the half space, we derive some monotonicity and symmetry results establishing that a bounded solution of (1) actually only depends on one variable. This is related to a conjecture of De Giorgi on the classification of solutions to some problems of the type (1) in the whole space.

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