Normalized solution to Kirchhoff-fractional system involving critical Choquard nonlinearity
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909778830163968 |
|---|---|
| author | Goel, Divya Gupta, Shilpa Rai, Asmita |
| author_facet | Goel, Divya Gupta, Shilpa Rai, Asmita |
| contents | In this article, we explore the fractional Kirchhoff-Choquard system given by
$$ \left\{ \begin{array}{lr} (a+b\int_{\mathbb{R}^N}|(-Δ)^{\frac{s}{2}} u|^2\;dx)(-Δ)^su=λ_1u+(I_μ*|v|^{2^*_{μ,s}})|u|^{2^*_{μ,s}-2}u +αp (I_μ*|v|^{q})|u|^{p-2}u \;\text{in}\;\mathbb{R}^N,\\ (a+b\int_{\mathbb{R}^N}|(-Δ)^{\frac{s}{2}} v|^2\;dx)(-Δ)^sv=λ_2v+ (I_μ*|u|^{2^*_{μ,s}})|v|^{2^*_{μ,s}-2}u +αq(I_μ*|u|^{p})|v|^{q-2}v \;\;\text{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2=d_1^2,\;\;\int_{\mathbb{R}^N}|v|^2=d_2^2. \end{array} \right. $$
where $N> 2s$, $s \in (0,1)$, $μ\in (0, N)$, $α\in\mathbb{R}$. Here, $I_μ:\mathbb{R}^N \to \mathbb{R}$ denotes the Riesz potential. We denote by $2_{μ,*}:=\frac{2N-μ}{N}$ and $\frac{2N-μ}{N-2s}:={2^*_{μ,s}}$, the lower and upper Hardy-Littlewood-Sobolev critical exponents, repectively, and assume that $2_{μ,*} < p,q< {2^*_{μ,s}}$. Our primary focus is on the existence of normalized solutions for the case $α>0$ in two scenarios: the $L^2$ subcritical case characterized by $22_{μ,*}<p + q < 4 + \frac{4s-2μ}{N}$ and $L^2$ supercritical associated with $4+\frac{8s-2μ}{N}< p + q < 2{2^*_{μ,s}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solution to Kirchhoff-fractional system involving critical Choquard nonlinearity Goel, Divya Gupta, Shilpa Rai, Asmita Analysis of PDEs 35B33, 35J50, 35J60, 35R11, 45K05 In this article, we explore the fractional Kirchhoff-Choquard system given by $$ \left\{ \begin{array}{lr} (a+b\int_{\mathbb{R}^N}|(-Δ)^{\frac{s}{2}} u|^2\;dx)(-Δ)^su=λ_1u+(I_μ*|v|^{2^*_{μ,s}})|u|^{2^*_{μ,s}-2}u +αp (I_μ*|v|^{q})|u|^{p-2}u \;\text{in}\;\mathbb{R}^N,\\ (a+b\int_{\mathbb{R}^N}|(-Δ)^{\frac{s}{2}} v|^2\;dx)(-Δ)^sv=λ_2v+ (I_μ*|u|^{2^*_{μ,s}})|v|^{2^*_{μ,s}-2}u +αq(I_μ*|u|^{p})|v|^{q-2}v \;\;\text{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2=d_1^2,\;\;\int_{\mathbb{R}^N}|v|^2=d_2^2. \end{array} \right. $$ where $N> 2s$, $s \in (0,1)$, $μ\in (0, N)$, $α\in\mathbb{R}$. Here, $I_μ:\mathbb{R}^N \to \mathbb{R}$ denotes the Riesz potential. We denote by $2_{μ,*}:=\frac{2N-μ}{N}$ and $\frac{2N-μ}{N-2s}:={2^*_{μ,s}}$, the lower and upper Hardy-Littlewood-Sobolev critical exponents, repectively, and assume that $2_{μ,*} < p,q< {2^*_{μ,s}}$. Our primary focus is on the existence of normalized solutions for the case $α>0$ in two scenarios: the $L^2$ subcritical case characterized by $22_{μ,*}<p + q < 4 + \frac{4s-2μ}{N}$ and $L^2$ supercritical associated with $4+\frac{8s-2μ}{N}< p + q < 2{2^*_{μ,s}}$. |
| title | Normalized solution to Kirchhoff-fractional system involving critical Choquard nonlinearity |
| topic | Analysis of PDEs 35B33, 35J50, 35J60, 35R11, 45K05 |
| url | https://arxiv.org/abs/2509.07597 |