Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities
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| Format: | Preprint |
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2022
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| _version_ | 1866911962338689024 |
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| author | Zuo, Jiabin Rădulescu, Vicenţiu D. |
| author_facet | Zuo, Jiabin Rădulescu, Vicenţiu D. |
| contents | In this paper, we investigate the following fractional Sobolev critical nonlinear Schrödinger (NLS) coupled systems: \begin{equation*} \left\{\begin{array}{lll} (-Δ)^{s} u=μ_{1} u+|u|^{2^{*}_{s}-2}u+η_{1}|u|^{p-2}u+γα|u|^{α-2}u|v|^β ~ \text{in}~ \mathbb{R}^{N},\\ (-Δ)^{s} v=μ_{2} v+|v|^{2^{*}_{s}-2}v+η_{2}|v|^{q-2}v+γβ|u|^α|v|^{β-2}v ~~\text{in}~ \mathbb{R}^{N},\\ \|u\|^{2}_{L^{2}}=m_{1}^{2} ~\text{and}~ \|v\|^{2}_{L^{2}}=m_{2}^{2}, \end{array}\right. \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $N={3,4}$, $s\in(0,1)$, $μ_{1}, μ_{2}\in\mathbb{R}$ are unknown constants, which will appear as Lagrange multipliers, $2^{*}_{s}$ is the fractional Sobolev critical index, $η_{1}, η_{2}, γ, m_{1}, m_{2}>0$, $α>1, β>1$, $p, q, α+β\in(2+4s/N,2^{*}_{s}]$. Firstly, if $p, q, α+β<2^{*}_{s}$, we obtain the existence of positive normalized solution when $γ$ is big enough. Secondly, if $p=q=α+β=2^{*}_{s}$, we show that nonexistence of positive normalized solution. The main ideas and methods of this paper are scaling transformation, classification discussion and concentration-compactness principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_13051 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities Zuo, Jiabin Rădulescu, Vicenţiu D. Analysis of PDEs 35J50, 35J60, 35B33 In this paper, we investigate the following fractional Sobolev critical nonlinear Schrödinger (NLS) coupled systems: \begin{equation*} \left\{\begin{array}{lll} (-Δ)^{s} u=μ_{1} u+|u|^{2^{*}_{s}-2}u+η_{1}|u|^{p-2}u+γα|u|^{α-2}u|v|^β ~ \text{in}~ \mathbb{R}^{N},\\ (-Δ)^{s} v=μ_{2} v+|v|^{2^{*}_{s}-2}v+η_{2}|v|^{q-2}v+γβ|u|^α|v|^{β-2}v ~~\text{in}~ \mathbb{R}^{N},\\ \|u\|^{2}_{L^{2}}=m_{1}^{2} ~\text{and}~ \|v\|^{2}_{L^{2}}=m_{2}^{2}, \end{array}\right. \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $N={3,4}$, $s\in(0,1)$, $μ_{1}, μ_{2}\in\mathbb{R}$ are unknown constants, which will appear as Lagrange multipliers, $2^{*}_{s}$ is the fractional Sobolev critical index, $η_{1}, η_{2}, γ, m_{1}, m_{2}>0$, $α>1, β>1$, $p, q, α+β\in(2+4s/N,2^{*}_{s}]$. Firstly, if $p, q, α+β<2^{*}_{s}$, we obtain the existence of positive normalized solution when $γ$ is big enough. Secondly, if $p=q=α+β=2^{*}_{s}$, we show that nonexistence of positive normalized solution. The main ideas and methods of this paper are scaling transformation, classification discussion and concentration-compactness principle. |
| title | Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities |
| topic | Analysis of PDEs 35J50, 35J60, 35B33 |
| url | https://arxiv.org/abs/2206.13051 |