Construction of solutions for the critical polyharmonic equation with competing potentials
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913454230601728 |
|---|---|
| author | Chen, Wenjing Wang, Zexi |
| author_facet | Chen, Wenjing Wang, Zexi |
| contents | 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 $N>4m+1$, $m\in \mathbb{N}^+$, $m^*=\frac{2N}{N-2m}$, $V(|y'|,y'')$ and $Q(|y'|,y'')$ are bounded nonnegative functions in $\mathbb{R}^+\times \mathbb{R}^{N-3}$. By using the reduction argument and local Pohouzaev identities, we prove that if $Q(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $Q(r_0,y_0'')>0$, $D^αQ(r_0,y_0'')=0$ for any $|α|\leq 2m-1$ and $B_1V(r_0,y_0'')-B_2\sum\limits_{|α|=2m}D^αQ(r_0,y_0'')\int_{\mathbb{R}^N}y^αU_{0,1}^{m^*}dy>0$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00007 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of solutions for the critical polyharmonic equation with competing potentials Chen, Wenjing Wang, Zexi Analysis of PDEs 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 $N>4m+1$, $m\in \mathbb{N}^+$, $m^*=\frac{2N}{N-2m}$, $V(|y'|,y'')$ and $Q(|y'|,y'')$ are bounded nonnegative functions in $\mathbb{R}^+\times \mathbb{R}^{N-3}$. By using the reduction argument and local Pohouzaev identities, we prove that if $Q(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $Q(r_0,y_0'')>0$, $D^αQ(r_0,y_0'')=0$ for any $|α|\leq 2m-1$ and $B_1V(r_0,y_0'')-B_2\sum\limits_{|α|=2m}D^αQ(r_0,y_0'')\int_{\mathbb{R}^N}y^αU_{0,1}^{m^*}dy>0$, 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. |
| title | Construction of solutions for the critical polyharmonic equation with competing potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.00007 |