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Zeitschrift für Analysis und ihre Anwendungen


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Volume 4, Issue 4, 1985, pp. 353–362
DOI: 10.4171/ZAA/158

Published online: 1985-08-31

Bernstein-Sätze nullter Ordnung und Liouville-Sätze für eine Klasse elliptischer Gleichungen

Jens Frehse[1]

(1) Universität Bonn, Germany

Subject of § 1 are certain second order nonlinear partial differential equations $Lu = 0$ which allow a so called zero order Bernstein theorem: If $u$ is a solution which is defined in all of $\mathbb R^n$ then $u$ is constant. In § 2 Liouville theorems for powers of certain linear elliptic operators $L$ of second order are presented, this means that solutions of $L^mu = 0$ which are defined and bounded in all of $\mathbb R^n$ must be constant. A connection to the hyperbolic equation $\varphi_{tt} + L\varphi = 0$ is shown.

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Frehse Jens: Bernstein-Sätze nullter Ordnung und Liouville-Sätze für eine Klasse elliptischer Gleichungen. Z. Anal. Anwend. 4 (1985), 353-362. doi: 10.4171/ZAA/158