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

In this paper, we study the existence of weak solutions for the boundary-value problem
\begin{equation} \label{TriLuyen1: DG 1} \Delta_{\gamma}u+g(x,u)=0 \quad\text{in}\ \ \Omega,\qquad u=u_0 \quad\text{on}\ \ \partial \Omega, \end{equation}
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 type
$$ {{\Delta }_{\gamma }}u=\sum\limits_{j=1}^{N}{{{\partial }_{{{x}_{j}}}} (\gamma _{j}^{2}{{\partial }_{{{x}_{j}}}}u ),\qquad {{\partial }_{{{x}_{j}}}}u =\frac{\partial u}{\partial {{x}_{j}}}},\qquad \gamma = (\gamma_1, \gamma_2, \dots, \gamma_N). $$
We use the sub-super solution and variational methods.