Existence and local uniqueness of bubbling solutions for the Grushin critical problem

In this paper, we study the following Grushin critical problem $$ -\Delta u(x)=\Phi(x)\frac{u^{\frac{N}{N-2}}(x)} {|y|},\,\,\,\,u>0,\,\,\,\text{in}\,\,\,\mathbb R^{N}, $$ where $x=(y,z)\in\mathbb R^{k}\times \mathbb R^{N-k},N\geq 5,\Phi(x)$ is positive and periodic in its the $\bar{k}$ variables $(z_{1},...,z_{\bar{k}}),1\leq \bar{k} < \frac{N-2}{2}.$ Under some suitable conditions on $\Phi(x)$ near its critical point, we prove that the problem above has solutions with infinitely many bubbles. Moreover, we also show that the bubbling solutions obtained in our existence result are locally unique. Our result implies that some bubbling solutions preserve the symmetry from the potential $\Phi(x).$