  We revisit the following nonlinear critical elliptic equation
\begin{equation*}
  -\Delta u+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\; u>0\;\;\;\hbox{ in } \mathbb{R}^N,
\end{equation*} where $N\geq 5.$ There seems to be no results about the
periodicity of bubbling solutions. Here we try to investigate some related
problems. Assuming that $Q(y)$ is periodic in $y_1$ with period 1 and has a
local minimum at 0 satisfying $Q(0)=0,$ we prove the existence and local
uniqueness of infinitely many bubbling solutions of the problem above. This
local uniqueness result implies that some bubbling solutions preserve the
symmetry of the potential function $Q(y),$ i.e. the bubbling solution whose
blow-up set is $\{(jL,0,...,0):j=0,\pm 1, \pm 2,..., \pm m\}$ must be periodic
in $y_{1}$ provided that $L$ is large enough, where $m$ is the number of the
bubbles which is large enough but independent of $L.$ Moreover, we also show a
non-existence of this bubbling solutions for the problem above if the local
minimum of $Q(y)$ does not equal to zero.

Analysis of PDEs