Existence of Positive Solutions for Quasilinear Elliptic Systems with Sobolev Critical Exponents

In this paper, we consider the existence of positive solutions to the following problem ⎪⎪⎨ ⎪⎪⎩ −div(|∇u|p−2∇u) = ∂F ∂u (u,v)+ ε p−1g(x) in Ω, −div(|∇v|q−2∇v) = ∂F ∂v (u,v)+ εq−1h(x) in Ω, u,v > 0 in Ω, u = v = 0 on ∂Ω, where Ω is a bounded smooth domain in RN ; F ∈C1((R+)2,R+) is positively homogeneous of degree μ ; g,h ∈C1(Ω)\{0} ; and ε is a positive parameter. Using sub-supersolution method and comparison principle, we prove the existence of positive solutions for the above problem. Mathematics subject classification (2010): 35J25, 35J60.