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

We prove a conjecture which was recently formulated by Maia, Montefusco, Pellacci saying that minimal energy solutions of the saturated nonlinear Schrödinger system \begin{align*} - Δu + λ_1 u &= \frac{αu(αu^2+βv^2)}{1+s(αu^2+βv^2)} \qquad\text{in }\mathbb{R}^n, \newline - Δv + λ_2 v &= \frac{βv(αu^2+βv^2)}{1+s(αu^2+βv^2)}\qquad\text{in }\mathbb{R}^n \end{align*} are necessarily semitrivial whenever $α,β,λ_1,λ_2>0$ and $0<s<\max\{\fracα{λ_1},\fracβ{λ_2}\}$ except for the symmetric case $λ_1=λ_2,α=β$. Moreover it is shown that for most parameter samples $α,β,λ_1,λ_2$ there are infinitely many branches containing seminodal solutions which bifurcate from a semitrivial solution curve parametrized by $s$.