Quantitative linear nondegeneracy of approximate solutions to strongly competitive Gross-Pitaevskii systems in general domains in $N\geq 1$ dimensions

TL;DR

This paper investigates the nondegeneracy and invertibility of linearized operators around approximate solutions to strongly coupled Gross-Pitaevskii systems, providing insights into the stability and persistence of solutions in general domains.

Contribution

It establishes the linear nondegeneracy of approximate solutions to strongly coupled Gross-Pitaevskii systems in arbitrary domains, extending previous results beyond symmetric or two-dimensional cases.

Findings

Linearization around approximate solutions has no kernel under certain conditions.

Provides estimates for the inverse of the linearized operator in weighted norms.

Results apply to general smooth bounded domains in any dimension.

Abstract

We consider strongly coupled competitive elliptic systems of Gross-Pitaevskii type that arise in the study of two-component Bose-Einstein condensates, in general smooth bounded domains of $\mathbb{R}^N$, $N\geq 1$. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. Under natural non-degeneracy assumptions on a solution of the limit problem, we show that the linearization of the Gross-Pitaevskii system around a 'sufficiently good' approximate solution does not have a kernel and obtain an estimate for its inverse with respect to carefully chosen weighted norms. Our motivation is the study of the persistence of solutions of the limit scalar problem for large values of the coupling parameter which is known only in two dimensions or if the domain has radial symmetry.