NONEXISTENCE RESULTS AND EXISTENCE THEOREMS OF POSITIVE SOLUTIONS OF DIRICHLET PROBLEMS FOR A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS OF SECOND ORDER

We consider the following boundary problem of semilinear elliptic systems: Assume that each fl is a homogeneous function of degree p and when ul>0, l=1, 2, m sum from l=1 to N（fl）> 0. In this paper, we prove that:1. If Ω is a pre-starshaped domain and （A） satisfies the following conditions;i)and when ul>0, F>0;ii) Either when p=n+2/n-2, λ（?）[0, λ1) or when p>n+2/n-2,λ（?）（0, λ1） then （A） has no solution.2. Suppose the following conditions hold:i) fl（u,u,u）=kup,l=1,2,…, m. k>0;ii) p=n+2/n-2 either when n≥4, λ∈ （0, λ1） or when n=3, λ∈（λ1/4, λ1） and Ω is a ball, then （A） is solvable.3. We give many families of examples which satisfy all conditions above. These facts show there really exists a big class of these elliptic systems