Some critical quasilinear elliptic problems with mixed Dirichlet–Neumann boundary conditions: Relation with Sobolev and Hardy–Sobolev optimal constants

In this paper we deal with the following mixed Dirichlet–Neumann elliptic problemsequation(1){−div(|x|−pγ|∇u|p−2∇u)=λup−1|x|p(γ+1)+ur|x|(r+1)γ,u>0inΩ,u=0onΣ1,|x|−pγ|∇u|p−2∂u∂ν=0onΣ2 where Ω⊂RNΩ⊂RN (N⩾3N⩾3) is a bounded domain such that 0∈Ω0∈Ω and with different choices of the parameters 1