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A gradient bound and a Liouville theorem for nonlinear Poisson equations. (English) Zbl 0612.35051
It is proved that a bounded solution in $$R^ n$$ of the equation $$\Delta u=f(u)$$, where $$f=F'$$ with F positive and $$F(u(x_ 0))=0$$ for some $$x_ 0\in {\mathbb{R}}^ n$$, is constant in $${\mathbb{R}}^ n$$. No convexity is assumed about F as in a well known theorem of J. Serrin [Proc. Lond. Math. Soc., III. Ser. 24, 348-366 (1972; Zbl 0229.35035)]. The method of proof is to establish a gradient bound $$| Du(x)| \leq 2F(u)$$.
Reviewer: G.Gudmundsdottir

##### MSC:
 35J60 Nonlinear elliptic equations 35B35 Stability in context of PDEs 35B45 A priori estimates in context of PDEs
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##### References:
 [1] and , Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin, 1977. [2] Leibbrandt, Phys. Rev. B 15 pp 3353– (1977) [3] Modica, Boll. Un. Mat. Ital. (5) 17-B pp 614– (1980) [4] Peletier, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 5 pp 65– (1978) [5] Serrin, Proc. London Math. Soc. 24 pp 348– (1972) [6] Maximum Principles and Their Applications, Mathematics in Science and Engineering, vol. 157, Academic Press, New York, 1981.
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