Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915639192453120 |
|---|---|
| author | Cannone, Alessandro Cingolani, Silvia Yang, Minbo Zhao, Shunneng |
| author_facet | Cannone, Alessandro Cingolani, Silvia Yang, Minbo Zhao, Shunneng |
| contents | In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents \begin{equation*}
-Δu=(|x|^{-(n-2)}\ast u^{p-ε})u^{p-1-ε}\quad \mbox{in}~~Ω,~~ u=0\quad \mbox{on}~~\partialΩ,
\end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $ε>0$ is a small parameter and $p=\frac{n+2}{n-2}$ is $\mathcal{D}^{1,2}$ energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first $(n+2)$-eigenvalues and their corresponding eigenfunctions, and examine the qualitative behavior of the eigenpairs $(λ_{i,ε}, v_{i,ε})$ to the linearied problem of the above nonlocal equations for $i=1,\cdots,n+2$. As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21372 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent Cannone, Alessandro Cingolani, Silvia Yang, Minbo Zhao, Shunneng Analysis of PDEs In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents \begin{equation*} -Δu=(|x|^{-(n-2)}\ast u^{p-ε})u^{p-1-ε}\quad \mbox{in}~~Ω,~~ u=0\quad \mbox{on}~~\partialΩ, \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $ε>0$ is a small parameter and $p=\frac{n+2}{n-2}$ is $\mathcal{D}^{1,2}$ energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first $(n+2)$-eigenvalues and their corresponding eigenfunctions, and examine the qualitative behavior of the eigenpairs $(λ_{i,ε}, v_{i,ε})$ to the linearied problem of the above nonlocal equations for $i=1,\cdots,n+2$. As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting. |
| title | Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.21372 |