Competition reaction–absorption in some elliptic and parabolic problems related to the Caffarelli–Kohn–Nirenberg inequalities

We consider the following elliptic problem:equation(1){−div(|x|−2γ∇u)+u|x|2(γ+1)=f(x,u),u⩾0 in Ω,u|∂Ω=0 and the corresponding parabolic versionequation(2){ut−div(|x|−2γ∇u)+u|x|2(γ+1)=f(x,u),u⩾0 in Ω×(0,T),u(x,t)=0for (x,t)∈∂Ω×(0,T),u(x,0)=φ(x)if x∈Ω,Ω⊂RNΩ⊂RN is a smooth bounded domain with 0∈Ω0∈Ω and −∞<γ