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

【摘要】 <正> We consider the following boundary problem of semilinear elliptic systems: Assume that each f^l is a homogeneous function of degree p and when u^l>0, l=1, 2, m sum from l=1 to N（f^l）> 0. In this paper, we prove that:1. If Ω is a pre-starshaped domain and （A） satisfies the following conditions;i)and when u^l>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) f^l（u,u,u）=ku^p,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更多还原

【Abstract】 We consider the following boundary problem of semilinear elliptic systems: Assume that each f^l is a homogeneous function of degree p and when u^l>0, l=1, 2, m sum from l=1 to N（f^l）> 0. In this paper, we prove that:1. If Ω is a pre-starshaped domain and （A） satisfies the following conditions;i)and when u^l>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) f^l（u,u,u）=ku^p,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更多还原