DOI: 10.1515/dema-2025-0288 ISSN: 2391-4661

Normalized solutions for a class of fractional Choquard equations with mixed nonlinearities

Shaoxiong Chen, Zhipeng Yang, Xi Zhang
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Abstract

In this paper we study the following fractional Choquard equation with mixed nonlinearities:

( − Δ ) s u = λ u + α I μ ∗ | u | q | u | q − 2 u + I μ ∗ | u | p | u | p − 2 u , x ∈ R N , ∫ R N | u | 2 d x = c 2 > 0 . $$\begin{cases}{\left(-{\Delta}\right)}^{s}u=\lambda u+\alpha \left({I}_{\mu }\ast \vert u{\vert }^{q}\right)\vert u{\vert }^{q-2}u+\left({I}_{\mu }\ast \vert u{\vert }^{p}\right)\vert u{\vert }^{p-2}u,\quad x\in {\mathbb{R}}^{N},\hfill \\ \underset{{\mathbb{R}}^{N}}{\int }\vert u{\vert }^{2} \mathrm{d}x={c}^{2}{ >}0.\hfill \end{cases}$$

Here N > 2 s , s ∈ (0, 1), μ ∈ (0, N ), and the exponents satisfy

2 N − μ N < q < p < 2 N − μ N − 2 s , $$\frac{2N-\mu }{N}{< }q{< }p{< }\frac{2N-\mu }{N-2s},$$
while α > 0 is a real parameter,
λ ∈ R $\lambda \in \mathbb{R}$
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 in the corresponding parameter regimes.

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