Holder bounds and regularity of emerging free boundaries for strongly competing Schrodinger equations with nontrivial grouping

We study interior regularity issues for systems of elliptic equations of the type -Delta u(i) = f(i,beta)(x) - beta Sigma(j not equal i) a(ij)u(i)vertical bar u(i)vertical bar(p-1)vertical bar u(j)vertical bar(p+1) set in domains Omega subset of R-N, for N >= 1. The paper is devoted to the derivation of C-0,C- 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 -> + infinity. 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 -> +infinity, 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. (C) 2015 Elsevier Ltd. All rights reserved.