Modica, Luciano A gradient bound and a Liouville theorem for nonlinear Poisson equations. (English) Zbl 0612.35051 Commun. Pure Appl. Math. 38, 679-684 (1985). 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 Cited in 4 ReviewsCited in 58 Documents MSC: 35J60 Nonlinear elliptic equations 35B35 Stability in context of PDEs 35B45 A priori estimates in context of PDEs Keywords:Liouville theorem; nonlinear Poisson equations; bounded solution; gradient bound Citations:Zbl 0229.35035 PDF BibTeX XML Cite \textit{L. Modica}, Commun. Pure Appl. Math. 38, 679--684 (1985; Zbl 0612.35051) Full Text: DOI OpenURL 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. 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.