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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.01476 |
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| _version_ | 1866915612326887424 |
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| author | Chen, Yongpeng Yang, Zhipeng Zhang, Jianjun |
| author_facet | Chen, Yongpeng Yang, Zhipeng Zhang, Jianjun |
| contents | We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$ (-Δ)^s u+V(εx)u=λu+\left(I_α*|u|^q\right)|u|^{q-2} u+\left(I_α*|u|^p\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N, $$ subject to the constraint $$ \int_{\mathbb{R}^N}|u|^2 \mathrm{d}x=a>0, $$ where $N>2 s, s \in(0,1), α\in(0, N), \frac{N+α}{N}<q<\frac{N+2 s+α}{N}<p\leq \frac{N+α}{N-2 s}$, $ε>0$ is a parameter, and $λ\in \mathbb{R}$ serves as an unknown parameter acting as a Lagrange multiplier. By employing the Lusternik-Schnirelmann category theory, we estimate the number of normalized solutions to this problem by virtue of the category of the set of minimum points of the potential function $V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01476 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of multiple normalized solutions for a class of fractional Choquard equations with mixed nonlinearities Chen, Yongpeng Yang, Zhipeng Zhang, Jianjun Analysis of PDEs 35A15, 35B40, 35J20 We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$ (-Δ)^s u+V(εx)u=λu+\left(I_α*|u|^q\right)|u|^{q-2} u+\left(I_α*|u|^p\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N, $$ subject to the constraint $$ \int_{\mathbb{R}^N}|u|^2 \mathrm{d}x=a>0, $$ where $N>2 s, s \in(0,1), α\in(0, N), \frac{N+α}{N}<q<\frac{N+2 s+α}{N}<p\leq \frac{N+α}{N-2 s}$, $ε>0$ is a parameter, and $λ\in \mathbb{R}$ serves as an unknown parameter acting as a Lagrange multiplier. By employing the Lusternik-Schnirelmann category theory, we estimate the number of normalized solutions to this problem by virtue of the category of the set of minimum points of the potential function $V$. |
| title | On the existence of multiple normalized solutions for a class of fractional Choquard equations with mixed nonlinearities |
| topic | Analysis of PDEs 35A15, 35B40, 35J20 |
| url | https://arxiv.org/abs/2411.01476 |