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