Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents

In this paper, we study the existence and multiplicity of positive solutions to the following system −Δu=∂F∂u(u,v)+εg(x), −Δv=∂F∂v(u,v)+εh(x) in ΩΩ; u,v>0u,v>0 in ΩΩ; and u=v=0u=v=0 on ∂Ω∂Ω, where ΩΩ is a bounded smooth domain in RNRN; F∈C1((R+)2,R+)F∈C1((R+)2,R+) is positively homogeneous of degree μμ; g,h∈C1(Ω¯)∖{0}; and εε is a positive parameter. Using sub–supersolution method, we prove the existence of positive solutions for the above problem. By means of the variational approach, we prove the multiplicity of positive solutions for the above problem with μ∈(2,2∗]μ∈(2,2∗].