A gradient bound and a Liouville theorem for nonlinear Poisson equations. (English) Zbl 0612.35051

Modica, Luciano

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

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

References:

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[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.