Boundedness of solutions of the Ginzburg–Landau system involving a subelliptic operator

The aim of this paper is to study the boundedness of solutions of the Ginzburg–Landau system
\begin{equation*} \begin{cases} \partial_t u -\Delta_\lambda u = u - u^3 - \gamma uv^2 & \text{in } \mathbb{R}\times \mathbb{R}^N, \\ \partial_t v -\Delta_\lambda v = v - v^3 - \gamma u^2v & \text{in }\mathbb{R}\times \mathbb{R}^N, \end{cases} \end{equation*}
where $\gamma>0$ and $\Delta_{\lambda}$ is the subelliptic operator
\begin{equation*} \sum_{i=1}^N \partial_{x_i}(\lambda_i^2\partial_{x_i}). \end{equation*}
In the stationary case, where the solutions are independent of the time variable, our result can be seen as an extension of some results in [A. Farina, B. Sciunzi, and N. Soave, Commun. Contemp. Math. 22 (5), Article no. 1950044 (2020)] from the Laplace operator to the subelliptic operator $\Delta_{\lambda}$.