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On the solutions of Liouville systems. (English) Zbl 0902.35039

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 S. Chanillo and 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 B. Gidas, W. M. Ni and 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.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J50 Variational methods for elliptic systems
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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