Multiple sign-changing and semi-nodal normalized solutions for a Gross-Pitaevskii type system on bounded domain: 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} &-Δu_j + λ_j u_j = \sum_{k=1 }^mβ_{kj} u_k^2 u_j, \quad u_j \in H_0^1(Ω), &\int_Ωu_j^2dx = c_j, \quad j = 1,2,\cdots,m. \end{aligned} \right. \end{equation} Here, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N=3,4$) is a bounded domain. The constants ${\beta }_{kj}\mathrm{\ne }0$ 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 $\vec{c}=(c_1,\dots ,c_m)\to \vec{0}$ we obtain some bifurcation results. Note that when $N=4$, the system is of Sobolev critical.