Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping

We study interior regularity issues for systems of elliptic equations of the type −Δui=fi,β(x)−β∑j≠iaijui|ui|p−1|uj|p+1 set in domains Ω⊂RNΩ⊂RN, for N⩾1N⩾1. The paper is devoted to the derivation of C0,αC0,α estimates that are uniform in the competition parameter β>0β>0, as well as to the regularity of the limiting free-boundary problem obtained for β→+∞β→+∞.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 aijaij are only non-negative, and thus may vanish for specific couples (i,j)(i,j). As a main consequence, in the limit β→+∞β→+∞, 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>0p>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.