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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.15864 |
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| _version_ | 1866913747529891840 |
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| author | Zhang, Yuxuan Chang, Xiaojun Chen, Lin |
| author_facet | Zhang, Yuxuan Chang, Xiaojun Chen, Lin |
| contents | Consider the Neumann problem: \begin{eqnarray*} \begin{cases}
&-Δu-\fracμ{|x|^2}u +λu =|u|^{q-2}u+|u|^{p-2}u ~~~\mbox{in}~~\mathbb{R}_+^N,~N\ge3,
&\frac{\partial u}{\partial ν}=0 ~~ \mbox{on}~~ \partial\mathbb{R}_+^N \end{cases} \end{eqnarray*} with the prescribed mass:
\begin{equation*}
\int_{\mathbb{R}_+^N}|u|^2 dx=a>0,
\end{equation*} where $\mathbb{R}_+^N$ denotes the upper half-space in $\mathbb{R}^N$, $\frac{1}{|x|^2}$ is the Hardy potential, $2<q<2+\frac{4}{N}<p<2^*$, $μ>0$, $ν$ stands for the outward unit normal vector to $\partial \mathbb{R}_+^N$, and $λ$ appears as a Lagrange multiplier. Firstly, by applying Ekeland's variational principle, we establish the existence of normalized solutions that correspond to local minima of the associated energy functional. Furthermore, we find a second normalized solution of mountain pass type by employing a parameterized minimax principle that incorporates Morse index information. Our analysis relies on a Hardy inequality in $H^1(\mathbb{R}_+^N)$, as well as a Pohozaev identity involving the Hardy potential on $\mathbb{R}_+^N$. This work provides a variational framework for investigating the existence of normalized solutions to the Hardy type system within a half-space, and our approach is flexible, allowing it to be adapted to handle more general nonlinearities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15864 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Standing waves with prescribed mass for NLS equations with Hardy potential in the half-space under Neumman boundary condition Zhang, Yuxuan Chang, Xiaojun Chen, Lin Analysis of PDEs Consider the Neumann problem: \begin{eqnarray*} \begin{cases} &-Δu-\fracμ{|x|^2}u +λu =|u|^{q-2}u+|u|^{p-2}u ~~~\mbox{in}~~\mathbb{R}_+^N,~N\ge3, &\frac{\partial u}{\partial ν}=0 ~~ \mbox{on}~~ \partial\mathbb{R}_+^N \end{cases} \end{eqnarray*} with the prescribed mass: \begin{equation*} \int_{\mathbb{R}_+^N}|u|^2 dx=a>0, \end{equation*} where $\mathbb{R}_+^N$ denotes the upper half-space in $\mathbb{R}^N$, $\frac{1}{|x|^2}$ is the Hardy potential, $2<q<2+\frac{4}{N}<p<2^*$, $μ>0$, $ν$ stands for the outward unit normal vector to $\partial \mathbb{R}_+^N$, and $λ$ appears as a Lagrange multiplier. Firstly, by applying Ekeland's variational principle, we establish the existence of normalized solutions that correspond to local minima of the associated energy functional. Furthermore, we find a second normalized solution of mountain pass type by employing a parameterized minimax principle that incorporates Morse index information. Our analysis relies on a Hardy inequality in $H^1(\mathbb{R}_+^N)$, as well as a Pohozaev identity involving the Hardy potential on $\mathbb{R}_+^N$. This work provides a variational framework for investigating the existence of normalized solutions to the Hardy type system within a half-space, and our approach is flexible, allowing it to be adapted to handle more general nonlinearities. |
| title | Standing waves with prescribed mass for NLS equations with Hardy potential in the half-space under Neumman boundary condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.15864 |