Existence of Infinitely Many Solutions for $\Delta_\gamma $-Laplace Problems

In this article, we study the existence of infinitely many solutions for the boundary–value problem
\begin{gather*} -\Delta_\gamma u+a(x)u=f(x,u) \quad \text{in}\ \ \Omega, \qquad u=0 \quad\text{on}\ \ \partial\Omega, \end{gather*}
where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$ ($N \ge 2$) and $\Delta_{\gamma}$ is a subelliptic operator of the form
$$ \Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \big(\gamma_j^2 \partial_{x_j} \big), \qquad \partial_{x_j}: =\frac{\partial }{\partial x_{j}},\quad \gamma = (\gamma_1, \gamma_2, \dots, \gamma_N). $$
Our main tools are the local linking and the symmetric mountain pass theorem in critical point theory.