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

Abstract

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 $L^{m}u=0$ which are defined and bounded in all of ${\mathbb{R}}^n$ must be constant. A connection to the hyperbolic equation ${\phi }_{tt}+L\phi =0$ is shown.