Normalized solutions for a nonlinear Dirac equation
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916746941693952 |
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| author | Zelati, Vittorio Coti Nolasco, Margherita |
| author_facet | Zelati, Vittorio Coti Nolasco, Margherita |
| contents | We prove the existence of a normalized, stationary solution $Ψ\colon \mathbb{R}^{3} \to \mathbb{C}^{4}$ with frequency $w > 0$ of the nonlinear Dirac equation. The result covers the case in which the nonlinearity is the gradient of a function of the form \begin{equation*} F(Ψ) = a|(Ψ, γ^{0}Ψ)|^{\fracα{2}} + b|(Ψ, γ^{1}γ^{2} γ^{3} Ψ)|^{\fracα{2}} \end{equation*} with $α\in (2,\frac{8}{3}]$, $b \geq 0$ and $a > 0$ sufficiently small. Here $γ^{i}$, $i = 0,\ldots, 3$ are the $4 \times 4$ Dirac's matrices.
We find the solution as a critical point of a suitable functional restricted to the unit sphere in $L^{2}$, and $w$ turns out to be the corresponding Lagrange multiplier. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07512 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for a nonlinear Dirac equation Zelati, Vittorio Coti Nolasco, Margherita Analysis of PDEs 35Q40 35P30 47J10 49J35 We prove the existence of a normalized, stationary solution $Ψ\colon \mathbb{R}^{3} \to \mathbb{C}^{4}$ with frequency $w > 0$ of the nonlinear Dirac equation. The result covers the case in which the nonlinearity is the gradient of a function of the form \begin{equation*} F(Ψ) = a|(Ψ, γ^{0}Ψ)|^{\fracα{2}} + b|(Ψ, γ^{1}γ^{2} γ^{3} Ψ)|^{\fracα{2}} \end{equation*} with $α\in (2,\frac{8}{3}]$, $b \geq 0$ and $a > 0$ sufficiently small. Here $γ^{i}$, $i = 0,\ldots, 3$ are the $4 \times 4$ Dirac's matrices. We find the solution as a critical point of a suitable functional restricted to the unit sphere in $L^{2}$, and $w$ turns out to be the corresponding Lagrange multiplier. |
| title | Normalized solutions for a nonlinear Dirac equation |
| topic | Analysis of PDEs 35Q40 35P30 47J10 49J35 |
| url | https://arxiv.org/abs/2310.07512 |