Normalized solutions for nonlinear Schr\"odinger systems

TL;DR

This paper establishes the existence of normalized solutions for coupled nonlinear Schrödinger systems modeling ultracold quantum gases, using variational methods for Sobolev subcritical and supercritical nonlinearities.

Contribution

It introduces a new variational framework to find normalized solutions with prescribed mass in coupled Schrödinger systems, including supercritical cases.

Findings

Existence of solutions for Sobolev subcritical nonlinearities.

Solutions include cases where nonlinearities are L^2-supercritical.

Lagrange multipliers appear as unknowns in the solutions.

Abstract

We consider the existence of \emph{normalized} solutions in $H^1(\R^N) \times H^1(\R^N)$ for systems of nonlinear Schr\"odinger equations which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz one is led to coupled systems of elliptic equations of the form \[ \left\{ \begin{aligned} -\De u_1 &= \la_1u_1 + f_1(u_1)+\pa_1F(u_1,u_2),\\ -\De u_2 &= \la_2u_2 + f_2(u_2)+\pa_2F(u_1,u_2),\\ u_1,u_2&\in H^1(\R^N),\ N\ge2, \end{aligned} \right. \] and we are looking for solutions satisfying \[ \int_{\R^N}|u_1|^2 = a_1,\quad \int_{\R^N}|u_2|^2 = a_2 \] where $a_1>0$ and $a_2>0$ are prescribed. In the system $\la_1$ and $\la_2$ are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e.\ $f_i(u_i)=\mu_i|u_i|^{p_i-1}u_i$, $F(u_1,u_2)=\be|u_1|^{r_1}|u_2|^{r_2}$, with positive constants $\be, \mu_i, p_i, r_i$. The exponents are Sobolev subcritical but may be $L^2$-supercritical: $p_1,p_2,r_1+r_2\in]2,2^*[\,\setminus\left\{2+\frac4N\right\}$.