Normalized solutions for a class of fractional Choquard equations with mixed nonlinearities
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912773431099392 |
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| author | Chen, Shaoxiong Yang, Zhipeng Zhang, Xi |
| author_facet | Chen, Shaoxiong Yang, Zhipeng Zhang, Xi |
| contents | In this paper we study the following fractional Choquard equation with mixed nonlinearities:
\[
\left\{
\begin{array}{l}
(-Δ)^s u = λu + α\left( I_μ* |u|^q \right) |u|^{q-2} u + \left( I_μ* |u|^p \right) |u|^{p-2} u, \quad x \in \mathbb{R}^N, \\[4pt]
\displaystyle \int_{\mathbb{R}^N} |u|^2 \,\mathrm{d}x = c^2 > 0.
\end{array}
\right.
\]
Here $N > 2s$, $s \in (0,1)$, $μ\in (0, N)$, and the exponents satisfy
\[
\frac{2N - μ}{N} < q < p < \frac{2N - μ}{N - 2s},
\]
while $α> 0$ is a sufficiently small parameter, $λ\in \mathbb{R}$ is the Lagrange multiplier associated with the mass constraint, and $I_μ$ denotes the Riesz potential. We establish existence and multiplicity results for normalized solutions and, in addition, prove the existence of ground state normalized solutions for $α$ in a suitable range. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16438 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for a class of fractional Choquard equations with mixed nonlinearities Chen, Shaoxiong Yang, Zhipeng Zhang, Xi Analysis of PDEs 35A15, 35B40, 35J20 In this paper we study the following fractional Choquard equation with mixed nonlinearities: \[ \left\{ \begin{array}{l} (-Δ)^s u = λu + α\left( I_μ* |u|^q \right) |u|^{q-2} u + \left( I_μ* |u|^p \right) |u|^{p-2} u, \quad x \in \mathbb{R}^N, \\[4pt] \displaystyle \int_{\mathbb{R}^N} |u|^2 \,\mathrm{d}x = c^2 > 0. \end{array} \right. \] Here $N > 2s$, $s \in (0,1)$, $μ\in (0, N)$, and the exponents satisfy \[ \frac{2N - μ}{N} < q < p < \frac{2N - μ}{N - 2s}, \] while $α> 0$ is a sufficiently small parameter, $λ\in \mathbb{R}$ is the Lagrange multiplier associated with the mass constraint, and $I_μ$ denotes the Riesz potential. We establish existence and multiplicity results for normalized solutions and, in addition, prove the existence of ground state normalized solutions for $α$ in a suitable range. |
| title | Normalized solutions for a class of fractional Choquard equations with mixed nonlinearities |
| topic | Analysis of PDEs 35A15, 35B40, 35J20 |
| url | https://arxiv.org/abs/2512.16438 |