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 $$\mathbb{R}^2$$ , $$\Omega \cap \{x=0\}\ne \emptyset$$ , $$k >0$$ , $$f(x,y,0)=0$$ .