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


35J60 Nonlinear elliptic equations
35B35 Stability in context of PDEs
35B45 A priori estimates in context of PDEs


Zbl 0229.35035
Full Text: DOI


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