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 \begin{gather} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial y^2}=|x|^{2k}u^p+f(x,y,u) \quad$\text{ in }$Ω, \ u=0 \quad$\text{ on }$\partialΩ,\label{equ0} \end{gather} where $\mathrm{\Omega }$ is a bounded domain with smooth boundary in ${\mathbb{R}}^2,\mathrm{\Omega }\cap \{x=0\}\mathrm{\ne }\mathrm{\varnothing },$ $k\in \mathbb{N},$ $f(x,y,0)=0,$ and $p=(4+5k)\mathrm{/}k$ 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]