Multiple sign-changing and semi-nodal normalized solutions for a Gross-Pitaevskii type system on bounded domains: the $L^2$-supercritical case

In this paper we investigate the existence of multiple sign-changing and semi-nodal normalized solutions for an $m$-coupled elliptic system of the Gross-Pitaevskii type:
\begin{equation}
\left\{
\begin{aligned}
&-\Delta u_j + \lambda_j u_j = \sum_{k=1 }^m\beta_{kj} u_k^2 u_j, \quad u_j \in H_0^1(\Omega),
&\int_\Omega u_j^2dx = c_j, \quad j = 1,2,\cdots,m.
\end{aligned}
\right.
\end{equation}
Here, $\Omega \subset \mathbb{R}^N$ ($N = 3,4$) is a bounded domain. The constants $\beta_{kj} \neq 0$ and $c_j > 0$ are prescribed constants, while $\lambda_1, \cdots, \lambda_m$ are unknown and appear as Lagrange multipliers. This is the first result in the literature on the existence and multiplicity of sign-changing and semi-nodal normalized solutions of couple Schrödinger system in all regimes of $\beta_{kj}$. The main tool which we use is a new skill of vector linking and this article attempts for the first time to use linking method to search for solutions of a coupled system. Particularly, to obtain semi-nodal normalized solutions, we introduce partial vector linking which is new up to our knowledge. Moreover, by investigating the limit process as $\vec{c}=(c_1,\ldots,c_m) \to \vec{0}$ we obtain some bifurcation results. Note that when $N=4$, the system is of Sobolev critical.