Multiple positive solutions of nonhomogeneous elliptic systems involving critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to the following elliptic system: -Δu=[2α/(α+β)]u|u|α-2|v|β+εf(x),-Δv=[2β/(α+β)]|u|αv|v|β-2+εg(x)inΩ; u,v>0inΩ; and u=v=0on∂Ω(*), where ΩΩ is a bounded domain in RN(N⩾3)RN(N⩾3); f,g∈C1(Ω¯); and α,β>1α,β>1. For the subcritical and critical cases, we prove that problem (*)(*) has at least two solutions for any ε∈(0,ε0)ε∈(0,ε0) and has no solution for any ε>ε*ε>ε*, ε0⩽ε*ε0⩽ε*. In the supercritical case, we find that the existence of solutions to problem (*)(*) is closely related to the existence of nonnegative solutions for two linear elliptic problems.