Construction of solutions for the critical polyharmonic equation with competing potentials

In this paper, we consider the following critical polyharmonic equation \begin{align*}%\label{abs} ( -Δ)^m u+V(|y’|,y’‘)u=Q(|y’|,y’')u^{m^-1},\quad u>0, \quad y=(y’,y’')\in $\mathbb{R}$^3\times $\mathbb{R}$^{N-3}, \end{align} where $N>4m+1$, $m\in {\mathbb{N}}^{+}$, $m^{\ast }=\frac{2N}{N-2m}$, $V(\mathrm{\mid }{y}^{\mathrm{\prime }}\mathrm{\mid },y^{\mathrm{\prime }\mathrm{\prime }})$ and $Q(\mathrm{\mid }{y}^{\mathrm{\prime }}\mathrm{\mid },y^{\mathrm{\prime }\mathrm{\prime }})$ are bounded nonnegative functions in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N-3}$. By using the reduction argument and local Pohouzaev identities, we prove that if $Q(r,y^{\mathrm{\prime }\mathrm{\prime }})$ has a stable critical point $(r_0,y_0^{\mathrm{\prime }\mathrm{\prime }})$ with $r_0>0$, $Q(r_0,y_0^{\mathrm{\prime }\mathrm{\prime }})>0$, $D^{\alpha }Q(r_0,y_0^{\mathrm{\prime }\mathrm{\prime }})=0$ for any $\mathrm{\mid }\alpha \mathrm{\mid }\le 2m-1$ and $B_{1}V(r_0,y_0^{\mathrm{\prime }\mathrm{\prime }})-B_2\sum _{\mathrm{\mid }\alpha \mathrm{\mid }=2m}{D}^{\alpha }Q(r_0,y_0^{\mathrm{\prime }\mathrm{\prime }}){\int }_{{\mathbb{R}}^N}{y}^{\alpha }{U}_{0,1}^{m^{\ast }}dy>0$, then the above problem has a family of solutions concentrated at points lying on the top and the bottom circles of a cylinder, where $B_1$ and $B_2$ are positive constants that will be given later.