Normalized solutions to at least mass critical problems: singular polyharmonic equations and related curl-curl problems
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914893040451584 |
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| author | Bieganowski, Bartosz Mederski, Jarosław Schino, Jacopo |
| author_facet | Bieganowski, Bartosz Mederski, Jarosław Schino, Jacopo |
| contents | We are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-Δ)^m u+\fracμ{|y|^{2m}}u + λu = g(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} |u|^2 \, dx = ρ> 0, \end{cases} \end{equation*} in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the $L^2$-ball. Moreover, we find also a solution to the related curl-curl problem \begin{equation*}
\begin{cases}
\nabla\times\nabla\times\mathbf{U}+λ\mathbf{U}=f(\mathbf{U}), \quad x \in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|\mathbf{U}|^2\,dx=ρ,
\end{cases} \end{equation*} which arises from the system of Maxwell equations and is of great importance in nonlinear optics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_12361 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to at least mass critical problems: singular polyharmonic equations and related curl-curl problems Bieganowski, Bartosz Mederski, Jarosław Schino, Jacopo Analysis of PDEs 35J20, 35J35, 35R11, 35Q55, 78M30 We are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-Δ)^m u+\fracμ{|y|^{2m}}u + λu = g(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} |u|^2 \, dx = ρ> 0, \end{cases} \end{equation*} in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the $L^2$-ball. Moreover, we find also a solution to the related curl-curl problem \begin{equation*} \begin{cases} \nabla\times\nabla\times\mathbf{U}+λ\mathbf{U}=f(\mathbf{U}), \quad x \in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|\mathbf{U}|^2\,dx=ρ, \end{cases} \end{equation*} which arises from the system of Maxwell equations and is of great importance in nonlinear optics. |
| title | Normalized solutions to at least mass critical problems: singular polyharmonic equations and related curl-curl problems |
| topic | Analysis of PDEs 35J20, 35J35, 35R11, 35Q55, 78M30 |
| url | https://arxiv.org/abs/2212.12361 |