On the existence of normalized solutions to a class of fractional Choquard equation with potentials
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| Format: | Preprint |
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2025
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| _version_ | 1866913084002533376 |
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| author | Chen, Yongpeng Yang, Zhipeng Zhang, Jianjun |
| author_facet | Chen, Yongpeng Yang, Zhipeng Zhang, Jianjun |
| contents | This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-Δ)^s u+V(x) u=λu+f(x)\left(I_α*\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_α*\left(g|u|^p\right)\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N $$ subject to the mass constraint $$ \int_{\mathbb{R}^N}|u|^2 d x=a>0, $$ where $N>2 s, s \in(0,1), α\in(0, N)$, and $\frac{N+α}{N} \leq q<p \leq \frac{N+α+2 s}{N}$. Here, the parameter $λ\in \mathbb{R}$ appears as an unknown Lagrange multiplier associated with the normalization condition. By employing variational methods under appropriate assumptions on the potentials $V(x), f(x)$, and $g(x)$, we establish several existence results for normalized solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_23363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the existence of normalized solutions to a class of fractional Choquard equation with potentials Chen, Yongpeng Yang, Zhipeng Zhang, Jianjun Analysis of PDEs 35A15, 35B40, 35J20 This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-Δ)^s u+V(x) u=λu+f(x)\left(I_α*\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_α*\left(g|u|^p\right)\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N $$ subject to the mass constraint $$ \int_{\mathbb{R}^N}|u|^2 d x=a>0, $$ where $N>2 s, s \in(0,1), α\in(0, N)$, and $\frac{N+α}{N} \leq q<p \leq \frac{N+α+2 s}{N}$. Here, the parameter $λ\in \mathbb{R}$ appears as an unknown Lagrange multiplier associated with the normalization condition. By employing variational methods under appropriate assumptions on the potentials $V(x), f(x)$, and $g(x)$, we establish several existence results for normalized solutions. |
| title | On the existence of normalized solutions to a class of fractional Choquard equation with potentials |
| topic | Analysis of PDEs 35A15, 35B40, 35J20 |
| url | https://arxiv.org/abs/2507.23363 |