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

Abstract

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 $\alpha ,\beta ,{\lambda }_1,{\lambda }_2>0$ 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 $s$.