Semilinear elliptic degenerate equations with critical exponent

In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation {gather} -∂^2 u/∂ x^2 -|x|²ᵏ∂^2 u/∂ y^2=|x|²ᵏu^p+f(x,y,u) in Ω, \ u=0 on ∂Ω,{equ0} {gather} where Ω is a bounded domain with smooth boundary in R², Ω ∩ =0\≠ ∅, k∈ N, $f(x,y,0)=0,$ and $p=(4+5k)/k$ is the critical exponent. Recently, the equation (1) was investigated in [12] for the subcritical case based on a new result obtained in [17] on embedding theorem of weighted Sobolev spaces. In the critical case considered in this paper we will essentially use the optimal functions and constants found in [17]