Sign-changing solutions of boundary value problems for semilinear ${\Delta }_{\gamma }$-Laplace equations

Abstract

In this article, we study the multiplicity of weak solutions to the boundary value problem

where $\Omega$ is a bounded domain with smooth boundary in ${\mathbb{R}}^N(N\ge 2),\alpha \in \mathbb{N},g(x,y,\xi ),f(x,y,\xi )$ are Carathéodory functions and $G_{\alpha }$ is the Grushin operator. We use the lower bounds of eigenvalues and an abstract theory on sign-changing solutions.