*Mathematical Methods in the Applied Sciences*The existence of nontrivial solutions for a critically coupled
Schr\”{o}dinger system in a bounded domain of
$\R^3$

In this paper, we consider the following coupled
Schr\”{o}dinger system with doubly critical exponents,
which can be seen as a counterpart of the Brezis-Nirenberg problem
$$\left\{%
\begin{array}{ll} -\Delta
u+\lambda_1 u=\mu_1 u^5+
\beta u^2v^3, &
\hbox{$x\in \Omega$},
\\ -\Delta
v+\lambda_2 v=\mu_2 v^5+
\beta v^2u^3, &
\hbox{$x\in \Omega$},
\\ u=v=0,&
\hbox{$x\in
\partial\Omega$},
\\ \end{array}%
\right.$$ where $\Omega$ is a ball in
$\R^3,$
$-\lambda_1(\Omega)<\lambda_1,\lambda_2<-\frac14\lambda_1(\Omega)$,
$\mu_1,\mu_2>0$ and
$\beta>0$. Here
$\lambda_1(\Omega)$ is the first
eigenvalue of $-\Delta$ with Dirichlet boundary
condition in $\Omega$. We show that the problem has at
least one nontrivial solution for all
$\beta>0$.

13 May 2021

13 May 2021

13 May 2021

31 May 2021