Bifurcations for a coupled Schrödinger system with multiple components

In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: - Δ u j + a u j = μ j u j 3 + β ∑ k ≠ j u k 2 u j , u j > 0 in Ω , u j = 0 on ∂ Ω , j = 1 , . . . , n . Here Ω ⊂ R N is a smooth and bounded domain, n ≥ 3 , a < - Λ 1 where Λ 1 is the principal eigenvalue of ( - Δ , H 0 1 ( Ω ) ) ; μ j and β are real constants. Using the positive and nondegenerate solution of the scalar equation - Δ ω - ω = - ω 3 , ω ∈ H 0 1 ( Ω ) , we construct a synchronized solution branch T ω . Then we find a sequence of local bifurcations with respect to T ω , and we find global bifurcation branches of partially synchronized solutions.