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 as to the regularity of the limiting free-boundary problem obtained for $\beta \to + \infty$. The main novelty of the problem under consideration resides in the non-trivial grouping of the densities: in particular, we assume that the interaction parameters $a_{ ij}$ are only non-negative, and thus may vanish for specific couples $(i,j)$. As a main consequence, in the limit $\beta \to +\infty$, densities do not segregate pairwise in general, but are grouped in classes which, in turn, form a mutually disjoint partition. Moreover, with respect to the literature, we consider more general forcing terms, sign-changing solutions, and an arbitrary $p>0$. In addition, we present a regularity theory of the emerging free-boundary, defined by the interface among different segregated groups. These equations are very common in the study of Bose-Einstein condensates and are of key importance for the analysis of optimal partition problems related to high order eigenvalues.
Categories: math.AP