Abstract:

This paper is concerned with the following nonlinear coupled system
$\left\{\begin{array}{l} -\Delta u_{1}+\omega_{1} u_{1}-\frac{1}{2} \Delta\left(u_{1}^{2}\right) u_{1}=\mu_{1}\left|u_{1}\right|^{p-1} u_{1}+\beta\left|u_{2}\right|^{\frac{p+1}{2}}\left|u_{1}\right|^{\frac{p-3}{2}} u_{1} \\ -\Delta u_{2}+\omega_{2} u_{2}-\frac{1}{2} \Delta\left(u_{2}^{2}\right) u_{2}=\mu_{2}\left|u_{2}\right|^{p-1} u_{2}+\beta\left|u_{1}\right|^{\frac{p+1}{2}}\left|u_{2}\right|^{\frac{p-3}{2}} u_{2} \\ \int_{\Omega}\left|u_{i}\right|^{2} \mathrm{~d} x=\rho_{i}, \quad i=1,2, \quad\left(u_{1}, u_{2}\right) \in H_{0}^{1}\left(\Omega ; \mathbb{R}^{2}\right) \end{array}\right.$
and linear coupled system
$\left\{\begin{array}{l} -\Delta u_{1}+\omega_{1} u_{1}-\frac{1}{2} \Delta\left(u_{1}^{2}\right) u_{1}=\mu_{1}\left|u_{1}\right|^{p-1} u_{1}+\beta u_{2} \\ -\Delta u_{2}+\omega_{2} u_{2}-\frac{1}{2} \Delta\left(u_{2}^{2}\right) u_{2}=\mu_{2}\left|u_{2}\right|^{p-1} u_{2}+\beta u_{1} \\ \int_{\Omega}\left|u_{i}\right|^{2} \mathrm{~d} x=\rho_{i}, \quad i=1,2, \quad\left(u_{1}, u_{2}\right) \in H_{0}^{1}\left(\Omega ; \mathbb{R}^{2}\right) \end{array}\right.$
where $\Omega\subset\mathbb R^N(N\geq1)$ is a bounded smooth domain, $\omega_i,\ \beta\in\mathbb R$, $\mu_i,\ \rho_i>0,\ i=1,2.$ Moreover, $p>1$ if $N=1,2$ and $1<p\leqslant\frac{3N+2}{N-2}$ if $N\geqslant3$. Using change of variables, on the one hand, we prove the existence and stability of normalized solutions in nonlinear coupled system and the limiting behavior of normalized solutions as $\beta\rightarrow -\infty$. On the other hand, we apply the minimization constraint technique to obtain the existence of normalized solutions for linear coupled system. Compared with some previous results, we extend the existing results to the quasilinear Schrödinger system and also obtain normalized solutions for the linear coupling case.

Key words: linear and nonlinear coupled, bounded domains, change of variables, normalized solution, limiting behavior.