New type of solutions for the critical polyharmonic equation

TL;DR

This paper introduces a new class of solutions for a critical polyharmonic equation with variable potential, demonstrating concentration phenomena on specific geometric structures under certain stability conditions.

Contribution

The authors develop a reduction method combined with Poho identities to establish existence of solutions concentrating on top and bottom circles of a cylinder.

Findings

Solutions concentrate on top and bottom circles of a cylinder.

Existence depends on stable critical points of a weighted potential.

New solution types are characterized by geometric concentration patterns.

Abstract

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.