journal of differential equations 140, 59105 (1997) On the Solutions of Liouville Systems M. Chipot Institut fur Angewandte Mathematik, Universitat ZurichIrchel, CH-8057 Zu rich, Switzerland and I. Shafrir and G. Wolansky Department of Mathematics, Technion, Haifa 32000, Israel Received October 10, 1996; revised April 11, 1997 1. INTRODUCTION In this paper we are concerned with the system analog of the classical Liouville equation [27] &2u=+Ve au (L) in a domain 0R 2 , where V is a positive function, a is a constant and + is an unprescribed positive constant. The solution of (L) is a pair [u, +] which satisfies the integral constraint + | 0 Ve au =M (1.1) 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 &2u i =+ i V i exp \ : n j =1 a i, j u j + , 1i n (LS) article no. DE973316 59 0022-039697 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.