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Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation. (English) Zbl 0854.35033
This paper mostly concerns radial symmetry of nonnegative classical solutions \(u\) to the equation (1) \(- \Delta u= f(u)\) in \(\mathbb{R}^n\) or in a sufficiently regular connected domain \(\Omega\subset \mathbb{R}^n\), \(n\geq 2\), where \(f\in C(\overline{\mathbb{R}}_+, \mathbb{R})\), \(f\) is locally Lipschitz in \(\mathbb{R}_+\), \(f(0)\leq 0\), and \(f\) is strictly decreasing in some interval \([0, a]\).
Theorem 1. If \(u\) has compact support in \(\mathbb{R}^n\), then any connected component of the set \(S= \{x: u(x)> 0\}\) is a ball on which \(u\) is radially symmetric with respect to its centre.
Theorem 2. If \(u\) solves an (appropriate) overdetermined boundary value problem in \(\Omega\) and \(u(x)> 0\), then \(\Omega\) is a ball and \(u\) is radially symmetric with respect to its centre.
Another result is a partial generalization of a symmetry theorem of Y. Li and W.-M. Ni [Commun. Partial Differ. Equations 18, No. 5-6, 1043-1054 (1993; Zbl 0788.35042)]. Applicability to the blowup set for a porous medium equation is indicated. Related results have been obtained recently by C. Gui [Commun. Pure Appl. Math. 48, No. 5, 471-500 (1995; Zbl 0827.35014)], H. G. Kaper, M. K. Kwong and Y. Li [Differ. Integral Equ. 6, No. 5, 1045-1056 (1993; Zbl 0799.35083)].

MSC:
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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