Construction of solutions for the critical polyharmonic equation with competing potentials

Abstract

In this paper, we consider the following critical polyharmonic equation \begin{align*}%\label{abs} ( -\Delta)^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 $N>4m+1$, $m\in \mathbb{N}^+$, $m^*=\frac{2N}{N-2m}$, $V(|y'|,y'')$ and $Q(|y'|,y'')$ are bounded nonnegative functions in $\mathbb{R}^+\times \mathbb{R}^{N-3}$. By using the reduction argument and local Poho\u{z}aev identities, we prove that if $Q(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $Q(r_0,y_0'')>0$, $D^\alpha Q(r_0,y_0'')=0$ for any $|\alpha|\leq 2m-1$ and $B_1V(r_0,y_0'')-B_2\sum\limits_{|\alpha|=2m}D^\alpha Q(r_0,y_0'')\int_{\mathbb{R}^N}y^\alpha U_{0,1}^{m^*}dy>0$, 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.