  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 }\Omega, \
  u=0 \quad\text{ on }\partial\Omega,
  \end{gather*} where $\Omega$ is a bounded domain with smooth boundary in
$\mathbb{R}^2, \Omega \cap \{x=0\}\ne \emptyset,$ $k >0,$ $f(x,y,0)=0. $
