Existence and non-existence of positive solutions of the scalar field system in R(n) .1.

Let n greater than or equal to 3 and Omega be either the entire space R(n) or a Euclidean ball in R(n). Consider the following boundary value problem [GRAPHICS] with homogeneous Dirichlet boundary data (replaced by u,v --> 0 as /x/ --> infinity when Omega = R(n)), where p > 1 and q > 1. In this paper, we investigate the question of existence and non-existence of solutions of (I) and prove that (I) admits a solution if and only if 1/p+1 + 1/q+1 > n-2/n. The existence on a ball and on R(n) are established by a variational approach and an approximation argument respectively. The Pohozaev identity is used to show non-existence on R(n).