Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent
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arXiv
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| Format: | Preprint |
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2025
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| author | Cingolani, Silvia Yang, Minbo Zhao, Shunneng |
| author_facet | Cingolani, Silvia Yang, Minbo Zhao, Shunneng |
| contents | In this paper, we study that the nearly critical nonlocal problem \begin{equation*} \left\lbrace \begin{aligned}
&-Δu=(|x|^{-{(n-2)}}\ast u^{p-ε})u^{p-1-ε} \quad \mbox{in}\quad Ω,
&u>0\quad \mbox{in}\quad\hspace{1mm} Ω,
&u=0\quad \mbox{on}\hspace{2.5mm}\partialΩ,
\end{aligned} \right. \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 energy-critical exponent.
We study the asymptotic behavior of least energy solutions as $ε\rightarrow0$. These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied. Finally, in order to further locate the blowing-up point $x_0$, we prove that $x_0$ is a global maximum point of the Robin's function of $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20136 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent Cingolani, Silvia Yang, Minbo Zhao, Shunneng Analysis of PDEs In this paper, we study that the nearly critical nonlocal problem \begin{equation*} \left\lbrace \begin{aligned} &-Δu=(|x|^{-{(n-2)}}\ast u^{p-ε})u^{p-1-ε} \quad \mbox{in}\quad Ω, &u>0\quad \mbox{in}\quad\hspace{1mm} Ω, &u=0\quad \mbox{on}\hspace{2.5mm}\partialΩ, \end{aligned} \right. \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 energy-critical exponent. We study the asymptotic behavior of least energy solutions as $ε\rightarrow0$. These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied. Finally, in order to further locate the blowing-up point $x_0$, we prove that $x_0$ is a global maximum point of the Robin's function of $Ω$. |
| title | Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.20136 |