Two Nontrivial Solutions of Boundary-Value Problems for Semilinear $\Delta_{\gamma}$-Differential Equations

In this paper, we study the existence of multiple solutions for the boundary-value problem
$$ \Delta_{\gamma} u+f(x,u)=0 \quad \text{in}\ \ \Omega, \qquad u=0 \quad\text{on}\ \ \partial \Omega, $$
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
$$ \Delta_\gamma u =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j}u \right),\qquad \partial_{x_j}u=\frac{\partial u}{\partial x_{j}},\quad \gamma = (\gamma_1, \gamma_2, \dots, \gamma_N). $$

We use the three critical point theorem.