Inthispaper,weconsiderthefollowingproblem{-Δu(x)+u(x)=λ(u~p(x)+h(x)),x∈R~N,u(x)∈h~1(R~N),u(x)>0,x∈R~N,(*)whereλ>0isaparameter,p=(N+2)/(N—2).Wewillprovethatthereexistsapositiveconstant0
λ*,auniquesolutionforλ=λ*.Furthermore,(*)possessesatleasttwopositivesolutionswhenλ∈(0,λ*)and3≤N≤5.ForN≥6,undersomemonotonicityconditionsofhweshowthatthereexistsaconstant0 
