New type of solutions for the critical polyharmonic equation

Abstract

In this paper, we consider the following critical polyharmonic equation \begin{align*}%\label{abs} ( -Δ)^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 $m^{\ast }=\frac{2N}{N-2m}$, $N>4m+1$, $m\in {\mathbb{N}}^{+}$, and $V(|y^{\prime }|,y^{\prime \prime })$ is a bounded nonnegative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N-3}$. By using the reduction argument and local Pohouzaev identities, we prove that if $r^{2m}V(r,y^{\prime \prime })$ has a stable critical point $(r_0,y_0^{\prime \prime })$ with $r_0>0$ and $V(r_0,y_0^{\prime \prime })>0$, 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.