Normalized solutions for a class of fractional Choquard equations with the HLS lower critical term and a nonlocal perturbation
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2026
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| author | Chen, Shaoxiong Kumar, Vishvesh Yang, Zhipeng Zhang, Xi |
| author_facet | Chen, Shaoxiong Kumar, Vishvesh Yang, Zhipeng Zhang, Xi |
| contents | In this paper, we study the mass-constrained fractional Choquard equation \( (-Δ)^s u = λu + α(I_μ* |u|^{\frac{2N-μ}{N}})|u|^{\frac{2N-μ}{N}-2}u + (I_μ* |u|^p)|u|^{p-2}u \) in \( \mathbb{R}^N \), under the constraint \( \int_{\mathbb{R}^N} |u|^2 \, dx = c^2 > 0 \), where \( N > 2s \), \( s \in (0,1) \), \( μ\in (0,N) \), \( α> 0 \), and \( 2 + \frac{2s-μ}{N} \le p < \frac{2N-μ}{N-2s} \). We first establish a nonexistence result in the \( L^2 \)-critical case \( p = 2 + \frac{2s-μ}{N} \). Then, in the \( L^2 \)-supercritical range, we prove the existence of normalized ground states in two complementary regimes determined by the quantity \( \mathcal{M}_1(c) \). Our approach is based on constrained variational methods, a min-max construction, and refined estimates for the associated fiber maps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12774 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for a class of fractional Choquard equations with the HLS lower critical term and a nonlocal perturbation Chen, Shaoxiong Kumar, Vishvesh Yang, Zhipeng Zhang, Xi Analysis of PDEs 35A15, 35B40, 35J20 In this paper, we study the mass-constrained fractional Choquard equation \( (-Δ)^s u = λu + α(I_μ* |u|^{\frac{2N-μ}{N}})|u|^{\frac{2N-μ}{N}-2}u + (I_μ* |u|^p)|u|^{p-2}u \) in \( \mathbb{R}^N \), under the constraint \( \int_{\mathbb{R}^N} |u|^2 \, dx = c^2 > 0 \), where \( N > 2s \), \( s \in (0,1) \), \( μ\in (0,N) \), \( α> 0 \), and \( 2 + \frac{2s-μ}{N} \le p < \frac{2N-μ}{N-2s} \). We first establish a nonexistence result in the \( L^2 \)-critical case \( p = 2 + \frac{2s-μ}{N} \). Then, in the \( L^2 \)-supercritical range, we prove the existence of normalized ground states in two complementary regimes determined by the quantity \( \mathcal{M}_1(c) \). Our approach is based on constrained variational methods, a min-max construction, and refined estimates for the associated fiber maps. |
| title | Normalized solutions for a class of fractional Choquard equations with the HLS lower critical term and a nonlocal perturbation |
| topic | Analysis of PDEs 35A15, 35B40, 35J20 |
| url | https://arxiv.org/abs/2604.12774 |