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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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##### References:
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