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 problems ⎧⎪ ⎪⎪⎪ ⎨⎪⎪⎩ −div |x|−pγ |∇u|p−2∇u = λ up−1 |x|p(γ+1) + ur |x|(r+1)γ , u>0 inΩ, u =0 onΣ1, |x|−pγ |∇u|p−2 ∂u ∂ν =0 onΣ2 (1) where Ω ⊂ RN (N 3) is a bounded domain such that 0 ∈ Ω and with different choices of the parameters 1