The Existence Of Positive Solution For The Problem Of Elliptic Systems With Critical Exponent

In this paper, we study the following Dirichlet problem with Sobolev critical exponent where a,β≥ 1, κ+β= 2*:=2N/(N-2)(N≥ 3) and Ω is a bounded domain in RN(N≥ 3) with (?)Ω C2. Suppose λ is the first eigenvalue of operator -△ in H10(Ω), when 0<λ1,λ2< λ, by applying the Mountain Pass Theorem, we prove that the above problem exists positive solution when the parameters N and λ1, λ2 are in some ranges. On the other hand, when λ1, λ2< 0 or λ1, λ2> λ, no nontrivial solutions exist.