Minimal Energy Solutions and Infinitely Many Bifurcating Branches for a Class of Saturated Nonlinear Schrödinger Systems

We prove a conjecture recently formulated by Maia, Montefusco and Pellacci saying that minimal energy solutions of the saturated nonlinear Schrödinger system$\left\{\begin{aligned} \displaystyle-\Delta u+\lambda_{1}u&\displaystyle=\frac% {\alpha u(\alpha u^{2}+\beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&% \displaystyle\text{in }\mathbb{R}^{n},\\ \displaystyle-\Delta v+\lambda_{2}v&\displaystyle=\frac{\beta v(\alpha u^{2}+% \beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&\displaystyle\text{in }\mathbb{R% }^{n}\end{aligned}\right.$are necessarily semitrivial whenever ${\alpha,\hskip 0.5pt\beta,\hskip 0.5pt\lambda_{1},\hskip 0.5pt\lambda_{2}>0}$ and ${0<s<\max\{\alpha/\lambda_{1},\hskip 0.5pt\beta/\lambda_{2}\}}$ 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.