Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation

Abstract

In this paper, we study the existence of multiple solutions for the boundary value problem
\begin{equation}
\Delta_{\gamma} u+f(x,u)=0 \quad \mbox{ in } \Omega, \quad \quad
u=0 \quad \mbox{ on } \partial \Omega, \notag
\end{equation}
where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2)$
and $\Delta_{\gamma}$ is the subelliptic operator of the type $$
\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \ \partial_{x_j}
=\frac{\partial }{\partial x_{j}}, \gamma = (\gamma_1, \gamma_2, ..., \gamma_N),
$$
the nonlinearity $f(x , \xi)$ is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.