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 $\Phi(x).$

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Grushin关键问题的冒泡解决方案的存在性和局部唯一性

在本文中，我们研究以下Grushin关键问题$$-\ Delta u（x）= \ Phi（x）\ frac {u ^ {\ frac {N} {N-2}}（x）} {| y |}，\，\，\，\，u> 0，\，\，\，\ text {in} \，\，\，\ mathbb R ^ {N}，$$其中$ x =（y，z ）\ in \ mathbb R ^ {k} \ times \ mathbb R ^ {Nk}，N \ geq 5，\ Phi（x）$在其$ \ bar {k} $变量$（z_ { 1}，...，z _ {\ bar {k}}），1 \ leq \ bar {k} <\ frac {N-2} {2}。$在$ \ Phi（x）$上的某些合适条件下在接近临界点时，我们证明上述问题的解决方案具有无限多个气泡。而且，我们还表明，在我们存在的结果中获得的冒泡解在局部是唯一的。我们的结果表明，一些冒泡的解保留了潜在的\\ Phi（x）。$的对称性。