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We study regularity issues for systems of elliptic equations of the type \[  -\Delta u_i=f_{i,\beta}(x)-\beta \sum_{j\neq i} a_{ij} u_i|u_i|^{p-1}|u_j|^{p+1} \] set in domains $\Omega \subset \mathbb{R}^N$, for $N\geq 1$. The paper is devoted to the derivation of $\mathcal{C}^{0,\alpha}$estimates that are uniform in the competition parameter $\beta > 0$, as well asto the regularity of the limiting free-boundary problem obtained for $\beta \to+ \infty$.  The main novelty of the problem under consideration resides in thenon-trivial grouping of the densities: in particular, we assume that theinteraction parameters $a_{ij}$ are only non-negative, and thus may vanish forspecific couples $(i,j)$. As a main consequence, in the limit $\beta \to+\infty$, densities do not segregate pairwise in general, but are grouped inclasses which, in turn, form a mutually disjoint partition. Moreover, withrespect to the literature, we consider more general forcing terms,sign-changing solutions, and an arbitrary $p>0$. In addition, we present aregularity theory of the emerging free-boundary, defined by the interface amongdifferent segregated groups.  These equations are very common in the study of Bose-Einstein condensates andare of key importance for the analysis of optimal partition problems related tohigh order eigenvalues.

H\"older bounds and regularity of emerging free boundaries for strongly competing Schr\"odinger equations with nontrivial grouping