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} ( -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 $N>4m+1$, $m\in {\mathbb{N}}^{+}$, $m^{\ast }=\frac{2N}{N-2m}$, $V(|y^{\prime }|,y^{\prime \prime })$ and $Q(|y^{\prime }|,y^{\prime \prime })$ 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 $Q(r,y^{\prime \prime })$ has a stable critical point $(r_0,y_0^{\prime \prime })$ with $r_0>0$, $Q(r_0,y_0^{\prime \prime })>0$, $D^{\hat{I}}\pm Q(r_0,y_0^{\prime \prime })=0$ for any $|\hat{I}\pm |\le 2m-1$ and $B_{1}V(r_0,y_0^{\prime \prime })-B_2\sum _{|\hat{I}\pm |=2m}{D}^{\hat{I}}\pm Q(r_0,y_0^{\prime \prime }){\int }_{{\mathbb{R}}^N}{y}^{\hat{I}}\pm U_{0,1}^{m^{\ast }}dy>0$, 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.

arXiv admin note: substantial text overlap with arXiv:2407.00353