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

TL;DR

This paper proves that minimal energy solutions of a class of saturated nonlinear Schrödinger systems are semitrivial under certain conditions, and demonstrates the existence of infinitely many bifurcating branches of solutions for most parameters.

Contribution

It confirms a conjecture about the nature of minimal energy solutions and reveals the existence of infinitely many bifurcating solution branches in the system.

Findings

Minimal energy solutions are semitrivial under specified conditions.

Existence of infinitely many bifurcating solution branches for most parameters.

Bifurcations occur from semitrivial solution curves parametrized by s.

Abstract

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 $s$.