On maximum principles and Liouville theorems for quasilinear elliptic equations and systems. (English) Zbl 0543.35031

Kawohl, Bernhard

Commentat. Math. Univ. Carol. 24, 647-655 (1983).

Let \(\vec u\) be an \({\mathbb{R}}^ N\)-valued function, which is a solution (weak, bounded) of a system of uniformly strongly elliptic quasilinear differential equations with divergence form, in a domain with Lipschitz continuous boundary in \({\mathbb{R}}^ n\). A strong maximum principle for \(| \vec u|\) as well as a maximum principle for the components of \(\vec u\) are obtained under certain assumptions by simply applying the classical maximum principle for single equation and elementary calculations. In case \(n=2\), a Liouville theorem for \(| \vec u|\) under weaker assumptions is observed.

Reviewer: K.Chang

MSC:

35J60	Nonlinear elliptic equations
35B50	Maximum principles in context of PDEs
35B35	Stability in context of PDEs

Keywords:

quasilinear differential equations; divergence form; strong maximum principle; Liouville theorem