2025/06/27 by Tianhao Liu, Linjie Song, Liu, Tianhao +6
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Nonlinear Waves and Solitons #math.AP
paper · pdf · doi:10.48550/arxiv.2506.22152
openalex publication_date 2025/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: \ \beginaligned amp;-Δuj + λj uj = ∑k=1 mβkj uk2 uj, uj ∈ H01(Ω), amp;∫Ωuj2dx = cj, j = 1,2,⋯,m. \endaligned . Here, Ω⊂ ℝN (N = 3,4) is a bounded domain. The constants βkj ≠ 0 and cj > 0 are prescribed constants, while λ1, ⋯, λm 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 βkj. 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 c=(c1,…,cm) → 0 we obtain some bifurcation results. Note that when N=4, the system is of Sobolev critical.