  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.
