Bifurcations of elliptic systems with linear couplings

Abstract Consider the elliptic system with linearly coupled terms − Δ u = λ v + f 1 ( u , v ) , i n Ω , − Δ v = μ u + f 2 ( u , v ) , i n Ω , u = 0 , v = 0 , o n ∂ Ω , where λ , μ ∈ R are constants and Ω ⊂ R N is a smooth bounded domain. We study the local and global bifurcations with respect to T 0 ≔ { ( ( λ , μ ) , ( 0 , 0 ) ) } ⊂ R 2 × X , where X is a proper Banach space. Our results are of particular interest for obtaining nontrivial solutions in the case λ ≠ μ .