In this paper, we consider the following critical polyharmonic equation \begin{ align*}%\label{ abs} ( -\Delta)^m u+V(|y'|,y'')u=u^{ m^*-1},\quad u>0, \quad y=(y',y'')\in \mathbb{ R}^3\times \mathbb{ R}^{ N-3}, \end{ align*} where $m^*=\frac{ 2N}{ N-2m}$, $N>4m+1$, $m\in \mathbb{ N}^+$, and $V(|y'|,y'')$ is a bounded nonnegative function in $\mathbb{ R}^+\times \mathbb{ R}^{ N-3}$. By using the reduction argument and local Poho\u{ z}aev identities, we prove that if $r^{ 2m}V(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$ and $V(r_0,y_0'')>0$, then the above problem has a new type of solutions, which concentrate at points lying on the top and the bottom circles of a cylinder.
Categories: math.AP