zbMATH — the first resource for mathematics

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)].

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI
[1] Berestycki H., Bo. Soc. Bras. Mat. 22 (1) pp 1– (1991) · Zbl 0784.35025
[2] Chen X.Y, J. Diff. Equations 78 pp 16– (1986)
[3] C. Cortazar, M. Elgueta, P. Felmer. ”On a sernilinear elliptic problem in IRN with a non-Lipschitzian non-linearity” To appear in Advances in Partial Differential Equations.
[4] Galaktionov V., J. Diff. Equations 101 pp 66– (1993) · Zbl 0802.35065
[5] D. Gilbarg and N. Trudinger, ” Eliptic Partial Differential Equations of Second Order”. Springer Verlag, 2th Edition, 1982. · Zbl 0691.35001
[6] DOI: 10.1007/BF01221125 · Zbl 0425.35020
[7] DOI: 10.1002/cpa.3160480502 · Zbl 0827.35014
[8] Herrero M.A, Ann. Inst. Henri Poinar4, Analyse Non lineaire 10 (2) pp 131– (1993)
[9] Kaper H.G., Differential Integral Equations 6 (5) pp 1045– (1993)
[10] Li Y., Cornm. in P.D.E. 18 (5) pp 1043– (1993) · Zbl 0788.35042
[11] Samarskii A.A., Walter dc Gruyter, (1995)
[12] DOI: 10.1007/BF00250468 · Zbl 0222.31007
[13] Velázquez. J.J.L., Comm. in P.D.E. 17 (9) pp 1597– (1992) · Zbl 0765.73059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.