On the solutions of Liouville systems (Q1370987)

In the case when \(\Omega \) is a bounded domain in \({\mathbb R}^{2}\), necessary and sufficient conditions for the existence of solutions of the Liouville system are obtained, using Pokhozaev's identity and a variational argument based on a dual formulation of the problem. In the case when \(\Omega ={\mathbb{R}}^{2}\), some results on the symmetry of entire solutions and on a ``Liouville solution'' as a ``special'' solution, both recently studied by \textit{S. Chanillo} and \textit{M. K.-H. Kiessling} [Geom. Func. Anal. 5, 924-947 (1995; Zbl 0858.35035)], are improved. The authors provide completely new proofs based on the moving plane method, developed by \textit{B. Gidas, W. M. Ni} and \textit{L. Nirenberg} [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)]. The main results are illustrated by suitable examples, the proofs are presented in detail, and a few interesting references on applications of the Liouville systems in various fields of Physics, Chemistry and Ecology are given.