Normalized solutions to nonlinear Schrödinger equations with competing Hartree-type nonlinearities
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916465107533824 |
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| author | Bhimani, Divyang Gou, Tianxiang Hajaiej, Hichem |
| author_facet | Bhimani, Divyang Gou, Tianxiang Hajaiej, Hichem |
| contents | In this paper, we consider solutions to the following nonlinear Schrödinger equation with competing Hartree-type nonlinearities, $$ -Δu + λu=\left(|x|^{-γ_1} \ast |u|^2\right) u - \left(|x|^{-γ_2} \ast |u|^2\right) u\quad \mbox{in} \,\, \R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<γ_2 < γ_1 <\min\{N, 4\}$ and $λ\in \R$ appearing as Lagrange multiplier is unknown. First we establish the existence of ground states in the mass subcritical, critical and supercritical cases. Then we consider the well-posedness and dynamical behaviors of solutions to the Cauchy problem for the associated time-dependent equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_00429 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to nonlinear Schrödinger equations with competing Hartree-type nonlinearities Bhimani, Divyang Gou, Tianxiang Hajaiej, Hichem Analysis of PDEs 35J20, 35J60 In this paper, we consider solutions to the following nonlinear Schrödinger equation with competing Hartree-type nonlinearities, $$ -Δu + λu=\left(|x|^{-γ_1} \ast |u|^2\right) u - \left(|x|^{-γ_2} \ast |u|^2\right) u\quad \mbox{in} \,\, \R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<γ_2 < γ_1 <\min\{N, 4\}$ and $λ\in \R$ appearing as Lagrange multiplier is unknown. First we establish the existence of ground states in the mass subcritical, critical and supercritical cases. Then we consider the well-posedness and dynamical behaviors of solutions to the Cauchy problem for the associated time-dependent equations. |
| title | Normalized solutions to nonlinear Schrödinger equations with competing Hartree-type nonlinearities |
| topic | Analysis of PDEs 35J20, 35J60 |
| url | https://arxiv.org/abs/2209.00429 |