Nontrivial solutions of the Dirichlet problems for semilinear degenerate elliptic equations

In this article, we study the existence of nontrivial weak solutions of the boundary value problem
$$ -\frac{\partial^2 u}{\partial x^2} -|x|^{2k}\frac{\partial^2 u}{\partial y^2}=f(x,y,u)\text{ in }\Omega, \quad u=0 \text{ on }\partial\Omega, $$
where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^2$, $\Omega \cap \{x=0\}\ne \varnothing$, $k >0$, $f(x,y,0)=0$.