On the Solutions of Liouville Systems

Abstract

for some prescribed M>0, together with the Dirichlet boundary condition u=0 on 0. The Liouville equation finds applications in various fields of Physics and Mathematics. To name a few, it represents the Newtonian potential of a cluster of self-gravitating mass distribution where a>0 [1, 6, 36, 37], the electric potential induced by charge carriers in the theory of electrolytes where a<0 [31], and maximal entropy solutions of the incompressible Euler equation [23, 9]. In the field of Differential Geometry it stands for the problem of finding a metric corresponding to a prescribed Gaussian curvature [3, 10]. The system-analog of (L) takes the form