Normalized solutions for a coupled fractional Schrödinger system in low dimensions
2020
We consider the following coupled fractional Schrodinger system:
$$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \vert ^{2p-2}u+ \beta \vert v \vert ^{p} \vert u \vert ^{p-2}u, \\ (-\Delta )^{s}v+\lambda _{2}v=\mu _{2} \vert v \vert ^{2p-2}v+\beta \vert u \vert ^{p} \vert v \vert ^{p-2}v \end{cases}\displaystyle \quad \text{in } {\mathbb{R}^{N}}, $$
with
$0< s<1$
,
$2s< N\le 4s$
and
$1+\frac{2s}{N}< p<\frac{N}{N-2s}$
, under the following constraint:
$$ \int _{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx=a_{1}^{2} \quad \text{and}\quad \int _{ \mathbb{R}^{N}} \vert v \vert ^{2}\,dx=a_{2}^{2}. $$
Assuming that the parameters
$\mu _{1}$
,
$\mu _{2}$
,
$a_{1}$
,
$a_{2}$
are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter
$\beta >0$
.
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