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 $s$.