Ground states of the planar nonlinear Schrödinger--Newton system with a point interaction
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916037431132160 |
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| author | Ramos, Gustavo de Paula |
| author_facet | Ramos, Gustavo de Paula |
| contents | We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schrödinger--Newton system with a point interaction: \[ \begin{cases} - Δ_αu = w u + βu |u|^{p - 2} &\text{on} ~ \mathbb{R}^2; \\ - Δw = 2 π|u|^2 &\text{on} ~ \mathbb{R}^2; \\ \|u\|_{L^2}^2 = c, \end{cases} \] where $p > 2$; $α, β\in \mathbb{R}$ and $- Δ_α$ denotes the Laplacian of point interaction with scattering length $(- 2 πα)^{- 1}$. Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem \[ \mathrm{i} ψ' (t) =
- Δ_αψ(t) - (\log |\cdot| \ast |ψ(t)|^2) ψ(t) - βψ(t) |ψ(t)|^{p - 2}. \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ground states of the planar nonlinear Schrödinger--Newton system with a point interaction Ramos, Gustavo de Paula Analysis of PDEs We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schrödinger--Newton system with a point interaction: \[ \begin{cases} - Δ_αu = w u + βu |u|^{p - 2} &\text{on} ~ \mathbb{R}^2; \\ - Δw = 2 π|u|^2 &\text{on} ~ \mathbb{R}^2; \\ \|u\|_{L^2}^2 = c, \end{cases} \] where $p > 2$; $α, β\in \mathbb{R}$ and $- Δ_α$ denotes the Laplacian of point interaction with scattering length $(- 2 πα)^{- 1}$. Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem \[ \mathrm{i} ψ' (t) = - Δ_αψ(t) - (\log |\cdot| \ast |ψ(t)|^2) ψ(t) - βψ(t) |ψ(t)|^{p - 2}. \] |
| title | Ground states of the planar nonlinear Schrödinger--Newton system with a point interaction |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.18202 |