Nontrivial solutions to the Dirichlet problems for semilinear degenerate elliptic equations

In this article, we study the existence of non-trivial weak solutions for the following boundary-value problem \begin{gather*} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial y^2}=f(x,y,u) \quad$\text{ in }$Ω, \ u=0 \quad$\text{ on }$\partialΩ, \end{gather*} where $\mathrm{\Omega }$ is a bounded domain with smooth boundary in ${\mathbb{R}}^2,\mathrm{\Omega }\cap \{x=0\}\mathrm{\ne }\mathrm{\varnothing },$ $k>0,$ $f(x,y,0)=0.$