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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2402.12934 |
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| author | Pan, Kefan Wen, Shixin Yang, Jing |
| author_facet | Pan, Kefan Wen, Shixin Yang, Jing |
| contents | In this paper, we are concerned with the critical Hartree equation \begin{equation*} \begin{cases} -Δu=\left(\displaystyle{\displaystyle{\int_Ω}}\frac{u^{2^{*}_μ}(y)}{|x-y|^μ}dy\right)u^{2^{*}_μ-1}+\varepsilon u,\quad u>0,\quad &\text{in $Ω$,}\\ u=0,\quad &\text{on $\partialΩ$,} \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ ($N\geq 5$) is a smooth bounded domain, $μ\in (0,4)$ and $2^{*}_μ=\frac{2N-μ}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under a non-degeneracy condition on the critical point $x_0\inΩ$ of the Robin function $R(x)$, we perform that for $\varepsilon>0$ sufficiently small, the Morse index of the blow-up solutions $u_\varepsilon$ concentrating at $x_0$ can be computed in terms of the negative eigenvalues of the Hessian matrix $D^{2}R(x)$ at $x_0$. Compared with the usual local cases, our problem is non-local due to the nonlinearity with Hartree-type, and several difficulties arise and new estimates of the eigenpairs $\{\left(λ_{i,\varepsilon},v_{i,\varepsilon}\right)\}$ to the associated linearized problem at $u_{\varepsilon}$ should be introduced. To our knowledge, this seems to be the first paper to consider the qualitative analysis of a Hartree type Brézis-Nirenberg problem and our results extend the works established by M. Grossi et al in \cite{GP} and F. Takahashi in \cite{Ta3} to the non-local case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12934 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Qualitative analysis to an eigenvalue problem of the Hartree type Brézis-Nirenberg problem Pan, Kefan Wen, Shixin Yang, Jing Analysis of PDEs In this paper, we are concerned with the critical Hartree equation \begin{equation*} \begin{cases} -Δu=\left(\displaystyle{\displaystyle{\int_Ω}}\frac{u^{2^{*}_μ}(y)}{|x-y|^μ}dy\right)u^{2^{*}_μ-1}+\varepsilon u,\quad u>0,\quad &\text{in $Ω$,}\\ u=0,\quad &\text{on $\partialΩ$,} \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ ($N\geq 5$) is a smooth bounded domain, $μ\in (0,4)$ and $2^{*}_μ=\frac{2N-μ}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under a non-degeneracy condition on the critical point $x_0\inΩ$ of the Robin function $R(x)$, we perform that for $\varepsilon>0$ sufficiently small, the Morse index of the blow-up solutions $u_\varepsilon$ concentrating at $x_0$ can be computed in terms of the negative eigenvalues of the Hessian matrix $D^{2}R(x)$ at $x_0$. Compared with the usual local cases, our problem is non-local due to the nonlinearity with Hartree-type, and several difficulties arise and new estimates of the eigenpairs $\{\left(λ_{i,\varepsilon},v_{i,\varepsilon}\right)\}$ to the associated linearized problem at $u_{\varepsilon}$ should be introduced. To our knowledge, this seems to be the first paper to consider the qualitative analysis of a Hartree type Brézis-Nirenberg problem and our results extend the works established by M. Grossi et al in \cite{GP} and F. Takahashi in \cite{Ta3} to the non-local case. |
| title | Qualitative analysis to an eigenvalue problem of the Hartree type Brézis-Nirenberg problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.12934 |