Minimal energy solutions and infinitely many bifurcating branches for a class of saturated nonlinear Schr\"odinger systems

We prove a conjecture which was recently formulated by Maia, Montefusco, Pellacci saying that minimal energy solutions of the saturated nonlinear Schr\"odinger system \begin{align*} - \Delta u + \lambda_1 u &= \frac{\alpha u(\alpha u^2+\beta v^2)}{1+s(\alpha u^2+\beta v^2)} \qquad\text{in }\mathbb{R}^n, \newline - \Delta v + \lambda_2 v &= \frac{\beta v(\alpha u^2+\beta v^2)}{1+s(\alpha u^2+\beta v^2)}\qquad\text{in }\mathbb{R}^n \end{align*} are necessarily semitrivial whenever $\alpha,\beta,\lambda_1,\lambda_2>0$ and $0<s<\max\{\frac{\alpha}{\lambda_1},\frac{\beta}{\lambda_2}\}$ except for the symmetric case $\lambda_1=\lambda_2,\alpha=\beta$. Moreover it is shown that for most parameter samples $\alpha,\beta,\lambda_1,\lambda_2$ there are infinitely many branches containing seminodal solutions which bifurcate from a semitrivial solution curve parametrized by $s$.