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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.10067 |
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| _version_ | 1866909426888212480 |
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| author | Azzolini, Antonio Pomponio, Alessio Secchi, Simone |
| author_facet | Azzolini, Antonio Pomponio, Alessio Secchi, Simone |
| contents | In this paper we study the embedding properties for the weighted Sobolev space $H^1_V(\mathbb{R}^N)$ into the Lebesgue weighted space $L^τ_W(\mathbb{R}^N)$. Here $V$ and $W$ are diverging weight functions.
The different behaviour of $V$ with respect to $W$ at infinity plays a crucial role. Particular attention is paid to the case $V=W$. This situation is very delicate since it depends strongly on the dimension and, in particular, $N=2$ is somewhat a limit case. As an application, an existence result for a planar nonlinear Schrödinger equation in presence of coercive potentials is provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10067 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the embedding of weighted Sobolev spaces with applications to a planar nonlinear Schrödinger equation Azzolini, Antonio Pomponio, Alessio Secchi, Simone Analysis of PDEs In this paper we study the embedding properties for the weighted Sobolev space $H^1_V(\mathbb{R}^N)$ into the Lebesgue weighted space $L^τ_W(\mathbb{R}^N)$. Here $V$ and $W$ are diverging weight functions. The different behaviour of $V$ with respect to $W$ at infinity plays a crucial role. Particular attention is paid to the case $V=W$. This situation is very delicate since it depends strongly on the dimension and, in particular, $N=2$ is somewhat a limit case. As an application, an existence result for a planar nonlinear Schrödinger equation in presence of coercive potentials is provided. |
| title | On the embedding of weighted Sobolev spaces with applications to a planar nonlinear Schrödinger equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.10067 |