Nontrivial solutions to the Dirichlet problems for semilinear degenerate elliptic equations

Abstract

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 $\Omega$ is a bounded domain with smooth boundary in ${\mathbb{R}}^2,\Omega \cap \{x=0\}\ne \varnothing ,$ $k>0,$ $f(x,y,0)=0.$

14 pages