  \begin{abstract} We consider the following poly-harmonic equations with
critical exponents: \begin{equation}\label{P} (-\Delta)^m u
=K(y)u^{\frac{N+2m}{N-2m}},\;\;\; u>0\;\;\;\hbox{in} \mathbb{R}^N,
\end{equation} where $N> 2m+2,m\in\mathbb{N}_{+}, K(y)$ is positive and
periodic in its first $k$ variables $(y_1,\cdots, y_k)$, $1\leq
k<\frac{N-2m}{2}$. Under some conditions on $K(y)$ near its critical point, we
prove not only that problem~\eqref{P} admits solutions with infinitely many
bubbles, but also that the bubbling solutions obtained in our existence result
are locally unique. This local uniqueness result implies that some bubbling
solutions preserve the symmetry of the scalar curvature $K(y).$
