Minimal energy solutions and infinitely many bifurcating branches for a class of saturated nonlinear Schrödinger systems

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We prove a conjecture which was recently formulated by Maia, Montefusco, Pellacci saying that minimal energy solutions of the saturated nonlinear Schrödinger system align* - Δu + λ₁ u &= αu(αu²+βv²)/1+s(αu²+βv²) Rⁿ, - Δv + λ₂ v &= βv(αu²+βv²)/1+s(αu²+βv²) Rⁿ align* are necessarily semitrivial whenever α,β,λ₁,λ₂>0 and 0<s<{αλ₁,βλ₂} except for the symmetric case λ₁=λ₂,α=β. Moreover it is shown that for most parameter samples α,β,λ₁,λ₂ there are infinitely many branches containing seminodal solutions which bifurcate from a semitrivial solution curve parametrized by s.