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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.00922 |
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| _version_ | 1866915645661118464 |
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| author | Aikyn, Yergen Chen, Yongpeng Ruzhansky, Michael Yang, Zhipeng |
| author_facet | Aikyn, Yergen Chen, Yongpeng Ruzhansky, Michael Yang, Zhipeng |
| contents | We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-Δ)^s u+V(\varepsilon x)u=λu+\big(I_α*|u|^{p}\big)|u|^{p-2}u+\big(I_α*|u|^{q}\big)|u|^{q-2}u\quad\text{in }\mathbb{R}^N,\] where $N>2s$, $0<s<1$, $(N-4s)^+<α<N$, and the nonlocal exponents satisfy \[\frac{N+2s+α}{N}< q< p=\frac{N+α}{N-2s},\] so that both nonlinearities are $L^2$-supercritical and the $p$ term has upper critical growth of Hartree type. Under standard assumptions on the slowly varying potential $V$, we develop a constrained variational approach on the $L^2$-sphere, based on a truncation-penalization of the critical term in the energy functional, to overcome the lack of compactness. We prove that, for all sufficiently small $\varepsilon>0$, the problem admits at least $\mathrm{cat}_{M_δ}(M)$ distinct normalized solutions, where $M$ is the set of global minima of $V$ and these solutions concentrate near $M$ as $\varepsilon\to0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicity of normalized solutions to the upper critical fractional Choquard equation with $L^2$-supercritical perturbation Aikyn, Yergen Chen, Yongpeng Ruzhansky, Michael Yang, Zhipeng Analysis of PDEs 35A15, 35B40, 35J20 We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-Δ)^s u+V(\varepsilon x)u=λu+\big(I_α*|u|^{p}\big)|u|^{p-2}u+\big(I_α*|u|^{q}\big)|u|^{q-2}u\quad\text{in }\mathbb{R}^N,\] where $N>2s$, $0<s<1$, $(N-4s)^+<α<N$, and the nonlocal exponents satisfy \[\frac{N+2s+α}{N}< q< p=\frac{N+α}{N-2s},\] so that both nonlinearities are $L^2$-supercritical and the $p$ term has upper critical growth of Hartree type. Under standard assumptions on the slowly varying potential $V$, we develop a constrained variational approach on the $L^2$-sphere, based on a truncation-penalization of the critical term in the energy functional, to overcome the lack of compactness. We prove that, for all sufficiently small $\varepsilon>0$, the problem admits at least $\mathrm{cat}_{M_δ}(M)$ distinct normalized solutions, where $M$ is the set of global minima of $V$ and these solutions concentrate near $M$ as $\varepsilon\to0$. |
| title | Multiplicity of normalized solutions to the upper critical fractional Choquard equation with $L^2$-supercritical perturbation |
| topic | Analysis of PDEs 35A15, 35B40, 35J20 |
| url | https://arxiv.org/abs/2512.00922 |