Abstract
Consider the system
\left\{\begin{aligned} \displaystyle-\Delta u_{i}+\mu_{i}u_{i}&\displaystyle=%
\nu_{i}u_{i}^{2^{*}-1}+\beta\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}^{\frac{2^{*}}%
{2}}u_{i}^{\frac{2^{*}}{2}-1}+\lambda\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}&&%
\displaystyle\phantom{}\text{in}\ \Omega,\\
\displaystyle u_{i}&\displaystyle>0&&\displaystyle\phantom{}\text{in}\ \Omega,%
\\
\displaystyle u_{i}&\displaystyle=0&&\displaystyle\phantom{}\text{on}\ %
\partial\Omega,\quad i=1,2,\ldots,k,\end{aligned}\right.
where
{k\geq 2}
,
{\Omega\subset\mathbb{R}^{N}}
(
{N\geq 3}
) is a bounded domain,
{2^{*}=\frac{2N}{N-2}}
,
{\mu_{i}\in\mathbb{R}}
and
{\nu_{i}>0}
are constants, and
{\beta,\lambda>0}
are parameters.
By showing a unique result of the limit system, we prove existence and nonexistence results of ground states to this system by variational methods, which generalize the results in [7, 18].
Concentration behaviors of ground states for
{\beta,\lambda}
are also established.
Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case